The Stable theorem atlas

This is the complete interactive reading surface for the Stable native Peano arithmetic library. It is generated from the same checked snapshot used by the current library tests and catalog—not copied by hand. The released training corpus remains explicitly frozen at its earlier 247-theorem checkpoint. Search by mathematical idea, filter by domain, focus a theorem to move backward to prerequisites or forward to its clients, and expand any card to read the exact first-order statement and the complete authored tactic recipe.

What “actual proof” means here

Every checked card embeds its full authored tactic body plus the explicit use imports that reconstruct its dependencies. Replay produces one closed self-contained certificate, which the independent kernel checks from the empty context. The much larger certificate tree is identified by its hash and metrics rather than pasted as tens of thousands of constructor nodes.

Stable is not the same as hosted deployment

These 432 theorems are the canonical Stable repository edition. Whether a hosted Peano Lab has deployed this exact commit is a separate operational state and does not downgrade the library to Alpha. See Alpha and Stable library editions.

432checked native theorems

1,982,360structural proof occurrences

57,692self-contained Cuts

1explicit language boundary

Snapshot root: 4d02dc439d53533e8992a471b26ee34059fb6001f822041e42c56b2cc0a7a079 Generated card digest: 1036b44c204887ae37c4e5d4fd78954078b357847c13c98e0582f13a7d94cf99

Search all names, statements, summaries, and prerequisites Domain All domains (432) Equality (10) Addition (12) Multiplication (19) Order (29) Divisibility (20) Congruence & CRT (58) Division (13) GCD, Bézout & coprimality (45) Primes (13) β sequences, products & FTA (76) Quadratic reciprocity campaign (137) Status Checked + boundary Checked only (432) Language boundary (1) Clear filters

433 entries

Back-and-forth navigator

Focus one theorem

Selected theorem zero_add — Zero is a left identity for addition add_succ_left — Successor moves through addition on the left add_comm — Addition is commutative add_assoc — Addition is associative mul_zero_left — Zero is a left annihilator for multiplication mul_succ_left — Successor moves through multiplication on the left mul_comm — Multiplication is commutative mul_add — Multiplication distributes over right addition mul_assoc — Multiplication is associative one_mul — One is a left identity for multiplication mul_one — One is a right identity for multiplication add_mul — Left addition distributes through multiplication succ_ne_zero — A successor is not zero succ_injective — Successor is injective le_refl — Order is reflexive le_trans — Order is transitive no_succ_add_fixed — Adding a positive successor has no fixed point drop_add_prefix_from_fixed — Drop an additive prefix from a fixed-point equation antisymm_from_witnesses — Opposing additive witnesses force equality le_antisymm — Order is antisymmetric le_total — Natural-number order is total add_eq_zero_right — A zero sum has a zero right summand mul_eq_zero — A zero product has a zero factor eq_symm — Equality is symmetric eq_trans — Equality is transitive succ_congr — Successor preserves equality zero_or_succ — Every natural is zero or a successor nonzero_is_succ — Every nonzero natural has a predecessor add_congr — Addition preserves equality in both arguments mul_congr — Multiplication preserves equality in both arguments add_right_cancel — Addition cancels on the right add_left_cancel — Addition cancels on the left zero_le — Zero is below every natural number le_succ_self — Every natural is below its successor le_zero — Only zero lies below zero one_le_of_ne_zero — Every nonzero natural is at least one ne_zero_of_one_le — A natural at least one is nonzero le_add_left — A number is below a left-added summand le_add_right — A number is below itself plus a right summand add_le_add_right — Adding the same right summand preserves order add_le_add_left — Adding the same left summand preserves order succ_le_succ — Successor preserves order le_of_succ_le_succ — Successor reflects order le_succ — Order is preserved by raising the upper bound lt_to_le — Strict order entails weak order add_le_cancel_right — Cancel a common right summand from order lt_irrefl_expanded — Strict order is irreflexive le_eq_or_lt — A weak inequality is equality or strict inequality lt_of_lt_of_le — Strict order followed by weak order lt_of_le_of_lt — Weak order followed by strict order lt_trans — Strict order is transitive le_or_lt — Weak-or-reverse-strict comparison lt_trichotomy — Natural strict-order trichotomy lt_not_le — Strict order excludes reverse weak order le_not_lt — Weak order excludes reverse strict order lt_not_eq_add_middle — A strict bound cannot occur inside an additive decomposition of the lower term mul_le_mul_left — Left multiplication preserves order mul_le_mul_right — Right multiplication preserves order mul_lt_mul_succ_left_nonzero — A nonzero left factor strictly separates successor products division_remainder_succ — Division with remainder for successor divisors division_remainder_exists — Division with remainder for every nonzero divisor remainder_bound_step — Advance a strict remainder bound by one step division_block_upper — A bounded remainder lies below the next divisor block positive_quotient_gap_impossible — A positive quotient gap contradicts bounded remainders remainder_unique_same_quotient — Remainders are unique at a fixed quotient division_remainder_unique — Quotient and remainder are unique zero_remainder_implies_multiple — Zero remainder implies divisibility multiple_has_zero_remainder — A multiple has a bounded zero remainder add_eq_zero_left — A zero sum has a zero left summand add_eq_zero_components — Both summands of a zero sum are zero mul_eq_one_components — Both factors of a product equal to one are one mul_ne_zero — A product of nonzero naturals is nonzero mul_left_cancel_nonzero — A nonzero factor cancels on the left mul_right_cancel_nonzero — A nonzero factor cancels on the right two_large_factors_impossible — Two is not a product of two successors of successors prime_two — Two is prime multiple_zero — Every natural number divides zero one_multiple — One divides every natural number multiple_refl — Divisibility is reflexive multiple_add — A common divisor divides a sum multiple_mul_right — A divisor remains a divisor after right multiplication multiple_mul_left — A divisor remains a divisor after left multiplication multiple_trans — Divisibility is transitive divisor_le_nonzero — A divisor of a nonzero natural is bounded divisor_one — Every divisor of one is one multiple_antisymm — Divisibility is antisymmetric factor_difference — A common-factor difference retains the factor divides_remainder — A common divisor divides a Euclidean remainder divides_linear_step — A common divisor divides a Euclidean linear step not_multiple_pointwise — A nonmultiple differs from every candidate multiple not_multiple_from_pointwise — Pointwise inequality excludes every multiple witness is_gcd_zero_right — Every natural is the gcd of itself and zero is_gcd_symm — The relational gcd specification is symmetric is_gcd_dvd_left — A relational gcd divides the left input is_gcd_dvd_right — A relational gcd divides the right input is_gcd_greatest — Every common divisor divides a relational gcd is_gcd_of_dvd — A divisor is the gcd of itself and its multiple is_gcd_unique — The relational greatest common divisor is unique is_gcd_euclid_forward — Euclidean gcd invariance in the forward direction is_gcd_euclid_backward — Euclidean gcd invariance in the backward direction gcd_exists_up_to — A greatest common divisor exists below a fixed bound gcd_exists_relational — A greatest common divisor exists coprime_symm — Coprimality is symmetric coprime_one_right — Every natural number is coprime to one coprime_one_left — One is coprime to every natural number coprime_to_is_gcd_one — Coprimality constructs a relational gcd equal to one is_gcd_one_to_coprime — A relational gcd equal to one implies coprimality add_permute_outer — The outer terms of two additive pairs may be permuted balanced_bezout_euclid_step — Balanced Bézout coefficients cross one Euclidean step gcd_balanced_bezout_exists_up_to — A gcd with balanced Bézout witnesses exists below a bound gcd_balanced_bezout_exists — A gcd has balanced natural Bézout coefficients balanced_combination_scale_right — A balanced combination may be scaled on the right common_divisor_divides_balanced_result — A common divisor divides a balanced-combination result coprime_balanced_bezout — Coprime inputs have a balanced Bézout identity gauss_coprime_cancel — Gauss cancellation for coprime factors eq_decidable — Equality of naturals is decidable multiple_decidable_nonzero — Divisibility by a nonzero natural is decidable multiple_decidable — Divisibility of naturals is decidable factor_property_succ — A bounded factor property extends across one boundary factor_search_up_to — Bounded nontrivial factor search prime_or_composite — Every nontrivial natural is prime or composite prime_nonzero — Every prime natural is nonzero prime_decidable — Primality of naturals is decidable factor_nonzero_left — A left factor of a nonzero product is nonzero proper_factor_lt — A proper factor is strictly smaller than its product prime_divisor_exists_up_to — Bounded prime-divisor existence prime_divisor_exists — Every nontrivial natural has a prime divisor prime_divisor_eq_one_or_self — Every divisor of a prime is one or the prime euclid_prime_dvd_product — Euclid's lemma mod_eq_refl — Balanced natural congruence is reflexive mod_eq_symm — Balanced natural congruence is symmetric mod_eq_trans — Balanced natural congruence is transitive mod_eq_add — Balanced congruence respects addition mod_eq_mul_right — Balanced congruence is stable under right multiplication mod_eq_mul_left — Balanced congruence is stable under left multiplication mod_eq_mul — Balanced congruence respects multiplication remainder_decomposition_to_mod_eq — A quotient-remainder decomposition gives congruence mod_eq_bounded_unique — Bounded congruent representatives are equal mod_eq_to_remainder_decomposition — A bounded congruent representative is a directed remainder beta_modulus_nonzero — Every Gödel-beta modulus is nonzero beta_at_self_of_bound — A bounded value is its own Gödel-beta residue beta_at_exists — Every Gödel-beta position has a decoded value beta_at_unique — A Gödel-beta decoded value is unique beta_at_exists_unique — Every Gödel-beta position has exactly one decoded value beta_at_to_mod_eq — A Gödel-beta value is congruent to its code beta_at_of_mod_eq_bound — A bounded congruent value is a Gödel-beta value dvd_to_mod_zero — A multiple is congruent to zero add_residue — Absorb a known residue decomposition into a multiple add_residue_lift — Lift an additive residue decomposition square_decomp — Expand the square of a quotient-residue decomposition square_residue_lift — Lift a residue decomposition through squaring square_residue_witness — Package a square residue as an existential witness bezout_mod_left — Balanced Bézout identity projected modulo the left modulus bezout_mod_right — Balanced Bézout identity projected modulo the right modulus mod_eq_predecessor_cancel — Successor-predecessor cancellation in balanced congruence binary_crt — Constructive binary Chinese remainder theorem binary_crt_remainders — Binary CRT with directed remainder witnesses binary_crt_beta_pair — A single Gödel-beta code realizes two bounded values beta_modulus_coprime_base — A Gödel-beta modulus is coprime to its base common_divisor_beta_moduli_divides_gap_times_c — A common beta-modulus divisor divides the scaled index gap beta_moduli_coprime_of_gap_dvd — Divisible index gaps give coprime Gödel-beta moduli binary_crt_beta_pair_of_gap_dvd — A divisible index gap supplies a two-position Gödel-beta code bounded_common_multiple_step — A bounded common multiple extends by one bounded_common_multiple_exists — Every finite initial interval has a nonzero common multiple beta_moduli_coprime_of_lt_bounded_common_multiple — Ordered bounded Gödel-beta moduli are coprime beta_moduli_pairwise_coprime_bounded — A bounded Gödel-beta prefix has pairwise coprime moduli bounded_beta_moduli_pairwise_coprime_exists — Every finite Gödel-beta prefix admits a pairwise-coprime base coprime_mul_left — Coprimality is closed under multiplication on the left coprime_mul_right — Coprimality is closed under multiplication on the right mod_eq_of_mod_eq_multiple — Congruence descends along modulus divisibility binary_crt_fold_step — One CRT fold step preserves all accumulated congruences right_factor_divides_product — The right factor divides a product beta_accumulated_product_step — Extend the accumulated beta-modulus product invariant beta_crt_prefix_congruence_step — Extend decoded-prefix congruence by one beta position beta_crt_prefix_invariant_step — One full bounded beta-prefix CRT invariant step bounded_beta_crt_prefix_invariant — Bounded beta-prefix CRT invariant bounded_beta_crt_for_existing_code — Bounded CRT witness for an existing beta code prime_unbounded — There is a prime larger than every given natural beta_value_le_code — A decoded beta value is bounded by its code base_le_beta_modulus — The beta base is bounded by every beta modulus le_scaled_nonzero — Nonzero scaling is monotone scaled_bounded_common_multiple — Scaling preserves bounded common multiples beta_value_lt_scaled_base — Old decoded values fit the scaled target base new_value_lt_scaled_base — An appended value fits the scaled target base beta_exclusive_accumulated_product_step — Extend an exclusive beta-modulus product invariant beta_exclusive_recode_congruence_step — Extend exclusive cross-base beta congruences beta_exclusive_recode_invariant_step — One exclusive beta-recoding invariant step bounded_beta_exclusive_recode_invariant — Bounded exclusive beta-recoding invariant beta_prefix_extend — Recode and append one exact beta-prefix value beta_prefix_product_trace_exists — Exact beta-coded prefix-product trace exists beta_product_exists — Relational finite beta product exists beta_product_functional — Finite beta Product is functional beta_product_exists_unique — A unique finite beta Product exists beta_product_zero — The empty beta Product is one beta_product_succ_decompose — Decompose a successor beta Product beta_product_succ_append — Append one factor to a beta Product beta_product_transport_prefix — Transport Product across an extensionally preserved factor prefix beta_factor_prefix_product_append — Extend a factor prefix and Product together all_prime_empty — AllPrime holds on the empty beta prefix all_prime_succ_intro — Extend AllPrime across one beta entry all_prime_succ_elim_prefix — Project AllPrime to the previous beta prefix all_prime_succ_elim_last — Extract the last prime from AllPrime all_prime_transport — Transport AllPrime across beta recoding sorted_empty — Adjacent Sorted holds on the empty beta prefix sorted_singleton — Adjacent Sorted holds on a singleton beta prefix sorted_succ_intro — Extend adjacent Sorted across one beta entry sorted_succ_elim_prefix — Project adjacent Sorted to the previous beta prefix sorted_succ_elim_last — Extract the last adjacent order witness sorted_transport — Transport adjacent Sorted across beta recoding beta_prefix_extend_all_prime — Recode a beta prefix and preserve AllPrime beta_prefix_extend_sorted_singleton — Recode an empty prefix into a sorted singleton beta_prefix_extend_sorted_succ — Recode a nonempty prefix and preserve adjacent Sorted beta_canonical_append_empty — Canonical first-prime append beta_canonical_append_succ — Canonical ordered-prime append prime_divides_decidable — Prime divisibility is constructively decidable greatest_prime_divisor_search — Bounded greatest-prime-divisor search greatest_prime_divisor_exists — A greatest prime divisor exists greatest_prime_divisor_quotient_bound — Prime divisors of the quotient are bounded greatest_prime_divisor_descent — Greatest-prime-divisor quotient descent beta_factor_divides_product — Every decoded factor divides its Product beta_canonical_append_general — Append one prime to any canonical beta prefix beta_canonical_last_factor_bound — The last canonical factor obeys prime-divisor bounds prime_factorization_exists_up_to — Bounded prime-factorization existence prime_factorization_existence — Existence of a finite prime factorization beta_prime_divisor_product_member — A prime divisor of an AllPrime beta Product occurs as a factor beta_sorted_factor_le_last — Every sorted factor is bounded by the last factor beta_nonempty_all_prime_product_ne_one — A nonempty AllPrime beta Product is not one beta_all_prime_product_one_iff_length_zero — An AllPrime beta Product is one exactly at length zero beta_canonical_last_factors_equal — Canonical factorizations have equal last factors beta_canonical_product_cancel_last — Cancel the common last factor of canonical Products prime_factorization_uniqueness_by_length — Prime-factorization uniqueness by length prime_factorization_uniqueness — Uniqueness of finite prime factorization fundamental_theorem_of_arithmetic — Fundamental theorem of arithmetic prime_three — Three is prime two_prime_product_uniqueness — A product of two primes is unique up to order fourth_power_regroup — Regroup a fourth power as a square of a square mod5_residue_complete — Every natural has one of five residues modulo five mod5_nonzero_residue_cases — A nonmultiple of five has a nonzero residue mod5_square_residue_one — Squaring residue one modulo five preserves residue one mod5_square_residue_two — Squaring residue two modulo five gives residue four mod5_square_residue_three — Squaring residue three modulo five gives residue four mod5_square_residue_four — Squaring residue four modulo five gives residue one mod5_fourth_power_residue_one — Fourth powers preserve residue one modulo five mod5_fourth_power_residue_two — Fourth powers of residue two are one modulo five mod5_fourth_power_residue_three — Fourth powers of residue three are one modulo five mod5_fourth_power_residue_four — Fourth powers of residue four are one modulo five mod5_fourth_power_one — A nonmultiple of five has fourth power one modulo five parity_cases — Parity cases even_odd_exclusive_pointwise — Even odd exclusive pointwise even_not_odd — Even not odd odd_not_even — Odd not even successor_odd_of_even — Successor odd of even successor_even_of_odd — Successor even of odd even_add_even — Even add even even_add_odd — Even add odd odd_add_even — Odd add even odd_add_odd — Odd add odd even_mul_left — Even mul left mul_double_right — Mul double right even_mul_right — Even mul right odd_mul_odd — Odd mul odd odd_half_unique — Odd half unique odd_half_exists_unique — Odd half exists unique four_mul_eq_double_double — Four mul eq double double odd_mod4_cases — Odd mod4 cases mod4_one_three_exclusive_pointwise — Mod4 one three exclusive pointwise mod4_one_is_odd — Mod4 one is odd mod4_three_is_odd — Mod4 three is odd prime_ne_two_is_odd — Prime ne two is odd mod_eq_decidable_from_remainders — Mod eq decidable from remainders mod_eq_decidable_nonzero — Mod eq decidable nonzero quadratic_residue_search_up_to — Quadratic residue search up to quadratic_residue_bounded_decidable_nonzero — Quadratic residue bounded decidable nonzero quadratic_residue_bounded_equiv — Quadratic residue bounded equiv quadratic_residue_decidable_nonzero — Quadratic residue decidable nonzero beta_repeat_empty — Beta repeat empty beta_repeat_succ_extend — Beta repeat succ extend beta_repeat_exists — Beta repeat exists beta_repeat_entry_eq — Beta repeat entry eq beta_repeat_transport_entry — Beta repeat transport entry pow_exists — Pow exists pow_zero — Pow zero pow_functional — Pow functional pow_successor_decompose — Pow successor decompose beta_range_empty — Beta range empty beta_range_succ_extend — Beta range succ extend beta_range_exists — Beta range exists beta_range_entry_eq — Beta range entry eq beta_range_transport_entry — Beta range transport entry beta_prefix_sum_trace_exists — Beta prefix sum trace exists beta_sum_exists — Beta sum exists beta_sum_trace_functional — Beta sum trace functional beta_sum_functional — Beta sum functional beta_sum_exists_unique — Beta sum exists unique beta_sum_zero — Beta sum zero beta_sum_succ_decompose — Beta sum succ decompose beta_product_pointwise_mod_congruent — Beta product pointwise mod congruent beta_sum_pointwise_mod_congruent — Beta sum pointwise mod congruent all_bits_zero — All bits zero all_bits_prefix_succ — All bits prefix succ all_bits_last_succ — All bits last succ bit_count_exists — Bit count exists bit_count_functional — Bit count functional bit_count_zero — Bit count zero bit_count_succ_decompose — Bit count succ decompose bit_count_bounded — Bit count bounded prime_coprime_or_divides — Prime coprime or divides prime_not_divides_coprime — Prime not divides coprime distinct_primes_coprime — Distinct primes coprime coprime_balanced_mod_inverse — Coprime balanced mod inverse coprime_mod_inverse — Coprime mod inverse mod_eq_cancel_coprime — Mod eq cancel coprime prime_mod_inverse — Prime mod inverse prime_mod_cancel — Prime mod cancel factorial_exists — Factorial exists factorial_functional — Factorial functional factorial_zero — Factorial zero factorial_succ_decompose — Factorial succ decompose pow_one_from_zero_successor — Pow one from zero successor pow_one — Pow one pow_successor_pair_mul — Pow successor pair mul pow_mod_congruent — Pow mod congruent lt_three_cases — Lt three cases lt_five_cases — Lt five cases lt_seven_cases — Lt seven cases bounded_square_mod3_classify — Bounded square mod3 classify bounded_square_mod5_classify — Bounded square mod5 classify bounded_square_mod7_classify — Bounded square mod7 classify qres_mod3_zero — Qres mod3 zero qres_mod3_one — Qres mod3 one qres_mod5_zero — Qres mod5 zero qres_mod5_one — Qres mod5 one qres_mod5_four — Qres mod5 four qres_mod7_zero — Qres mod7 zero qres_mod7_one — Qres mod7 one qres_mod7_two — Qres mod7 two qres_mod7_four — Qres mod7 four qres_mod3_canonical_iff — Qres mod3 canonical iff qres_mod5_canonical_iff — Qres mod5 canonical iff qres_mod7_canonical_iff — Qres mod7 canonical iff not_qres_mod3_two — Not qres mod3 two not_qres_mod5_two — Not qres mod5 two not_qres_mod5_three — Not qres mod5 three not_qres_mod7_three — Not qres mod7 three not_qres_mod7_five — Not qres mod7 five not_qres_mod7_six — Not qres mod7 six pow_two_from_one_successor — Pow two from one successor pow_two — Pow two pow_add — Pow add pow_mul_exp — Pow mul exp predecessor_square_mod_one — Predecessor square mod one even_successor_to_odd — Even successor to odd odd_successor_to_even — Odd successor to even pow_predecessor_parity_mod — Pow predecessor parity mod beta_range_injective — Beta range injective beta_half_range_entry_bounds — Beta half range entry bounds beta_half_range_mod_eq_value — Beta half range mod eq value beta_half_range_mod_injective — Beta half range mod injective finite_surjective_zero — Finite surjective zero finite_injective_prefix_succ — Finite injective prefix succ finite_lt_succ_eq_or_lt — Finite lt succ eq or lt finite_bounded_entry_lt — Finite bounded entry lt beta_prefix_replace_exists — Beta prefix replace exists beta_prefix_swap_last_from_entries — Beta prefix swap last from entries beta_prefix_swap_last_exists — Beta prefix swap last exists beta_prefix_swap_last_reflect — Beta prefix swap last reflect finite_swap_last_bounded — Finite swap last bounded finite_swap_last_injective — Finite swap last injective finite_swap_last_surjective_back — Finite swap last surjective back finite_contains_decidable — Finite contains decidable finite_bounded_prefix_without_top — Finite bounded prefix without top finite_bounded_last_succ — Finite bounded last succ finite_surjective_succ_intro — Finite surjective succ intro finite_last_is_top_from_prefix_surjective — Finite last is top from prefix surjective finite_surjective_succ_from_prefix — Finite surjective succ from prefix finite_no_top_successor_gate — Finite no top successor gate finite_bounded_injective_surjective — Finite bounded injective surjective beta_prefix_replace_reflect — Beta prefix replace reflect beta_product_replace_balance — Beta product replace balance beta_product_swap_last_invariant — Beta product swap last invariant finite_fixed_last_prefix_bounded — Finite fixed last prefix bounded beta_reindex_alignment_swap_last — Beta reindex alignment swap last prime_is_succ_succ — Prime is succ succ prime_bounded_nonzero_mod_inverse — Prime bounded nonzero mod inverse canonical_remainder_exists — Canonical remainder existence canonical_remainder_functional — Canonical remainder functionality canonical_remainder_zero_impossible — Canonical remainder excludes modulus zero canonical_remainder_exists_unique — Unique canonical remainder canonical_remainders_characterize_mod_eq — Canonical remainders characterize congruence bounded_mod_inverse_unique — Uniqueness of bounded modular inverses coprime_bounded_mod_inverse — Coprime residues have bounded modular inverses mod_inverse_implies_coprime — A modular inverse implies coprimality coprime_iff_unique_bounded_mod_inverse — Coprimality iff unique bounded modular inverse is_lcm_multiple_left — A relational LCM is a multiple of its left input is_lcm_multiple_right — A relational LCM is a multiple of its right input is_lcm_least — A relational LCM divides every common multiple is_lcm_symm — Symmetry of relational LCM is_lcm_unique — Uniqueness of relational LCM is_lcm_zero_right — LCM with zero on the right is_lcm_zero_left — LCM with zero on the left balanced_bezout_one_implies_coprime — Balanced Bezout result one implies coprimality coprime_product_is_lcm — The product of coprime naturals is an LCM is_lcm_scale_nonzero — Nonzero scaling preserves relational LCM balanced_bezout_cancel_gcd — Cancel a nonzero gcd from balanced Bezout gcd_zero_inputs — A zero gcd forces zero inputs gcd_lcm_compatible_exists — Compatible gcd and LCM witnesses exist lcm_exists_relational — Relational LCM exists canonical_lcm_exists_unique — Canonical LCM exists uniquely gcd_lcm_product — The gcd-LCM product identity mod_eq_zero_iff_eq — Mod eq zero iff eq mod_eq_scale — Mod eq scale crt_solution_pair_congruent — Crt solution pair congruent crt_common_solution_implies_gcd_compatible — Crt common solution implies gcd compatible crt_incompatibility_obstructs_solution — Crt incompatibility obstructs solution is_gcd_quotients_coprime_nonzero — Is gcd quotients coprime nonzero mod_eq_common_remainder_decomposition — Mod eq common remainder decomposition crt_scaled_common_remainder_lift — Crt scaled common remainder lift generalized_binary_crt_sufficient_nonzero — Generalized binary crt sufficient nonzero generalized_binary_crt_sufficient_zero_left — Generalized binary crt sufficient zero left generalized_binary_crt_sufficient_zero_right — Generalized binary crt sufficient zero right generalized_binary_crt_sufficient — Generalized binary crt sufficient generalized_binary_crt_solvable_iff — Generalized binary crt solvable iff mod_eq_ordered_gap_multiple — Mod eq ordered gap multiple mod_eq_lcm_merge — Mod eq lcm merge mod_eq_lcm_iff_pair — Mod eq lcm iff pair crt_solution_class_iff_lcm — Crt solution class iff lcm crt_solution_unique_lcm_zero — Crt solution unique lcm zero crt_solution_canonical_remainder_nonzero — Crt solution canonical remainder nonzero generalized_binary_crt_canonical_boundary — Generalized binary crt canonical boundary mod_eq_decidable — Mod eq decidable generalized_binary_crt_solution_or_obstruction — Generalized binary crt solution or obstruction generalized_binary_crt_total_decision — Generalized binary crt total decision Neighborhood depth 1 hop 2 hops 3 hops 4 hops

zero_addZero is a left identity for addition Addition checked

Zero is a left identity for addition; unlike PA3, this needs induction.

Prerequisites

none — a root theorem

Used directly by

add_comm mul_one le_refl drop_add_prefix_from_fixed le_succ_self le_eq_or_lt division_remainder_succ division_remainder_unique factor_difference gcd_balanced_bezout_exists_up_to beta_at_self_of_bound dvd_to_mod_zero binary_crt common_divisor_beta_moduli_divides_gap_times_c quadratic_residue_search_up_to quadratic_residue_bounded_equiv factorial_succ_decompose predecessor_square_mod_one beta_half_range_entry_bounds mod_inverse_implies_coprime

Exact expanded PA statement

Copy statement

forall n. 0 + n = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. 0 + n = n
induction n
simp
simp [IH]
qed

Occurrences

17

Distinct objects

17

Reused references

0

Depth

8

Cuts

0

Certificate SHA-256

5f9cb3146e4a…

add_succ_leftSuccessor moves through addition on the left Addition checked

A successor can move through addition on the left.

Prerequisites

none — a root theorem

Used directly by

add_comm drop_add_prefix_from_fixed le_succ_self le_succ lt_to_le lt_irrefl_expanded le_eq_or_lt lt_trans lt_not_eq_add_middle division_remainder_succ common_divisor_beta_moduli_divides_gap_times_c prime_unbounded odd_add_even odd_add_odd odd_mul_odd quadratic_residue_search_up_to bit_count_bounded factorial_succ_decompose predecessor_square_mod_one beta_half_range_entry_bounds

Exact expanded PA statement

Copy statement

forall n m. S n + m = S (n + m)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. S n + m = S (n + m)
intro n
induction m
simp
simp [IH]
qed

Occurrences

27

Distinct objects

25

Reused references

2

Depth

10

Cuts

0

Certificate SHA-256

a551da8ef511…

add_commAddition is commutative Addition checked

Addition is commutative.

Prerequisites

zero_add add_succ_left

Used directly by

mul_succ_left add_left_cancel le_add_right add_le_add_left mul_lt_mul_succ_left_nonzero division_block_upper positive_quotient_gap_impossible add_eq_zero_left factor_difference add_permute_outer balanced_bezout_euclid_step common_divisor_divides_balanced_result mod_eq_trans mod_eq_add remainder_decomposition_to_mod_eq mod_eq_bounded_unique mod_eq_to_remainder_decomposition add_residue_lift bezout_mod_left mod_eq_predecessor_cancel beta_moduli_coprime_of_lt_bounded_common_multiple prime_unbounded mod5_square_residue_two odd_add_odd predecessor_square_mod_one mod_eq_ordered_gap_multiple

Exact expanded PA statement

Copy statement

forall n m. n + m = m + n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. n + m = m + n
use zero_add
use add_succ_left
intro n
induction m
simp [zero_add]
simp [add_succ_left, IH]
qed

Occurrences

73

Distinct objects

67

Reused references

6

Depth

12

Cuts

2

Certificate SHA-256

281109de8768…

add_assocAddition is associative Addition checked

Addition is associative.

Prerequisites

none — a root theorem

Used directly by

mul_succ_left mul_add le_trans antisymm_from_witnesses add_le_add_right add_le_add_left add_le_cancel_right lt_of_le_of_lt lt_trans division_block_upper positive_quotient_gap_impossible factor_difference add_permute_outer balanced_bezout_euclid_step mod_eq_trans add_residue add_residue_lift square_decomp bezout_mod_left bezout_mod_right mod_eq_predecessor_cancel mod5_square_residue_one mod5_square_residue_two even_add_odd odd_add_odd odd_mul_odd bit_count_bounded coprime_balanced_mod_inverse predecessor_square_mod_one beta_half_range_entry_bounds mod_eq_ordered_gap_multiple

Exact expanded PA statement

Copy statement

forall n m k. (n + m) + k = n + (m + k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m k. (n + m) + k = n + (m + k)
intro n
intro m
induction k
simp
simp [IH]
qed

Occurrences

33

Distinct objects

30

Reused references

3

Depth

12

Cuts

0

Certificate SHA-256

a21351de86c8…

mul_zero_leftZero is a left annihilator for multiplication Multiplication checked

Zero annihilates multiplication on the left.

Prerequisites

none — a root theorem

Used directly by

mul_comm mul_eq_one_components prime_two multiple_antisymm multiple_decidable factor_search_up_to prime_nonzero factor_nonzero_left prime_divisor_exists_up_to beta_at_self_of_bound prime_three beta_half_range_entry_bounds gcd_zero_inputs mod_eq_zero_iff_eq generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall n. 0 * n = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. 0 * n = 0
induction n
simp
simp [IH]
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

8

Cuts

0

Certificate SHA-256

fb469fdd0b52…

mul_succ_leftSuccessor moves through multiplication on the left Multiplication checked

A successor can move through multiplication on the left.

Prerequisites

add_comm add_assoc

Used directly by

mul_comm two_large_factors_impossible mod_eq_predecessor_cancel base_le_beta_modulus prime_three mod5_square_residue_one mod5_square_residue_two predecessor_square_mod_one beta_half_range_entry_bounds

Exact expanded PA statement

Copy statement

forall n m. S n * m = n * m + m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. S n * m = n * m + m
use add_comm
use add_assoc
intro n
induction m
simp
specialize add_comm n
specialize add_comm m
simp [IH, add_comm, add_assoc]
qed

Occurrences

172

Distinct objects

155

Reused references

17

Depth

22

Cuts

4

Certificate SHA-256

b57ad3ce119f…

mul_commMultiplication is commutative Multiplication checked

Multiplication is commutative.

Prerequisites

mul_zero_left mul_succ_left

Used directly by

add_mul mul_right_cancel_nonzero multiple_mul_left balanced_combination_scale_right mod_eq_mul_left remainder_decomposition_to_mod_eq mod_eq_to_remainder_decomposition beta_at_exists beta_at_unique square_decomp binary_crt beta_modulus_coprime_base beta_moduli_coprime_of_gap_dvd bounded_common_multiple_step right_factor_divides_product greatest_prime_divisor_quotient_bound greatest_prime_divisor_descent beta_factor_divides_product prime_factorization_exists_up_to two_prime_product_uniqueness mul_double_right mod_eq_decidable_nonzero quadratic_residue_bounded_equiv coprime_mod_inverse mod_eq_cancel_coprime beta_product_replace_balance prime_bounded_nonzero_mod_inverse canonical_remainders_characterize_mod_eq bounded_mod_inverse_unique coprime_bounded_mod_inverse coprime_product_is_lcm gcd_lcm_compatible_exists mod_eq_common_remainder_decomposition mod_eq_lcm_merge crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall n m. n * m = m * n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. n * m = m * n
use mul_zero_left
use mul_succ_left
intro n
induction m
simp [mul_zero_left]
simp [IH, mul_succ_left]
qed

Occurrences

222

Distinct objects

200

Reused references

22

Depth

24

Cuts

6

Certificate SHA-256

c3efa368cf22…

mul_addMultiplication distributes over right addition Multiplication checked

Multiplication distributes over addition on the right.

Prerequisites

add_assoc

Used directly by

mul_assoc add_mul mul_le_mul_left positive_quotient_gap_impossible multiple_add divides_linear_step balanced_bezout_euclid_step common_divisor_divides_balanced_result mod_eq_trans mod_eq_add add_residue add_residue_lift square_decomp binary_crt mod5_square_residue_one mod5_square_residue_two even_add_even even_add_odd odd_add_even odd_add_odd odd_mul_odd odd_mod4_cases mod4_three_is_odd coprime_mod_inverse balanced_bezout_cancel_gcd mod_eq_scale

Exact expanded PA statement

Copy statement

forall n m k. n * (m + k) = n * m + n * k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m k. n * (m + k) = n * m + n * k
use add_assoc
intro n
intro m
induction k
simp
simp [IH, add_assoc]
qed

Occurrences

77

Distinct objects

70

Reused references

7

Depth

16

Cuts

1

Certificate SHA-256

2a65ee30f1db…

mul_assocMultiplication is associative Multiplication checked

Multiplication is associative.

Prerequisites

mul_add

Used directly by

multiple_mul_right multiple_trans multiple_antisymm divides_remainder divides_linear_step balanced_bezout_euclid_step balanced_combination_scale_right common_divisor_divides_balanced_result mod_eq_mul_right square_decomp binary_crt mod_eq_of_mod_eq_multiple fourth_power_regroup mod5_square_residue_one mod5_square_residue_two mod5_fourth_power_residue_one mod5_fourth_power_residue_two mod5_fourth_power_residue_three mod5_fourth_power_residue_four mod5_fourth_power_one even_mul_left mul_double_right four_mul_eq_double_double coprime_mod_inverse mod_eq_cancel_coprime pow_add beta_product_replace_balance bounded_mod_inverse_unique coprime_product_is_lcm is_lcm_scale_nonzero balanced_bezout_cancel_gcd gcd_lcm_compatible_exists mod_eq_scale is_gcd_quotients_coprime_nonzero

Exact expanded PA statement

Copy statement

forall n m k. (n * m) * k = n * (m * k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m k. (n * m) * k = n * (m * k)
use mul_add
intro n
intro m
induction k
simp
simp [IH, mul_add]
qed

Occurrences

116

Distinct objects

104

Reused references

12

Depth

17

Cuts

2

Certificate SHA-256

4784291080cf…

one_mulOne is a left identity for multiplication Multiplication checked

One is a left identity for multiplication.

Prerequisites

none — a root theorem

Used directly by

mul_eq_one_components one_multiple coprime_to_is_gcd_one gauss_coprime_cancel le_scaled_nonzero beta_prefix_product_trace_exists beta_canonical_append_empty prime_factorization_exists_up_to beta_prefix_sum_trace_exists pow_one_from_zero_successor pow_predecessor_parity_mod bounded_mod_inverse_unique

Exact expanded PA statement

Copy statement

forall n. 1 * n = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. 1 * n = n
induction n
simp
simp [IH]
qed

Occurrences

26

Distinct objects

26

Reused references

0

Depth

9

Cuts

0

Certificate SHA-256

09b763d9af1e…

mul_oneOne is a right identity for multiplication Multiplication checked

One is a right identity for multiplication.

Prerequisites

zero_add

Used directly by

multiple_refl multiple_antisymm coprime_to_is_gcd_one proper_factor_lt prime_divisor_eq_one_or_self binary_crt prime_unbounded mod_eq_cancel_coprime pow_add predecessor_square_mod_one bounded_mod_inverse_unique balanced_bezout_cancel_gcd is_gcd_quotients_coprime_nonzero

Exact expanded PA statement

Copy statement

forall n. n * 1 = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n * 1 = n
use zero_add
intro n
simp [zero_add]
qed

Occurrences

33

Distinct objects

33

Reused references

0

Depth

9

Cuts

1

Certificate SHA-256

c7e423c814db…

add_mulLeft addition distributes through multiplication Multiplication checked

Multiplication distributes over addition on the left.

Prerequisites

mul_comm mul_add

Used directly by

mul_le_mul_right balanced_bezout_euclid_step balanced_combination_scale_right mod_eq_mul_right square_decomp common_divisor_beta_moduli_divides_gap_times_c odd_mul_odd

Exact expanded PA statement

Copy statement

forall n m k. (n + m) * k = n * k + m * k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m k. (n + m) * k = n * k + m * k
use mul_comm
use mul_add
intro n
intro m
intro k
simp [mul_comm, mul_add]
qed

Occurrences

326

Distinct objects

263

Reused references

31

Depth

25

Cuts

9

Certificate SHA-256

436c20edef86…

succ_ne_zeroA successor is not zero Equality checked

No successor is zero (the reusable PA1 lemma).

Prerequisites

none — a root theorem

Used directly by

mul_left_cancel_nonzero factor_search_up_to prime_nonzero beta_modulus_nonzero bounded_common_multiple_step bounded_common_multiple_exists bounded_beta_exclusive_recode_invariant beta_prefix_product_trace_exists all_prime_empty sorted_empty sorted_singleton beta_factor_divides_product prime_factorization_exists_up_to beta_all_prime_product_one_iff_length_zero prime_factorization_uniqueness_by_length beta_repeat_empty beta_range_empty beta_prefix_sum_trace_exists all_bits_zero qres_mod3_canonical_iff qres_mod5_canonical_iff qres_mod7_canonical_iff finite_surjective_zero beta_prefix_replace_exists finite_contains_decidable beta_product_replace_balance prime_bounded_nonzero_mod_inverse canonical_remainder_zero_impossible

Exact expanded PA statement

Copy statement

forall n. ~(S n = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(S n = 0)
apply PA1
qed

Occurrences

1

Distinct objects

1

Reused references

0

Depth

1

Cuts

0

Certificate SHA-256

885b0b7ed12a…

succ_injectiveSuccessor is injective Equality checked

Successor is injective (the reusable PA2 lemma).

Prerequisites

none — a root theorem

Used directly by

not_qres_mod3_two not_qres_mod5_two not_qres_mod5_three not_qres_mod7_three not_qres_mod7_five not_qres_mod7_six

Exact expanded PA statement

Copy statement

forall n m. S n = S m -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. S n = S m -> n = m
apply PA2
qed

Occurrences

1

Distinct objects

1

Reused references

0

Depth

1

Cuts

0

Certificate SHA-256

2ff195ad0d0d…

le_reflOrder is reflexive Order checked

The defined order is reflexive; zero is its witness.

Prerequisites

zero_add

Used directly by

gcd_exists_relational gcd_balanced_bezout_exists factor_search_up_to prime_divisor_exists beta_crt_prefix_congruence_step bounded_beta_crt_for_existing_code beta_exclusive_recode_congruence_step beta_prefix_extend beta_prefix_product_trace_exists beta_product_functional beta_product_succ_decompose beta_product_succ_append beta_factor_prefix_product_append all_prime_succ_elim_last sorted_succ_elim_last beta_prefix_extend_sorted_succ beta_canonical_append_succ greatest_prime_divisor_search beta_canonical_append_general beta_canonical_last_factor_bound prime_factorization_exists_up_to prime_factorization_existence beta_prime_divisor_product_member beta_sorted_factor_le_last beta_nonempty_all_prime_product_ne_one beta_canonical_last_factors_equal pow_successor_decompose beta_prefix_sum_trace_exists beta_sum_trace_functional beta_sum_succ_decompose beta_product_pointwise_mod_congruent beta_sum_pointwise_mod_congruent all_bits_last_succ factorial_succ_decompose beta_prefix_swap_last_from_entries finite_swap_last_bounded finite_swap_last_injective finite_swap_last_surjective_back finite_contains_decidable finite_bounded_last_succ finite_surjective_succ_intro finite_last_is_top_from_prefix_surjective finite_bounded_injective_surjective beta_product_replace_balance beta_product_swap_last_invariant finite_fixed_last_prefix_bounded

Exact expanded PA statement

Copy statement

forall n. n <= n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n <= n
use zero_add
intro n
exists 0
simp [zero_add]
qed

Occurrences

25

Distinct objects

25

Reused references

0

Depth

9

Cuts

1

Certificate SHA-256

78852db8db26…

le_transOrder is transitive Order checked

Order witnesses compose by addition, so the defined order is transitive.

Prerequisites

add_assoc

Used directly by

lt_of_lt_of_le beta_moduli_coprime_of_lt_bounded_common_multiple bounded_beta_crt_prefix_invariant beta_value_lt_scaled_base new_value_lt_scaled_base beta_exclusive_accumulated_product_step bounded_beta_exclusive_recode_invariant beta_sorted_factor_le_last

Exact expanded PA statement

Copy statement

forall n m k. n <= m -> m <= k -> n <= k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m k. n <= m -> m <= k -> n <= k
use add_assoc
intro n
intro m
intro k
intro h_nm
intro h_mk
cases h_nm
cases h_mk
exists x1 + x
simp [add_assoc, h_nm_witness, h_mk_witness]
qed

Occurrences

57

Distinct objects

54

Reused references

3

Depth

15

Cuts

1

Certificate SHA-256

eca4525bf3b2…

no_succ_add_fixedAdding a positive successor has no fixed point Addition checked

Adding a positive successor cannot leave a natural number fixed.

Prerequisites

none — a root theorem

Used directly by

drop_add_prefix_from_fixed lt_irrefl_expanded

Exact expanded PA statement

Copy statement

forall p n. S p + n = n -> false

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p n. S p + n = n -> false
intro p
induction n
intro h
apply PA1
rewrite PA3 at h
exact h
intro h
apply IH
apply PA2
rewrite PA4 at h
exact h
qed

Occurrences

30

Distinct objects

30

Reused references

0

Depth

12

Cuts

0

Certificate SHA-256

3cfcff019a80…

drop_add_prefix_from_fixedDrop an additive prefix from a fixed-point equation Addition checked

A fixed-point equation remains fixed after dropping an additive prefix.

Prerequisites

zero_add add_succ_left no_succ_add_fixed

Used directly by

antisymm_from_witnesses

Exact expanded PA statement

Copy statement

forall a b n. (b + a) + n = n -> a + n = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b n. (b + a) + n = n -> a + n = n
use zero_add
use add_succ_left
use no_succ_add_fixed
intro a
induction b
intro n
intro h
specialize zero_add a
rewrite zero_add at h
exact h
intro n
intro h
exfalso
specialize no_succ_add_fixed (b + a)
specialize no_succ_add_fixed n
apply no_succ_add_fixed
specialize add_succ_left b
specialize add_succ_left a
rewrite add_succ_left at h
exact h
qed

Occurrences

120

Distinct objects

116

Reused references

4

Depth

22

Cuts

3

Certificate SHA-256

ebf03686ffd7…

antisymm_from_witnessesOpposing additive witnesses force equality Order checked

Opposing additive witnesses force equality.

Prerequisites

add_assoc drop_add_prefix_from_fixed

Used directly by

le_antisymm

Exact expanded PA statement

Copy statement

forall a b n m. a + n = m -> b + m = n -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b n m. a + n = m -> b + m = n -> n = m
use add_assoc
use drop_add_prefix_from_fixed
intro a
intro b
intro n
intro m
intro h_anm
intro h_bmn
symm
rewrite <- h_anm
specialize drop_add_prefix_from_fixed a
specialize drop_add_prefix_from_fixed b
specialize drop_add_prefix_from_fixed n
apply drop_add_prefix_from_fixed
specialize add_assoc b
specialize add_assoc a
specialize add_assoc n
rewrite add_assoc
rewrite h_anm
rewrite h_bmn
refl
qed

Occurrences

200

Distinct objects

191

Reused references

9

Depth

28

Cuts

5

Certificate SHA-256

b62cd9302971…

le_antisymmOrder is antisymmetric Order checked

The witness-defined order is antisymmetric.

Prerequisites

antisymm_from_witnesses

Used directly by

beta_canonical_last_factors_equal

Exact expanded PA statement

Copy statement

forall n m. n <= m -> m <= n -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. n <= m -> m <= n -> n = m
use antisymm_from_witnesses
intro n
intro m
intro h_nm
intro h_mn
cases h_nm
cases h_mn
apply antisymm_from_witnesses
exact h_nm_witness
exact h_mn_witness
qed

Occurrences

218

Distinct objects

209

Reused references

9

Depth

29

Cuts

6

Certificate SHA-256

e0c6f0ae6bc7…

le_totalNatural-number order is total Order checked

Every pair of natural numbers is comparable in the defined order.

Prerequisites

none — a root theorem

Used directly by

division_remainder_unique mod_eq_lcm_merge

Exact expanded PA statement

Copy statement

forall n m. n <= m \/ m <= n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. n <= m \/ m <= n
induction n
intro m
left
exists m
simp
induction m
right
exists (S n)
simp
specialize IH m
cases IH
cases IH_left
left
exists x
rewrite PA4
congr
exact IH_left_witness
cases IH_right
right
exists x
rewrite PA4
congr
exact IH_right_witness
qed

Occurrences

49

Distinct objects

48

Reused references

1

Depth

17

Cuts

0

Certificate SHA-256

13db746df298…

add_eq_zero_rightA zero sum has a zero right summand Addition checked

A sum equal to zero has zero as its right addend.

Prerequisites

none — a root theorem

Used directly by

mul_eq_zero le_zero add_eq_zero_left add_eq_zero_components mul_eq_one_components factor_difference bounded_beta_exclusive_recode_invariant beta_prefix_product_trace_exists all_prime_empty beta_factor_divides_product prime_factorization_exists_up_to prime_factorization_uniqueness_by_length quadratic_residue_search_up_to beta_repeat_empty beta_range_empty beta_prefix_sum_trace_exists all_bits_zero finite_surjective_zero beta_prefix_replace_exists finite_contains_decidable beta_product_replace_balance

Exact expanded PA statement

Copy statement

forall a b. a + b = 0 -> b = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a + b = 0 -> b = 0
intro a
induction b
intro h
refl
intro h
exfalso
apply PA1
rewrite PA4 at h
exact h
qed

Occurrences

19

Distinct objects

19

Reused references

0

Depth

12

Cuts

0

Certificate SHA-256

e9b362b69b8a…

mul_eq_zeroA zero product has a zero factor Multiplication checked

Zero products have a zero factor: the 23-entry core capstone.

Prerequisites

add_eq_zero_right

Used directly by

mul_ne_zero mul_left_cancel_nonzero two_large_factors_impossible bounded_common_multiple_step prime_three

Exact expanded PA statement

Copy statement

forall n m. n * m = 0 -> n = 0 \/ m = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n m. n * m = 0 -> n = 0 \/ m = 0
use add_eq_zero_right
intro n
induction m
intro h
right
refl
intro h
left
specialize add_eq_zero_right (n * m)
specialize add_eq_zero_right n
apply add_eq_zero_right
rewrite PA6 at h
exact h
qed

Occurrences

47

Distinct objects

47

Reused references

0

Depth

17

Cuts

1

Certificate SHA-256

ff49ed484000…

eq_symmEquality is symmetric Equality checked

Equality is symmetric.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. a = b -> b = a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a = b -> b = a
intro a
intro b
intro h
symm
exact h
qed

Occurrences

5

Distinct objects

5

Reused references

0

Depth

5

Cuts

0

Certificate SHA-256

a7f8dc7f3c54…

eq_transEquality is transitive Equality checked

Equality is transitive.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. a = b -> b = c -> a = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. a = b -> b = c -> a = c
intro a
intro b
intro c
intro hab
intro hbc
trans b
exact hab
exact hbc
qed

Occurrences

8

Distinct objects

8

Reused references

0

Depth

7

Cuts

0

Certificate SHA-256

b883c2beef2b…

succ_congrSuccessor preserves equality Equality checked

Successor preserves equality.

Prerequisites

none — a root theorem

Used directly by

prime_factorization_uniqueness_by_length

Exact expanded PA statement

Copy statement

forall a b. a = b -> S a = S b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a = b -> S a = S b
intro a
intro b
intro h
congr
exact h
qed

Occurrences

5

Distinct objects

5

Reused references

0

Depth

5

Cuts

0

Certificate SHA-256

50f243457746…

zero_or_succEvery natural is zero or a successor Equality checked

Every natural is either zero or the successor of a natural.

Prerequisites

none — a root theorem

Used directly by

le_eq_or_lt division_remainder_succ division_remainder_exists division_remainder_unique multiple_has_zero_remainder multiple_antisymm bounded_common_multiple_step beta_canonical_append_general prime_three

Exact expanded PA statement

Copy statement

forall n. n = 0 \/ exists k. n = S k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n = 0 \/ exists k. n = S k
induction n
left
refl
right
exists n
refl
qed

Occurrences

8

Distinct objects

8

Reused references

0

Depth

6

Cuts

0

Certificate SHA-256

c045c0f7e88d…

nonzero_is_succEvery nonzero natural has a predecessor Equality checked

Every nonzero natural has a predecessor.

Prerequisites

none — a root theorem

Used directly by

binary_crt prime_unbounded prime_factorization_uniqueness_by_length quadratic_residue_bounded_decidable_nonzero coprime_mod_inverse prime_is_succ_succ

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> exists k. n = S k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> exists k. n = S k
induction n
intro h
exfalso
apply h
refl
intro h
exists n
refl
qed

Occurrences

11

Distinct objects

11

Reused references

0

Depth

6

Cuts

0

Certificate SHA-256

ce59c977bbdf…

add_congrAddition preserves equality in both arguments Equality checked

Addition preserves equality in both arguments.

Prerequisites

none — a root theorem

Used directly by

beta_sum_trace_functional

Exact expanded PA statement

Copy statement

forall a b c d. a = b -> c = d -> a + c = b + d

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c d. a = b -> c = d -> a + c = b + d
intro a
intro b
intro c
intro d
intro hab
intro hcd
congr
exact hab
exact hcd
qed

Occurrences

9

Distinct objects

9

Reused references

0

Depth

8

Cuts

0

Certificate SHA-256

36c4025bd91f…

mul_congrMultiplication preserves equality in both arguments Equality checked

Multiplication preserves equality in both arguments.

Prerequisites

none — a root theorem

Used directly by

beta_product_functional

Exact expanded PA statement

Copy statement

forall a b c d. a = b -> c = d -> a * c = b * d

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c d. a = b -> c = d -> a * c = b * d
intro a
intro b
intro c
intro d
intro hab
intro hcd
congr
exact hab
exact hcd
qed

Occurrences

9

Distinct objects

9

Reused references

0

Depth

8

Cuts

0

Certificate SHA-256

a55a97cc7dfa…

add_right_cancelAddition cancels on the right Addition checked

A common right addend can be cancelled.

Prerequisites

none — a root theorem

Used directly by

add_left_cancel add_le_cancel_right mul_left_cancel_nonzero factor_difference odd_half_unique

Exact expanded PA statement

Copy statement

forall a b c. a + c = b + c -> a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. a + c = b + c -> a = b
intro a
intro b
induction c
intro h
rewrite PA3 at h
rewrite PA3 at h
exact h
intro h
apply IH
apply PA2
rewrite PA4 at h
rewrite PA4 at h
exact h
qed

Occurrences

41

Distinct objects

41

Reused references

0

Depth

15

Cuts

0

Certificate SHA-256

ea6126f3e712…

add_left_cancelAddition cancels on the left Addition checked

A common left addend can be cancelled.

Prerequisites

add_comm add_right_cancel

Used directly by

positive_quotient_gap_impossible remainder_unique_same_quotient division_remainder_unique beta_range_injective mod_eq_ordered_gap_multiple

Exact expanded PA statement

Copy statement

forall a b c. a + b = a + c -> b = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. a + b = a + c -> b = c
use add_comm
use add_right_cancel
intro a
intro b
intro c
intro h
specialize add_right_cancel b
specialize add_right_cancel c
specialize add_right_cancel a
apply add_right_cancel
trans a + b
apply add_comm
trans a + c
exact h
apply add_comm
qed

Occurrences

143

Distinct objects

137

Reused references

6

Depth

18

Cuts

4

Certificate SHA-256

891b97905c1d…

zero_leZero is below every natural number Order checked

Zero is below every natural number.

Prerequisites

none — a root theorem

Used directly by

beta_prefix_product_trace_exists beta_product_succ_append beta_factor_prefix_product_append beta_prefix_sum_trace_exists

Exact expanded PA statement

Copy statement

forall n. 0 <= n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. 0 <= n
intro n
exists n
rewrite PA3
refl
qed

Occurrences

7

Distinct objects

7

Reused references

0

Depth

6

Cuts

0

Certificate SHA-256

7ba583081e4b…

le_succ_selfEvery natural is below its successor Order checked

Every natural number is below its successor.

Prerequisites

zero_add add_succ_left

Used directly by

beta_accumulated_product_step bounded_beta_crt_prefix_invariant beta_exclusive_accumulated_product_step bounded_beta_exclusive_recode_invariant

Exact expanded PA statement

Copy statement

forall n. n <= S n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n <= S n
use zero_add
use add_succ_left
intro n
exists 1
simp [add_succ_left, zero_add]
qed

Occurrences

58

Distinct objects

54

Reused references

4

Depth

12

Cuts

2

Certificate SHA-256

1bdf6c9f2c04…

le_zeroOnly zero lies below zero Order checked

Only zero is less than or equal to zero.

Prerequisites

add_eq_zero_right

Used directly by

gcd_exists_up_to gcd_balanced_bezout_exists_up_to factor_search_up_to prime_divisor_exists_up_to bounded_beta_crt_prefix_invariant sorted_empty sorted_singleton greatest_prime_divisor_search prime_factorization_exists_up_to beta_sorted_factor_le_last lt_three_cases

Exact expanded PA statement

Copy statement

forall n. n <= 0 -> n = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n <= 0 -> n = 0
use add_eq_zero_right
intro n
intro h
cases h
apply add_eq_zero_right
exact h_witness
qed

Occurrences

29

Distinct objects

29

Reused references

0

Depth

13

Cuts

1

Certificate SHA-256

7674b5d583d8…

one_le_of_ne_zeroEvery nonzero natural is at least one Order checked

Every nonzero natural is at least one.

Prerequisites

none — a root theorem

Used directly by

divisor_le_nonzero le_scaled_nonzero

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> 1 <= n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> 1 <= n
induction n
intro h
exfalso
apply h
refl
intro h
exists n
simp
qed

Occurrences

20

Distinct objects

20

Reused references

0

Depth

10

Cuts

0

Certificate SHA-256

627f1a9c62d5…

ne_zero_of_one_leA natural at least one is nonzero Order checked

A natural at least one is nonzero.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. 1 <= n -> ~(n = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. 1 <= n -> ~(n = 0)
intro n
intro h
intro hn
cases h
rewrite hn at h_witness
apply PA1
rewrite PA4 at h_witness
exact h_witness
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

12

Cuts

0

Certificate SHA-256

9a743f0b80bd…

le_add_leftA number is below a left-added summand Order checked

Adding on the left produces an explicit order witness.

Prerequisites

none — a root theorem

Used directly by

base_le_beta_modulus new_value_lt_scaled_base

Exact expanded PA statement

Copy statement

forall a b. exists k. k + a = b + a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists k. k + a = b + a
intro a
intro b
exists b
refl
qed

Occurrences

4

Distinct objects

4

Reused references

0

Depth

4

Cuts

0

Certificate SHA-256

87a93b4692ab…

le_add_rightA number is below itself plus a right summand Order checked

Adding on the right produces an explicit order witness.

Prerequisites

add_comm

Used directly by

beta_value_lt_scaled_base

Exact expanded PA statement

Copy statement

forall a b. exists k. k + a = a + b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists k. k + a = a + b
use add_comm
intro a
intro b
exists b
apply add_comm
qed

Occurrences

80

Distinct objects

74

Reused references

6

Depth

13

Cuts

3

Certificate SHA-256

a7c973720d50…

add_le_add_rightAdding the same right summand preserves order Order checked

Adding the same right summand preserves the witness-defined order.

Prerequisites

add_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + a = b) -> exists r. r + (a + c) = b + c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + a = b) -> exists r. r + (a + c) = b + c
use add_assoc
intro a
intro b
intro c
intro h
cases h
exists x
trans (x + a) + c
symm
apply add_assoc
congr
exact h_witness
refl
qed

Occurrences

50

Distinct objects

47

Reused references

3

Depth

13

Cuts

1

Certificate SHA-256

b9f1f17b67a5…

add_le_add_leftAdding the same left summand preserves order Order checked

Adding the same left summand preserves the witness-defined order.

Prerequisites

add_assoc add_comm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + a = b) -> exists r. r + (c + a) = c + b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + a = b) -> exists r. r + (c + a) = c + b
use add_assoc
use add_comm
intro a
intro b
intro c
intro h
cases h
exists x
trans (x + c) + a
symm
apply add_assoc
trans (c + x) + a
congr
apply add_comm
refl
trans c + (x + a)
apply add_assoc
congr
refl
exact h_witness
qed

Occurrences

135

Distinct objects

124

Reused references

11

Depth

15

Cuts

4

Certificate SHA-256

2c818a03f8f1…

succ_le_succSuccessor preserves order Order checked

Successor preserves the witness-defined order.

Prerequisites

none — a root theorem

Used directly by

beta_value_lt_scaled_base new_value_lt_scaled_base beta_prefix_product_trace_exists beta_product_succ_append beta_factor_prefix_product_append sorted_succ_intro beta_prefix_sum_trace_exists

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + a = b) -> exists r. r + S a = S b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + a = b) -> exists r. r + S a = S b
intro a
intro b
intro h
cases h
exists x
rewrite PA4
congr
exact h_witness
qed

Occurrences

13

Distinct objects

13

Reused references

0

Depth

10

Cuts

0

Certificate SHA-256

72f0bef7672b…

le_of_succ_le_succSuccessor reflects order Order checked

Successor order reflects to the underlying naturals.

Prerequisites

none — a root theorem

Used directly by

gcd_exists_up_to gcd_balanced_bezout_exists_up_to factor_property_succ prime_divisor_exists_up_to beta_accumulated_product_step beta_crt_prefix_congruence_step beta_exclusive_accumulated_product_step beta_exclusive_recode_congruence_step beta_prefix_product_trace_exists beta_product_succ_append beta_factor_prefix_product_append all_prime_succ_intro sorted_singleton sorted_succ_intro greatest_prime_divisor_search beta_factor_divides_product prime_factorization_exists_up_to beta_sorted_factor_le_last prime_factorization_uniqueness_by_length beta_repeat_succ_extend beta_range_succ_extend beta_prefix_sum_trace_exists lt_three_cases lt_five_cases lt_seven_cases finite_lt_succ_eq_or_lt

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + S a = S b) -> exists r. r + a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + S a = S b) -> exists r. r + a = b
intro a
intro b
intro h
cases h
exists x
apply PA2
trans x + S a
symm
apply PA4
exact h_witness
qed

Occurrences

16

Distinct objects

16

Reused references

0

Depth

11

Cuts

0

Certificate SHA-256

f4da1b80d84f…

le_succOrder is preserved by raising the upper bound Order checked

A weak inequality remains true after raising its upper bound by one.

Prerequisites

add_succ_left

Used directly by

factor_search_up_to base_le_beta_modulus beta_product_functional beta_product_succ_decompose all_prime_succ_elim_prefix sorted_succ_elim_prefix greatest_prime_divisor_search beta_prime_divisor_product_member pow_successor_decompose beta_sum_trace_functional beta_sum_succ_decompose beta_product_pointwise_mod_congruent beta_sum_pointwise_mod_congruent all_bits_prefix_succ factorial_succ_decompose finite_injective_prefix_succ beta_prefix_swap_last_from_entries finite_swap_last_bounded finite_swap_last_injective finite_swap_last_surjective_back finite_contains_decidable finite_bounded_prefix_without_top finite_surjective_succ_intro finite_last_is_top_from_prefix_surjective finite_bounded_injective_surjective beta_product_replace_balance beta_product_swap_last_invariant finite_fixed_last_prefix_bounded

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + a = b) -> exists r. r + a = S b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + a = b) -> exists r. r + a = S b
use add_succ_left
intro a
intro b
intro h
cases h
exists S x
trans S (x + a)
apply add_succ_left
congr
exact h_witness
qed

Occurrences

40

Distinct objects

38

Reused references

2

Depth

11

Cuts

1

Certificate SHA-256

acb1d2bedd29…

lt_to_leStrict order entails weak order Order checked

A witnessed strict inequality entails the corresponding weak inequality.

Prerequisites

add_succ_left

Used directly by

beta_exclusive_accumulated_product_step beta_exclusive_recode_congruence_step sorted_transport

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + S a = b) -> exists r. r + a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + S a = b) -> exists r. r + a = b
use add_succ_left
intro a
intro b
intro h
cases h
exists S x
trans x + S a
trans S (x + a)
apply add_succ_left
symm
apply PA4
exact h_witness
qed

Occurrences

44

Distinct objects

42

Reused references

2

Depth

12

Cuts

1

Certificate SHA-256

04cba73adcda…

add_le_cancel_rightCancel a common right summand from order Order checked

A common right summand can be cancelled from an order comparison.

Prerequisites

add_assoc add_right_cancel

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + (a + c) = b + c) -> exists r. r + a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + (a + c) = b + c) -> exists r. r + a = b
use add_assoc
use add_right_cancel
intro a
intro b
intro c
intro h
cases h
exists x
specialize add_right_cancel (x + a)
specialize add_right_cancel b
specialize add_right_cancel c
apply add_right_cancel
trans x + (a + c)
apply add_assoc
exact h_witness
qed

Occurrences

103

Distinct objects

100

Reused references

3

Depth

20

Cuts

2

Certificate SHA-256

cec17a619915…

lt_irrefl_expandedStrict order is irreflexive Order checked

No natural is strictly below itself, with strict order fully expanded.

Prerequisites

add_succ_left no_succ_add_fixed

Used directly by

beta_accumulated_product_step beta_exclusive_accumulated_product_step beta_prefix_swap_last_from_entries finite_last_is_top_from_prefix_surjective finite_bounded_injective_surjective beta_product_replace_balance beta_product_swap_last_invariant finite_fixed_last_prefix_bounded

Exact expanded PA statement

Copy statement

forall n. ~(exists k. k + S n = n)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(exists k. k + S n = n)
use add_succ_left
use no_succ_add_fixed
intro n
intro h
cases h
specialize no_succ_add_fixed x
specialize no_succ_add_fixed n
apply no_succ_add_fixed
trans x + S n
trans S (x + n)
apply add_succ_left
symm
apply PA4
exact h_witness
qed

Occurrences

83

Distinct objects

81

Reused references

2

Depth

16

Cuts

2

Certificate SHA-256

fb85108bee59…

le_eq_or_ltA weak inequality is equality or strict inequality Order checked

A witnessed inequality is either equality or a witnessed strict inequality.

Prerequisites

zero_or_succ zero_add add_succ_left

Used directly by

remainder_bound_step gcd_exists_up_to gcd_balanced_bezout_exists_up_to factor_property_succ proper_factor_lt beta_accumulated_product_step beta_crt_prefix_congruence_step beta_exclusive_accumulated_product_step beta_exclusive_recode_congruence_step beta_prefix_product_trace_exists beta_product_succ_append beta_factor_prefix_product_append all_prime_succ_intro sorted_succ_intro greatest_prime_divisor_search beta_factor_divides_product prime_factorization_exists_up_to beta_sorted_factor_le_last prime_factorization_uniqueness_by_length quadratic_residue_search_up_to beta_repeat_succ_extend beta_range_succ_extend beta_prefix_sum_trace_exists lt_three_cases lt_five_cases lt_seven_cases finite_lt_succ_eq_or_lt

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + a = b) -> a = b \/ exists k. k + S a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + a = b) -> a = b \/ exists k. k + S a = b
use zero_or_succ
use zero_add
use add_succ_left
intro a
intro b
intro h
cases h
specialize zero_or_succ x
cases zero_or_succ
left
rewrite zero_or_succ_left at h_witness
specialize zero_add a
rewrite zero_add at h_witness
exact h_witness
cases zero_or_succ_right
right
exists x1
trans S x1 + a
trans S (x1 + a)
apply PA4
symm
apply add_succ_left
rewrite <- zero_or_succ_right_witness
exact h_witness
qed

Occurrences

98

Distinct objects

94

Reused references

4

Depth

19

Cuts

3

Certificate SHA-256

59ff9abf5aa8…

lt_of_lt_of_leStrict order followed by weak order Order checked

Strict order followed by weak order remains strict.

Prerequisites

le_trans

Used directly by

prime_divisor_exists_up_to

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + S a = b) -> (exists k. k + b = c) -> exists k. k + S a = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + S a = b) -> (exists k. k + b = c) -> exists k. k + S a = c
use le_trans
intro a
intro b
intro c
intro hab
intro hbc
specialize le_trans (S a)
specialize le_trans b
specialize le_trans c
apply le_trans
exact hab
exact hbc
qed

Occurrences

80

Distinct objects

77

Reused references

3

Depth

16

Cuts

2

Certificate SHA-256

ef269788128f…

lt_of_le_of_ltWeak order followed by strict order Order checked

Weak order followed by strict order remains strict.

Prerequisites

add_assoc

Used directly by

beta_accumulated_product_step beta_half_range_entry_bounds

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + a = b) -> (exists k. k + S b = c) -> exists k. k + S a = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + a = b) -> (exists k. k + S b = c) -> exists k. k + S a = c
use add_assoc
intro a
intro b
intro c
intro hab
intro hbc
cases hab
cases hbc
exists x1 + x
trans x1 + (x + S a)
apply add_assoc
rewrite PA4
rewrite hab_witness
exact hbc_witness
qed

Occurrences

58

Distinct objects

55

Reused references

3

Depth

15

Cuts

1

Certificate SHA-256

f3e188673abc…

lt_transStrict order is transitive Order checked

Strict order is transitive.

Prerequisites

add_assoc add_succ_left

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + S a = b) -> (exists k. k + S b = c) -> exists k. k + S a = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + S a = b) -> (exists k. k + S b = c) -> exists k. k + S a = c
use add_assoc
use add_succ_left
intro a
intro b
intro c
intro hab
intro hbc
cases hab
cases hbc
exists x1 + S x
trans x1 + (S x + S a)
apply add_assoc
trans x1 + S (x + S a)
congr
refl
apply add_succ_left
trans x1 + S b
congr
refl
congr
exact hab_witness
exact hbc_witness
qed

Occurrences

89

Distinct objects

82

Reused references

7

Depth

16

Cuts

2

Certificate SHA-256

83ea214a0810…

le_or_ltWeak-or-reverse-strict comparison Order checked

Any two naturals satisfy weak order in one direction or strict order in the other.

Prerequisites

none — a root theorem

Used directly by

prime_unbounded

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + a = b) \/ exists k. k + S b = a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + a = b) \/ exists k. k + S b = a
induction a
intro b
left
exists b
apply PA3
induction b
right
exists a
trans S (a + 0)
apply PA4
congr
apply PA3
specialize IH b
cases IH
left
cases IH_left
exists x
rewrite PA4
congr
exact IH_left_witness
right
cases IH_right
exists x
rewrite PA4
congr
exact IH_right_witness
qed

Occurrences

48

Distinct objects

48

Reused references

0

Depth

17

Cuts

0

Certificate SHA-256

8317909b51ce…

lt_trichotomyNatural strict-order trichotomy Order checked

Two naturals are equal or strictly ordered in exactly one displayed direction.

Prerequisites

none — a root theorem

Used directly by

beta_moduli_pairwise_coprime_bounded

Exact expanded PA statement

Copy statement

forall a b. a = b \/ ((exists k. k + S a = b) \/ exists k. k + S b = a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a = b \/ ((exists k. k + S a = b) \/ exists k. k + S b = a)
induction a
induction b
left
refl
right
left
exists b
trans S (b + 0)
apply PA4
congr
apply PA3
induction b
right
right
exists a
trans S (a + 0)
apply PA4
congr
apply PA3
specialize IH b
cases IH
left
congr
exact IH_left
cases IH_right
right
left
cases IH_right_left
exists x
rewrite PA4
congr
exact IH_right_left_witness
right
right
cases IH_right_right
exists x
rewrite PA4
congr
exact IH_right_right_witness
qed

Occurrences

66

Distinct objects

66

Reused references

0

Depth

19

Cuts

0

Certificate SHA-256

abdf421beb86…

lt_not_leStrict order excludes reverse weak order Order checked

A strict inequality excludes the reverse weak inequality.

Prerequisites

none — a root theorem

Used directly by

le_not_lt prime_bounded_nonzero_mod_inverse

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + S a = b) -> ~ (exists k. k + b = a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + S a = b) -> ~ (exists k. k + b = a)
have hbad : forall z u v. ~(z = u + (v + S z))
intro z
induction z
intro u
intro v
intro hz
apply PA1
symm
rewrite PA4 at hz
rewrite PA4 at hz
exact hz
intro u
intro v
intro hz
specialize IH u
specialize IH v
apply IH
apply PA2
rewrite PA4 at hz
rewrite PA4 at hz
exact hz
intro a
intro b
intro hab
intro hba
cases hab
cases hba
specialize hbad a
specialize hbad x1
specialize hbad x
apply hbad
symm
rewrite <- hab_witness at hba_witness
exact hba_witness
qed

Occurrences

56

Distinct objects

56

Reused references

0

Depth

23

Cuts

0

Certificate SHA-256

e410e57fb272…

le_not_ltWeak order excludes reverse strict order Order checked

A weak inequality excludes strict inequality in the reverse direction.

Prerequisites

lt_not_le

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. (exists k. k + a = b) -> ~ (exists k. k + S b = a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists k. k + a = b) -> ~ (exists k. k + S b = a)
use lt_not_le
intro a
intro b
intro hab
intro hba
specialize lt_not_le b
specialize lt_not_le a
apply lt_not_le
exact hba
exact hab
qed

Occurrences

74

Distinct objects

74

Reused references

0

Depth

24

Cuts

1

Certificate SHA-256

d9e660d3f46d…

lt_not_eq_add_middleA strict bound cannot occur inside an additive decomposition of the lower term Order checked

A strict upper bound prevents the lower term from containing that bound as an additive middle block.

Prerequisites

add_succ_left

Used directly by

positive_quotient_gap_impossible

Exact expanded PA statement

Copy statement

forall r m a b. (exists k. k + S r = m) -> ~(r = (a + m) + b)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall r m a b. (exists k. k + S r = m) -> ~(r = (a + m) + b)
use add_succ_left
have hbad : forall z a w b. ~(z = (a + (w + S z)) + b)
intro z
induction z
intro a
intro w
intro b
intro hz
apply PA1
symm
rewrite PA4 at hz
rewrite PA4 at hz
specialize add_succ_left (a + (w + 0))
specialize add_succ_left b
rewrite add_succ_left at hz
exact hz
intro a
intro w
intro b
intro hz
specialize IH a
specialize IH w
specialize IH b
apply IH
apply PA2
rewrite PA4 at hz
rewrite PA4 at hz
specialize add_succ_left (a + (w + S z))
specialize add_succ_left b
rewrite add_succ_left at hz
exact hz
intro r
intro m
intro a
intro b
intro hr
intro heq
cases hr
specialize hbad r
specialize hbad a
specialize hbad x
specialize hbad b
apply hbad
rewrite <- hr_witness at heq
exact heq
qed

Occurrences

95

Distinct objects

93

Reused references

2

Depth

28

Cuts

1

Certificate SHA-256

77ec046fb0e4…

mul_le_mul_leftLeft multiplication preserves order Multiplication checked

Left multiplication preserves the witness-defined order.

Prerequisites

mul_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + a = b) -> exists r. r + c * a = c * b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + a = b) -> exists r. r + c * a = c * b
use mul_add
intro a
intro b
intro c
intro h
cases h
exists c * x
trans c * (x + a)
symm
apply mul_add
congr
refl
exact h_witness
qed

Occurrences

94

Distinct objects

87

Reused references

7

Depth

17

Cuts

2

Certificate SHA-256

44b7ebda990f…

mul_le_mul_rightRight multiplication preserves order Multiplication checked

Right multiplication preserves the witness-defined order.

Prerequisites

add_mul

Used directly by

le_scaled_nonzero

Exact expanded PA statement

Copy statement

forall a b c. (exists k. k + a = b) -> exists r. r + a * c = b * c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. (exists k. k + a = b) -> exists r. r + a * c = b * c
use add_mul
intro a
intro b
intro c
intro h
cases h
exists x * c
trans (x + a) * c
symm
apply add_mul
congr
exact h_witness
refl
qed

Occurrences

343

Distinct objects

280

Reused references

31

Depth

26

Cuts

10

Certificate SHA-256

71720fa700b3…

mul_lt_mul_succ_left_nonzeroA nonzero left factor strictly separates successor products Multiplication checked

Multiplication by a nonzero left factor strictly increases across a successor step.

Prerequisites

add_comm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall c a. ~(c = 0) -> exists r. r + S (c * a) = c * S a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c a. ~(c = 0) -> exists r. r + S (c * a) = c * S a
use add_comm
induction c
intro a
intro hc
exfalso
apply hc
refl
intro a
intro hc
exists c
trans S (c + S c * a)
apply PA4
trans S (S c * a + c)
congr
apply add_comm
trans S c * a + S c
symm
apply PA4
symm
apply PA6
qed

Occurrences

104

Distinct objects

98

Reused references

6

Depth

14

Cuts

3

Certificate SHA-256

9a2a8ae4e6ea…

division_remainder_succDivision with remainder for successor divisors Division checked

Every dividend has a quotient and bounded remainder for a successor divisor.

Prerequisites

zero_add add_succ_left zero_or_succ

Used directly by

division_remainder_exists

Exact expanded PA statement

Copy statement

forall d n. exists q r. n = S d * q + r /\ S r <= S d

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d n. exists q r. n = S d * q + r /\ S r <= S d
use zero_add
use add_succ_left
use zero_or_succ
intro d
induction n
exists 0
exists 0
split
simp
exists d
simp
cases IH
cases IH_witness
cases IH_witness_witness
cases IH_witness_witness_right
specialize zero_or_succ x2
cases zero_or_succ
rewrite zero_or_succ_left at IH_witness_witness_right_witness
specialize zero_add S x1
rewrite zero_add at IH_witness_witness_right_witness
have hrd : x1 = d
apply PA2
exact IH_witness_witness_right_witness
exists S x
exists 0
split
rewrite IH_witness_witness_left
rewrite hrd
simp
exists d
simp
cases zero_or_succ_right
exists x
exists S x1
split
rewrite IH_witness_witness_left
simp
exists x3
rewrite <- IH_witness_witness_right_witness
rewrite zero_or_succ_right_witness
simp [add_succ_left]
qed

Occurrences

179

Distinct objects

164

Reused references

15

Depth

26

Cuts

3

Certificate SHA-256

1b331ac3a9c4…

division_remainder_existsDivision with remainder for every nonzero divisor Division checked

Every positive divisor admits a quotient and a strictly bounded remainder.

Prerequisites

zero_or_succ division_remainder_succ

Used directly by

gcd_exists_up_to gcd_balanced_bezout_exists_up_to multiple_decidable_nonzero mod_eq_to_remainder_decomposition beta_at_exists mod_eq_decidable_nonzero quadratic_residue_bounded_equiv prime_bounded_nonzero_mod_inverse canonical_remainder_exists mod_eq_common_remainder_decomposition crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall m n. ~(m = 0) -> exists q r. n = m * q + r /\ S r <= m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. ~(m = 0) -> exists q r. n = m * q + r /\ S r <= m
use zero_or_succ
use division_remainder_succ
intro m
intro n
intro hm
specialize zero_or_succ m
cases zero_or_succ
exfalso
apply hm
exact zero_or_succ_left
cases zero_or_succ_right
specialize division_remainder_succ x
specialize division_remainder_succ n
rewrite zero_or_succ_right_witness
rewrite zero_or_succ_right_witness
exact division_remainder_succ
qed

Occurrences

219

Distinct objects

196

Reused references

16

Depth

28

Cuts

5

Certificate SHA-256

9642451ab71c…

remainder_bound_stepAdvance a strict remainder bound by one step Division checked

A bounded remainder either reaches the divisor or remains bounded after successor.

Prerequisites

le_eq_or_lt

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall r d. (exists k. k + S r = d) -> S r = d \/ exists k. k + S (S r) = d

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall r d. (exists k. k + S r = d) -> S r = d \/ exists k. k + S (S r) = d
use le_eq_or_lt
intro r
intro d
intro h
specialize le_eq_or_lt (S r)
specialize le_eq_or_lt d
apply le_eq_or_lt
exact h
qed

Occurrences

113

Distinct objects

109

Reused references

4

Depth

20

Cuts

4

Certificate SHA-256

715004086fe6…

division_block_upperA bounded remainder lies below the next divisor block Division checked

A bounded remainder keeps its decomposition below the next divisor block.

Prerequisites

add_assoc add_comm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall d q r. (exists k. k + S r = d) -> exists k. k + S (d * q + r) = d * S q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d q r. (exists k. k + S r = d) -> exists k. k + S (d * q + r) = d * S q
use add_assoc
use add_comm
intro d
intro q
intro r
intro h
cases h
exists x
trans S (x + (d * q + r))
apply PA4
trans S ((x + d * q) + r)
congr
symm
apply add_assoc
trans S ((d * q + x) + r)
congr
congr
apply add_comm
refl
trans S (d * q + (x + r))
congr
apply add_assoc
trans d * q + S (x + r)
symm
apply PA4
trans d * q + (x + S r)
congr
refl
symm
apply PA4
trans d * q + d
congr
refl
exact h_witness
symm
apply PA6
qed

Occurrences

159

Distinct objects

148

Reused references

11

Depth

19

Cuts

4

Certificate SHA-256

a723be287d1c…

positive_quotient_gap_impossibleA positive quotient gap contradicts bounded remainders Division checked

A positive gap between quotients makes two bounded-remainder decompositions unequal.

Prerequisites

add_comm add_assoc mul_add add_left_cancel lt_not_eq_add_middle

Used directly by

division_remainder_unique

Exact expanded PA statement

Copy statement

forall m q q2 r s k. (exists z. z + S r = m) -> S k + q = q2 -> ~(m * q + r = m * q2 + s)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m q q2 r s k. (exists z. z + S r = m) -> S k + q = q2 -> ~(m * q + r = m * q2 + s)
use add_comm
use add_assoc
use mul_add
use add_left_cancel
use lt_not_eq_add_middle
intro m
intro q
intro q2
intro r
intro s
intro k
intro hr
intro hgap
intro heq
specialize lt_not_eq_add_middle r
specialize lt_not_eq_add_middle m
specialize lt_not_eq_add_middle (m * k)
specialize lt_not_eq_add_middle s
apply lt_not_eq_add_middle
exact hr
specialize add_left_cancel (m * q)
specialize add_left_cancel r
specialize add_left_cancel ((m * k + m) + s)
apply add_left_cancel
trans m * q2 + s
exact heq
rewrite <- hgap
specialize add_comm S k
specialize add_comm q
rewrite add_comm
specialize mul_add m
specialize mul_add q
specialize mul_add S k
rewrite mul_add
rewrite PA6
specialize add_assoc (m * q)
specialize add_assoc (m * k + m)
specialize add_assoc s
apply add_assoc
qed

Occurrences

517

Distinct objects

369

Reused references

18

Depth

52

Cuts

13

Certificate SHA-256

de44a14fab4f…

remainder_unique_same_quotientRemainders are unique at a fixed quotient Division checked

Equal decompositions with the same quotient have equal remainders.

Prerequisites

add_left_cancel

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall d q r s. d * q + r = d * q + s -> r = s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d q r s. d * q + r = d * q + s -> r = s
use add_left_cancel
intro d
intro q
intro r
intro s
intro h
specialize add_left_cancel (d * q)
specialize add_left_cancel r
specialize add_left_cancel s
apply add_left_cancel
exact h
qed

Occurrences

164

Distinct objects

158

Reused references

6

Depth

19

Cuts

5

Certificate SHA-256

45700cc00c1d…

division_remainder_uniqueQuotient and remainder are unique Division checked

Bounded quotient-remainder decompositions have unique quotients and remainders.

Prerequisites

zero_add le_total zero_or_succ add_left_cancel positive_quotient_gap_impossible

Used directly by

multiple_decidable_nonzero mod_eq_bounded_unique beta_at_unique even_odd_exclusive_pointwise mod4_one_three_exclusive_pointwise canonical_remainder_functional

Exact expanded PA statement

Copy statement

forall m n q r q2 r2. n = m * q + r -> (exists k. k + S r = m) -> n = m * q2 + r2 -> (exists k. k + S r2 = m) -> q = q2 /\ r = r2

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n q r q2 r2. n = m * q + r -> (exists k. k + S r = m) -> n = m * q2 + r2 -> (exists k. k + S r2 = m) -> q = q2 /\ r = r2
use zero_add
use le_total
use zero_or_succ
use add_left_cancel
use positive_quotient_gap_impossible
intro m
intro n
intro q
intro r
intro q2
intro r2
intro h1
intro hr
intro h2
intro hr2
have hsum : m * q + r = m * q2 + r2
trans n
symm
exact h1
exact h2
specialize le_total q
specialize le_total q2
cases le_total
cases le_total_left
specialize zero_or_succ x
cases zero_or_succ
rewrite zero_or_succ_left at le_total_left_witness
specialize zero_add q
rewrite zero_add at le_total_left_witness
split
exact le_total_left_witness
specialize add_left_cancel (m * q)
specialize add_left_cancel r
specialize add_left_cancel r2
apply add_left_cancel
rewrite <- le_total_left_witness at hsum
exact hsum
cases zero_or_succ_right
exfalso
specialize positive_quotient_gap_impossible m
specialize positive_quotient_gap_impossible q
specialize positive_quotient_gap_impossible q2
specialize positive_quotient_gap_impossible r
specialize positive_quotient_gap_impossible r2
specialize positive_quotient_gap_impossible x1
apply positive_quotient_gap_impossible
exact hr
rewrite zero_or_succ_right_witness at le_total_left_witness
exact le_total_left_witness
exact hsum
cases le_total_right
specialize zero_or_succ x
cases zero_or_succ
rewrite zero_or_succ_left at le_total_right_witness
specialize zero_add q2
rewrite zero_add at le_total_right_witness
split
symm
exact le_total_right_witness
specialize add_left_cancel (m * q)
specialize add_left_cancel r
specialize add_left_cancel r2
apply add_left_cancel
rewrite le_total_right_witness at hsum
exact hsum
cases zero_or_succ_right
exfalso
specialize positive_quotient_gap_impossible m
specialize positive_quotient_gap_impossible q2
specialize positive_quotient_gap_impossible q
specialize positive_quotient_gap_impossible r2
specialize positive_quotient_gap_impossible r
specialize positive_quotient_gap_impossible x1
apply positive_quotient_gap_impossible
exact hr2
rewrite zero_or_succ_right_witness at le_total_right_witness
exact le_total_right_witness
symm
exact hsum
qed

Occurrences

854

Distinct objects

544

Reused references

22

Depth

57

Cuts

22

Certificate SHA-256

1c8ee6b98aae…

zero_remainder_implies_multipleZero remainder implies divisibility Division checked

A quotient decomposition with zero remainder supplies a divisibility witness.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n q. n = m * q + 0 -> exists k. n = m * k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n q. n = m * q + 0 -> exists k. n = m * k
intro m
intro n
intro q
intro h
exists q
rewrite h
simp
qed

Occurrences

13

Distinct objects

13

Reused references

0

Depth

10

Cuts

0

Certificate SHA-256

983bdc1e759a…

multiple_has_zero_remainderA multiple has a bounded zero remainder Division checked

Every multiple of a nonzero divisor has a bounded zero-remainder decomposition.

Prerequisites

zero_or_succ

Used directly by

multiple_decidable_nonzero

Exact expanded PA statement

Copy statement

forall m n. ~(m = 0) -> (exists q. n = m * q) -> exists q r. (n = m * q + r /\ r = 0) /\ S r <= m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. ~(m = 0) -> (exists q. n = m * q) -> exists q r. (n = m * q + r /\ r = 0) /\ S r <= m
use zero_or_succ
intro m
intro n
intro hm
intro hd
cases hd
specialize zero_or_succ m
cases zero_or_succ
exfalso
apply hm
exact zero_or_succ_left
cases zero_or_succ_right
exists x
exists 0
split
split
rewrite hd_witness
simp
refl
exists x1
rewrite zero_or_succ_right_witness
simp
qed

Occurrences

54

Distinct objects

53

Reused references

1

Depth

20

Cuts

1

Certificate SHA-256

2dd49a512758…

add_eq_zero_leftA zero sum has a zero left summand Addition checked

A sum equal to zero has zero as its left addend.

Prerequisites

add_comm add_eq_zero_right

Used directly by

add_eq_zero_components two_large_factors_impossible bounded_common_multiple_exists prime_three

Exact expanded PA statement

Copy statement

forall a b. a + b = 0 -> a = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a + b = 0 -> a = 0
use add_comm
use add_eq_zero_right
intro a
intro b
intro h
specialize add_eq_zero_right b
specialize add_eq_zero_right a
apply add_eq_zero_right
trans a + b
apply add_comm
exact h
qed

Occurrences

112

Distinct objects

106

Reused references

6

Depth

14

Cuts

4

Certificate SHA-256

546a7553991f…

add_eq_zero_componentsBoth summands of a zero sum are zero Addition checked

Both summands of a zero sum are zero.

Prerequisites

add_eq_zero_left add_eq_zero_right

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. a + b = 0 -> a = 0 /\ b = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a + b = 0 -> a = 0 /\ b = 0
use add_eq_zero_left
use add_eq_zero_right
intro a
intro b
intro h
split
apply add_eq_zero_left
exact h
apply add_eq_zero_right
exact h
qed

Occurrences

147

Distinct objects

122

Reused references

7

Depth

15

Cuts

6

Certificate SHA-256

5ebeb330038f…

mul_eq_one_componentsBoth factors of a product equal to one are one Multiplication checked

A product is one only when both natural factors are one.

Prerequisites

mul_zero_left add_eq_zero_right one_mul

Used directly by

divisor_one multiple_antisymm prime_three

Exact expanded PA statement

Copy statement

forall a b. a * b = 1 -> a = 1 /\ b = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a * b = 1 -> a = 1 /\ b = 1
use mul_zero_left
use add_eq_zero_right
use one_mul
intro a
induction a
intro b
intro h
specialize mul_zero_left b
rewrite mul_zero_left at h
exfalso
apply PA1
symm
exact h
intro b
induction b
intro h
rewrite PA5 at h
exfalso
apply PA1
symm
exact h
intro h
rewrite PA6 at h
rewrite PA4 at h
have hz : S a * b + a = 0
apply PA2
exact h
specialize add_eq_zero_right (S a * b)
specialize add_eq_zero_right a
have ha0 : a = 0
apply add_eq_zero_right
exact hz
split
congr
exact ha0
rewrite ha0 at hz
rewrite ha0 at hz
rewrite PA3 at hz
specialize one_mul b
rewrite one_mul at hz
congr
exact hz
qed

Occurrences

176

Distinct objects

164

Reused references

12

Depth

27

Cuts

3

Certificate SHA-256

a6ffdfad0473…

mul_ne_zeroA product of nonzero naturals is nonzero Multiplication checked

A product of two nonzero naturals is nonzero.

Prerequisites

mul_eq_zero

Used directly by

mul_left_cancel_nonzero beta_accumulated_product_step beta_exclusive_accumulated_product_step

Exact expanded PA statement

Copy statement

forall a b. ~(a = 0) -> ~(b = 0) -> ~(a * b = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. ~(a = 0) -> ~(b = 0) -> ~(a * b = 0)
use mul_eq_zero
intro a
intro b
intro ha
intro hb
intro hab
specialize mul_eq_zero a
specialize mul_eq_zero b
have hz : a = 0 \/ b = 0
apply mul_eq_zero
exact hab
cases hz
apply ha
exact hz_left
apply hb
exact hz_right
qed

Occurrences

65

Distinct objects

65

Reused references

0

Depth

18

Cuts

2

Certificate SHA-256

645eefcefbe2…

mul_left_cancel_nonzeroA nonzero factor cancels on the left Multiplication checked

A nonzero common left factor can be cancelled.

Prerequisites

mul_eq_zero mul_ne_zero add_right_cancel succ_ne_zero

Used directly by

mul_right_cancel_nonzero multiple_antisymm factor_search_up_to proper_factor_lt two_prime_product_uniqueness odd_half_unique is_lcm_scale_nonzero balanced_bezout_cancel_gcd is_gcd_quotients_coprime_nonzero

Exact expanded PA statement

Copy statement

forall a b c. ~(a = 0) -> a * b = a * c -> b = c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. ~(a = 0) -> a * b = a * c -> b = c
use mul_eq_zero
use mul_ne_zero
use add_right_cancel
use succ_ne_zero
intro a
induction b
intro c
intro ha
intro h
have hz : a * c = 0
symm
rewrite PA5 at h
exact h
have factors : a = 0 \/ c = 0
specialize mul_eq_zero a
specialize mul_eq_zero c
apply mul_eq_zero
exact hz
cases factors
exfalso
apply ha
exact factors_left
symm
exact factors_right
intro c
induction c
intro ha
intro h
exfalso
specialize mul_ne_zero a
specialize mul_ne_zero (S b)
apply mul_ne_zero
exact ha
specialize succ_ne_zero b
exact succ_ne_zero
rewrite PA5 at h
exact h
intro ha
intro h
congr
apply IH
exact ha
apply add_right_cancel
rewrite PA6 at h
rewrite PA6 at h
exact h
qed

Occurrences

224

Distinct objects

177

Reused references

1

Depth

23

Cuts

7

Certificate SHA-256

6ac15cdf9d96…

mul_right_cancel_nonzeroA nonzero factor cancels on the right Multiplication checked

A nonzero common right factor can be cancelled.

Prerequisites

mul_comm mul_left_cancel_nonzero

Used directly by

beta_canonical_product_cancel_last

Exact expanded PA statement

Copy statement

forall a b c. ~(c = 0) -> a * c = b * c -> a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c. ~(c = 0) -> a * c = b * c -> a = b
use mul_comm
use mul_left_cancel_nonzero
intro a
intro b
intro c
intro hc
intro h
specialize mul_left_cancel_nonzero c
specialize mul_left_cancel_nonzero a
specialize mul_left_cancel_nonzero b
apply mul_left_cancel_nonzero
exact hc
trans a * c
apply mul_comm
trans b * c
exact h
apply mul_comm
qed

Occurrences

478

Distinct objects

409

Reused references

23

Depth

25

Cuts

15

Certificate SHA-256

7c4fbfcfbc65…

two_large_factors_impossibleTwo is not a product of two successors of successors Multiplication checked

Two naturals at least two cannot multiply to two.

Prerequisites

mul_succ_left add_eq_zero_left mul_eq_zero

Used directly by

prime_two

Exact expanded PA statement

Copy statement

forall a b. ~(2 = S (S a) * S (S b))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. ~(2 = S (S a) * S (S b))
use mul_succ_left
use add_eq_zero_left
use mul_eq_zero
intro a
intro b
intro h
specialize mul_succ_left (S a)
specialize mul_succ_left (S (S b))
rewrite mul_succ_left at h
rewrite PA4 at h
rewrite PA4 at h
have hzero : 0 = S a * S (S b) + b
apply PA2
apply PA2
exact h
have hzero' : S a * S (S b) + b = 0
symm
exact hzero
have hprod : S a * S (S b) = 0
apply add_eq_zero_left
exact hzero'
specialize mul_eq_zero (S a)
specialize mul_eq_zero (S (S b))
have hz : S a = 0 \/ S (S b) = 0
apply mul_eq_zero
exact hprod
cases hz
apply PA1
exact hz_left
apply PA1
exact hz_right
qed

Occurrences

376

Distinct objects

267

Reused references

19

Depth

23

Cuts

12

Certificate SHA-256

33d33a9acbc7…

prime_twoTwo is prime Primes checked

Two is prime in the expanded first-order prime predicate.

Prerequisites

mul_zero_left two_large_factors_impossible

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(2 = 1) /\ forall a b. 2 = a * b -> a = 1 \/ b = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(2 = 1) /\ forall a b. 2 = a * b -> a = 1 \/ b = 1
use mul_zero_left
use two_large_factors_impossible
split
intro h
have h10 : 1 = 0
apply PA2
exact h
apply PA1
exact h10
intro a
induction b
intro h
rewrite PA5 at h
exfalso
apply PA1
exact h
induction b
intro h
right
refl
induction a
intro h
specialize mul_zero_left (S (S b))
rewrite mul_zero_left at h
exfalso
apply PA1
exact h
induction a
intro h
left
refl
intro h
exfalso
specialize two_large_factors_impossible a
specialize two_large_factors_impossible b
apply two_large_factors_impossible
exact h
qed

Occurrences

456

Distinct objects

344

Reused references

22

Depth

25

Cuts

14

Certificate SHA-256

6f30b33256b8…

multiple_zeroEvery natural number divides zero Divisibility checked

Zero is a multiple of every natural number.

Prerequisites

none — a root theorem

Used directly by

is_gcd_zero_right is_lcm_zero_right

Exact expanded PA statement

Copy statement

forall a. exists q. 0 = a * q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. exists q. 0 = a * q
intro a
exists 0
rewrite PA5
refl
qed

Occurrences

7

Distinct objects

7

Reused references

0

Depth

6

Cuts

0

Certificate SHA-256

2304f77f5c8a…

one_multipleOne divides every natural number Divisibility checked

Every natural number is a multiple of one.

Prerequisites

one_mul

Used directly by

coprime_to_is_gcd_one

Exact expanded PA statement

Copy statement

forall n. exists q. n = 1 * q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. exists q. n = 1 * q
use one_mul
intro n
exists n
symm
apply one_mul
qed

Occurrences

32

Distinct objects

32

Reused references

0

Depth

10

Cuts

1

Certificate SHA-256

3493ac871515…

multiple_reflDivisibility is reflexive Divisibility checked

Every natural number is a multiple of itself.

Prerequisites

mul_one

Used directly by

is_gcd_zero_right is_gcd_of_dvd gcd_exists_up_to gauss_coprime_cancel prime_divisor_exists_up_to beta_moduli_coprime_of_gap_dvd bounded_beta_crt_prefix_invariant

Exact expanded PA statement

Copy statement

forall a. exists q. a = a * q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. exists q. a = a * q
use mul_one
intro a
exists 1
symm
apply mul_one
qed

Occurrences

39

Distinct objects

39

Reused references

0

Depth

10

Cuts

2

Certificate SHA-256

f8eb406ea24d…

multiple_addA common divisor divides a sum Divisibility checked

Multiples of a fixed number are closed under addition.

Prerequisites

mul_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a n m. (exists q. n = a * q) -> (exists r. m = a * r) -> exists s. n + m = a * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a n m. (exists q. n = a * q) -> (exists r. m = a * r) -> exists s. n + m = a * s
use mul_add
intro a
intro n
intro m
intro hn
intro hm
cases hn
cases hm
exists x + x1
rewrite hn_witness
rewrite hm_witness
symm
apply mul_add
qed

Occurrences

99

Distinct objects

92

Reused references

7

Depth

17

Cuts

2

Certificate SHA-256

40c1927b5346…

multiple_mul_rightA divisor remains a divisor after right multiplication Divisibility checked

A right multiple of a multiple remains a multiple.

Prerequisites

mul_assoc

Used directly by

multiple_mul_left bounded_common_multiple_step beta_accumulated_product_step scaled_bounded_common_multiple beta_exclusive_accumulated_product_step beta_factor_divides_product

Exact expanded PA statement

Copy statement

forall a n m. (exists q. n = a * q) -> exists s. n * m = a * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a n m. (exists q. n = a * q) -> exists s. n * m = a * s
use mul_assoc
intro a
intro n
intro m
intro hn
cases hn
exists x * m
rewrite hn_witness
apply mul_assoc
qed

Occurrences

131

Distinct objects

119

Reused references

12

Depth

18

Cuts

3

Certificate SHA-256

e2b7b5396400…

multiple_mul_leftA divisor remains a divisor after left multiplication Divisibility checked

A left multiple of a multiple remains a multiple.

Prerequisites

mul_comm multiple_mul_right

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a n m. (exists q. n = a * q) -> exists s. m * n = a * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a n m. (exists q. n = a * q) -> exists s. m * n = a * s
use mul_comm
use multiple_mul_right
intro a
intro n
intro m
intro hn
suffices hswap : m * n = n * m
rewrite hswap
apply multiple_mul_right
exact hn
apply mul_comm
qed

Occurrences

370

Distinct objects

304

Reused references

34

Depth

25

Cuts

11

Certificate SHA-256

532e24fbf506…

multiple_transDivisibility is transitive Divisibility checked

The multiple relation is transitive.

Prerequisites

mul_assoc

Used directly by

prime_divisor_exists_up_to beta_moduli_coprime_of_gap_dvd coprime_mul_left greatest_prime_divisor_quotient_bound

Exact expanded PA statement

Copy statement

forall a b n. (exists q. n = a * q) -> (exists r. a = b * r) -> exists s. n = b * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b n. (exists q. n = a * q) -> (exists r. a = b * r) -> exists s. n = b * s
use mul_assoc
intro a
intro b
intro n
intro hn
intro hab
cases hn
cases hab
exists x1 * x
rewrite hn_witness
rewrite hab_witness
apply mul_assoc
qed

Occurrences

137

Distinct objects

125

Reused references

12

Depth

18

Cuts

3

Certificate SHA-256

8cb32c63f8ac…

divisor_le_nonzeroA divisor of a nonzero natural is bounded Divisibility checked

A divisor of a nonzero natural is bounded by that natural.

Prerequisites

one_le_of_ne_zero

Used directly by

prime_or_composite proper_factor_lt greatest_prime_divisor_exists prime_bounded_nonzero_mod_inverse

Exact expanded PA statement

Copy statement

forall d n. ~(n = 0) -> (exists q. n = d * q) -> exists k. k + d = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d n. ~(n = 0) -> (exists q. n = d * q) -> exists k. k + d = n
use one_le_of_ne_zero
intro d
intro n
intro hn
intro hd
cases hd
have hq : ~(x = 0)
intro hx
apply hn
trans d * x
exact hd_witness
rewrite hx
apply PA5
specialize one_le_of_ne_zero x
have h1q : exists k. k + 1 = x
apply one_le_of_ne_zero
exact hq
cases h1q
have hs : S x1 = x
trans x1 + 1
simp
exact h1q_witness
exists d * x1
trans d * S x1
symm
apply PA6
trans d * x
congr
refl
exact hs
symm
exact hd_witness
qed

Occurrences

64

Distinct objects

62

Reused references

2

Depth

17

Cuts

1

Certificate SHA-256

b96a5e2d2843…

divisor_oneEvery divisor of one is one Divisibility checked

Every natural divisor of one equals one.

Prerequisites

mul_eq_one_components

Used directly by

coprime_one_right coprime_one_left is_gcd_one_to_coprime beta_modulus_coprime_base prime_unbounded beta_prime_divisor_product_member beta_nonempty_all_prime_product_ne_one mod_inverse_implies_coprime balanced_bezout_one_implies_coprime is_gcd_quotients_coprime_nonzero

Exact expanded PA statement

Copy statement

forall d. (exists y. 1 = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d. (exists y. 1 = d * y) -> d = 1
use mul_eq_one_components
intro d
intro h
cases h
specialize mul_eq_one_components d
specialize mul_eq_one_components x
have parts : d = 1 /\ x = 1
apply mul_eq_one_components
symm
exact h_witness
cases parts
exact parts_left
qed

Occurrences

188

Distinct objects

176

Reused references

12

Depth

28

Cuts

4

Certificate SHA-256

ef2c18d807f3…

multiple_antisymmDivisibility is antisymmetric Divisibility checked

Mutual divisibility is antisymmetric over natural numbers.

Prerequisites

zero_or_succ mul_zero_left mul_assoc mul_one mul_left_cancel_nonzero mul_eq_one_components

Used directly by

is_gcd_unique is_lcm_unique

Exact expanded PA statement

Copy statement

forall a b. (exists x. b = a * x) -> (exists y. a = b * y) -> a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists x. b = a * x) -> (exists y. a = b * y) -> a = b
use zero_or_succ
use mul_zero_left
use mul_assoc
use mul_one
use mul_left_cancel_nonzero
use mul_eq_one_components
intro a
intro b
intro hab
intro hba
cases hab
cases hba
specialize zero_or_succ a
cases zero_or_succ
rewrite zero_or_succ_left
rewrite zero_or_succ_left at hab_witness
specialize mul_zero_left x
rewrite mul_zero_left at hab_witness
symm
exact hab_witness
cases zero_or_succ_right
have ha : ~(a = 0)
intro ha0
rewrite zero_or_succ_right_witness at ha0
apply PA1
exact ha0
have hcycle : a = a * (x * x1)
trans b * x1
exact hba_witness
trans (a * x) * x1
congr
exact hab_witness
refl
apply mul_assoc
specialize mul_left_cancel_nonzero a
specialize mul_left_cancel_nonzero 1
specialize mul_left_cancel_nonzero (x * x1)
have hunit : 1 = x * x1
apply mul_left_cancel_nonzero
exact ha
specialize mul_one a
trans a
apply mul_one
exact hcycle
specialize mul_eq_one_components x
specialize mul_eq_one_components x1
have hparts : x = 1 /\ x1 = 1
apply mul_eq_one_components
symm
exact hunit
cases hparts
symm
trans a * x
exact hab_witness
rewrite hparts_left
apply mul_one
qed

Occurrences

646

Distinct objects

527

Reused references

35

Depth

33

Cuts

19

Certificate SHA-256

2b8e9547aaac…

factor_differenceA common-factor difference retains the factor Divisibility checked

A common-factor difference is itself a multiple of that factor.

Prerequisites

zero_add add_eq_zero_right add_right_cancel add_assoc add_comm

Used directly by

divides_remainder common_divisor_divides_balanced_result mod_eq_ordered_gap_multiple

Exact expanded PA statement

Copy statement

forall c u v r. c * u = c * v + r -> exists w. r = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c u v r. c * u = c * v + r -> exists w. r = c * w
use zero_add
use add_eq_zero_right
use add_right_cancel
use add_assoc
use add_comm
intro c
induction u
intro v
intro r
intro h
rewrite PA5 at h
have hr : r = 0
apply add_eq_zero_right
symm
exact h
exists 0
rewrite hr
rewrite PA5
refl
intro v
induction v
intro r
intro h
exists S u
rewrite PA5 at h
specialize zero_add r
rewrite zero_add at h
symm
exact h
intro r
intro h
have hred : c * u = c * v + r
specialize add_right_cancel (c * u)
specialize add_right_cancel (c * v + r)
specialize add_right_cancel c
apply add_right_cancel
rewrite PA6 at h
rewrite PA6 at h
trans (c * v + c) + r
exact h
trans c * v + (c + r)
apply add_assoc
trans c * v + (r + c)
congr
refl
apply add_comm
symm
apply add_assoc
specialize IH v
specialize IH r
apply IH
exact hred
qed

Occurrences

265

Distinct objects

237

Reused references

12

Depth

26

Cuts

7

Certificate SHA-256

c26302d91bcb…

divides_remainderA common divisor divides a Euclidean remainder Divisibility checked

A common divisor of a dividend and divisor also divides the remainder.

Prerequisites

mul_assoc factor_difference

Used directly by

is_gcd_euclid_forward is_gcd_euclid_backward beta_modulus_coprime_base common_divisor_beta_moduli_divides_gap_times_c prime_unbounded

Exact expanded PA statement

Copy statement

forall c a b q r. (exists u. a = c * u) -> (exists v. b = c * v) -> a = b * q + r -> exists w. r = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c a b q r. (exists u. a = c * u) -> (exists v. b = c * v) -> a = b * q + r -> exists w. r = c * w
use mul_assoc
use factor_difference
intro c
intro a
intro b
intro q
intro r
intro ha
intro hb
intro h
cases ha
cases hb
specialize factor_difference c
specialize factor_difference x
specialize factor_difference (x1 * q)
specialize factor_difference r
apply factor_difference
trans a
symm
exact ha_witness
trans b * q + r
exact h
congr
rewrite hb_witness
apply mul_assoc
refl
qed

Occurrences

427

Distinct objects

357

Reused references

22

Depth

29

Cuts

11

Certificate SHA-256

2bd033084870…

divides_linear_stepA common divisor divides a Euclidean linear step Divisibility checked

A common divisor of a divisor and remainder divides their Euclidean linear step.

Prerequisites

mul_assoc mul_add

Used directly by

is_gcd_euclid_forward is_gcd_euclid_backward

Exact expanded PA statement

Copy statement

forall c b q r. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. b * q + r = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c b q r. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. b * q + r = c * w
use mul_assoc
use mul_add
intro c
intro b
intro q
intro r
intro hb
intro hr
cases hb
cases hr
exists x * q + x1
rewrite hb_witness
rewrite hr_witness
trans c * (x * q) + c * x1
congr
apply mul_assoc
refl
symm
apply mul_add
qed

Occurrences

224

Distinct objects

135

Reused references

13

Depth

19

Cuts

5

Certificate SHA-256

faf479e85693…

not_multiple_pointwiseA nonmultiple differs from every candidate multiple Divisibility checked

Turn a negated existential multiple into pointwise inequalities.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a n. ~(exists q. n = a * q) -> forall q. ~(n = a * q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a n. ~(exists q. n = a * q) -> forall q. ~(n = a * q)
intro a
intro n
intro h
intro q
intro hq
apply h
exists q
exact hq
qed

Occurrences

9

Distinct objects

9

Reused references

0

Depth

8

Cuts

0

Certificate SHA-256

fd88abf61d9c…

not_multiple_from_pointwisePointwise inequality excludes every multiple witness Divisibility checked

Reconstruct a negated existential from pointwise inequalities.

Prerequisites

none — a root theorem

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a n. (forall q. ~(n = a * q)) -> ~(exists q. n = a * q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a n. (forall q. ~(n = a * q)) -> ~(exists q. n = a * q)
intro a
intro n
intro h
intro hex
cases hex
specialize h x
apply h
exact hex_witness
qed

Occurrences

13

Distinct objects

13

Reused references

0

Depth

9

Cuts

0

Certificate SHA-256

53691b6629a8…

is_gcd_zero_rightEvery natural is the gcd of itself and zero GCD, Bézout & coprimality checked

Every natural is the relational gcd of itself and zero.

Prerequisites

multiple_refl multiple_zero

Used directly by

gcd_balanced_bezout_exists_up_to generalized_binary_crt_sufficient_zero_left generalized_binary_crt_sufficient_zero_right

Exact expanded PA statement

Copy statement

forall a. (((exists x. a = a * x) /\ (exists y. 0 = a * y)) /\ forall c. (exists u. a = c * u) -> (exists v. 0 = c * v) -> exists w. a = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (((exists x. a = a * x) /\ (exists y. 0 = a * y)) /\ forall c. (exists u. a = c * u) -> (exists v. 0 = c * v) -> exists w. a = c * w)
use multiple_refl
use multiple_zero
intro a
split
split
specialize multiple_refl a
exact multiple_refl
specialize multiple_zero a
exact multiple_zero
intro c
intro ha
intro hz
exact ha
qed

Occurrences

65

Distinct objects

65

Reused references

0

Depth

11

Cuts

4

Certificate SHA-256

1d964354c1b3…

is_gcd_symmThe relational gcd specification is symmetric GCD, Bézout & coprimality checked

The expanded relational greatest-common-divisor specification is symmetric.

Prerequisites

none — a root theorem

Used directly by

generalized_binary_crt_sufficient_zero_left

Exact expanded PA statement

Copy statement

forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> ((exists x. b = g * x) /\ (exists y. a = g * y)) /\ forall c. (exists u. b = c * u) -> (exists v. a = c * v) -> exists w. g = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> ((exists x. b = g * x) /\ (exists y. a = g * y)) /\ forall c. (exists u. b = c * u) -> (exists v. a = c * v) -> exists w. g = c * w
intro g
intro a
intro b
intro h
cases h
cases h_left
split
split
exact h_left_right
exact h_left_left
intro c
intro hb
intro ha
specialize h_right c
apply h_right
exact ha
exact hb
qed

Occurrences

36

Distinct objects

36

Reused references

0

Depth

21

Cuts

0

Certificate SHA-256

bf63a0e526c3…

is_gcd_dvd_leftA relational gcd divides the left input GCD, Bézout & coprimality checked

A relational greatest common divisor divides its left input.

Prerequisites

none — a root theorem

Used directly by

crt_common_solution_implies_gcd_compatible generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> exists x. a = g * x

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> exists x. a = g * x
intro g
intro a
intro b
intro h
cases h
cases h_left
exact h_left_left
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

13

Cuts

0

Certificate SHA-256

848a8df1aab9…

is_gcd_dvd_rightA relational gcd divides the right input GCD, Bézout & coprimality checked

A relational greatest common divisor divides its right input.

Prerequisites

none — a root theorem

Used directly by

crt_common_solution_implies_gcd_compatible generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> exists y. b = g * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> exists y. b = g * y
intro g
intro a
intro b
intro h
cases h
cases h_left
exact h_left_right
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

13

Cuts

0

Certificate SHA-256

bdbcd5dadc77…

is_gcd_greatestEvery common divisor divides a relational gcd GCD, Bézout & coprimality checked

Every common divisor of the inputs divides their relational gcd.

Prerequisites

none — a root theorem

Used directly by

is_gcd_quotients_coprime_nonzero

Exact expanded PA statement

Copy statement

forall g a b c. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. g = d * w) -> (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b c. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. g = d * w) -> (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w
intro g
intro a
intro b
intro c
intro h
intro ha
intro hb
cases h
specialize h_right c
apply h_right
exact ha
exact hb
qed

Occurrences

24

Distinct objects

24

Reused references

0

Depth

16

Cuts

0

Certificate SHA-256

90f411bbe254…

is_gcd_of_dvdA divisor is the gcd of itself and its multiple GCD, Bézout & coprimality checked

If the left input divides the right, the left input is their relational gcd.

Prerequisites

multiple_refl

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. (exists y. b = a * y) -> (((exists x. a = a * x) /\ (exists y. b = a * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. a = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (exists y. b = a * y) -> (((exists x. a = a * x) /\ (exists y. b = a * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. a = c * w)
use multiple_refl
intro a
intro b
intro hd
split
split
specialize multiple_refl a
exact multiple_refl
exact hd
intro c
intro ha
intro hb
exact ha
qed

Occurrences

55

Distinct objects

55

Reused references

0

Depth

11

Cuts

3

Certificate SHA-256

3377fa1041e5…

is_gcd_uniqueThe relational greatest common divisor is unique GCD, Bézout & coprimality checked

The fully expanded relational greatest common divisor is unique.

Prerequisites

multiple_antisymm

Used directly by

gcd_lcm_product generalized_binary_crt_sufficient_zero_left generalized_binary_crt_sufficient_zero_right

Exact expanded PA statement

Copy statement

forall g h a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> (((exists x. a = h * x) /\ (exists y. b = h * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. h = c * w) -> g = h

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g h a b. (((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. g = c * w) -> (((exists x. a = h * x) /\ (exists y. b = h * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. h = c * w) -> g = h
use multiple_antisymm
intro g
intro h
intro a
intro b
intro hg
intro hh
cases hg
cases hg_left
cases hh
cases hh_left
specialize hg_right h
have hdg : exists w. g = h * w
apply hg_right
exact hh_left_left
exact hh_left_right
specialize hh_right g
have gdh : exists w. h = g * w
apply hh_right
exact hg_left_left
exact hg_left_right
specialize multiple_antisymm g
specialize multiple_antisymm h
apply multiple_antisymm
exact gdh
exact hdg
qed

Occurrences

680

Distinct objects

561

Reused references

35

Depth

34

Cuts

20

Certificate SHA-256

f903e7437550…

is_gcd_euclid_forwardEuclidean gcd invariance in the forward direction GCD, Bézout & coprimality checked

A relational gcd of divisor and remainder is a gcd of dividend and divisor.

Prerequisites

divides_remainder divides_linear_step

Used directly by

gcd_exists_up_to gcd_balanced_bezout_exists_up_to

Exact expanded PA statement

Copy statement

forall d a b q r. a = b * q + r -> (((exists x. b = d * x) /\ (exists y. r = d * y)) /\ forall c. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. d = c * w) -> (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d a b q r. a = b * q + r -> (((exists x. b = d * x) /\ (exists y. r = d * y)) /\ forall c. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. d = c * w) -> (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)
use divides_remainder
use divides_linear_step
intro d
intro a
intro b
intro q
intro r
intro hstep
intro hg
cases hg
cases hg_left
split
split
rewrite hstep
specialize divides_linear_step d
specialize divides_linear_step b
specialize divides_linear_step q
specialize divides_linear_step r
apply divides_linear_step
exact hg_left_left
exact hg_left_right
exact hg_left_left
intro c
intro hca
intro hcb
specialize hg_right c
apply hg_right
exact hcb
specialize divides_remainder c
specialize divides_remainder a
specialize divides_remainder b
specialize divides_remainder q
specialize divides_remainder r
apply divides_remainder
exact hca
exact hcb
exact hstep
qed

Occurrences

741

Distinct objects

478

Reused references

24

Depth

38

Cuts

18

Certificate SHA-256

3da415ea4ec6…

is_gcd_euclid_backwardEuclidean gcd invariance in the backward direction GCD, Bézout & coprimality checked

A relational gcd of dividend and divisor is a gcd of divisor and remainder.

Prerequisites

divides_remainder divides_linear_step

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall d a b q r. a = b * q + r -> (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) -> (((exists x. b = d * x) /\ (exists y. r = d * y)) /\ forall c. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. d = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d a b q r. a = b * q + r -> (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) -> (((exists x. b = d * x) /\ (exists y. r = d * y)) /\ forall c. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. d = c * w)
use divides_remainder
use divides_linear_step
intro d
intro a
intro b
intro q
intro r
intro hstep
intro hg
cases hg
cases hg_left
split
split
exact hg_left_right
specialize divides_remainder d
specialize divides_remainder a
specialize divides_remainder b
specialize divides_remainder q
specialize divides_remainder r
apply divides_remainder
exact hg_left_left
exact hg_left_right
exact hstep
intro c
intro hcb
intro hcr
specialize hg_right c
apply hg_right
rewrite hstep
specialize divides_linear_step c
specialize divides_linear_step b
specialize divides_linear_step q
specialize divides_linear_step r
apply divides_linear_step
exact hcb
exact hcr
exact hcb
qed

Occurrences

741

Distinct objects

478

Reused references

24

Depth

37

Cuts

18

Certificate SHA-256

bd24f6b65f7e…

gcd_exists_up_toA greatest common divisor exists below a fixed bound GCD, Bézout & coprimality checked

Bounded induction constructs a relational gcd whenever the right input is at most the bound.

Prerequisites

multiple_refl le_zero le_eq_or_lt le_of_succ_le_succ division_remainder_exists is_gcd_euclid_forward

Used directly by

gcd_exists_relational

Exact expanded PA statement

Copy statement

forall B b. (exists t. t + b = B) -> forall a. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B b. (exists t. t + b = B) -> forall a. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)
use multiple_refl
use le_zero
use le_eq_or_lt
use le_of_succ_le_succ
use division_remainder_exists
use is_gcd_euclid_forward
intro B
induction B
intro b
intro hb
intro a
have hb0 : b = 0
apply le_zero
exact hb
exists a
split
split
specialize multiple_refl a
exact multiple_refl
exists 0
trans 0
exact hb0
symm
apply PA5
intro c
intro hca
intro hcb
exact hca
intro b
intro hb
intro a
specialize le_eq_or_lt b
specialize le_eq_or_lt (S B)
have hsplit : b = S B \/ exists k. k + S b = S B
apply le_eq_or_lt
exact hb
cases hsplit
have hb0 : ~(b = 0)
intro hzero
apply PA1
trans b
symm
exact hsplit_left
exact hzero
have hdiv : exists q r. a = b * q + r /\ exists k. k + S r = b
apply division_remainder_exists
exact hb0
cases hdiv
cases hdiv_witness
cases hdiv_witness_witness
have hrB : exists k. k + x1 = B
apply le_of_succ_le_succ
rewrite hsplit_left at hdiv_witness_witness_right
exact hdiv_witness_witness_right
have hsmall : exists d. (((exists u. b = d * u) /\ (exists v. x1 = d * v)) /\ forall c. (exists s. b = c * s) -> (exists t. x1 = c * t) -> exists w. d = c * w)
specialize IH x1
have hall : forall z. exists d. (((exists u. z = d * u) /\ (exists v. x1 = d * v)) /\ forall c. (exists s. z = c * s) -> (exists t. x1 = c * t) -> exists w. d = c * w)
apply IH
exact hrB
specialize hall b
exact hall
cases hsmall
exists x2
specialize is_gcd_euclid_forward x2
specialize is_gcd_euclid_forward a
specialize is_gcd_euclid_forward b
specialize is_gcd_euclid_forward x
specialize is_gcd_euclid_forward x1
apply is_gcd_euclid_forward
exact hdiv_witness_witness_left
exact hsmall_witness
have hbB : exists k. k + b = B
apply le_of_succ_le_succ
exact hsplit_right
specialize IH b
have hall : forall z. exists d. (((exists u. z = d * u) /\ (exists v. b = d * v)) /\ forall c. (exists s. z = c * s) -> (exists t. b = c * t) -> exists w. d = c * w)
apply IH
exact hbB
specialize hall a
exact hall
qed

Occurrences

1,232

Distinct objects

814

Reused references

47

Depth

44

Cuts

35

Certificate SHA-256

ef027a9b219f…

gcd_exists_relationalA greatest common divisor exists GCD, Bézout & coprimality checked

Every pair of naturals has a relational greatest common divisor.

Prerequisites

le_refl gcd_exists_up_to

Used directly by

euclid_prime_dvd_product prime_coprime_or_divides generalized_binary_crt_total_decision

Exact expanded PA statement

Copy statement

forall a b. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)
use le_refl
use gcd_exists_up_to
intro a
intro b
specialize gcd_exists_up_to b
specialize gcd_exists_up_to b
have hbb : exists t. t + b = b
apply le_refl
have hall : forall z. exists d. (((exists x. z = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. z = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)
apply gcd_exists_up_to
exact hbb
specialize hall a
exact hall
qed

Occurrences

1,268

Distinct objects

833

Reused references

48

Depth

46

Cuts

38

Certificate SHA-256

ee2e36ed07e4…

coprime_symmCoprimality is symmetric GCD, Bézout & coprimality checked

Coprimality in its expanded common-divisor form is symmetric.

Prerequisites

none — a root theorem

Used directly by

coprime_mul_right prime_mod_cancel

Exact expanded PA statement

Copy statement

forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> forall c. (exists u. b = c * u) -> (exists v. a = c * v) -> c = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> forall c. (exists u. b = c * u) -> (exists v. a = c * v) -> c = 1
intro a
intro b
intro h
intro c
intro hb
intro ha
specialize h c
apply h
exact ha
exact hb
qed

Occurrences

15

Distinct objects

15

Reused references

0

Depth

11

Cuts

0

Certificate SHA-256

3f127d01f834…

coprime_one_rightEvery natural number is coprime to one GCD, Bézout & coprimality checked

Every natural is coprime to one in the expanded common-divisor relation.

Prerequisites

divisor_one

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a d. (exists x. a = d * x) -> (exists y. 1 = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a d. (exists x. a = d * x) -> (exists y. 1 = d * y) -> d = 1
use divisor_one
intro a
intro d
intro ha
intro h1
specialize divisor_one d
apply divisor_one
exact h1
qed

Occurrences

200

Distinct objects

188

Reused references

12

Depth

29

Cuts

5

Certificate SHA-256

904cdc493146…

coprime_one_leftOne is coprime to every natural number GCD, Bézout & coprimality checked

One is coprime to every natural in the expanded common-divisor relation.

Prerequisites

divisor_one

Used directly by

bounded_beta_exclusive_recode_invariant

Exact expanded PA statement

Copy statement

forall a d. (exists x. 1 = d * x) -> (exists y. a = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a d. (exists x. 1 = d * x) -> (exists y. a = d * y) -> d = 1
use divisor_one
intro a
intro d
intro h1
intro ha
specialize divisor_one d
apply divisor_one
exact h1
qed

Occurrences

200

Distinct objects

188

Reused references

12

Depth

29

Cuts

5

Certificate SHA-256

6bb2edf69b5e…

coprime_to_is_gcd_oneCoprimality constructs a relational gcd equal to one GCD, Bézout & coprimality checked

Expanded coprimality constructs the relational gcd witness one.

Prerequisites

one_multiple one_mul mul_one

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> (((exists x. a = 1 * x) /\ (exists y. b = 1 * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. 1 = c * w)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> (((exists x. a = 1 * x) /\ (exists y. b = 1 * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. 1 = c * w)
use one_multiple
use one_mul
use mul_one
intro a
intro b
intro h
split
split
specialize one_multiple a
exact one_multiple
exists b
symm
apply one_mul
intro c
intro ha
intro hb
specialize h c
have hc : c = 1
apply h
exact ha
exact hb
exists 1
rewrite hc
symm
apply mul_one
qed

Occurrences

120

Distinct objects

90

Reused references

5

Depth

17

Cuts

5

Certificate SHA-256

530efb1368fb…

is_gcd_one_to_coprimeA relational gcd equal to one implies coprimality GCD, Bézout & coprimality checked

A relational gcd witness one implies expanded coprimality.

Prerequisites

divisor_one

Used directly by

euclid_prime_dvd_product prime_coprime_or_divides

Exact expanded PA statement

Copy statement

forall a b. (((exists x. a = 1 * x) /\ (exists y. b = 1 * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. 1 = d * w) -> forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> c = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (((exists x. a = 1 * x) /\ (exists y. b = 1 * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. 1 = d * w) -> forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> c = 1
use divisor_one
intro a
intro b
intro hg
intro c
intro ha
intro hb
cases hg
specialize hg_right c
have hd : exists w. 1 = c * w
apply hg_right
exact ha
exact hb
specialize divisor_one c
apply divisor_one
exact hd
qed

Occurrences

205

Distinct objects

193

Reused references

12

Depth

29

Cuts

5

Certificate SHA-256

5f7782d1cc3c…

add_permute_outerThe outer terms of two additive pairs may be permuted Addition checked

Permute the outer entries of two additive pairs.

Prerequisites

add_assoc add_comm

Used directly by

balanced_bezout_euclid_step mod_eq_add

Exact expanded PA statement

Copy statement

forall a b c d. (a + b) + (c + d) = (c + b) + (a + d)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b c d. (a + b) + (c + d) = (c + b) + (a + d)
use add_assoc
use add_comm
intro a
intro b
intro c
intro d
trans a + (b + (c + d))
apply add_assoc
trans a + ((b + c) + d)
congr
refl
symm
apply add_assoc
trans a + ((c + b) + d)
congr
refl
congr
apply add_comm
refl
trans (a + (c + b)) + d
symm
apply add_assoc
trans ((c + b) + a) + d
congr
apply add_comm
refl
apply add_assoc
qed

Occurrences

149

Distinct objects

138

Reused references

11

Depth

15

Cuts

4

Certificate SHA-256

897b1a1c5b72…

balanced_bezout_euclid_stepBalanced Bézout coefficients cross one Euclidean step GCD, Bézout & coprimality checked

Transport balanced natural Bezout coefficients across one Euclidean division step.

Prerequisites

add_assoc add_comm mul_add mul_assoc add_mul add_permute_outer

Used directly by

gcd_balanced_bezout_exists_up_to

Exact expanded PA statement

Copy statement

forall a b q r d xp yp xn yn. a = b * q + r -> b * xp + r * yp = d + (b * xn + r * yn) -> a * yp + b * (xp + q * yn) = d + (a * yn + b * (xn + q * yp))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b q r d xp yp xn yn. a = b * q + r -> b * xp + r * yp = d + (b * xn + r * yn) -> a * yp + b * (xp + q * yn) = d + (a * yn + b * (xn + q * yp))
use add_assoc
use add_comm
use mul_add
use mul_assoc
use add_mul
use add_permute_outer
intro a
intro b
intro q
intro r
intro d
intro xp
intro yp
intro xn
intro yn
intro hab
intro hbez
rewrite hab
trans ((b * q) * yp + r * yp) + b * (xp + q * yn)
congr
apply add_mul
refl
trans ((b * q) * yp + r * yp) + (b * xp + b * (q * yn))
congr
refl
apply mul_add
trans ((b * q) * yp + r * yp) + (b * xp + (b * q) * yn)
congr
refl
congr
refl
symm
apply mul_assoc
trans (b * xp + r * yp) + ((b * q) * yp + (b * q) * yn)
apply add_permute_outer
trans (b * xp + r * yp) + ((b * q) * yn + (b * q) * yp)
congr
refl
apply add_comm
trans (d + (b * xn + r * yn)) + ((b * q) * yn + (b * q) * yp)
congr
exact hbez
refl
trans d + ((b * xn + r * yn) + ((b * q) * yn + (b * q) * yp))
apply add_assoc
trans d + (((b * q) * yn + r * yn) + (b * xn + (b * q) * yp))
congr
refl
apply add_permute_outer
trans d + ((b * q + r) * yn + (b * xn + (b * q) * yp))
congr
refl
congr
symm
apply add_mul
refl
trans d + ((b * q + r) * yn + (b * xn + b * (q * yp)))
congr
refl
congr
refl
congr
refl
apply mul_assoc
congr
refl
congr
congr
symm
exact hab
refl
symm
apply mul_add
qed

Occurrences

880

Distinct objects

446

Reused references

42

Depth

35

Cuts

24

Certificate SHA-256

feae8ba0dfcb…

gcd_balanced_bezout_exists_up_toA gcd with balanced Bézout witnesses exists below a bound GCD, Bézout & coprimality checked

Bounded Euclidean descent simultaneously constructs a relational gcd and balanced natural Bezout witnesses.

Prerequisites

zero_add le_zero le_eq_or_lt le_of_succ_le_succ division_remainder_exists is_gcd_zero_right is_gcd_euclid_forward balanced_bezout_euclid_step

Used directly by

gcd_balanced_bezout_exists

Exact expanded PA statement

Copy statement

forall B b. (exists t. t + b = B) -> forall a. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B b. (exists t. t + b = B) -> forall a. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))
use zero_add
use le_zero
use le_eq_or_lt
use le_of_succ_le_succ
use division_remainder_exists
use is_gcd_zero_right
use is_gcd_euclid_forward
use balanced_bezout_euclid_step
intro B
induction B
intro b
intro hb
intro a
have hb0 : b = 0
apply le_zero
exact hb
exists a
split
rewrite hb0
rewrite hb0
specialize is_gcd_zero_right a
exact is_gcd_zero_right
exists 1
exists 0
exists 0
exists 0
rewrite hb0
simp [zero_add]
intro b
intro hb
intro a
specialize le_eq_or_lt b
specialize le_eq_or_lt (S B)
have hsplit : b = S B \/ exists k. k + S b = S B
apply le_eq_or_lt
exact hb
cases hsplit
have hb0 : ~(b = 0)
intro hzero
apply PA1
trans b
symm
exact hsplit_left
exact hzero
have hdiv : exists q r. a = b * q + r /\ exists k. k + S r = b
apply division_remainder_exists
exact hb0
cases hdiv
cases hdiv_witness
cases hdiv_witness_witness
have hrB : exists k. k + x1 = B
apply le_of_succ_le_succ
rewrite hsplit_left at hdiv_witness_witness_right
exact hdiv_witness_witness_right
have hsmall : exists d. ((((exists u. b = d * u) /\ (exists v. x1 = d * v)) /\ forall c. (exists s. b = c * s) -> (exists t. x1 = c * t) -> exists w. d = c * w) /\ exists xp yp xn yn. b * xp + x1 * yp = d + (b * xn + x1 * yn))
specialize IH x1
have hall : forall z. exists d. ((((exists u. z = d * u) /\ (exists v. x1 = d * v)) /\ forall c. (exists s. z = c * s) -> (exists t. x1 = c * t) -> exists w. d = c * w) /\ exists xp yp xn yn. z * xp + x1 * yp = d + (z * xn + x1 * yn))
apply IH
exact hrB
specialize hall b
exact hall
cases hsmall
cases hsmall_witness
cases hsmall_witness_right
cases hsmall_witness_right_witness
cases hsmall_witness_right_witness_witness
cases hsmall_witness_right_witness_witness_witness
exists x2
split
apply is_gcd_euclid_forward
exact hdiv_witness_witness_left
exact hsmall_witness_left
exists x4
exists x3 + x * x6
exists x6
exists x5 + x * x4
apply balanced_bezout_euclid_step
exact hdiv_witness_witness_left
exact hsmall_witness_right_witness_witness_witness_witness
have hbB : exists k. k + b = B
apply le_of_succ_le_succ
exact hsplit_right
specialize IH b
have hall : forall z. exists d. ((((exists u. z = d * u) /\ (exists v. b = d * v)) /\ forall c. (exists s. z = c * s) -> (exists t. b = c * t) -> exists w. d = c * w) /\ exists xp yp xn yn. z * xp + b * yp = d + (z * xn + b * yn))
apply IH
exact hbB
specialize hall a
exact hall
qed

Occurrences

2,233

Distinct objects

1,187

Reused references

80

Depth

45

Cuts

63

Certificate SHA-256

2e611f8aee45…

gcd_balanced_bezout_existsA gcd has balanced natural Bézout coefficients GCD, Bézout & coprimality checked

Every pair has a relational gcd together with balanced natural Bezout witnesses.

Prerequisites

le_refl gcd_balanced_bezout_exists_up_to

Used directly by

coprime_balanced_bezout gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall a b. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))
use le_refl
use gcd_balanced_bezout_exists_up_to
intro a
intro b
specialize gcd_balanced_bezout_exists_up_to b
specialize gcd_balanced_bezout_exists_up_to b
have hbb : exists t. t + b = b
apply le_refl
have hall : forall z. exists d. ((((exists x. z = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. z = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. z * xp + b * yp = d + (z * xn + b * yn))
apply gcd_balanced_bezout_exists_up_to
exact hbb
specialize hall a
exact hall
qed

Occurrences

2,269

Distinct objects

1,206

Reused references

81

Depth

47

Cuts

66

Certificate SHA-256

0c34da8ea071…

balanced_combination_scale_rightA balanced combination may be scaled on the right GCD, Bézout & coprimality checked

Scale a balanced natural combination on the right.

Prerequisites

mul_assoc mul_comm add_mul

Used directly by

gauss_coprime_cancel

Exact expanded PA statement

Copy statement

forall a b d xp yp xn yn z. a * xp + b * yp = d + (a * xn + b * yn) -> a * (xp * z) + (b * z) * yp = d * z + (a * (xn * z) + (b * z) * yn)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b d xp yp xn yn z. a * xp + b * yp = d + (a * xn + b * yn) -> a * (xp * z) + (b * z) * yp = d * z + (a * (xn * z) + (b * z) * yn)
use mul_assoc
use mul_comm
use add_mul
intro a
intro b
intro d
intro xp
intro yp
intro xn
intro yn
intro z
intro h
trans (a * xp) * z + (b * z) * yp
congr
symm
apply mul_assoc
refl
trans (a * xp) * z + (b * yp) * z
congr
refl
trans b * (z * yp)
apply mul_assoc
trans b * (yp * z)
congr
refl
apply mul_comm
symm
apply mul_assoc
trans (a * xp + b * yp) * z
symm
apply add_mul
trans (d + (a * xn + b * yn)) * z
congr
exact h
refl
trans d * z + (a * xn + b * yn) * z
apply add_mul
trans d * z + ((a * xn) * z + (b * yn) * z)
congr
refl
apply add_mul
trans d * z + (a * (xn * z) + (b * yn) * z)
congr
refl
congr
apply mul_assoc
refl
congr
refl
congr
refl
trans b * (yn * z)
apply mul_assoc
trans b * (z * yn)
congr
refl
apply mul_comm
symm
apply mul_assoc
qed

Occurrences

754

Distinct objects

387

Reused references

38

Depth

28

Cuts

20

Certificate SHA-256

dfb696fa22b5…

common_divisor_divides_balanced_resultA common divisor divides a balanced-combination result GCD, Bézout & coprimality checked

Every common divisor of two inputs divides the result of a balanced natural combination.

Prerequisites

mul_assoc mul_add add_comm factor_difference

Used directly by

gauss_coprime_cancel mod_inverse_implies_coprime balanced_bezout_one_implies_coprime

Exact expanded PA statement

Copy statement

forall c a b d xp yp xn yn. (exists u. a = c * u) -> (exists v. b = c * v) -> a * xp + b * yp = d + (a * xn + b * yn) -> exists w. d = c * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c a b d xp yp xn yn. (exists u. a = c * u) -> (exists v. b = c * v) -> a * xp + b * yp = d + (a * xn + b * yn) -> exists w. d = c * w
use mul_assoc
use mul_add
use add_comm
use factor_difference
intro c
intro a
intro b
intro d
intro xp
intro yp
intro xn
intro yn
intro ha
intro hb
intro h
cases ha
cases hb
specialize factor_difference c
specialize factor_difference (x * xp + x1 * yp)
specialize factor_difference (x * xn + x1 * yn)
specialize factor_difference d
apply factor_difference
trans c * (x * xp) + c * (x1 * yp)
apply mul_add
trans (c * x) * xp + (c * x1) * yp
congr
symm
apply mul_assoc
symm
apply mul_assoc
trans a * xp + b * yp
rewrite ha_witness
rewrite hb_witness
refl
trans d + (a * xn + b * yn)
exact h
trans (a * xn + b * yn) + d
apply add_comm
trans ((c * x) * xn + (c * x1) * yn) + d
rewrite ha_witness
rewrite hb_witness
refl
trans (c * (x * xn) + c * (x1 * yn)) + d
congr
congr
apply mul_assoc
apply mul_assoc
refl
congr
symm
apply mul_add
refl
qed

Occurrences

626

Distinct objects

406

Reused references

24

Depth

39

Cuts

16

Certificate SHA-256

b8a6860c9469…

coprime_balanced_bezoutCoprime inputs have a balanced Bézout identity GCD, Bézout & coprimality checked

Coprime inputs admit balanced natural Bezout coefficients with result one.

Prerequisites

gcd_balanced_bezout_exists

Used directly by

gauss_coprime_cancel binary_crt coprime_balanced_mod_inverse

Exact expanded PA statement

Copy statement

forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> exists xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> exists xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn)
use gcd_balanced_bezout_exists
intro a
intro b
intro hcop
have hgb : exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))
apply gcd_balanced_bezout_exists
cases hgb
cases hgb_witness
cases hgb_witness_left
cases hgb_witness_left_left
have hd : x = 1
specialize hcop x
apply hcop
exact hgb_witness_left_left_left
exact hgb_witness_left_left_right
cases hgb_witness_right
cases hgb_witness_right_witness
cases hgb_witness_right_witness_witness
cases hgb_witness_right_witness_witness_witness
exists x1
exists x2
exists x3
exists x4
rewrite hd at hgb_witness_right_witness_witness_witness_witness
exact hgb_witness_right_witness_witness_witness_witness
qed

Occurrences

2,304

Distinct objects

1,241

Reused references

81

Depth

48

Cuts

67

Certificate SHA-256

342339e0e63d…

gauss_coprime_cancelGauss cancellation for coprime factors GCD, Bézout & coprimality checked

Cancel a coprime factor from a divisibility witness (Gauss cancellation).

Prerequisites

multiple_refl one_mul coprime_balanced_bezout balanced_combination_scale_right common_divisor_divides_balanced_result

Used directly by

euclid_prime_dvd_product beta_moduli_coprime_of_gap_dvd coprime_mul_left coprime_product_is_lcm

Exact expanded PA statement

Copy statement

forall a b z. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> (exists q. b * z = a * q) -> exists w. z = a * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b z. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> (exists q. b * z = a * q) -> exists w. z = a * w
use multiple_refl
use one_mul
use coprime_balanced_bezout
use balanced_combination_scale_right
use common_divisor_divides_balanced_result
intro a
intro b
intro z
intro hcop
intro hdiv
have hbez : exists xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn)
apply coprime_balanced_bezout
exact hcop
cases hbez
cases hbez_witness
cases hbez_witness_witness
cases hbez_witness_witness_witness
have hscaled : a * (x * z) + (b * z) * x1 = 1 * z + (a * (x2 * z) + (b * z) * x3)
apply balanced_combination_scale_right
exact hbez_witness_witness_witness_witness
specialize one_mul z
rewrite one_mul at hscaled
specialize common_divisor_divides_balanced_result a
specialize common_divisor_divides_balanced_result a
specialize common_divisor_divides_balanced_result (b * z)
specialize common_divisor_divides_balanced_result z
specialize common_divisor_divides_balanced_result (x * z)
specialize common_divisor_divides_balanced_result x1
specialize common_divisor_divides_balanced_result (x2 * z)
specialize common_divisor_divides_balanced_result x3
apply common_divisor_divides_balanced_result
specialize multiple_refl a
exact multiple_refl
exact hdiv
exact hscaled
qed

Occurrences

3,800

Distinct objects

1,499

Reused references

93

Depth

51

Cuts

110

Certificate SHA-256

763e42141ac2…

eq_decidableEquality of naturals is decidable Equality checked

Equality of natural numbers is constructively decidable.

Prerequisites

none — a root theorem

Used directly by

multiple_decidable_nonzero multiple_decidable factor_search_up_to prime_decidable prime_factorization_exists_up_to mod_eq_decidable_from_remainders beta_prefix_swap_last_reflect finite_swap_last_bounded finite_contains_decidable beta_prefix_replace_reflect gcd_lcm_compatible_exists generalized_binary_crt_sufficient generalized_binary_crt_canonical_boundary mod_eq_decidable

Exact expanded PA statement

Copy statement

forall a b. a = b \/ ~(a = b)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. a = b \/ ~(a = b)
intro a
induction a
intro b
induction b
left
refl
right
intro h
apply PA1
symm
exact h
intro b
induction b
right
intro h
apply PA1
exact h
specialize IH b
cases IH
left
congr
exact IH_left
right
intro h
apply IH_right
apply PA2
exact h
qed

Occurrences

48

Distinct objects

48

Reused references

0

Depth

20

Cuts

0

Certificate SHA-256

d89ef3447e28…

multiple_decidable_nonzeroDivisibility by a nonzero natural is decidable Divisibility checked

Divisibility by a nonzero natural is constructively decidable.

Prerequisites

eq_decidable division_remainder_exists multiple_has_zero_remainder division_remainder_unique

Used directly by

multiple_decidable factor_search_up_to

Exact expanded PA statement

Copy statement

forall d n. ~(d = 0) -> (exists q. n = d * q) \/ ~(exists q. n = d * q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d n. ~(d = 0) -> (exists q. n = d * q) \/ ~(exists q. n = d * q)
use eq_decidable
use division_remainder_exists
use multiple_has_zero_remainder
use division_remainder_unique
intro d
intro n
intro hd
have hdiv : exists q r. n = d * q + r /\ S r <= d
apply division_remainder_exists
exact hd
cases hdiv
cases hdiv_witness
cases hdiv_witness_witness
specialize eq_decidable x1
specialize eq_decidable 0
have hr : x1 = 0 \/ ~(x1 = 0)
apply eq_decidable
cases hr
left
exists x
rewrite hr_left at hdiv_witness_witness_left
rewrite PA3 at hdiv_witness_witness_left
exact hdiv_witness_witness_left
right
intro hmul
have hzero : exists q r. ((n = d * q + r /\ r = 0) /\ S r <= d)
apply multiple_has_zero_remainder
exact hd
exact hmul
cases hzero
cases hzero_witness
cases hzero_witness_witness
cases hzero_witness_witness_left
have huniq : x = x2 /\ x1 = x3
apply division_remainder_unique
exact hdiv_witness_witness_left
exact hdiv_witness_witness_right
exact hzero_witness_witness_left_left
exact hzero_witness_witness_right
cases huniq
apply hr_right
trans x3
exact huniq_right
exact hzero_witness_witness_left_right
qed

Occurrences

1,242

Distinct objects

848

Reused references

43

Depth

61

Cuts

32

Certificate SHA-256

e472d09fa365…

multiple_decidableDivisibility of naturals is decidable Divisibility checked

Divisibility of natural numbers is constructively decidable, including the zero divisor case.

Prerequisites

mul_zero_left eq_decidable multiple_decidable_nonzero

Used directly by

prime_divides_decidable

Exact expanded PA statement

Copy statement

forall d n. (exists q. n = d * q) \/ ~(exists q. n = d * q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d n. (exists q. n = d * q) \/ ~(exists q. n = d * q)
use mul_zero_left
use eq_decidable
use multiple_decidable_nonzero
intro d
intro n
specialize eq_decidable d
specialize eq_decidable 0
have hd : d = 0 \/ ~(d = 0)
apply eq_decidable
cases hd
specialize eq_decidable_before n
specialize eq_decidable_before 0
have hn : n = 0 \/ ~(n = 0)
apply eq_decidable_before
cases hn
left
exists 0
trans 0
exact hn_left
symm
rewrite hd_left
apply mul_zero_left
right
intro hmultiple
cases hmultiple
apply hn_right
trans d * x
exact hmultiple_witness
rewrite hd_left
apply mul_zero_left
apply multiple_decidable_nonzero
exact hd_right
qed

Occurrences

1,352

Distinct objects

907

Reused references

47

Depth

64

Cuts

35

Certificate SHA-256

27aa9616c05f…

factor_property_succA bounded factor property extends across one boundary Primes checked

Extend a bounded prime factor-pair property by checking the new boundary.

Prerequisites

le_eq_or_lt le_of_succ_le_succ

Used directly by

factor_search_up_to

Exact expanded PA statement

Copy statement

forall B n. (forall c d. (exists k. k + c = B) -> n = c * d -> c = 1 \/ d = 1) -> (forall d. n = S B * d -> S B = 1 \/ d = 1) -> forall c d. (exists k. k + c = S B) -> n = c * d -> c = 1 \/ d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B n. (forall c d. (exists k. k + c = B) -> n = c * d -> c = 1 \/ d = 1) -> (forall d. n = S B * d -> S B = 1 \/ d = 1) -> forall c d. (exists k. k + c = S B) -> n = c * d -> c = 1 \/ d = 1
use le_eq_or_lt
use le_of_succ_le_succ
intro B
intro n
intro hprev
intro hboundary
intro c
intro d
intro hc
intro hfac
specialize le_eq_or_lt c
specialize le_eq_or_lt (S B)
have hsplit : c = S B \/ exists k. k + S c = S B
apply le_eq_or_lt
exact hc
cases hsplit
rewrite hsplit_left
specialize hboundary d
apply hboundary
rewrite <- hsplit_left
exact hfac
have hcB : exists k. k + c = B
apply le_of_succ_le_succ
exact hsplit_right
specialize hprev c
specialize hprev d
apply hprev
exact hcB
exact hfac
qed

Occurrences

150

Distinct objects

146

Reused references

4

Depth

20

Cuts

5

Certificate SHA-256

0d6bbb6b6480…

factor_search_up_toBounded nontrivial factor search Primes checked

Constructively decide whether a nonzero natural has a bounded nontrivial factor pair.

Prerequisites

mul_zero_left succ_ne_zero le_zero le_refl le_succ mul_left_cancel_nonzero eq_decidable multiple_decidable_nonzero factor_property_succ

Used directly by

prime_or_composite

Exact expanded PA statement

Copy statement

forall B n. ~(n = 0) -> ((forall c d. (exists k. k + c = B) -> n = c * d -> c = 1 \/ d = 1) \/ exists c d. ((((exists k. k + c = B) /\ ~(c = 1)) /\ ~(d = 1)) /\ n = c * d))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B n. ~(n = 0) -> ((forall c d. (exists k. k + c = B) -> n = c * d -> c = 1 \/ d = 1) \/ exists c d. ((((exists k. k + c = B) /\ ~(c = 1)) /\ ~(d = 1)) /\ n = c * d))
use mul_zero_left
use succ_ne_zero
use le_zero
use le_refl
use le_succ
use mul_left_cancel_nonzero
use eq_decidable
use multiple_decidable_nonzero
use factor_property_succ
intro B
induction B
intro n
intro hn
left
intro c
intro d
intro hc
intro hfac
have hc0 : c = 0
apply le_zero
exact hc
exfalso
apply hn
trans c * d
exact hfac
rewrite hc0
apply mul_zero_left
intro n
intro hn
specialize IH n
have hprev : (forall c d. (exists k. k + c = B) -> n = c * d -> c = 1 \/ d = 1) \/ exists c d. ((((exists k. k + c = B) /\ ~(c = 1)) /\ ~(d = 1)) /\ n = c * d)
apply IH
exact hn
cases hprev
have hs0 : ~(S B = 0)
specialize succ_ne_zero B
exact succ_ne_zero
specialize multiple_decidable_nonzero (S B)
specialize multiple_decidable_nonzero n
have hdiv : (exists q. n = S B * q) \/ ~(exists q. n = S B * q)
apply multiple_decidable_nonzero
exact hs0
cases hdiv
cases hdiv_left
specialize eq_decidable (S B)
specialize eq_decidable 1
have hc1 : S B = 1 \/ ~(S B = 1)
apply eq_decidable
cases hc1
left
apply factor_property_succ
exact hprev_left
intro d
intro hboundary
left
exact hc1_left
specialize eq_decidable_before x
specialize eq_decidable_before 1
have hq1 : x = 1 \/ ~(x = 1)
apply eq_decidable_before
cases hq1
left
apply factor_property_succ
exact hprev_left
intro d
intro hboundary
right
trans x
apply mul_left_cancel_nonzero
exact hs0
trans n
symm
exact hboundary
exact hdiv_left_witness
exact hq1_left
right
exists S B
exists x
split
split
split
apply le_refl
exact hc1_right
exact hq1_right
exact hdiv_left_witness
left
apply factor_property_succ
exact hprev_left
intro d
intro hboundary
exfalso
apply hdiv_right
exists d
exact hboundary
right
cases hprev_right
cases hprev_right_witness
cases hprev_right_witness_witness
cases hprev_right_witness_witness_left
cases hprev_right_witness_witness_left_left
exists x
exists x1
split
split
split
apply le_succ
exact hprev_right_witness_witness_left_left_left
exact hprev_right_witness_witness_left_left_right
exact hprev_right_witness_witness_left_right
exact hprev_right_witness_witness_right
qed

Occurrences

1,925

Distinct objects

1,276

Reused references

56

Depth

69

Cuts

56

Certificate SHA-256

b678af35c4db…

prime_or_compositeEvery nontrivial natural is prime or composite Primes checked

Every nonzero nonunit natural is constructively prime or has a nontrivial factor pair.

Prerequisites

divisor_le_nonzero factor_search_up_to

Used directly by

prime_decidable prime_divisor_exists_up_to

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> ~(n = 1) -> ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ exists c d. ((~(c = 1) /\ ~(d = 1)) /\ n = c * d))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> ~(n = 1) -> ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ exists c d. ((~(c = 1) /\ ~(d = 1)) /\ n = c * d))
use divisor_le_nonzero
use factor_search_up_to
intro n
intro hn0
intro hn1
specialize factor_search_up_to n
specialize factor_search_up_to n
have hsearch : (forall c d. (exists k. k + c = n) -> n = c * d -> c = 1 \/ d = 1) \/ exists c d. ((((exists k. k + c = n) /\ ~(c = 1)) /\ ~(d = 1)) /\ n = c * d)
apply factor_search_up_to
exact hn0
cases hsearch
left
split
exact hn1
intro c
intro d
intro hfac
specialize hsearch_left c
specialize hsearch_left d
apply hsearch_left
specialize divisor_le_nonzero c
specialize divisor_le_nonzero n
apply divisor_le_nonzero
exact hn0
exists d
exact hfac
exact hfac
right
cases hsearch_right
cases hsearch_right_witness
cases hsearch_right_witness_witness
cases hsearch_right_witness_witness_left
cases hsearch_right_witness_witness_left_left
exists x
exists x1
split
split
exact hsearch_right_witness_witness_left_left_right
exact hsearch_right_witness_witness_left_right
exact hsearch_right_witness_witness_right
qed

Occurrences

2,038

Distinct objects

1,385

Reused references

60

Depth

71

Cuts

59

Certificate SHA-256

4f3834b266fc…

prime_nonzeroEvery prime natural is nonzero Primes checked

Every prime natural is nonzero.

Prerequisites

mul_zero_left succ_ne_zero

Used directly by

prime_decidable prime_unbounded greatest_prime_divisor_search beta_canonical_product_cancel_last two_prime_product_uniqueness prime_mod_inverse prime_mod_cancel prime_is_succ_succ prime_bounded_nonzero_mod_inverse

Exact expanded PA statement

Copy statement

forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)
use mul_zero_left
use succ_ne_zero
intro p
intro hp
intro hp0
cases hp
specialize hp_right 0
specialize hp_right 0
have hunit : 0 = 1 \/ 0 = 1
apply hp_right
rewrite hp0
symm
apply mul_zero_left
cases hunit
specialize succ_ne_zero 0
apply succ_ne_zero
symm
exact hunit_left
specialize succ_ne_zero 0
apply succ_ne_zero
symm
exact hunit_right
qed

Occurrences

49

Distinct objects

49

Reused references

0

Depth

11

Cuts

2

Certificate SHA-256

28430940e08d…

prime_decidablePrimality of naturals is decidable Primes checked

Primality of every natural number is constructively decidable.

Prerequisites

eq_decidable prime_or_composite prime_nonzero

Used directly by

prime_divides_decidable

Exact expanded PA statement

Copy statement

forall n. ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ ~((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ ~((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1)))
use eq_decidable
use prime_or_composite
use prime_nonzero
intro n
specialize eq_decidable n
specialize eq_decidable 0
have hn0 : n = 0 \/ ~(n = 0)
apply eq_decidable
cases hn0
right
intro hp
specialize prime_nonzero n
apply prime_nonzero
exact hp
exact hn0_left
specialize eq_decidable_before n
specialize eq_decidable_before 1
have hn1 : n = 1 \/ ~(n = 1)
apply eq_decidable_before
cases hn1
right
intro hp
cases hp
apply hp_left
exact hn1_left
specialize prime_or_composite n
have hkind : ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ exists c d. ((~(c = 1) /\ ~(d = 1)) /\ n = c * d))
apply prime_or_composite
exact hn0_right
exact hn1_right
cases hkind
left
exact hkind_left
right
intro hp
cases hp
cases hkind_right
cases hkind_right_witness
cases hkind_right_witness_witness
cases hkind_right_witness_witness_left
specialize hp_right x
specialize hp_right x1
have hunit : x = 1 \/ x1 = 1
apply hp_right
exact hkind_right_witness_witness_right
cases hunit
apply hkind_right_witness_witness_left_left
exact hunit_left
apply hkind_right_witness_witness_left_right
exact hunit_right
qed

Occurrences

2,194

Distinct objects

1,471

Reused references

63

Depth

73

Cuts

64

Certificate SHA-256

80a5f7837c8e…

factor_nonzero_leftA left factor of a nonzero product is nonzero Multiplication checked

The left factor of a nonzero product is nonzero.

Prerequisites

mul_zero_left

Used directly by

greatest_prime_divisor_descent

Exact expanded PA statement

Copy statement

forall n c d. ~(n = 0) -> n = c * d -> ~(c = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n c d. ~(n = 0) -> n = c * d -> ~(c = 0)
use mul_zero_left
intro n
intro c
intro d
intro hn
intro hfac
intro hc
apply hn
trans c * d
exact hfac
rewrite hc
apply mul_zero_left
qed

Occurrences

37

Distinct objects

37

Reused references

0

Depth

12

Cuts

1

Certificate SHA-256

623e666aaba6…

proper_factor_ltA proper factor is strictly smaller than its product Primes checked

A factor with a nonunit cofactor is strictly smaller than a nonzero product.

Prerequisites

divisor_le_nonzero le_eq_or_lt mul_left_cancel_nonzero mul_one

Used directly by

prime_divisor_exists_up_to greatest_prime_divisor_descent

Exact expanded PA statement

Copy statement

forall n c d. ~(n = 0) -> n = c * d -> ~(d = 1) -> exists k. k + S c = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n c d. ~(n = 0) -> n = c * d -> ~(d = 1) -> exists k. k + S c = n
use divisor_le_nonzero
use le_eq_or_lt
use mul_left_cancel_nonzero
use mul_one
intro n
intro c
intro d
intro hn
intro hfactor
intro hd
have hle : exists k. k + c = n
specialize divisor_le_nonzero c
specialize divisor_le_nonzero n
apply divisor_le_nonzero
exact hn
exists d
exact hfactor
have hcases : c = n \/ exists k. k + S c = n
specialize le_eq_or_lt c
specialize le_eq_or_lt n
apply le_eq_or_lt
exact hle
cases hcases
exfalso
apply hd
have hc : ~(c = 0)
intro hc0
apply hn
trans c
symm
exact hcases_left
exact hc0
specialize mul_left_cancel_nonzero c
specialize mul_left_cancel_nonzero d
specialize mul_left_cancel_nonzero 1
apply mul_left_cancel_nonzero
exact hc
trans n
symm
exact hfactor
trans c
symm
exact hcases_left
symm
specialize mul_one c
exact mul_one
exact hcases_right
qed

Occurrences

468

Distinct objects

396

Reused references

10

Depth

26

Cuts

16

Certificate SHA-256

22b2140fa2bd…

prime_divisor_exists_up_toBounded prime-divisor existence Primes checked

Bounded strong induction constructs a prime divisor of every nonzero nonunit natural.

Prerequisites

mul_zero_left le_zero lt_of_lt_of_le le_of_succ_le_succ multiple_refl multiple_trans prime_or_composite proper_factor_lt

Used directly by

prime_divisor_exists

Exact expanded PA statement

Copy statement

forall B n. (exists t. t + n = B) -> ~(n = 0) -> ~(n = 1) -> exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. n = p * k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B n. (exists t. t + n = B) -> ~(n = 0) -> ~(n = 1) -> exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. n = p * k)
use mul_zero_left
use le_zero
use lt_of_lt_of_le
use le_of_succ_le_succ
use multiple_refl
use multiple_trans
use prime_or_composite
use proper_factor_lt
intro B
induction B
intro n
intro hnB
intro hn0
intro hn1
exfalso
apply hn0
apply le_zero
exact hnB
intro n
intro hnB
intro hn0
intro hn1
specialize prime_or_composite n
have hpc : (~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ exists c d. ((~(c = 1) /\ ~(d = 1)) /\ n = c * d)
apply prime_or_composite
exact hn0
exact hn1
cases hpc
exists n
split
exact hpc_left
apply multiple_refl
cases hpc_right
cases hpc_right_witness
cases hpc_right_witness_witness
cases hpc_right_witness_witness_left
have hc0 : ~(x = 0)
intro hc
apply hn0
trans x * x1
exact hpc_right_witness_witness_right
rewrite hc
apply mul_zero_left
have hcn : exists k. k + S x = n
specialize proper_factor_lt n
specialize proper_factor_lt x
specialize proper_factor_lt x1
apply proper_factor_lt
exact hn0
exact hpc_right_witness_witness_right
exact hpc_right_witness_witness_left_right
have hcSB : exists k. k + S x = S B
specialize lt_of_lt_of_le x
specialize lt_of_lt_of_le n
specialize lt_of_lt_of_le (S B)
apply lt_of_lt_of_le
exact hcn
exact hnB
have hcB : exists k. k + x = B
apply le_of_succ_le_succ
exact hcSB
specialize IH x
have hp : exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. x = p * k)
apply IH
exact hcB
exact hc0
exact hpc_right_witness_witness_left_left
cases hp
cases hp_witness
exists x2
split
exact hp_witness_left
specialize multiple_trans x
specialize multiple_trans x2
specialize multiple_trans n
apply multiple_trans
exists x1
exact hpc_right_witness_witness_right
exact hp_witness_right
qed

Occurrences

2,931

Distinct objects

1,659

Reused references

77

Depth

78

Cuts

91

Certificate SHA-256

1258bfd21a67…

prime_divisor_existsEvery nontrivial natural has a prime divisor Primes checked

Every nonzero nonunit natural has a prime divisor.

Prerequisites

le_refl prime_divisor_exists_up_to

Used directly by

prime_unbounded greatest_prime_divisor_exists

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> ~(n = 1) -> exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. n = p * k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> ~(n = 1) -> exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. n = p * k)
use le_refl
use prime_divisor_exists_up_to
intro n
intro hn0
intro hn1
specialize prime_divisor_exists_up_to n
specialize prime_divisor_exists_up_to n
apply prime_divisor_exists_up_to
apply le_refl
exact hn0
exact hn1
qed

Occurrences

2,977

Distinct objects

1,680

Reused references

78

Depth

80

Cuts

94

Certificate SHA-256

fab66d905009…

prime_divisor_eq_one_or_selfEvery divisor of a prime is one or the prime Primes checked

Every divisor of a prime is one or the prime itself.

Prerequisites

mul_one

Used directly by

euclid_prime_dvd_product beta_prime_divisor_product_member two_prime_product_uniqueness prime_coprime_or_divides distinct_primes_coprime

Exact expanded PA statement

Copy statement

forall p g. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists x. p = g * x) -> g = 1 \/ p = g

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p g. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists x. p = g * x) -> g = 1 \/ p = g
use mul_one
intro p
intro g
intro hp
intro hdiv
cases hp
cases hdiv
specialize hp_right g
specialize hp_right x
have hfactor : g = 1 \/ x = 1
apply hp_right
exact hdiv_witness
cases hfactor
left
exact hfactor_left
right
trans g * x
exact hdiv_witness
rewrite hfactor_right
apply mul_one
qed

Occurrences

57

Distinct objects

57

Reused references

0

Depth

12

Cuts

2

Certificate SHA-256

e7e3017fe0ec…

euclid_prime_dvd_productEuclid's lemma Primes checked

A prime dividing a product divides at least one factor (Euclid's lemma).

Prerequisites

prime_divisor_eq_one_or_self gcd_exists_relational is_gcd_one_to_coprime gauss_coprime_cancel

Used directly by

beta_prime_divisor_product_member two_prime_product_uniqueness

Exact expanded PA statement

Copy statement

forall p a b. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists k. a * b = p * k) -> (exists u. a = p * u) \/ exists v. b = p * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a b. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists k. a * b = p * k) -> (exists u. a = p * u) \/ exists v. b = p * v
use prime_divisor_eq_one_or_self
use gcd_exists_relational
use is_gcd_one_to_coprime
use gauss_coprime_cancel
intro p
intro a
intro b
intro hp
intro hab
have hg : exists g. (((exists x. p = g * x) /\ (exists y. a = g * y)) /\ forall c. (exists u. p = c * u) -> (exists v. a = c * v) -> exists w. g = c * w)
apply gcd_exists_relational
cases hg
have hgfull : (((exists u. p = x * u) /\ (exists v. a = x * v)) /\ forall c. (exists s. p = c * s) -> (exists t. a = c * t) -> exists w. x = c * w)
exact hg_witness
cases hg_witness
cases hg_witness_left
have hfactor : x = 1 \/ p = x
specialize prime_divisor_eq_one_or_self p
specialize prime_divisor_eq_one_or_self x
apply prime_divisor_eq_one_or_self
exact hp
exact hg_witness_left_left
cases hfactor
right
apply gauss_coprime_cancel
have hcop : forall d. (exists u. p = d * u) -> (exists v. a = d * v) -> d = 1
apply is_gcd_one_to_coprime
have hg1 : (((exists u. p = 1 * u) /\ (exists v. a = 1 * v)) /\ forall c. (exists s. p = c * s) -> (exists t. a = c * t) -> exists w. 1 = c * w)
rewrite <- hfactor_left
rewrite <- hfactor_left
rewrite <- hfactor_left
exact hgfull
exact hg1
exact hcop
exact hab
left
cases hg_witness_left_right
exists x1
rewrite hfactor_right
exact hg_witness_left_right_witness
qed

Occurrences

5,382

Distinct objects

1,806

Reused references

113

Depth

55

Cuts

159

Certificate SHA-256

8db2385c53a7…

mod_eq_reflBalanced natural congruence is reflexive Congruence & CRT checked

Balanced natural congruence is reflexive.

Prerequisites

none — a root theorem

Used directly by

binary_crt beta_product_pointwise_mod_congruent beta_sum_pointwise_mod_congruent coprime_mod_inverse pow_mod_congruent pow_predecessor_parity_mod crt_scaled_common_remainder_lift generalized_binary_crt_sufficient_zero_left generalized_binary_crt_sufficient_zero_right

Exact expanded PA statement

Copy statement

forall m a. exists u v. a + m * u = a + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a. exists u v. a + m * u = a + m * v
intro m
intro a
exists 0
exists 0
refl
qed

Occurrences

5

Distinct objects

5

Reused references

0

Depth

5

Cuts

0

Certificate SHA-256

48a2acf45d57…

mod_eq_symmBalanced natural congruence is symmetric Congruence & CRT checked

Balanced natural congruence is symmetric.

Prerequisites

none — a root theorem

Used directly by

mod_eq_decidable_from_remainders quadratic_residue_bounded_equiv mod_eq_cancel_coprime prime_bounded_nonzero_mod_inverse canonical_remainders_characterize_mod_eq bounded_mod_inverse_unique coprime_bounded_mod_inverse crt_solution_pair_congruent crt_common_solution_implies_gcd_compatible mod_eq_common_remainder_decomposition generalized_binary_crt_sufficient_zero_right mod_eq_lcm_merge crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall m a b. (exists u v. a + m * u = b + m * v) -> exists r s. b + m * r = a + m * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b. (exists u v. a + m * u = b + m * v) -> exists r s. b + m * r = a + m * s
intro m
intro a
intro b
intro h
cases h
cases h_witness
exists x1
exists x
symm
exact h_witness_witness
qed

Occurrences

12

Distinct objects

12

Reused references

0

Depth

10

Cuts

0

Certificate SHA-256

136980f4284d…

mod_eq_transBalanced natural congruence is transitive Congruence & CRT checked

Balanced natural congruence is transitive.

Prerequisites

add_assoc add_comm mul_add

Used directly by

mod_eq_mul mod_eq_to_remainder_decomposition binary_crt binary_crt_fold_step mod_eq_decidable_from_remainders quadratic_residue_bounded_equiv coprime_mod_inverse mod_eq_cancel_coprime bounded_square_mod3_classify bounded_square_mod5_classify bounded_square_mod7_classify pow_predecessor_parity_mod prime_bounded_nonzero_mod_inverse canonical_remainders_characterize_mod_eq bounded_mod_inverse_unique coprime_bounded_mod_inverse crt_solution_pair_congruent crt_common_solution_implies_gcd_compatible mod_eq_common_remainder_decomposition crt_solution_class_iff_lcm

Exact expanded PA statement

Copy statement

forall m a b c. (exists u v. a + m * u = b + m * v) -> (exists r s. b + m * r = c + m * s) -> exists x y. a + m * x = c + m * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b c. (exists u v. a + m * u = b + m * v) -> (exists r s. b + m * r = c + m * s) -> exists x y. a + m * x = c + m * y
use add_assoc
use add_comm
use mul_add
intro m
intro a
intro b
intro c
intro hab
intro hbc
cases hab
cases hab_witness
cases hbc
cases hbc_witness
exists x + x2
exists x3 + x1
trans a + (m * x + m * x2)
congr
refl
apply mul_add
trans (a + m * x) + m * x2
symm
apply add_assoc
trans (b + m * x1) + m * x2
congr
exact hab_witness_witness
refl
trans b + (m * x1 + m * x2)
apply add_assoc
trans b + (m * x2 + m * x1)
congr
refl
apply add_comm
trans (b + m * x2) + m * x1
symm
apply add_assoc
trans (c + m * x3) + m * x1
congr
exact hbc_witness_witness
refl
trans c + (m * x3 + m * x1)
apply add_assoc
congr
refl
symm
apply mul_add
qed

Occurrences

252

Distinct objects

204

Reused references

16

Depth

29

Cuts

6

Certificate SHA-256

052be6f7213b…

mod_eq_addBalanced congruence respects addition Congruence & CRT checked

Balanced natural congruence respects addition.

Prerequisites

mul_add add_comm add_permute_outer

Used directly by

binary_crt beta_sum_pointwise_mod_congruent coprime_mod_inverse crt_scaled_common_remainder_lift

Exact expanded PA statement

Copy statement

forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a + c) + m * x = (b + d) + m * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a + c) + m * x = (b + d) + m * y
use mul_add
use add_comm
use add_permute_outer
intro m
intro a
intro b
intro c
intro d
intro hab
intro hcd
cases hab
cases hab_witness
cases hcd
cases hcd_witness
exists x2 + x
exists x3 + x1
trans (a + c) + (m * x2 + m * x)
congr
refl
apply mul_add
trans (m * x2 + c) + (a + m * x)
apply add_permute_outer
trans (c + m * x2) + (a + m * x)
congr
apply add_comm
refl
trans (a + m * x) + (c + m * x2)
apply add_comm
trans (b + m * x1) + (d + m * x3)
congr
exact hab_witness_witness
exact hcd_witness_witness
trans (d + m * x3) + (b + m * x1)
apply add_comm
trans (m * x3 + d) + (b + m * x1)
congr
apply add_comm
refl
trans (b + d) + (m * x3 + m * x1)
symm
apply add_permute_outer
congr
refl
symm
apply mul_add
qed

Occurrences

370

Distinct objects

249

Reused references

17

Depth

30

Cuts

10

Certificate SHA-256

49e4d310fb28…

mod_eq_mul_rightBalanced congruence is stable under right multiplication Congruence & CRT checked

Balanced congruence is preserved by multiplication on the right.

Prerequisites

add_mul mul_assoc

Used directly by

mod_eq_mul_left mod_eq_mul mod_eq_cancel_coprime bounded_mod_inverse_unique

Exact expanded PA statement

Copy statement

forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (a * c) + m * r = (b * c) + m * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (a * c) + m * r = (b * c) + m * s
use add_mul
use mul_assoc
intro m
intro a
intro b
intro c
intro h
cases h
cases h_witness
exists x * c
exists x1 * c
trans a * c + (m * x) * c
congr
refl
symm
apply mul_assoc
trans (a + m * x) * c
symm
apply add_mul
trans (b + m * x1) * c
congr
exact h_witness_witness
refl
trans b * c + (m * x1) * c
apply add_mul
congr
refl
apply mul_assoc
qed

Occurrences

484

Distinct objects

339

Reused references

37

Depth

26

Cuts

13

Certificate SHA-256

4994212781f9…

mod_eq_mul_leftBalanced congruence is stable under left multiplication Congruence & CRT checked

Balanced congruence is preserved by multiplication on the left.

Prerequisites

mod_eq_mul_right mul_comm

Used directly by

mod_eq_mul binary_crt mod_eq_cancel_coprime prime_bounded_nonzero_mod_inverse bounded_mod_inverse_unique coprime_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (c * a) + m * r = (c * b) + m * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (c * a) + m * r = (c * b) + m * s
use mod_eq_mul_right
use mul_comm
intro m
intro a
intro b
intro c
intro h
specialize mod_eq_mul_right m
specialize mod_eq_mul_right a
specialize mod_eq_mul_right b
specialize mod_eq_mul_right c
have hr : exists r s. (a * c) + m * r = (b * c) + m * s
apply mod_eq_mul_right
exact h
cases hr
cases hr_witness
exists x
exists x1
trans a * c + m * x
congr
apply mul_comm
refl
trans b * c + m * x1
exact hr_witness_witness
congr
apply mul_comm
refl
qed

Occurrences

738

Distinct objects

371

Reused references

38

Depth

27

Cuts

21

Certificate SHA-256

84e29f574f4e…

mod_eq_mulBalanced congruence respects multiplication Congruence & CRT checked

Balanced natural congruence respects multiplication.

Prerequisites

mod_eq_mul_right mod_eq_mul_left mod_eq_trans

Used directly by

beta_product_pointwise_mod_congruent pow_mod_congruent pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a * c) + m * x = (b * d) + m * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a * c) + m * x = (b * d) + m * y
use mod_eq_mul_right
use mod_eq_mul_left
use mod_eq_trans
intro m
intro a
intro b
intro c
intro d
intro hab
intro hcd
have hacbc : exists r s. (a * c) + m * r = (b * c) + m * s
specialize mod_eq_mul_right m
specialize mod_eq_mul_right a
specialize mod_eq_mul_right b
specialize mod_eq_mul_right c
apply mod_eq_mul_right
exact hab
have hbcbd : exists r s. (b * c) + m * r = (b * d) + m * s
specialize mod_eq_mul_left m
specialize mod_eq_mul_left c
specialize mod_eq_mul_left d
specialize mod_eq_mul_left b
apply mod_eq_mul_left
exact hcd
specialize mod_eq_trans m
specialize mod_eq_trans (a * c)
specialize mod_eq_trans (b * c)
specialize mod_eq_trans (b * d)
apply mod_eq_trans
exact hacbc
exact hbcbd
qed

Occurrences

1,505

Distinct objects

471

Reused references

42

Depth

32

Cuts

43

Certificate SHA-256

660f19930d52…

remainder_decomposition_to_mod_eqA quotient-remainder decomposition gives congruence Congruence & CRT checked

A directed quotient/remainder equation gives balanced congruence to its remainder.

Prerequisites

add_comm mul_comm

Used directly by

beta_at_to_mod_eq mod_eq_decidable_from_remainders quadratic_residue_bounded_equiv prime_bounded_nonzero_mod_inverse canonical_remainders_characterize_mod_eq coprime_bounded_mod_inverse mod_eq_common_remainder_decomposition mod_eq_lcm_merge crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * v
use add_comm
use mul_comm
intro m
intro b
intro q
intro x
intro h
exists 0
exists q
rewrite PA5
rewrite PA3
trans q * m + x
exact h
trans x + q * m
apply add_comm
congr
refl
apply mul_comm
qed

Occurrences

323

Distinct objects

228

Reused references

23

Depth

26

Cuts

10

Certificate SHA-256

acea3f60f7fa…

mod_eq_bounded_uniqueBounded congruent representatives are equal Congruence & CRT checked

Two balanced-congruent values below the same modulus are equal.

Prerequisites

add_comm division_remainder_unique

Used directly by

mod_eq_to_remainder_decomposition mod_eq_decidable_from_remainders bounded_square_mod3_classify bounded_square_mod5_classify bounded_square_mod7_classify beta_half_range_mod_eq_value prime_bounded_nonzero_mod_inverse canonical_remainders_characterize_mod_eq bounded_mod_inverse_unique crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall m a b. (exists ha. ha + S a = m) -> (exists hb. hb + S b = m) -> (exists u v. a + m * u = b + m * v) -> a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b. (exists ha. ha + S a = m) -> (exists hb. hb + S b = m) -> (exists u v. a + m * u = b + m * v) -> a = b
use add_comm
use division_remainder_unique
intro m
intro a
intro b
intro ha
intro hb
intro hab
cases hab
cases hab_witness
have hda : a + m * x = m * x + a
apply add_comm
have hdb : a + m * x = m * x1 + b
trans b + m * x1
exact hab_witness_witness
apply add_comm
specialize division_remainder_unique m
specialize division_remainder_unique (a + m * x)
specialize division_remainder_unique x
specialize division_remainder_unique a
specialize division_remainder_unique x1
specialize division_remainder_unique b
have huniq : x = x1 /\ a = b
apply division_remainder_unique
exact hda
exact ha
exact hdb
exact hb
cases huniq
exact huniq_right
qed

Occurrences

961

Distinct objects

578

Reused references

23

Depth

59

Cuts

26

Certificate SHA-256

9bdbf756db7d…

mod_eq_to_remainder_decompositionA bounded congruent representative is a directed remainder Congruence & CRT checked

A bounded balanced residue has a directed quotient/remainder witness.

Prerequisites

division_remainder_exists add_comm mul_comm mod_eq_trans mod_eq_bounded_unique

Used directly by

beta_at_of_mod_eq_bound binary_crt_remainders mod_eq_common_remainder_decomposition

Exact expanded PA statement

Copy statement

forall m b x. ~(m = 0) -> (exists h. h + S x = m) -> (exists u v. b + m * u = x + m * v) -> exists q. b = q * m + x

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m b x. ~(m = 0) -> (exists h. h + S x = m) -> (exists u v. b + m * u = x + m * v) -> exists q. b = q * m + x
use division_remainder_exists
use add_comm
use mul_comm
use mod_eq_trans
use mod_eq_bounded_unique
intro m
intro b
intro x
intro hm
intro hx
intro hbx
have hdiv : exists q r. b = m * q + r /\ exists h. h + S r = m
specialize division_remainder_exists m
specialize division_remainder_exists b
apply division_remainder_exists
exact hm
cases hdiv
cases hdiv_witness
cases hdiv_witness_witness
have hremb : exists u v. x2 + m * u = b + m * v
exists x1
exists 0
trans m * x1 + x2
apply add_comm
trans b
symm
exact hdiv_witness_witness_left
symm
rewrite PA5
apply PA3
have hremx : exists u v. x2 + m * u = x + m * v
specialize mod_eq_trans m
specialize mod_eq_trans x2
specialize mod_eq_trans b
specialize mod_eq_trans x
apply mod_eq_trans
exact hremb
exact hbx
have hrx : x2 = x
specialize mod_eq_bounded_unique m
specialize mod_eq_bounded_unique x2
specialize mod_eq_bounded_unique x
apply mod_eq_bounded_unique
exact hdiv_witness_witness_right
exact hx
exact hremx
exists x1
trans m * x1 + x2
exact hdiv_witness_witness_left
trans x1 * m + x2
congr
apply mul_comm
refl
congr
refl
exact hrx
qed

Occurrences

1,793

Distinct objects

962

Reused references

59

Depth

64

Cuts

50

Certificate SHA-256

a92fbaccbecc…

beta_modulus_nonzeroEvery Gödel-beta modulus is nonzero β sequences, products & FTA checked

Every Gödel-beta decoding modulus is nonzero.

Prerequisites

succ_ne_zero

Used directly by

beta_at_exists beta_at_of_mod_eq_bound binary_crt_beta_pair beta_accumulated_product_step beta_crt_prefix_congruence_step bounded_beta_crt_prefix_invariant beta_exclusive_accumulated_product_step beta_exclusive_recode_congruence_step beta_prefix_extend

Exact expanded PA statement

Copy statement

forall c i. ~(S ((S i) * c) = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i. ~(S ((S i) * c) = 0)
use succ_ne_zero
intro c
intro i
specialize succ_ne_zero ((S i) * c)
exact succ_ne_zero
qed

Occurrences

9

Distinct objects

9

Reused references

0

Depth

6

Cuts

1

Certificate SHA-256

e8ce620074f6…

beta_at_self_of_boundA bounded value is its own Gödel-beta residue β sequences, products & FTA checked

A value below a Gödel-beta modulus decodes to itself when used as the code.

Prerequisites

mul_zero_left zero_add

Used directly by

beta_prefix_product_trace_exists prime_factorization_exists_up_to beta_prefix_sum_trace_exists

Exact expanded PA statement

Copy statement

forall c i x. (exists h. h + S x = S ((S i) * c)) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. x = q * S ((S i) * c) + x)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i x. (exists h. h + S x = S ((S i) * c)) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. x = q * S ((S i) * c) + x)
use mul_zero_left
use zero_add
intro c
intro i
intro x
intro hx
split
exact hx
exists 0
specialize mul_zero_left (S ((S i) * c))
rewrite mul_zero_left
specialize zero_add x
rewrite zero_add
refl
qed

Occurrences

62

Distinct objects

61

Reused references

1

Depth

16

Cuts

2

Certificate SHA-256

b563c1208a1c…

beta_at_existsEvery Gödel-beta position has a decoded value β sequences, products & FTA checked

Every Gödel-beta position has a bounded decoded residue.

Prerequisites

beta_modulus_nonzero mul_comm division_remainder_exists

Used directly by

beta_at_exists_unique beta_crt_prefix_congruence_step beta_exclusive_recode_congruence_step beta_prefix_product_trace_exists beta_product_exists prime_factorization_exists_up_to beta_prefix_sum_trace_exists beta_sum_exists beta_prefix_replace_exists beta_prefix_swap_last_exists beta_prefix_swap_last_reflect finite_contains_decidable beta_prefix_replace_reflect

Exact expanded PA statement

Copy statement

forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)
use beta_modulus_nonzero
use mul_comm
use division_remainder_exists
intro b
intro c
intro i
have hm0 : ~(S ((S i) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero i
exact beta_modulus_nonzero
specialize division_remainder_exists (S ((S i) * c))
specialize division_remainder_exists b
have hdiv : exists q r. b = S ((S i) * c) * q + r /\ exists h. h + S r = S ((S i) * c)
apply division_remainder_exists
exact hm0
cases hdiv
cases hdiv_witness
cases hdiv_witness_witness
exists x1
split
exact hdiv_witness_witness_right
exists x
trans S ((S i) * c) * x + x1
exact hdiv_witness_witness_left
congr
apply mul_comm
refl
qed

Occurrences

479

Distinct objects

392

Reused references

38

Depth

31

Cuts

15

Certificate SHA-256

967de23f5a2e…

beta_at_uniqueA Gödel-beta decoded value is unique β sequences, products & FTA checked

The decoded residue at a Gödel-beta position is unique.

Prerequisites

mul_comm division_remainder_unique

Used directly by

beta_at_exists_unique beta_crt_prefix_congruence_step beta_exclusive_recode_congruence_step beta_product_functional beta_product_zero beta_product_succ_decompose beta_factor_divides_product prime_factorization_exists_up_to beta_prime_divisor_product_member beta_sorted_factor_le_last beta_canonical_last_factors_equal beta_canonical_product_cancel_last prime_factorization_uniqueness_by_length beta_repeat_entry_eq beta_range_entry_eq beta_sum_trace_functional beta_sum_zero beta_sum_succ_decompose bit_count_succ_decompose finite_bounded_entry_lt beta_prefix_replace_exists beta_prefix_swap_last_reflect finite_contains_decidable finite_bounded_injective_surjective beta_prefix_replace_reflect beta_product_replace_balance beta_product_swap_last_invariant beta_reindex_alignment_swap_last

Exact expanded PA statement

Copy statement

forall b c i x y. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i x y. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y
use mul_comm
use division_remainder_unique
intro b
intro c
intro i
intro x
intro y
intro hx
intro hy
cases hx
cases hy
cases hx_right
cases hy_right
have hdx : b = S ((S i) * c) * x1 + x
trans x1 * S ((S i) * c) + x
exact hx_right_witness
congr
apply mul_comm
refl
have hdy : b = S ((S i) * c) * x2 + y
trans x2 * S ((S i) * c) + y
exact hy_right_witness
congr
apply mul_comm
refl
specialize division_remainder_unique (S ((S i) * c))
specialize division_remainder_unique b
specialize division_remainder_unique x1
specialize division_remainder_unique x
specialize division_remainder_unique x2
specialize division_remainder_unique y
have huniq : x1 = x2 /\ x = y
apply division_remainder_unique
exact hdx
exact hx_left
exact hdy
exact hy_left
cases huniq
exact huniq_right
qed

Occurrences

1,121

Distinct objects

692

Reused references

37

Depth

59

Cuts

30

Certificate SHA-256

891320ec0873…

beta_at_exists_uniqueEvery Gödel-beta position has exactly one decoded value β sequences, products & FTA checked

Every Gödel-beta position has exactly one decoded residue.

Prerequisites

beta_at_exists beta_at_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c i. exists x. (((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) /\ forall y. ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i. exists x. (((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) /\ forall y. ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y)
use beta_at_exists
use beta_at_unique
intro b
intro c
intro i
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists i
have hx : exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)
apply beta_at_exists
cases hx
exists x
split
exact hx_witness
intro y
intro hy
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique i
specialize beta_at_unique x
specialize beta_at_unique y
apply beta_at_unique
exact hx_witness
exact hy
qed

Occurrences

1,625

Distinct objects

901

Reused references

55

Depth

61

Cuts

47

Certificate SHA-256

37ae2a410d20…

beta_at_to_mod_eqA Gödel-beta value is congruent to its code β sequences, products & FTA checked

Every expanded Gödel-beta value is balanced-congruent to its code.

Prerequisites

remainder_decomposition_to_mod_eq

Used directly by

bounded_beta_crt_prefix_invariant

Exact expanded PA statement

Copy statement

forall b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v
use remainder_decomposition_to_mod_eq
intro b
intro c
intro i
intro x
intro hat
cases hat
cases hat_right
specialize remainder_decomposition_to_mod_eq (S ((S i) * c))
specialize remainder_decomposition_to_mod_eq b
specialize remainder_decomposition_to_mod_eq x1
specialize remainder_decomposition_to_mod_eq x
apply remainder_decomposition_to_mod_eq
exact hat_right_witness
qed

Occurrences

358

Distinct objects

263

Reused references

23

Depth

27

Cuts

11

Certificate SHA-256

dd174002d470…

beta_at_of_mod_eq_boundA bounded congruent value is a Gödel-beta value β sequences, products & FTA checked

A bounded value congruent to a code is its expanded Gödel-beta value.

Prerequisites

beta_modulus_nonzero mod_eq_to_remainder_decomposition

Used directly by

binary_crt_beta_pair beta_prefix_extend

Exact expanded PA statement

Copy statement

forall b c i x. (exists h. h + S x = S ((S i) * c)) -> (exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i x. (exists h. h + S x = S ((S i) * c)) -> (exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)
use beta_modulus_nonzero
use mod_eq_to_remainder_decomposition
intro b
intro c
intro i
intro x
intro hx
intro hmod
split
exact hx
specialize mod_eq_to_remainder_decomposition (S ((S i) * c))
specialize mod_eq_to_remainder_decomposition b
specialize mod_eq_to_remainder_decomposition x
apply mod_eq_to_remainder_decomposition
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero i
exact beta_modulus_nonzero
exact hx
exact hmod
qed

Occurrences

1,839

Distinct objects

1,008

Reused references

59

Depth

66

Cuts

53

Certificate SHA-256

e4a2ead06ca6…

dvd_to_mod_zeroA multiple is congruent to zero Congruence & CRT checked

A multiple is balanced-congruent to zero.

Prerequisites

zero_add

Used directly by

binary_crt

Exact expanded PA statement

Copy statement

forall m a. (exists k. a = m * k) -> exists u v. a + m * u = 0 + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a. (exists k. a = m * k) -> exists u v. a + m * u = 0 + m * v
use zero_add
intro m
intro a
intro h
cases h
exists 0
exists x
rewrite h_witness
simp [zero_add]
qed

Occurrences

41

Distinct objects

40

Reused references

1

Depth

14

Cuts

1

Certificate SHA-256

fa1d8f3ac2a8…

add_residueAbsorb a known residue decomposition into a multiple Congruence & CRT checked

Absorb a second quotient into an existing residue equation.

Prerequisites

add_assoc mul_add

Used directly by

square_residue_lift

Exact expanded PA statement

Copy statement

forall a u c r s. r = a * c + s -> a * u + r = a * (u + c) + s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a u c r s. r = a * c + s -> a * u + r = a * (u + c) + s
use add_assoc
use mul_add
intro a
intro u
intro c
intro r
intro s
intro hr
trans a * u + (a * c + s)
congr
refl
exact hr
trans (a * u + a * c) + s
symm
apply add_assoc
congr
symm
apply mul_add
refl
qed

Occurrences

135

Distinct objects

95

Reused references

8

Depth

18

Cuts

3

Certificate SHA-256

2b9ca2734b62…

add_residue_liftLift an additive residue decomposition Congruence & CRT checked

Combine two quotient-and-residue summands and normalize the residue.

Prerequisites

add_comm add_assoc mul_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a x u r q t c s. x = a * q + t -> r + t = a * c + s -> (a * u + r) + x = a * ((u + q) + c) + s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a x u r q t c s. x = a * q + t -> r + t = a * c + s -> (a * u + r) + x = a * ((u + q) + c) + s
use add_comm
use add_assoc
use mul_add
intro a
intro x
intro u
intro r
intro q
intro t
intro c
intro s
intro hx
intro hr
trans (a * u + r) + (a * q + t)
congr
refl
exact hx
trans a * u + (r + (a * q + t))
apply add_assoc
trans a * u + ((r + a * q) + t)
congr
refl
symm
apply add_assoc
trans a * u + ((a * q + r) + t)
congr
refl
congr
apply add_comm
refl
trans a * u + (a * q + (r + t))
congr
refl
apply add_assoc
trans (a * u + a * q) + (r + t)
symm
apply add_assoc
trans a * (u + q) + (r + t)
congr
symm
apply mul_add
refl
trans a * (u + q) + (a * c + s)
congr
refl
exact hr
trans (a * (u + q) + a * c) + s
symm
apply add_assoc
congr
symm
apply mul_add
refl
qed

Occurrences

259

Distinct objects

211

Reused references

16

Depth

28

Cuts

6

Certificate SHA-256

7eb051f21c6a…

square_decompExpand the square of a quotient-residue decomposition Congruence & CRT checked

Expand a square while retaining an explicit quotient and remainder.

Prerequisites

add_assoc mul_comm mul_add add_mul mul_assoc

Used directly by

square_residue_lift

Exact expanded PA statement

Copy statement

forall a z q r. z = a * q + r -> z * z = a * (q * z + r * q) + r * r

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a z q r. z = a * q + r -> z * z = a * (q * z + r * q) + r * r
use add_assoc
use mul_comm
use mul_add
use add_mul
use mul_assoc
intro a
intro z
intro q
intro r
intro hz
trans (a * q + r) * z
congr
exact hz
refl
trans (a * q) * z + r * z
apply add_mul
trans a * (q * z) + r * z
congr
apply mul_assoc
refl
trans a * (q * z) + r * (a * q + r)
congr
refl
congr
refl
exact hz
trans a * (q * z) + (r * (a * q) + r * r)
congr
refl
apply mul_add
trans a * (q * z) + (a * (r * q) + r * r)
congr
refl
congr
trans (r * a) * q
symm
apply mul_assoc
trans (a * r) * q
congr
apply mul_comm
refl
apply mul_assoc
refl
trans (a * (q * z) + a * (r * q)) + r * r
symm
apply add_assoc
congr
symm
apply mul_add
refl
qed

Occurrences

847

Distinct objects

370

Reused references

40

Depth

29

Cuts

23

Certificate SHA-256

504cec489222…

square_residue_liftLift a residue decomposition through squaring Congruence & CRT checked

Lift one quotient-and-remainder equation through squaring.

Prerequisites

square_decomp add_residue

Used directly by

square_residue_witness mod5_square_residue_three mod5_square_residue_four mod5_fourth_power_one

Exact expanded PA statement

Copy statement

forall a z q r c s. z = a * q + r -> r * r = a * c + s -> z * z = a * ((q * z + r * q) + c) + s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a z q r c s. z = a * q + r -> r * r = a * c + s -> z * z = a * ((q * z + r * q) + c) + s
use square_decomp
use add_residue
intro a
intro z
intro q
intro r
intro c
intro s
intro hz
intro hrc
trans a * (q * z + r * q) + r * r
apply square_decomp
exact hz
apply add_residue
exact hrc
qed

Occurrences

1,008

Distinct objects

421

Reused references

42

Depth

30

Cuts

28

Certificate SHA-256

efdeed4ea296…

square_residue_witnessPackage a square residue as an existential witness Congruence & CRT checked

Existential wrapper for the generic square-residue lift.

Prerequisites

square_residue_lift

Used directly by

quadratic_residue_bounded_equiv

Exact expanded PA statement

Copy statement

forall a z q r c s. z = a * q + r -> r * r = a * c + s -> exists w. z * z = a * w + s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a z q r c s. z = a * q + r -> r * r = a * c + s -> exists w. z * z = a * w + s
use square_residue_lift
intro a
intro z
intro q
intro r
intro c
intro s
intro hz
intro hrc
exists (q * z + r * q) + c
apply square_residue_lift
exact hz
exact hrc
qed

Occurrences

1,029

Distinct objects

442

Reused references

42

Depth

31

Cuts

29

Certificate SHA-256

3246695c5c67…

bezout_mod_leftBalanced Bézout identity projected modulo the left modulus Congruence & CRT checked

A balanced Bezout identity selects the right coefficient modulo the left modulus.

Prerequisites

add_assoc add_comm

Used directly by

binary_crt

Exact expanded PA statement

Copy statement

forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * v
use add_assoc
use add_comm
intro m
intro n
intro xp
intro yp
intro xn
intro yn
intro h
exists xp
exists xn
trans m * xp + n * yp
apply add_comm
trans 1 + (m * xn + n * yn)
exact h
trans 1 + (n * yn + m * xn)
congr
refl
apply add_comm
symm
apply add_assoc
qed

Occurrences

134

Distinct objects

123

Reused references

11

Depth

19

Cuts

4

Certificate SHA-256

0ba24c738b19…

bezout_mod_rightBalanced Bézout identity projected modulo the right modulus Congruence & CRT checked

A balanced Bezout identity selects the left coefficient modulo the right modulus.

Prerequisites

add_assoc

Used directly by

binary_crt

Exact expanded PA statement

Copy statement

forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. m * xp + n * u = (1 + m * xn) + n * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. m * xp + n * u = (1 + m * xn) + n * v
use add_assoc
intro m
intro n
intro xp
intro yp
intro xn
intro yn
intro h
exists yp
exists yn
trans 1 + (m * xn + n * yn)
exact h
symm
apply add_assoc
qed

Occurrences

50

Distinct objects

47

Reused references

3

Depth

16

Cuts

1

Certificate SHA-256

2b31fb490cc7…

mod_eq_predecessor_cancelSuccessor-predecessor cancellation in balanced congruence Congruence & CRT checked

The predecessor of a successor acts as minus one in balanced congruence.

Prerequisites

add_assoc add_comm mul_succ_left

Used directly by

binary_crt coprime_mod_inverse

Exact expanded PA statement

Copy statement

forall k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * v
use add_assoc
use add_comm
use mul_succ_left
intro k
intro a
intro z
exists 0
exists z
rewrite PA5
rewrite PA3
specialize mul_succ_left k
specialize mul_succ_left z
rewrite mul_succ_left
trans a + (z + k * z)
apply add_assoc
congr
refl
apply add_comm
qed

Occurrences

315

Distinct objects

192

Reused references

19

Depth

25

Cuts

9

Certificate SHA-256

35e5bcc42164…

binary_crtConstructive binary Chinese remainder theorem Congruence & CRT checked

Constructive binary CRT for positive coprime natural moduli using balanced congruence.

Prerequisites

nonzero_is_succ coprime_balanced_bezout bezout_mod_left bezout_mod_right mod_eq_mul_left mul_add mul_one dvd_to_mod_zero mul_assoc mul_comm mod_eq_add mod_eq_refl mod_eq_trans mod_eq_predecessor_cancel zero_add

Used directly by

binary_crt_remainders binary_crt_beta_pair binary_crt_fold_step crt_scaled_common_remainder_lift

Exact expanded PA statement

Copy statement

forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> exists x. (exists u v. x + m * u = a + m * v) /\ (exists r s. x + n * r = b + n * s)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> exists x. (exists u v. x + m * u = a + m * v) /\ (exists r s. x + n * r = b + n * s)
use nonzero_is_succ
use coprime_balanced_bezout
use bezout_mod_left
use bezout_mod_right
use mod_eq_mul_left
use mul_add
use mul_one
use dvd_to_mod_zero
use mul_assoc
use mul_comm
use mod_eq_add
use mod_eq_refl
use mod_eq_trans
use mod_eq_predecessor_cancel
use zero_add
intro m
intro n
intro a
intro b
intro hm
intro hn
intro hcop
have hms : exists k. m = S k
specialize nonzero_is_succ m
apply nonzero_is_succ
exact hm
have hns : exists k. n = S k
specialize nonzero_is_succ n
apply nonzero_is_succ
exact hn
have hbez : exists xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn)
specialize coprime_balanced_bezout m
specialize coprime_balanced_bezout n
apply coprime_balanced_bezout
exact hcop
cases hms
cases hns
cases hbez
cases hbez_witness
cases hbez_witness_witness
cases hbez_witness_witness_witness
have hbl : exists u v. n * x3 + m * u = (1 + n * x5) + m * v
specialize bezout_mod_left m
specialize bezout_mod_left n
specialize bezout_mod_left x2
specialize bezout_mod_left x3
specialize bezout_mod_left x4
specialize bezout_mod_left x5
apply bezout_mod_left
exact hbez_witness_witness_witness_witness
have hbr : exists u v. m * x2 + n * u = (1 + m * x4) + n * v
specialize bezout_mod_right m
specialize bezout_mod_right n
specialize bezout_mod_right x2
specialize bezout_mod_right x3
specialize bezout_mod_right x4
specialize bezout_mod_right x5
apply bezout_mod_right
exact hbez_witness_witness_witness_witness
have hal0 : exists u v. (a * (n * x3)) + m * u = (a * (1 + n * x5)) + m * v
specialize mod_eq_mul_left m
specialize mod_eq_mul_left (n * x3)
specialize mod_eq_mul_left (1 + n * x5)
specialize mod_eq_mul_left a
apply mod_eq_mul_left
exact hbl
have hal : exists u v. (a * (n * x3)) + m * u = (a + a * (n * x5)) + m * v
have haexpand : a * (1 + n * x5) = a + a * (n * x5)
trans a * 1 + a * (n * x5)
apply mul_add
congr
apply mul_one
refl
rewrite <- haexpand
exact hal0
have hbm : exists u v. (b * (m * x2)) + m * u = 0 + m * v
apply dvd_to_mod_zero
exists b * x2
trans (b * m) * x2
symm
apply mul_assoc
trans (m * b) * x2
congr
apply mul_comm
refl
apply mul_assoc
have hym : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = ((a + a * (n * x5)) + 0) + m * v
specialize mod_eq_add m
specialize mod_eq_add (a * (n * x3))
specialize mod_eq_add (a + a * (n * x5))
specialize mod_eq_add (b * (m * x2))
specialize mod_eq_add 0
apply mod_eq_add
exact hal
exact hbm
have hym_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = (a + a * (n * x5)) + m * v
have hym_zero : (a + a * (n * x5)) + 0 = a + a * (n * x5)
rewrite PA3
refl
rewrite <- hym_zero
exact hym
have hkm : exists u v. (x * (a * (n * x5))) + m * u = (x * (a * (n * x5))) + m * v
specialize mod_eq_refl m
specialize mod_eq_refl (x * (a * (n * x5)))
apply mod_eq_refl
have hymk : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = ((a + a * (n * x5)) + (x * (a * (n * x5)))) + m * v
specialize mod_eq_add m
specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))
specialize mod_eq_add (a + a * (n * x5))
specialize mod_eq_add (x * (a * (n * x5)))
specialize mod_eq_add (x * (a * (n * x5)))
apply mod_eq_add
exact hym_norm
exact hkm
have hcancelm : exists u v. ((a + a * (n * x5)) + x * (a * (n * x5))) + S x * u = a + S x * v
specialize mod_eq_predecessor_cancel x
specialize mod_eq_predecessor_cancel a
specialize mod_eq_predecessor_cancel (a * (n * x5))
apply mod_eq_predecessor_cancel
rewrite <- hms_witness at hcancelm
rewrite <- hms_witness at hcancelm
have hbasem : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = a + m * v
specialize mod_eq_trans m
specialize mod_eq_trans (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))
specialize mod_eq_trans ((a + a * (n * x5)) + (x * (a * (n * x5))))
specialize mod_eq_trans a
apply mod_eq_trans
exact hymk
exact hcancelm
have hknznm : exists u v. (x1 * (b * (m * x4))) + m * u = 0 + m * v
apply dvd_to_mod_zero
exists x1 * (b * x4)
trans x1 * ((b * m) * x4)
congr
refl
symm
apply mul_assoc
trans x1 * ((m * b) * x4)
congr
refl
congr
apply mul_comm
refl
trans x1 * (m * (b * x4))
congr
refl
apply mul_assoc
trans (x1 * m) * (b * x4)
symm
apply mul_assoc
trans (m * x1) * (b * x4)
congr
apply mul_comm
refl
apply mul_assoc
have hfinalm0 : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = (a + 0) + m * v
specialize mod_eq_add m
specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))
specialize mod_eq_add a
specialize mod_eq_add (x1 * (b * (m * x4)))
specialize mod_eq_add 0
apply mod_eq_add
exact hbasem
exact hknznm
have hazerom : exists u v. (a + 0) + m * u = a + m * v
exists 0
exists 0
simp
have hfinalm : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = a + m * v
specialize mod_eq_trans m
specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))
specialize mod_eq_trans (a + 0)
specialize mod_eq_trans a
apply mod_eq_trans
exact hfinalm0
exact hazerom
have hbn0 : exists u v. (b * (m * x2)) + n * u = (b * (1 + m * x4)) + n * v
specialize mod_eq_mul_left n
specialize mod_eq_mul_left (m * x2)
specialize mod_eq_mul_left (1 + m * x4)
specialize mod_eq_mul_left b
apply mod_eq_mul_left
exact hbr
have hbn : exists u v. (b * (m * x2)) + n * u = (b + b * (m * x4)) + n * v
have hbexpand : b * (1 + m * x4) = b + b * (m * x4)
trans b * 1 + b * (m * x4)
apply mul_add
congr
apply mul_one
refl
rewrite <- hbexpand
exact hbn0
have han : exists u v. (a * (n * x3)) + n * u = 0 + n * v
apply dvd_to_mod_zero
exists a * x3
trans (a * n) * x3
symm
apply mul_assoc
trans (n * a) * x3
congr
apply mul_comm
refl
apply mul_assoc
have hyn0 : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (0 + (b + b * (m * x4))) + n * v
specialize mod_eq_add n
specialize mod_eq_add (a * (n * x3))
specialize mod_eq_add 0
specialize mod_eq_add (b * (m * x2))
specialize mod_eq_add (b + b * (m * x4))
apply mod_eq_add
exact han
exact hbn
have hyn_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (b + b * (m * x4)) + n * v
have hyn_zero : 0 + (b + b * (m * x4)) = b + b * (m * x4)
specialize zero_add (b + b * (m * x4))
exact zero_add
rewrite <- hyn_zero
exact hyn0
have hkmz : exists u v. (x * (a * (n * x5))) + n * u = 0 + n * v
apply dvd_to_mod_zero
exists x * (a * x5)
trans x * ((a * n) * x5)
congr
refl
symm
apply mul_assoc
trans x * ((n * a) * x5)
congr
refl
congr
apply mul_comm
refl
trans x * (n * (a * x5))
congr
refl
apply mul_assoc
trans (x * n) * (a * x5)
symm
apply mul_assoc
trans (n * x) * (a * x5)
congr
apply mul_comm
refl
apply mul_assoc
have hyn1 : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = ((b + b * (m * x4)) + 0) + n * v
specialize mod_eq_add n
specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))
specialize mod_eq_add (b + b * (m * x4))
specialize mod_eq_add (x * (a * (n * x5)))
specialize mod_eq_add 0
apply mod_eq_add
exact hyn_norm
exact hkmz
have hyn1_norm : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = (b + b * (m * x4)) + n * v
have hyn1_zero : (b + b * (m * x4)) + 0 = b + b * (m * x4)
rewrite PA3
refl
rewrite <- hyn1_zero
exact hyn1
have hkn : exists u v. (x1 * (b * (m * x4))) + n * u = (x1 * (b * (m * x4))) + n * v
specialize mod_eq_refl n
specialize mod_eq_refl (x1 * (b * (m * x4)))
apply mod_eq_refl
have hynk : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = ((b + b * (m * x4)) + (x1 * (b * (m * x4)))) + n * v
specialize mod_eq_add n
specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))
specialize mod_eq_add (b + b * (m * x4))
specialize mod_eq_add (x1 * (b * (m * x4)))
specialize mod_eq_add (x1 * (b * (m * x4)))
apply mod_eq_add
exact hyn1_norm
exact hkn
have hcanceln : exists u v. ((b + b * (m * x4)) + x1 * (b * (m * x4))) + S x1 * u = b + S x1 * v
specialize mod_eq_predecessor_cancel x1
specialize mod_eq_predecessor_cancel b
specialize mod_eq_predecessor_cancel (b * (m * x4))
apply mod_eq_predecessor_cancel
rewrite <- hns_witness at hcanceln
rewrite <- hns_witness at hcanceln
have hfinaln : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = b + n * v
specialize mod_eq_trans n
specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))
specialize mod_eq_trans ((b + b * (m * x4)) + (x1 * (b * (m * x4))))
specialize mod_eq_trans b
apply mod_eq_trans
exact hynk
exact hcanceln
exists (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))
split
exact hfinalm
exact hfinaln
qed

Occurrences

5,044

Distinct objects

1,929

Reused references

109

Depth

51

Cuts

144

Certificate SHA-256

fd5384ba933f…

binary_crt_remaindersBinary CRT with directed remainder witnesses Congruence & CRT checked

Binary CRT with bounded residues exposed as directed quotient/remainder equations.

Prerequisites

binary_crt mod_eq_to_remainder_decomposition

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> (exists ha. ha + S a = m) -> (exists hb. hb + S b = n) -> exists x q r. (x = q * m + a /\ x = r * n + b)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> (exists ha. ha + S a = m) -> (exists hb. hb + S b = n) -> exists x q r. (x = q * m + a /\ x = r * n + b)
use binary_crt
use mod_eq_to_remainder_decomposition
intro m
intro n
intro a
intro b
intro hm
intro hn
intro hcop
intro ha
intro hb
have hcrt : exists x. (exists u v. x + m * u = a + m * v) /\ (exists r s. x + n * r = b + n * s)
specialize binary_crt m
specialize binary_crt n
specialize binary_crt a
specialize binary_crt b
apply binary_crt
exact hm
exact hn
exact hcop
cases hcrt
cases hcrt_witness
have hma : exists q. x = q * m + a
specialize mod_eq_to_remainder_decomposition m
specialize mod_eq_to_remainder_decomposition x
specialize mod_eq_to_remainder_decomposition a
apply mod_eq_to_remainder_decomposition
exact hm
exact ha
exact hcrt_witness_left
have hnb : exists q. x = q * n + b
specialize mod_eq_to_remainder_decomposition n
specialize mod_eq_to_remainder_decomposition x
specialize mod_eq_to_remainder_decomposition b
apply mod_eq_to_remainder_decomposition
exact hn
exact hb
exact hcrt_witness_right
cases hma
cases hnb
exists x
exists x1
exists x2
split
exact hma_witness
exact hnb_witness
qed

Occurrences

6,890

Distinct objects

2,442

Reused references

125

Depth

66

Cuts

196

Certificate SHA-256

f3432e4bdae4…

binary_crt_beta_pairA single Gödel-beta code realizes two bounded values β sequences, products & FTA checked

Pairwise coprime beta moduli admit a single code realizing two bounded beta values.

Prerequisites

beta_modulus_nonzero binary_crt beta_at_of_mod_eq_bound

Used directly by

binary_crt_beta_pair_of_gap_dvd

Exact expanded PA statement

Copy statement

forall c i j a b. (forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (exists ha. ha + S a = S ((S i) * c)) -> (exists hb. hb + S b = S ((S j) * c)) -> exists code. (((exists ha. ha + S a = S ((S i) * c)) /\ exists q. code = q * S ((S i) * c) + a) /\ ((exists hb. hb + S b = S ((S j) * c)) /\ exists r. code = r * S ((S j) * c) + b))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i j a b. (forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (exists ha. ha + S a = S ((S i) * c)) -> (exists hb. hb + S b = S ((S j) * c)) -> exists code. (((exists ha. ha + S a = S ((S i) * c)) /\ exists q. code = q * S ((S i) * c) + a) /\ ((exists hb. hb + S b = S ((S j) * c)) /\ exists r. code = r * S ((S j) * c) + b))
use beta_modulus_nonzero
use binary_crt
use beta_at_of_mod_eq_bound
intro c
intro i
intro j
intro a
intro b
intro hcop
intro ha
intro hb
have hmi : ~(S ((S i) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero i
exact beta_modulus_nonzero
have hmj : ~(S ((S j) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero j
exact beta_modulus_nonzero
have hcrt : exists code. (exists u v. code + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ (exists r s. code + S ((S j) * c) * r = b + S ((S j) * c) * s)
specialize binary_crt (S ((S i) * c))
specialize binary_crt (S ((S j) * c))
specialize binary_crt a
specialize binary_crt b
apply binary_crt
exact hmi
exact hmj
exact hcop
cases hcrt
cases hcrt_witness
have hati : ((exists h. h + S a = S ((S i) * c)) /\ exists q. x = q * S ((S i) * c) + a)
specialize beta_at_of_mod_eq_bound x
specialize beta_at_of_mod_eq_bound c
specialize beta_at_of_mod_eq_bound i
specialize beta_at_of_mod_eq_bound a
apply beta_at_of_mod_eq_bound
exact ha
exact hcrt_witness_left
have hatj : ((exists h. h + S b = S ((S j) * c)) /\ exists q. x = q * S ((S j) * c) + b)
specialize beta_at_of_mod_eq_bound x
specialize beta_at_of_mod_eq_bound c
specialize beta_at_of_mod_eq_bound j
specialize beta_at_of_mod_eq_bound b
apply beta_at_of_mod_eq_bound
exact hb
exact hcrt_witness_right
exists x
split
exact hati
exact hatj
qed

Occurrences

6,941

Distinct objects

2,484

Reused references

126

Depth

69

Cuts

201

Certificate SHA-256

d0c6658e1cbb…

beta_modulus_coprime_baseA Gödel-beta modulus is coprime to its base β sequences, products & FTA checked

Every beta-shaped successor modulus is coprime to its base c.

Prerequisites

divides_remainder divisor_one mul_comm

Used directly by

beta_moduli_coprime_of_gap_dvd

Exact expanded PA statement

Copy statement

forall c k d. (exists u. S (k * c) = d * u) -> (exists v. c = d * v) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c k d. (exists u. S (k * c) = d * u) -> (exists v. c = d * v) -> d = 1
use divides_remainder
use divisor_one
use mul_comm
intro c
intro k
intro d
intro hm
intro hc
have hstep : S (k * c) = c * k + 1
simp [mul_comm]
have h1 : exists w. 1 = d * w
specialize divides_remainder d
specialize divides_remainder (S (k * c))
specialize divides_remainder c
specialize divides_remainder k
specialize divides_remainder 1
apply divides_remainder
exact hm
exact hc
exact hstep
specialize divisor_one d
apply divisor_one
exact h1
qed

Occurrences

874

Distinct objects

630

Reused references

54

Depth

30

Cuts

24

Certificate SHA-256

f7e3c800ecbe…

common_divisor_beta_moduli_divides_gap_times_cA common beta-modulus divisor divides the scaled index gap β sequences, products & FTA checked

A common divisor of two ordered beta moduli divides the index gap times c.

Prerequisites

divides_remainder add_succ_left add_mul zero_add

Used directly by

beta_moduli_coprime_of_gap_dvd

Exact expanded PA statement

Copy statement

forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * w
use divides_remainder
use add_succ_left
use add_mul
use zero_add
intro c
intro i
intro j
intro gap
intro d
intro hij
intro hmi
intro hmj
have hstep : S ((S j) * c) = S ((S i) * c) * 1 + gap * c
rewrite hij
specialize add_succ_left i
specialize add_succ_left gap
rewrite <- add_succ_left
simp [add_mul, zero_add]
symm
specialize add_succ_left_before (S i * c)
specialize add_succ_left_before (gap * c)
exact add_succ_left_before
specialize divides_remainder d
specialize divides_remainder (S ((S j) * c))
specialize divides_remainder (S ((S i) * c))
specialize divides_remainder 1
specialize divides_remainder (gap * c)
apply divides_remainder
exact hmj
exact hmi
exact hstep
qed

Occurrences

855

Distinct objects

540

Reused references

45

Depth

30

Cuts

24

Certificate SHA-256

42a60bc7d8fe…

beta_moduli_coprime_of_gap_dvdDivisible index gaps give coprime Gödel-beta moduli β sequences, products & FTA checked

Beta moduli at an additive index gap dividing c are coprime.

Prerequisites

beta_modulus_coprime_base common_divisor_beta_moduli_divides_gap_times_c multiple_trans multiple_refl gauss_coprime_cancel mul_comm

Used directly by

binary_crt_beta_pair_of_gap_dvd beta_moduli_coprime_of_lt_bounded_common_multiple

Exact expanded PA statement

Copy statement

forall c i j gap. j = i + gap -> (exists k. c = gap * k) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i j gap. j = i + gap -> (exists k. c = gap * k) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
use beta_modulus_coprime_base
use common_divisor_beta_moduli_divides_gap_times_c
use multiple_trans
use multiple_refl
use gauss_coprime_cancel
use mul_comm
intro c
intro i
intro j
intro gap
intro hij
intro hgapc
intro d
intro hmi
intro hmj
have hcopdc : forall e. (exists u. d = e * u) -> (exists v. c = e * v) -> e = 1
intro e
intro hed
intro hec
have hmei : exists u. S ((S i) * c) = e * u
specialize multiple_trans d
specialize multiple_trans e
specialize multiple_trans (S ((S i) * c))
apply multiple_trans
exact hmi
exact hed
specialize beta_modulus_coprime_base c
specialize beta_modulus_coprime_base (S i)
specialize beta_modulus_coprime_base e
apply beta_modulus_coprime_base
exact hmei
exact hec
have hgapprod : exists w. gap * c = d * w
specialize common_divisor_beta_moduli_divides_gap_times_c c
specialize common_divisor_beta_moduli_divides_gap_times_c i
specialize common_divisor_beta_moduli_divides_gap_times_c j
specialize common_divisor_beta_moduli_divides_gap_times_c gap
specialize common_divisor_beta_moduli_divides_gap_times_c d
apply common_divisor_beta_moduli_divides_gap_times_c
exact hij
exact hmi
exact hmj
cases hgapprod
have hdivgap : exists w. gap = d * w
specialize gauss_coprime_cancel d
specialize gauss_coprime_cancel c
specialize gauss_coprime_cancel gap
apply gauss_coprime_cancel
exact hcopdc
exists x
trans gap * c
apply mul_comm
exact hgapprod_witness
have hdc : exists w. c = d * w
specialize multiple_trans gap
specialize multiple_trans d
specialize multiple_trans c
apply multiple_trans
exact hgapc
exact hdivgap
specialize hcopdc d
apply hcopdc
specialize multiple_refl d
exact multiple_refl
exact hdc
qed

Occurrences

6,007

Distinct objects

1,803

Reused references

119

Depth

56

Cuts

175

Certificate SHA-256

e31f87a1af34…

binary_crt_beta_pair_of_gap_dvdA divisible index gap supplies a two-position Gödel-beta code β sequences, products & FTA checked

Discharge the beta-pair CRT coprimality premise from a divisible index gap.

Prerequisites

beta_moduli_coprime_of_gap_dvd binary_crt_beta_pair

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall c i j gap a b. j = i + gap -> (exists k. c = gap * k) -> (exists ha. ha + S a = S ((S i) * c)) -> (exists hb. hb + S b = S ((S j) * c)) -> exists code. (((exists ha. ha + S a = S ((S i) * c)) /\ exists q. code = q * S ((S i) * c) + a) /\ ((exists hb. hb + S b = S ((S j) * c)) /\ exists r. code = r * S ((S j) * c) + b))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i j gap a b. j = i + gap -> (exists k. c = gap * k) -> (exists ha. ha + S a = S ((S i) * c)) -> (exists hb. hb + S b = S ((S j) * c)) -> exists code. (((exists ha. ha + S a = S ((S i) * c)) /\ exists q. code = q * S ((S i) * c) + a) /\ ((exists hb. hb + S b = S ((S j) * c)) /\ exists r. code = r * S ((S j) * c) + b))
use beta_moduli_coprime_of_gap_dvd
use binary_crt_beta_pair
intro c
intro i
intro j
intro gap
intro a
intro b
intro hij
intro hgapc
intro ha
intro hb
have hcop : forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize beta_moduli_coprime_of_gap_dvd c
specialize beta_moduli_coprime_of_gap_dvd i
specialize beta_moduli_coprime_of_gap_dvd j
specialize beta_moduli_coprime_of_gap_dvd gap
apply beta_moduli_coprime_of_gap_dvd
exact hij
exact hgapc
specialize binary_crt_beta_pair c
specialize binary_crt_beta_pair i
specialize binary_crt_beta_pair j
specialize binary_crt_beta_pair a
specialize binary_crt_beta_pair b
apply binary_crt_beta_pair
exact hcop
exact ha
exact hb
qed

Occurrences

12,980

Distinct objects

3,078

Reused references

165

Depth

71

Cuts

378

Certificate SHA-256

79ec20e402f2…

bounded_common_multiple_stepA bounded common multiple extends by one Divisibility checked

Extend a nonzero common multiple through the next positive natural.

Prerequisites

mul_eq_zero succ_ne_zero zero_or_succ multiple_mul_right mul_comm

Used directly by

bounded_common_multiple_exists

Exact expanded PA statement

Copy statement

forall B c. ~(c = 0) -> (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> exists c2. (~(c2 = 0) /\ forall t. (exists h. S t + S h = S (S B)) -> exists k. c2 = S t * k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B c. ~(c = 0) -> (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> exists c2. (~(c2 = 0) /\ forall t. (exists h. S t + S h = S (S B)) -> exists k. c2 = S t * k)
use mul_eq_zero
use succ_ne_zero
use zero_or_succ
use multiple_mul_right
use mul_comm
intro B
intro c
intro hc
intro hall
exists c * S B
split
have hSB : ~(S B = 0)
specialize succ_ne_zero B
exact succ_ne_zero
have hprod : ~(c * S B = 0)
intro hzero
have hz : c = 0 \/ S B = 0
specialize mul_eq_zero c
specialize mul_eq_zero (S B)
apply mul_eq_zero
exact hzero
cases hz
apply hc
exact hz_left
apply hSB
exact hz_right
exact hprod
intro t
intro ht
cases ht
specialize zero_or_succ x
cases zero_or_succ
rewrite zero_or_succ_left at ht_witness
have hteq : S t = S B
rewrite PA4 at ht_witness
rewrite PA3 at ht_witness
apply PA2
exact ht_witness
exists c
rewrite hteq
apply mul_comm
cases zero_or_succ_right
have hprev : exists h. S t + S h = S B
exists x1
rewrite zero_or_succ_right_witness at ht_witness
rewrite PA4 at ht_witness
apply PA2
exact ht_witness
have hdivc : exists k. c = S t * k
specialize hall t
apply hall
exact hprev
specialize multiple_mul_right (S t)
specialize multiple_mul_right c
specialize multiple_mul_right (S B)
apply multiple_mul_right
exact hdivc
qed

Occurrences

483

Distinct objects

417

Reused references

34

Depth

29

Cuts

15

Certificate SHA-256

aa455c44508f…

bounded_common_multiple_existsEvery finite initial interval has a nonzero common multiple Divisibility checked

Every finite initial interval has a nonzero common-multiple surrogate.

Prerequisites

bounded_common_multiple_step succ_ne_zero add_eq_zero_left

Used directly by

bounded_beta_moduli_pairwise_coprime_exists prime_unbounded beta_prefix_extend

Exact expanded PA statement

Copy statement

forall B. exists c. (~(c = 0) /\ forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B. exists c. (~(c = 0) /\ forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k)
use bounded_common_multiple_step
use succ_ne_zero
use add_eq_zero_left
intro B
induction B
exists 1
split
specialize succ_ne_zero 0
exact succ_ne_zero
intro t
intro ht
cases ht
exfalso
have hz : S t + x = 0
rewrite PA4 at ht_witness
apply PA2
exact ht_witness
have hst0 : S t = 0
specialize add_eq_zero_left (S t)
specialize add_eq_zero_left x
apply add_eq_zero_left
exact hz
specialize succ_ne_zero t
apply succ_ne_zero
exact hst0
cases IH
cases IH_witness
specialize bounded_common_multiple_step B
specialize bounded_common_multiple_step x
apply bounded_common_multiple_step
exact IH_witness_left
exact IH_witness_right
qed

Occurrences

640

Distinct objects

481

Reused references

37

Depth

30

Cuts

22

Certificate SHA-256

ccb71e803625…

beta_moduli_coprime_of_lt_bounded_common_multipleOrdered bounded Gödel-beta moduli are coprime β sequences, products & FTA checked

Ordered bounded indices have coprime beta moduli when c is a common multiple of the bounded positive gaps.

Prerequisites

beta_moduli_coprime_of_gap_dvd add_comm le_trans

Used directly by

beta_moduli_pairwise_coprime_bounded bounded_beta_crt_prefix_invariant

Exact expanded PA statement

Copy statement

forall B c i j. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> (exists g. g + S i = j) -> (exists h. h + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B c i j. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> (exists g. g + S i = j) -> (exists h. h + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
use beta_moduli_coprime_of_gap_dvd
use add_comm
use le_trans
intro B
intro c
intro i
intro j
intro hcm
intro hlt
intro hjB
intro d
intro hdi
intro hdj
cases hlt
have hij : j = i + S x
symm
trans x + S i
simp [add_comm]
exact hlt_witness
have hgaple : exists r. r + S x = B
specialize le_trans (S x)
specialize le_trans j
specialize le_trans B
apply le_trans
exists i
symm
exact hij
exact hjB
cases hgaple
have hgapbound : exists h. S x + S h = S B
exists x1
rewrite PA4
congr
trans x1 + S x
apply add_comm
exact hgaple_witness
have hgapdvd : exists k. c = S x * k
specialize hcm x
apply hcm
exact hgapbound
have hcop : forall e. (exists u. S ((S i) * c) = e * u) -> (exists v. S ((S j) * c) = e * v) -> e = 1
specialize beta_moduli_coprime_of_gap_dvd c
specialize beta_moduli_coprime_of_gap_dvd i
specialize beta_moduli_coprime_of_gap_dvd j
specialize beta_moduli_coprime_of_gap_dvd (S x)
apply beta_moduli_coprime_of_gap_dvd
exact hij
exact hgapdvd
specialize hcop d
apply hcop
exact hdi
exact hdj
qed

Occurrences

6,227

Distinct objects

1,913

Reused references

125

Depth

57

Cuts

181

Certificate SHA-256

9d97203f2164…

beta_moduli_pairwise_coprime_boundedA bounded Gödel-beta prefix has pairwise coprime moduli β sequences, products & FTA checked

Distinct indices in a bounded prefix have pairwise coprime beta moduli under a bounded common-multiple invariant.

Prerequisites

lt_trichotomy beta_moduli_coprime_of_lt_bounded_common_multiple

Used directly by

bounded_beta_moduli_pairwise_coprime_exists beta_accumulated_product_step beta_exclusive_accumulated_product_step

Exact expanded PA statement

Copy statement

forall B c. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> forall i j. ~(i = j) -> (exists hi. hi + i = B) -> (exists hj. hj + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B c. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> forall i j. ~(i = j) -> (exists hi. hi + i = B) -> (exists hj. hj + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
use lt_trichotomy
use beta_moduli_coprime_of_lt_bounded_common_multiple
intro B
intro c
intro hcm
intro i
intro j
intro hne
intro hiB
intro hjB
intro d
intro hdi
intro hdj
specialize lt_trichotomy i
specialize lt_trichotomy j
cases lt_trichotomy
exfalso
apply hne
exact lt_trichotomy_left
cases lt_trichotomy_right
have hcopij : forall e. (exists u. S ((S i) * c) = e * u) -> (exists v. S ((S j) * c) = e * v) -> e = 1
specialize beta_moduli_coprime_of_lt_bounded_common_multiple B
specialize beta_moduli_coprime_of_lt_bounded_common_multiple c
specialize beta_moduli_coprime_of_lt_bounded_common_multiple i
specialize beta_moduli_coprime_of_lt_bounded_common_multiple j
apply beta_moduli_coprime_of_lt_bounded_common_multiple
exact hcm
exact lt_trichotomy_right_left
exact hjB
specialize hcopij d
apply hcopij
exact hdi
exact hdj
have hcopji : forall e. (exists u. S ((S j) * c) = e * u) -> (exists v. S ((S i) * c) = e * v) -> e = 1
specialize beta_moduli_coprime_of_lt_bounded_common_multiple B
specialize beta_moduli_coprime_of_lt_bounded_common_multiple c
specialize beta_moduli_coprime_of_lt_bounded_common_multiple j
specialize beta_moduli_coprime_of_lt_bounded_common_multiple i
apply beta_moduli_coprime_of_lt_bounded_common_multiple
exact hcm
exact lt_trichotomy_right_right
exact hiB
specialize hcopji d
apply hcopji
exact hdj
exact hdi
qed

Occurrences

6,348

Distinct objects

2,034

Reused references

125

Depth

59

Cuts

183

Certificate SHA-256

67f94b655541…

bounded_beta_moduli_pairwise_coprime_existsEvery finite Gödel-beta prefix admits a pairwise-coprime base β sequences, products & FTA checked

Every finite beta-modulus prefix admits a nonzero base making its distinct positions pairwise coprime.

Prerequisites

bounded_common_multiple_exists beta_moduli_pairwise_coprime_bounded

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall B. exists c. (~(c = 0) /\ forall i j. ~(i = j) -> (exists hi. hi + i = B) -> (exists hj. hj + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B. exists c. (~(c = 0) /\ forall i j. ~(i = j) -> (exists hi. hi + i = B) -> (exists hj. hj + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)
use bounded_common_multiple_exists
use beta_moduli_pairwise_coprime_bounded
intro B
specialize bounded_common_multiple_exists B
cases bounded_common_multiple_exists
cases bounded_common_multiple_exists_witness
exists x
split
exact bounded_common_multiple_exists_witness_left
specialize beta_moduli_pairwise_coprime_bounded B
specialize beta_moduli_pairwise_coprime_bounded x
apply beta_moduli_pairwise_coprime_bounded
exact bounded_common_multiple_exists_witness_right
qed

Occurrences

7,019

Distinct objects

2,247

Reused references

132

Depth

61

Cuts

207

Certificate SHA-256

6d8ad65d4fb5…

coprime_mul_leftCoprimality is closed under multiplication on the left GCD, Bézout & coprimality checked

Coprimality with a fixed right operand is closed under multiplication on the left.

Prerequisites

multiple_trans gauss_coprime_cancel

Used directly by

coprime_mul_right beta_accumulated_product_step beta_exclusive_accumulated_product_step

Exact expanded PA statement

Copy statement

forall a b n. (forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1) -> (forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1) -> forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b n. (forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1) -> (forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1) -> forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1
use multiple_trans
use gauss_coprime_cancel
intro a
intro b
intro n
intro han
intro hbn
intro d
intro hab
intro hdn
have hda : forall e. (exists u. d = e * u) -> (exists v. a = e * v) -> e = 1
intro e
intro hed
intro hea
have hen : exists q. n = e * q
specialize multiple_trans d
specialize multiple_trans e
specialize multiple_trans n
apply multiple_trans
exact hdn
exact hed
specialize han e
apply han
exact hea
exact hen
have hdb : exists w. b = d * w
specialize gauss_coprime_cancel d
specialize gauss_coprime_cancel a
specialize gauss_coprime_cancel b
apply gauss_coprime_cancel
exact hda
exact hab
specialize hbn d
apply hbn
exact hdb
exact hdn
qed

Occurrences

3,975

Distinct objects

1,558

Reused references

94

Depth

53

Cuts

115

Certificate SHA-256

51b298808927…

coprime_mul_rightCoprimality is closed under multiplication on the right GCD, Bézout & coprimality checked

Coprimality with a fixed left operand is closed under multiplication on the right.

Prerequisites

coprime_mul_left coprime_symm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n a b. (forall d. (exists x. n = d * x) -> (exists y. a = d * y) -> d = 1) -> (forall d. (exists x. n = d * x) -> (exists y. b = d * y) -> d = 1) -> forall d. (exists x. n = d * x) -> (exists y. a * b = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n a b. (forall d. (exists x. n = d * x) -> (exists y. a = d * y) -> d = 1) -> (forall d. (exists x. n = d * x) -> (exists y. b = d * y) -> d = 1) -> forall d. (exists x. n = d * x) -> (exists y. a * b = d * y) -> d = 1
use coprime_mul_left
use coprime_symm
intro n
intro a
intro b
intro hna
intro hnb
have han : forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1
specialize coprime_symm n
specialize coprime_symm a
apply coprime_symm
exact hna
have hbn : forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1
specialize coprime_symm n
specialize coprime_symm b
apply coprime_symm
exact hnb
have habn : forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1
specialize coprime_mul_left a
specialize coprime_mul_left b
specialize coprime_mul_left n
apply coprime_mul_left
exact han
exact hbn
specialize coprime_symm (a * b)
specialize coprime_symm n
apply coprime_symm
exact habn
qed

Occurrences

4,017

Distinct objects

1,600

Reused references

94

Depth

54

Cuts

117

Certificate SHA-256

8a3a38d99c52…

mod_eq_of_mod_eq_multipleCongruence descends along modulus divisibility Congruence & CRT checked

Balanced congruence descends from a multiple modulus to every divisor modulus.

Prerequisites

mul_assoc

Used directly by

binary_crt_fold_step crt_common_solution_implies_gcd_compatible mod_eq_lcm_iff_pair

Exact expanded PA statement

Copy statement

forall m P x a. (exists k. P = m * k) -> (exists u v. x + P * u = a + P * v) -> exists r s. x + m * r = a + m * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m P x a. (exists k. P = m * k) -> (exists u v. x + P * u = a + P * v) -> exists r s. x + m * r = a + m * s
use mul_assoc
intro m
intro P
intro x
intro a
intro hdiv
intro hmod
cases hdiv
cases hmod
cases hmod_witness
rewrite hdiv_witness at hmod_witness_witness
rewrite hdiv_witness at hmod_witness_witness
exists x1 * x2
exists x1 * x3
trans x + (m * x1) * x2
congr
refl
symm
apply mul_assoc
trans a + (m * x1) * x3
exact hmod_witness_witness
congr
refl
apply mul_assoc
qed

Occurrences

157

Distinct objects

145

Reused references

12

Depth

23

Cuts

3

Certificate SHA-256

f3d604756157…

binary_crt_fold_stepOne CRT fold step preserves all accumulated congruences Congruence & CRT checked

One binary CRT extension preserves every old congruence whose modulus divides the accumulated product.

Prerequisites

binary_crt mod_eq_of_mod_eq_multiple mod_eq_trans

Used directly by

beta_crt_prefix_congruence_step beta_exclusive_recode_congruence_step beta_prefix_extend

Exact expanded PA statement

Copy statement

forall P n x b. ~(P = 0) -> ~(n = 0) -> (forall d. (exists u. P = d * u) -> (exists v. n = d * v) -> d = 1) -> exists z. ((forall m a. (exists k. P = m * k) -> (exists u v. x + m * u = a + m * v) -> exists r s. z + m * r = a + m * s) /\ exists q r. z + n * q = b + n * r)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall P n x b. ~(P = 0) -> ~(n = 0) -> (forall d. (exists u. P = d * u) -> (exists v. n = d * v) -> d = 1) -> exists z. ((forall m a. (exists k. P = m * k) -> (exists u v. x + m * u = a + m * v) -> exists r s. z + m * r = a + m * s) /\ exists q r. z + n * q = b + n * r)
use binary_crt
use mod_eq_of_mod_eq_multiple
use mod_eq_trans
intro P
intro n
intro x
intro b
intro hP
intro hn
intro hcop
have hcrt : exists z. (exists u v. z + P * u = x + P * v) /\ (exists q r. z + n * q = b + n * r)
specialize binary_crt P
specialize binary_crt n
specialize binary_crt x
specialize binary_crt b
apply binary_crt
exact hP
exact hn
exact hcop
cases hcrt
cases hcrt_witness
exists x1
split
intro m
intro a
intro hmP
intro hxa
have hzx : exists u v. x1 + m * u = x + m * v
specialize mod_eq_of_mod_eq_multiple m
specialize mod_eq_of_mod_eq_multiple P
specialize mod_eq_of_mod_eq_multiple x1
specialize mod_eq_of_mod_eq_multiple x
apply mod_eq_of_mod_eq_multiple
exact hmP
exact hcrt_witness_left
specialize mod_eq_trans m
specialize mod_eq_trans x1
specialize mod_eq_trans x
specialize mod_eq_trans a
apply mod_eq_trans
exact hzx
exact hxa
exact hcrt_witness_right
qed

Occurrences

5,501

Distinct objects

2,018

Reused references

111

Depth

52

Cuts

156

Certificate SHA-256

f65c05b0db8a…

right_factor_divides_productThe right factor divides a product Divisibility checked

The right factor divides a product.

Prerequisites

mul_comm

Used directly by

beta_accumulated_product_step beta_exclusive_accumulated_product_step

Exact expanded PA statement

Copy statement

forall a b. exists k. a * b = b * k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists k. a * b = b * k
use mul_comm
intro a
intro b
exists a
apply mul_comm
qed

Occurrences

229

Distinct objects

207

Reused references

22

Depth

25

Cuts

7

Certificate SHA-256

91ff76024f69…

beta_accumulated_product_stepExtend the accumulated beta-modulus product invariant β sequences, products & FTA checked

Extend the accumulated beta-modulus product invariant by one bounded position.

Prerequisites

mul_ne_zero right_factor_divides_product beta_modulus_nonzero le_eq_or_lt le_of_succ_le_succ multiple_mul_right le_succ_self lt_of_le_of_lt lt_irrefl_expanded beta_moduli_pairwise_coprime_bounded coprime_mul_left

Used directly by

beta_crt_prefix_invariant_step

Exact expanded PA statement

Copy statement

forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. P * S ((S (S k)) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. P * S ((S (S k)) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))
use mul_ne_zero
use right_factor_divides_product
use beta_modulus_nonzero
use le_eq_or_lt
use le_of_succ_le_succ
use multiple_mul_right
use le_succ_self
use lt_of_le_of_lt
use lt_irrefl_expanded
use beta_moduli_pairwise_coprime_bounded
use coprime_mul_left
intro N
intro c
intro k
intro P
intro hcm
intro hkN
intro hP
intro hdiv
intro hfuture
have hnew : ~(S ((S (S k)) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero (S k)
exact beta_modulus_nonzero
split
specialize mul_ne_zero P
specialize mul_ne_zero (S ((S (S k)) * c))
intro hzero
apply mul_ne_zero
exact hP
exact hnew
exact hzero
split
intro i
intro hi
have hsplit : i = S k \/ exists r. r + S i = S k
specialize le_eq_or_lt i
specialize le_eq_or_lt (S k)
apply le_eq_or_lt
exact hi
cases hsplit
rewrite hsplit_left
specialize right_factor_divides_product P
specialize right_factor_divides_product (S ((S (S k)) * c))
exact right_factor_divides_product
have hik : exists r. r + i = k
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ k
apply le_of_succ_le_succ
exact hsplit_right
have hiP : exists q. P = S ((S i) * c) * q
specialize hdiv i
apply hdiv
exact hik
specialize multiple_mul_right (S ((S i) * c))
specialize multiple_mul_right P
specialize multiple_mul_right (S ((S (S k)) * c))
apply multiple_mul_right
exact hiP
intro j
intro hSkj
intro hjN
have hkj : exists r. r + S k = j
have hkSk : exists r. r + k = S k
specialize le_succ_self k
exact le_succ_self
specialize lt_of_le_of_lt k
specialize lt_of_le_of_lt (S k)
specialize lt_of_le_of_lt j
apply lt_of_le_of_lt
exact hkSk
exact hSkj
have hPj : forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize hfuture j
apply hfuture
exact hkj
exact hjN
have hneq : ~(S k = j)
intro heq
rewrite <- heq at hSkj
specialize lt_irrefl_expanded (S k)
apply lt_irrefl_expanded
exact hSkj
have hpairs : forall i j. ~(i = j) -> (exists hi. hi + i = N) -> (exists hj. hj + j = N) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize beta_moduli_pairwise_coprime_bounded N
specialize beta_moduli_pairwise_coprime_bounded c
apply beta_moduli_pairwise_coprime_bounded
exact hcm
have hnewj : forall d. (exists u. S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize hpairs (S k)
specialize hpairs j
apply hpairs
exact hneq
exact hkN
exact hjN
specialize coprime_mul_left P
specialize coprime_mul_left (S ((S (S k)) * c))
specialize coprime_mul_left (S ((S j) * c))
apply coprime_mul_left
exact hPj
exact hnewj
qed

Occurrences

11,174

Distinct objects

2,348

Reused references

136

Depth

69

Cuts

330

Certificate SHA-256

6da6cc49d4d9…

beta_crt_prefix_congruence_stepExtend decoded-prefix congruence by one beta position β sequences, products & FTA checked

Extend balanced congruence to one more decoded beta position using the CRT fold step.

Prerequisites

beta_modulus_nonzero le_refl binary_crt_fold_step beta_at_exists beta_at_unique le_eq_or_lt le_of_succ_le_succ

Used directly by

beta_crt_prefix_invariant_step

Exact expanded PA statement

Copy statement

forall N c b k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v
use beta_modulus_nonzero
use le_refl
use binary_crt_fold_step
use beta_at_exists
use beta_at_unique
use le_eq_or_lt
use le_of_succ_le_succ
intro N
intro c
intro b
intro k
intro P
intro z
intro hkN
intro hP
intro hdiv
intro hcong
intro hfuture
have hnew : ~(S ((S (S k)) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero (S k)
exact beta_modulus_nonzero
have hstrict : exists g. g + S k = S k
specialize le_refl (S k)
exact le_refl
have hcop : forall d. (exists u. P = d * u) -> (exists v. S ((S (S k)) * c) = d * v) -> d = 1
specialize hfuture (S k)
apply hfuture
exact hstrict
exact hkN
have hvalue : exists a. ((exists h. h + S a = S ((S (S k)) * c)) /\ exists q. b = q * S ((S (S k)) * c) + a)
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists (S k)
exact beta_at_exists
cases hvalue
have hfold : exists z2. ((forall m a. (exists q. P = m * q) -> (exists u v. z + m * u = a + m * v) -> exists r s. z2 + m * r = a + m * s) /\ exists q r. z2 + S ((S (S k)) * c) * q = x + S ((S (S k)) * c) * r)
specialize binary_crt_fold_step P
specialize binary_crt_fold_step (S ((S (S k)) * c))
specialize binary_crt_fold_step z
specialize binary_crt_fold_step x
apply binary_crt_fold_step
exact hP
exact hnew
exact hcop
cases hfold
cases hfold_witness
exists x1
intro i
intro a
intro hi
intro hati
have hsplit : i = S k \/ exists r. r + S i = S k
specialize le_eq_or_lt i
specialize le_eq_or_lt (S k)
apply le_eq_or_lt
exact hi
cases hsplit
have hati_new : ((exists h. h + S a = S ((S (S k)) * c)) /\ exists q. b = q * S ((S (S k)) * c) + a)
rewrite <- hsplit_left
rewrite <- hsplit_left
exact hati
have haeq : a = x
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique (S k)
specialize beta_at_unique a
specialize beta_at_unique x
apply beta_at_unique
exact hati_new
exact hvalue_witness
rewrite hsplit_left
rewrite hsplit_left
rewrite haeq
exact hfold_witness_right
have hik : exists r. r + i = k
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ k
apply le_of_succ_le_succ
exact hsplit_right
have hmiP : exists q. P = S ((S i) * c) * q
specialize hdiv i
apply hdiv
exact hik
have hzold : exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v
specialize hcong i
specialize hcong a
apply hcong
exact hik
exact hati
specialize hfold_witness_left (S ((S i) * c))
specialize hfold_witness_left a
apply hfold_witness_left
exact hmiP
exact hzold
qed

Occurrences

7,352

Distinct objects

2,564

Reused references

129

Depth

64

Cuts

213

Certificate SHA-256

39d62088be96…

beta_crt_prefix_invariant_stepOne full bounded beta-prefix CRT invariant step β sequences, products & FTA checked

One concrete induction step for the bounded beta-prefix CRT invariant.

Prerequisites

beta_accumulated_product_step beta_crt_prefix_congruence_step

Used directly by

bounded_beta_crt_prefix_invariant

Exact expanded PA statement

Copy statement

forall N c b k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. P * S ((S (S k)) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. P * S ((S (S k)) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
use beta_accumulated_product_step
use beta_crt_prefix_congruence_step
intro N
intro c
intro b
intro k
intro P
intro z
intro hcm
intro hkN
intro hP
intro hdiv
intro hcong
intro hfuture
have hproduct : (~(P * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. P * S ((S (S k)) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))
specialize beta_accumulated_product_step N
specialize beta_accumulated_product_step c
specialize beta_accumulated_product_step k
specialize beta_accumulated_product_step P
apply beta_accumulated_product_step
exact hcm
exact hkN
exact hP
exact hdiv
exact hfuture
have hcodes : exists z2. forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v
specialize beta_crt_prefix_congruence_step N
specialize beta_crt_prefix_congruence_step c
specialize beta_crt_prefix_congruence_step b
specialize beta_crt_prefix_congruence_step k
specialize beta_crt_prefix_congruence_step P
specialize beta_crt_prefix_congruence_step z
apply beta_crt_prefix_congruence_step
exact hkN
exact hP
exact hdiv
exact hcong
exact hfuture
cases hcodes
cases hproduct
cases hproduct_right
exists x
split
exact hproduct_left
split
exact hproduct_right_left
split
exact hcodes_witness
exact hproduct_right_right
qed

Occurrences

18,613

Distinct objects

3,749

Reused references

186

Depth

70

Cuts

545

Certificate SHA-256

9cb74ed1e6aa…

bounded_beta_crt_prefix_invariantBounded beta-prefix CRT invariant β sequences, products & FTA checked

Concrete ordinary-induction fold of the beta CRT invariant through every bounded prefix.

Prerequisites

beta_modulus_nonzero le_zero multiple_refl beta_at_to_mod_eq beta_moduli_coprime_of_lt_bounded_common_multiple le_succ_self le_trans beta_crt_prefix_invariant_step

Used directly by

bounded_beta_crt_for_existing_code

Exact expanded PA statement

Copy statement

forall N c b. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
use beta_modulus_nonzero
use le_zero
use multiple_refl
use beta_at_to_mod_eq
use beta_moduli_coprime_of_lt_bounded_common_multiple
use le_succ_self
use le_trans
use beta_crt_prefix_invariant_step
intro N
intro c
intro b
intro hcm
induction k
intro hkN
exists S ((S 0) * c)
exists b
split
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero 0
exact beta_modulus_nonzero
split
intro i
intro hi
have hi0 : i = 0
specialize le_zero i
apply le_zero
exact hi
specialize multiple_refl (S ((S 0) * c))
rewrite hi0
exact multiple_refl
split
intro i
intro a
intro hi
intro hati
specialize beta_at_to_mod_eq b
specialize beta_at_to_mod_eq c
specialize beta_at_to_mod_eq i
specialize beta_at_to_mod_eq a
apply beta_at_to_mod_eq
exact hati
intro j
intro h0j
intro hjN
specialize beta_moduli_coprime_of_lt_bounded_common_multiple N
specialize beta_moduli_coprime_of_lt_bounded_common_multiple c
specialize beta_moduli_coprime_of_lt_bounded_common_multiple 0
specialize beta_moduli_coprime_of_lt_bounded_common_multiple j
apply beta_moduli_coprime_of_lt_bounded_common_multiple
exact hcm
exact h0j
exact hjN
intro hkN
have hkprev : exists h. h + k = N
have hkstep : exists h. h + k = S k
specialize le_succ_self k
exact le_succ_self
specialize le_trans k
specialize le_trans (S k)
specialize le_trans N
apply le_trans
exact hkstep
exact hkN
have hprev : exists P z. (~(P = 0) /\ ((forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
apply IH
exact hkprev
cases hprev
cases hprev_witness
cases hprev_witness_witness
cases hprev_witness_witness_right
cases hprev_witness_witness_right_right
have hnext : exists z2. (~(x * S ((S (S k)) * c) = 0) /\ ((forall i. (exists h. h + i = S k) -> exists q. x * S ((S (S k)) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = S k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S (S k) = j) -> (exists h. h + j = N) -> forall d. (exists u. x * S ((S (S k)) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
specialize beta_crt_prefix_invariant_step N
specialize beta_crt_prefix_invariant_step c
specialize beta_crt_prefix_invariant_step b
specialize beta_crt_prefix_invariant_step k
specialize beta_crt_prefix_invariant_step x
specialize beta_crt_prefix_invariant_step x1
apply beta_crt_prefix_invariant_step
exact hcm
exact hkN
exact hprev_witness_witness_left
exact hprev_witness_witness_right_left
exact hprev_witness_witness_right_right_left
exact hprev_witness_witness_right_right_right
cases hnext
exists x * S ((S (S k)) * c)
exists x2
exact hnext_witness
qed

Occurrences

25,496

Distinct objects

3,918

Reused references

194

Depth

78

Cuts

752

Certificate SHA-256

ef961db40c76…

bounded_beta_crt_for_existing_codeBounded CRT witness for an existing beta code β sequences, products & FTA checked

Fold-produced bounded CRT witness for residues decoded from an already supplied beta code.

Prerequisites

bounded_beta_crt_prefix_invariant le_refl

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall N c b. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> exists z. forall i a. (exists h. h + i = N) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> exists z. forall i a. (exists h. h + i = N) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v
use bounded_beta_crt_prefix_invariant
use le_refl
intro N
intro c
intro b
intro hcm
have hfold : exists P z. (~(P = 0) /\ ((forall i. (exists h. h + i = N) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = N) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S N = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
have hall : forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + i = k) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
specialize bounded_beta_crt_prefix_invariant N
specialize bounded_beta_crt_prefix_invariant c
specialize bounded_beta_crt_prefix_invariant b
apply bounded_beta_crt_prefix_invariant
exact hcm
specialize hall N
apply hall
specialize le_refl N
exact le_refl
cases hfold
cases hfold_witness
cases hfold_witness_witness
cases hfold_witness_witness_right
cases hfold_witness_witness_right_right
exists x1
exact hfold_witness_witness_right_right_left
qed

Occurrences

25,545

Distinct objects

3,942

Reused references

195

Depth

79

Cuts

755

Certificate SHA-256

8d9242fe5e07…

prime_unboundedThere is a prime larger than every given natural Primes checked

Euclid's common-multiple argument constructs a prime above every natural.

Prerequisites

bounded_common_multiple_exists prime_divisor_exists prime_nonzero nonzero_is_succ add_succ_left add_comm divides_remainder divisor_one mul_one le_or_lt

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. exists p. (exists k. k + S n = p) /\ (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. exists p. (exists k. k + S n = p) /\ (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1)
use bounded_common_multiple_exists
use prime_divisor_exists
use prime_nonzero
use nonzero_is_succ
use add_succ_left
use add_comm
use divides_remainder
use divisor_one
use mul_one
use le_or_lt
intro n
specialize bounded_common_multiple_exists n
have hcommon : exists c. (~(c = 0) /\ forall t. (exists h. S t + S h = S n) -> exists k. c = S t * k)
exact bounded_common_multiple_exists
cases hcommon
cases hcommon_witness
have hsucc_not_one : ~(S x = 1)
intro hsucc_one
apply hcommon_witness_left
apply PA2
exact hsucc_one
have hprime_divisor : exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ exists k. S x = p * k)
specialize prime_divisor_exists (S x)
apply prime_divisor_exists
intro hsucc_zero
apply PA1
exact hsucc_zero
exact hsucc_not_one
cases hprime_divisor
cases hprime_divisor_witness
exists x1
split
have horder : (exists k. k + x1 = n) \/ exists k. k + S n = x1
specialize le_or_lt x1
specialize le_or_lt n
exact le_or_lt
cases horder
exfalso
have hp_nonzero : ~(x1 = 0)
intro hp_zero
specialize prime_nonzero x1
apply prime_nonzero
exact hprime_divisor_witness_left
exact hp_zero
have hp_succ : exists t. x1 = S t
specialize nonzero_is_succ x1
apply nonzero_is_succ
exact hp_nonzero
cases hp_succ
have hp_bounded : exists h. S x2 + S h = S n
cases horder_left
exists x3
rewrite hp_succ_witness at horder_left_witness
trans S (x2 + S x3)
apply add_succ_left
trans S (S (x2 + x3))
congr
apply PA4
trans S (S (x3 + x2))
congr
congr
apply add_comm
trans S (x3 + S x2)
congr
symm
apply PA4
congr
exact horder_left_witness
have hp_divides_common : exists k. x = x1 * k
specialize hcommon_witness_right x2
have hpred_divides : exists k. x = S x2 * k
apply hcommon_witness_right
exact hp_bounded
rewrite hp_succ_witness
exact hpred_divides
have hp_divides_one : exists w. 1 = x1 * w
specialize divides_remainder x1
specialize divides_remainder (S x)
specialize divides_remainder x
specialize divides_remainder 1
specialize divides_remainder 1
apply divides_remainder
exact hprime_divisor_witness_right
exact hp_divides_common
specialize mul_one x
rewrite mul_one
simp
cases hprime_divisor_witness_left
apply hprime_divisor_witness_left_left
specialize divisor_one x1
apply divisor_one
exact hp_divides_one
exact horder_right
exact hprime_divisor_witness_left
qed

Occurrences

4,595

Distinct objects

2,387

Reused references

126

Depth

82

Cuts

146

Certificate SHA-256

8a44fb2d207c…

beta_value_le_codeA decoded beta value is bounded by its code β sequences, products & FTA checked

Every decoded beta value is at most its code.

Prerequisites

none — a root theorem

Used directly by

beta_value_lt_scaled_base

Exact expanded PA statement

Copy statement

forall b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists h. h + x = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists h. h + x = b
intro b
intro c
intro i
intro x
intro hat
cases hat
cases hat_right
exists x1 * S ((S i) * c)
symm
exact hat_right_witness
qed

Occurrences

18

Distinct objects

18

Reused references

0

Depth

13

Cuts

0

Certificate SHA-256

f037744e9dd7…

base_le_beta_modulusThe beta base is bounded by every beta modulus β sequences, products & FTA checked

A beta base is at most every beta modulus over that base.

Prerequisites

le_add_left mul_succ_left le_succ

Used directly by

beta_value_lt_scaled_base new_value_lt_scaled_base

Exact expanded PA statement

Copy statement

forall c i. exists h. h + c = S ((S i) * c)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall c i. exists h. h + c = S ((S i) * c)
use le_add_left
use mul_succ_left
use le_succ
intro c
intro i
have hproduct : exists h. h + c = S i * c
specialize mul_succ_left i
specialize mul_succ_left c
rewrite mul_succ_left
specialize le_add_left c
specialize le_add_left (i * c)
exact le_add_left
specialize le_succ c
specialize le_succ (S i * c)
apply le_succ
exact hproduct
qed

Occurrences

233

Distinct objects

189

Reused references

18

Depth

24

Cuts

8

Certificate SHA-256

cdc7fe6d2ef2…

le_scaled_nonzeroNonzero scaling is monotone β sequences, products & FTA checked

Scaling by a nonzero natural does not decrease a natural.

Prerequisites

one_le_of_ne_zero mul_le_mul_right one_mul

Used directly by

beta_value_lt_scaled_base new_value_lt_scaled_base

Exact expanded PA statement

Copy statement

forall C B. ~(C = 0) -> exists h. h + B = C * B

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall C B. ~(C = 0) -> exists h. h + B = C * B
use one_le_of_ne_zero
use mul_le_mul_right
use one_mul
intro C
intro B
intro hC
have h1C : exists h. h + 1 = C
specialize one_le_of_ne_zero C
apply one_le_of_ne_zero
exact hC
have hscaled : exists h. h + 1 * B = C * B
specialize mul_le_mul_right 1
specialize mul_le_mul_right C
specialize mul_le_mul_right B
apply mul_le_mul_right
exact h1C
specialize one_mul B
rewrite one_mul at hscaled
exact hscaled
qed

Occurrences

407

Distinct objects

338

Reused references

37

Depth

28

Cuts

13

Certificate SHA-256

d68bec41b222…

scaled_bounded_common_multipleScaling preserves bounded common multiples β sequences, products & FTA checked

A right multiple of a bounded common multiple remains such a common multiple.

Prerequisites

multiple_mul_right

Used directly by

beta_prefix_extend

Exact expanded PA statement

Copy statement

forall N C B. (forall t. (exists h. S t + S h = S N) -> exists q. C = S t * q) -> forall t. (exists h. S t + S h = S N) -> exists q. C * B = S t * q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N C B. (forall t. (exists h. S t + S h = S N) -> exists q. C = S t * q) -> forall t. (exists h. S t + S h = S N) -> exists q. C * B = S t * q
use multiple_mul_right
intro N
intro C
intro B
intro hcm
intro t
intro ht
have htC : exists q. C = S t * q
specialize hcm t
apply hcm
exact ht
specialize multiple_mul_right (S t)
specialize multiple_mul_right C
specialize multiple_mul_right B
apply multiple_mul_right
exact htC
qed

Occurrences

147

Distinct objects

135

Reused references

12

Depth

19

Cuts

4

Certificate SHA-256

c4ce3a408de7…

beta_value_lt_scaled_baseOld decoded values fit the scaled target base β sequences, products & FTA checked

An old beta value fits every modulus after a constructive scaled-base rebase.

Prerequisites

beta_value_le_code le_add_right succ_le_succ le_trans le_scaled_nonzero base_le_beta_modulus

Used directly by

beta_prefix_extend

Exact expanded PA statement

Copy statement

forall b c i x C s j. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ~(C = 0) -> exists h. h + S x = S ((S j) * (C * S (b + s)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i x C s j. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ~(C = 0) -> exists h. h + S x = S ((S j) * (C * S (b + s)))
use beta_value_le_code
use le_add_right
use succ_le_succ
use le_trans
use le_scaled_nonzero
use base_le_beta_modulus
intro b
intro c
intro i
intro x
intro C
intro s
intro j
intro hat
intro hC
have hxb : exists h. h + x = b
specialize beta_value_le_code b
specialize beta_value_le_code c
specialize beta_value_le_code i
specialize beta_value_le_code x
apply beta_value_le_code
exact hat
have hbs : exists h. h + b = b + s
specialize le_add_right b
specialize le_add_right s
exact le_add_right
have hxs : exists h. h + x = b + s
specialize le_trans x
specialize le_trans b
specialize le_trans (b + s)
apply le_trans
exact hxb
exact hbs
have hsx : exists h. h + S x = S (b + s)
specialize succ_le_succ x
specialize succ_le_succ (b + s)
apply succ_le_succ
exact hxs
have hscale : exists h. h + S (b + s) = C * S (b + s)
specialize le_scaled_nonzero C
specialize le_scaled_nonzero (S (b + s))
apply le_scaled_nonzero
exact hC
have hxbase : exists h. h + S x = C * S (b + s)
specialize le_trans (S x)
specialize le_trans (S (b + s))
specialize le_trans (C * S (b + s))
apply le_trans
exact hsx
exact hscale
have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))
specialize base_le_beta_modulus (C * S (b + s))
specialize base_le_beta_modulus j
exact base_le_beta_modulus
specialize le_trans (S x)
specialize le_trans (C * S (b + s))
specialize le_trans (S ((S j) * (C * S (b + s))))
apply le_trans
exact hxbase
exact hmod
qed

Occurrences

863

Distinct objects

489

Reused references

41

Depth

33

Cuts

31

Certificate SHA-256

05aeb5bdd1f8…

new_value_lt_scaled_baseAn appended value fits the scaled target base β sequences, products & FTA checked

The appended value fits every modulus after the same constructive scaled-base rebase.

Prerequisites

le_add_left succ_le_succ le_scaled_nonzero le_trans base_le_beta_modulus

Used directly by

beta_prefix_extend

Exact expanded PA statement

Copy statement

forall b s C j. ~(C = 0) -> exists h. h + S s = S ((S j) * (C * S (b + s)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b s C j. ~(C = 0) -> exists h. h + S s = S ((S j) * (C * S (b + s)))
use le_add_left
use succ_le_succ
use le_scaled_nonzero
use le_trans
use base_le_beta_modulus
intro b
intro s
intro C
intro j
intro hC
have hsb : exists h. h + s = b + s
specialize le_add_left s
specialize le_add_left b
exact le_add_left
have hss : exists h. h + S s = S (b + s)
specialize succ_le_succ s
specialize succ_le_succ (b + s)
apply succ_le_succ
exact hsb
have hscale : exists h. h + S (b + s) = C * S (b + s)
specialize le_scaled_nonzero C
specialize le_scaled_nonzero (S (b + s))
apply le_scaled_nonzero
exact hC
have hsbase : exists h. h + S s = C * S (b + s)
specialize le_trans (S s)
specialize le_trans (S (b + s))
specialize le_trans (C * S (b + s))
apply le_trans
exact hss
exact hscale
have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))
specialize base_le_beta_modulus (C * S (b + s))
specialize base_le_beta_modulus j
exact base_le_beta_modulus
specialize le_trans (S s)
specialize le_trans (C * S (b + s))
specialize le_trans (S ((S j) * (C * S (b + s))))
apply le_trans
exact hsbase
exact hmod
qed

Occurrences

751

Distinct objects

446

Reused references

41

Depth

31

Cuts

27

Certificate SHA-256

08fcf3a5d804…

beta_exclusive_accumulated_product_stepExtend an exclusive beta-modulus product invariant β sequences, products & FTA checked

Extend the accumulated target-modulus product for an exclusive prefix.

Prerequisites

mul_ne_zero right_factor_divides_product beta_modulus_nonzero le_of_succ_le_succ le_eq_or_lt multiple_mul_right le_succ_self le_trans lt_to_le lt_irrefl_expanded beta_moduli_pairwise_coprime_bounded coprime_mul_left

Used directly by

beta_exclusive_recode_invariant_step

Exact expanded PA statement

Copy statement

forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))
use mul_ne_zero
use right_factor_divides_product
use beta_modulus_nonzero
use le_of_succ_le_succ
use le_eq_or_lt
use multiple_mul_right
use le_succ_self
use le_trans
use lt_to_le
use lt_irrefl_expanded
use beta_moduli_pairwise_coprime_bounded
use coprime_mul_left
intro N
intro c
intro k
intro P
intro hcm
intro hkN
intro hP
intro hdiv
intro hfuture
have hnew : ~(S ((S k) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero k
exact beta_modulus_nonzero
split
specialize mul_ne_zero P
specialize mul_ne_zero (S ((S k) * c))
intro hzero
apply mul_ne_zero
exact hP
exact hnew
exact hzero
split
intro i
intro hi
have hik : exists r. r + i = k
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ k
apply le_of_succ_le_succ
exact hi
have hsplit : i = k \/ exists r. r + S i = k
specialize le_eq_or_lt i
specialize le_eq_or_lt k
apply le_eq_or_lt
exact hik
cases hsplit
rewrite hsplit_left
specialize right_factor_divides_product P
specialize right_factor_divides_product (S ((S k) * c))
exact right_factor_divides_product
have hiP : exists q. P = S ((S i) * c) * q
specialize hdiv i
apply hdiv
exact hsplit_right
specialize multiple_mul_right (S ((S i) * c))
specialize multiple_mul_right P
specialize multiple_mul_right (S ((S k) * c))
apply multiple_mul_right
exact hiP
intro j
intro hSkj
intro hjN
have hkj : exists r. r + k = j
have hkSk : exists r. r + k = S k
specialize le_succ_self k
exact le_succ_self
specialize le_trans k
specialize le_trans (S k)
specialize le_trans j
apply le_trans
exact hkSk
exact hSkj
have hPj : forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize hfuture j
apply hfuture
exact hkj
exact hjN
have hneq : ~(k = j)
intro heq
rewrite <- heq at hSkj
specialize lt_irrefl_expanded k
apply lt_irrefl_expanded
exact hSkj
have hkbound : exists r. r + k = N
specialize lt_to_le k
specialize lt_to_le N
apply lt_to_le
exact hkN
have hpairs : forall i j. ~(i = j) -> (exists hi. hi + i = N) -> (exists hj. hj + j = N) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize beta_moduli_pairwise_coprime_bounded N
specialize beta_moduli_pairwise_coprime_bounded c
apply beta_moduli_pairwise_coprime_bounded
exact hcm
have hnewj : forall d. (exists u. S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1
specialize hpairs k
specialize hpairs j
apply hpairs
exact hneq
exact hkbound
exact hjN
specialize coprime_mul_left P
specialize coprime_mul_left (S ((S k) * c))
specialize coprime_mul_left (S ((S j) * c))
apply coprime_mul_left
exact hPj
exact hnewj
qed

Occurrences

11,222

Distinct objects

2,345

Reused references

137

Depth

70

Cuts

332

Certificate SHA-256

1833bd103225…

beta_exclusive_recode_congruence_stepExtend exclusive cross-base beta congruences β sequences, products & FTA checked

Add the next source value to a target-base CRT code for an exclusive prefix.

Prerequisites

beta_modulus_nonzero le_refl lt_to_le binary_crt_fold_step beta_at_exists beta_at_unique le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_exclusive_recode_invariant_step

Exact expanded PA statement

Copy statement

forall N c b e k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b e k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v
use beta_modulus_nonzero
use le_refl
use lt_to_le
use binary_crt_fold_step
use beta_at_exists
use beta_at_unique
use le_of_succ_le_succ
use le_eq_or_lt
intro N
intro c
intro b
intro e
intro k
intro P
intro z
intro hkN
intro hP
intro hdiv
intro hcong
intro hfuture
have hnew : ~(S ((S k) * c) = 0)
specialize beta_modulus_nonzero c
specialize beta_modulus_nonzero k
exact beta_modulus_nonzero
have hkbound : exists h. h + k = N
specialize lt_to_le k
specialize lt_to_le N
apply lt_to_le
exact hkN
have hcop : forall d. (exists u. P = d * u) -> (exists v. S ((S k) * c) = d * v) -> d = 1
specialize hfuture k
apply hfuture
specialize le_refl k
exact le_refl
exact hkbound
have hvalue : exists a. ((exists h. h + S a = S ((S k) * e)) /\ exists q. b = q * S ((S k) * e) + a)
specialize beta_at_exists b
specialize beta_at_exists e
specialize beta_at_exists k
exact beta_at_exists
cases hvalue
have hfold : exists z2. ((forall m a. (exists q. P = m * q) -> (exists u v. z + m * u = a + m * v) -> exists r s. z2 + m * r = a + m * s) /\ exists q r. z2 + S ((S k) * c) * q = x + S ((S k) * c) * r)
specialize binary_crt_fold_step P
specialize binary_crt_fold_step (S ((S k) * c))
specialize binary_crt_fold_step z
specialize binary_crt_fold_step x
apply binary_crt_fold_step
exact hP
exact hnew
exact hcop
cases hfold
cases hfold_witness
exists x1
intro i
intro a
intro hi
intro hati
have hik : exists r. r + i = k
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ k
apply le_of_succ_le_succ
exact hi
have hsplit : i = k \/ exists r. r + S i = k
specialize le_eq_or_lt i
specialize le_eq_or_lt k
apply le_eq_or_lt
exact hik
cases hsplit
have hati_new : ((exists h. h + S a = S ((S k) * e)) /\ exists q. b = q * S ((S k) * e) + a)
rewrite <- hsplit_left
rewrite <- hsplit_left
exact hati
have haeq : a = x
specialize beta_at_unique b
specialize beta_at_unique e
specialize beta_at_unique k
specialize beta_at_unique a
specialize beta_at_unique x
apply beta_at_unique
exact hati_new
exact hvalue_witness
rewrite hsplit_left
rewrite hsplit_left
rewrite haeq
exact hfold_witness_right
have hmiP : exists q. P = S ((S i) * c) * q
specialize hdiv i
apply hdiv
exact hsplit_right
have hzold : exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v
specialize hcong i
specialize hcong a
apply hcong
exact hsplit_right
exact hati
specialize hfold_witness_left (S ((S i) * c))
specialize hfold_witness_left a
apply hfold_witness_left
exact hmiP
exact hzold
qed

Occurrences

7,398

Distinct objects

2,583

Reused references

130

Depth

65

Cuts

215

Certificate SHA-256

282847b6bbba…

beta_exclusive_recode_invariant_stepOne exclusive beta-recoding invariant step β sequences, products & FTA checked

Combine modulus-product and cross-base congruence updates for an exclusive prefix.

Prerequisites

beta_exclusive_accumulated_product_step beta_exclusive_recode_congruence_step

Used directly by

bounded_beta_exclusive_recode_invariant

Exact expanded PA statement

Copy statement

forall N c b e k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b e k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
use beta_exclusive_accumulated_product_step
use beta_exclusive_recode_congruence_step
intro N
intro c
intro b
intro e
intro k
intro P
intro z
intro hcm
intro hkN
intro hP
intro hdiv
intro hcong
intro hfuture
have hproduct : (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))
specialize beta_exclusive_accumulated_product_step N
specialize beta_exclusive_accumulated_product_step c
specialize beta_exclusive_accumulated_product_step k
specialize beta_exclusive_accumulated_product_step P
apply beta_exclusive_accumulated_product_step
exact hcm
exact hkN
exact hP
exact hdiv
exact hfuture
have hcodes : exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v
specialize beta_exclusive_recode_congruence_step N
specialize beta_exclusive_recode_congruence_step c
specialize beta_exclusive_recode_congruence_step b
specialize beta_exclusive_recode_congruence_step e
specialize beta_exclusive_recode_congruence_step k
specialize beta_exclusive_recode_congruence_step P
specialize beta_exclusive_recode_congruence_step z
apply beta_exclusive_recode_congruence_step
exact hkN
exact hP
exact hdiv
exact hcong
exact hfuture
cases hcodes
cases hproduct
cases hproduct_right
exists x
split
exact hproduct_left
split
exact hproduct_right_left
split
exact hcodes_witness
exact hproduct_right_right
qed

Occurrences

18,709

Distinct objects

3,750

Reused references

188

Depth

71

Cuts

549

Certificate SHA-256

1057ae7a6b5e…

bounded_beta_exclusive_recode_invariantBounded exclusive beta-recoding invariant β sequences, products & FTA checked

Fold an empty-based, exclusive beta prefix into another base with append readiness.

Prerequisites

succ_ne_zero add_eq_zero_right coprime_one_left le_succ_self le_trans beta_exclusive_recode_invariant_step

Used directly by

beta_prefix_extend

Exact expanded PA statement

Copy statement

forall N c b e. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall N c b e. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
use succ_ne_zero
use add_eq_zero_right
use coprime_one_left
use le_succ_self
use le_trans
use beta_exclusive_recode_invariant_step
intro N
intro c
intro b
intro e
intro hcm
induction k
intro hkN
exists 1
exists 0
split
specialize succ_ne_zero 0
exact succ_ne_zero
split
intro i
intro hi
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
split
intro i
intro a
intro hi
intro hati
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
intro j
intro h0j
intro hjN
intro d
intro h1
intro hm
specialize coprime_one_left (S ((S j) * c))
specialize coprime_one_left d
apply coprime_one_left
exact h1
exact hm
intro hkN
have hkprev : exists h. h + k = N
have hkstep : exists h. h + k = S k
specialize le_succ_self k
exact le_succ_self
specialize le_trans k
specialize le_trans (S k)
specialize le_trans N
apply le_trans
exact hkstep
exact hkN
have hprev : exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
apply IH
exact hkprev
cases hprev
cases hprev_witness
cases hprev_witness_witness
cases hprev_witness_witness_right
cases hprev_witness_witness_right_right
have hnext : exists z2. (~(x * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. x * S ((S k) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. x * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))
specialize beta_exclusive_recode_invariant_step N
specialize beta_exclusive_recode_invariant_step c
specialize beta_exclusive_recode_invariant_step b
specialize beta_exclusive_recode_invariant_step e
specialize beta_exclusive_recode_invariant_step k
specialize beta_exclusive_recode_invariant_step x
specialize beta_exclusive_recode_invariant_step x1
apply beta_exclusive_recode_invariant_step
exact hcm
exact hkN
exact hprev_witness_witness_left
exact hprev_witness_witness_right_left
exact hprev_witness_witness_right_right_left
exact hprev_witness_witness_right_right_right
cases hnext
exists x * S ((S k) * c)
exists x2
exact hnext_witness
qed

Occurrences

19,155

Distinct objects

3,873

Reused references

193

Depth

77

Cuts

563

Certificate SHA-256

2c9541dd49fb…

beta_prefix_extendRecode and append one exact beta-prefix value β sequences, products & FTA checked

Rebase an arbitrary decoded prefix and append one exact natural value.

Prerequisites

bounded_common_multiple_exists scaled_bounded_common_multiple bounded_beta_exclusive_recode_invariant le_refl beta_modulus_nonzero binary_crt_fold_step new_value_lt_scaled_base beta_value_lt_scaled_base beta_at_of_mod_eq_bound

Used directly by

beta_prefix_product_trace_exists beta_product_succ_append beta_factor_prefix_product_append beta_prefix_extend_all_prime beta_prefix_extend_sorted_singleton beta_prefix_extend_sorted_succ beta_repeat_succ_extend beta_range_succ_extend beta_prefix_sum_trace_exists beta_prefix_replace_exists

Exact expanded PA statement

Copy statement

forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists q. z = q * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. z = q * S ((S i) * c) + a))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists q. z = q * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. z = q * S ((S i) * c) + a))
use bounded_common_multiple_exists
use scaled_bounded_common_multiple
use bounded_beta_exclusive_recode_invariant
use le_refl
use beta_modulus_nonzero
use binary_crt_fold_step
use new_value_lt_scaled_base
use beta_value_lt_scaled_base
use beta_at_of_mod_eq_bound
intro k
intro b
intro e
intro s
have hC : exists C. (~(C = 0) /\ forall t. (exists h. S t + S h = S k) -> exists q. C = S t * q)
specialize bounded_common_multiple_exists k
exact bounded_common_multiple_exists
cases hC
cases hC_witness
have hcm2 : forall t. (exists h. S t + S h = S k) -> exists q. x * S (b + s) = S t * q
specialize scaled_bounded_common_multiple k
specialize scaled_bounded_common_multiple x
specialize scaled_bounded_common_multiple (S (b + s))
apply scaled_bounded_common_multiple
exact hC_witness_right
have hall : forall n. (exists h. h + n = k) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = n) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\ ((forall i a. (exists h. h + S i = n) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\ forall j. (exists g. g + n = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))
specialize bounded_beta_exclusive_recode_invariant k
specialize bounded_beta_exclusive_recode_invariant (x * S (b + s))
specialize bounded_beta_exclusive_recode_invariant b
specialize bounded_beta_exclusive_recode_invariant e
apply bounded_beta_exclusive_recode_invariant
exact hcm2
have hinv : exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\ forall j. (exists g. g + k = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))
specialize hall k
apply hall
specialize le_refl k
exact le_refl
cases hinv
cases hinv_witness
cases hinv_witness_witness
cases hinv_witness_witness_right
cases hinv_witness_witness_right_right
have hcop : forall d. (exists u. x1 = d * u) -> (exists v. S ((S k) * (x * S (b + s))) = d * v) -> d = 1
specialize hinv_witness_witness_right_right_right k
apply hinv_witness_witness_right_right_right
specialize le_refl k
exact le_refl
specialize le_refl k
exact le_refl
have hnew0 : ~(S ((S k) * (x * S (b + s))) = 0)
specialize beta_modulus_nonzero (x * S (b + s))
specialize beta_modulus_nonzero k
exact beta_modulus_nonzero
have hfold : exists z2. ((forall m a. (exists q. x1 = m * q) -> (exists u v. x2 + m * u = a + m * v) -> exists r t. z2 + m * r = a + m * t) /\ exists q r. z2 + S ((S k) * (x * S (b + s))) * q = s + S ((S k) * (x * S (b + s))) * r)
specialize binary_crt_fold_step x1
specialize binary_crt_fold_step (S ((S k) * (x * S (b + s))))
specialize binary_crt_fold_step x2
specialize binary_crt_fold_step s
apply binary_crt_fold_step
exact hinv_witness_witness_left
exact hnew0
exact hcop
cases hfold
cases hfold_witness
exists x3
exists x * S (b + s)
split
specialize beta_at_of_mod_eq_bound x3
specialize beta_at_of_mod_eq_bound (x * S (b + s))
specialize beta_at_of_mod_eq_bound k
specialize beta_at_of_mod_eq_bound s
apply beta_at_of_mod_eq_bound
specialize new_value_lt_scaled_base b
specialize new_value_lt_scaled_base s
specialize new_value_lt_scaled_base x
specialize new_value_lt_scaled_base k
apply new_value_lt_scaled_base
exact hC_witness_left
exact hfold_witness_right
intro i
intro a
intro hi
intro hati
have hmi : exists q. x1 = S ((S i) * (x * S (b + s))) * q
specialize hinv_witness_witness_right_left i
apply hinv_witness_witness_right_left
exact hi
have hzold : exists u v. x2 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v
specialize hinv_witness_witness_right_right_left i
specialize hinv_witness_witness_right_right_left a
apply hinv_witness_witness_right_right_left
exact hi
exact hati
have hznew : exists u v. x3 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v
specialize hfold_witness_left (S ((S i) * (x * S (b + s))))
specialize hfold_witness_left a
apply hfold_witness_left
exact hmi
exact hzold
specialize beta_at_of_mod_eq_bound x3
specialize beta_at_of_mod_eq_bound (x * S (b + s))
specialize beta_at_of_mod_eq_bound i
specialize beta_at_of_mod_eq_bound a
apply beta_at_of_mod_eq_bound
specialize beta_value_lt_scaled_base b
specialize beta_value_lt_scaled_base e
specialize beta_value_lt_scaled_base i
specialize beta_value_lt_scaled_base a
specialize beta_value_lt_scaled_base x
specialize beta_value_lt_scaled_base s
specialize beta_value_lt_scaled_base i
apply beta_value_lt_scaled_base
exact hati
exact hC_witness_left
exact hznew
qed

Occurrences

29,057

Distinct objects

4,508

Reused references

225

Depth

80

Cuts

867

Certificate SHA-256

511d3bde3fc4…

beta_prefix_product_trace_existsExact beta-coded prefix-product trace exists β sequences, products & FTA checked

Every decoded beta factor prefix admits a beta-coded exact prefix-product trace.

Prerequisites

beta_at_self_of_bound add_eq_zero_right succ_ne_zero beta_at_exists beta_prefix_extend zero_le succ_le_succ le_refl le_of_succ_le_succ le_eq_or_lt one_mul

Used directly by

beta_product_exists

Exact expanded PA statement

Copy statement

forall b c l. exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p))))
use beta_at_self_of_bound
use add_eq_zero_right
use succ_ne_zero
use beta_at_exists
use beta_prefix_extend
use zero_le
use succ_le_succ
use le_refl
use le_of_succ_le_succ
use le_eq_or_lt
use one_mul
intro b
intro c
induction l
exists 1
exists 1
split
specialize beta_at_self_of_bound 1
specialize beta_at_self_of_bound 0
specialize beta_at_self_of_bound 1
apply beta_at_self_of_bound
specialize one_mul 1
rewrite one_mul
specialize le_refl 2
exact le_refl
intro i
intro hi
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
have htrace : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p))))
apply IH
cases htrace
cases htrace_witness
cases htrace_witness_witness
have hfactor : exists p. ((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p)
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists l
exact beta_at_exists
cases hfactor
have hlast : exists r. ((exists h. h + S r = S ((S l) * x1)) /\ exists q. x = q * S ((S l) * x1) + r)
specialize beta_at_exists x
specialize beta_at_exists x1
specialize beta_at_exists l
exact beta_at_exists
cases hlast
have hext : exists z v. (((exists h. h + S (x3 * x2) = S ((S (S l)) * v)) /\ exists q. z = q * S ((S (S l)) * v) + (x3 * x2)) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + a) -> ((exists h. h + S a = S ((S i) * v)) /\ exists q. z = q * S ((S i) * v) + a))
specialize beta_prefix_extend (S l)
specialize beta_prefix_extend x
specialize beta_prefix_extend x1
specialize beta_prefix_extend (x3 * x2)
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
exists x4
exists x5
split
specialize hext_witness_witness_right 0
specialize hext_witness_witness_right 1
apply hext_witness_witness_right
have h0 : exists h. h + S 0 = S l
have hzero : exists h. h + 0 = l
specialize zero_le l
exact zero_le
specialize succ_le_succ 0
specialize succ_le_succ l
apply succ_le_succ
exact hzero
exact h0
exact htrace_witness_witness_left
intro i
intro hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists x2
exists x3
exists x3 * x2
split
rewrite hsplit_left
rewrite hsplit_left
exact hfactor_witness
split
rewrite hsplit_left
rewrite hsplit_left
specialize hext_witness_witness_right l
specialize hext_witness_witness_right x3
apply hext_witness_witness_right
specialize le_refl (S l)
exact le_refl
exact hlast_witness
split
rewrite hsplit_left
rewrite hsplit_left
exact hext_witness_witness_left
refl
have hold : exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + r) /\ (((exists h. h + S s = S ((S (S i)) * x1)) /\ exists q. x = q * S ((S (S i)) * x1) + s) /\ s = r * p)))
specialize htrace_witness_witness_right i
apply htrace_witness_witness_right
exact hsplit_right
cases hold
cases hold_witness
cases hold_witness_witness
cases hold_witness_witness_witness
cases hold_witness_witness_witness_right
cases hold_witness_witness_witness_right_right
exists x6
exists x7
exists x8
split
exact hold_witness_witness_witness_left
split
specialize hext_witness_witness_right i
specialize hext_witness_witness_right x7
apply hext_witness_witness_right
exact hi
exact hold_witness_witness_witness_right_left
split
specialize hext_witness_witness_right (S i)
specialize hext_witness_witness_right x8
apply hext_witness_witness_right
specialize succ_le_succ (S i)
specialize succ_le_succ l
apply succ_le_succ
exact hsplit_right
exact hold_witness_witness_witness_right_right_left
exact hold_witness_witness_witness_right_right_right
qed

Occurrences

29,981

Distinct objects

4,717

Reused references

235

Depth

85

Cuts

899

Certificate SHA-256

0dd19ccc0b06…

beta_product_existsRelational finite beta product exists β sequences, products & FTA checked

Every finite decoded beta prefix has an exact relational product and a coded trace.

Prerequisites

beta_prefix_product_trace_exists beta_at_exists

Used directly by

beta_product_exists_unique pow_exists factorial_exists

Exact expanded PA statement

Copy statement

forall b c l. exists n u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists n u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))
use beta_prefix_product_trace_exists
use beta_at_exists
intro b
intro c
intro l
have htrace : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p))))
specialize beta_prefix_product_trace_exists b
specialize beta_prefix_product_trace_exists c
specialize beta_prefix_product_trace_exists l
exact beta_prefix_product_trace_exists
cases htrace
cases htrace_witness
cases htrace_witness_witness
have hterminal : exists n. ((exists h. h + S n = S ((S l) * x1)) /\ exists q. x = q * S ((S l) * x1) + n)
specialize beta_at_exists x
specialize beta_at_exists x1
specialize beta_at_exists l
exact beta_at_exists
cases hterminal
exists x2
exists x
exists x1
split
exact htrace_witness_witness_left
split
exact hterminal_witness
exact htrace_witness_witness_right
qed

Occurrences

30,487

Distinct objects

4,744

Reused references

236

Depth

86

Cuts

916

Certificate SHA-256

6656c8e6b7f3…

beta_product_functionalFinite beta Product is functional β sequences, products & FTA checked

The fully expanded beta-coded Product relation is functional in its terminal product.

Prerequisites

beta_at_unique le_refl le_succ mul_congr

Used directly by

beta_product_exists_unique pow_functional factorial_functional beta_product_replace_balance

Exact expanded PA statement

Copy statement

forall b c l n u v m w d. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))) -> (((exists h. h + S 1 = S ((S 0) * d)) /\ exists q. w = q * S ((S 0) * d) + 1) /\ (((exists h. h + S m = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + m) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * d)) /\ exists q. w = q * S ((S i) * d) + r) /\ (((exists h. h + S s = S ((S S i) * d)) /\ exists q. w = q * S ((S S i) * d) + s) /\ s = r * p))))) -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n u v m w d. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))) -> (((exists h. h + S 1 = S ((S 0) * d)) /\ exists q. w = q * S ((S 0) * d) + 1) /\ (((exists h. h + S m = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + m) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * d)) /\ exists q. w = q * S ((S i) * d) + r) /\ (((exists h. h + S s = S ((S S i) * d)) /\ exists q. w = q * S ((S S i) * d) + s) /\ s = r * p))))) -> n = m
use beta_at_unique
use le_refl
use le_succ
use mul_congr
intro b
intro c
induction l
intro n
intro u
intro v
intro m
intro w
intro d
intro h1
intro h2
cases h1
cases h1_right
cases h2
cases h2_right
have hn : n = 1
specialize beta_at_unique u
specialize beta_at_unique v
specialize beta_at_unique 0
specialize beta_at_unique n
specialize beta_at_unique 1
apply beta_at_unique
exact h1_right_left
exact h1_left
have hm : m = 1
specialize beta_at_unique w
specialize beta_at_unique d
specialize beta_at_unique 0
specialize beta_at_unique m
specialize beta_at_unique 1
apply beta_at_unique
exact h2_right_left
exact h2_left
trans 1
exact hn
symm
exact hm
intro n
intro u
intro v
intro m
intro w
intro d
intro h1
intro h2
cases h1
cases h1_right
cases h2
cases h2_right
have hstep1 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ (((exists h. h + S s = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + s) /\ s = r * p)))
specialize h1_right_right l
apply h1_right_right
specialize le_refl (S l)
exact le_refl
cases hstep1
cases hstep1_witness
cases hstep1_witness_witness
cases hstep1_witness_witness_witness
cases hstep1_witness_witness_witness_right
cases hstep1_witness_witness_witness_right_right
have hstep2 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) /\ (((exists h. h + S r = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + r) /\ (((exists h. h + S s = S ((S S l) * d)) /\ exists q. w = q * S ((S S l) * d) + s) /\ s = r * p)))
specialize h2_right_right l
apply h2_right_right
specialize le_refl (S l)
exact le_refl
cases hstep2
cases hstep2_witness
cases hstep2_witness_witness
cases hstep2_witness_witness_witness
cases hstep2_witness_witness_witness_right
cases hstep2_witness_witness_witness_right_right
have hn : n = x2
specialize beta_at_unique u
specialize beta_at_unique v
specialize beta_at_unique (S l)
specialize beta_at_unique n
specialize beta_at_unique x2
apply beta_at_unique
exact h1_right_left
exact hstep1_witness_witness_witness_right_right_left
have hm : m = x5
specialize beta_at_unique w
specialize beta_at_unique d
specialize beta_at_unique (S l)
specialize beta_at_unique m
specialize beta_at_unique x5
apply beta_at_unique
exact h2_right_left
exact hstep2_witness_witness_witness_right_right_left
have hp : x = x3
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique x3
apply beta_at_unique
exact hstep1_witness_witness_witness_left
exact hstep2_witness_witness_witness_left
have hprod1 : (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S x1 = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + x1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))
split
exact h1_left
split
exact hstep1_witness_witness_witness_right_left
intro i
intro hi
specialize h1_right_right i
apply h1_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
have hprod2 : (((exists h. h + S 1 = S ((S 0) * d)) /\ exists q. w = q * S ((S 0) * d) + 1) /\ (((exists h. h + S x4 = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + x4) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * d)) /\ exists q. w = q * S ((S i) * d) + r) /\ (((exists h. h + S s = S ((S S i) * d)) /\ exists q. w = q * S ((S S i) * d) + s) /\ s = r * p)))))
split
exact h2_left
split
exact hstep2_witness_witness_witness_right_left
intro i
intro hi
specialize h2_right_right i
apply h2_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
have hprev : x1 = x4
specialize IH x1
specialize IH u
specialize IH v
specialize IH x4
specialize IH w
specialize IH d
apply IH
exact hprod1
exact hprod2
have hmul : x1 * x = x4 * x3
specialize mul_congr x1
specialize mul_congr x4
specialize mul_congr x
specialize mul_congr x3
apply mul_congr
exact hprev
exact hp
trans x2
exact hn
trans x1 * x
exact hstep1_witness_witness_witness_right_right_right
trans x4 * x3
exact hmul
trans x5
symm
exact hstep2_witness_witness_witness_right_right_right
symm
exact hm
qed

Occurrences

1,382

Distinct objects

909

Reused references

39

Depth

60

Cuts

36

Certificate SHA-256

12a5727a4593…

beta_product_exists_uniqueA unique finite beta Product exists β sequences, products & FTA checked

Every finite decoded beta prefix has exactly one relational product value.

Prerequisites

beta_product_exists beta_product_functional

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l. exists n. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) /\ forall m. (exists w d. (((exists h. h + S 1 = S ((S 0) * d)) /\ exists q. w = q * S ((S 0) * d) + 1) /\ (((exists h. h + S m = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + m) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * d)) /\ exists q. w = q * S ((S i) * d) + r) /\ (((exists h. h + S s = S ((S S i) * d)) /\ exists q. w = q * S ((S S i) * d) + s) /\ s = r * p)))))) -> n = m)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists n. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) /\ forall m. (exists w d. (((exists h. h + S 1 = S ((S 0) * d)) /\ exists q. w = q * S ((S 0) * d) + 1) /\ (((exists h. h + S m = S ((S l) * d)) /\ exists q. w = q * S ((S l) * d) + m) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * d)) /\ exists q. w = q * S ((S i) * d) + r) /\ (((exists h. h + S s = S ((S S i) * d)) /\ exists q. w = q * S ((S S i) * d) + s) /\ s = r * p)))))) -> n = m)
use beta_product_exists
use beta_product_functional
intro b
intro c
intro l
have hex : exists n u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))
specialize beta_product_exists b
specialize beta_product_exists c
specialize beta_product_exists l
exact beta_product_exists
cases hex
cases hex_witness
cases hex_witness_witness
exists x
split
exists x1
exists x2
exact hex_witness_witness_witness
intro m
intro hm
cases hm
cases hm_witness
specialize beta_product_functional b
specialize beta_product_functional c
specialize beta_product_functional l
specialize beta_product_functional x
specialize beta_product_functional x1
specialize beta_product_functional x2
specialize beta_product_functional m
specialize beta_product_functional x3
specialize beta_product_functional x4
apply beta_product_functional
exact hex_witness_witness_witness
exact hm_witness_witness
qed

Occurrences

31,908

Distinct objects

4,979

Reused references

239

Depth

87

Cuts

954

Certificate SHA-256

7a2e26c9d930…

beta_product_zeroThe empty beta Product is one β sequences, products & FTA checked

The product of an empty decoded prefix is one.

Prerequisites

beta_at_unique

Used directly by

beta_prime_divisor_product_member beta_all_prime_product_one_iff_length_zero prime_factorization_uniqueness_by_length pow_zero beta_product_pointwise_mod_congruent factorial_zero

Exact expanded PA statement

Copy statement

forall b c n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + n) /\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> n = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + n) /\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> n = 1
use beta_at_unique
intro b
intro c
intro n
intro hproduct
cases hproduct
cases hproduct_witness
cases hproduct_witness_witness
cases hproduct_witness_witness_right
specialize beta_at_unique x
specialize beta_at_unique x1
specialize beta_at_unique 0
specialize beta_at_unique n
specialize beta_at_unique 1
apply beta_at_unique
exact hproduct_witness_witness_right_left
exact hproduct_witness_witness_left
qed

Occurrences

1,171

Distinct objects

742

Reused references

37

Depth

60

Cuts

31

Certificate SHA-256

63088511aac2…

beta_product_succ_decomposeDecompose a successor beta Product β sequences, products & FTA checked

A successor product decomposes into its prefix product and final decoded factor.

Prerequisites

le_refl le_succ beta_at_unique

Used directly by

beta_factor_divides_product beta_prime_divisor_product_member beta_canonical_product_cancel_last pow_successor_decompose beta_product_pointwise_mod_congruent factorial_succ_decompose beta_product_replace_balance beta_product_swap_last_invariant

Exact expanded PA statement

Copy statement

forall b c l n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists p r. (((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = r * p))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists p r. (((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = r * p))
use le_refl
use le_succ
use beta_at_unique
intro b
intro c
intro l
intro n
intro hproduct
cases hproduct
cases hproduct_witness
cases hproduct_witness_witness
cases hproduct_witness_witness_right
have hstep : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) /\ (((exists h. h + S r = S ((S l) * x1)) /\ exists q. x = q * S ((S l) * x1) + r) /\ (((exists h. h + S s = S ((S S l) * x1)) /\ exists q. x = q * S ((S S l) * x1) + s) /\ s = r * p)))
specialize hproduct_witness_witness_right_right l
apply hproduct_witness_witness_right_right
specialize le_refl (S l)
exact le_refl
cases hstep
cases hstep_witness
cases hstep_witness_witness
cases hstep_witness_witness_witness
cases hstep_witness_witness_witness_right
cases hstep_witness_witness_witness_right_right
have hn : n = x4
specialize beta_at_unique x
specialize beta_at_unique x1
specialize beta_at_unique (S l)
specialize beta_at_unique n
specialize beta_at_unique x4
apply beta_at_unique
exact hproduct_witness_witness_right_left
exact hstep_witness_witness_witness_right_right_left
exists x2
exists x3
split
exact hstep_witness_witness_witness_left
split
exists x
exists x1
split
exact hproduct_witness_witness_left
split
exact hstep_witness_witness_witness_right_left
intro i
intro hi
specialize hproduct_witness_witness_right_right i
apply hproduct_witness_witness_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
trans x4
exact hn
exact hstep_witness_witness_witness_right_right_right
qed

Occurrences

1,257

Distinct objects

784

Reused references

39

Depth

62

Cuts

35

Certificate SHA-256

2a5502bdcf73…

beta_product_succ_appendAppend one factor to a beta Product β sequences, products & FTA checked

Append one decoded factor to an existing fully expanded Product witness.

Prerequisites

beta_prefix_extend zero_le succ_le_succ le_refl le_of_succ_le_succ le_eq_or_lt

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l r p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S (r * p) = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + (r * p)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l r p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + p) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S (r * p) = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + (r * p)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))
use beta_prefix_extend
use zero_le
use succ_le_succ
use le_refl
use le_of_succ_le_succ
use le_eq_or_lt
intro b
intro c
intro l
intro r
intro p
intro hproduct
intro hp
cases hproduct
cases hproduct_witness
cases hproduct_witness_witness
cases hproduct_witness_witness_right
have hext : exists z d. (((exists h. h + S (r * p) = S ((S S l) * d)) /\ exists q. z = q * S ((S S l) * d) + (r * p)) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + a) -> ((exists h. h + S a = S ((S i) * d)) /\ exists q. z = q * S ((S i) * d) + a))
specialize beta_prefix_extend (S l)
specialize beta_prefix_extend x
specialize beta_prefix_extend x1
specialize beta_prefix_extend (r * p)
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
exists x2
exists x3
split
specialize hext_witness_witness_right 0
specialize hext_witness_witness_right 1
apply hext_witness_witness_right
have hzero : exists h. h + 0 = l
specialize zero_le l
exact zero_le
specialize succ_le_succ 0
specialize succ_le_succ l
apply succ_le_succ
exact hzero
exact hproduct_witness_witness_left
split
exact hext_witness_witness_left
intro i
intro hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists p
exists r
exists r * p
split
rewrite hsplit_left
rewrite hsplit_left
exact hp
split
rewrite hsplit_left
rewrite hsplit_left
specialize hext_witness_witness_right l
specialize hext_witness_witness_right r
apply hext_witness_witness_right
specialize le_refl (S l)
exact le_refl
exact hproduct_witness_witness_right_left
split
rewrite hsplit_left
rewrite hsplit_left
exact hext_witness_witness_left
refl
have hold : exists p0 r0 s0. (((exists h. h + S p0 = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p0) /\ (((exists h. h + S r0 = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + r0) /\ (((exists h. h + S s0 = S ((S S i) * x1)) /\ exists q. x = q * S ((S S i) * x1) + s0) /\ s0 = r0 * p0)))
specialize hproduct_witness_witness_right_right i
apply hproduct_witness_witness_right_right
exact hsplit_right
cases hold
cases hold_witness
cases hold_witness_witness
cases hold_witness_witness_witness
cases hold_witness_witness_witness_right
cases hold_witness_witness_witness_right_right
exists x4
exists x5
exists x6
split
exact hold_witness_witness_witness_left
split
specialize hext_witness_witness_right i
specialize hext_witness_witness_right x5
apply hext_witness_witness_right
exact hi
exact hold_witness_witness_witness_right_left
split
specialize hext_witness_witness_right (S i)
specialize hext_witness_witness_right x6
apply hext_witness_witness_right
specialize succ_le_succ (S i)
specialize succ_le_succ l
apply succ_le_succ
exact hsplit_right
exact hold_witness_witness_witness_right_right_left
exact hold_witness_witness_witness_right_right_right
qed

Occurrences

29,360

Distinct objects

4,659

Reused references

229

Depth

81

Cuts

877

Certificate SHA-256

ffa7ace29cd7…

beta_product_transport_prefixTransport Product across an extensionally preserved factor prefix β sequences, products & FTA checked

One-way extensional factor-prefix preservation transports Product without changing its trace.

Prerequisites

none — a root theorem

Used directly by

beta_factor_prefix_product_append pow_functional factorial_functional beta_product_replace_balance

Exact expanded PA statement

Copy statement

forall b c z e l n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + a)) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z e l n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + a)) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))
intro b
intro c
intro z
intro e
intro l
intro n
intro hproduct
intro hpres
cases hproduct
cases hproduct_witness
cases hproduct_witness_witness
cases hproduct_witness_witness_right
exists x
exists x1
split
exact hproduct_witness_witness_left
split
exact hproduct_witness_witness_right_left
intro i
intro hi
have hstep : exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + r) /\ (((exists h. h + S s = S ((S S i) * x1)) /\ exists q. x = q * S ((S S i) * x1) + s) /\ s = r * p)))
specialize hproduct_witness_witness_right_right i
apply hproduct_witness_witness_right_right
exact hi
cases hstep
cases hstep_witness
cases hstep_witness_witness
cases hstep_witness_witness_witness
cases hstep_witness_witness_witness_right
cases hstep_witness_witness_witness_right_right
exists x2
exists x3
exists x4
split
specialize hpres i
specialize hpres x2
apply hpres
exact hi
exact hstep_witness_witness_witness_left
split
exact hstep_witness_witness_witness_right_left
split
exact hstep_witness_witness_witness_right_right_left
exact hstep_witness_witness_witness_right_right_right
qed

Occurrences

59

Distinct objects

59

Reused references

0

Depth

29

Cuts

0

Certificate SHA-256

4c40f0a22c7a…

beta_factor_prefix_product_appendExtend a factor prefix and Product together β sequences, products & FTA checked

Extend a factor prefix by p and simultaneously append p to its exact Product.

Prerequisites

beta_prefix_extend beta_product_transport_prefix zero_le succ_le_succ le_refl le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_canonical_append_empty beta_canonical_append_succ beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c l r p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists z e. (((exists h. h + S p = S ((S l) * e)) /\ exists q. z = q * S ((S l) * e) + p) /\ ((forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + a)) /\ (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S (r * p) = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + (r * p)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l r p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists z e. (((exists h. h + S p = S ((S l) * e)) /\ exists q. z = q * S ((S l) * e) + p) /\ ((forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + a)) /\ (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S (r * p) = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + (r * p)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p))))))))
use beta_prefix_extend
use beta_product_transport_prefix
use zero_le
use succ_le_succ
use le_refl
use le_of_succ_le_succ
use le_eq_or_lt
intro b
intro c
intro l
intro r
intro p
intro hproduct
have hfactor_ext : exists z e. (((exists h. h + S p = S ((S l) * e)) /\ exists q. z = q * S ((S l) * e) + p) /\ forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. z = q * S ((S i) * e) + a))
specialize beta_prefix_extend l
specialize beta_prefix_extend b
specialize beta_prefix_extend c
specialize beta_prefix_extend p
exact beta_prefix_extend
cases hfactor_ext
cases hfactor_ext_witness
cases hfactor_ext_witness_witness
have htransport : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))
specialize beta_product_transport_prefix b
specialize beta_product_transport_prefix c
specialize beta_product_transport_prefix x
specialize beta_product_transport_prefix x1
specialize beta_product_transport_prefix l
specialize beta_product_transport_prefix r
apply beta_product_transport_prefix
exact hproduct
exact hfactor_ext_witness_witness_right
cases htransport
cases htransport_witness
cases htransport_witness_witness
cases htransport_witness_witness_right
have htrace_ext : exists u v. (((exists h. h + S (r * p) = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + (r * p)) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * x3)) /\ exists q. x2 = q * S ((S i) * x3) + a) -> ((exists h. h + S a = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + a))
specialize beta_prefix_extend (S l)
specialize beta_prefix_extend x2
specialize beta_prefix_extend x3
specialize beta_prefix_extend (r * p)
exact beta_prefix_extend
cases htrace_ext
cases htrace_ext_witness
cases htrace_ext_witness_witness
exists x
exists x1
split
exact hfactor_ext_witness_witness_left
split
exact hfactor_ext_witness_witness_right
exists x4
exists x5
split
specialize htrace_ext_witness_witness_right 0
specialize htrace_ext_witness_witness_right 1
apply htrace_ext_witness_witness_right
have hzero : exists h. h + 0 = l
specialize zero_le l
exact zero_le
specialize succ_le_succ 0
specialize succ_le_succ l
apply succ_le_succ
exact hzero
exact htransport_witness_witness_left
split
exact htrace_ext_witness_witness_left
intro i
intro hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists p
exists r
exists r * p
split
rewrite hsplit_left
rewrite hsplit_left
exact hfactor_ext_witness_witness_left
split
rewrite hsplit_left
rewrite hsplit_left
specialize htrace_ext_witness_witness_right l
specialize htrace_ext_witness_witness_right r
apply htrace_ext_witness_witness_right
specialize le_refl (S l)
exact le_refl
exact htransport_witness_witness_right_left
split
rewrite hsplit_left
rewrite hsplit_left
exact htrace_ext_witness_witness_left
refl
have hold : exists p0 r0 s0. (((exists h. h + S p0 = S ((S i) * x1)) /\ exists q. x = q * S ((S i) * x1) + p0) /\ (((exists h. h + S r0 = S ((S i) * x3)) /\ exists q. x2 = q * S ((S i) * x3) + r0) /\ (((exists h. h + S s0 = S ((S S i) * x3)) /\ exists q. x2 = q * S ((S S i) * x3) + s0) /\ s0 = r0 * p0)))
specialize htransport_witness_witness_right_right i
apply htransport_witness_witness_right_right
exact hsplit_right
cases hold
cases hold_witness
cases hold_witness_witness
cases hold_witness_witness_witness
cases hold_witness_witness_witness_right
cases hold_witness_witness_witness_right_right
exists x6
exists x7
exists x8
split
exact hold_witness_witness_witness_left
split
specialize htrace_ext_witness_witness_right i
specialize htrace_ext_witness_witness_right x7
apply htrace_ext_witness_witness_right
exact hi
exact hold_witness_witness_witness_right_left
split
specialize htrace_ext_witness_witness_right (S i)
specialize htrace_ext_witness_witness_right x8
apply htrace_ext_witness_witness_right
specialize succ_le_succ (S i)
specialize succ_le_succ l
apply succ_le_succ
exact hsplit_right
exact hold_witness_witness_witness_right_right_left
exact hold_witness_witness_witness_right_right_right
qed

Occurrences

29,447

Distinct objects

4,746

Reused references

229

Depth

81

Cuts

878

Certificate SHA-256

ee15cbf39071…

all_prime_emptyAllPrime holds on the empty beta prefix β sequences, products & FTA checked

The fully expanded AllPrime predicate holds vacuously on the empty prefix.

Prerequisites

add_eq_zero_right succ_ne_zero

Used directly by

beta_canonical_append_empty prime_factorization_exists_up_to

Exact expanded PA statement

Copy statement

forall b c. (forall i. (exists h. h + S i = 0) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c. (forall i. (exists h. h + S i = 0) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
use add_eq_zero_right
use succ_ne_zero
intro b
intro c
intro i
intro hi
exfalso
cases hi
have hzero : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hzero
qed

Occurrences

37

Distinct objects

37

Reused references

0

Depth

13

Cuts

2

Certificate SHA-256

17228ed7fa9b…

all_prime_succ_introExtend AllPrime across one beta entry β sequences, products & FTA checked

Append one explicitly decoded prime to an AllPrime prefix.

Prerequisites

le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_prefix_extend_all_prime beta_canonical_append_empty beta_canonical_append_succ beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c l p. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)) -> (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l p. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)) -> (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
use le_of_succ_le_succ
use le_eq_or_lt
intro b
intro c
intro l
intro p
intro hprev
intro hlast
cases hlast
intro i
intro hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists p
split
rewrite hsplit_left
rewrite hsplit_left
exact hlast_left
exact hlast_right
specialize hprev i
apply hprev
exact hsplit_right
qed

Occurrences

150

Distinct objects

146

Reused references

4

Depth

21

Cuts

5

Certificate SHA-256

7905f0e448fd…

all_prime_succ_elim_prefixProject AllPrime to the previous beta prefix β sequences, products & FTA checked

Restrict an AllPrime successor prefix to its old prefix.

Prerequisites

le_succ

Used directly by

beta_prime_divisor_product_member beta_canonical_product_cancel_last

Exact expanded PA statement

Copy statement

forall b c l. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
use le_succ
intro b
intro c
intro l
intro hall
intro i
intro hi
specialize hall i
apply hall
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
qed

Occurrences

64

Distinct objects

62

Reused references

2

Depth

16

Cuts

2

Certificate SHA-256

0126509667dc…

all_prime_succ_elim_lastExtract the last prime from AllPrime β sequences, products & FTA checked

Extract the final decoded prime from an AllPrime successor prefix.

Prerequisites

le_refl

Used directly by

beta_canonical_append_general beta_canonical_last_factor_bound beta_prime_divisor_product_member beta_nonempty_all_prime_product_ne_one beta_canonical_last_factors_equal beta_canonical_product_cancel_last

Exact expanded PA statement

Copy statement

forall b c l. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))
use le_refl
intro b
intro c
intro l
intro hall
specialize hall l
apply hall
specialize le_refl (S l)
exact le_refl
qed

Occurrences

41

Distinct objects

41

Reused references

0

Depth

11

Cuts

2

Certificate SHA-256

a36af944a569…

all_prime_transportTransport AllPrime across beta recoding β sequences, products & FTA checked

Transport AllPrime across pointwise value-preserving beta recoding.

Prerequisites

none — a root theorem

Used directly by

beta_prefix_extend_all_prime beta_canonical_append_succ beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c z d l. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p)) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d l. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p)) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
intro b
intro c
intro z
intro d
intro l
intro hall
intro htransport
intro i
intro hi
have hp : exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))
specialize hall i
apply hall
exact hi
cases hp
cases hp_witness
exists x
split
specialize htransport i
specialize htransport x
apply htransport
exact hi
exact hp_witness_left
exact hp_witness_right
qed

Occurrences

26

Distinct objects

26

Reused references

0

Depth

17

Cuts

0

Certificate SHA-256

4abed2d820d8…

sorted_emptyAdjacent Sorted holds on the empty beta prefix β sequences, products & FTA checked

The fully expanded Sorted predicate holds vacuously on the empty prefix.

Prerequisites

le_zero succ_ne_zero

Used directly by

prime_factorization_exists_up_to

Exact expanded PA statement

Copy statement

forall b c. (forall i. (exists h. h + S (S i) = 0) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c. (forall i. (exists h. h + S (S i) = 0) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
use le_zero
use succ_ne_zero
intro b
intro c
intro i
intro hi
exfalso
have hzero : S (S i) = 0
specialize le_zero (S (S i))
apply le_zero
exact hi
specialize succ_ne_zero (S i)
apply succ_ne_zero
exact hzero
qed

Occurrences

44

Distinct objects

44

Reused references

0

Depth

14

Cuts

3

Certificate SHA-256

a3dc87fe88e5…

sorted_singletonAdjacent Sorted holds on a singleton beta prefix β sequences, products & FTA checked

The fully expanded Sorted predicate holds vacuously on every singleton prefix.

Prerequisites

le_of_succ_le_succ le_zero succ_ne_zero

Used directly by

beta_prefix_extend_sorted_singleton beta_canonical_append_empty beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c. (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c. (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
use le_of_succ_le_succ
use le_zero
use succ_ne_zero
intro b
intro c
intro i
intro hi
exfalso
have hzero_le : exists h. h + S i = 0
specialize le_of_succ_le_succ (S i)
specialize le_of_succ_le_succ 0
apply le_of_succ_le_succ
exact hi
have hzero : S i = 0
specialize le_zero (S i)
apply le_zero
exact hzero_le
specialize succ_ne_zero i
apply succ_ne_zero
exact hzero
qed

Occurrences

65

Distinct objects

65

Reused references

0

Depth

15

Cuts

4

Certificate SHA-256

844e77f2d4bb…

sorted_succ_introExtend adjacent Sorted across one beta entry β sequences, products & FTA checked

Append one ordered adjacent pair to a nonempty Sorted prefix.

Prerequisites

le_of_succ_le_succ le_eq_or_lt succ_le_succ

Used directly by

beta_prefix_extend_sorted_succ beta_canonical_append_succ beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c l p q. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> ((exists h. h + S q = S ((S S l) * c)) /\ exists w. b = w * S ((S S l) * c) + q) -> (exists h. h + p = q) -> (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l p q. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> ((exists h. h + S q = S ((S S l) * c)) /\ exists w. b = w * S ((S S l) * c) + q) -> (exists h. h + p = q) -> (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
use le_of_succ_le_succ
use le_eq_or_lt
use succ_le_succ
intro b
intro c
intro l
intro p
intro q
intro hprev
intro hp
intro hq
intro hpq
intro i
intro hi
have hisl : exists h. h + S i = S l
specialize le_of_succ_le_succ (S i)
specialize le_of_succ_le_succ (S l)
apply le_of_succ_le_succ
exact hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hisl
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists p
exists q
split
rewrite hsplit_left
rewrite hsplit_left
exact hp
split
rewrite hsplit_left
rewrite hsplit_left
exact hq
exact hpq
specialize hprev i
apply hprev
specialize succ_le_succ (S i)
specialize succ_le_succ l
apply succ_le_succ
exact hsplit_right
qed

Occurrences

182

Distinct objects

178

Reused references

4

Depth

23

Cuts

6

Certificate SHA-256

8c5d249ae99a…

sorted_succ_elim_prefixProject adjacent Sorted to the previous beta prefix β sequences, products & FTA checked

Restrict a Sorted successor prefix to its old prefix.

Prerequisites

le_succ

Used directly by

beta_sorted_factor_le_last beta_canonical_product_cancel_last

Exact expanded PA statement

Copy statement

forall b c l. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
use le_succ
intro b
intro c
intro l
intro hsorted
intro i
intro hi
specialize hsorted i
apply hsorted
specialize le_succ (S (S i))
specialize le_succ l
apply le_succ
exact hi
qed

Occurrences

64

Distinct objects

62

Reused references

2

Depth

16

Cuts

2

Certificate SHA-256

1fe322c40f9f…

sorted_succ_elim_lastExtract the last adjacent order witness β sequences, products & FTA checked

Extract the final adjacent ordered pair from a Sorted prefix of length at least two.

Prerequisites

le_refl

Used directly by

beta_sorted_factor_le_last

Exact expanded PA statement

Copy statement

forall b c l. (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> exists p q. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (((exists h. h + S q = S ((S S l) * c)) /\ exists w. b = w * S ((S S l) * c) + q) /\ (exists h. h + p = q)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> exists p q. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (((exists h. h + S q = S ((S S l) * c)) /\ exists w. b = w * S ((S S l) * c) + q) /\ (exists h. h + p = q)))
use le_refl
intro b
intro c
intro l
intro hsorted
specialize hsorted l
apply hsorted
specialize le_refl (S (S l))
exact le_refl
qed

Occurrences

41

Distinct objects

41

Reused references

0

Depth

11

Cuts

2

Certificate SHA-256

96d1e00a0d95…

sorted_transportTransport adjacent Sorted across beta recoding β sequences, products & FTA checked

Transport Sorted across pointwise value-preserving beta recoding.

Prerequisites

lt_to_le

Used directly by

beta_prefix_extend_sorted_succ beta_canonical_append_succ beta_canonical_append_general

Exact expanded PA statement

Copy statement

forall b c z d l. (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (forall i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p)) -> (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p) /\ (((exists h. h + S q = S ((S S i) * d)) /\ exists w. z = w * S ((S S i) * d) + q) /\ (exists h. h + p = q))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d l. (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (forall i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p)) -> (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S p = S ((S i) * d)) /\ exists w. z = w * S ((S i) * d) + p) /\ (((exists h. h + S q = S ((S S i) * d)) /\ exists w. z = w * S ((S S i) * d) + q) /\ (exists h. h + p = q))))
use lt_to_le
intro b
intro c
intro z
intro d
intro l
intro hsorted
intro htransport
intro i
intro hi
have hpq : exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))
specialize hsorted i
apply hsorted
exact hi
cases hpq
cases hpq_witness
cases hpq_witness_witness
cases hpq_witness_witness_right
have hii : exists h. h + S i = l
specialize lt_to_le (S i)
specialize lt_to_le l
apply lt_to_le
exact hi
exists x
exists x1
split
specialize htransport i
specialize htransport x
apply htransport
exact hii
exact hpq_witness_witness_left
split
specialize htransport (S i)
specialize htransport x1
apply htransport
exact hi
exact hpq_witness_witness_right_left
exact hpq_witness_witness_right_right
qed

Occurrences

89

Distinct objects

87

Reused references

2

Depth

21

Cuts

2

Certificate SHA-256

a9cf8c461f4f…

beta_prefix_extend_all_primeRecode a beta prefix and preserve AllPrime β sequences, products & FTA checked

Recode a beta prefix, preserve all old primes, and append one exact prime.

Prerequisites

beta_prefix_extend all_prime_transport all_prime_succ_intro

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall k b e s. (forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> exists z c. ((((exists h. h + S s = S ((S k) * c)) /\ exists w. z = w * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k b e s. (forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> exists z c. ((((exists h. h + S s = S ((S k) * c)) /\ exists w. z = w * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))))
use beta_prefix_extend
use all_prime_transport
use all_prime_succ_intro
intro k
intro b
intro e
intro s
intro hall
intro hs
have hext : exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists w. z = w * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a))
specialize beta_prefix_extend k
specialize beta_prefix_extend b
specialize beta_prefix_extend e
specialize beta_prefix_extend s
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
have hnew : (forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_transport b
specialize all_prime_transport e
specialize all_prime_transport x
specialize all_prime_transport x1
specialize all_prime_transport k
apply all_prime_transport
exact hall
exact hext_witness_witness_right
exists x
exists x1
split
split
exact hext_witness_witness_left
exact hext_witness_witness_right
specialize all_prime_succ_intro x
specialize all_prime_succ_intro x1
specialize all_prime_succ_intro k
specialize all_prime_succ_intro s
apply all_prime_succ_intro
exact hnew
split
exact hext_witness_witness_left
exact hs
qed

Occurrences

29,280

Distinct objects

4,617

Reused references

227

Depth

81

Cuts

875

Certificate SHA-256

c99644a6fbf8…

beta_prefix_extend_sorted_singletonRecode an empty prefix into a sorted singleton β sequences, products & FTA checked

Append the first value to an empty code; the resulting singleton is Sorted.

Prerequisites

beta_prefix_extend sorted_singleton

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b e s. exists z c. ((((exists h. h + S s = S ((S 0) * c)) /\ exists w. z = w * S ((S 0) * c) + s) /\ forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. z = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b e s. exists z c. ((((exists h. h + S s = S ((S 0) * c)) /\ exists w. z = w * S ((S 0) * c) + s) /\ forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. z = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))
use beta_prefix_extend
use sorted_singleton
intro b
intro e
intro s
have hext : exists z c. (((exists h. h + S s = S ((S 0) * c)) /\ exists w. z = w * S ((S 0) * c) + s) /\ forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a))
specialize beta_prefix_extend 0
specialize beta_prefix_extend b
specialize beta_prefix_extend e
specialize beta_prefix_extend s
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
exists x
exists x1
split
split
exact hext_witness_witness_left
exact hext_witness_witness_right
specialize sorted_singleton x
specialize sorted_singleton x1
exact sorted_singleton
qed

Occurrences

29,146

Distinct objects

4,551

Reused references

228

Depth

81

Cuts

873

Certificate SHA-256

5750ecab9c2a…

beta_prefix_extend_sorted_succRecode a nonempty prefix and preserve adjacent Sorted β sequences, products & FTA checked

Recode a nonempty Sorted prefix and append a value above its former last value.

Prerequisites

beta_prefix_extend sorted_transport sorted_succ_intro le_refl

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall l b e s p. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. b = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * e)) /\ exists w. b = w * S ((S l) * e) + p) -> (exists h. h + p = s) -> exists z c. ((((exists h. h + S s = S ((S S l) * c)) /\ exists w. z = w * S ((S S l) * c) + s) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. z = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l b e s p. (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. b = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * e)) /\ exists w. b = w * S ((S l) * e) + p) -> (exists h. h + p = s) -> exists z c. ((((exists h. h + S s = S ((S S l) * c)) /\ exists w. z = w * S ((S S l) * c) + s) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a)) /\ (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. z = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))
use beta_prefix_extend
use sorted_transport
use sorted_succ_intro
use le_refl
intro l
intro b
intro e
intro s
intro p
intro hsorted
intro hp
intro hps
have hext : exists z c. (((exists h. h + S s = S ((S S l) * c)) /\ exists w. z = w * S ((S S l) * c) + s) /\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. b = w * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. z = w * S ((S i) * c) + a))
specialize beta_prefix_extend (S l)
specialize beta_prefix_extend b
specialize beta_prefix_extend e
specialize beta_prefix_extend s
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
have hnewsorted : (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S q = S ((S S i) * x1)) /\ exists w. x = w * S ((S S i) * x1) + q) /\ (exists h. h + p = q))))
specialize sorted_transport b
specialize sorted_transport e
specialize sorted_transport x
specialize sorted_transport x1
specialize sorted_transport (S l)
apply sorted_transport
exact hsorted
exact hext_witness_witness_right
have hnewp : ((exists h. h + S p = S ((S l) * x1)) /\ exists w. x = w * S ((S l) * x1) + p)
specialize hext_witness_witness_right l
specialize hext_witness_witness_right p
apply hext_witness_witness_right
specialize le_refl (S l)
exact le_refl
exact hp
exists x
exists x1
split
split
exact hext_witness_witness_left
exact hext_witness_witness_right
specialize sorted_succ_intro x
specialize sorted_succ_intro x1
specialize sorted_succ_intro l
specialize sorted_succ_intro p
specialize sorted_succ_intro s
apply sorted_succ_intro
exact hnewsorted
exact hnewp
exact hext_witness_witness_left
exact hps
qed

Occurrences

29,414

Distinct objects

4,669

Reused references

230

Depth

81

Cuts

880

Certificate SHA-256

eb120eb89de0…

beta_canonical_append_emptyCanonical first-prime append β sequences, products & FTA checked

Append the first prime using the canonical shared-code factor/Product append helper.

Prerequisites

beta_factor_prefix_product_append all_prime_empty all_prime_succ_intro sorted_singleton one_mul

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c s. (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists z e. (((exists h. h + S s = S ((S 0) * e)) /\ exists w. z = w * S ((S 0) * e) + s) /\ ((forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S 1) * v)) /\ exists w. u = w * S ((S 1) * v) + s) /\ forall i. (exists h. h + S i = 1) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = 1) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c s. (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists z e. (((exists h. h + S s = S ((S 0) * e)) /\ exists w. z = w * S ((S 0) * e) + s) /\ ((forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S 1) * v)) /\ exists w. u = w * S ((S 1) * v) + s) /\ forall i. (exists h. h + S i = 1) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = 1) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = 1) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))
use beta_factor_prefix_product_append
use all_prime_empty
use all_prime_succ_intro
use sorted_singleton
use one_mul
intro b
intro c
intro s
intro hs
intro hproduct
have hext : exists z e. (((exists h. h + S s = S ((S 0) * e)) /\ exists w. z = w * S ((S 0) * e) + s) /\ ((forall i a. (exists h. h + S i = 0) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (1 * s) = S ((S 1) * v)) /\ exists w. u = w * S ((S 1) * v) + (1 * s)) /\ forall i. (exists h. h + S i = 1) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p))))))))
specialize beta_factor_prefix_product_append b
specialize beta_factor_prefix_product_append c
specialize beta_factor_prefix_product_append 0
specialize beta_factor_prefix_product_append 1
specialize beta_factor_prefix_product_append s
apply beta_factor_prefix_product_append
exact hproduct
cases hext
cases hext_witness
cases hext_witness_witness
cases hext_witness_witness_right
have hproduct_s : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S 1) * v)) /\ exists w. u = w * S ((S 1) * v) + s) /\ forall i. (exists h. h + S i = 1) -> exists p r s. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))
specialize one_mul s
rewrite one_mul at hext_witness_witness_right_right
rewrite one_mul at hext_witness_witness_right_right
exact hext_witness_witness_right_right
have hall : (forall i. (exists h. h + S i = 1) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_intro x
specialize all_prime_succ_intro x1
specialize all_prime_succ_intro 0
specialize all_prime_succ_intro s
apply all_prime_succ_intro
specialize all_prime_empty x
specialize all_prime_empty x1
exact all_prime_empty
split
exact hext_witness_witness_left
exact hs
exists x
exists x1
split
exact hext_witness_witness_left
split
exact hext_witness_witness_right_left
split
exact hproduct_s
split
exact hall
specialize sorted_singleton x
specialize sorted_singleton x1
exact sorted_singleton
qed

Occurrences

29,783

Distinct objects

4,876

Reused references

237

Depth

82

Cuts

894

Certificate SHA-256

e89975bab8db…

beta_canonical_append_succCanonical ordered-prime append β sequences, products & FTA checked

Append one ordered prime using the canonical shared-code factor/Product append helper.

Prerequisites

beta_factor_prefix_product_append all_prime_transport all_prime_succ_intro sorted_transport sorted_succ_intro le_refl

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall l b c n s p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> (exists h. h + p = s) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> exists z e. (((exists h. h + S s = S ((S S l) * e)) /\ exists w. z = w * S ((S S l) * e) + s) /\ ((forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S (S l)) * v)) /\ exists w. u = w * S ((S S (S l)) * v) + (n * s)) /\ forall i. (exists h. h + S i = S (S l)) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = S (S l)) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l b c n s p. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> (exists h. h + p = s) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> exists z e. (((exists h. h + S s = S ((S S l) * e)) /\ exists w. z = w * S ((S S l) * e) + s) /\ ((forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S (S l)) * v)) /\ exists w. u = w * S ((S S (S l)) * v) + (n * s)) /\ forall i. (exists h. h + S i = S (S l)) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = S (S l)) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))
use beta_factor_prefix_product_append
use all_prime_transport
use all_prime_succ_intro
use sorted_transport
use sorted_succ_intro
use le_refl
intro l
intro b
intro c
intro n
intro s
intro p
intro hproduct
intro hall
intro hsorted
intro hp
intro hps
intro hs
have hext : exists z e. (((exists h. h + S s = S ((S S l) * e)) /\ exists w. z = w * S ((S S l) * e) + s) /\ ((forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S (S l)) * v)) /\ exists w. u = w * S ((S S (S l)) * v) + (n * s)) /\ forall i. (exists h. h + S i = S (S l)) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p))))))))
specialize beta_factor_prefix_product_append b
specialize beta_factor_prefix_product_append c
specialize beta_factor_prefix_product_append (S l)
specialize beta_factor_prefix_product_append n
specialize beta_factor_prefix_product_append s
apply beta_factor_prefix_product_append
exact hproduct
cases hext
cases hext_witness
cases hext_witness_witness
cases hext_witness_witness_right
have hnew_all_prefix : (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_transport b
specialize all_prime_transport c
specialize all_prime_transport x
specialize all_prime_transport x1
specialize all_prime_transport (S l)
apply all_prime_transport
exact hall
exact hext_witness_witness_right_left
have hnew_all : (forall i. (exists h. h + S i = S (S l)) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_intro x
specialize all_prime_succ_intro x1
specialize all_prime_succ_intro (S l)
specialize all_prime_succ_intro s
apply all_prime_succ_intro
exact hnew_all_prefix
split
exact hext_witness_witness_left
exact hs
have hnew_sorted_prefix : (forall i. (exists h. h + S (S i) = S l) -> exists p q. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S q = S ((S S i) * x1)) /\ exists w. x = w * S ((S S i) * x1) + q) /\ (exists h. h + p = q))))
specialize sorted_transport b
specialize sorted_transport c
specialize sorted_transport x
specialize sorted_transport x1
specialize sorted_transport (S l)
apply sorted_transport
exact hsorted
exact hext_witness_witness_right_left
have hnewp : ((exists h. h + S p = S ((S l) * x1)) /\ exists w. x = w * S ((S l) * x1) + p)
specialize hext_witness_witness_right_left l
specialize hext_witness_witness_right_left p
apply hext_witness_witness_right_left
specialize le_refl (S l)
exact le_refl
exact hp
have hnew_sorted : (forall i. (exists h. h + S (S i) = S (S l)) -> exists p q. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S q = S ((S S i) * x1)) /\ exists w. x = w * S ((S S i) * x1) + q) /\ (exists h. h + p = q))))
specialize sorted_succ_intro x
specialize sorted_succ_intro x1
specialize sorted_succ_intro l
specialize sorted_succ_intro p
specialize sorted_succ_intro s
apply sorted_succ_intro
exact hnew_sorted_prefix
exact hnewp
exact hext_witness_witness_left
exact hps
exists x
exists x1
split
exact hext_witness_witness_left
split
exact hext_witness_witness_right_left
split
exact hext_witness_witness_right_right
split
exact hnew_all
exact hnew_sorted
qed

Occurrences

30,020

Distinct objects

5,009

Reused references

236

Depth

82

Cuts

898

Certificate SHA-256

618288aae5fa…

prime_divides_decidablePrime divisibility is constructively decidable β sequences, products & FTA checked

The concrete property of being a prime divisor is constructively decidable.

Prerequisites

prime_decidable multiple_decidable

Used directly by

greatest_prime_divisor_search

Exact expanded PA statement

Copy statement

forall p n. ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) \/ ~(((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p n. ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) \/ ~(((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)))
use prime_decidable
use multiple_decidable
intro p
intro n
have hp : (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) \/ ~((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))
specialize prime_decidable p
exact prime_decidable
cases hp
have hd : (exists k. n = p * k) \/ ~((exists k. n = p * k))
specialize multiple_decidable p
specialize multiple_decidable n
exact multiple_decidable
cases hd
left
split
exact hp_left
exact hd_left
right
intro hpd
cases hpd
apply hd_right
exact hpd_right
right
intro hpd
cases hpd
apply hp_right
exact hpd_left
qed

Occurrences

3,573

Distinct objects

1,539

Reused references

66

Depth

74

Cuts

101

Certificate SHA-256

44c96dfc214f…

greatest_prime_divisor_searchBounded greatest-prime-divisor search β sequences, products & FTA checked

Bounded search either excludes all prime divisors or returns a greatest one in the bound.

Prerequisites

prime_nonzero le_zero prime_divides_decidable le_refl le_eq_or_lt le_of_succ_le_succ le_succ

Used directly by

greatest_prime_divisor_exists

Exact expanded PA statement

Copy statement

forall B n. ((forall r. (exists h. h + r = B) -> ~(((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)))) \/ exists p. (((exists h. h + p = B) /\ ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k))) /\ forall r. (exists h. h + r = B) -> ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B n. ((forall r. (exists h. h + r = B) -> ~(((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)))) \/ exists p. (((exists h. h + p = B) /\ ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k))) /\ forall r. (exists h. h + r = B) -> ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)))
use prime_nonzero
use le_zero
use prime_divides_decidable
use le_refl
use le_eq_or_lt
use le_of_succ_le_succ
use le_succ
induction B
intro n
left
intro r
intro hr
intro hpd
cases hpd
have hr0 : r = 0
specialize le_zero r
apply le_zero
exact hr
specialize prime_nonzero r
apply prime_nonzero
exact hpd_left
exact hr0
intro n
have hboundary : ((~(S B = 1) /\ forall a d. S B = a * d -> a = 1 \/ d = 1) /\ (exists k. n = S B * k)) \/ ~(((~(S B = 1) /\ forall a d. S B = a * d -> a = 1 \/ d = 1) /\ (exists k. n = S B * k)))
specialize prime_divides_decidable (S B)
specialize prime_divides_decidable n
exact prime_divides_decidable
cases hboundary
right
exists S B
split
split
specialize le_refl (S B)
exact le_refl
exact hboundary_left
intro r
intro hr
intro hpd
exact hr
have hprev : ((forall r. (exists h. h + r = B) -> ~(((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)))) \/ exists p. (((exists h. h + p = B) /\ ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k))) /\ forall r. (exists h. h + r = B) -> ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)))
specialize IH n
exact IH
cases hprev
left
intro r
intro hr
intro hpd
have hsplit : r = S B \/ exists h. h + S r = S B
specialize le_eq_or_lt r
specialize le_eq_or_lt (S B)
apply le_eq_or_lt
exact hr
cases hsplit
apply hboundary_right
rewrite hsplit_left at hpd
rewrite hsplit_left at hpd
rewrite hsplit_left at hpd
exact hpd
specialize hprev_left r
apply hprev_left
specialize le_of_succ_le_succ r
specialize le_of_succ_le_succ B
apply le_of_succ_le_succ
exact hsplit_right
exact hpd
right
cases hprev_right
cases hprev_right_witness
cases hprev_right_witness_left
exists x
split
split
specialize le_succ x
specialize le_succ B
apply le_succ
exact hprev_right_witness_left_left
exact hprev_right_witness_left_right
intro r
intro hr
intro hpd
have hsplit2 : r = S B \/ exists h. h + S r = S B
specialize le_eq_or_lt r
specialize le_eq_or_lt (S B)
apply le_eq_or_lt
exact hr
cases hsplit2
exfalso
apply hboundary_right
rewrite hsplit2_left at hpd
rewrite hsplit2_left at hpd
rewrite hsplit2_left at hpd
exact hpd
specialize hprev_right_witness_right r
apply hprev_right_witness_right
specialize le_of_succ_le_succ r
specialize le_of_succ_le_succ B
apply le_of_succ_le_succ
exact hsplit2_right
exact hpd
qed

Occurrences

3,949

Distinct objects

1,658

Reused references

72

Depth

77

Cuts

116

Certificate SHA-256

6ea4e6870350…

greatest_prime_divisor_existsA greatest prime divisor exists β sequences, products & FTA checked

Every nonzero nonunit has a greatest prime divisor in the ordinary natural order.

Prerequisites

prime_divisor_exists divisor_le_nonzero greatest_prime_divisor_search

Used directly by

greatest_prime_divisor_descent

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> ~(n = 1) -> exists p. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) /\ forall r. ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> ~(n = 1) -> exists p. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) /\ forall r. ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p))
use prime_divisor_exists
use divisor_le_nonzero
use greatest_prime_divisor_search
intro n
intro hn0
intro hn1
have hsome : exists p. ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k))
specialize prime_divisor_exists n
apply prime_divisor_exists
exact hn0
exact hn1
cases hsome
cases hsome_witness
have hxle : (exists h. h + x = n)
specialize divisor_le_nonzero x
specialize divisor_le_nonzero n
apply divisor_le_nonzero
exact hn0
exact hsome_witness_right
have hsearch : ((forall r. (exists h. h + r = n) -> ~(((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)))) \/ exists p. (((exists h. h + p = n) /\ ((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k))) /\ forall r. (exists h. h + r = n) -> ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)))
specialize greatest_prime_divisor_search n
specialize greatest_prime_divisor_search n
exact greatest_prime_divisor_search
cases hsearch
exfalso
specialize hsearch_left x
apply hsearch_left
exact hxle
split
exact hsome_witness_left
exact hsome_witness_right
cases hsearch_right
cases hsearch_right_witness
cases hsearch_right_witness_left
exists x1
split
exact hsearch_right_witness_left_right
intro r
intro hrpd
cases hrpd
specialize hsearch_right_witness_right r
apply hsearch_right_witness_right
specialize divisor_le_nonzero r
specialize divisor_le_nonzero n
apply divisor_le_nonzero
exact hn0
exact hrpd_right
split
exact hrpd_left
exact hrpd_right
qed

Occurrences

7,052

Distinct objects

2,015

Reused references

92

Depth

81

Cuts

214

Certificate SHA-256

4ee7dca9d7ad…

greatest_prime_divisor_quotient_boundPrime divisors of the quotient are bounded β sequences, products & FTA checked

Every prime divisor of the quotient by a greatest prime divisor is bounded by that divisor.

Prerequisites

mul_comm multiple_trans

Used directly by

greatest_prime_divisor_descent

Exact expanded PA statement

Copy statement

forall n p q. n = p * q -> (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) /\ forall r. ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)) -> forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. q = r * k) -> (exists h. h + r = p)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n p q. n = p * q -> (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) /\ forall r. ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p)) -> forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. q = r * k) -> (exists h. h + r = p)
use mul_comm
use multiple_trans
intro n
intro p
intro q
intro hfactor
intro hgreat
intro r
intro hrprime
intro hrdq
cases hgreat
have hqdn : (exists k. n = q * k)
exists p
trans p * q
exact hfactor
apply mul_comm
have hrdn : (exists k. n = r * k)
specialize multiple_trans q
specialize multiple_trans r
specialize multiple_trans n
apply multiple_trans
exact hqdn
exact hrdq
specialize hgreat_right r
apply hgreat_right
split
exact hrprime
exact hrdn
qed

Occurrences

388

Distinct objects

322

Reused references

34

Depth

25

Cuts

11

Certificate SHA-256

3c331ed30837…

greatest_prime_divisor_descentGreatest-prime-divisor quotient descent β sequences, products & FTA checked

Choose a greatest prime factor with a nonzero strict quotient and the exact append-order bound.

Prerequisites

greatest_prime_divisor_exists mul_comm factor_nonzero_left proper_factor_lt greatest_prime_divisor_quotient_bound

Used directly by

prime_factorization_exists_up_to

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> ~(n = 1) -> exists p q. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ n = p * q) /\ (~(q = 0) /\ ((exists h. h + S q = n) /\ forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. q = r * k) -> (exists h. h + r = p))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> ~(n = 1) -> exists p q. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ n = p * q) /\ (~(q = 0) /\ ((exists h. h + S q = n) /\ forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. q = r * k) -> (exists h. h + r = p))))
use greatest_prime_divisor_exists
use mul_comm
use factor_nonzero_left
use proper_factor_lt
use greatest_prime_divisor_quotient_bound
intro n
intro hn0
intro hn1
have hgreat : exists p. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ (exists k. n = p * k)) /\ forall r. ((~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) /\ (exists k. n = r * k)) -> (exists h. h + r = p))
specialize greatest_prime_divisor_exists n
apply greatest_prime_divisor_exists
exact hn0
exact hn1
cases hgreat
cases hgreat_witness
cases hgreat_witness_left
cases hgreat_witness_left_left
cases hgreat_witness_left_right
have hqp : n = x1 * x
trans x * x1
exact hgreat_witness_left_right_witness
apply mul_comm
have hq0 : ~(x1 = 0)
intro hx10
specialize factor_nonzero_left n
specialize factor_nonzero_left x1
specialize factor_nonzero_left x
apply factor_nonzero_left
exact hn0
exact hqp
exact hx10
have hqlt : (exists h. h + S x1 = n)
specialize proper_factor_lt n
specialize proper_factor_lt x1
specialize proper_factor_lt x
apply proper_factor_lt
exact hn0
exact hqp
exact hgreat_witness_left_left_left
have hbound : forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. x1 = r * k) -> (exists h. h + r = x)
specialize greatest_prime_divisor_quotient_bound n
specialize greatest_prime_divisor_quotient_bound x
specialize greatest_prime_divisor_quotient_bound x1
apply greatest_prime_divisor_quotient_bound
exact hgreat_witness_left_right_witness
split
split
split
exact hgreat_witness_left_left_left
exact hgreat_witness_left_left_right
exists x1
exact hgreat_witness_left_right_witness
exact hgreat_witness_right
exists x
exists x1
split
split
split
exact hgreat_witness_left_left_left
exact hgreat_witness_left_left_right
exact hgreat_witness_left_right_witness
split
exact hq0
split
exact hqlt
exact hbound
qed

Occurrences

8,256

Distinct objects

2,234

Reused references

109

Depth

82

Cuts

253

Certificate SHA-256

136a861b3f30…

beta_factor_divides_productEvery decoded factor divides its Product β sequences, products & FTA checked

Every decoded factor inside an exact beta Product divides its terminal product.

Prerequisites

add_eq_zero_right succ_ne_zero beta_product_succ_decompose le_of_succ_le_succ le_eq_or_lt beta_at_unique mul_comm multiple_mul_right

Used directly by

beta_canonical_append_general beta_canonical_last_factor_bound beta_nonempty_all_prime_product_ne_one beta_canonical_last_factors_equal

Exact expanded PA statement

Copy statement

forall b c l n i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists q. n = p * q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n i p. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists q. n = p * q
use add_eq_zero_right
use succ_ne_zero
use beta_product_succ_decompose
use le_of_succ_le_succ
use le_eq_or_lt
use beta_at_unique
use mul_comm
use multiple_mul_right
intro b
intro c
induction l
intro n
intro i
intro p
intro hi
intro hp
intro hproduct
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
intro n
intro i
intro p
intro hi
intro hp
intro hproduct
have hdecomp : exists a r. (((exists h. h + S a = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + a) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = r * a))
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose l
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hproduct
cases hdecomp
cases hdecomp_witness
cases hdecomp_witness_witness
cases hdecomp_witness_witness_right
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
have hpa : p = x
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique p
specialize beta_at_unique x
apply beta_at_unique
rewrite hsplit_left at hp
rewrite hsplit_left at hp
exact hp
exact hdecomp_witness_witness_left
exists x1
trans x1 * x
exact hdecomp_witness_witness_right_right
rewrite hpa
apply mul_comm
have hidiv : exists q. x1 = p * q
specialize IH x1
specialize IH i
specialize IH p
apply IH
exact hsplit_right
exact hp
exact hdecomp_witness_witness_right_left
have hmul : exists q. x1 * x = p * q
specialize multiple_mul_right p
specialize multiple_mul_right x1
specialize multiple_mul_right x
apply multiple_mul_right
exact hidiv
cases hmul
exists x2
trans x1 * x
exact hdecomp_witness_witness_right_right
exact hmul_witness
qed

Occurrences

2,970

Distinct objects

1,020

Reused references

50

Depth

65

Cuts

85

Certificate SHA-256

1584e62c8f35…

beta_canonical_append_generalAppend one prime to any canonical beta prefix β sequences, products & FTA checked

Append one prime to any canonical prefix using one shared beta append certificate.

Prerequisites

beta_factor_prefix_product_append all_prime_transport all_prime_succ_intro zero_or_succ sorted_singleton sorted_transport all_prime_succ_elim_last beta_factor_divides_product le_refl sorted_succ_intro

Used directly by

prime_factorization_exists_up_to

Exact expanded PA statement

Copy statement

forall l b c n s. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> (forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. n = r * k) -> (exists h. h + r = s)) -> exists z e. (((exists h. h + S s = S ((S l) * e)) /\ exists w. z = w * S ((S l) * e) + s) /\ ((forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + (n * s)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l b c n s. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (~(s = 1) /\ forall a d. s = a * d -> a = 1 \/ d = 1) -> (forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. n = r * k) -> (exists h. h + r = s)) -> exists z e. (((exists h. h + S s = S ((S l) * e)) /\ exists w. z = w * S ((S l) * e) + s) /\ ((forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + (n * s)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))
use beta_factor_prefix_product_append
use all_prime_transport
use all_prime_succ_intro
use zero_or_succ
use sorted_singleton
use sorted_transport
use all_prime_succ_elim_last
use beta_factor_divides_product
use le_refl
use sorted_succ_intro
intro l
intro b
intro c
intro n
intro s
intro hproduct
intro hall
intro hsorted
intro hs
intro hbound
have hext : exists z e. (((exists h. h + S s = S ((S l) * e)) /\ exists w. z = w * S ((S l) * e) + s) /\ ((forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (n * s) = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + (n * s)) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p))))))))
specialize beta_factor_prefix_product_append b
specialize beta_factor_prefix_product_append c
specialize beta_factor_prefix_product_append l
specialize beta_factor_prefix_product_append n
specialize beta_factor_prefix_product_append s
apply beta_factor_prefix_product_append
exact hproduct
cases hext
cases hext_witness
cases hext_witness_witness
cases hext_witness_witness_right
have hnew_all_prefix : (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_transport b
specialize all_prime_transport c
specialize all_prime_transport x
specialize all_prime_transport x1
specialize all_prime_transport l
apply all_prime_transport
exact hall
exact hext_witness_witness_right_left
have hnew_all : (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_intro x
specialize all_prime_succ_intro x1
specialize all_prime_succ_intro l
specialize all_prime_succ_intro s
apply all_prime_succ_intro
exact hnew_all_prefix
split
exact hext_witness_witness_left
exact hs
have hnew_sorted_prefix : (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S q = S ((S S i) * x1)) /\ exists w. x = w * S ((S S i) * x1) + q) /\ (exists h. h + p = q))))
specialize sorted_transport b
specialize sorted_transport c
specialize sorted_transport x
specialize sorted_transport x1
specialize sorted_transport l
apply sorted_transport
exact hsorted
exact hext_witness_witness_right_left
have hlen : l = 0 \/ exists k. l = S k
specialize zero_or_succ l
exact zero_or_succ
have hnew_sorted : (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * x1)) /\ exists w. x = w * S ((S i) * x1) + p) /\ (((exists h. h + S q = S ((S S i) * x1)) /\ exists w. x = w * S ((S S i) * x1) + q) /\ (exists h. h + p = q))))
cases hlen
rewrite hlen_left
specialize sorted_singleton x
specialize sorted_singleton x1
exact sorted_singleton
cases hlen_right
rewrite hlen_right_witness
rewrite hlen_right_witness at hall
rewrite hlen_right_witness at hnew_sorted_prefix
rewrite hlen_right_witness at hproduct
rewrite hlen_right_witness at hproduct
rewrite hlen_right_witness at hproduct
have hlast : exists r. (((exists h. h + S r = S ((S x2) * c)) /\ exists w. b = w * S ((S x2) * c) + r) /\ (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last x2
apply all_prime_succ_elim_last
exact hall
cases hlast
cases hlast_witness
have hrdiv : (exists k. n = x3 * k)
specialize beta_factor_divides_product b
specialize beta_factor_divides_product c
specialize beta_factor_divides_product (S x2)
specialize beta_factor_divides_product n
specialize beta_factor_divides_product x2
specialize beta_factor_divides_product x3
apply beta_factor_divides_product
specialize le_refl (S x2)
exact le_refl
exact hlast_witness_left
exact hproduct
have hrs : (exists h. h + x3 = s)
specialize hbound x3
apply hbound
exact hlast_witness_right
exact hrdiv
have hklt : (exists h. h + S x2 = l)
rewrite hlen_right_witness
specialize le_refl (S x2)
exact le_refl
have hnew_last : ((exists h. h + S x3 = S ((S x2) * x1)) /\ exists w. x = w * S ((S x2) * x1) + x3)
specialize hext_witness_witness_right_left x2
specialize hext_witness_witness_right_left x3
apply hext_witness_witness_right_left
exact hklt
exact hlast_witness_left
have hnew_append : ((exists h. h + S s = S ((S S x2) * x1)) /\ exists w. x = w * S ((S S x2) * x1) + s)
rewrite <- hlen_right_witness
rewrite <- hlen_right_witness
exact hext_witness_witness_left
specialize sorted_succ_intro x
specialize sorted_succ_intro x1
specialize sorted_succ_intro x2
specialize sorted_succ_intro x3
specialize sorted_succ_intro s
apply sorted_succ_intro
exact hnew_sorted_prefix
exact hnew_last
exact hnew_append
exact hrs
exists x
exists x1
split
exact hext_witness_witness_left
split
exact hext_witness_witness_right_left
split
exact hext_witness_witness_right_right
split
exact hnew_all
exact hnew_sorted
qed

Occurrences

33,165

Distinct objects

5,281

Reused references

251

Depth

82

Cuts

993

Certificate SHA-256

72e9862443ac…

beta_canonical_last_factor_boundThe last canonical factor obeys prime-divisor bounds β sequences, products & FTA checked

The last factor of a nonempty AllPrime Product obeys any bound on prime divisors of the product.

Prerequisites

all_prime_succ_elim_last beta_factor_divides_product le_refl

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l n s. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. n = r * k) -> (exists h. h + r = s)) -> exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (exists h. h + p = s))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n s. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists q. u = q * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. n = r * k) -> (exists h. h + r = s)) -> exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (exists h. h + p = s))
use all_prime_succ_elim_last
use beta_factor_divides_product
use le_refl
intro b
intro c
intro l
intro n
intro s
intro hall
intro hproduct
intro hbound
have hlast : exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last l
apply all_prime_succ_elim_last
exact hall
cases hlast
cases hlast_witness
have hpdiv : (exists k. n = x * k)
specialize beta_factor_divides_product b
specialize beta_factor_divides_product c
specialize beta_factor_divides_product (S l)
specialize beta_factor_divides_product n
specialize beta_factor_divides_product l
specialize beta_factor_divides_product x
apply beta_factor_divides_product
specialize le_refl (S l)
exact le_refl
exact hlast_witness_left
exact hproduct
have hps : (exists h. h + x = s)
specialize hbound x
apply hbound
exact hlast_witness_right
exact hpdiv
exists x
split
exact hlast_witness_left
exact hps
qed

Occurrences

3,079

Distinct objects

1,079

Reused references

52

Depth

67

Cuts

91

Certificate SHA-256

0c23f5d9bd89…

prime_factorization_exists_up_toBounded prime-factorization existence β sequences, products & FTA checked

Bounded induction with one consolidated canonical append dependency.

Prerequisites

le_zero le_eq_or_lt le_of_succ_le_succ eq_decidable succ_ne_zero beta_at_exists beta_at_self_of_bound beta_at_unique one_mul le_refl add_eq_zero_right all_prime_empty sorted_empty greatest_prime_divisor_descent beta_canonical_append_general mul_comm

Used directly by

prime_factorization_existence

Exact expanded PA statement

Copy statement

forall B n. (exists h. h + n = B) -> ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B n. (exists h. h + n = B) -> ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))
use le_zero
use le_eq_or_lt
use le_of_succ_le_succ
use eq_decidable
use succ_ne_zero
use beta_at_exists
use beta_at_self_of_bound
use beta_at_unique
use one_mul
use le_refl
use add_eq_zero_right
use all_prime_empty
use sorted_empty
use greatest_prime_divisor_descent
use beta_canonical_append_general
use mul_comm
induction B
intro n
intro hnB
intro hn0
exfalso
apply hn0
specialize le_zero n
apply le_zero
exact hnB
intro n
intro hnB
intro hn0
have hsplit : n = S B \/ exists h. h + S n = S B
specialize le_eq_or_lt n
specialize le_eq_or_lt (S B)
apply le_eq_or_lt
exact hnB
cases hsplit
rewrite hsplit_left
rewrite hsplit_left
have honecase : S B = 1 \/ ~(S B = 1)
specialize eq_decidable (S B)
specialize eq_decidable 1
apply eq_decidable
cases honecase
rewrite honecase_left
rewrite honecase_left
have hsome : exists x. ((exists h. h + S x = S ((S 0) * 1)) /\ exists w. 1 = w * S ((S 0) * 1) + x)
specialize beta_at_exists 1
specialize beta_at_exists 1
specialize beta_at_exists 0
exact beta_at_exists
cases hsome
have hone : ((exists h. h + S 1 = S ((S 0) * 1)) /\ exists w. 1 = w * S ((S 0) * 1) + 1)
specialize beta_at_self_of_bound 1
specialize beta_at_self_of_bound 0
specialize beta_at_self_of_bound 1
apply beta_at_self_of_bound
specialize one_mul 1
rewrite one_mul
specialize le_refl 2
exact le_refl
have hx1 : x = 1
specialize beta_at_unique 1
specialize beta_at_unique 1
specialize beta_at_unique 0
specialize beta_at_unique x
specialize beta_at_unique 1
apply beta_at_unique
exact hsome_witness
exact hone
rewrite hx1 at hsome_witness
rewrite hx1 at hsome_witness
exists 0
exists 0
exists 0
split
exists 1
exists 1
split
exact hsome_witness
split
exact hsome_witness
intro i
intro hi
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x1
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
split
specialize all_prime_empty 0
specialize all_prime_empty 0
exact all_prime_empty
specialize sorted_empty 0
specialize sorted_empty 0
exact sorted_empty
have hdesc : exists p q. (((~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) /\ S B = p * q) /\ (~(q = 0) /\ ((exists h. h + S q = S B) /\ forall r. (~(r = 1) /\ forall a d. r = a * d -> a = 1 \/ d = 1) -> (exists k. q = r * k) -> (exists h. h + r = p))))
specialize greatest_prime_divisor_descent (S B)
apply greatest_prime_divisor_descent
specialize succ_ne_zero B
exact succ_ne_zero
exact honecase_right
cases hdesc
cases hdesc_witness
cases hdesc_witness_witness
cases hdesc_witness_witness_left
cases hdesc_witness_witness_right
cases hdesc_witness_witness_right_right
have hqB : exists h. h + x1 = B
specialize le_of_succ_le_succ x1
specialize le_of_succ_le_succ B
apply le_of_succ_le_succ
exact hdesc_witness_witness_right_right_left
have hqfac : exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S x1 = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + x1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))
specialize IH x1
apply IH
exact hqB
exact hdesc_witness_witness_right_left
cases hqfac
cases hqfac_witness
cases hqfac_witness_witness
cases hqfac_witness_witness_witness
cases hqfac_witness_witness_witness_right
have happend : exists z e. (((exists h. h + S x = S ((S x2) * e)) /\ exists w. z = w * S ((S x2) * e) + x) /\ ((forall i a. (exists h. h + S i = x2) -> ((exists h. h + S a = S ((S i) * x4)) /\ exists w. x3 = w * S ((S i) * x4) + a) -> ((exists h. h + S a = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + a)) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (x1 * x) = S ((S S x2) * v)) /\ exists w. u = w * S ((S S x2) * v) + (x1 * x)) /\ forall i. (exists h. h + S i = S x2) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = S x2) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = S x2) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))))
specialize beta_canonical_append_general x2
specialize beta_canonical_append_general x3
specialize beta_canonical_append_general x4
specialize beta_canonical_append_general x1
specialize beta_canonical_append_general x
apply beta_canonical_append_general
exact hqfac_witness_witness_witness_left
exact hqfac_witness_witness_witness_right_left
exact hqfac_witness_witness_witness_right_right
exact hdesc_witness_witness_left_left
exact hdesc_witness_witness_right_right_right
cases happend
cases happend_witness
cases happend_witness_witness
cases happend_witness_witness_right
cases happend_witness_witness_right_right
cases happend_witness_witness_right_right_right
have hmul : x1 * x = S B
trans x * x1
apply mul_comm
symm
exact hdesc_witness_witness_left_right
have hcap : exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S (x1 * x) = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + (x1 * x)) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))
exists S x2
exists x5
exists x6
split
exact happend_witness_witness_right_right_left
split
exact happend_witness_witness_right_right_right_left
exact happend_witness_witness_right_right_right_right
rewrite hmul at hcap
rewrite hmul at hcap
exact hcap
have hnB0 : exists h. h + n = B
specialize le_of_succ_le_succ n
specialize le_of_succ_le_succ B
apply le_of_succ_le_succ
exact hsplit_right
specialize IH n
apply IH
exact hnB0
exact hn0
qed

Occurrences

43,927

Distinct objects

6,780

Reused references

319

Depth

97

Cuts

1,325

Certificate SHA-256

ba5730d9fb3e…

prime_factorization_existenceExistence of a finite prime factorization β sequences, products & FTA checked

Every nonzero natural has a canonical factorization using consolidated append.

Prerequisites

prime_factorization_exists_up_to le_refl

Used directly by

fundamental_theorem_of_arithmetic

Exact expanded PA statement

Copy statement

forall n. ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))
use prime_factorization_exists_up_to
use le_refl
intro n
intro hn0
specialize prime_factorization_exists_up_to n
specialize prime_factorization_exists_up_to n
apply prime_factorization_exists_up_to
specialize le_refl n
exact le_refl
exact hn0
qed

Occurrences

43,973

Distinct objects

6,801

Reused references

320

Depth

98

Cuts

1,328

Certificate SHA-256

f25aa6adb2b3…

beta_prime_divisor_product_memberA prime divisor of an AllPrime beta Product occurs as a factor β sequences, products & FTA checked

A prime divisor of an AllPrime beta Product occurs as one of its decoded factors.

Prerequisites

beta_product_zero divisor_one beta_product_succ_decompose all_prime_succ_elim_prefix all_prime_succ_elim_last euclid_prime_dvd_product prime_divisor_eq_one_or_self beta_at_unique le_succ le_refl

Used directly by

beta_canonical_last_factors_equal

Exact expanded PA statement

Copy statement

forall b c l n p. (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (exists k. n = p * k) -> exists i. ((exists h. h + S i = l) /\ ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n p. (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1) -> (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (exists k. n = p * k) -> exists i. ((exists h. h + S i = l) /\ ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p))
use beta_product_zero
use divisor_one
use beta_product_succ_decompose
use all_prime_succ_elim_prefix
use all_prime_succ_elim_last
use euclid_prime_dvd_product
use prime_divisor_eq_one_or_self
use beta_at_unique
use le_succ
use le_refl
intro b
intro c
induction l
intro n
intro p
intro hpprime
intro hall
intro hproduct
intro hpdiv
have hn1 : n = 1
specialize beta_product_zero b
specialize beta_product_zero c
specialize beta_product_zero n
apply beta_product_zero
exact hproduct
cases hpprime
exfalso
apply hpprime_left
have hp1 : p = 1
specialize divisor_one p
apply divisor_one
rewrite hn1 at hpdiv
exact hpdiv
exact hp1
intro n
intro p
intro hpprime
intro hall
intro hproduct
intro hpdiv
have hdecomp : exists a r. (((exists h. h + S a = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + a) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = r * a))
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose l
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hproduct
cases hdecomp
cases hdecomp_witness
cases hdecomp_witness_witness
cases hdecomp_witness_witness_right
have hprefix : (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_elim_prefix b
specialize all_prime_succ_elim_prefix c
specialize all_prime_succ_elim_prefix l
apply all_prime_succ_elim_prefix
exact hall
have hlast : exists t. (((exists h. h + S t = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + t) /\ (~(t = 1) /\ forall a d. t = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last l
apply all_prime_succ_elim_last
exact hall
have hpd : (exists k. x1 * x = p * k)
rewrite hdecomp_witness_witness_right_right at hpdiv
exact hpdiv
have hsplit : (exists k. x1 = p * k) \/ (exists k. x = p * k)
specialize euclid_prime_dvd_product p
specialize euclid_prime_dvd_product x1
specialize euclid_prime_dvd_product x
apply euclid_prime_dvd_product
exact hpprime
exact hpd
cases hsplit
have hmember : exists i. ((exists h. h + S i = l) /\ ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p))
specialize IH x1
specialize IH p
apply IH
exact hpprime
exact hprefix
exact hdecomp_witness_witness_right_left
exact hsplit_left
cases hmember
cases hmember_witness
exists x2
split
specialize le_succ (S x2)
specialize le_succ l
apply le_succ
exact hmember_witness_left
exact hmember_witness_right
cases hlast
cases hlast_witness
have hlast_eq : x = x2
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique x2
apply beta_at_unique
exact hdecomp_witness_witness_left
exact hlast_witness_left
have hxprime : (~(x = 1) /\ forall a d. x = a * d -> a = 1 \/ d = 1)
rewrite hlast_eq
rewrite hlast_eq
exact hlast_witness_right
have hpone_or_eq : p = 1 \/ x = p
specialize prime_divisor_eq_one_or_self x
specialize prime_divisor_eq_one_or_self p
apply prime_divisor_eq_one_or_self
exact hxprime
exact hsplit_right
cases hpone_or_eq
exfalso
cases hpprime
apply hpprime_left
exact hpone_or_eq_left
exists l
split
specialize le_refl (S l)
exact le_refl
rewrite hpone_or_eq_right at hdecomp_witness_witness_left
rewrite hpone_or_eq_right at hdecomp_witness_witness_left
exact hdecomp_witness_witness_left
qed

Occurrences

9,499

Distinct objects

2,538

Reused references

135

Depth

67

Cuts

277

Certificate SHA-256

7723a9186a8a…

beta_sorted_factor_le_lastEvery sorted factor is bounded by the last factor β sequences, products & FTA checked

Every factor in a nonempty adjacent-sorted beta prefix is at most its last factor.

Prerequisites

le_of_succ_le_succ le_zero beta_at_unique le_refl le_eq_or_lt sorted_succ_elim_prefix sorted_succ_elim_last le_trans

Used directly by

beta_canonical_last_factors_equal

Exact expanded PA statement

Copy statement

forall b c l i p q. (exists h. h + S i = S l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + q) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists h. h + p = q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l i p q. (exists h. h + S i = S l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + q) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists h. h + p = q)
use le_of_succ_le_succ
use le_zero
use beta_at_unique
use le_refl
use le_eq_or_lt
use sorted_succ_elim_prefix
use sorted_succ_elim_last
use le_trans
intro b
intro c
induction l
intro i
intro p
intro q
intro hi
intro hip
intro hql
intro hsorted
have hile0 : (exists h. h + i = 0)
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ 0
apply le_of_succ_le_succ
exact hi
have hi0 : i = 0
specialize le_zero i
apply le_zero
exact hile0
have hpq : p = q
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique 0
specialize beta_at_unique p
specialize beta_at_unique q
apply beta_at_unique
rewrite hi0 at hip
rewrite hi0 at hip
exact hip
exact hql
rewrite hpq
specialize le_refl q
exact le_refl
intro i
intro p
intro q
intro hi
intro hip
intro hql
intro hsorted
have hile : (exists h. h + i = S l)
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ (S l)
apply le_of_succ_le_succ
exact hi
have hsplit : i = S l \/ (exists h. h + S i = S l)
specialize le_eq_or_lt i
specialize le_eq_or_lt (S l)
apply le_eq_or_lt
exact hile
cases hsplit
have hpq : p = q
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique (S l)
specialize beta_at_unique p
specialize beta_at_unique q
apply beta_at_unique
rewrite hsplit_left at hip
rewrite hsplit_left at hip
exact hip
exact hql
rewrite hpq
specialize le_refl q
exact le_refl
have hprefix : (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
specialize sorted_succ_elim_prefix b
specialize sorted_succ_elim_prefix c
specialize sorted_succ_elim_prefix (S l)
apply sorted_succ_elim_prefix
exact hsorted
have hadj : exists a d. (((exists h. h + S a = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + a) /\ (((exists h. h + S d = S ((S S l) * c)) /\ exists w. b = w * S ((S S l) * c) + d) /\ (exists h. h + a = d)))
specialize sorted_succ_elim_last b
specialize sorted_succ_elim_last c
specialize sorted_succ_elim_last l
apply sorted_succ_elim_last
exact hsorted
cases hadj
cases hadj_witness
cases hadj_witness_witness
cases hadj_witness_witness_right
have hdq : x1 = q
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique (S l)
specialize beta_at_unique x1
specialize beta_at_unique q
apply beta_at_unique
exact hadj_witness_witness_right_left
exact hql
have hpa : (exists h. h + p = x)
specialize IH i
specialize IH p
specialize IH x
apply IH
exact hsplit_right
exact hip
exact hadj_witness_witness_left
exact hprefix
specialize le_trans p
specialize le_trans x
specialize le_trans q
apply le_trans
exact hpa
rewrite hdq at hadj_witness_witness_right_right
exact hadj_witness_witness_right_right
qed

Occurrences

1,587

Distinct objects

1,004

Reused references

44

Depth

62

Cuts

48

Certificate SHA-256

02ee4e476dae…

beta_nonempty_all_prime_product_ne_oneA nonempty AllPrime beta Product is not one β sequences, products & FTA checked

A nonempty product of prime decoded factors cannot have terminal value one.

Prerequisites

all_prime_succ_elim_last beta_factor_divides_product le_refl divisor_one

Used directly by

beta_all_prime_product_one_iff_length_zero prime_factorization_uniqueness_by_length

Exact expanded PA statement

Copy statement

forall b c l n. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ~(n = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ~(n = 1)
use all_prime_succ_elim_last
use beta_factor_divides_product
use le_refl
use divisor_one
intro b
intro c
intro l
intro n
intro hall
intro hproduct
intro hn1
have hlast : exists p. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last l
apply all_prime_succ_elim_last
exact hall
cases hlast
cases hlast_witness
have hpdiv : (exists k. n = x * k)
specialize beta_factor_divides_product b
specialize beta_factor_divides_product c
specialize beta_factor_divides_product (S l)
specialize beta_factor_divides_product n
specialize beta_factor_divides_product l
specialize beta_factor_divides_product x
apply beta_factor_divides_product
specialize le_refl (S l)
exact le_refl
exact hlast_witness_left
exact hproduct
have hp1 : x = 1
specialize divisor_one x
apply divisor_one
rewrite hn1 at hpdiv
exact hpdiv
cases hlast_witness_right
apply hlast_witness_right_left
exact hp1
qed

Occurrences

3,266

Distinct objects

1,213

Reused references

67

Depth

67

Cuts

96

Certificate SHA-256

9cde473a6b8f…

beta_all_prime_product_one_iff_length_zeroAn AllPrime beta Product is one exactly at length zero β sequences, products & FTA checked

Under AllPrime, a beta Product is one exactly when its encoded factor length is zero.

Prerequisites

beta_product_zero beta_nonempty_all_prime_product_ne_one succ_ne_zero

Used directly by

prime_factorization_uniqueness_by_length

Exact expanded PA statement

Copy statement

forall b c l n. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((n = 1 -> l = 0) /\ (l = 0 -> n = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((n = 1 -> l = 0) /\ (l = 0 -> n = 1))
use beta_product_zero
use beta_nonempty_all_prime_product_ne_one
use succ_ne_zero
intro b
intro c
induction l
intro n
intro hall
intro hproduct
split
intro hn1
refl
intro hl0
specialize beta_product_zero b
specialize beta_product_zero c
specialize beta_product_zero n
apply beta_product_zero
exact hproduct
intro n
intro hall
intro hproduct
split
intro hn1
exfalso
specialize beta_nonempty_all_prime_product_ne_one b
specialize beta_nonempty_all_prime_product_ne_one c
specialize beta_nonempty_all_prime_product_ne_one l
specialize beta_nonempty_all_prime_product_ne_one n
apply beta_nonempty_all_prime_product_ne_one
exact hall
exact hproduct
exact hn1
intro hl0
exfalso
specialize succ_ne_zero l
apply succ_ne_zero
exact hl0
qed

Occurrences

4,506

Distinct objects

1,331

Reused references

69

Depth

69

Cuts

130

Certificate SHA-256

6edf708691c6…

beta_canonical_last_factors_equalCanonical factorizations have equal last factors β sequences, products & FTA checked

Two nonempty sorted AllPrime Products of the same number have equal last factors.

Prerequisites

all_prime_succ_elim_last beta_at_unique beta_factor_divides_product le_refl beta_prime_divisor_product_member beta_sorted_factor_le_last le_antisymm

Used directly by

beta_canonical_product_cancel_last

Exact expanded PA statement

Copy statement

forall b c z e l k n p q. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S k) * v)) /\ exists w. u = w * S ((S S k) * v) + n) /\ forall i. (exists h. h + S i = S k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S q = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + q) -> p = q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z e l k n p q. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) -> (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S k) * v)) /\ exists w. u = w * S ((S S k) * v) + n) /\ forall i. (exists h. h + S i = S k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> ((exists h. h + S q = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + q) -> p = q
use all_prime_succ_elim_last
use beta_at_unique
use beta_factor_divides_product
use le_refl
use beta_prime_divisor_product_member
use beta_sorted_factor_le_last
use le_antisymm
intro b
intro c
intro z
intro e
intro l
intro k
intro n
intro p
intro q
intro hall1
intro hsort1
intro hprod1
intro hat1
intro hall2
intro hsort2
intro hprod2
intro hat2
have hlast1 : exists t. (((exists h. h + S t = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + t) /\ (~(t = 1) /\ forall a d. t = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last l
apply all_prime_succ_elim_last
exact hall1
cases hlast1
cases hlast1_witness
have hpeq1 : p = x
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique p
specialize beta_at_unique x
apply beta_at_unique
exact hat1
exact hlast1_witness_left
have hpprime : (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)
rewrite hpeq1
rewrite hpeq1
exact hlast1_witness_right
have hlast2 : exists t. (((exists h. h + S t = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + t) /\ (~(t = 1) /\ forall a d. t = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last z
specialize all_prime_succ_elim_last e
specialize all_prime_succ_elim_last k
apply all_prime_succ_elim_last
exact hall2
cases hlast2
cases hlast2_witness
have hqeq2 : q = x1
specialize beta_at_unique z
specialize beta_at_unique e
specialize beta_at_unique k
specialize beta_at_unique q
specialize beta_at_unique x1
apply beta_at_unique
exact hat2
exact hlast2_witness_left
have hqprime : (~(q = 1) /\ forall a d. q = a * d -> a = 1 \/ d = 1)
rewrite hqeq2
rewrite hqeq2
exact hlast2_witness_right
have hpdiv : (exists k. n = p * k)
specialize beta_factor_divides_product b
specialize beta_factor_divides_product c
specialize beta_factor_divides_product (S l)
specialize beta_factor_divides_product n
specialize beta_factor_divides_product l
specialize beta_factor_divides_product p
apply beta_factor_divides_product
specialize le_refl (S l)
exact le_refl
exact hat1
exact hprod1
have hqdiv : (exists k. n = q * k)
specialize beta_factor_divides_product z
specialize beta_factor_divides_product e
specialize beta_factor_divides_product (S k)
specialize beta_factor_divides_product n
specialize beta_factor_divides_product k
specialize beta_factor_divides_product q
apply beta_factor_divides_product
specialize le_refl (S k)
exact le_refl
exact hat2
exact hprod2
have hpmember2 : exists i. ((exists h. h + S i = S k) /\ ((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p))
specialize beta_prime_divisor_product_member z
specialize beta_prime_divisor_product_member e
specialize beta_prime_divisor_product_member (S k)
specialize beta_prime_divisor_product_member n
specialize beta_prime_divisor_product_member p
apply beta_prime_divisor_product_member
exact hpprime
exact hall2
exact hprod2
exact hpdiv
cases hpmember2
cases hpmember2_witness
have hpq : (exists h. h + p = q)
specialize beta_sorted_factor_le_last z
specialize beta_sorted_factor_le_last e
specialize beta_sorted_factor_le_last k
specialize beta_sorted_factor_le_last x2
specialize beta_sorted_factor_le_last p
specialize beta_sorted_factor_le_last q
apply beta_sorted_factor_le_last
exact hpmember2_witness_left
exact hpmember2_witness_right
exact hat2
exact hsort2
have hqmember1 : exists i. ((exists h. h + S i = S l) /\ ((exists h. h + S q = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + q))
specialize beta_prime_divisor_product_member b
specialize beta_prime_divisor_product_member c
specialize beta_prime_divisor_product_member (S l)
specialize beta_prime_divisor_product_member n
specialize beta_prime_divisor_product_member q
apply beta_prime_divisor_product_member
exact hqprime
exact hall1
exact hprod1
exact hqdiv
cases hqmember1
cases hqmember1_witness
have hqp : (exists h. h + q = p)
specialize beta_sorted_factor_le_last b
specialize beta_sorted_factor_le_last c
specialize beta_sorted_factor_le_last l
specialize beta_sorted_factor_le_last x3
specialize beta_sorted_factor_le_last q
specialize beta_sorted_factor_le_last p
apply beta_sorted_factor_le_last
exact hqmember1_witness_left
exact hqmember1_witness_right
exact hat1
exact hsort1
specialize le_antisymm p
specialize le_antisymm q
apply le_antisymm
exact hpq
exact hqp
qed

Occurrences

15,648

Distinct objects

3,187

Reused references

156

Depth

72

Cuts

456

Certificate SHA-256

930f4810f83f…

beta_canonical_product_cancel_lastCancel the common last factor of canonical Products β sequences, products & FTA checked

Cancel the common last prime from two nonempty canonical beta Products and return canonical prefixes.

Prerequisites

beta_product_succ_decompose beta_canonical_last_factors_equal all_prime_succ_elim_last beta_at_unique prime_nonzero mul_right_cancel_nonzero all_prime_succ_elim_prefix sorted_succ_elim_prefix

Used directly by

prime_factorization_uniqueness_by_length

Exact expanded PA statement

Copy statement

forall b c z e l k n. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S k) * v)) /\ exists w. u = w * S ((S S k) * v) + n) /\ forall i. (exists h. h + S i = S k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (((exists h. h + S p = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + p) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S k) * v)) /\ exists w. u = w * S ((S k) * v) + s) /\ forall i. (exists h. h + S i = k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ (n = r * p /\ (n = s * p /\ r = s))))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z e l k n. (forall i. (exists h. h + S i = S l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S l) * v)) /\ exists w. u = w * S ((S S l) * v) + n) /\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> (forall i. (exists h. h + S i = S k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) -> (forall i. (exists h. h + S S i = S k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S S k) * v)) /\ exists w. u = w * S ((S S k) * v) + n) /\ forall i. (exists h. h + S i = S k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) -> exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (((exists h. h + S p = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + p) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S k) * v)) /\ exists w. u = w * S ((S k) * v) + s) /\ forall i. (exists h. h + S i = k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ (n = r * p /\ (n = s * p /\ r = s))))))))))
use beta_product_succ_decompose
use beta_canonical_last_factors_equal
use all_prime_succ_elim_last
use beta_at_unique
use prime_nonzero
use mul_right_cancel_nonzero
use all_prime_succ_elim_prefix
use sorted_succ_elim_prefix
intro b
intro c
intro z
intro e
intro l
intro k
intro n
intro hall1
intro hsort1
intro hprod1
intro hall2
intro hsort2
intro hprod2
have hdec1 : exists p r. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = r * p))
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose l
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hprod1
cases hdec1
cases hdec1_witness
cases hdec1_witness_witness
cases hdec1_witness_witness_right
have hdec2 : exists p s. (((exists h. h + S p = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + p) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S k) * v)) /\ exists w. u = w * S ((S k) * v) + s) /\ forall i. (exists h. h + S i = k) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ n = s * p))
specialize beta_product_succ_decompose z
specialize beta_product_succ_decompose e
specialize beta_product_succ_decompose k
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hprod2
cases hdec2
cases hdec2_witness
cases hdec2_witness_witness
cases hdec2_witness_witness_right
have hfactor_eq : x = x2
specialize beta_canonical_last_factors_equal b
specialize beta_canonical_last_factors_equal c
specialize beta_canonical_last_factors_equal z
specialize beta_canonical_last_factors_equal e
specialize beta_canonical_last_factors_equal l
specialize beta_canonical_last_factors_equal k
specialize beta_canonical_last_factors_equal n
specialize beta_canonical_last_factors_equal x
specialize beta_canonical_last_factors_equal x2
apply beta_canonical_last_factors_equal
exact hall1
exact hsort1
exact hprod1
exact hdec1_witness_witness_left
exact hall2
exact hsort2
exact hprod2
exact hdec2_witness_witness_left
have hshared_at2 : ((exists h. h + S x = S ((S k) * e)) /\ exists w. z = w * S ((S k) * e) + x)
rewrite hfactor_eq
rewrite hfactor_eq
exact hdec2_witness_witness_left
have hshared_prod2 : n = x3 * x
rewrite hfactor_eq
exact hdec2_witness_witness_right_right
have hlastprime : exists t. (((exists h. h + S t = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + t) /\ (~(t = 1) /\ forall a d. t = a * d -> a = 1 \/ d = 1))
specialize all_prime_succ_elim_last b
specialize all_prime_succ_elim_last c
specialize all_prime_succ_elim_last l
apply all_prime_succ_elim_last
exact hall1
cases hlastprime
cases hlastprime_witness
have hfactor_last : x = x4
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique x4
apply beta_at_unique
exact hdec1_witness_witness_left
exact hlastprime_witness_left
have hprime : (~(x = 1) /\ forall a d. x = a * d -> a = 1 \/ d = 1)
rewrite hfactor_last
rewrite hfactor_last
exact hlastprime_witness_right
have hnonzero : ~(x = 0)
intro hxzero
specialize prime_nonzero x
apply prime_nonzero
exact hprime
exact hxzero
have hmuleq : x1 * x = x3 * x
trans n
symm
exact hdec1_witness_witness_right_right
exact hshared_prod2
have hprefix_eq : x1 = x3
specialize mul_right_cancel_nonzero x1
specialize mul_right_cancel_nonzero x3
specialize mul_right_cancel_nonzero x
apply mul_right_cancel_nonzero
exact hnonzero
exact hmuleq
have hallprefix1 : (forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_elim_prefix b
specialize all_prime_succ_elim_prefix c
specialize all_prime_succ_elim_prefix l
apply all_prime_succ_elim_prefix
exact hall1
have hsortprefix1 : (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))
specialize sorted_succ_elim_prefix b
specialize sorted_succ_elim_prefix c
specialize sorted_succ_elim_prefix l
apply sorted_succ_elim_prefix
exact hsort1
have hallprefix2 : (forall i. (exists h. h + S i = k) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1)))
specialize all_prime_succ_elim_prefix z
specialize all_prime_succ_elim_prefix e
specialize all_prime_succ_elim_prefix k
apply all_prime_succ_elim_prefix
exact hall2
have hsortprefix2 : (forall i. (exists h. h + S S i = k) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. z = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. z = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))
specialize sorted_succ_elim_prefix z
specialize sorted_succ_elim_prefix e
specialize sorted_succ_elim_prefix k
apply sorted_succ_elim_prefix
exact hsort2
exists x
exists x1
exists x3
split
exact hdec1_witness_witness_left
split
exact hshared_at2
split
exact hallprefix1
split
exact hsortprefix1
split
exact hdec1_witness_witness_right_left
split
exact hallprefix2
split
exact hsortprefix2
split
exact hdec2_witness_witness_right_left
split
exact hdec1_witness_witness_right_right
split
exact hshared_prod2
exact hprefix_eq
qed

Occurrences

18,993

Distinct objects

3,633

Reused references

168

Depth

74

Cuts

552

Certificate SHA-256

7670e0b1e59f…

prime_factorization_uniqueness_by_lengthPrime-factorization uniqueness by length β sequences, products & FTA checked

Strengthened induction on the first canonical factorization length.

Prerequisites

beta_product_zero beta_all_prime_product_one_iff_length_zero add_eq_zero_right succ_ne_zero beta_nonempty_all_prime_product_ne_one nonzero_is_succ beta_canonical_product_cancel_last succ_congr le_of_succ_le_succ le_eq_or_lt beta_at_unique

Used directly by

prime_factorization_uniqueness

Exact expanded PA statement

Copy statement

forall l n b c m d e. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))) -> ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists w. u = w * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = m) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + q) -> p = q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l n b c m d e. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))) -> ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists w. u = w * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = m) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + q) -> p = q)
use beta_product_zero
use beta_all_prime_product_one_iff_length_zero
use add_eq_zero_right
use succ_ne_zero
use beta_nonempty_all_prime_product_ne_one
use nonzero_is_succ
use beta_canonical_product_cancel_last
use succ_congr
use le_of_succ_le_succ
use le_eq_or_lt
use beta_at_unique
induction l
intro n
intro b
intro c
intro m
intro d
intro e
intro hcan1
intro hcan2
cases hcan1
cases hcan1_right
cases hcan2
cases hcan2_right
have hn1 : n = 1
specialize beta_product_zero b
specialize beta_product_zero c
specialize beta_product_zero n
apply beta_product_zero
exact hcan1_left
have hiff2 : ((n = 1 -> m = 0) /\ (m = 0 -> n = 1))
specialize beta_all_prime_product_one_iff_length_zero d
specialize beta_all_prime_product_one_iff_length_zero e
specialize beta_all_prime_product_one_iff_length_zero m
specialize beta_all_prime_product_one_iff_length_zero n
apply beta_all_prime_product_one_iff_length_zero
exact hcan2_right_left
exact hcan2_left
cases hiff2
have hm0 : m = 0
apply hiff2_left
exact hn1
split
symm
exact hm0
intro i
intro p
intro q
intro hi
intro hat1
intro hat2
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
intro n
intro b
intro c
intro m
intro d
intro e
intro hcan1
intro hcan2
cases hcan1
cases hcan1_right
cases hcan2
cases hcan2_right
have hiff2 : ((n = 1 -> m = 0) /\ (m = 0 -> n = 1))
specialize beta_all_prime_product_one_iff_length_zero d
specialize beta_all_prime_product_one_iff_length_zero e
specialize beta_all_prime_product_one_iff_length_zero m
specialize beta_all_prime_product_one_iff_length_zero n
apply beta_all_prime_product_one_iff_length_zero
exact hcan2_right_left
exact hcan2_left
cases hiff2
have hmne : ~(m = 0)
intro hm0
have hn1 : n = 1
apply hiff2_right
exact hm0
specialize beta_nonempty_all_prime_product_ne_one b
specialize beta_nonempty_all_prime_product_ne_one c
specialize beta_nonempty_all_prime_product_ne_one l
specialize beta_nonempty_all_prime_product_ne_one n
apply beta_nonempty_all_prime_product_ne_one
exact hcan1_right_left
exact hcan1_left
exact hn1
have hmsucc : exists k. m = S k
specialize nonzero_is_succ m
apply nonzero_is_succ
exact hmne
cases hmsucc
rewrite hmsucc_witness at hcan2_right_left
rewrite hmsucc_witness at hcan2_right_right
rewrite hmsucc_witness at hcan2_left
rewrite hmsucc_witness at hcan2_left
rewrite hmsucc_witness at hcan2_left
have hcancel : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\ exists w. b = w * S ((S l) * c) + p) /\ (((exists h. h + S p = S ((S x) * e)) /\ exists w. d = w * S ((S x) * e) + p) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S r = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + r) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = x) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ ((forall i. (exists h. h + S S i = x) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q)))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S s = S ((S x) * v)) /\ exists w. u = w * S ((S x) * v) + s) /\ forall i. (exists h. h + S i = x) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ (n = r * p /\ (n = s * p /\ r = s))))))))))
specialize beta_canonical_product_cancel_last b
specialize beta_canonical_product_cancel_last c
specialize beta_canonical_product_cancel_last d
specialize beta_canonical_product_cancel_last e
specialize beta_canonical_product_cancel_last l
specialize beta_canonical_product_cancel_last x
specialize beta_canonical_product_cancel_last n
apply beta_canonical_product_cancel_last
exact hcan1_right_left
exact hcan1_right_right
exact hcan1_left
exact hcan2_right_left
exact hcan2_right_right
exact hcan2_left
cases hcancel
cases hcancel_witness
cases hcancel_witness_witness
cases hcancel_witness_witness_witness
cases hcancel_witness_witness_witness_right
cases hcancel_witness_witness_witness_right_right
cases hcancel_witness_witness_witness_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right_right_right_right_right
cases hcancel_witness_witness_witness_right_right_right_right_right_right_right_right_right
have hprod2prefix_same : (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S x2 = S ((S x) * v)) /\ exists w. u = w * S ((S x) * v) + x2) /\ forall i. (exists h. h + S i = x) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p))))))
rewrite hcancel_witness_witness_witness_right_right_right_right_right_right_right_right_right_right
rewrite hcancel_witness_witness_witness_right_right_right_right_right_right_right_right_right_right
exact hcancel_witness_witness_witness_right_right_right_right_right_right_right_left
have hprefixcan1 : ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S x2 = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + x2) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q))))))
split
exact hcancel_witness_witness_witness_right_right_right_right_left
split
exact hcancel_witness_witness_witness_right_right_left
exact hcancel_witness_witness_witness_right_right_right_left
have hprefixcan2 : ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S x2 = S ((S x) * v)) /\ exists w. u = w * S ((S x) * v) + x2) /\ forall i. (exists h. h + S i = x) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = x) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = x) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))
split
exact hprod2prefix_same
split
exact hcancel_witness_witness_witness_right_right_right_right_right_left
exact hcancel_witness_witness_witness_right_right_right_right_right_right_left
have hrec : (l = x /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + q) -> p = q)
specialize IH x2
specialize IH b
specialize IH c
specialize IH x
specialize IH d
specialize IH e
apply IH
exact hprefixcan1
exact hprefixcan2
cases hrec
split
have hslen : S l = S x
specialize succ_congr l
specialize succ_congr x
apply succ_congr
exact hrec_left
trans S x
exact hslen
symm
exact hmsucc_witness
intro i
intro p
intro q
intro hi
intro hat1
intro hat2
have hile : (exists h. h + i = l)
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ (exists h. h + S i = l)
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hile
cases hsplit
have hp_last : p = x1
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique p
specialize beta_at_unique x1
apply beta_at_unique
rewrite hsplit_left at hat1
rewrite hsplit_left at hat1
exact hat1
exact hcancel_witness_witness_witness_left
have hq_last : q = x1
specialize beta_at_unique d
specialize beta_at_unique e
specialize beta_at_unique x
specialize beta_at_unique q
specialize beta_at_unique x1
apply beta_at_unique
rewrite hsplit_left at hat2
rewrite hsplit_left at hat2
rewrite hrec_left at hat2
rewrite hrec_left at hat2
exact hat2
exact hcancel_witness_witness_witness_right_left
trans x1
exact hp_last
symm
exact hq_last
specialize hrec_right i
specialize hrec_right p
specialize hrec_right q
apply hrec_right
exact hsplit_right
exact hat1
exact hat2
qed

Occurrences

29,739

Distinct objects

4,291

Reused references

181

Depth

81

Cuts

853

Certificate SHA-256

c12e77244e85…

prime_factorization_uniquenessUniqueness of finite prime factorization β sequences, products & FTA checked

Exact catalog finite prime-factorization uniqueness target.

Prerequisites

prime_factorization_uniqueness_by_length

Used directly by

fundamental_theorem_of_arithmetic

Exact expanded PA statement

Copy statement

forall n l b c m d e. (((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists w. u = w * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = m) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + q) -> p = q)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n l b c m d e. (((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists w. u = w * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = l) -> exists p q. (((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) /\ (((exists h. h + S q = S ((S S i) * c)) /\ exists w. b = w * S ((S S i) * c) + q) /\ (exists h. h + p = q)))))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists w. u = w * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists w. u = w * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists w. u = w * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists w. u = w * S ((S S i) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S S i = m) -> exists p q. (((exists h. h + S p = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + p) /\ (((exists h. h + S q = S ((S S i) * e)) /\ exists w. d = w * S ((S S i) * e) + q) /\ (exists h. h + p = q))))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S p = S ((S i) * c)) /\ exists w. b = w * S ((S i) * c) + p) -> ((exists h. h + S q = S ((S i) * e)) /\ exists w. d = w * S ((S i) * e) + q) -> p = q)
use prime_factorization_uniqueness_by_length
intro n
intro l
intro b
intro c
intro m
intro d
intro e
intro hboth
cases hboth
specialize prime_factorization_uniqueness_by_length l
specialize prime_factorization_uniqueness_by_length n
specialize prime_factorization_uniqueness_by_length b
specialize prime_factorization_uniqueness_by_length c
specialize prime_factorization_uniqueness_by_length m
specialize prime_factorization_uniqueness_by_length d
specialize prime_factorization_uniqueness_by_length e
apply prime_factorization_uniqueness_by_length
exact hboth_left
exact hboth_right
qed

Occurrences

29,789

Distinct objects

4,341

Reused references

181

Depth

82

Cuts

854

Certificate SHA-256

2d7a7156431d…

fundamental_theorem_of_arithmeticFundamental theorem of arithmetic β sequences, products & FTA checked

Native beta-coded prime-factorization existence and uniqueness in first-order PA; no conventional list primitive and no DNE.

Prerequisites

prime_factorization_existence prime_factorization_uniqueness

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

(forall n. ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * c)) /\ exists w. b = w * S ((S (S i)) * c) + q) /\ exists h. h + p = q)))))) /\ (forall n l b c m d e. (((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * c)) /\ exists w. b = w * S ((S (S i)) * c) + q) /\ exists h. h + p = q))))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists q. u = q * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. d = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S (p) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = m) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * e)) /\ exists w. d = w * S ((S (S i)) * e) + q) /\ exists h. h + p = q)))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) -> ((exists h. h + S (q) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + q) -> p = q))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove (forall n. ~(n = 0) -> exists l b c. ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * c)) /\ exists w. b = w * S ((S (S i)) * c) + q) /\ exists h. h + p = q)))))) /\ (forall n l b c m d e. (((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = l) -> exists p. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = l) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * c)) /\ exists w. b = w * S ((S (S i)) * c) + q) /\ exists h. h + p = q))))) /\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S m) * v)) /\ exists q. u = q * S ((S m) * v) + n) /\ forall i. (exists h. h + S i = m) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\ exists q. d = q * S ((S i) * e) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))) /\ ((forall i. (exists h. h + S i = m) -> exists p. (((exists h. h + S (p) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + p) /\ (~(p = 1) /\ forall a d. p = a * d -> a = 1 \/ d = 1))) /\ (forall i. (exists h. h + S (S i) = m) -> exists p q. (((exists h. h + S (p) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + p) /\ (((exists h. h + S (q) = S ((S (S i)) * e)) /\ exists w. d = w * S ((S (S i)) * e) + q) /\ exists h. h + p = q)))))) -> (l = m /\ forall i p q. (exists h. h + S i = l) -> ((exists h. h + S (p) = S ((S (i)) * c)) /\ exists w. b = w * S ((S (i)) * c) + p) -> ((exists h. h + S (q) = S ((S (i)) * e)) /\ exists w. d = w * S ((S (i)) * e) + q) -> p = q))
use prime_factorization_existence
use prime_factorization_uniqueness
split
exact prime_factorization_existence
exact prime_factorization_uniqueness
qed

Occurrences

73,767

Distinct objects

8,701

Reused references

377

Depth

99

Cuts

2,184

Certificate SHA-256

fd978f59bf3b…

prime_threeThree is prime Primes checked

Three is prime in the expanded first-order prime predicate.

Prerequisites

mul_succ_left mul_eq_one_components add_eq_zero_left mul_eq_zero mul_zero_left zero_or_succ

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(3 = 1) /\ forall a b. 3 = a * b -> a = 1 \/ b = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(3 = 1) /\ forall a b. 3 = a * b -> a = 1 \/ b = 1
use mul_succ_left
use mul_eq_one_components
use add_eq_zero_left
use mul_eq_zero
use mul_zero_left
use zero_or_succ
have hlarge : forall x y. ~(3 = S (S x) * S (S y))
intro x
intro y
intro hlarge_eq
specialize mul_succ_left (S x)
specialize mul_succ_left (S (S y))
rewrite mul_succ_left at hlarge_eq
rewrite PA4 at hlarge_eq
rewrite PA4 at hlarge_eq
have hone : 1 = S x * S (S y) + y
apply PA2
apply PA2
exact hlarge_eq
have hysplit : y = 0 \/ exists z. y = S z
specialize zero_or_succ y
exact zero_or_succ
cases hysplit
rewrite hysplit_left at hone
rewrite hysplit_left at hone
rewrite PA3 at hone
have hprod_one : S x * S (S 0) = 1
symm
exact hone
have hcomponents : S x = 1 /\ S (S 0) = 1
specialize mul_eq_one_components (S x)
specialize mul_eq_one_components (S (S 0))
apply mul_eq_one_components
exact hprod_one
cases hcomponents
have h10 : 1 = 0
apply PA2
exact hcomponents_right
apply PA1
exact h10
cases hysplit_right
rewrite hysplit_right_witness at hone
rewrite hysplit_right_witness at hone
rewrite PA4 at hone
have hzero : 0 = S x * S (S (S x1)) + x1
apply PA2
exact hone
have hsumzero : S x * S (S (S x1)) + x1 = 0
symm
exact hzero
have hprodzero : S x * S (S (S x1)) = 0
specialize add_eq_zero_left (S x * S (S (S x1)))
specialize add_eq_zero_left x1
apply add_eq_zero_left
exact hsumzero
have hfactors : S x = 0 \/ S (S (S x1)) = 0
specialize mul_eq_zero (S x)
specialize mul_eq_zero (S (S (S x1)))
apply mul_eq_zero
exact hprodzero
cases hfactors
apply PA1
exact hfactors_left
apply PA1
exact hfactors_right
split
intro h31
have h20 : 2 = 0
apply PA2
exact h31
apply PA1
exact h20
intro a
induction b
intro hab
rewrite PA5 at hab
exfalso
apply PA1
exact hab
induction b
intro hab
right
refl
induction a
intro hab
specialize mul_zero_left (S (S b))
rewrite mul_zero_left at hab
exfalso
apply PA1
exact hab
induction a
intro hab
left
refl
intro hab
exfalso
specialize hlarge a
specialize hlarge b
apply hlarge
exact hab
qed

Occurrences

691

Distinct objects

522

Reused references

41

Depth

43

Cuts

18

Certificate SHA-256

e6386e0d41a9…

two_prime_product_uniquenessA product of two primes is unique up to order β sequences, products & FTA checked

A product of two primes determines its factors up to swapping them.

Prerequisites

euclid_prime_dvd_product prime_divisor_eq_one_or_self prime_nonzero mul_left_cancel_nonzero mul_comm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p q r s. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> (~(q = 1) /\ forall c d. q = c * d -> c = 1 \/ d = 1) -> (~(r = 1) /\ forall e f. r = e * f -> e = 1 \/ f = 1) -> (~(s = 1) /\ forall g h. s = g * h -> g = 1 \/ h = 1) -> p * q = r * s -> (p = r /\ q = s) \/ (p = s /\ q = r)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p q r s. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> (~(q = 1) /\ forall c d. q = c * d -> c = 1 \/ d = 1) -> (~(r = 1) /\ forall e f. r = e * f -> e = 1 \/ f = 1) -> (~(s = 1) /\ forall g h. s = g * h -> g = 1 \/ h = 1) -> p * q = r * s -> (p = r /\ q = s) \/ (p = s /\ q = r)
use euclid_prime_dvd_product
use prime_divisor_eq_one_or_self
use prime_nonzero
use mul_left_cancel_nonzero
use mul_comm
intro p
intro q
intro r
intro s
intro hp
cases hp
intro hq
intro hr
intro hs
intro hproduct
have hpdiv : exists k. r * s = p * k
exists q
symm
exact hproduct
have hcases : (exists u. r = p * u) \/ exists v. s = p * v
specialize euclid_prime_dvd_product p
specialize euclid_prime_dvd_product r
specialize euclid_prime_dvd_product s
apply euclid_prime_dvd_product
split
exact hp_left
exact hp_right
exact hpdiv
cases hcases
have hpr : p = 1 \/ r = p
specialize prime_divisor_eq_one_or_self r
specialize prime_divisor_eq_one_or_self p
apply prime_divisor_eq_one_or_self
exact hr
exact hcases_left
cases hpr
exfalso
apply hp_left
exact hpr_left
left
split
symm
exact hpr_right
specialize mul_left_cancel_nonzero r
specialize mul_left_cancel_nonzero q
specialize mul_left_cancel_nonzero s
apply mul_left_cancel_nonzero
intro hrzero
specialize prime_nonzero r
apply prime_nonzero
exact hr
exact hrzero
rewrite <- hpr_right at hproduct
exact hproduct
have hps : p = 1 \/ s = p
specialize prime_divisor_eq_one_or_self s
specialize prime_divisor_eq_one_or_self p
apply prime_divisor_eq_one_or_self
exact hs
exact hcases_right
cases hps
exfalso
apply hp_left
exact hps_left
right
split
symm
exact hps_right
specialize mul_left_cancel_nonzero s
specialize mul_left_cancel_nonzero q
specialize mul_left_cancel_nonzero r
apply mul_left_cancel_nonzero
intro hszero
specialize prime_nonzero s
apply prime_nonzero
exact hs
exact hszero
rewrite <- hps_right at hproduct
specialize mul_comm r
specialize mul_comm s
rewrite mul_comm at hproduct
exact hproduct
qed

Occurrences

6,035

Distinct objects

2,051

Reused references

120

Depth

56

Cuts

181

Certificate SHA-256

bb1cd0a1af5d…

fourth_power_regroupRegroup a fourth power as a square of a square Multiplication checked

Regroup a fourth power as the square of a square.

Prerequisites

mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. n * n * n * n = (n * n) * (n * n)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. n * n * n * n = (n * n) * (n * n)
use mul_assoc
intro n
apply mul_assoc
qed

Occurrences

122

Distinct objects

110

Reused references

12

Depth

18

Cuts

3

Certificate SHA-256

d49403a5bc73…

mod5_residue_completeEvery natural has one of five residues modulo five Congruence & CRT checked

Every natural number has one of the five canonical residues modulo five.

Prerequisites

none — a root theorem

Used directly by

mod5_nonzero_residue_cases mod5_fourth_power_one

Exact expanded PA statement

Copy statement

forall n. exists q. n = 5 * q \/ n = 5 * q + 1 \/ n = 5 * q + 2 \/ n = 5 * q + 3 \/ n = 5 * q + 4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. exists q. n = 5 * q \/ n = 5 * q + 1 \/ n = 5 * q + 2 \/ n = 5 * q + 3 \/ n = 5 * q + 4
induction n
exists 0
left
left
left
left
norm_num
cases IH
cases IH_witness
cases IH_witness_left
cases IH_witness_left_left
cases IH_witness_left_left_left
exists x
left
left
left
right
rewrite IH_witness_left_left_left_left
simp
exists x
left
left
right
rewrite IH_witness_left_left_left_right
simp
exists x
left
right
rewrite IH_witness_left_left_right
simp
exists x
right
rewrite IH_witness_left_right
simp
exists x + 1
left
left
left
left
rewrite IH_witness_right
simp
qed

Occurrences

248

Distinct objects

214

Reused references

34

Depth

28

Cuts

0

Certificate SHA-256

f89dfde45b60…

mod5_nonzero_residue_casesA nonmultiple of five has a nonzero residue Congruence & CRT checked

A nonmultiple of five lies in one of the four nonzero residue classes.

Prerequisites

mod5_residue_complete

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. ~(exists q. n = 5 * q) -> exists q. n = 5 * q + 1 \/ n = 5 * q + 2 \/ n = 5 * q + 3 \/ n = 5 * q + 4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(exists q. n = 5 * q) -> exists q. n = 5 * q + 1 \/ n = 5 * q + 2 \/ n = 5 * q + 3 \/ n = 5 * q + 4
use mod5_residue_complete
intro n
intro h
specialize mod5_residue_complete n
cases mod5_residue_complete
cases mod5_residue_complete_witness
cases mod5_residue_complete_witness_left
cases mod5_residue_complete_witness_left_left
cases mod5_residue_complete_witness_left_left_left
exfalso
apply h
exists x
exact mod5_residue_complete_witness_left_left_left_left
exists x
left
left
left
exact mod5_residue_complete_witness_left_left_left_right
exists x
left
left
right
exact mod5_residue_complete_witness_left_left_right
exists x
left
right
exact mod5_residue_complete_witness_left_right
exists x
right
exact mod5_residue_complete_witness_right
qed

Occurrences

287

Distinct objects

253

Reused references

34

Depth

29

Cuts

1

Certificate SHA-256

b9e7df484004…

mod5_square_residue_oneSquaring residue one modulo five preserves residue one Congruence & CRT checked

Squaring residue one modulo five preserves residue one.

Prerequisites

add_assoc mul_add mul_assoc mul_succ_left

Used directly by

mod5_fourth_power_residue_one mod5_fourth_power_residue_four

Exact expanded PA statement

Copy statement

forall z q. z = 5 * q + 1 -> z * z = 5 * (q * z + q) + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall z q. z = 5 * q + 1 -> z * z = 5 * (q * z + q) + 1
use add_assoc
use mul_add
use mul_assoc
use mul_succ_left
intro z
intro q
intro hz
trans (5 * q + 1) * z
congr
exact hz
refl
trans S(5 * q) * z
congr
simp
refl
trans (5 * q) * z + z
apply mul_succ_left
trans 5 * (q * z) + z
congr
apply mul_assoc
refl
trans 5 * (q * z) + (5 * q + 1)
congr
refl
exact hz
trans (5 * (q * z) + 5 * q) + 1
symm
apply add_assoc
congr
symm
apply mul_add
refl
qed

Occurrences

450

Distinct objects

277

Reused references

33

Depth

26

Cuts

11

Certificate SHA-256

f84410e6ab51…

mod5_square_residue_twoSquaring residue two modulo five gives residue four Congruence & CRT checked

Squaring residue two modulo five gives residue four.

Prerequisites

add_comm add_assoc mul_add mul_assoc mul_succ_left

Used directly by

mod5_fourth_power_residue_two

Exact expanded PA statement

Copy statement

forall z q. z = 5 * q + 2 -> z * z = 5 * ((q * z + q) + q) + 4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall z q. z = 5 * q + 2 -> z * z = 5 * ((q * z + q) + q) + 4
use add_comm
use add_assoc
use mul_add
use mul_assoc
use mul_succ_left
intro z
intro q
intro hz
trans (5 * q + 2) * z
congr
exact hz
refl
trans S(S(5 * q)) * z
congr
simp
refl
trans S(5 * q) * z + z
apply mul_succ_left
trans ((5 * q) * z + z) + z
congr
apply mul_succ_left
refl
trans (5 * (q * z) + z) + z
congr
congr
apply mul_assoc
refl
refl
trans (5 * (q * z) + (5 * q + 2)) + (5 * q + 2)
congr
congr
refl
exact hz
exact hz
simp [add_comm, add_assoc, mul_add]
qed

Occurrences

641

Distinct objects

373

Reused references

56

Depth

42

Cuts

14

Certificate SHA-256

5e6df4f87532…

mod5_square_residue_threeSquaring residue three modulo five gives residue four Congruence & CRT checked

Squaring residue three modulo five gives residue four.

Prerequisites

square_residue_lift

Used directly by

mod5_fourth_power_residue_three

Exact expanded PA statement

Copy statement

forall z q. z = 5 * q + 3 -> z * z = 5 * ((q * z + 3 * q) + 1) + 4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall z q. z = 5 * q + 3 -> z * z = 5 * ((q * z + 3 * q) + 1) + 4
use square_residue_lift
intro z
intro q
intro hz
apply square_residue_lift
exact hz
norm_num
qed

Occurrences

1,185

Distinct objects

598

Reused references

42

Depth

31

Cuts

29

Certificate SHA-256

a427916ee3e7…

mod5_square_residue_fourSquaring residue four modulo five gives residue one Congruence & CRT checked

Squaring residue four modulo five gives residue one.

Prerequisites

square_residue_lift

Used directly by

mod5_fourth_power_residue_two mod5_fourth_power_residue_three mod5_fourth_power_residue_four

Exact expanded PA statement

Copy statement

forall z q. z = 5 * q + 4 -> z * z = 5 * ((q * z + 4 * q) + 3) + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall z q. z = 5 * q + 4 -> z * z = 5 * ((q * z + 4 * q) + 3) + 1
use square_residue_lift
intro z
intro q
intro hz
apply square_residue_lift
exact hz
norm_num
qed

Occurrences

1,283

Distinct objects

696

Reused references

42

Depth

31

Cuts

29

Certificate SHA-256

271a2d490759…

mod5_fourth_power_residue_oneFourth powers preserve residue one modulo five Congruence & CRT checked

A number congruent to one modulo five has fourth power congruent to one.

Prerequisites

mod5_square_residue_one mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall q. exists w. (5 * q + 1) * (5 * q + 1) * (5 * q + 1) * (5 * q + 1) = 5 * w + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall q. exists w. (5 * q + 1) * (5 * q + 1) * (5 * q + 1) * (5 * q + 1) = 5 * w + 1
use mod5_square_residue_one
use mul_assoc
intro q
exists (q * (5 * q + 1) + q) * ((5 * q + 1) * (5 * q + 1)) + (q * (5 * q + 1) + q)
trans ((5 * q + 1) * (5 * q + 1)) * ((5 * q + 1) * (5 * q + 1))
apply mul_assoc
apply mod5_square_residue_one
apply mod5_square_residue_one
refl
qed

Occurrences

584

Distinct objects

295

Reused references

34

Depth

27

Cuts

15

Certificate SHA-256

bdde252c1412…

mod5_fourth_power_residue_twoFourth powers of residue two are one modulo five Congruence & CRT checked

A number congruent to two modulo five has fourth power congruent to one.

Prerequisites

mod5_square_residue_two mod5_square_residue_four mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall q. exists w. (5 * q + 2) * (5 * q + 2) * (5 * q + 2) * (5 * q + 2) = 5 * w + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall q. exists w. (5 * q + 2) * (5 * q + 2) * (5 * q + 2) * (5 * q + 2) = 5 * w + 1
use mod5_square_residue_two
use mod5_square_residue_four
use mul_assoc
intro q
exists ((((q * (5 * q + 2) + q) + q) * ((5 * q + 2) * (5 * q + 2)) + 4 * ((q * (5 * q + 2) + q) + q)) + 3)
trans ((5 * q + 2) * (5 * q + 2)) * ((5 * q + 2) * (5 * q + 2))
apply mul_assoc
apply mod5_square_residue_four
apply mod5_square_residue_two
refl
qed

Occurrences

2,059

Distinct objects

861

Reused references

72

Depth

43

Cuts

48

Certificate SHA-256

3ec61a64b955…

mod5_fourth_power_residue_threeFourth powers of residue three are one modulo five Congruence & CRT checked

A number congruent to three modulo five has fourth power congruent to one.

Prerequisites

mod5_square_residue_three mod5_square_residue_four mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall q. exists w. (5 * q + 3) * (5 * q + 3) * (5 * q + 3) * (5 * q + 3) = 5 * w + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall q. exists w. (5 * q + 3) * (5 * q + 3) * (5 * q + 3) * (5 * q + 3) = 5 * w + 1
use mod5_square_residue_three
use mod5_square_residue_four
use mul_assoc
intro q
exists (((((q * (5 * q + 3) + 3 * q) + 1) * ((5 * q + 3) * (5 * q + 3)) + 4 * ((q * (5 * q + 3) + 3 * q) + 1)) + 3))
trans ((5 * q + 3) * (5 * q + 3)) * ((5 * q + 3) * (5 * q + 3))
apply mul_assoc
apply mod5_square_residue_four
apply mod5_square_residue_three
refl
qed

Occurrences

2,603

Distinct objects

892

Reused references

44

Depth

33

Cuts

63

Certificate SHA-256

d070cbf4af5a…

mod5_fourth_power_residue_fourFourth powers of residue four are one modulo five Congruence & CRT checked

A number congruent to four modulo five has fourth power congruent to one.

Prerequisites

mod5_square_residue_four mod5_square_residue_one mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall q. exists w. (5 * q + 4) * (5 * q + 4) * (5 * q + 4) * (5 * q + 4) = 5 * w + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall q. exists w. (5 * q + 4) * (5 * q + 4) * (5 * q + 4) * (5 * q + 4) = 5 * w + 1
use mod5_square_residue_four
use mod5_square_residue_one
use mul_assoc
intro q
exists ((q * (5 * q + 4) + 4 * q) + 3) * ((5 * q + 4) * (5 * q + 4)) + ((q * (5 * q + 4) + 4 * q) + 3)
trans ((5 * q + 4) * (5 * q + 4)) * ((5 * q + 4) * (5 * q + 4))
apply mul_assoc
apply mod5_square_residue_one
apply mod5_square_residue_four
refl
qed

Occurrences

1,868

Distinct objects

765

Reused references

49

Depth

32

Cuts

45

Certificate SHA-256

9b506838edf8…

mod5_fourth_power_oneA nonmultiple of five has fourth power one modulo five Congruence & CRT checked

A fourth power of a nonmultiple of five is one modulo five.

Prerequisites

mod5_residue_complete square_residue_lift mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. ~(exists x. n = 5 * x) -> exists x. n * n * n * n = 5 * x + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. ~(exists x. n = 5 * x) -> exists x. n * n * n * n = 5 * x + 1
use mod5_residue_complete
use square_residue_lift
use mul_assoc
intro n
intro h
specialize mod5_residue_complete n
cases mod5_residue_complete
cases mod5_residue_complete_witness
cases mod5_residue_complete_witness_left
cases mod5_residue_complete_witness_left_left
cases mod5_residue_complete_witness_left_left_left
exfalso
apply h
exists x
exact mod5_residue_complete_witness_left_left_left_left
have hsquare : n * n = 5 * ((x * n + 1 * x) + 0) + 1
apply square_residue_lift
exact mod5_residue_complete_witness_left_left_left_right
norm_num
exists ((((x * n + 1 * x) + 0) * (n * n) + 1 * ((x * n + 1 * x) + 0)) + 0)
trans (n * n) * (n * n)
apply mul_assoc
apply square_residue_lift
exact hsquare
norm_num
have hsquare : n * n = 5 * ((x * n + 2 * x) + 0) + 4
apply square_residue_lift
exact mod5_residue_complete_witness_left_left_right
norm_num
exists ((((x * n + 2 * x) + 0) * (n * n) + 4 * ((x * n + 2 * x) + 0)) + 3)
trans (n * n) * (n * n)
apply mul_assoc
apply square_residue_lift
exact hsquare
norm_num
have hsquare : n * n = 5 * ((x * n + 3 * x) + 1) + 4
apply square_residue_lift
exact mod5_residue_complete_witness_left_right
norm_num
exists ((((x * n + 3 * x) + 1) * (n * n) + 4 * ((x * n + 3 * x) + 1)) + 3)
trans (n * n) * (n * n)
apply mul_assoc
apply square_residue_lift
exact hsquare
norm_num
have hsquare : n * n = 5 * ((x * n + 4 * x) + 3) + 1
apply square_residue_lift
exact mod5_residue_complete_witness_right
norm_num
exists ((((x * n + 4 * x) + 3) * (n * n) + 1 * ((x * n + 4 * x) + 3)) + 0)
trans (n * n) * (n * n)
apply mul_assoc
apply square_residue_lift
exact hsquare
norm_num
qed

Occurrences

2,675

Distinct objects

1,935

Reused references

80

Depth

38

Cuts

33

Certificate SHA-256

67a58b050043…

parity_casesParity cases Quadratic reciprocity campaign checked

Every natural has a constructive even-or-odd witness.

Prerequisites

none — a root theorem

Used directly by

odd_mod4_cases prime_ne_two_is_odd even_successor_to_odd odd_successor_to_even

Exact expanded PA statement

Copy statement

forall n. exists k. n = 2 * k \/ n = 2 * k + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. exists k. n = 2 * k \/ n = 2 * k + 1
induction n
exists 0
left
norm_num
cases IH
cases IH_witness
exists x
right
rewrite IH_witness_left
simp
exists x + 1
left
rewrite IH_witness_right
simp
qed

Occurrences

80

Distinct objects

73

Reused references

7

Depth

19

Cuts

0

Certificate SHA-256

958d38a5bc3e…

even_odd_exclusive_pointwiseEven odd exclusive pointwise Quadratic reciprocity campaign checked

An even and an odd decomposition of the same natural are incompatible.

Prerequisites

division_remainder_unique

Used directly by

even_not_odd odd_not_even

Exact expanded PA statement

Copy statement

forall n a b. n = 2 * a -> n = 2 * b + 1 -> false

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n a b. n = 2 * a -> n = 2 * b + 1 -> false
use division_remainder_unique
intro n
intro a
intro b
intro he
intro ho
have hu : a = b /\ 0 = 1
specialize division_remainder_unique 2
specialize division_remainder_unique n
specialize division_remainder_unique a
specialize division_remainder_unique 0
specialize division_remainder_unique b
specialize division_remainder_unique 1
apply division_remainder_unique
rewrite he
simp
exists 1
norm_num
exact ho
exists 0
norm_num
cases hu
apply PA1
symm
exact hu_right
qed

Occurrences

928

Distinct objects

617

Reused references

23

Depth

58

Cuts

23

Certificate SHA-256

77e109bc6645…

even_not_oddEven not odd Quadratic reciprocity campaign checked

No even natural is odd.

Prerequisites

even_odd_exclusive_pointwise

Used directly by

even_successor_to_odd

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 2 * a) -> ~(exists b. n = 2 * b + 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 2 * a) -> ~(exists b. n = 2 * b + 1)
use even_odd_exclusive_pointwise
intro n
intro he
intro ho
cases he
cases ho
specialize even_odd_exclusive_pointwise n
specialize even_odd_exclusive_pointwise x
specialize even_odd_exclusive_pointwise x1
apply even_odd_exclusive_pointwise
exact he_witness
exact ho_witness
qed

Occurrences

953

Distinct objects

642

Reused references

23

Depth

59

Cuts

24

Certificate SHA-256

70a1576b3f52…

odd_not_evenOdd not even Quadratic reciprocity campaign checked

No odd natural is even.

Prerequisites

even_odd_exclusive_pointwise

Used directly by

odd_successor_to_even pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall n. (exists b. n = 2 * b + 1) -> ~(exists a. n = 2 * a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists b. n = 2 * b + 1) -> ~(exists a. n = 2 * a)
use even_odd_exclusive_pointwise
intro n
intro ho
intro he
cases ho
cases he
specialize even_odd_exclusive_pointwise n
specialize even_odd_exclusive_pointwise x1
specialize even_odd_exclusive_pointwise x
apply even_odd_exclusive_pointwise
exact he_witness
exact ho_witness
qed

Occurrences

953

Distinct objects

642

Reused references

23

Depth

59

Cuts

24

Certificate SHA-256

9d87768cd216…

successor_odd_of_evenSuccessor odd of even Quadratic reciprocity campaign checked

The successor of an even natural is odd.

Prerequisites

none — a root theorem

Used directly by

even_successor_to_odd

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 2 * a) -> exists b. S n = 2 * b + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 2 * a) -> exists b. S n = 2 * b + 1
intro n
intro he
cases he
exists x
rewrite he_witness
simp
qed

Occurrences

18

Distinct objects

18

Reused references

0

Depth

10

Cuts

0

Certificate SHA-256

60be8e49bc3a…

successor_even_of_oddSuccessor even of odd Quadratic reciprocity campaign checked

The successor of an odd natural is even.

Prerequisites

none — a root theorem

Used directly by

odd_successor_to_even

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 2 * a + 1) -> exists b. S n = 2 * b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 2 * a + 1) -> exists b. S n = 2 * b
intro n
intro ho
cases ho
exists x + 1
rewrite ho_witness
simp
qed

Occurrences

46

Distinct objects

41

Reused references

5

Depth

16

Cuts

0

Certificate SHA-256

f2a63e2bd60e…

even_add_evenEven add even Quadratic reciprocity campaign checked

The sum of two even naturals is even.

Prerequisites

mul_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a) -> (exists b. n = 2 * b) -> exists c. m + n = 2 * c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a) -> (exists b. n = 2 * b) -> exists c. m + n = 2 * c
use mul_add
intro m
intro n
intro hm
intro hn
cases hm
cases hn
exists x + x1
rewrite hm_witness
rewrite hn_witness
simp [mul_add]
qed

Occurrences

100

Distinct objects

93

Reused references

7

Depth

17

Cuts

2

Certificate SHA-256

6ac093483969…

even_add_oddEven add odd Quadratic reciprocity campaign checked

An even natural plus an odd natural is odd.

Prerequisites

mul_add add_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a) -> (exists b. n = 2 * b + 1) -> exists c. m + n = 2 * c + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a) -> (exists b. n = 2 * b + 1) -> exists c. m + n = 2 * c + 1
use mul_add
use add_assoc
intro m
intro n
intro hm
intro hn
cases hm
cases hn
exists x + x1
rewrite hm_witness
rewrite hn_witness
simp [mul_add, add_assoc]
qed

Occurrences

157

Distinct objects

112

Reused references

13

Depth

22

Cuts

3

Certificate SHA-256

5abbdaf62582…

odd_add_evenOdd add even Quadratic reciprocity campaign checked

An odd natural plus an even natural is odd.

Prerequisites

mul_add add_succ_left

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b) -> exists c. m + n = 2 * c + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b) -> exists c. m + n = 2 * c + 1
use mul_add
use add_succ_left
intro m
intro n
intro hm
intro hn
cases hm
cases hn
exists x + x1
rewrite hm_witness
rewrite hn_witness
simp [mul_add, add_succ_left]
qed

Occurrences

151

Distinct objects

136

Reused references

15

Depth

21

Cuts

3

Certificate SHA-256

51c7f3101d48…

odd_add_oddOdd add odd Quadratic reciprocity campaign checked

The sum of two odd naturals is even.

Prerequisites

mul_add add_succ_left add_assoc add_comm

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b + 1) -> exists c. m + n = 2 * c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b + 1) -> exists c. m + n = 2 * c
use mul_add
use add_succ_left
use add_assoc
use add_comm
intro m
intro n
intro hm
intro hn
cases hm
cases hn
exists x + x1 + 1
rewrite hm_witness
rewrite hn_witness
simp [mul_add, add_assoc, add_comm]
qed

Occurrences

302

Distinct objects

214

Reused references

30

Depth

32

Cuts

7

Certificate SHA-256

f00b8b27dc6f…

even_mul_leftEven mul left Quadratic reciprocity campaign checked

A product with an even left factor is even.

Prerequisites

mul_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a) -> exists c. m * n = 2 * c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a) -> exists c. m * n = 2 * c
use mul_assoc
intro m
intro n
intro hm
cases hm
exists x * n
rewrite hm_witness
apply mul_assoc
qed

Occurrences

130

Distinct objects

118

Reused references

12

Depth

18

Cuts

3

Certificate SHA-256

eaa49b843794…

mul_double_rightMul double right Quadratic reciprocity campaign checked

Doubling commutes through multiplication on the right.

Prerequisites

mul_assoc mul_comm

Used directly by

even_mul_right odd_mul_odd

Exact expanded PA statement

Copy statement

forall m b. m * (2 * b) = 2 * (m * b)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m b. m * (2 * b) = 2 * (m * b)
use mul_assoc
use mul_comm
intro m
intro b
trans (2 * b) * m
apply mul_comm
trans 2 * (b * m)
apply mul_assoc
congr
refl
apply mul_comm
qed

Occurrences

356

Distinct objects

290

Reused references

34

Depth

26

Cuts

10

Certificate SHA-256

609eaec9eea9…

even_mul_rightEven mul right Quadratic reciprocity campaign checked

A product with an even right factor is even.

Prerequisites

mul_double_right

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists b. n = 2 * b) -> exists c. m * n = 2 * c

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists b. n = 2 * b) -> exists c. m * n = 2 * c
use mul_double_right
intro m
intro n
intro hn
cases hn
exists m * x
rewrite hn_witness
apply mul_double_right
qed

Occurrences

369

Distinct objects

303

Reused references

34

Depth

27

Cuts

11

Certificate SHA-256

857e963b17a3…

odd_mul_oddOdd mul odd Quadratic reciprocity campaign checked

The product of two odd naturals is odd.

Prerequisites

mul_add add_mul add_assoc add_succ_left mul_double_right

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b + 1) -> exists c. m * n = 2 * c + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b + 1) -> exists c. m * n = 2 * c + 1
use mul_add
use add_mul
use add_assoc
use add_succ_left
use mul_double_right
intro m
intro n
intro hm
intro hn
cases hm
cases hn
exists (2 * x + 1) * x1 + x
rewrite hm_witness
rewrite hn_witness
simp [mul_add, add_mul, add_assoc, add_succ_left]
congr
congr
apply mul_double_right
refl
qed

Occurrences

897

Distinct objects

383

Reused references

51

Depth

31

Cuts

25

Certificate SHA-256

4749c2e868fc…

odd_half_uniqueOdd half unique Quadratic reciprocity campaign checked

The half witness in an odd decomposition is unique.

Prerequisites

add_right_cancel mul_left_cancel_nonzero

Used directly by

odd_half_exists_unique

Exact expanded PA statement

Copy statement

forall n a b. n = 2 * a + 1 -> n = 2 * b + 1 -> a = b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n a b. n = 2 * a + 1 -> n = 2 * b + 1 -> a = b
use add_right_cancel
use mul_left_cancel_nonzero
intro n
intro a
intro b
intro ha
intro hb
have hs : 2 * a + 1 = 2 * b + 1
trans n
symm
exact ha
exact hb
have hm : 2 * a = 2 * b
specialize add_right_cancel (2 * a)
specialize add_right_cancel (2 * b)
specialize add_right_cancel 1
apply add_right_cancel
exact hs
specialize mul_left_cancel_nonzero 2
specialize mul_left_cancel_nonzero a
specialize mul_left_cancel_nonzero b
apply mul_left_cancel_nonzero
intro htwo
apply PA1
exact htwo
exact hm
qed

Occurrences

292

Distinct objects

204

Reused references

2

Depth

25

Cuts

9

Certificate SHA-256

e21d6292387c…

odd_half_exists_uniqueOdd half exists unique Quadratic reciprocity campaign checked

Every odd natural has a unique half witness.

Prerequisites

odd_half_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 2 * a + 1) -> exists h. n = 2 * h + 1 /\ forall k. n = 2 * k + 1 -> h = k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 2 * a + 1) -> exists h. n = 2 * h + 1 /\ forall k. n = 2 * k + 1 -> h = k
use odd_half_unique
intro n
intro ho
cases ho
exists x
split
exact ho_witness
intro k
intro hk
specialize odd_half_unique n
specialize odd_half_unique x
specialize odd_half_unique k
apply odd_half_unique
exact ho_witness
exact hk
qed

Occurrences

319

Distinct objects

231

Reused references

2

Depth

26

Cuts

10

Certificate SHA-256

7fa50bde2df8…

four_mul_eq_double_doubleFour mul eq double double Quadratic reciprocity campaign checked

Multiplication by four is iterated doubling.

Prerequisites

mul_assoc

Used directly by

odd_mod4_cases mod4_one_is_odd mod4_three_is_odd

Exact expanded PA statement

Copy statement

forall a. 4 * a = 2 * (2 * a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. 4 * a = 2 * (2 * a)
use mul_assoc
intro a
trans (2 * 2) * a
congr
norm_num
refl
apply mul_assoc
qed

Occurrences

178

Distinct objects

166

Reused references

12

Depth

19

Cuts

3

Certificate SHA-256

3b9071b49ae4…

odd_mod4_casesOdd mod4 cases Quadratic reciprocity campaign checked

Every odd natural is congruent to one or three modulo four.

Prerequisites

parity_cases mul_add four_mul_eq_double_double

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. (exists h. n = 2 * h + 1) -> (exists a. n = 4 * a + 1) \/ exists b. n = 4 * b + 3

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists h. n = 2 * h + 1) -> (exists a. n = 4 * a + 1) \/ exists b. n = 4 * b + 3
use parity_cases
use mul_add
use four_mul_eq_double_double
intro n
intro ho
cases ho
specialize parity_cases x
cases parity_cases
cases parity_cases_witness
left
exists x1
rewrite ho_witness
rewrite parity_cases_witness_left
simp [mul_add]
congr
symm
apply four_mul_eq_double_double
right
exists x1
rewrite ho_witness
rewrite parity_cases_witness_right
simp [mul_add]
congr
congr
congr
symm
apply four_mul_eq_double_double
qed

Occurrences

450

Distinct objects

335

Reused references

39

Depth

32

Cuts

7

Certificate SHA-256

99e28f734f8b…

mod4_one_three_exclusive_pointwiseMod4 one three exclusive pointwise Quadratic reciprocity campaign checked

Residues one and three modulo four cannot describe the same natural.

Prerequisites

division_remainder_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n a b. n = 4 * a + 1 -> n = 4 * b + 3 -> false

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n a b. n = 4 * a + 1 -> n = 4 * b + 3 -> false
use division_remainder_unique
intro n
intro a
intro b
intro h1
intro h3
have hu : a = b /\ 1 = 3
specialize division_remainder_unique 4
specialize division_remainder_unique n
specialize division_remainder_unique a
specialize division_remainder_unique 1
specialize division_remainder_unique b
specialize division_remainder_unique 3
apply division_remainder_unique
exact h1
exists 2
norm_num
exact h3
exists 0
norm_num
cases hu
have hz : 0 = 2
apply PA2
exact hu_right
apply PA1
symm
exact hz
qed

Occurrences

944

Distinct objects

634

Reused references

22

Depth

58

Cuts

23

Certificate SHA-256

8ab417671cd8…

mod4_one_is_oddMod4 one is odd Quadratic reciprocity campaign checked

A natural congruent to one modulo four is odd.

Prerequisites

four_mul_eq_double_double

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 4 * a + 1) -> exists h. n = 2 * h + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 4 * a + 1) -> exists h. n = 2 * h + 1
use four_mul_eq_double_double
intro n
intro h
cases h
exists 2 * x
rewrite h_witness
congr
apply four_mul_eq_double_double
refl
qed

Occurrences

191

Distinct objects

179

Reused references

12

Depth

20

Cuts

4

Certificate SHA-256

100ca77514b0…

mod4_three_is_oddMod4 three is odd Quadratic reciprocity campaign checked

A natural congruent to three modulo four is odd.

Prerequisites

mul_add four_mul_eq_double_double

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. (exists a. n = 4 * a + 3) -> exists h. n = 2 * h + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. n = 4 * a + 3) -> exists h. n = 2 * h + 1
use mul_add
use four_mul_eq_double_double
intro n
intro h
cases h
exists 2 * x + 1
rewrite h_witness
simp [mul_add]
congr
congr
congr
apply four_mul_eq_double_double
qed

Occurrences

326

Distinct objects

225

Reused references

25

Depth

24

Cuts

6

Certificate SHA-256

ecc3cdd64943…

prime_ne_two_is_oddPrime ne two is odd Quadratic reciprocity campaign checked

Every prime other than two is odd.

Prerequisites

parity_cases

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 2) -> exists h. p = 2 * h + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 2) -> exists h. p = 2 * h + 1
use parity_cases
intro p
intro hp
intro hne
cases hp
specialize parity_cases p
cases parity_cases
cases parity_cases_witness
have hfac : 2 = 1 \/ x = 1
specialize hp_right 2
specialize hp_right x
apply hp_right
exact parity_cases_witness_left
cases hfac
exfalso
apply PA1
apply PA2
exact hfac_left
exfalso
apply hne
trans 2 * x
exact parity_cases_witness_left
rewrite hfac_right
simp
exists x
exact parity_cases_witness_right
qed

Occurrences

139

Distinct objects

128

Reused references

11

Depth

20

Cuts

1

Certificate SHA-256

438d56815002…

mod_eq_decidable_from_remaindersMod eq decidable from remainders Quadratic reciprocity campaign checked

Canonical bounded remainders constructively decide congruence.

Prerequisites

eq_decidable remainder_decomposition_to_mod_eq mod_eq_symm mod_eq_trans mod_eq_bounded_unique

Used directly by

mod_eq_decidable_nonzero

Exact expanded PA statement

Copy statement

forall p a b qa ra qb rb. a = qa * p + ra -> (exists ha. ha + S ra = p) -> b = qb * p + rb -> (exists hb. hb + S rb = p) -> (exists u v. a + p * u = b + p * v) \/ ~(exists u v. a + p * u = b + p * v)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a b qa ra qb rb. a = qa * p + ra -> (exists ha. ha + S ra = p) -> b = qb * p + rb -> (exists hb. hb + S rb = p) -> (exists u v. a + p * u = b + p * v) \/ ~(exists u v. a + p * u = b + p * v)
use eq_decidable
use remainder_decomposition_to_mod_eq
use mod_eq_symm
use mod_eq_trans
use mod_eq_bounded_unique
intro p
intro a
intro b
intro qa
intro ra
intro qb
intro rb
intro ha
intro hra
intro hb
intro hrb
specialize eq_decidable ra
specialize eq_decidable rb
cases eq_decidable
left
have har : exists u v. a + p * u = ra + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq a
specialize remainder_decomposition_to_mod_eq qa
specialize remainder_decomposition_to_mod_eq ra
apply remainder_decomposition_to_mod_eq
exact ha
have hbr : exists u v. b + p * u = rb + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq b
specialize remainder_decomposition_to_mod_eq qb
specialize remainder_decomposition_to_mod_eq rb
apply remainder_decomposition_to_mod_eq
exact hb
have hbra : exists u v. b + p * u = ra + p * v
rewrite eq_decidable_left
exact hbr
have hrab : exists u v. ra + p * u = b + p * v
specialize mod_eq_symm p
specialize mod_eq_symm b
specialize mod_eq_symm ra
apply mod_eq_symm
exact hbra
specialize mod_eq_trans p
specialize mod_eq_trans a
specialize mod_eq_trans ra
specialize mod_eq_trans b
apply mod_eq_trans
exact har
exact hrab
right
intro hab
apply eq_decidable_right
specialize mod_eq_bounded_unique p
specialize mod_eq_bounded_unique ra
specialize mod_eq_bounded_unique rb
apply mod_eq_bounded_unique
exact hra
exact hrb
have har : exists u v. a + p * u = ra + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq a
specialize remainder_decomposition_to_mod_eq qa
specialize remainder_decomposition_to_mod_eq ra
apply remainder_decomposition_to_mod_eq
exact ha
have hra_a : exists u v. ra + p * u = a + p * v
specialize mod_eq_symm p
specialize mod_eq_symm a
specialize mod_eq_symm ra
apply mod_eq_symm
exact har
have hra_b : exists u v. ra + p * u = b + p * v
specialize mod_eq_trans p
specialize mod_eq_trans ra
specialize mod_eq_trans a
specialize mod_eq_trans b
apply mod_eq_trans
exact hra_a
exact hab
have hbr : exists u v. b + p * u = rb + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq b
specialize remainder_decomposition_to_mod_eq qb
specialize remainder_decomposition_to_mod_eq rb
apply remainder_decomposition_to_mod_eq
exact hb
specialize mod_eq_trans p
specialize mod_eq_trans ra
specialize mod_eq_trans b
specialize mod_eq_trans rb
apply mod_eq_trans
exact hra_b
exact hbr
qed

Occurrences

1,693

Distinct objects

935

Reused references

42

Depth

64

Cuts

47

Certificate SHA-256

954dd3079783…

mod_eq_decidable_nonzeroMod eq decidable nonzero Quadratic reciprocity campaign checked

Balanced congruence is constructively decidable at nonzero modulus.

Prerequisites

division_remainder_exists mul_comm mod_eq_decidable_from_remainders

Used directly by

quadratic_residue_search_up_to mod_eq_decidable

Exact expanded PA statement

Copy statement

forall p a b. ~(p = 0) -> (exists qr_u_dec_yes qr_v_dec_yes. a + p * qr_u_dec_yes = b + p * qr_v_dec_yes) \/ ~(exists qr_u_dec_no qr_v_dec_no. a + p * qr_u_dec_no = b + p * qr_v_dec_no)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a b. ~(p = 0) -> (exists qr_u_dec_yes qr_v_dec_yes. a + p * qr_u_dec_yes = b + p * qr_v_dec_yes) \/ ~(exists qr_u_dec_no qr_v_dec_no. a + p * qr_u_dec_no = b + p * qr_v_dec_no)
use division_remainder_exists
use mul_comm
use mod_eq_decidable_from_remainders
intro p
intro a
intro b
intro hp
have hda : exists qa ra. a = p * qa + ra /\ exists ha. ha + S ra = p
specialize division_remainder_exists p
specialize division_remainder_exists a
apply division_remainder_exists
exact hp
cases hda
cases hda_witness
cases hda_witness_witness
have hdb : exists qb rb. b = p * qb + rb /\ exists hb. hb + S rb = p
specialize division_remainder_exists p
specialize division_remainder_exists b
apply division_remainder_exists
exact hp
cases hdb
cases hdb_witness
cases hdb_witness_witness
have ha_comm : a = x * p + x1
trans p * x + x1
exact hda_witness_witness_left
congr
apply mul_comm
refl
have hb_comm : b = x2 * p + x3
trans p * x2 + x3
exact hdb_witness_witness_left
congr
apply mul_comm
refl
specialize mod_eq_decidable_from_remainders p
specialize mod_eq_decidable_from_remainders a
specialize mod_eq_decidable_from_remainders b
specialize mod_eq_decidable_from_remainders x
specialize mod_eq_decidable_from_remainders x1
specialize mod_eq_decidable_from_remainders x2
specialize mod_eq_decidable_from_remainders x3
apply mod_eq_decidable_from_remainders
exact ha_comm
exact hda_witness_witness_right
exact hb_comm
exact hdb_witness_witness_right
qed

Occurrences

2,189

Distinct objects

1,136

Reused references

60

Depth

67

Cuts

61

Certificate SHA-256

dacd77a7c266…

quadratic_residue_search_up_toQuadratic residue search up to Quadratic reciprocity campaign checked

Inclusive bounded search constructively decides square congruence.

Prerequisites

mod_eq_decidable_nonzero add_eq_zero_right le_eq_or_lt add_succ_left zero_add

Used directly by

quadratic_residue_bounded_decidable_nonzero

Exact expanded PA statement

Copy statement

forall B p a. ~(p = 0) -> ((exists qr_x_search_yes. (exists qr_bound_search_yes. qr_bound_search_yes + qr_x_search_yes = B) /\ (exists qr_u_search_yes qr_v_search_yes. qr_x_search_yes * qr_x_search_yes + p * qr_u_search_yes = a + p * qr_v_search_yes)) \/ (forall qr_x_search_no. (exists qr_bound_search_no. qr_bound_search_no + qr_x_search_no = B) -> ~(exists qr_u_search_no qr_v_search_no. qr_x_search_no * qr_x_search_no + p * qr_u_search_no = a + p * qr_v_search_no)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall B p a. ~(p = 0) -> ((exists qr_x_search_yes. (exists qr_bound_search_yes. qr_bound_search_yes + qr_x_search_yes = B) /\ (exists qr_u_search_yes qr_v_search_yes. qr_x_search_yes * qr_x_search_yes + p * qr_u_search_yes = a + p * qr_v_search_yes)) \/ (forall qr_x_search_no. (exists qr_bound_search_no. qr_bound_search_no + qr_x_search_no = B) -> ~(exists qr_u_search_no qr_v_search_no. qr_x_search_no * qr_x_search_no + p * qr_u_search_no = a + p * qr_v_search_no)))
use mod_eq_decidable_nonzero
use add_eq_zero_right
use le_eq_or_lt
use add_succ_left
use zero_add
induction B
intro p
intro a
intro hp
have hdec : (exists u v. 0 * 0 + p * u = a + p * v) \/ ~(exists u v. 0 * 0 + p * u = a + p * v)
specialize mod_eq_decidable_nonzero p
specialize mod_eq_decidable_nonzero (0 * 0)
specialize mod_eq_decidable_nonzero a
apply mod_eq_decidable_nonzero
exact hp
cases hdec
left
exists 0
split
exists 0
simp
exact hdec_left
right
intro x
intro hx
intro hs
cases hx
have hx0 : x = 0
specialize add_eq_zero_right x1
specialize add_eq_zero_right x
apply add_eq_zero_right
exact hx_witness
apply hdec_right
rewrite hx0 at hs
rewrite hx0 at hs
exact hs
intro p
intro a
intro hp
have hprev : (exists x. (exists h. h + x = B) /\ exists u v. x * x + p * u = a + p * v) \/ (forall x. (exists h. h + x = B) -> ~(exists u v. x * x + p * u = a + p * v))
specialize IH p
specialize IH a
apply IH
exact hp
cases hprev
left
cases hprev_left
cases hprev_left_witness
exists x
split
cases hprev_left_witness_left
exists S x1
trans S (x1 + x)
apply add_succ_left
congr
exact hprev_left_witness_left_witness
exact hprev_left_witness_right
have hdec : (exists u v. (S B) * (S B) + p * u = a + p * v) \/ ~(exists u v. (S B) * (S B) + p * u = a + p * v)
specialize mod_eq_decidable_nonzero p
specialize mod_eq_decidable_nonzero ((S B) * (S B))
specialize mod_eq_decidable_nonzero a
apply mod_eq_decidable_nonzero
exact hp
cases hdec
left
exists S B
split
exists 0
simp
congr
specialize zero_add B
exact zero_add
exact hdec_left
right
intro x
intro hx
intro hs
specialize le_eq_or_lt x
specialize le_eq_or_lt (S B)
have hcases : x = S B \/ exists k. k + S x = S B
apply le_eq_or_lt
exact hx
cases hcases
apply hdec_right
rewrite hcases_left at hs
rewrite hcases_left at hs
exact hs
apply hprev_right
cases hcases_right
exists x1
apply PA2
trans x1 + S x
symm
apply PA4
exact hcases_right_witness
exact hs
qed

Occurrences

2,480

Distinct objects

1,329

Reused references

67

Depth

68

Cuts

69

Certificate SHA-256

e459beb71367…

quadratic_residue_bounded_decidable_nonzeroQuadratic residue bounded decidable nonzero Quadratic reciprocity campaign checked

A nonzero modulus admits a finite constructive residue search.

Prerequisites

nonzero_is_succ quadratic_residue_search_up_to

Used directly by

quadratic_residue_decidable_nonzero

Exact expanded PA statement

Copy statement

forall p a. ~(p = 0) -> (exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded) \/ ~(exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. ~(p = 0) -> (exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded) \/ ~(exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded)
use nonzero_is_succ
use quadratic_residue_search_up_to
intro p
intro a
intro hp
have hs : exists B. p = S B
specialize nonzero_is_succ p
apply nonzero_is_succ
exact hp
cases hs
have hsearch : (exists r. (exists h. h + r = x) /\ exists u v. r * r + p * u = a + p * v) \/ (forall r. (exists h. h + r = x) -> ~(exists u v. r * r + p * u = a + p * v))
specialize quadratic_residue_search_up_to x
specialize quadratic_residue_search_up_to p
specialize quadratic_residue_search_up_to a
apply quadratic_residue_search_up_to
exact hp
cases hsearch
left
cases hsearch_left
cases hsearch_left_witness
exists x1
split
cases hsearch_left_witness_left
exists x2
trans S (x2 + x1)
apply PA4
trans S x
congr
exact hsearch_left_witness_left_witness
symm
exact hs_witness
exact hsearch_left_witness_right
right
intro hbounded
cases hbounded
cases hbounded_witness
specialize hsearch_right x1
apply hsearch_right
cases hbounded_witness_left
exists x2
apply PA2
trans x2 + S x1
symm
apply PA4
trans p
exact hbounded_witness_left_witness
exact hs_witness
exact hbounded_witness_right
qed

Occurrences

2,554

Distinct objects

1,403

Reused references

67

Depth

70

Cuts

71

Certificate SHA-256

228002501cce…

quadratic_residue_bounded_equivQuadratic residue bounded equiv Quadratic reciprocity campaign checked

Every square witness has an equivalent canonical bounded root.

Prerequisites

division_remainder_exists square_residue_witness remainder_decomposition_to_mod_eq mod_eq_symm mod_eq_trans mul_comm zero_add

Used directly by

quadratic_residue_decidable_nonzero qres_mod3_canonical_iff qres_mod5_canonical_iff qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

forall p a. ~(p = 0) -> (((exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded) -> (exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded)) /\ ((exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded) -> (exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. ~(p = 0) -> (((exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded) -> (exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded)) /\ ((exists qr_x_bounded. (exists qr_h_bounded. qr_h_bounded + S qr_x_bounded = p) /\ exists qr_u_bounded qr_v_bounded. qr_x_bounded * qr_x_bounded + p * qr_u_bounded = a + p * qr_v_bounded) -> (exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded)))
use division_remainder_exists
use square_residue_witness
use remainder_decomposition_to_mod_eq
use mod_eq_symm
use mod_eq_trans
use mul_comm
use zero_add
intro p
intro a
intro hp
split
intro hunbounded
cases hunbounded
have hdivision : exists q r. x = p * q + r /\ exists h. h + S r = p
specialize division_remainder_exists p
specialize division_remainder_exists x
apply division_remainder_exists
exact hp
cases hdivision
cases hdivision_witness
cases hdivision_witness_witness
have hsquare : exists w. x * x = p * w + x2 * x2
specialize square_residue_witness p
specialize square_residue_witness x
specialize square_residue_witness x1
specialize square_residue_witness x2
specialize square_residue_witness 0
specialize square_residue_witness (x2 * x2)
apply square_residue_witness
exact hdivision_witness_witness_left
simp
specialize zero_add (x2 * x2)
symm
exact zero_add
cases hsquare
have hdecomp : x * x = x3 * p + x2 * x2
trans p * x3 + x2 * x2
exact hsquare_witness
congr
apply mul_comm
refl
have hxr : exists u v. x * x + p * u = x2 * x2 + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq (x * x)
specialize remainder_decomposition_to_mod_eq x3
specialize remainder_decomposition_to_mod_eq (x2 * x2)
apply remainder_decomposition_to_mod_eq
exact hdecomp
have hrx : exists u v. x2 * x2 + p * u = x * x + p * v
specialize mod_eq_symm p
specialize mod_eq_symm (x * x)
specialize mod_eq_symm (x2 * x2)
apply mod_eq_symm
exact hxr
have hra : exists u v. x2 * x2 + p * u = a + p * v
specialize mod_eq_trans p
specialize mod_eq_trans (x2 * x2)
specialize mod_eq_trans (x * x)
specialize mod_eq_trans a
apply mod_eq_trans
exact hrx
exact hunbounded_witness
exists x2
split
exact hdivision_witness_witness_right
exact hra
intro hbounded
cases hbounded
cases hbounded_witness
exists x
exact hbounded_witness_right
qed

Occurrences

2,151

Distinct objects

781

Reused references

66

Depth

34

Cuts

63

Certificate SHA-256

5b36bdde6d12…

quadratic_residue_decidable_nonzeroQuadratic residue decidable nonzero Quadratic reciprocity campaign checked

Quadratic residuosity is constructively decidable at nonzero modulus.

Prerequisites

quadratic_residue_bounded_decidable_nonzero quadratic_residue_bounded_equiv

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p a. ~(p = 0) -> (exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded) \/ ~(exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. ~(p = 0) -> (exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded) \/ ~(exists qr_x_unbounded. exists qr_u_unbounded qr_v_unbounded. qr_x_unbounded * qr_x_unbounded + p * qr_u_unbounded = a + p * qr_v_unbounded)
use quadratic_residue_bounded_decidable_nonzero
use quadratic_residue_bounded_equiv
intro p
intro a
intro hp
have hequiv : (((exists x. exists u v. x * x + p * u = a + p * v) -> (exists r. (exists h. h + S r = p) /\ exists u v. r * r + p * u = a + p * v)) /\ ((exists r. (exists h. h + S r = p) /\ exists u v. r * r + p * u = a + p * v) -> (exists x. exists u v. x * x + p * u = a + p * v)))
specialize quadratic_residue_bounded_equiv p
specialize quadratic_residue_bounded_equiv a
apply quadratic_residue_bounded_equiv
exact hp
cases hequiv
have hdec : (exists r. (exists h. h + S r = p) /\ exists u v. r * r + p * u = a + p * v) \/ ~(exists r. (exists h. h + S r = p) /\ exists u v. r * r + p * u = a + p * v)
specialize quadratic_residue_bounded_decidable_nonzero p
specialize quadratic_residue_bounded_decidable_nonzero a
apply quadratic_residue_bounded_decidable_nonzero
exact hp
cases hdec
left
apply hequiv_right
exact hdec_left
right
intro hunbounded
apply hdec_right
apply hequiv_left
exact hunbounded
qed

Occurrences

4,737

Distinct objects

1,715

Reused references

89

Depth

71

Cuts

136

Certificate SHA-256

3d53eba6ed8f…

beta_repeat_emptyBeta repeat empty Quadratic reciprocity campaign checked

Every constant beta prefix of length zero is vacuously Repeat.

Prerequisites

add_eq_zero_right succ_ne_zero

Used directly by

beta_repeat_exists

Exact expanded PA statement

Copy statement

forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a))))
use add_eq_zero_right
use succ_ne_zero
intro b
intro c
intro a
intro l
intro hl
intro i
intro hi
rewrite hl at hi
exfalso
cases hi
have hsi : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi
qed

Occurrences

42

Distinct objects

42

Reused references

0

Depth

16

Cuts

2

Certificate SHA-256

ef25ee646a84…

beta_repeat_succ_extendBeta repeat succ extend Quadratic reciprocity campaign checked

Recode a constant prefix and append one more copy of its value.

Prerequisites

beta_prefix_extend le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_repeat_exists

Exact expanded PA statement

Copy statement

forall b c a l sl. sl = S l -> (forall ff_i_before. (exists ff_lt_before_bound. ff_lt_before_bound + S ff_i_before = l) -> (((exists ff_h_before_decoded. ff_h_before_decoded + S (a) = S ((S (ff_i_before)) * c)) /\ exists ff_q_before_decoded. b = ff_q_before_decoded * S ((S (ff_i_before)) * c) + (a)))) -> exists z d. (forall ff_i_after. (exists ff_lt_after_bound. ff_lt_after_bound + S ff_i_after = sl) -> (((exists ff_h_after_decoded. ff_h_after_decoded + S (a) = S ((S (ff_i_after)) * d)) /\ exists ff_q_after_decoded. z = ff_q_after_decoded * S ((S (ff_i_after)) * d) + (a))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l sl. sl = S l -> (forall ff_i_before. (exists ff_lt_before_bound. ff_lt_before_bound + S ff_i_before = l) -> (((exists ff_h_before_decoded. ff_h_before_decoded + S (a) = S ((S (ff_i_before)) * c)) /\ exists ff_q_before_decoded. b = ff_q_before_decoded * S ((S (ff_i_before)) * c) + (a)))) -> exists z d. (forall ff_i_after. (exists ff_lt_after_bound. ff_lt_after_bound + S ff_i_after = sl) -> (((exists ff_h_after_decoded. ff_h_after_decoded + S (a) = S ((S (ff_i_after)) * d)) /\ exists ff_q_after_decoded. z = ff_q_after_decoded * S ((S (ff_i_after)) * d) + (a))))
use beta_prefix_extend
use le_of_succ_le_succ
use le_eq_or_lt
intro b
intro c
intro a
intro l
intro sl
intro hsl
intro hrepeat
specialize beta_prefix_extend l
specialize beta_prefix_extend b
specialize beta_prefix_extend c
specialize beta_prefix_extend a
cases beta_prefix_extend
cases beta_prefix_extend_witness
cases beta_prefix_extend_witness_witness
exists x
exists x1
intro i
intro hi
rewrite hsl at hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
rewrite hsplit_left
rewrite hsplit_left
exact beta_prefix_extend_witness_witness_left
specialize beta_prefix_extend_witness_witness_right i
specialize beta_prefix_extend_witness_witness_right a
apply beta_prefix_extend_witness_witness_right
exact hsplit_right
specialize hrepeat i
apply hrepeat
exact hsplit_right
qed

Occurrences

29,224

Distinct objects

4,561

Reused references

227

Depth

81

Cuts

873

Certificate SHA-256

fa75837000d6…

beta_repeat_existsBeta repeat exists Quadratic reciprocity campaign checked

Every value and length admit a beta-coded constant prefix.

Prerequisites

beta_repeat_empty beta_repeat_succ_extend

Used directly by

pow_exists

Exact expanded PA statement

Copy statement

forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a))))
use beta_repeat_empty
use beta_repeat_succ_extend
intro a
induction l
exists 0
exists 0
specialize beta_repeat_empty 0
specialize beta_repeat_empty 0
specialize beta_repeat_empty a
specialize beta_repeat_empty 0
apply beta_repeat_empty
refl
cases IH
cases IH_witness
specialize beta_repeat_succ_extend x
specialize beta_repeat_succ_extend x1
specialize beta_repeat_succ_extend a
specialize beta_repeat_succ_extend l
specialize beta_repeat_succ_extend (S l)
apply beta_repeat_succ_extend
refl
exact IH_witness_witness
qed

Occurrences

29,322

Distinct objects

4,639

Reused references

229

Depth

83

Cuts

877

Certificate SHA-256

b7e496b360e7…

beta_repeat_entry_eqBeta repeat entry eq Quadratic reciprocity campaign checked

Every decoded entry of a Repeat prefix equals its repeated value.

Prerequisites

beta_at_unique

Used directly by

beta_repeat_transport_entry pow_successor_decompose

Exact expanded PA statement

Copy statement

forall b c a l i x. (forall ff_i_entry. (exists ff_lt_entry_bound. ff_lt_entry_bound + S ff_i_entry = l) -> (((exists ff_h_entry_decoded. ff_h_entry_decoded + S (a) = S ((S (ff_i_entry)) * c)) /\ exists ff_q_entry_decoded. b = ff_q_entry_decoded * S ((S (ff_i_entry)) * c) + (a)))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_x. ff_h_entry_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_x. b = ff_q_entry_x * S ((S (i)) * c) + (x))) -> x = a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l i x. (forall ff_i_entry. (exists ff_lt_entry_bound. ff_lt_entry_bound + S ff_i_entry = l) -> (((exists ff_h_entry_decoded. ff_h_entry_decoded + S (a) = S ((S (ff_i_entry)) * c)) /\ exists ff_q_entry_decoded. b = ff_q_entry_decoded * S ((S (ff_i_entry)) * c) + (a)))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_x. ff_h_entry_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_x. b = ff_q_entry_x * S ((S (i)) * c) + (x))) -> x = a
use beta_at_unique
intro b
intro c
intro a
intro l
intro i
intro x
intro hrepeat
intro hi
intro hx
have ha : ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a)
specialize hrepeat i
apply hrepeat
exact hi
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique i
specialize beta_at_unique x
specialize beta_at_unique a
apply beta_at_unique
exact hx
exact ha
qed

Occurrences

1,144

Distinct objects

715

Reused references

37

Depth

60

Cuts

31

Certificate SHA-256

d54a91085710…

beta_repeat_transport_entryBeta repeat transport entry Quadratic reciprocity campaign checked

Repeat prefixes with one value preserve every decoded entry extensionally.

Prerequisites

beta_repeat_entry_eq

Used directly by

pow_functional

Exact expanded PA statement

Copy statement

forall b c z d a l. (forall ff_i_transport_l. (exists ff_lt_transport_l_bound. ff_lt_transport_l_bound + S ff_i_transport_l = l) -> (((exists ff_h_transport_l_decoded. ff_h_transport_l_decoded + S (a) = S ((S (ff_i_transport_l)) * c)) /\ exists ff_q_transport_l_decoded. b = ff_q_transport_l_decoded * S ((S (ff_i_transport_l)) * c) + (a)))) -> (forall ff_i_transport_r. (exists ff_lt_transport_r_bound. ff_lt_transport_r_bound + S ff_i_transport_r = l) -> (((exists ff_h_transport_r_decoded. ff_h_transport_r_decoded + S (a) = S ((S (ff_i_transport_r)) * d)) /\ exists ff_q_transport_r_decoded. z = ff_q_transport_r_decoded * S ((S (ff_i_transport_r)) * d) + (a)))) -> forall i x. (exists h. h + S i = l) -> (((exists ff_h_transport_x. ff_h_transport_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_transport_x. b = ff_q_transport_x * S ((S (i)) * c) + (x))) -> (((exists ff_h_transport_y. ff_h_transport_y + S (x) = S ((S (i)) * d)) /\ exists ff_q_transport_y. z = ff_q_transport_y * S ((S (i)) * d) + (x)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d a l. (forall ff_i_transport_l. (exists ff_lt_transport_l_bound. ff_lt_transport_l_bound + S ff_i_transport_l = l) -> (((exists ff_h_transport_l_decoded. ff_h_transport_l_decoded + S (a) = S ((S (ff_i_transport_l)) * c)) /\ exists ff_q_transport_l_decoded. b = ff_q_transport_l_decoded * S ((S (ff_i_transport_l)) * c) + (a)))) -> (forall ff_i_transport_r. (exists ff_lt_transport_r_bound. ff_lt_transport_r_bound + S ff_i_transport_r = l) -> (((exists ff_h_transport_r_decoded. ff_h_transport_r_decoded + S (a) = S ((S (ff_i_transport_r)) * d)) /\ exists ff_q_transport_r_decoded. z = ff_q_transport_r_decoded * S ((S (ff_i_transport_r)) * d) + (a)))) -> forall i x. (exists h. h + S i = l) -> (((exists ff_h_transport_x. ff_h_transport_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_transport_x. b = ff_q_transport_x * S ((S (i)) * c) + (x))) -> (((exists ff_h_transport_y. ff_h_transport_y + S (x) = S ((S (i)) * d)) /\ exists ff_q_transport_y. z = ff_q_transport_y * S ((S (i)) * d) + (x)))
use beta_repeat_entry_eq
intro b
intro c
intro z
intro d
intro a
intro l
intro hleft
intro hright
intro i
intro x
intro hi
intro hx
have hxa : x = a
specialize beta_repeat_entry_eq b
specialize beta_repeat_entry_eq c
specialize beta_repeat_entry_eq a
specialize beta_repeat_entry_eq l
specialize beta_repeat_entry_eq i
specialize beta_repeat_entry_eq x
apply beta_repeat_entry_eq
exact hleft
exact hi
exact hx
rewrite hxa
rewrite hxa
specialize hright i
apply hright
exact hi
qed

Occurrences

1,191

Distinct objects

762

Reused references

37

Depth

61

Cuts

32

Certificate SHA-256

42179ced89f7…

pow_existsPow exists Quadratic reciprocity campaign checked

Every base and exponent have a relational finite-product power.

Prerequisites

beta_repeat_exists beta_product_exists

Used directly by

pow_mul_exp

Exact expanded PA statement

Copy statement

forall a e. exists n. (exists ff_b_x ff_c_x. ((forall ff_i_x_repeat. (exists ff_lt_x_repeat_bound. ff_lt_x_repeat_bound + S ff_i_x_repeat = e) -> (((exists ff_h_x_repeat_decoded. ff_h_x_repeat_decoded + S (a) = S ((S (ff_i_x_repeat)) * ff_c_x)) /\ exists ff_q_x_repeat_decoded. ff_b_x = ff_q_x_repeat_decoded * S ((S (ff_i_x_repeat)) * ff_c_x) + (a)))) /\ (exists ff_u_x_product ff_v_x_product. ((((exists ff_h_x_product_start. ff_h_x_product_start + S (1) = S ((S (0)) * ff_v_x_product)) /\ exists ff_q_x_product_start. ff_u_x_product = ff_q_x_product_start * S ((S (0)) * ff_v_x_product) + (1))) /\ ((((exists ff_h_x_product_terminal. ff_h_x_product_terminal + S (n) = S ((S (e)) * ff_v_x_product)) /\ exists ff_q_x_product_terminal. ff_u_x_product = ff_q_x_product_terminal * S ((S (e)) * ff_v_x_product) + (n))) /\ forall ff_i_x_product. (exists ff_lt_x_product_bound. ff_lt_x_product_bound + S ff_i_x_product = e) -> exists ff_p_x_product ff_r_x_product ff_s_x_product. ((((exists ff_h_x_product_factor. ff_h_x_product_factor + S (ff_p_x_product) = S ((S (ff_i_x_product)) * ff_c_x)) /\ exists ff_q_x_product_factor. ff_b_x = ff_q_x_product_factor * S ((S (ff_i_x_product)) * ff_c_x) + (ff_p_x_product))) /\ ((((exists ff_h_x_product_partial. ff_h_x_product_partial + S (ff_r_x_product) = S ((S (ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_partial. ff_u_x_product = ff_q_x_product_partial * S ((S (ff_i_x_product)) * ff_v_x_product) + (ff_r_x_product))) /\ ((((exists ff_h_x_product_successor. ff_h_x_product_successor + S (ff_s_x_product) = S ((S (S ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_successor. ff_u_x_product = ff_q_x_product_successor * S ((S (S ff_i_x_product)) * ff_v_x_product) + (ff_s_x_product))) /\ ff_s_x_product = ff_r_x_product * ff_p_x_product))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e. exists n. (exists ff_b_x ff_c_x. ((forall ff_i_x_repeat. (exists ff_lt_x_repeat_bound. ff_lt_x_repeat_bound + S ff_i_x_repeat = e) -> (((exists ff_h_x_repeat_decoded. ff_h_x_repeat_decoded + S (a) = S ((S (ff_i_x_repeat)) * ff_c_x)) /\ exists ff_q_x_repeat_decoded. ff_b_x = ff_q_x_repeat_decoded * S ((S (ff_i_x_repeat)) * ff_c_x) + (a)))) /\ (exists ff_u_x_product ff_v_x_product. ((((exists ff_h_x_product_start. ff_h_x_product_start + S (1) = S ((S (0)) * ff_v_x_product)) /\ exists ff_q_x_product_start. ff_u_x_product = ff_q_x_product_start * S ((S (0)) * ff_v_x_product) + (1))) /\ ((((exists ff_h_x_product_terminal. ff_h_x_product_terminal + S (n) = S ((S (e)) * ff_v_x_product)) /\ exists ff_q_x_product_terminal. ff_u_x_product = ff_q_x_product_terminal * S ((S (e)) * ff_v_x_product) + (n))) /\ forall ff_i_x_product. (exists ff_lt_x_product_bound. ff_lt_x_product_bound + S ff_i_x_product = e) -> exists ff_p_x_product ff_r_x_product ff_s_x_product. ((((exists ff_h_x_product_factor. ff_h_x_product_factor + S (ff_p_x_product) = S ((S (ff_i_x_product)) * ff_c_x)) /\ exists ff_q_x_product_factor. ff_b_x = ff_q_x_product_factor * S ((S (ff_i_x_product)) * ff_c_x) + (ff_p_x_product))) /\ ((((exists ff_h_x_product_partial. ff_h_x_product_partial + S (ff_r_x_product) = S ((S (ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_partial. ff_u_x_product = ff_q_x_product_partial * S ((S (ff_i_x_product)) * ff_v_x_product) + (ff_r_x_product))) /\ ((((exists ff_h_x_product_successor. ff_h_x_product_successor + S (ff_s_x_product) = S ((S (S ff_i_x_product)) * ff_v_x_product)) /\ exists ff_q_x_product_successor. ff_u_x_product = ff_q_x_product_successor * S ((S (S ff_i_x_product)) * ff_v_x_product) + (ff_s_x_product))) /\ ff_s_x_product = ff_r_x_product * ff_p_x_product))))))))
use beta_repeat_exists
use beta_product_exists
intro a
intro e
have hrepeat : exists b c. (forall i. (exists h. h + S i = e) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a))
specialize beta_repeat_exists a
specialize beta_repeat_exists e
exact beta_repeat_exists
cases hrepeat
cases hrepeat_witness
specialize beta_product_exists x
specialize beta_product_exists x1
specialize beta_product_exists e
cases beta_product_exists
cases beta_product_exists_witness
cases beta_product_exists_witness_witness
exists x2
exists x
exists x1
split
exact hrepeat_witness_witness
exists x3
exists x4
exact beta_product_exists_witness_witness_witness
qed

Occurrences

59,836

Distinct objects

4,902

Reused references

241

Depth

88

Cuts

1,795

Certificate SHA-256

37a53929d638…

pow_zeroPow zero Quadratic reciprocity campaign checked

The relational zeroth power is one.

Prerequisites

beta_product_zero

Used directly by

pow_one_from_zero_successor pow_mod_congruent pow_add pow_mul_exp pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall a e n. e = 0 -> (exists ff_b_z ff_c_z. ((forall ff_i_z_repeat. (exists ff_lt_z_repeat_bound. ff_lt_z_repeat_bound + S ff_i_z_repeat = e) -> (((exists ff_h_z_repeat_decoded. ff_h_z_repeat_decoded + S (a) = S ((S (ff_i_z_repeat)) * ff_c_z)) /\ exists ff_q_z_repeat_decoded. ff_b_z = ff_q_z_repeat_decoded * S ((S (ff_i_z_repeat)) * ff_c_z) + (a)))) /\ (exists ff_u_z_product ff_v_z_product. ((((exists ff_h_z_product_start. ff_h_z_product_start + S (1) = S ((S (0)) * ff_v_z_product)) /\ exists ff_q_z_product_start. ff_u_z_product = ff_q_z_product_start * S ((S (0)) * ff_v_z_product) + (1))) /\ ((((exists ff_h_z_product_terminal. ff_h_z_product_terminal + S (n) = S ((S (e)) * ff_v_z_product)) /\ exists ff_q_z_product_terminal. ff_u_z_product = ff_q_z_product_terminal * S ((S (e)) * ff_v_z_product) + (n))) /\ forall ff_i_z_product. (exists ff_lt_z_product_bound. ff_lt_z_product_bound + S ff_i_z_product = e) -> exists ff_p_z_product ff_r_z_product ff_s_z_product. ((((exists ff_h_z_product_factor. ff_h_z_product_factor + S (ff_p_z_product) = S ((S (ff_i_z_product)) * ff_c_z)) /\ exists ff_q_z_product_factor. ff_b_z = ff_q_z_product_factor * S ((S (ff_i_z_product)) * ff_c_z) + (ff_p_z_product))) /\ ((((exists ff_h_z_product_partial. ff_h_z_product_partial + S (ff_r_z_product) = S ((S (ff_i_z_product)) * ff_v_z_product)) /\ exists ff_q_z_product_partial. ff_u_z_product = ff_q_z_product_partial * S ((S (ff_i_z_product)) * ff_v_z_product) + (ff_r_z_product))) /\ ((((exists ff_h_z_product_successor. ff_h_z_product_successor + S (ff_s_z_product) = S ((S (S ff_i_z_product)) * ff_v_z_product)) /\ exists ff_q_z_product_successor. ff_u_z_product = ff_q_z_product_successor * S ((S (S ff_i_z_product)) * ff_v_z_product) + (ff_s_z_product))) /\ ff_s_z_product = ff_r_z_product * ff_p_z_product)))))))) -> n = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e n. e = 0 -> (exists ff_b_z ff_c_z. ((forall ff_i_z_repeat. (exists ff_lt_z_repeat_bound. ff_lt_z_repeat_bound + S ff_i_z_repeat = e) -> (((exists ff_h_z_repeat_decoded. ff_h_z_repeat_decoded + S (a) = S ((S (ff_i_z_repeat)) * ff_c_z)) /\ exists ff_q_z_repeat_decoded. ff_b_z = ff_q_z_repeat_decoded * S ((S (ff_i_z_repeat)) * ff_c_z) + (a)))) /\ (exists ff_u_z_product ff_v_z_product. ((((exists ff_h_z_product_start. ff_h_z_product_start + S (1) = S ((S (0)) * ff_v_z_product)) /\ exists ff_q_z_product_start. ff_u_z_product = ff_q_z_product_start * S ((S (0)) * ff_v_z_product) + (1))) /\ ((((exists ff_h_z_product_terminal. ff_h_z_product_terminal + S (n) = S ((S (e)) * ff_v_z_product)) /\ exists ff_q_z_product_terminal. ff_u_z_product = ff_q_z_product_terminal * S ((S (e)) * ff_v_z_product) + (n))) /\ forall ff_i_z_product. (exists ff_lt_z_product_bound. ff_lt_z_product_bound + S ff_i_z_product = e) -> exists ff_p_z_product ff_r_z_product ff_s_z_product. ((((exists ff_h_z_product_factor. ff_h_z_product_factor + S (ff_p_z_product) = S ((S (ff_i_z_product)) * ff_c_z)) /\ exists ff_q_z_product_factor. ff_b_z = ff_q_z_product_factor * S ((S (ff_i_z_product)) * ff_c_z) + (ff_p_z_product))) /\ ((((exists ff_h_z_product_partial. ff_h_z_product_partial + S (ff_r_z_product) = S ((S (ff_i_z_product)) * ff_v_z_product)) /\ exists ff_q_z_product_partial. ff_u_z_product = ff_q_z_product_partial * S ((S (ff_i_z_product)) * ff_v_z_product) + (ff_r_z_product))) /\ ((((exists ff_h_z_product_successor. ff_h_z_product_successor + S (ff_s_z_product) = S ((S (S ff_i_z_product)) * ff_v_z_product)) /\ exists ff_q_z_product_successor. ff_u_z_product = ff_q_z_product_successor * S ((S (S ff_i_z_product)) * ff_v_z_product) + (ff_s_z_product))) /\ ff_s_z_product = ff_r_z_product * ff_p_z_product)))))))) -> n = 1
use beta_product_zero
intro a
intro e
intro n
intro he
intro hpow
rewrite he at hpow
rewrite he at hpow
rewrite he at hpow
rewrite he at hpow
cases hpow
cases hpow_witness
cases hpow_witness_witness
specialize beta_product_zero x
specialize beta_product_zero x1
specialize beta_product_zero n
apply beta_product_zero
exact hpow_witness_witness_right
qed

Occurrences

1,224

Distinct objects

795

Reused references

37

Depth

61

Cuts

32

Certificate SHA-256

f3e5705226dd…

pow_functionalPow functional Quadratic reciprocity campaign checked

Relational powers have a unique natural value.

Prerequisites

beta_repeat_transport_entry beta_product_transport_prefix beta_product_functional

Used directly by

pow_successor_pair_mul pow_add

Exact expanded PA statement

Copy statement

forall a e n m. (exists ff_b_l ff_c_l. ((forall ff_i_l_repeat. (exists ff_lt_l_repeat_bound. ff_lt_l_repeat_bound + S ff_i_l_repeat = e) -> (((exists ff_h_l_repeat_decoded. ff_h_l_repeat_decoded + S (a) = S ((S (ff_i_l_repeat)) * ff_c_l)) /\ exists ff_q_l_repeat_decoded. ff_b_l = ff_q_l_repeat_decoded * S ((S (ff_i_l_repeat)) * ff_c_l) + (a)))) /\ (exists ff_u_l_product ff_v_l_product. ((((exists ff_h_l_product_start. ff_h_l_product_start + S (1) = S ((S (0)) * ff_v_l_product)) /\ exists ff_q_l_product_start. ff_u_l_product = ff_q_l_product_start * S ((S (0)) * ff_v_l_product) + (1))) /\ ((((exists ff_h_l_product_terminal. ff_h_l_product_terminal + S (n) = S ((S (e)) * ff_v_l_product)) /\ exists ff_q_l_product_terminal. ff_u_l_product = ff_q_l_product_terminal * S ((S (e)) * ff_v_l_product) + (n))) /\ forall ff_i_l_product. (exists ff_lt_l_product_bound. ff_lt_l_product_bound + S ff_i_l_product = e) -> exists ff_p_l_product ff_r_l_product ff_s_l_product. ((((exists ff_h_l_product_factor. ff_h_l_product_factor + S (ff_p_l_product) = S ((S (ff_i_l_product)) * ff_c_l)) /\ exists ff_q_l_product_factor. ff_b_l = ff_q_l_product_factor * S ((S (ff_i_l_product)) * ff_c_l) + (ff_p_l_product))) /\ ((((exists ff_h_l_product_partial. ff_h_l_product_partial + S (ff_r_l_product) = S ((S (ff_i_l_product)) * ff_v_l_product)) /\ exists ff_q_l_product_partial. ff_u_l_product = ff_q_l_product_partial * S ((S (ff_i_l_product)) * ff_v_l_product) + (ff_r_l_product))) /\ ((((exists ff_h_l_product_successor. ff_h_l_product_successor + S (ff_s_l_product) = S ((S (S ff_i_l_product)) * ff_v_l_product)) /\ exists ff_q_l_product_successor. ff_u_l_product = ff_q_l_product_successor * S ((S (S ff_i_l_product)) * ff_v_l_product) + (ff_s_l_product))) /\ ff_s_l_product = ff_r_l_product * ff_p_l_product)))))))) -> (exists ff_b_r ff_c_r. ((forall ff_i_r_repeat. (exists ff_lt_r_repeat_bound. ff_lt_r_repeat_bound + S ff_i_r_repeat = e) -> (((exists ff_h_r_repeat_decoded. ff_h_r_repeat_decoded + S (a) = S ((S (ff_i_r_repeat)) * ff_c_r)) /\ exists ff_q_r_repeat_decoded. ff_b_r = ff_q_r_repeat_decoded * S ((S (ff_i_r_repeat)) * ff_c_r) + (a)))) /\ (exists ff_u_r_product ff_v_r_product. ((((exists ff_h_r_product_start. ff_h_r_product_start + S (1) = S ((S (0)) * ff_v_r_product)) /\ exists ff_q_r_product_start. ff_u_r_product = ff_q_r_product_start * S ((S (0)) * ff_v_r_product) + (1))) /\ ((((exists ff_h_r_product_terminal. ff_h_r_product_terminal + S (m) = S ((S (e)) * ff_v_r_product)) /\ exists ff_q_r_product_terminal. ff_u_r_product = ff_q_r_product_terminal * S ((S (e)) * ff_v_r_product) + (m))) /\ forall ff_i_r_product. (exists ff_lt_r_product_bound. ff_lt_r_product_bound + S ff_i_r_product = e) -> exists ff_p_r_product ff_r_r_product ff_s_r_product. ((((exists ff_h_r_product_factor. ff_h_r_product_factor + S (ff_p_r_product) = S ((S (ff_i_r_product)) * ff_c_r)) /\ exists ff_q_r_product_factor. ff_b_r = ff_q_r_product_factor * S ((S (ff_i_r_product)) * ff_c_r) + (ff_p_r_product))) /\ ((((exists ff_h_r_product_partial. ff_h_r_product_partial + S (ff_r_r_product) = S ((S (ff_i_r_product)) * ff_v_r_product)) /\ exists ff_q_r_product_partial. ff_u_r_product = ff_q_r_product_partial * S ((S (ff_i_r_product)) * ff_v_r_product) + (ff_r_r_product))) /\ ((((exists ff_h_r_product_successor. ff_h_r_product_successor + S (ff_s_r_product) = S ((S (S ff_i_r_product)) * ff_v_r_product)) /\ exists ff_q_r_product_successor. ff_u_r_product = ff_q_r_product_successor * S ((S (S ff_i_r_product)) * ff_v_r_product) + (ff_s_r_product))) /\ ff_s_r_product = ff_r_r_product * ff_p_r_product)))))))) -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e n m. (exists ff_b_l ff_c_l. ((forall ff_i_l_repeat. (exists ff_lt_l_repeat_bound. ff_lt_l_repeat_bound + S ff_i_l_repeat = e) -> (((exists ff_h_l_repeat_decoded. ff_h_l_repeat_decoded + S (a) = S ((S (ff_i_l_repeat)) * ff_c_l)) /\ exists ff_q_l_repeat_decoded. ff_b_l = ff_q_l_repeat_decoded * S ((S (ff_i_l_repeat)) * ff_c_l) + (a)))) /\ (exists ff_u_l_product ff_v_l_product. ((((exists ff_h_l_product_start. ff_h_l_product_start + S (1) = S ((S (0)) * ff_v_l_product)) /\ exists ff_q_l_product_start. ff_u_l_product = ff_q_l_product_start * S ((S (0)) * ff_v_l_product) + (1))) /\ ((((exists ff_h_l_product_terminal. ff_h_l_product_terminal + S (n) = S ((S (e)) * ff_v_l_product)) /\ exists ff_q_l_product_terminal. ff_u_l_product = ff_q_l_product_terminal * S ((S (e)) * ff_v_l_product) + (n))) /\ forall ff_i_l_product. (exists ff_lt_l_product_bound. ff_lt_l_product_bound + S ff_i_l_product = e) -> exists ff_p_l_product ff_r_l_product ff_s_l_product. ((((exists ff_h_l_product_factor. ff_h_l_product_factor + S (ff_p_l_product) = S ((S (ff_i_l_product)) * ff_c_l)) /\ exists ff_q_l_product_factor. ff_b_l = ff_q_l_product_factor * S ((S (ff_i_l_product)) * ff_c_l) + (ff_p_l_product))) /\ ((((exists ff_h_l_product_partial. ff_h_l_product_partial + S (ff_r_l_product) = S ((S (ff_i_l_product)) * ff_v_l_product)) /\ exists ff_q_l_product_partial. ff_u_l_product = ff_q_l_product_partial * S ((S (ff_i_l_product)) * ff_v_l_product) + (ff_r_l_product))) /\ ((((exists ff_h_l_product_successor. ff_h_l_product_successor + S (ff_s_l_product) = S ((S (S ff_i_l_product)) * ff_v_l_product)) /\ exists ff_q_l_product_successor. ff_u_l_product = ff_q_l_product_successor * S ((S (S ff_i_l_product)) * ff_v_l_product) + (ff_s_l_product))) /\ ff_s_l_product = ff_r_l_product * ff_p_l_product)))))))) -> (exists ff_b_r ff_c_r. ((forall ff_i_r_repeat. (exists ff_lt_r_repeat_bound. ff_lt_r_repeat_bound + S ff_i_r_repeat = e) -> (((exists ff_h_r_repeat_decoded. ff_h_r_repeat_decoded + S (a) = S ((S (ff_i_r_repeat)) * ff_c_r)) /\ exists ff_q_r_repeat_decoded. ff_b_r = ff_q_r_repeat_decoded * S ((S (ff_i_r_repeat)) * ff_c_r) + (a)))) /\ (exists ff_u_r_product ff_v_r_product. ((((exists ff_h_r_product_start. ff_h_r_product_start + S (1) = S ((S (0)) * ff_v_r_product)) /\ exists ff_q_r_product_start. ff_u_r_product = ff_q_r_product_start * S ((S (0)) * ff_v_r_product) + (1))) /\ ((((exists ff_h_r_product_terminal. ff_h_r_product_terminal + S (m) = S ((S (e)) * ff_v_r_product)) /\ exists ff_q_r_product_terminal. ff_u_r_product = ff_q_r_product_terminal * S ((S (e)) * ff_v_r_product) + (m))) /\ forall ff_i_r_product. (exists ff_lt_r_product_bound. ff_lt_r_product_bound + S ff_i_r_product = e) -> exists ff_p_r_product ff_r_r_product ff_s_r_product. ((((exists ff_h_r_product_factor. ff_h_r_product_factor + S (ff_p_r_product) = S ((S (ff_i_r_product)) * ff_c_r)) /\ exists ff_q_r_product_factor. ff_b_r = ff_q_r_product_factor * S ((S (ff_i_r_product)) * ff_c_r) + (ff_p_r_product))) /\ ((((exists ff_h_r_product_partial. ff_h_r_product_partial + S (ff_r_r_product) = S ((S (ff_i_r_product)) * ff_v_r_product)) /\ exists ff_q_r_product_partial. ff_u_r_product = ff_q_r_product_partial * S ((S (ff_i_r_product)) * ff_v_r_product) + (ff_r_r_product))) /\ ((((exists ff_h_r_product_successor. ff_h_r_product_successor + S (ff_s_r_product) = S ((S (S ff_i_r_product)) * ff_v_r_product)) /\ exists ff_q_r_product_successor. ff_u_r_product = ff_q_r_product_successor * S ((S (S ff_i_r_product)) * ff_v_r_product) + (ff_s_r_product))) /\ ff_s_r_product = ff_r_r_product * ff_p_r_product)))))))) -> n = m
use beta_repeat_transport_entry
use beta_product_transport_prefix
use beta_product_functional
intro a
intro e
intro n
intro m
intro hn
intro hm
cases hn
cases hn_witness
cases hn_witness_witness
cases hm
cases hm_witness
cases hm_witness_witness
have htransport : exists ff_u_transport ff_v_transport. ((((exists ff_h_transport_start. ff_h_transport_start + S (1) = S ((S (0)) * ff_v_transport)) /\ exists ff_q_transport_start. ff_u_transport = ff_q_transport_start * S ((S (0)) * ff_v_transport) + (1))) /\ ((((exists ff_h_transport_terminal. ff_h_transport_terminal + S (n) = S ((S (e)) * ff_v_transport)) /\ exists ff_q_transport_terminal. ff_u_transport = ff_q_transport_terminal * S ((S (e)) * ff_v_transport) + (n))) /\ forall ff_i_transport. (exists ff_lt_transport_bound. ff_lt_transport_bound + S ff_i_transport = e) -> exists ff_p_transport ff_r_transport ff_s_transport. ((((exists ff_h_transport_factor. ff_h_transport_factor + S (ff_p_transport) = S ((S (ff_i_transport)) * x3)) /\ exists ff_q_transport_factor. x2 = ff_q_transport_factor * S ((S (ff_i_transport)) * x3) + (ff_p_transport))) /\ ((((exists ff_h_transport_partial. ff_h_transport_partial + S (ff_r_transport) = S ((S (ff_i_transport)) * ff_v_transport)) /\ exists ff_q_transport_partial. ff_u_transport = ff_q_transport_partial * S ((S (ff_i_transport)) * ff_v_transport) + (ff_r_transport))) /\ ((((exists ff_h_transport_successor. ff_h_transport_successor + S (ff_s_transport) = S ((S (S ff_i_transport)) * ff_v_transport)) /\ exists ff_q_transport_successor. ff_u_transport = ff_q_transport_successor * S ((S (S ff_i_transport)) * ff_v_transport) + (ff_s_transport))) /\ ff_s_transport = ff_r_transport * ff_p_transport)))))
specialize beta_product_transport_prefix x
specialize beta_product_transport_prefix x1
specialize beta_product_transport_prefix x2
specialize beta_product_transport_prefix x3
specialize beta_product_transport_prefix e
specialize beta_product_transport_prefix n
apply beta_product_transport_prefix
exact hn_witness_witness_right
intro i
intro p
intro hi
intro hp
specialize beta_repeat_transport_entry x
specialize beta_repeat_transport_entry x1
specialize beta_repeat_transport_entry x2
specialize beta_repeat_transport_entry x3
specialize beta_repeat_transport_entry a
specialize beta_repeat_transport_entry e
have hentries : forall i p. (exists h. h + S i = e) -> (((exists ff_h_pow_transport_l. ff_h_pow_transport_l + S (p) = S ((S (i)) * x1)) /\ exists ff_q_pow_transport_l. x = ff_q_pow_transport_l * S ((S (i)) * x1) + (p))) -> (((exists ff_h_pow_transport_r. ff_h_pow_transport_r + S (p) = S ((S (i)) * x3)) /\ exists ff_q_pow_transport_r. x2 = ff_q_pow_transport_r * S ((S (i)) * x3) + (p)))
apply beta_repeat_transport_entry
exact hn_witness_witness_left
exact hm_witness_witness_left
specialize hentries i
specialize hentries p
apply hentries
exact hi
exact hp
cases htransport
cases htransport_witness
cases hm_witness_witness_right
cases hm_witness_witness_right_witness
specialize beta_product_functional x2
specialize beta_product_functional x3
specialize beta_product_functional e
specialize beta_product_functional n
specialize beta_product_functional x4
specialize beta_product_functional x5
specialize beta_product_functional m
specialize beta_product_functional x6
specialize beta_product_functional x7
apply beta_product_functional
exact htransport_witness_witness
exact hm_witness_witness_right_witness_witness
qed

Occurrences

2,705

Distinct objects

1,111

Reused references

40

Depth

63

Cuts

71

Certificate SHA-256

93be6fa1bf46…

pow_successor_decomposePow successor decompose Quadratic reciprocity campaign checked

A successor relational power is its predecessor power times the base.

Prerequisites

beta_product_succ_decompose beta_repeat_entry_eq le_refl le_succ

Used directly by

pow_one_from_zero_successor pow_successor_pair_mul pow_mod_congruent pow_two_from_one_successor pow_add pow_mul_exp pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall a e se n. se = S e -> (exists ff_b_s ff_c_s. ((forall ff_i_s_repeat. (exists ff_lt_s_repeat_bound. ff_lt_s_repeat_bound + S ff_i_s_repeat = se) -> (((exists ff_h_s_repeat_decoded. ff_h_s_repeat_decoded + S (a) = S ((S (ff_i_s_repeat)) * ff_c_s)) /\ exists ff_q_s_repeat_decoded. ff_b_s = ff_q_s_repeat_decoded * S ((S (ff_i_s_repeat)) * ff_c_s) + (a)))) /\ (exists ff_u_s_product ff_v_s_product. ((((exists ff_h_s_product_start. ff_h_s_product_start + S (1) = S ((S (0)) * ff_v_s_product)) /\ exists ff_q_s_product_start. ff_u_s_product = ff_q_s_product_start * S ((S (0)) * ff_v_s_product) + (1))) /\ ((((exists ff_h_s_product_terminal. ff_h_s_product_terminal + S (n) = S ((S (se)) * ff_v_s_product)) /\ exists ff_q_s_product_terminal. ff_u_s_product = ff_q_s_product_terminal * S ((S (se)) * ff_v_s_product) + (n))) /\ forall ff_i_s_product. (exists ff_lt_s_product_bound. ff_lt_s_product_bound + S ff_i_s_product = se) -> exists ff_p_s_product ff_r_s_product ff_s_s_product. ((((exists ff_h_s_product_factor. ff_h_s_product_factor + S (ff_p_s_product) = S ((S (ff_i_s_product)) * ff_c_s)) /\ exists ff_q_s_product_factor. ff_b_s = ff_q_s_product_factor * S ((S (ff_i_s_product)) * ff_c_s) + (ff_p_s_product))) /\ ((((exists ff_h_s_product_partial. ff_h_s_product_partial + S (ff_r_s_product) = S ((S (ff_i_s_product)) * ff_v_s_product)) /\ exists ff_q_s_product_partial. ff_u_s_product = ff_q_s_product_partial * S ((S (ff_i_s_product)) * ff_v_s_product) + (ff_r_s_product))) /\ ((((exists ff_h_s_product_successor. ff_h_s_product_successor + S (ff_s_s_product) = S ((S (S ff_i_s_product)) * ff_v_s_product)) /\ exists ff_q_s_product_successor. ff_u_s_product = ff_q_s_product_successor * S ((S (S ff_i_s_product)) * ff_v_s_product) + (ff_s_s_product))) /\ ff_s_s_product = ff_r_s_product * ff_p_s_product)))))))) -> exists r. (exists ff_b_p ff_c_p. ((forall ff_i_p_repeat. (exists ff_lt_p_repeat_bound. ff_lt_p_repeat_bound + S ff_i_p_repeat = e) -> (((exists ff_h_p_repeat_decoded. ff_h_p_repeat_decoded + S (a) = S ((S (ff_i_p_repeat)) * ff_c_p)) /\ exists ff_q_p_repeat_decoded. ff_b_p = ff_q_p_repeat_decoded * S ((S (ff_i_p_repeat)) * ff_c_p) + (a)))) /\ (exists ff_u_p_product ff_v_p_product. ((((exists ff_h_p_product_start. ff_h_p_product_start + S (1) = S ((S (0)) * ff_v_p_product)) /\ exists ff_q_p_product_start. ff_u_p_product = ff_q_p_product_start * S ((S (0)) * ff_v_p_product) + (1))) /\ ((((exists ff_h_p_product_terminal. ff_h_p_product_terminal + S (r) = S ((S (e)) * ff_v_p_product)) /\ exists ff_q_p_product_terminal. ff_u_p_product = ff_q_p_product_terminal * S ((S (e)) * ff_v_p_product) + (r))) /\ forall ff_i_p_product. (exists ff_lt_p_product_bound. ff_lt_p_product_bound + S ff_i_p_product = e) -> exists ff_p_p_product ff_r_p_product ff_s_p_product. ((((exists ff_h_p_product_factor. ff_h_p_product_factor + S (ff_p_p_product) = S ((S (ff_i_p_product)) * ff_c_p)) /\ exists ff_q_p_product_factor. ff_b_p = ff_q_p_product_factor * S ((S (ff_i_p_product)) * ff_c_p) + (ff_p_p_product))) /\ ((((exists ff_h_p_product_partial. ff_h_p_product_partial + S (ff_r_p_product) = S ((S (ff_i_p_product)) * ff_v_p_product)) /\ exists ff_q_p_product_partial. ff_u_p_product = ff_q_p_product_partial * S ((S (ff_i_p_product)) * ff_v_p_product) + (ff_r_p_product))) /\ ((((exists ff_h_p_product_successor. ff_h_p_product_successor + S (ff_s_p_product) = S ((S (S ff_i_p_product)) * ff_v_p_product)) /\ exists ff_q_p_product_successor. ff_u_p_product = ff_q_p_product_successor * S ((S (S ff_i_p_product)) * ff_v_p_product) + (ff_s_p_product))) /\ ff_s_p_product = ff_r_p_product * ff_p_p_product)))))))) /\ n = r * a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e se n. se = S e -> (exists ff_b_s ff_c_s. ((forall ff_i_s_repeat. (exists ff_lt_s_repeat_bound. ff_lt_s_repeat_bound + S ff_i_s_repeat = se) -> (((exists ff_h_s_repeat_decoded. ff_h_s_repeat_decoded + S (a) = S ((S (ff_i_s_repeat)) * ff_c_s)) /\ exists ff_q_s_repeat_decoded. ff_b_s = ff_q_s_repeat_decoded * S ((S (ff_i_s_repeat)) * ff_c_s) + (a)))) /\ (exists ff_u_s_product ff_v_s_product. ((((exists ff_h_s_product_start. ff_h_s_product_start + S (1) = S ((S (0)) * ff_v_s_product)) /\ exists ff_q_s_product_start. ff_u_s_product = ff_q_s_product_start * S ((S (0)) * ff_v_s_product) + (1))) /\ ((((exists ff_h_s_product_terminal. ff_h_s_product_terminal + S (n) = S ((S (se)) * ff_v_s_product)) /\ exists ff_q_s_product_terminal. ff_u_s_product = ff_q_s_product_terminal * S ((S (se)) * ff_v_s_product) + (n))) /\ forall ff_i_s_product. (exists ff_lt_s_product_bound. ff_lt_s_product_bound + S ff_i_s_product = se) -> exists ff_p_s_product ff_r_s_product ff_s_s_product. ((((exists ff_h_s_product_factor. ff_h_s_product_factor + S (ff_p_s_product) = S ((S (ff_i_s_product)) * ff_c_s)) /\ exists ff_q_s_product_factor. ff_b_s = ff_q_s_product_factor * S ((S (ff_i_s_product)) * ff_c_s) + (ff_p_s_product))) /\ ((((exists ff_h_s_product_partial. ff_h_s_product_partial + S (ff_r_s_product) = S ((S (ff_i_s_product)) * ff_v_s_product)) /\ exists ff_q_s_product_partial. ff_u_s_product = ff_q_s_product_partial * S ((S (ff_i_s_product)) * ff_v_s_product) + (ff_r_s_product))) /\ ((((exists ff_h_s_product_successor. ff_h_s_product_successor + S (ff_s_s_product) = S ((S (S ff_i_s_product)) * ff_v_s_product)) /\ exists ff_q_s_product_successor. ff_u_s_product = ff_q_s_product_successor * S ((S (S ff_i_s_product)) * ff_v_s_product) + (ff_s_s_product))) /\ ff_s_s_product = ff_r_s_product * ff_p_s_product)))))))) -> exists r. (exists ff_b_p ff_c_p. ((forall ff_i_p_repeat. (exists ff_lt_p_repeat_bound. ff_lt_p_repeat_bound + S ff_i_p_repeat = e) -> (((exists ff_h_p_repeat_decoded. ff_h_p_repeat_decoded + S (a) = S ((S (ff_i_p_repeat)) * ff_c_p)) /\ exists ff_q_p_repeat_decoded. ff_b_p = ff_q_p_repeat_decoded * S ((S (ff_i_p_repeat)) * ff_c_p) + (a)))) /\ (exists ff_u_p_product ff_v_p_product. ((((exists ff_h_p_product_start. ff_h_p_product_start + S (1) = S ((S (0)) * ff_v_p_product)) /\ exists ff_q_p_product_start. ff_u_p_product = ff_q_p_product_start * S ((S (0)) * ff_v_p_product) + (1))) /\ ((((exists ff_h_p_product_terminal. ff_h_p_product_terminal + S (r) = S ((S (e)) * ff_v_p_product)) /\ exists ff_q_p_product_terminal. ff_u_p_product = ff_q_p_product_terminal * S ((S (e)) * ff_v_p_product) + (r))) /\ forall ff_i_p_product. (exists ff_lt_p_product_bound. ff_lt_p_product_bound + S ff_i_p_product = e) -> exists ff_p_p_product ff_r_p_product ff_s_p_product. ((((exists ff_h_p_product_factor. ff_h_p_product_factor + S (ff_p_p_product) = S ((S (ff_i_p_product)) * ff_c_p)) /\ exists ff_q_p_product_factor. ff_b_p = ff_q_p_product_factor * S ((S (ff_i_p_product)) * ff_c_p) + (ff_p_p_product))) /\ ((((exists ff_h_p_product_partial. ff_h_p_product_partial + S (ff_r_p_product) = S ((S (ff_i_p_product)) * ff_v_p_product)) /\ exists ff_q_p_product_partial. ff_u_p_product = ff_q_p_product_partial * S ((S (ff_i_p_product)) * ff_v_p_product) + (ff_r_p_product))) /\ ((((exists ff_h_p_product_successor. ff_h_p_product_successor + S (ff_s_p_product) = S ((S (S ff_i_p_product)) * ff_v_p_product)) /\ exists ff_q_p_product_successor. ff_u_p_product = ff_q_p_product_successor * S ((S (S ff_i_p_product)) * ff_v_p_product) + (ff_s_p_product))) /\ ff_s_p_product = ff_r_p_product * ff_p_p_product)))))))) /\ n = r * a
use beta_product_succ_decompose
use beta_repeat_entry_eq
use le_refl
use le_succ
intro a
intro e
intro se
intro n
intro hse
intro hpow
rewrite hse at hpow
rewrite hse at hpow
rewrite hse at hpow
rewrite hse at hpow
cases hpow
cases hpow_witness
cases hpow_witness_witness
have hdecomp : exists p r. (((exists ff_h_pow_succ_factor. ff_h_pow_succ_factor + S (p) = S ((S (e)) * x1)) /\ exists ff_q_pow_succ_factor. x = ff_q_pow_succ_factor * S ((S (e)) * x1) + (p))) /\ ((exists ff_u_pow_succ_prefix ff_v_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_start. ff_h_pow_succ_prefix_start + S (1) = S ((S (0)) * ff_v_pow_succ_prefix)) /\ exists ff_q_pow_succ_prefix_start. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_start * S ((S (0)) * ff_v_pow_succ_prefix) + (1))) /\ ((((exists ff_h_pow_succ_prefix_terminal. ff_h_pow_succ_prefix_terminal + S (r) = S ((S (e)) * ff_v_pow_succ_prefix)) /\ exists ff_q_pow_succ_prefix_terminal. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_terminal * S ((S (e)) * ff_v_pow_succ_prefix) + (r))) /\ forall ff_i_pow_succ_prefix. (exists ff_lt_pow_succ_prefix_bound. ff_lt_pow_succ_prefix_bound + S ff_i_pow_succ_prefix = e) -> exists ff_p_pow_succ_prefix ff_r_pow_succ_prefix ff_s_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_factor. ff_h_pow_succ_prefix_factor + S (ff_p_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * x1)) /\ exists ff_q_pow_succ_prefix_factor. x = ff_q_pow_succ_prefix_factor * S ((S (ff_i_pow_succ_prefix)) * x1) + (ff_p_pow_succ_prefix))) /\ ((((exists ff_h_pow_succ_prefix_partial. ff_h_pow_succ_prefix_partial + S (ff_r_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\ exists ff_q_pow_succ_prefix_partial. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_partial * S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_r_pow_succ_prefix))) /\ ((((exists ff_h_pow_succ_prefix_successor. ff_h_pow_succ_prefix_successor + S (ff_s_pow_succ_prefix) = S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\ exists ff_q_pow_succ_prefix_successor. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_successor * S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_s_pow_succ_prefix))) /\ ff_s_pow_succ_prefix = ff_r_pow_succ_prefix * ff_p_pow_succ_prefix)))))) /\ n = r * p)
specialize beta_product_succ_decompose x
specialize beta_product_succ_decompose x1
specialize beta_product_succ_decompose e
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hpow_witness_witness_right
cases hdecomp
cases hdecomp_witness
cases hdecomp_witness_witness
cases hdecomp_witness_witness_right
have hpa : x2 = a
specialize beta_repeat_entry_eq x
specialize beta_repeat_entry_eq x1
specialize beta_repeat_entry_eq a
specialize beta_repeat_entry_eq (S e)
specialize beta_repeat_entry_eq e
specialize beta_repeat_entry_eq x2
apply beta_repeat_entry_eq
exact hpow_witness_witness_left
specialize le_refl (S e)
exact le_refl
exact hdecomp_witness_witness_left
exists x3
split
exists x
exists x1
split
intro i
intro hi
specialize hpow_witness_witness_left i
apply hpow_witness_witness_left
specialize le_succ (S i)
specialize le_succ e
apply le_succ
exact hi
exact hdecomp_witness_witness_right_left
trans x3 * x2
exact hdecomp_witness_witness_right_right
rewrite hpa
refl
qed

Occurrences

2,541

Distinct objects

882

Reused references

42

Depth

63

Cuts

72

Certificate SHA-256

70defcae97b2…

beta_range_emptyBeta range empty Quadratic reciprocity campaign checked

Every consecutive beta range of length zero is vacuous.

Prerequisites

add_eq_zero_right succ_ne_zero

Used directly by

beta_range_exists

Exact expanded PA statement

Copy statement

forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a + ff_i_empty) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a + ff_i_empty))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a + ff_i_empty) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a + ff_i_empty))))
use add_eq_zero_right
use succ_ne_zero
intro b
intro c
intro a
intro l
intro hl
intro i
intro hi
rewrite hl at hi
exfalso
cases hi
have hsi : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi
qed

Occurrences

42

Distinct objects

42

Reused references

0

Depth

16

Cuts

2

Certificate SHA-256

d7a576b1da6f…

beta_range_succ_extendBeta range succ extend Quadratic reciprocity campaign checked

Recode a consecutive prefix and append its next value.

Prerequisites

beta_prefix_extend le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_range_exists

Exact expanded PA statement

Copy statement

forall b c a l sl. sl = S l -> (forall ff_i_before. (exists ff_lt_before_bound. ff_lt_before_bound + S ff_i_before = l) -> (((exists ff_h_before_decoded. ff_h_before_decoded + S (a + ff_i_before) = S ((S (ff_i_before)) * c)) /\ exists ff_q_before_decoded. b = ff_q_before_decoded * S ((S (ff_i_before)) * c) + (a + ff_i_before)))) -> exists z d. (forall ff_i_after. (exists ff_lt_after_bound. ff_lt_after_bound + S ff_i_after = sl) -> (((exists ff_h_after_decoded. ff_h_after_decoded + S (a + ff_i_after) = S ((S (ff_i_after)) * d)) /\ exists ff_q_after_decoded. z = ff_q_after_decoded * S ((S (ff_i_after)) * d) + (a + ff_i_after))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l sl. sl = S l -> (forall ff_i_before. (exists ff_lt_before_bound. ff_lt_before_bound + S ff_i_before = l) -> (((exists ff_h_before_decoded. ff_h_before_decoded + S (a + ff_i_before) = S ((S (ff_i_before)) * c)) /\ exists ff_q_before_decoded. b = ff_q_before_decoded * S ((S (ff_i_before)) * c) + (a + ff_i_before)))) -> exists z d. (forall ff_i_after. (exists ff_lt_after_bound. ff_lt_after_bound + S ff_i_after = sl) -> (((exists ff_h_after_decoded. ff_h_after_decoded + S (a + ff_i_after) = S ((S (ff_i_after)) * d)) /\ exists ff_q_after_decoded. z = ff_q_after_decoded * S ((S (ff_i_after)) * d) + (a + ff_i_after))))
use beta_prefix_extend
use le_of_succ_le_succ
use le_eq_or_lt
intro b
intro c
intro a
intro l
intro sl
intro hsl
intro hrange
specialize beta_prefix_extend l
specialize beta_prefix_extend b
specialize beta_prefix_extend c
specialize beta_prefix_extend (a + l)
cases beta_prefix_extend
cases beta_prefix_extend_witness
cases beta_prefix_extend_witness_witness
exists x
exists x1
intro i
intro hi
rewrite hsl at hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
rewrite hsplit_left
rewrite hsplit_left
rewrite hsplit_left
rewrite hsplit_left
exact beta_prefix_extend_witness_witness_left
specialize beta_prefix_extend_witness_witness_right i
specialize beta_prefix_extend_witness_witness_right (a + i)
apply beta_prefix_extend_witness_witness_right
exact hsplit_right
specialize hrange i
apply hrange
exact hsplit_right
qed

Occurrences

29,230

Distinct objects

4,567

Reused references

227

Depth

81

Cuts

873

Certificate SHA-256

379e8d001340…

beta_range_existsBeta range exists Quadratic reciprocity campaign checked

Every start and length admit a beta-coded consecutive range.

Prerequisites

beta_range_empty beta_range_succ_extend

Used directly by

factorial_exists

Exact expanded PA statement

Copy statement

forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a + ff_i_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a + ff_i_r))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a + ff_i_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a + ff_i_r))))
use beta_range_empty
use beta_range_succ_extend
intro a
induction l
exists 0
exists 0
specialize beta_range_empty 0
specialize beta_range_empty 0
specialize beta_range_empty a
specialize beta_range_empty 0
apply beta_range_empty
refl
cases IH
cases IH_witness
specialize beta_range_succ_extend x
specialize beta_range_succ_extend x1
specialize beta_range_succ_extend a
specialize beta_range_succ_extend l
specialize beta_range_succ_extend (S l)
apply beta_range_succ_extend
refl
exact IH_witness_witness
qed

Occurrences

29,328

Distinct objects

4,645

Reused references

229

Depth

83

Cuts

877

Certificate SHA-256

a1cbdd3326cc…

beta_range_entry_eqBeta range entry eq Quadratic reciprocity campaign checked

A decoded entry of a Range prefix is its start plus its index.

Prerequisites

beta_at_unique

Used directly by

beta_range_transport_entry factorial_succ_decompose beta_range_injective beta_half_range_entry_bounds

Exact expanded PA statement

Copy statement

forall b c a l i x. (forall ff_i_entry. (exists ff_lt_entry_bound. ff_lt_entry_bound + S ff_i_entry = l) -> (((exists ff_h_entry_decoded. ff_h_entry_decoded + S (a + ff_i_entry) = S ((S (ff_i_entry)) * c)) /\ exists ff_q_entry_decoded. b = ff_q_entry_decoded * S ((S (ff_i_entry)) * c) + (a + ff_i_entry)))) -> (exists h. h + S i = l) -> (((exists ff_h_range_entry_x. ff_h_range_entry_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_entry_x. b = ff_q_range_entry_x * S ((S (i)) * c) + (x))) -> x = a + i

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l i x. (forall ff_i_entry. (exists ff_lt_entry_bound. ff_lt_entry_bound + S ff_i_entry = l) -> (((exists ff_h_entry_decoded. ff_h_entry_decoded + S (a + ff_i_entry) = S ((S (ff_i_entry)) * c)) /\ exists ff_q_entry_decoded. b = ff_q_entry_decoded * S ((S (ff_i_entry)) * c) + (a + ff_i_entry)))) -> (exists h. h + S i = l) -> (((exists ff_h_range_entry_x. ff_h_range_entry_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_entry_x. b = ff_q_range_entry_x * S ((S (i)) * c) + (x))) -> x = a + i
use beta_at_unique
intro b
intro c
intro a
intro l
intro i
intro x
intro hrange
intro hi
intro hx
have ha : ((exists h. h + S (a + i) = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + (a + i))
specialize hrange i
apply hrange
exact hi
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique i
specialize beta_at_unique x
specialize beta_at_unique (a + i)
apply beta_at_unique
exact hx
exact ha
qed

Occurrences

1,144

Distinct objects

715

Reused references

37

Depth

60

Cuts

31

Certificate SHA-256

0dd30bba2db7…

beta_range_transport_entryBeta range transport entry Quadratic reciprocity campaign checked

Two Range codes preserve every decoded entry extensionally.

Prerequisites

beta_range_entry_eq

Used directly by

factorial_functional

Exact expanded PA statement

Copy statement

forall b c z d a l. (forall ff_i_transport_l. (exists ff_lt_transport_l_bound. ff_lt_transport_l_bound + S ff_i_transport_l = l) -> (((exists ff_h_transport_l_decoded. ff_h_transport_l_decoded + S (a + ff_i_transport_l) = S ((S (ff_i_transport_l)) * c)) /\ exists ff_q_transport_l_decoded. b = ff_q_transport_l_decoded * S ((S (ff_i_transport_l)) * c) + (a + ff_i_transport_l)))) -> (forall ff_i_transport_r. (exists ff_lt_transport_r_bound. ff_lt_transport_r_bound + S ff_i_transport_r = l) -> (((exists ff_h_transport_r_decoded. ff_h_transport_r_decoded + S (a + ff_i_transport_r) = S ((S (ff_i_transport_r)) * d)) /\ exists ff_q_transport_r_decoded. z = ff_q_transport_r_decoded * S ((S (ff_i_transport_r)) * d) + (a + ff_i_transport_r)))) -> forall i x. (exists h. h + S i = l) -> (((exists ff_h_range_transport_x. ff_h_range_transport_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_transport_x. b = ff_q_range_transport_x * S ((S (i)) * c) + (x))) -> (((exists ff_h_range_transport_y. ff_h_range_transport_y + S (x) = S ((S (i)) * d)) /\ exists ff_q_range_transport_y. z = ff_q_range_transport_y * S ((S (i)) * d) + (x)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d a l. (forall ff_i_transport_l. (exists ff_lt_transport_l_bound. ff_lt_transport_l_bound + S ff_i_transport_l = l) -> (((exists ff_h_transport_l_decoded. ff_h_transport_l_decoded + S (a + ff_i_transport_l) = S ((S (ff_i_transport_l)) * c)) /\ exists ff_q_transport_l_decoded. b = ff_q_transport_l_decoded * S ((S (ff_i_transport_l)) * c) + (a + ff_i_transport_l)))) -> (forall ff_i_transport_r. (exists ff_lt_transport_r_bound. ff_lt_transport_r_bound + S ff_i_transport_r = l) -> (((exists ff_h_transport_r_decoded. ff_h_transport_r_decoded + S (a + ff_i_transport_r) = S ((S (ff_i_transport_r)) * d)) /\ exists ff_q_transport_r_decoded. z = ff_q_transport_r_decoded * S ((S (ff_i_transport_r)) * d) + (a + ff_i_transport_r)))) -> forall i x. (exists h. h + S i = l) -> (((exists ff_h_range_transport_x. ff_h_range_transport_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_transport_x. b = ff_q_range_transport_x * S ((S (i)) * c) + (x))) -> (((exists ff_h_range_transport_y. ff_h_range_transport_y + S (x) = S ((S (i)) * d)) /\ exists ff_q_range_transport_y. z = ff_q_range_transport_y * S ((S (i)) * d) + (x)))
use beta_range_entry_eq
intro b
intro c
intro z
intro d
intro a
intro l
intro hleft
intro hright
intro i
intro x
intro hi
intro hx
have hxa : x = a + i
specialize beta_range_entry_eq b
specialize beta_range_entry_eq c
specialize beta_range_entry_eq a
specialize beta_range_entry_eq l
specialize beta_range_entry_eq i
specialize beta_range_entry_eq x
apply beta_range_entry_eq
exact hleft
exact hi
exact hx
rewrite hxa
rewrite hxa
specialize hright i
apply hright
exact hi
qed

Occurrences

1,191

Distinct objects

762

Reused references

37

Depth

61

Cuts

32

Certificate SHA-256

fcaeff825005…

beta_prefix_sum_trace_existsBeta prefix sum trace exists Quadratic reciprocity campaign checked

Every decoded beta prefix admits an exact beta-coded prefix-sum trace.

Prerequisites

beta_at_self_of_bound add_eq_zero_right succ_ne_zero beta_at_exists beta_prefix_extend zero_le succ_le_succ le_refl le_of_succ_le_succ le_eq_or_lt one_mul

Used directly by

beta_sum_exists

Exact expanded PA statement

Copy statement

forall b c l. exists fs_u_trace fs_v_trace. ((((exists fs_h_trace_start. fs_h_trace_start + S (0) = S ((S (0)) * fs_v_trace)) /\ exists fs_q_trace_start. fs_u_trace = fs_q_trace_start * S ((S (0)) * fs_v_trace) + (0))) /\ forall fs_i_trace_steps. (exists fs_lt_trace_steps_bound. fs_lt_trace_steps_bound + S fs_i_trace_steps = l) -> exists fs_a_trace_steps fs_r_trace_steps fs_s_trace_steps. ((((exists fs_h_trace_steps_summand. fs_h_trace_steps_summand + S (fs_a_trace_steps) = S ((S (fs_i_trace_steps)) * c)) /\ exists fs_q_trace_steps_summand. b = fs_q_trace_steps_summand * S ((S (fs_i_trace_steps)) * c) + (fs_a_trace_steps))) /\ ((((exists fs_h_trace_steps_partial. fs_h_trace_steps_partial + S (fs_r_trace_steps) = S ((S (fs_i_trace_steps)) * fs_v_trace)) /\ exists fs_q_trace_steps_partial. fs_u_trace = fs_q_trace_steps_partial * S ((S (fs_i_trace_steps)) * fs_v_trace) + (fs_r_trace_steps))) /\ ((((exists fs_h_trace_steps_successor. fs_h_trace_steps_successor + S (fs_s_trace_steps) = S ((S (S fs_i_trace_steps)) * fs_v_trace)) /\ exists fs_q_trace_steps_successor. fs_u_trace = fs_q_trace_steps_successor * S ((S (S fs_i_trace_steps)) * fs_v_trace) + (fs_s_trace_steps))) /\ fs_s_trace_steps = fs_r_trace_steps + fs_a_trace_steps))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists fs_u_trace fs_v_trace. ((((exists fs_h_trace_start. fs_h_trace_start + S (0) = S ((S (0)) * fs_v_trace)) /\ exists fs_q_trace_start. fs_u_trace = fs_q_trace_start * S ((S (0)) * fs_v_trace) + (0))) /\ forall fs_i_trace_steps. (exists fs_lt_trace_steps_bound. fs_lt_trace_steps_bound + S fs_i_trace_steps = l) -> exists fs_a_trace_steps fs_r_trace_steps fs_s_trace_steps. ((((exists fs_h_trace_steps_summand. fs_h_trace_steps_summand + S (fs_a_trace_steps) = S ((S (fs_i_trace_steps)) * c)) /\ exists fs_q_trace_steps_summand. b = fs_q_trace_steps_summand * S ((S (fs_i_trace_steps)) * c) + (fs_a_trace_steps))) /\ ((((exists fs_h_trace_steps_partial. fs_h_trace_steps_partial + S (fs_r_trace_steps) = S ((S (fs_i_trace_steps)) * fs_v_trace)) /\ exists fs_q_trace_steps_partial. fs_u_trace = fs_q_trace_steps_partial * S ((S (fs_i_trace_steps)) * fs_v_trace) + (fs_r_trace_steps))) /\ ((((exists fs_h_trace_steps_successor. fs_h_trace_steps_successor + S (fs_s_trace_steps) = S ((S (S fs_i_trace_steps)) * fs_v_trace)) /\ exists fs_q_trace_steps_successor. fs_u_trace = fs_q_trace_steps_successor * S ((S (S fs_i_trace_steps)) * fs_v_trace) + (fs_s_trace_steps))) /\ fs_s_trace_steps = fs_r_trace_steps + fs_a_trace_steps))))
use beta_at_self_of_bound
use add_eq_zero_right
use succ_ne_zero
use beta_at_exists
use beta_prefix_extend
use zero_le
use succ_le_succ
use le_refl
use le_of_succ_le_succ
use le_eq_or_lt
use one_mul
intro b
intro c
induction l
exists 0
exists 1
split
specialize beta_at_self_of_bound 1
specialize beta_at_self_of_bound 0
specialize beta_at_self_of_bound 0
apply beta_at_self_of_bound
specialize one_mul 1
rewrite one_mul
specialize succ_le_succ 0
specialize succ_le_succ (S 0)
apply succ_le_succ
specialize zero_le (S 0)
exact zero_le
intro i
intro hi
exfalso
cases hi
have hsi0 : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi0
have htrace : exists fs_u_induction fs_v_induction. ((((exists fs_h_induction_start. fs_h_induction_start + S (0) = S ((S (0)) * fs_v_induction)) /\ exists fs_q_induction_start. fs_u_induction = fs_q_induction_start * S ((S (0)) * fs_v_induction) + (0))) /\ forall fs_i_induction_steps. (exists fs_lt_induction_steps_bound. fs_lt_induction_steps_bound + S fs_i_induction_steps = l) -> exists fs_a_induction_steps fs_r_induction_steps fs_s_induction_steps. ((((exists fs_h_induction_steps_summand. fs_h_induction_steps_summand + S (fs_a_induction_steps) = S ((S (fs_i_induction_steps)) * c)) /\ exists fs_q_induction_steps_summand. b = fs_q_induction_steps_summand * S ((S (fs_i_induction_steps)) * c) + (fs_a_induction_steps))) /\ ((((exists fs_h_induction_steps_partial. fs_h_induction_steps_partial + S (fs_r_induction_steps) = S ((S (fs_i_induction_steps)) * fs_v_induction)) /\ exists fs_q_induction_steps_partial. fs_u_induction = fs_q_induction_steps_partial * S ((S (fs_i_induction_steps)) * fs_v_induction) + (fs_r_induction_steps))) /\ ((((exists fs_h_induction_steps_successor. fs_h_induction_steps_successor + S (fs_s_induction_steps) = S ((S (S fs_i_induction_steps)) * fs_v_induction)) /\ exists fs_q_induction_steps_successor. fs_u_induction = fs_q_induction_steps_successor * S ((S (S fs_i_induction_steps)) * fs_v_induction) + (fs_s_induction_steps))) /\ fs_s_induction_steps = fs_r_induction_steps + fs_a_induction_steps))))
apply IH
cases htrace
cases htrace_witness
cases htrace_witness_witness
have hfactor : exists p. ((exists fs_h_trace_factor. fs_h_trace_factor + S (p) = S ((S (l)) * c)) /\ exists fs_q_trace_factor. b = fs_q_trace_factor * S ((S (l)) * c) + (p))
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists l
exact beta_at_exists
cases hfactor
have hlast : exists r. ((exists fs_h_trace_last. fs_h_trace_last + S (r) = S ((S (l)) * x1)) /\ exists fs_q_trace_last. x = fs_q_trace_last * S ((S (l)) * x1) + (r))
specialize beta_at_exists x
specialize beta_at_exists x1
specialize beta_at_exists l
exact beta_at_exists
cases hlast
have hext : exists z v. ((((exists fs_h_trace_extension. fs_h_trace_extension + S (x3 + x2) = S ((S (S l)) * v)) /\ exists fs_q_trace_extension. z = fs_q_trace_extension * S ((S (S l)) * v) + (x3 + x2))) /\ forall i a. (exists h. h + S i = S l) -> (((exists fs_h_trace_old. fs_h_trace_old + S (a) = S ((S (i)) * x1)) /\ exists fs_q_trace_old. x = fs_q_trace_old * S ((S (i)) * x1) + (a))) -> (((exists fs_h_trace_new. fs_h_trace_new + S (a) = S ((S (i)) * v)) /\ exists fs_q_trace_new. z = fs_q_trace_new * S ((S (i)) * v) + (a))))
specialize beta_prefix_extend (S l)
specialize beta_prefix_extend x
specialize beta_prefix_extend x1
specialize beta_prefix_extend (x3 + x2)
exact beta_prefix_extend
cases hext
cases hext_witness
cases hext_witness_witness
exists x4
exists x5
split
specialize hext_witness_witness_right 0
specialize hext_witness_witness_right 0
apply hext_witness_witness_right
have h0 : exists h. h + S 0 = S l
have hzero : exists h. h + 0 = l
specialize zero_le l
exact zero_le
specialize succ_le_succ 0
specialize succ_le_succ l
apply succ_le_succ
exact hzero
exact h0
exact htrace_witness_witness_left
intro i
intro hi
have hil : exists h. h + i = l
specialize le_of_succ_le_succ i
specialize le_of_succ_le_succ l
apply le_of_succ_le_succ
exact hi
have hsplit : i = l \/ exists h. h + S i = l
specialize le_eq_or_lt i
specialize le_eq_or_lt l
apply le_eq_or_lt
exact hil
cases hsplit
exists x2
exists x3
exists x3 + x2
split
rewrite hsplit_left
rewrite hsplit_left
exact hfactor_witness
split
rewrite hsplit_left
rewrite hsplit_left
specialize hext_witness_witness_right l
specialize hext_witness_witness_right x3
apply hext_witness_witness_right
specialize le_refl (S l)
exact le_refl
exact hlast_witness
split
rewrite hsplit_left
rewrite hsplit_left
exact hext_witness_witness_left
refl
have hold : exists p r s. ((((exists fs_h_trace_hold_factor. fs_h_trace_hold_factor + S (p) = S ((S (i)) * c)) /\ exists fs_q_trace_hold_factor. b = fs_q_trace_hold_factor * S ((S (i)) * c) + (p))) /\ ((((exists fs_h_trace_hold_partial. fs_h_trace_hold_partial + S (r) = S ((S (i)) * x1)) /\ exists fs_q_trace_hold_partial. x = fs_q_trace_hold_partial * S ((S (i)) * x1) + (r))) /\ ((((exists fs_h_trace_hold_successor. fs_h_trace_hold_successor + S (s) = S ((S (S i)) * x1)) /\ exists fs_q_trace_hold_successor. x = fs_q_trace_hold_successor * S ((S (S i)) * x1) + (s))) /\ s = r + p)))
specialize htrace_witness_witness_right i
apply htrace_witness_witness_right
exact hsplit_right
cases hold
cases hold_witness
cases hold_witness_witness
cases hold_witness_witness_witness
cases hold_witness_witness_witness_right
cases hold_witness_witness_witness_right_right
exists x6
exists x7
exists x8
split
exact hold_witness_witness_witness_left
split
specialize hext_witness_witness_right i
specialize hext_witness_witness_right x7
apply hext_witness_witness_right
exact hi
exact hold_witness_witness_witness_right_left
split
specialize hext_witness_witness_right (S i)
specialize hext_witness_witness_right x8
apply hext_witness_witness_right
specialize succ_le_succ (S i)
specialize succ_le_succ l
apply succ_le_succ
exact hsplit_right
exact hold_witness_witness_witness_right_right_left
exact hold_witness_witness_witness_right_right_right
qed

Occurrences

29,985

Distinct objects

4,721

Reused references

235

Depth

85

Cuts

899

Certificate SHA-256

d0c3456472f1…

beta_sum_existsBeta sum exists Quadratic reciprocity campaign checked

Every decoded beta prefix has a relational finite sum.

Prerequisites

beta_prefix_sum_trace_exists beta_at_exists

Used directly by

beta_sum_exists_unique bit_count_exists

Exact expanded PA statement

Copy statement

forall b c l. exists n. (exists ff_u_x ff_v_x. ((((exists ff_h_x_start. ff_h_x_start + S (0) = S ((S (0)) * ff_v_x)) /\ exists ff_q_x_start. ff_u_x = ff_q_x_start * S ((S (0)) * ff_v_x) + (0))) /\ ((((exists ff_h_x_terminal. ff_h_x_terminal + S (n) = S ((S (l)) * ff_v_x)) /\ exists ff_q_x_terminal. ff_u_x = ff_q_x_terminal * S ((S (l)) * ff_v_x) + (n))) /\ forall ff_i_x. (exists ff_lt_x_bound. ff_lt_x_bound + S ff_i_x = l) -> exists ff_a_x ff_r_x ff_s_x. ((((exists ff_h_x_summand. ff_h_x_summand + S (ff_a_x) = S ((S (ff_i_x)) * c)) /\ exists ff_q_x_summand. b = ff_q_x_summand * S ((S (ff_i_x)) * c) + (ff_a_x))) /\ ((((exists ff_h_x_partial. ff_h_x_partial + S (ff_r_x) = S ((S (ff_i_x)) * ff_v_x)) /\ exists ff_q_x_partial. ff_u_x = ff_q_x_partial * S ((S (ff_i_x)) * ff_v_x) + (ff_r_x))) /\ ((((exists ff_h_x_successor. ff_h_x_successor + S (ff_s_x) = S ((S (S ff_i_x)) * ff_v_x)) /\ exists ff_q_x_successor. ff_u_x = ff_q_x_successor * S ((S (S ff_i_x)) * ff_v_x) + (ff_s_x))) /\ ff_s_x = ff_r_x + ff_a_x))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists n. (exists ff_u_x ff_v_x. ((((exists ff_h_x_start. ff_h_x_start + S (0) = S ((S (0)) * ff_v_x)) /\ exists ff_q_x_start. ff_u_x = ff_q_x_start * S ((S (0)) * ff_v_x) + (0))) /\ ((((exists ff_h_x_terminal. ff_h_x_terminal + S (n) = S ((S (l)) * ff_v_x)) /\ exists ff_q_x_terminal. ff_u_x = ff_q_x_terminal * S ((S (l)) * ff_v_x) + (n))) /\ forall ff_i_x. (exists ff_lt_x_bound. ff_lt_x_bound + S ff_i_x = l) -> exists ff_a_x ff_r_x ff_s_x. ((((exists ff_h_x_summand. ff_h_x_summand + S (ff_a_x) = S ((S (ff_i_x)) * c)) /\ exists ff_q_x_summand. b = ff_q_x_summand * S ((S (ff_i_x)) * c) + (ff_a_x))) /\ ((((exists ff_h_x_partial. ff_h_x_partial + S (ff_r_x) = S ((S (ff_i_x)) * ff_v_x)) /\ exists ff_q_x_partial. ff_u_x = ff_q_x_partial * S ((S (ff_i_x)) * ff_v_x) + (ff_r_x))) /\ ((((exists ff_h_x_successor. ff_h_x_successor + S (ff_s_x) = S ((S (S ff_i_x)) * ff_v_x)) /\ exists ff_q_x_successor. ff_u_x = ff_q_x_successor * S ((S (S ff_i_x)) * ff_v_x) + (ff_s_x))) /\ ff_s_x = ff_r_x + ff_a_x))))))
use beta_prefix_sum_trace_exists
use beta_at_exists
intro b
intro c
intro l
have htrace : exists fs_u_exists_trace fs_v_exists_trace. ((((exists fs_h_exists_trace_start. fs_h_exists_trace_start + S (0) = S ((S (0)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_start. fs_u_exists_trace = fs_q_exists_trace_start * S ((S (0)) * fs_v_exists_trace) + (0))) /\ forall fs_i_exists_trace_steps. (exists fs_lt_exists_trace_steps_bound. fs_lt_exists_trace_steps_bound + S fs_i_exists_trace_steps = l) -> exists fs_a_exists_trace_steps fs_r_exists_trace_steps fs_s_exists_trace_steps. ((((exists fs_h_exists_trace_steps_summand. fs_h_exists_trace_steps_summand + S (fs_a_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * c)) /\ exists fs_q_exists_trace_steps_summand. b = fs_q_exists_trace_steps_summand * S ((S (fs_i_exists_trace_steps)) * c) + (fs_a_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_partial. fs_h_exists_trace_steps_partial + S (fs_r_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_partial. fs_u_exists_trace = fs_q_exists_trace_steps_partial * S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_r_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_successor. fs_h_exists_trace_steps_successor + S (fs_s_exists_trace_steps) = S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_successor. fs_u_exists_trace = fs_q_exists_trace_steps_successor * S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_s_exists_trace_steps))) /\ fs_s_exists_trace_steps = fs_r_exists_trace_steps + fs_a_exists_trace_steps))))
specialize beta_prefix_sum_trace_exists b
specialize beta_prefix_sum_trace_exists c
specialize beta_prefix_sum_trace_exists l
exact beta_prefix_sum_trace_exists
cases htrace
cases htrace_witness
cases htrace_witness_witness
have hterminal : exists n. ((exists fs_h_sum_terminal. fs_h_sum_terminal + S (n) = S ((S (l)) * x1)) /\ exists fs_q_sum_terminal. x = fs_q_sum_terminal * S ((S (l)) * x1) + (n))
specialize beta_at_exists x
specialize beta_at_exists x1
specialize beta_at_exists l
exact beta_at_exists
cases hterminal
exists x2
exists x
exists x1
split
exact htrace_witness_witness_left
split
exact hterminal_witness
exact htrace_witness_witness_right
qed

Occurrences

30,491

Distinct objects

4,748

Reused references

236

Depth

86

Cuts

916

Certificate SHA-256

b3c6bf491d9c…

beta_sum_trace_functionalBeta sum trace functional Quadratic reciprocity campaign checked

Two exact prefix-sum traces over one decoded prefix have equal endpoints.

Prerequisites

beta_at_unique le_refl le_succ add_congr

Used directly by

beta_sum_functional

Exact expanded PA statement

Copy statement

forall b c l n u v m w d. (((((exists fs_h_functional_left_start. fs_h_functional_left_start + S (0) = S ((S (0)) * v)) /\ exists fs_q_functional_left_start. u = fs_q_functional_left_start * S ((S (0)) * v) + (0))) /\ ((((exists fs_h_functional_left_terminal. fs_h_functional_left_terminal + S (n) = S ((S (l)) * v)) /\ exists fs_q_functional_left_terminal. u = fs_q_functional_left_terminal * S ((S (l)) * v) + (n))) /\ forall fs_i_functional_left_steps. (exists fs_lt_functional_left_steps_bound. fs_lt_functional_left_steps_bound + S fs_i_functional_left_steps = l) -> exists fs_a_functional_left_steps fs_r_functional_left_steps fs_s_functional_left_steps. ((((exists fs_h_functional_left_steps_summand. fs_h_functional_left_steps_summand + S (fs_a_functional_left_steps) = S ((S (fs_i_functional_left_steps)) * c)) /\ exists fs_q_functional_left_steps_summand. b = fs_q_functional_left_steps_summand * S ((S (fs_i_functional_left_steps)) * c) + (fs_a_functional_left_steps))) /\ ((((exists fs_h_functional_left_steps_partial. fs_h_functional_left_steps_partial + S (fs_r_functional_left_steps) = S ((S (fs_i_functional_left_steps)) * v)) /\ exists fs_q_functional_left_steps_partial. u = fs_q_functional_left_steps_partial * S ((S (fs_i_functional_left_steps)) * v) + (fs_r_functional_left_steps))) /\ ((((exists fs_h_functional_left_steps_successor. fs_h_functional_left_steps_successor + S (fs_s_functional_left_steps) = S ((S (S fs_i_functional_left_steps)) * v)) /\ exists fs_q_functional_left_steps_successor. u = fs_q_functional_left_steps_successor * S ((S (S fs_i_functional_left_steps)) * v) + (fs_s_functional_left_steps))) /\ fs_s_functional_left_steps = fs_r_functional_left_steps + fs_a_functional_left_steps)))))) -> (((((exists fs_h_functional_right_start. fs_h_functional_right_start + S (0) = S ((S (0)) * d)) /\ exists fs_q_functional_right_start. w = fs_q_functional_right_start * S ((S (0)) * d) + (0))) /\ ((((exists fs_h_functional_right_terminal. fs_h_functional_right_terminal + S (m) = S ((S (l)) * d)) /\ exists fs_q_functional_right_terminal. w = fs_q_functional_right_terminal * S ((S (l)) * d) + (m))) /\ forall fs_i_functional_right_steps. (exists fs_lt_functional_right_steps_bound. fs_lt_functional_right_steps_bound + S fs_i_functional_right_steps = l) -> exists fs_a_functional_right_steps fs_r_functional_right_steps fs_s_functional_right_steps. ((((exists fs_h_functional_right_steps_summand. fs_h_functional_right_steps_summand + S (fs_a_functional_right_steps) = S ((S (fs_i_functional_right_steps)) * c)) /\ exists fs_q_functional_right_steps_summand. b = fs_q_functional_right_steps_summand * S ((S (fs_i_functional_right_steps)) * c) + (fs_a_functional_right_steps))) /\ ((((exists fs_h_functional_right_steps_partial. fs_h_functional_right_steps_partial + S (fs_r_functional_right_steps) = S ((S (fs_i_functional_right_steps)) * d)) /\ exists fs_q_functional_right_steps_partial. w = fs_q_functional_right_steps_partial * S ((S (fs_i_functional_right_steps)) * d) + (fs_r_functional_right_steps))) /\ ((((exists fs_h_functional_right_steps_successor. fs_h_functional_right_steps_successor + S (fs_s_functional_right_steps) = S ((S (S fs_i_functional_right_steps)) * d)) /\ exists fs_q_functional_right_steps_successor. w = fs_q_functional_right_steps_successor * S ((S (S fs_i_functional_right_steps)) * d) + (fs_s_functional_right_steps))) /\ fs_s_functional_right_steps = fs_r_functional_right_steps + fs_a_functional_right_steps)))))) -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n u v m w d. (((((exists fs_h_functional_left_start. fs_h_functional_left_start + S (0) = S ((S (0)) * v)) /\ exists fs_q_functional_left_start. u = fs_q_functional_left_start * S ((S (0)) * v) + (0))) /\ ((((exists fs_h_functional_left_terminal. fs_h_functional_left_terminal + S (n) = S ((S (l)) * v)) /\ exists fs_q_functional_left_terminal. u = fs_q_functional_left_terminal * S ((S (l)) * v) + (n))) /\ forall fs_i_functional_left_steps. (exists fs_lt_functional_left_steps_bound. fs_lt_functional_left_steps_bound + S fs_i_functional_left_steps = l) -> exists fs_a_functional_left_steps fs_r_functional_left_steps fs_s_functional_left_steps. ((((exists fs_h_functional_left_steps_summand. fs_h_functional_left_steps_summand + S (fs_a_functional_left_steps) = S ((S (fs_i_functional_left_steps)) * c)) /\ exists fs_q_functional_left_steps_summand. b = fs_q_functional_left_steps_summand * S ((S (fs_i_functional_left_steps)) * c) + (fs_a_functional_left_steps))) /\ ((((exists fs_h_functional_left_steps_partial. fs_h_functional_left_steps_partial + S (fs_r_functional_left_steps) = S ((S (fs_i_functional_left_steps)) * v)) /\ exists fs_q_functional_left_steps_partial. u = fs_q_functional_left_steps_partial * S ((S (fs_i_functional_left_steps)) * v) + (fs_r_functional_left_steps))) /\ ((((exists fs_h_functional_left_steps_successor. fs_h_functional_left_steps_successor + S (fs_s_functional_left_steps) = S ((S (S fs_i_functional_left_steps)) * v)) /\ exists fs_q_functional_left_steps_successor. u = fs_q_functional_left_steps_successor * S ((S (S fs_i_functional_left_steps)) * v) + (fs_s_functional_left_steps))) /\ fs_s_functional_left_steps = fs_r_functional_left_steps + fs_a_functional_left_steps)))))) -> (((((exists fs_h_functional_right_start. fs_h_functional_right_start + S (0) = S ((S (0)) * d)) /\ exists fs_q_functional_right_start. w = fs_q_functional_right_start * S ((S (0)) * d) + (0))) /\ ((((exists fs_h_functional_right_terminal. fs_h_functional_right_terminal + S (m) = S ((S (l)) * d)) /\ exists fs_q_functional_right_terminal. w = fs_q_functional_right_terminal * S ((S (l)) * d) + (m))) /\ forall fs_i_functional_right_steps. (exists fs_lt_functional_right_steps_bound. fs_lt_functional_right_steps_bound + S fs_i_functional_right_steps = l) -> exists fs_a_functional_right_steps fs_r_functional_right_steps fs_s_functional_right_steps. ((((exists fs_h_functional_right_steps_summand. fs_h_functional_right_steps_summand + S (fs_a_functional_right_steps) = S ((S (fs_i_functional_right_steps)) * c)) /\ exists fs_q_functional_right_steps_summand. b = fs_q_functional_right_steps_summand * S ((S (fs_i_functional_right_steps)) * c) + (fs_a_functional_right_steps))) /\ ((((exists fs_h_functional_right_steps_partial. fs_h_functional_right_steps_partial + S (fs_r_functional_right_steps) = S ((S (fs_i_functional_right_steps)) * d)) /\ exists fs_q_functional_right_steps_partial. w = fs_q_functional_right_steps_partial * S ((S (fs_i_functional_right_steps)) * d) + (fs_r_functional_right_steps))) /\ ((((exists fs_h_functional_right_steps_successor. fs_h_functional_right_steps_successor + S (fs_s_functional_right_steps) = S ((S (S fs_i_functional_right_steps)) * d)) /\ exists fs_q_functional_right_steps_successor. w = fs_q_functional_right_steps_successor * S ((S (S fs_i_functional_right_steps)) * d) + (fs_s_functional_right_steps))) /\ fs_s_functional_right_steps = fs_r_functional_right_steps + fs_a_functional_right_steps)))))) -> n = m
use beta_at_unique
use le_refl
use le_succ
use add_congr
intro b
intro c
induction l
intro n
intro u
intro v
intro m
intro w
intro d
intro h1
intro h2
cases h1
cases h1_right
cases h2
cases h2_right
have hn : n = 0
specialize beta_at_unique u
specialize beta_at_unique v
specialize beta_at_unique 0
specialize beta_at_unique n
specialize beta_at_unique 0
apply beta_at_unique
exact h1_right_left
exact h1_left
have hm : m = 0
specialize beta_at_unique w
specialize beta_at_unique d
specialize beta_at_unique 0
specialize beta_at_unique m
specialize beta_at_unique 0
apply beta_at_unique
exact h2_right_left
exact h2_left
trans 0
exact hn
symm
exact hm
intro n
intro u
intro v
intro m
intro w
intro d
intro h1
intro h2
cases h1
cases h1_right
cases h2
cases h2_right
have hstep1 : exists a r s. ((((exists fs_h_functional_step1_factor. fs_h_functional_step1_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_functional_step1_factor. b = fs_q_functional_step1_factor * S ((S (l)) * c) + (a))) /\ ((((exists fs_h_functional_step1_partial. fs_h_functional_step1_partial + S (r) = S ((S (l)) * v)) /\ exists fs_q_functional_step1_partial. u = fs_q_functional_step1_partial * S ((S (l)) * v) + (r))) /\ ((((exists fs_h_functional_step1_successor. fs_h_functional_step1_successor + S (s) = S ((S (S l)) * v)) /\ exists fs_q_functional_step1_successor. u = fs_q_functional_step1_successor * S ((S (S l)) * v) + (s))) /\ s = r + a)))
specialize h1_right_right l
apply h1_right_right
specialize le_refl (S l)
exact le_refl
cases hstep1
cases hstep1_witness
cases hstep1_witness_witness
cases hstep1_witness_witness_witness
cases hstep1_witness_witness_witness_right
cases hstep1_witness_witness_witness_right_right
have hstep2 : exists a r s. ((((exists fs_h_functional_step2_factor. fs_h_functional_step2_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_functional_step2_factor. b = fs_q_functional_step2_factor * S ((S (l)) * c) + (a))) /\ ((((exists fs_h_functional_step2_partial. fs_h_functional_step2_partial + S (r) = S ((S (l)) * d)) /\ exists fs_q_functional_step2_partial. w = fs_q_functional_step2_partial * S ((S (l)) * d) + (r))) /\ ((((exists fs_h_functional_step2_successor. fs_h_functional_step2_successor + S (s) = S ((S (S l)) * d)) /\ exists fs_q_functional_step2_successor. w = fs_q_functional_step2_successor * S ((S (S l)) * d) + (s))) /\ s = r + a)))
specialize h2_right_right l
apply h2_right_right
specialize le_refl (S l)
exact le_refl
cases hstep2
cases hstep2_witness
cases hstep2_witness_witness
cases hstep2_witness_witness_witness
cases hstep2_witness_witness_witness_right
cases hstep2_witness_witness_witness_right_right
have hn : n = x2
specialize beta_at_unique u
specialize beta_at_unique v
specialize beta_at_unique (S l)
specialize beta_at_unique n
specialize beta_at_unique x2
apply beta_at_unique
exact h1_right_left
exact hstep1_witness_witness_witness_right_right_left
have hm : m = x5
specialize beta_at_unique w
specialize beta_at_unique d
specialize beta_at_unique (S l)
specialize beta_at_unique m
specialize beta_at_unique x5
apply beta_at_unique
exact h2_right_left
exact hstep2_witness_witness_witness_right_right_left
have ha : x = x3
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique x3
apply beta_at_unique
exact hstep1_witness_witness_witness_left
exact hstep2_witness_witness_witness_left
have hsum1 : ((((exists fs_h_functional_prefix1_start. fs_h_functional_prefix1_start + S (0) = S ((S (0)) * v)) /\ exists fs_q_functional_prefix1_start. u = fs_q_functional_prefix1_start * S ((S (0)) * v) + (0))) /\ ((((exists fs_h_functional_prefix1_terminal. fs_h_functional_prefix1_terminal + S (x1) = S ((S (l)) * v)) /\ exists fs_q_functional_prefix1_terminal. u = fs_q_functional_prefix1_terminal * S ((S (l)) * v) + (x1))) /\ forall fs_i_functional_prefix1_steps. (exists fs_lt_functional_prefix1_steps_bound. fs_lt_functional_prefix1_steps_bound + S fs_i_functional_prefix1_steps = l) -> exists fs_a_functional_prefix1_steps fs_r_functional_prefix1_steps fs_s_functional_prefix1_steps. ((((exists fs_h_functional_prefix1_steps_summand. fs_h_functional_prefix1_steps_summand + S (fs_a_functional_prefix1_steps) = S ((S (fs_i_functional_prefix1_steps)) * c)) /\ exists fs_q_functional_prefix1_steps_summand. b = fs_q_functional_prefix1_steps_summand * S ((S (fs_i_functional_prefix1_steps)) * c) + (fs_a_functional_prefix1_steps))) /\ ((((exists fs_h_functional_prefix1_steps_partial. fs_h_functional_prefix1_steps_partial + S (fs_r_functional_prefix1_steps) = S ((S (fs_i_functional_prefix1_steps)) * v)) /\ exists fs_q_functional_prefix1_steps_partial. u = fs_q_functional_prefix1_steps_partial * S ((S (fs_i_functional_prefix1_steps)) * v) + (fs_r_functional_prefix1_steps))) /\ ((((exists fs_h_functional_prefix1_steps_successor. fs_h_functional_prefix1_steps_successor + S (fs_s_functional_prefix1_steps) = S ((S (S fs_i_functional_prefix1_steps)) * v)) /\ exists fs_q_functional_prefix1_steps_successor. u = fs_q_functional_prefix1_steps_successor * S ((S (S fs_i_functional_prefix1_steps)) * v) + (fs_s_functional_prefix1_steps))) /\ fs_s_functional_prefix1_steps = fs_r_functional_prefix1_steps + fs_a_functional_prefix1_steps)))))
split
exact h1_left
split
exact hstep1_witness_witness_witness_right_left
intro i
intro hi
specialize h1_right_right i
apply h1_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
have hsum2 : ((((exists fs_h_functional_prefix2_start. fs_h_functional_prefix2_start + S (0) = S ((S (0)) * d)) /\ exists fs_q_functional_prefix2_start. w = fs_q_functional_prefix2_start * S ((S (0)) * d) + (0))) /\ ((((exists fs_h_functional_prefix2_terminal. fs_h_functional_prefix2_terminal + S (x4) = S ((S (l)) * d)) /\ exists fs_q_functional_prefix2_terminal. w = fs_q_functional_prefix2_terminal * S ((S (l)) * d) + (x4))) /\ forall fs_i_functional_prefix2_steps. (exists fs_lt_functional_prefix2_steps_bound. fs_lt_functional_prefix2_steps_bound + S fs_i_functional_prefix2_steps = l) -> exists fs_a_functional_prefix2_steps fs_r_functional_prefix2_steps fs_s_functional_prefix2_steps. ((((exists fs_h_functional_prefix2_steps_summand. fs_h_functional_prefix2_steps_summand + S (fs_a_functional_prefix2_steps) = S ((S (fs_i_functional_prefix2_steps)) * c)) /\ exists fs_q_functional_prefix2_steps_summand. b = fs_q_functional_prefix2_steps_summand * S ((S (fs_i_functional_prefix2_steps)) * c) + (fs_a_functional_prefix2_steps))) /\ ((((exists fs_h_functional_prefix2_steps_partial. fs_h_functional_prefix2_steps_partial + S (fs_r_functional_prefix2_steps) = S ((S (fs_i_functional_prefix2_steps)) * d)) /\ exists fs_q_functional_prefix2_steps_partial. w = fs_q_functional_prefix2_steps_partial * S ((S (fs_i_functional_prefix2_steps)) * d) + (fs_r_functional_prefix2_steps))) /\ ((((exists fs_h_functional_prefix2_steps_successor. fs_h_functional_prefix2_steps_successor + S (fs_s_functional_prefix2_steps) = S ((S (S fs_i_functional_prefix2_steps)) * d)) /\ exists fs_q_functional_prefix2_steps_successor. w = fs_q_functional_prefix2_steps_successor * S ((S (S fs_i_functional_prefix2_steps)) * d) + (fs_s_functional_prefix2_steps))) /\ fs_s_functional_prefix2_steps = fs_r_functional_prefix2_steps + fs_a_functional_prefix2_steps)))))
split
exact h2_left
split
exact hstep2_witness_witness_witness_right_left
intro i
intro hi
specialize h2_right_right i
apply h2_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
have hprev : x1 = x4
specialize IH x1
specialize IH u
specialize IH v
specialize IH x4
specialize IH w
specialize IH d
apply IH
exact hsum1
exact hsum2
have hadd : x1 + x = x4 + x3
specialize add_congr x1
specialize add_congr x4
specialize add_congr x
specialize add_congr x3
apply add_congr
exact hprev
exact ha
trans x2
exact hn
trans x1 + x
exact hstep1_witness_witness_witness_right_right_right
trans x4 + x3
exact hadd
trans x5
symm
exact hstep2_witness_witness_witness_right_right_right
symm
exact hm
qed

Occurrences

1,382

Distinct objects

909

Reused references

39

Depth

60

Cuts

36

Certificate SHA-256

239426e5d78c…

beta_sum_functionalBeta sum functional Quadratic reciprocity campaign checked

The relational finite sum has a unique natural value.

Prerequisites

beta_sum_trace_functional

Used directly by

beta_sum_exists_unique bit_count_functional

Exact expanded PA statement

Copy statement

forall b c l n m. (exists ff_u_l ff_v_l. ((((exists ff_h_l_start. ff_h_l_start + S (0) = S ((S (0)) * ff_v_l)) /\ exists ff_q_l_start. ff_u_l = ff_q_l_start * S ((S (0)) * ff_v_l) + (0))) /\ ((((exists ff_h_l_terminal. ff_h_l_terminal + S (n) = S ((S (l)) * ff_v_l)) /\ exists ff_q_l_terminal. ff_u_l = ff_q_l_terminal * S ((S (l)) * ff_v_l) + (n))) /\ forall ff_i_l. (exists ff_lt_l_bound. ff_lt_l_bound + S ff_i_l = l) -> exists ff_a_l ff_r_l ff_s_l. ((((exists ff_h_l_summand. ff_h_l_summand + S (ff_a_l) = S ((S (ff_i_l)) * c)) /\ exists ff_q_l_summand. b = ff_q_l_summand * S ((S (ff_i_l)) * c) + (ff_a_l))) /\ ((((exists ff_h_l_partial. ff_h_l_partial + S (ff_r_l) = S ((S (ff_i_l)) * ff_v_l)) /\ exists ff_q_l_partial. ff_u_l = ff_q_l_partial * S ((S (ff_i_l)) * ff_v_l) + (ff_r_l))) /\ ((((exists ff_h_l_successor. ff_h_l_successor + S (ff_s_l) = S ((S (S ff_i_l)) * ff_v_l)) /\ exists ff_q_l_successor. ff_u_l = ff_q_l_successor * S ((S (S ff_i_l)) * ff_v_l) + (ff_s_l))) /\ ff_s_l = ff_r_l + ff_a_l)))))) -> (exists ff_u_r ff_v_r. ((((exists ff_h_r_start. ff_h_r_start + S (0) = S ((S (0)) * ff_v_r)) /\ exists ff_q_r_start. ff_u_r = ff_q_r_start * S ((S (0)) * ff_v_r) + (0))) /\ ((((exists ff_h_r_terminal. ff_h_r_terminal + S (m) = S ((S (l)) * ff_v_r)) /\ exists ff_q_r_terminal. ff_u_r = ff_q_r_terminal * S ((S (l)) * ff_v_r) + (m))) /\ forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> exists ff_a_r ff_r_r ff_s_r. ((((exists ff_h_r_summand. ff_h_r_summand + S (ff_a_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_summand. b = ff_q_r_summand * S ((S (ff_i_r)) * c) + (ff_a_r))) /\ ((((exists ff_h_r_partial. ff_h_r_partial + S (ff_r_r) = S ((S (ff_i_r)) * ff_v_r)) /\ exists ff_q_r_partial. ff_u_r = ff_q_r_partial * S ((S (ff_i_r)) * ff_v_r) + (ff_r_r))) /\ ((((exists ff_h_r_successor. ff_h_r_successor + S (ff_s_r) = S ((S (S ff_i_r)) * ff_v_r)) /\ exists ff_q_r_successor. ff_u_r = ff_q_r_successor * S ((S (S ff_i_r)) * ff_v_r) + (ff_s_r))) /\ ff_s_r = ff_r_r + ff_a_r)))))) -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n m. (exists ff_u_l ff_v_l. ((((exists ff_h_l_start. ff_h_l_start + S (0) = S ((S (0)) * ff_v_l)) /\ exists ff_q_l_start. ff_u_l = ff_q_l_start * S ((S (0)) * ff_v_l) + (0))) /\ ((((exists ff_h_l_terminal. ff_h_l_terminal + S (n) = S ((S (l)) * ff_v_l)) /\ exists ff_q_l_terminal. ff_u_l = ff_q_l_terminal * S ((S (l)) * ff_v_l) + (n))) /\ forall ff_i_l. (exists ff_lt_l_bound. ff_lt_l_bound + S ff_i_l = l) -> exists ff_a_l ff_r_l ff_s_l. ((((exists ff_h_l_summand. ff_h_l_summand + S (ff_a_l) = S ((S (ff_i_l)) * c)) /\ exists ff_q_l_summand. b = ff_q_l_summand * S ((S (ff_i_l)) * c) + (ff_a_l))) /\ ((((exists ff_h_l_partial. ff_h_l_partial + S (ff_r_l) = S ((S (ff_i_l)) * ff_v_l)) /\ exists ff_q_l_partial. ff_u_l = ff_q_l_partial * S ((S (ff_i_l)) * ff_v_l) + (ff_r_l))) /\ ((((exists ff_h_l_successor. ff_h_l_successor + S (ff_s_l) = S ((S (S ff_i_l)) * ff_v_l)) /\ exists ff_q_l_successor. ff_u_l = ff_q_l_successor * S ((S (S ff_i_l)) * ff_v_l) + (ff_s_l))) /\ ff_s_l = ff_r_l + ff_a_l)))))) -> (exists ff_u_r ff_v_r. ((((exists ff_h_r_start. ff_h_r_start + S (0) = S ((S (0)) * ff_v_r)) /\ exists ff_q_r_start. ff_u_r = ff_q_r_start * S ((S (0)) * ff_v_r) + (0))) /\ ((((exists ff_h_r_terminal. ff_h_r_terminal + S (m) = S ((S (l)) * ff_v_r)) /\ exists ff_q_r_terminal. ff_u_r = ff_q_r_terminal * S ((S (l)) * ff_v_r) + (m))) /\ forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> exists ff_a_r ff_r_r ff_s_r. ((((exists ff_h_r_summand. ff_h_r_summand + S (ff_a_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_summand. b = ff_q_r_summand * S ((S (ff_i_r)) * c) + (ff_a_r))) /\ ((((exists ff_h_r_partial. ff_h_r_partial + S (ff_r_r) = S ((S (ff_i_r)) * ff_v_r)) /\ exists ff_q_r_partial. ff_u_r = ff_q_r_partial * S ((S (ff_i_r)) * ff_v_r) + (ff_r_r))) /\ ((((exists ff_h_r_successor. ff_h_r_successor + S (ff_s_r) = S ((S (S ff_i_r)) * ff_v_r)) /\ exists ff_q_r_successor. ff_u_r = ff_q_r_successor * S ((S (S ff_i_r)) * ff_v_r) + (ff_s_r))) /\ ff_s_r = ff_r_r + ff_a_r)))))) -> n = m
use beta_sum_trace_functional
intro b
intro c
intro l
intro n
intro m
intro hn
intro hm
cases hn
cases hn_witness
cases hm
cases hm_witness
specialize beta_sum_trace_functional b
specialize beta_sum_trace_functional c
specialize beta_sum_trace_functional l
specialize beta_sum_trace_functional n
specialize beta_sum_trace_functional x
specialize beta_sum_trace_functional x1
specialize beta_sum_trace_functional m
specialize beta_sum_trace_functional x2
specialize beta_sum_trace_functional x3
apply beta_sum_trace_functional
exact hn_witness_witness
exact hm_witness_witness
qed

Occurrences

1,439

Distinct objects

966

Reused references

39

Depth

61

Cuts

37

Certificate SHA-256

580611f0e6ce…

beta_sum_exists_uniqueBeta sum exists unique Quadratic reciprocity campaign checked

Every decoded beta prefix has exactly one relational finite sum.

Prerequisites

beta_sum_exists beta_sum_functional

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l. exists n. ((exists ff_u_unique_value ff_v_unique_value. ((((exists ff_h_unique_value_start. ff_h_unique_value_start + S (0) = S ((S (0)) * ff_v_unique_value)) /\ exists ff_q_unique_value_start. ff_u_unique_value = ff_q_unique_value_start * S ((S (0)) * ff_v_unique_value) + (0))) /\ ((((exists ff_h_unique_value_terminal. ff_h_unique_value_terminal + S (n) = S ((S (l)) * ff_v_unique_value)) /\ exists ff_q_unique_value_terminal. ff_u_unique_value = ff_q_unique_value_terminal * S ((S (l)) * ff_v_unique_value) + (n))) /\ forall ff_i_unique_value. (exists ff_lt_unique_value_bound. ff_lt_unique_value_bound + S ff_i_unique_value = l) -> exists ff_a_unique_value ff_r_unique_value ff_s_unique_value. ((((exists ff_h_unique_value_summand. ff_h_unique_value_summand + S (ff_a_unique_value) = S ((S (ff_i_unique_value)) * c)) /\ exists ff_q_unique_value_summand. b = ff_q_unique_value_summand * S ((S (ff_i_unique_value)) * c) + (ff_a_unique_value))) /\ ((((exists ff_h_unique_value_partial. ff_h_unique_value_partial + S (ff_r_unique_value) = S ((S (ff_i_unique_value)) * ff_v_unique_value)) /\ exists ff_q_unique_value_partial. ff_u_unique_value = ff_q_unique_value_partial * S ((S (ff_i_unique_value)) * ff_v_unique_value) + (ff_r_unique_value))) /\ ((((exists ff_h_unique_value_successor. ff_h_unique_value_successor + S (ff_s_unique_value) = S ((S (S ff_i_unique_value)) * ff_v_unique_value)) /\ exists ff_q_unique_value_successor. ff_u_unique_value = ff_q_unique_value_successor * S ((S (S ff_i_unique_value)) * ff_v_unique_value) + (ff_s_unique_value))) /\ ff_s_unique_value = ff_r_unique_value + ff_a_unique_value)))))) /\ forall m. (exists ff_u_unique_other ff_v_unique_other. ((((exists ff_h_unique_other_start. ff_h_unique_other_start + S (0) = S ((S (0)) * ff_v_unique_other)) /\ exists ff_q_unique_other_start. ff_u_unique_other = ff_q_unique_other_start * S ((S (0)) * ff_v_unique_other) + (0))) /\ ((((exists ff_h_unique_other_terminal. ff_h_unique_other_terminal + S (m) = S ((S (l)) * ff_v_unique_other)) /\ exists ff_q_unique_other_terminal. ff_u_unique_other = ff_q_unique_other_terminal * S ((S (l)) * ff_v_unique_other) + (m))) /\ forall ff_i_unique_other. (exists ff_lt_unique_other_bound. ff_lt_unique_other_bound + S ff_i_unique_other = l) -> exists ff_a_unique_other ff_r_unique_other ff_s_unique_other. ((((exists ff_h_unique_other_summand. ff_h_unique_other_summand + S (ff_a_unique_other) = S ((S (ff_i_unique_other)) * c)) /\ exists ff_q_unique_other_summand. b = ff_q_unique_other_summand * S ((S (ff_i_unique_other)) * c) + (ff_a_unique_other))) /\ ((((exists ff_h_unique_other_partial. ff_h_unique_other_partial + S (ff_r_unique_other) = S ((S (ff_i_unique_other)) * ff_v_unique_other)) /\ exists ff_q_unique_other_partial. ff_u_unique_other = ff_q_unique_other_partial * S ((S (ff_i_unique_other)) * ff_v_unique_other) + (ff_r_unique_other))) /\ ((((exists ff_h_unique_other_successor. ff_h_unique_other_successor + S (ff_s_unique_other) = S ((S (S ff_i_unique_other)) * ff_v_unique_other)) /\ exists ff_q_unique_other_successor. ff_u_unique_other = ff_q_unique_other_successor * S ((S (S ff_i_unique_other)) * ff_v_unique_other) + (ff_s_unique_other))) /\ ff_s_unique_other = ff_r_unique_other + ff_a_unique_other)))))) -> n = m)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. exists n. ((exists ff_u_unique_value ff_v_unique_value. ((((exists ff_h_unique_value_start. ff_h_unique_value_start + S (0) = S ((S (0)) * ff_v_unique_value)) /\ exists ff_q_unique_value_start. ff_u_unique_value = ff_q_unique_value_start * S ((S (0)) * ff_v_unique_value) + (0))) /\ ((((exists ff_h_unique_value_terminal. ff_h_unique_value_terminal + S (n) = S ((S (l)) * ff_v_unique_value)) /\ exists ff_q_unique_value_terminal. ff_u_unique_value = ff_q_unique_value_terminal * S ((S (l)) * ff_v_unique_value) + (n))) /\ forall ff_i_unique_value. (exists ff_lt_unique_value_bound. ff_lt_unique_value_bound + S ff_i_unique_value = l) -> exists ff_a_unique_value ff_r_unique_value ff_s_unique_value. ((((exists ff_h_unique_value_summand. ff_h_unique_value_summand + S (ff_a_unique_value) = S ((S (ff_i_unique_value)) * c)) /\ exists ff_q_unique_value_summand. b = ff_q_unique_value_summand * S ((S (ff_i_unique_value)) * c) + (ff_a_unique_value))) /\ ((((exists ff_h_unique_value_partial. ff_h_unique_value_partial + S (ff_r_unique_value) = S ((S (ff_i_unique_value)) * ff_v_unique_value)) /\ exists ff_q_unique_value_partial. ff_u_unique_value = ff_q_unique_value_partial * S ((S (ff_i_unique_value)) * ff_v_unique_value) + (ff_r_unique_value))) /\ ((((exists ff_h_unique_value_successor. ff_h_unique_value_successor + S (ff_s_unique_value) = S ((S (S ff_i_unique_value)) * ff_v_unique_value)) /\ exists ff_q_unique_value_successor. ff_u_unique_value = ff_q_unique_value_successor * S ((S (S ff_i_unique_value)) * ff_v_unique_value) + (ff_s_unique_value))) /\ ff_s_unique_value = ff_r_unique_value + ff_a_unique_value)))))) /\ forall m. (exists ff_u_unique_other ff_v_unique_other. ((((exists ff_h_unique_other_start. ff_h_unique_other_start + S (0) = S ((S (0)) * ff_v_unique_other)) /\ exists ff_q_unique_other_start. ff_u_unique_other = ff_q_unique_other_start * S ((S (0)) * ff_v_unique_other) + (0))) /\ ((((exists ff_h_unique_other_terminal. ff_h_unique_other_terminal + S (m) = S ((S (l)) * ff_v_unique_other)) /\ exists ff_q_unique_other_terminal. ff_u_unique_other = ff_q_unique_other_terminal * S ((S (l)) * ff_v_unique_other) + (m))) /\ forall ff_i_unique_other. (exists ff_lt_unique_other_bound. ff_lt_unique_other_bound + S ff_i_unique_other = l) -> exists ff_a_unique_other ff_r_unique_other ff_s_unique_other. ((((exists ff_h_unique_other_summand. ff_h_unique_other_summand + S (ff_a_unique_other) = S ((S (ff_i_unique_other)) * c)) /\ exists ff_q_unique_other_summand. b = ff_q_unique_other_summand * S ((S (ff_i_unique_other)) * c) + (ff_a_unique_other))) /\ ((((exists ff_h_unique_other_partial. ff_h_unique_other_partial + S (ff_r_unique_other) = S ((S (ff_i_unique_other)) * ff_v_unique_other)) /\ exists ff_q_unique_other_partial. ff_u_unique_other = ff_q_unique_other_partial * S ((S (ff_i_unique_other)) * ff_v_unique_other) + (ff_r_unique_other))) /\ ((((exists ff_h_unique_other_successor. ff_h_unique_other_successor + S (ff_s_unique_other) = S ((S (S ff_i_unique_other)) * ff_v_unique_other)) /\ exists ff_q_unique_other_successor. ff_u_unique_other = ff_q_unique_other_successor * S ((S (S ff_i_unique_other)) * ff_v_unique_other) + (ff_s_unique_other))) /\ ff_s_unique_other = ff_r_unique_other + ff_a_unique_other)))))) -> n = m)
use beta_sum_exists
use beta_sum_functional
intro b
intro c
intro l
specialize beta_sum_exists b
specialize beta_sum_exists c
specialize beta_sum_exists l
cases beta_sum_exists
exists x
split
exact beta_sum_exists_witness
intro m
intro hm
specialize beta_sum_functional b
specialize beta_sum_functional c
specialize beta_sum_functional l
specialize beta_sum_functional x
specialize beta_sum_functional m
apply beta_sum_functional
exact beta_sum_exists_witness
exact hm
qed

Occurrences

31,979

Distinct objects

5,050

Reused references

239

Depth

87

Cuts

955

Certificate SHA-256

c40396e72cc9…

beta_sum_zeroBeta sum zero Quadratic reciprocity campaign checked

The sum of an empty decoded prefix is zero.

Prerequisites

beta_at_unique

Used directly by

beta_sum_pointwise_mod_congruent bit_count_zero

Exact expanded PA statement

Copy statement

forall b c n. (exists fs_u_zero fs_v_zero. ((((exists fs_h_zero_body_start. fs_h_zero_body_start + S (0) = S ((S (0)) * fs_v_zero)) /\ exists fs_q_zero_body_start. fs_u_zero = fs_q_zero_body_start * S ((S (0)) * fs_v_zero) + (0))) /\ ((((exists fs_h_zero_body_terminal. fs_h_zero_body_terminal + S (n) = S ((S (0)) * fs_v_zero)) /\ exists fs_q_zero_body_terminal. fs_u_zero = fs_q_zero_body_terminal * S ((S (0)) * fs_v_zero) + (n))) /\ forall fs_i_zero_body_steps. (exists fs_lt_zero_body_steps_bound. fs_lt_zero_body_steps_bound + S fs_i_zero_body_steps = 0) -> exists fs_a_zero_body_steps fs_r_zero_body_steps fs_s_zero_body_steps. ((((exists fs_h_zero_body_steps_summand. fs_h_zero_body_steps_summand + S (fs_a_zero_body_steps) = S ((S (fs_i_zero_body_steps)) * c)) /\ exists fs_q_zero_body_steps_summand. b = fs_q_zero_body_steps_summand * S ((S (fs_i_zero_body_steps)) * c) + (fs_a_zero_body_steps))) /\ ((((exists fs_h_zero_body_steps_partial. fs_h_zero_body_steps_partial + S (fs_r_zero_body_steps) = S ((S (fs_i_zero_body_steps)) * fs_v_zero)) /\ exists fs_q_zero_body_steps_partial. fs_u_zero = fs_q_zero_body_steps_partial * S ((S (fs_i_zero_body_steps)) * fs_v_zero) + (fs_r_zero_body_steps))) /\ ((((exists fs_h_zero_body_steps_successor. fs_h_zero_body_steps_successor + S (fs_s_zero_body_steps) = S ((S (S fs_i_zero_body_steps)) * fs_v_zero)) /\ exists fs_q_zero_body_steps_successor. fs_u_zero = fs_q_zero_body_steps_successor * S ((S (S fs_i_zero_body_steps)) * fs_v_zero) + (fs_s_zero_body_steps))) /\ fs_s_zero_body_steps = fs_r_zero_body_steps + fs_a_zero_body_steps)))))) -> n = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n. (exists fs_u_zero fs_v_zero. ((((exists fs_h_zero_body_start. fs_h_zero_body_start + S (0) = S ((S (0)) * fs_v_zero)) /\ exists fs_q_zero_body_start. fs_u_zero = fs_q_zero_body_start * S ((S (0)) * fs_v_zero) + (0))) /\ ((((exists fs_h_zero_body_terminal. fs_h_zero_body_terminal + S (n) = S ((S (0)) * fs_v_zero)) /\ exists fs_q_zero_body_terminal. fs_u_zero = fs_q_zero_body_terminal * S ((S (0)) * fs_v_zero) + (n))) /\ forall fs_i_zero_body_steps. (exists fs_lt_zero_body_steps_bound. fs_lt_zero_body_steps_bound + S fs_i_zero_body_steps = 0) -> exists fs_a_zero_body_steps fs_r_zero_body_steps fs_s_zero_body_steps. ((((exists fs_h_zero_body_steps_summand. fs_h_zero_body_steps_summand + S (fs_a_zero_body_steps) = S ((S (fs_i_zero_body_steps)) * c)) /\ exists fs_q_zero_body_steps_summand. b = fs_q_zero_body_steps_summand * S ((S (fs_i_zero_body_steps)) * c) + (fs_a_zero_body_steps))) /\ ((((exists fs_h_zero_body_steps_partial. fs_h_zero_body_steps_partial + S (fs_r_zero_body_steps) = S ((S (fs_i_zero_body_steps)) * fs_v_zero)) /\ exists fs_q_zero_body_steps_partial. fs_u_zero = fs_q_zero_body_steps_partial * S ((S (fs_i_zero_body_steps)) * fs_v_zero) + (fs_r_zero_body_steps))) /\ ((((exists fs_h_zero_body_steps_successor. fs_h_zero_body_steps_successor + S (fs_s_zero_body_steps) = S ((S (S fs_i_zero_body_steps)) * fs_v_zero)) /\ exists fs_q_zero_body_steps_successor. fs_u_zero = fs_q_zero_body_steps_successor * S ((S (S fs_i_zero_body_steps)) * fs_v_zero) + (fs_s_zero_body_steps))) /\ fs_s_zero_body_steps = fs_r_zero_body_steps + fs_a_zero_body_steps)))))) -> n = 0
use beta_at_unique
intro b
intro c
intro n
intro hsum
cases hsum
cases hsum_witness
cases hsum_witness_witness
cases hsum_witness_witness_right
specialize beta_at_unique x
specialize beta_at_unique x1
specialize beta_at_unique 0
specialize beta_at_unique n
specialize beta_at_unique 0
apply beta_at_unique
exact hsum_witness_witness_right_left
exact hsum_witness_witness_left
qed

Occurrences

1,171

Distinct objects

742

Reused references

37

Depth

60

Cuts

31

Certificate SHA-256

f1bbae9a62a0…

beta_sum_succ_decomposeBeta sum succ decompose Quadratic reciprocity campaign checked

A successor sum decomposes into its prefix sum and final summand.

Prerequisites

le_refl le_succ beta_at_unique

Used directly by

beta_sum_pointwise_mod_congruent bit_count_succ_decompose

Exact expanded PA statement

Copy statement

forall b c l n. (exists fs_u_succ fs_v_succ. ((((exists fs_h_succ_body_start. fs_h_succ_body_start + S (0) = S ((S (0)) * fs_v_succ)) /\ exists fs_q_succ_body_start. fs_u_succ = fs_q_succ_body_start * S ((S (0)) * fs_v_succ) + (0))) /\ ((((exists fs_h_succ_body_terminal. fs_h_succ_body_terminal + S (n) = S ((S (S l)) * fs_v_succ)) /\ exists fs_q_succ_body_terminal. fs_u_succ = fs_q_succ_body_terminal * S ((S (S l)) * fs_v_succ) + (n))) /\ forall fs_i_succ_body_steps. (exists fs_lt_succ_body_steps_bound. fs_lt_succ_body_steps_bound + S fs_i_succ_body_steps = S l) -> exists fs_a_succ_body_steps fs_r_succ_body_steps fs_s_succ_body_steps. ((((exists fs_h_succ_body_steps_summand. fs_h_succ_body_steps_summand + S (fs_a_succ_body_steps) = S ((S (fs_i_succ_body_steps)) * c)) /\ exists fs_q_succ_body_steps_summand. b = fs_q_succ_body_steps_summand * S ((S (fs_i_succ_body_steps)) * c) + (fs_a_succ_body_steps))) /\ ((((exists fs_h_succ_body_steps_partial. fs_h_succ_body_steps_partial + S (fs_r_succ_body_steps) = S ((S (fs_i_succ_body_steps)) * fs_v_succ)) /\ exists fs_q_succ_body_steps_partial. fs_u_succ = fs_q_succ_body_steps_partial * S ((S (fs_i_succ_body_steps)) * fs_v_succ) + (fs_r_succ_body_steps))) /\ ((((exists fs_h_succ_body_steps_successor. fs_h_succ_body_steps_successor + S (fs_s_succ_body_steps) = S ((S (S fs_i_succ_body_steps)) * fs_v_succ)) /\ exists fs_q_succ_body_steps_successor. fs_u_succ = fs_q_succ_body_steps_successor * S ((S (S fs_i_succ_body_steps)) * fs_v_succ) + (fs_s_succ_body_steps))) /\ fs_s_succ_body_steps = fs_r_succ_body_steps + fs_a_succ_body_steps)))))) -> exists a r. (((exists fs_h_succ_factor. fs_h_succ_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_succ_factor. b = fs_q_succ_factor * S ((S (l)) * c) + (a))) /\ ((exists ff_u_prefix ff_v_prefix. ((((exists ff_h_prefix_start. ff_h_prefix_start + S (0) = S ((S (0)) * ff_v_prefix)) /\ exists ff_q_prefix_start. ff_u_prefix = ff_q_prefix_start * S ((S (0)) * ff_v_prefix) + (0))) /\ ((((exists ff_h_prefix_terminal. ff_h_prefix_terminal + S (r) = S ((S (l)) * ff_v_prefix)) /\ exists ff_q_prefix_terminal. ff_u_prefix = ff_q_prefix_terminal * S ((S (l)) * ff_v_prefix) + (r))) /\ forall ff_i_prefix. (exists ff_lt_prefix_bound. ff_lt_prefix_bound + S ff_i_prefix = l) -> exists ff_a_prefix ff_r_prefix ff_s_prefix. ((((exists ff_h_prefix_summand. ff_h_prefix_summand + S (ff_a_prefix) = S ((S (ff_i_prefix)) * c)) /\ exists ff_q_prefix_summand. b = ff_q_prefix_summand * S ((S (ff_i_prefix)) * c) + (ff_a_prefix))) /\ ((((exists ff_h_prefix_partial. ff_h_prefix_partial + S (ff_r_prefix) = S ((S (ff_i_prefix)) * ff_v_prefix)) /\ exists ff_q_prefix_partial. ff_u_prefix = ff_q_prefix_partial * S ((S (ff_i_prefix)) * ff_v_prefix) + (ff_r_prefix))) /\ ((((exists ff_h_prefix_successor. ff_h_prefix_successor + S (ff_s_prefix) = S ((S (S ff_i_prefix)) * ff_v_prefix)) /\ exists ff_q_prefix_successor. ff_u_prefix = ff_q_prefix_successor * S ((S (S ff_i_prefix)) * ff_v_prefix) + (ff_s_prefix))) /\ ff_s_prefix = ff_r_prefix + ff_a_prefix)))))) /\ n = r + a)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. (exists fs_u_succ fs_v_succ. ((((exists fs_h_succ_body_start. fs_h_succ_body_start + S (0) = S ((S (0)) * fs_v_succ)) /\ exists fs_q_succ_body_start. fs_u_succ = fs_q_succ_body_start * S ((S (0)) * fs_v_succ) + (0))) /\ ((((exists fs_h_succ_body_terminal. fs_h_succ_body_terminal + S (n) = S ((S (S l)) * fs_v_succ)) /\ exists fs_q_succ_body_terminal. fs_u_succ = fs_q_succ_body_terminal * S ((S (S l)) * fs_v_succ) + (n))) /\ forall fs_i_succ_body_steps. (exists fs_lt_succ_body_steps_bound. fs_lt_succ_body_steps_bound + S fs_i_succ_body_steps = S l) -> exists fs_a_succ_body_steps fs_r_succ_body_steps fs_s_succ_body_steps. ((((exists fs_h_succ_body_steps_summand. fs_h_succ_body_steps_summand + S (fs_a_succ_body_steps) = S ((S (fs_i_succ_body_steps)) * c)) /\ exists fs_q_succ_body_steps_summand. b = fs_q_succ_body_steps_summand * S ((S (fs_i_succ_body_steps)) * c) + (fs_a_succ_body_steps))) /\ ((((exists fs_h_succ_body_steps_partial. fs_h_succ_body_steps_partial + S (fs_r_succ_body_steps) = S ((S (fs_i_succ_body_steps)) * fs_v_succ)) /\ exists fs_q_succ_body_steps_partial. fs_u_succ = fs_q_succ_body_steps_partial * S ((S (fs_i_succ_body_steps)) * fs_v_succ) + (fs_r_succ_body_steps))) /\ ((((exists fs_h_succ_body_steps_successor. fs_h_succ_body_steps_successor + S (fs_s_succ_body_steps) = S ((S (S fs_i_succ_body_steps)) * fs_v_succ)) /\ exists fs_q_succ_body_steps_successor. fs_u_succ = fs_q_succ_body_steps_successor * S ((S (S fs_i_succ_body_steps)) * fs_v_succ) + (fs_s_succ_body_steps))) /\ fs_s_succ_body_steps = fs_r_succ_body_steps + fs_a_succ_body_steps)))))) -> exists a r. (((exists fs_h_succ_factor. fs_h_succ_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_succ_factor. b = fs_q_succ_factor * S ((S (l)) * c) + (a))) /\ ((exists ff_u_prefix ff_v_prefix. ((((exists ff_h_prefix_start. ff_h_prefix_start + S (0) = S ((S (0)) * ff_v_prefix)) /\ exists ff_q_prefix_start. ff_u_prefix = ff_q_prefix_start * S ((S (0)) * ff_v_prefix) + (0))) /\ ((((exists ff_h_prefix_terminal. ff_h_prefix_terminal + S (r) = S ((S (l)) * ff_v_prefix)) /\ exists ff_q_prefix_terminal. ff_u_prefix = ff_q_prefix_terminal * S ((S (l)) * ff_v_prefix) + (r))) /\ forall ff_i_prefix. (exists ff_lt_prefix_bound. ff_lt_prefix_bound + S ff_i_prefix = l) -> exists ff_a_prefix ff_r_prefix ff_s_prefix. ((((exists ff_h_prefix_summand. ff_h_prefix_summand + S (ff_a_prefix) = S ((S (ff_i_prefix)) * c)) /\ exists ff_q_prefix_summand. b = ff_q_prefix_summand * S ((S (ff_i_prefix)) * c) + (ff_a_prefix))) /\ ((((exists ff_h_prefix_partial. ff_h_prefix_partial + S (ff_r_prefix) = S ((S (ff_i_prefix)) * ff_v_prefix)) /\ exists ff_q_prefix_partial. ff_u_prefix = ff_q_prefix_partial * S ((S (ff_i_prefix)) * ff_v_prefix) + (ff_r_prefix))) /\ ((((exists ff_h_prefix_successor. ff_h_prefix_successor + S (ff_s_prefix) = S ((S (S ff_i_prefix)) * ff_v_prefix)) /\ exists ff_q_prefix_successor. ff_u_prefix = ff_q_prefix_successor * S ((S (S ff_i_prefix)) * ff_v_prefix) + (ff_s_prefix))) /\ ff_s_prefix = ff_r_prefix + ff_a_prefix)))))) /\ n = r + a)
use le_refl
use le_succ
use beta_at_unique
intro b
intro c
intro l
intro n
intro hsum
cases hsum
cases hsum_witness
cases hsum_witness_witness
cases hsum_witness_witness_right
have hstep : exists a r s. ((((exists fs_h_decomp_factor. fs_h_decomp_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_decomp_factor. b = fs_q_decomp_factor * S ((S (l)) * c) + (a))) /\ ((((exists fs_h_decomp_partial. fs_h_decomp_partial + S (r) = S ((S (l)) * x1)) /\ exists fs_q_decomp_partial. x = fs_q_decomp_partial * S ((S (l)) * x1) + (r))) /\ ((((exists fs_h_decomp_successor. fs_h_decomp_successor + S (s) = S ((S (S l)) * x1)) /\ exists fs_q_decomp_successor. x = fs_q_decomp_successor * S ((S (S l)) * x1) + (s))) /\ s = r + a)))
specialize hsum_witness_witness_right_right l
apply hsum_witness_witness_right_right
specialize le_refl (S l)
exact le_refl
cases hstep
cases hstep_witness
cases hstep_witness_witness
cases hstep_witness_witness_witness
cases hstep_witness_witness_witness_right
cases hstep_witness_witness_witness_right_right
have hn : n = x4
specialize beta_at_unique x
specialize beta_at_unique x1
specialize beta_at_unique (S l)
specialize beta_at_unique n
specialize beta_at_unique x4
apply beta_at_unique
exact hsum_witness_witness_right_left
exact hstep_witness_witness_witness_right_right_left
exists x2
exists x3
split
exact hstep_witness_witness_witness_left
split
exists x
exists x1
split
exact hsum_witness_witness_left
split
exact hstep_witness_witness_witness_right_left
intro i
intro hi
specialize hsum_witness_witness_right_right i
apply hsum_witness_witness_right_right
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
trans x4
exact hn
exact hstep_witness_witness_witness_right_right_right
qed

Occurrences

1,257

Distinct objects

784

Reused references

39

Depth

62

Cuts

35

Certificate SHA-256

810c7005959f…

beta_product_pointwise_mod_congruentBeta product pointwise mod congruent Quadratic reciprocity campaign checked

Pointwise congruent decoded prefixes have congruent finite products.

Prerequisites

beta_product_zero beta_product_succ_decompose le_succ le_refl mod_eq_refl mod_eq_mul

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m b c d e l n q. (forall i a z. (exists fc_h_pp. fc_h_pp + S i = l) -> (((exists ff_h_pp_la. ff_h_pp_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_pp_la. b = ff_q_pp_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_pp_ra. ff_h_pp_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_pp_ra. d = ff_q_pp_ra * S ((S (i)) * e) + (z))) -> exists fc_u_pp_me fc_v_pp_me. a + m * fc_u_pp_me = z + m * fc_v_pp_me) -> (exists ff_u_pl ff_v_pl. ((((exists ff_h_pl_start. ff_h_pl_start + S (1) = S ((S (0)) * ff_v_pl)) /\ exists ff_q_pl_start. ff_u_pl = ff_q_pl_start * S ((S (0)) * ff_v_pl) + (1))) /\ ((((exists ff_h_pl_terminal. ff_h_pl_terminal + S (n) = S ((S (l)) * ff_v_pl)) /\ exists ff_q_pl_terminal. ff_u_pl = ff_q_pl_terminal * S ((S (l)) * ff_v_pl) + (n))) /\ forall ff_i_pl. (exists ff_lt_pl_bound. ff_lt_pl_bound + S ff_i_pl = l) -> exists ff_p_pl ff_r_pl ff_s_pl. ((((exists ff_h_pl_factor. ff_h_pl_factor + S (ff_p_pl) = S ((S (ff_i_pl)) * c)) /\ exists ff_q_pl_factor. b = ff_q_pl_factor * S ((S (ff_i_pl)) * c) + (ff_p_pl))) /\ ((((exists ff_h_pl_partial. ff_h_pl_partial + S (ff_r_pl) = S ((S (ff_i_pl)) * ff_v_pl)) /\ exists ff_q_pl_partial. ff_u_pl = ff_q_pl_partial * S ((S (ff_i_pl)) * ff_v_pl) + (ff_r_pl))) /\ ((((exists ff_h_pl_successor. ff_h_pl_successor + S (ff_s_pl) = S ((S (S ff_i_pl)) * ff_v_pl)) /\ exists ff_q_pl_successor. ff_u_pl = ff_q_pl_successor * S ((S (S ff_i_pl)) * ff_v_pl) + (ff_s_pl))) /\ ff_s_pl = ff_r_pl * ff_p_pl)))))) -> (exists ff_u_pr ff_v_pr. ((((exists ff_h_pr_start. ff_h_pr_start + S (1) = S ((S (0)) * ff_v_pr)) /\ exists ff_q_pr_start. ff_u_pr = ff_q_pr_start * S ((S (0)) * ff_v_pr) + (1))) /\ ((((exists ff_h_pr_terminal. ff_h_pr_terminal + S (q) = S ((S (l)) * ff_v_pr)) /\ exists ff_q_pr_terminal. ff_u_pr = ff_q_pr_terminal * S ((S (l)) * ff_v_pr) + (q))) /\ forall ff_i_pr. (exists ff_lt_pr_bound. ff_lt_pr_bound + S ff_i_pr = l) -> exists ff_p_pr ff_r_pr ff_s_pr. ((((exists ff_h_pr_factor. ff_h_pr_factor + S (ff_p_pr) = S ((S (ff_i_pr)) * e)) /\ exists ff_q_pr_factor. d = ff_q_pr_factor * S ((S (ff_i_pr)) * e) + (ff_p_pr))) /\ ((((exists ff_h_pr_partial. ff_h_pr_partial + S (ff_r_pr) = S ((S (ff_i_pr)) * ff_v_pr)) /\ exists ff_q_pr_partial. ff_u_pr = ff_q_pr_partial * S ((S (ff_i_pr)) * ff_v_pr) + (ff_r_pr))) /\ ((((exists ff_h_pr_successor. ff_h_pr_successor + S (ff_s_pr) = S ((S (S ff_i_pr)) * ff_v_pr)) /\ exists ff_q_pr_successor. ff_u_pr = ff_q_pr_successor * S ((S (S ff_i_pr)) * ff_v_pr) + (ff_s_pr))) /\ ff_s_pr = ff_r_pr * ff_p_pr)))))) -> exists fc_u_pv fc_v_pv. n + m * fc_u_pv = q + m * fc_v_pv

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m b c d e l n q. (forall i a z. (exists fc_h_pp. fc_h_pp + S i = l) -> (((exists ff_h_pp_la. ff_h_pp_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_pp_la. b = ff_q_pp_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_pp_ra. ff_h_pp_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_pp_ra. d = ff_q_pp_ra * S ((S (i)) * e) + (z))) -> exists fc_u_pp_me fc_v_pp_me. a + m * fc_u_pp_me = z + m * fc_v_pp_me) -> (exists ff_u_pl ff_v_pl. ((((exists ff_h_pl_start. ff_h_pl_start + S (1) = S ((S (0)) * ff_v_pl)) /\ exists ff_q_pl_start. ff_u_pl = ff_q_pl_start * S ((S (0)) * ff_v_pl) + (1))) /\ ((((exists ff_h_pl_terminal. ff_h_pl_terminal + S (n) = S ((S (l)) * ff_v_pl)) /\ exists ff_q_pl_terminal. ff_u_pl = ff_q_pl_terminal * S ((S (l)) * ff_v_pl) + (n))) /\ forall ff_i_pl. (exists ff_lt_pl_bound. ff_lt_pl_bound + S ff_i_pl = l) -> exists ff_p_pl ff_r_pl ff_s_pl. ((((exists ff_h_pl_factor. ff_h_pl_factor + S (ff_p_pl) = S ((S (ff_i_pl)) * c)) /\ exists ff_q_pl_factor. b = ff_q_pl_factor * S ((S (ff_i_pl)) * c) + (ff_p_pl))) /\ ((((exists ff_h_pl_partial. ff_h_pl_partial + S (ff_r_pl) = S ((S (ff_i_pl)) * ff_v_pl)) /\ exists ff_q_pl_partial. ff_u_pl = ff_q_pl_partial * S ((S (ff_i_pl)) * ff_v_pl) + (ff_r_pl))) /\ ((((exists ff_h_pl_successor. ff_h_pl_successor + S (ff_s_pl) = S ((S (S ff_i_pl)) * ff_v_pl)) /\ exists ff_q_pl_successor. ff_u_pl = ff_q_pl_successor * S ((S (S ff_i_pl)) * ff_v_pl) + (ff_s_pl))) /\ ff_s_pl = ff_r_pl * ff_p_pl)))))) -> (exists ff_u_pr ff_v_pr. ((((exists ff_h_pr_start. ff_h_pr_start + S (1) = S ((S (0)) * ff_v_pr)) /\ exists ff_q_pr_start. ff_u_pr = ff_q_pr_start * S ((S (0)) * ff_v_pr) + (1))) /\ ((((exists ff_h_pr_terminal. ff_h_pr_terminal + S (q) = S ((S (l)) * ff_v_pr)) /\ exists ff_q_pr_terminal. ff_u_pr = ff_q_pr_terminal * S ((S (l)) * ff_v_pr) + (q))) /\ forall ff_i_pr. (exists ff_lt_pr_bound. ff_lt_pr_bound + S ff_i_pr = l) -> exists ff_p_pr ff_r_pr ff_s_pr. ((((exists ff_h_pr_factor. ff_h_pr_factor + S (ff_p_pr) = S ((S (ff_i_pr)) * e)) /\ exists ff_q_pr_factor. d = ff_q_pr_factor * S ((S (ff_i_pr)) * e) + (ff_p_pr))) /\ ((((exists ff_h_pr_partial. ff_h_pr_partial + S (ff_r_pr) = S ((S (ff_i_pr)) * ff_v_pr)) /\ exists ff_q_pr_partial. ff_u_pr = ff_q_pr_partial * S ((S (ff_i_pr)) * ff_v_pr) + (ff_r_pr))) /\ ((((exists ff_h_pr_successor. ff_h_pr_successor + S (ff_s_pr) = S ((S (S ff_i_pr)) * ff_v_pr)) /\ exists ff_q_pr_successor. ff_u_pr = ff_q_pr_successor * S ((S (S ff_i_pr)) * ff_v_pr) + (ff_s_pr))) /\ ff_s_pr = ff_r_pr * ff_p_pr)))))) -> exists fc_u_pv fc_v_pv. n + m * fc_u_pv = q + m * fc_v_pv
use beta_product_zero
use beta_product_succ_decompose
use le_succ
use le_refl
use mod_eq_refl
use mod_eq_mul
intro m
intro b
intro c
intro d
intro e
induction l
intro n
intro q
intro hpw
intro hn
intro hq
have hn1 : n = 1
specialize beta_product_zero b
specialize beta_product_zero c
specialize beta_product_zero n
apply beta_product_zero
exact hn
have hq1 : q = 1
specialize beta_product_zero d
specialize beta_product_zero e
specialize beta_product_zero q
apply beta_product_zero
exact hq
rewrite hn1
rewrite hq1
specialize mod_eq_refl m
specialize mod_eq_refl 1
exact mod_eq_refl
intro n
intro q
intro hpw
intro hn
intro hq
have hnd : exists a r. (((exists ff_h_pd_l_a. ff_h_pd_l_a + S (a) = S ((S (l)) * c)) /\ exists ff_q_pd_l_a. b = ff_q_pd_l_a * S ((S (l)) * c) + (a))) /\ ((exists ff_u_pd_l_p ff_v_pd_l_p. ((((exists ff_h_pd_l_p_start. ff_h_pd_l_p_start + S (1) = S ((S (0)) * ff_v_pd_l_p)) /\ exists ff_q_pd_l_p_start. ff_u_pd_l_p = ff_q_pd_l_p_start * S ((S (0)) * ff_v_pd_l_p) + (1))) /\ ((((exists ff_h_pd_l_p_terminal. ff_h_pd_l_p_terminal + S (r) = S ((S (l)) * ff_v_pd_l_p)) /\ exists ff_q_pd_l_p_terminal. ff_u_pd_l_p = ff_q_pd_l_p_terminal * S ((S (l)) * ff_v_pd_l_p) + (r))) /\ forall ff_i_pd_l_p. (exists ff_lt_pd_l_p_bound. ff_lt_pd_l_p_bound + S ff_i_pd_l_p = l) -> exists ff_p_pd_l_p ff_r_pd_l_p ff_s_pd_l_p. ((((exists ff_h_pd_l_p_factor. ff_h_pd_l_p_factor + S (ff_p_pd_l_p) = S ((S (ff_i_pd_l_p)) * c)) /\ exists ff_q_pd_l_p_factor. b = ff_q_pd_l_p_factor * S ((S (ff_i_pd_l_p)) * c) + (ff_p_pd_l_p))) /\ ((((exists ff_h_pd_l_p_partial. ff_h_pd_l_p_partial + S (ff_r_pd_l_p) = S ((S (ff_i_pd_l_p)) * ff_v_pd_l_p)) /\ exists ff_q_pd_l_p_partial. ff_u_pd_l_p = ff_q_pd_l_p_partial * S ((S (ff_i_pd_l_p)) * ff_v_pd_l_p) + (ff_r_pd_l_p))) /\ ((((exists ff_h_pd_l_p_successor. ff_h_pd_l_p_successor + S (ff_s_pd_l_p) = S ((S (S ff_i_pd_l_p)) * ff_v_pd_l_p)) /\ exists ff_q_pd_l_p_successor. ff_u_pd_l_p = ff_q_pd_l_p_successor * S ((S (S ff_i_pd_l_p)) * ff_v_pd_l_p) + (ff_s_pd_l_p))) /\ ff_s_pd_l_p = ff_r_pd_l_p * ff_p_pd_l_p)))))) /\ n = r * a)
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose l
specialize beta_product_succ_decompose n
apply beta_product_succ_decompose
exact hn
cases hnd
cases hnd_witness
cases hnd_witness_witness
cases hnd_witness_witness_right
have hqd : exists a r. (((exists ff_h_pd_r_a. ff_h_pd_r_a + S (a) = S ((S (l)) * e)) /\ exists ff_q_pd_r_a. d = ff_q_pd_r_a * S ((S (l)) * e) + (a))) /\ ((exists ff_u_pd_r_p ff_v_pd_r_p. ((((exists ff_h_pd_r_p_start. ff_h_pd_r_p_start + S (1) = S ((S (0)) * ff_v_pd_r_p)) /\ exists ff_q_pd_r_p_start. ff_u_pd_r_p = ff_q_pd_r_p_start * S ((S (0)) * ff_v_pd_r_p) + (1))) /\ ((((exists ff_h_pd_r_p_terminal. ff_h_pd_r_p_terminal + S (r) = S ((S (l)) * ff_v_pd_r_p)) /\ exists ff_q_pd_r_p_terminal. ff_u_pd_r_p = ff_q_pd_r_p_terminal * S ((S (l)) * ff_v_pd_r_p) + (r))) /\ forall ff_i_pd_r_p. (exists ff_lt_pd_r_p_bound. ff_lt_pd_r_p_bound + S ff_i_pd_r_p = l) -> exists ff_p_pd_r_p ff_r_pd_r_p ff_s_pd_r_p. ((((exists ff_h_pd_r_p_factor. ff_h_pd_r_p_factor + S (ff_p_pd_r_p) = S ((S (ff_i_pd_r_p)) * e)) /\ exists ff_q_pd_r_p_factor. d = ff_q_pd_r_p_factor * S ((S (ff_i_pd_r_p)) * e) + (ff_p_pd_r_p))) /\ ((((exists ff_h_pd_r_p_partial. ff_h_pd_r_p_partial + S (ff_r_pd_r_p) = S ((S (ff_i_pd_r_p)) * ff_v_pd_r_p)) /\ exists ff_q_pd_r_p_partial. ff_u_pd_r_p = ff_q_pd_r_p_partial * S ((S (ff_i_pd_r_p)) * ff_v_pd_r_p) + (ff_r_pd_r_p))) /\ ((((exists ff_h_pd_r_p_successor. ff_h_pd_r_p_successor + S (ff_s_pd_r_p) = S ((S (S ff_i_pd_r_p)) * ff_v_pd_r_p)) /\ exists ff_q_pd_r_p_successor. ff_u_pd_r_p = ff_q_pd_r_p_successor * S ((S (S ff_i_pd_r_p)) * ff_v_pd_r_p) + (ff_s_pd_r_p))) /\ ff_s_pd_r_p = ff_r_pd_r_p * ff_p_pd_r_p)))))) /\ q = r * a)
specialize beta_product_succ_decompose d
specialize beta_product_succ_decompose e
specialize beta_product_succ_decompose l
specialize beta_product_succ_decompose q
apply beta_product_succ_decompose
exact hq
cases hqd
cases hqd_witness
cases hqd_witness_witness
cases hqd_witness_witness_right
have hpw_prefix : forall i a z. (exists fc_h_pp_pre. fc_h_pp_pre + S i = l) -> (((exists ff_h_pp_pre_la. ff_h_pp_pre_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_pp_pre_la. b = ff_q_pp_pre_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_pp_pre_ra. ff_h_pp_pre_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_pp_pre_ra. d = ff_q_pp_pre_ra * S ((S (i)) * e) + (z))) -> exists fc_u_pp_pre_me fc_v_pp_pre_me. a + m * fc_u_pp_pre_me = z + m * fc_v_pp_pre_me
intro i
intro a
intro z
intro hi
intro ha
intro hz
specialize hpw i
specialize hpw a
specialize hpw z
apply hpw
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
exact ha
exact hz
have hprefix : exists u v. x1 + m * u = x3 + m * v
specialize IH x1
specialize IH x3
apply IH
exact hpw_prefix
exact hnd_witness_witness_right_left
exact hqd_witness_witness_right_left
have hentry : exists u v. x + m * u = x2 + m * v
specialize hpw l
specialize hpw x
specialize hpw x2
apply hpw
specialize le_refl (S l)
exact le_refl
exact hnd_witness_witness_left
exact hqd_witness_witness_left
have hfold : exists u v. (x1 * x) + m * u = (x3 * x2) + m * v
specialize mod_eq_mul m
specialize mod_eq_mul x1
specialize mod_eq_mul x3
specialize mod_eq_mul x
specialize mod_eq_mul x2
apply mod_eq_mul
exact hprefix
exact hentry
rewrite hnd_witness_witness_right_right
rewrite hqd_witness_witness_right_right
exact hfold
qed

Occurrences

4,129

Distinct objects

1,198

Reused references

57

Depth

64

Cuts

117

Certificate SHA-256

eade7d849228…

beta_sum_pointwise_mod_congruentBeta sum pointwise mod congruent Quadratic reciprocity campaign checked

Pointwise congruent decoded prefixes have congruent finite sums.

Prerequisites

beta_sum_zero beta_sum_succ_decompose le_succ le_refl mod_eq_refl mod_eq_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m b c d e l n q. (forall i a z. (exists fc_h_sp. fc_h_sp + S i = l) -> (((exists ff_h_sp_la. ff_h_sp_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_sp_la. b = ff_q_sp_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_sp_ra. ff_h_sp_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_sp_ra. d = ff_q_sp_ra * S ((S (i)) * e) + (z))) -> exists fc_u_sp_me fc_v_sp_me. a + m * fc_u_sp_me = z + m * fc_v_sp_me) -> (exists ff_u_sl ff_v_sl. ((((exists ff_h_sl_start. ff_h_sl_start + S (0) = S ((S (0)) * ff_v_sl)) /\ exists ff_q_sl_start. ff_u_sl = ff_q_sl_start * S ((S (0)) * ff_v_sl) + (0))) /\ ((((exists ff_h_sl_terminal. ff_h_sl_terminal + S (n) = S ((S (l)) * ff_v_sl)) /\ exists ff_q_sl_terminal. ff_u_sl = ff_q_sl_terminal * S ((S (l)) * ff_v_sl) + (n))) /\ forall ff_i_sl. (exists ff_lt_sl_bound. ff_lt_sl_bound + S ff_i_sl = l) -> exists ff_a_sl ff_r_sl ff_s_sl. ((((exists ff_h_sl_summand. ff_h_sl_summand + S (ff_a_sl) = S ((S (ff_i_sl)) * c)) /\ exists ff_q_sl_summand. b = ff_q_sl_summand * S ((S (ff_i_sl)) * c) + (ff_a_sl))) /\ ((((exists ff_h_sl_partial. ff_h_sl_partial + S (ff_r_sl) = S ((S (ff_i_sl)) * ff_v_sl)) /\ exists ff_q_sl_partial. ff_u_sl = ff_q_sl_partial * S ((S (ff_i_sl)) * ff_v_sl) + (ff_r_sl))) /\ ((((exists ff_h_sl_successor. ff_h_sl_successor + S (ff_s_sl) = S ((S (S ff_i_sl)) * ff_v_sl)) /\ exists ff_q_sl_successor. ff_u_sl = ff_q_sl_successor * S ((S (S ff_i_sl)) * ff_v_sl) + (ff_s_sl))) /\ ff_s_sl = ff_r_sl + ff_a_sl)))))) -> (exists ff_u_sr ff_v_sr. ((((exists ff_h_sr_start. ff_h_sr_start + S (0) = S ((S (0)) * ff_v_sr)) /\ exists ff_q_sr_start. ff_u_sr = ff_q_sr_start * S ((S (0)) * ff_v_sr) + (0))) /\ ((((exists ff_h_sr_terminal. ff_h_sr_terminal + S (q) = S ((S (l)) * ff_v_sr)) /\ exists ff_q_sr_terminal. ff_u_sr = ff_q_sr_terminal * S ((S (l)) * ff_v_sr) + (q))) /\ forall ff_i_sr. (exists ff_lt_sr_bound. ff_lt_sr_bound + S ff_i_sr = l) -> exists ff_a_sr ff_r_sr ff_s_sr. ((((exists ff_h_sr_summand. ff_h_sr_summand + S (ff_a_sr) = S ((S (ff_i_sr)) * e)) /\ exists ff_q_sr_summand. d = ff_q_sr_summand * S ((S (ff_i_sr)) * e) + (ff_a_sr))) /\ ((((exists ff_h_sr_partial. ff_h_sr_partial + S (ff_r_sr) = S ((S (ff_i_sr)) * ff_v_sr)) /\ exists ff_q_sr_partial. ff_u_sr = ff_q_sr_partial * S ((S (ff_i_sr)) * ff_v_sr) + (ff_r_sr))) /\ ((((exists ff_h_sr_successor. ff_h_sr_successor + S (ff_s_sr) = S ((S (S ff_i_sr)) * ff_v_sr)) /\ exists ff_q_sr_successor. ff_u_sr = ff_q_sr_successor * S ((S (S ff_i_sr)) * ff_v_sr) + (ff_s_sr))) /\ ff_s_sr = ff_r_sr + ff_a_sr)))))) -> exists fc_u_sv fc_v_sv. n + m * fc_u_sv = q + m * fc_v_sv

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m b c d e l n q. (forall i a z. (exists fc_h_sp. fc_h_sp + S i = l) -> (((exists ff_h_sp_la. ff_h_sp_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_sp_la. b = ff_q_sp_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_sp_ra. ff_h_sp_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_sp_ra. d = ff_q_sp_ra * S ((S (i)) * e) + (z))) -> exists fc_u_sp_me fc_v_sp_me. a + m * fc_u_sp_me = z + m * fc_v_sp_me) -> (exists ff_u_sl ff_v_sl. ((((exists ff_h_sl_start. ff_h_sl_start + S (0) = S ((S (0)) * ff_v_sl)) /\ exists ff_q_sl_start. ff_u_sl = ff_q_sl_start * S ((S (0)) * ff_v_sl) + (0))) /\ ((((exists ff_h_sl_terminal. ff_h_sl_terminal + S (n) = S ((S (l)) * ff_v_sl)) /\ exists ff_q_sl_terminal. ff_u_sl = ff_q_sl_terminal * S ((S (l)) * ff_v_sl) + (n))) /\ forall ff_i_sl. (exists ff_lt_sl_bound. ff_lt_sl_bound + S ff_i_sl = l) -> exists ff_a_sl ff_r_sl ff_s_sl. ((((exists ff_h_sl_summand. ff_h_sl_summand + S (ff_a_sl) = S ((S (ff_i_sl)) * c)) /\ exists ff_q_sl_summand. b = ff_q_sl_summand * S ((S (ff_i_sl)) * c) + (ff_a_sl))) /\ ((((exists ff_h_sl_partial. ff_h_sl_partial + S (ff_r_sl) = S ((S (ff_i_sl)) * ff_v_sl)) /\ exists ff_q_sl_partial. ff_u_sl = ff_q_sl_partial * S ((S (ff_i_sl)) * ff_v_sl) + (ff_r_sl))) /\ ((((exists ff_h_sl_successor. ff_h_sl_successor + S (ff_s_sl) = S ((S (S ff_i_sl)) * ff_v_sl)) /\ exists ff_q_sl_successor. ff_u_sl = ff_q_sl_successor * S ((S (S ff_i_sl)) * ff_v_sl) + (ff_s_sl))) /\ ff_s_sl = ff_r_sl + ff_a_sl)))))) -> (exists ff_u_sr ff_v_sr. ((((exists ff_h_sr_start. ff_h_sr_start + S (0) = S ((S (0)) * ff_v_sr)) /\ exists ff_q_sr_start. ff_u_sr = ff_q_sr_start * S ((S (0)) * ff_v_sr) + (0))) /\ ((((exists ff_h_sr_terminal. ff_h_sr_terminal + S (q) = S ((S (l)) * ff_v_sr)) /\ exists ff_q_sr_terminal. ff_u_sr = ff_q_sr_terminal * S ((S (l)) * ff_v_sr) + (q))) /\ forall ff_i_sr. (exists ff_lt_sr_bound. ff_lt_sr_bound + S ff_i_sr = l) -> exists ff_a_sr ff_r_sr ff_s_sr. ((((exists ff_h_sr_summand. ff_h_sr_summand + S (ff_a_sr) = S ((S (ff_i_sr)) * e)) /\ exists ff_q_sr_summand. d = ff_q_sr_summand * S ((S (ff_i_sr)) * e) + (ff_a_sr))) /\ ((((exists ff_h_sr_partial. ff_h_sr_partial + S (ff_r_sr) = S ((S (ff_i_sr)) * ff_v_sr)) /\ exists ff_q_sr_partial. ff_u_sr = ff_q_sr_partial * S ((S (ff_i_sr)) * ff_v_sr) + (ff_r_sr))) /\ ((((exists ff_h_sr_successor. ff_h_sr_successor + S (ff_s_sr) = S ((S (S ff_i_sr)) * ff_v_sr)) /\ exists ff_q_sr_successor. ff_u_sr = ff_q_sr_successor * S ((S (S ff_i_sr)) * ff_v_sr) + (ff_s_sr))) /\ ff_s_sr = ff_r_sr + ff_a_sr)))))) -> exists fc_u_sv fc_v_sv. n + m * fc_u_sv = q + m * fc_v_sv
use beta_sum_zero
use beta_sum_succ_decompose
use le_succ
use le_refl
use mod_eq_refl
use mod_eq_add
intro m
intro b
intro c
intro d
intro e
induction l
intro n
intro q
intro hpw
intro hn
intro hq
have hn0 : n = 0
specialize beta_sum_zero b
specialize beta_sum_zero c
specialize beta_sum_zero n
apply beta_sum_zero
exact hn
have hq0 : q = 0
specialize beta_sum_zero d
specialize beta_sum_zero e
specialize beta_sum_zero q
apply beta_sum_zero
exact hq
rewrite hn0
rewrite hq0
specialize mod_eq_refl m
specialize mod_eq_refl 0
exact mod_eq_refl
intro n
intro q
intro hpw
intro hn
intro hq
have hnd : exists a r. (((exists ff_h_sd_l_a. ff_h_sd_l_a + S (a) = S ((S (l)) * c)) /\ exists ff_q_sd_l_a. b = ff_q_sd_l_a * S ((S (l)) * c) + (a))) /\ ((exists ff_u_sd_l_p ff_v_sd_l_p. ((((exists ff_h_sd_l_p_start. ff_h_sd_l_p_start + S (0) = S ((S (0)) * ff_v_sd_l_p)) /\ exists ff_q_sd_l_p_start. ff_u_sd_l_p = ff_q_sd_l_p_start * S ((S (0)) * ff_v_sd_l_p) + (0))) /\ ((((exists ff_h_sd_l_p_terminal. ff_h_sd_l_p_terminal + S (r) = S ((S (l)) * ff_v_sd_l_p)) /\ exists ff_q_sd_l_p_terminal. ff_u_sd_l_p = ff_q_sd_l_p_terminal * S ((S (l)) * ff_v_sd_l_p) + (r))) /\ forall ff_i_sd_l_p. (exists ff_lt_sd_l_p_bound. ff_lt_sd_l_p_bound + S ff_i_sd_l_p = l) -> exists ff_a_sd_l_p ff_r_sd_l_p ff_s_sd_l_p. ((((exists ff_h_sd_l_p_summand. ff_h_sd_l_p_summand + S (ff_a_sd_l_p) = S ((S (ff_i_sd_l_p)) * c)) /\ exists ff_q_sd_l_p_summand. b = ff_q_sd_l_p_summand * S ((S (ff_i_sd_l_p)) * c) + (ff_a_sd_l_p))) /\ ((((exists ff_h_sd_l_p_partial. ff_h_sd_l_p_partial + S (ff_r_sd_l_p) = S ((S (ff_i_sd_l_p)) * ff_v_sd_l_p)) /\ exists ff_q_sd_l_p_partial. ff_u_sd_l_p = ff_q_sd_l_p_partial * S ((S (ff_i_sd_l_p)) * ff_v_sd_l_p) + (ff_r_sd_l_p))) /\ ((((exists ff_h_sd_l_p_successor. ff_h_sd_l_p_successor + S (ff_s_sd_l_p) = S ((S (S ff_i_sd_l_p)) * ff_v_sd_l_p)) /\ exists ff_q_sd_l_p_successor. ff_u_sd_l_p = ff_q_sd_l_p_successor * S ((S (S ff_i_sd_l_p)) * ff_v_sd_l_p) + (ff_s_sd_l_p))) /\ ff_s_sd_l_p = ff_r_sd_l_p + ff_a_sd_l_p)))))) /\ n = r + a)
specialize beta_sum_succ_decompose b
specialize beta_sum_succ_decompose c
specialize beta_sum_succ_decompose l
specialize beta_sum_succ_decompose n
apply beta_sum_succ_decompose
exact hn
cases hnd
cases hnd_witness
cases hnd_witness_witness
cases hnd_witness_witness_right
have hqd : exists a r. (((exists ff_h_sd_r_a. ff_h_sd_r_a + S (a) = S ((S (l)) * e)) /\ exists ff_q_sd_r_a. d = ff_q_sd_r_a * S ((S (l)) * e) + (a))) /\ ((exists ff_u_sd_r_p ff_v_sd_r_p. ((((exists ff_h_sd_r_p_start. ff_h_sd_r_p_start + S (0) = S ((S (0)) * ff_v_sd_r_p)) /\ exists ff_q_sd_r_p_start. ff_u_sd_r_p = ff_q_sd_r_p_start * S ((S (0)) * ff_v_sd_r_p) + (0))) /\ ((((exists ff_h_sd_r_p_terminal. ff_h_sd_r_p_terminal + S (r) = S ((S (l)) * ff_v_sd_r_p)) /\ exists ff_q_sd_r_p_terminal. ff_u_sd_r_p = ff_q_sd_r_p_terminal * S ((S (l)) * ff_v_sd_r_p) + (r))) /\ forall ff_i_sd_r_p. (exists ff_lt_sd_r_p_bound. ff_lt_sd_r_p_bound + S ff_i_sd_r_p = l) -> exists ff_a_sd_r_p ff_r_sd_r_p ff_s_sd_r_p. ((((exists ff_h_sd_r_p_summand. ff_h_sd_r_p_summand + S (ff_a_sd_r_p) = S ((S (ff_i_sd_r_p)) * e)) /\ exists ff_q_sd_r_p_summand. d = ff_q_sd_r_p_summand * S ((S (ff_i_sd_r_p)) * e) + (ff_a_sd_r_p))) /\ ((((exists ff_h_sd_r_p_partial. ff_h_sd_r_p_partial + S (ff_r_sd_r_p) = S ((S (ff_i_sd_r_p)) * ff_v_sd_r_p)) /\ exists ff_q_sd_r_p_partial. ff_u_sd_r_p = ff_q_sd_r_p_partial * S ((S (ff_i_sd_r_p)) * ff_v_sd_r_p) + (ff_r_sd_r_p))) /\ ((((exists ff_h_sd_r_p_successor. ff_h_sd_r_p_successor + S (ff_s_sd_r_p) = S ((S (S ff_i_sd_r_p)) * ff_v_sd_r_p)) /\ exists ff_q_sd_r_p_successor. ff_u_sd_r_p = ff_q_sd_r_p_successor * S ((S (S ff_i_sd_r_p)) * ff_v_sd_r_p) + (ff_s_sd_r_p))) /\ ff_s_sd_r_p = ff_r_sd_r_p + ff_a_sd_r_p)))))) /\ q = r + a)
specialize beta_sum_succ_decompose d
specialize beta_sum_succ_decompose e
specialize beta_sum_succ_decompose l
specialize beta_sum_succ_decompose q
apply beta_sum_succ_decompose
exact hq
cases hqd
cases hqd_witness
cases hqd_witness_witness
cases hqd_witness_witness_right
have hpw_prefix : forall i a z. (exists fc_h_sp_pre. fc_h_sp_pre + S i = l) -> (((exists ff_h_sp_pre_la. ff_h_sp_pre_la + S (a) = S ((S (i)) * c)) /\ exists ff_q_sp_pre_la. b = ff_q_sp_pre_la * S ((S (i)) * c) + (a))) -> (((exists ff_h_sp_pre_ra. ff_h_sp_pre_ra + S (z) = S ((S (i)) * e)) /\ exists ff_q_sp_pre_ra. d = ff_q_sp_pre_ra * S ((S (i)) * e) + (z))) -> exists fc_u_sp_pre_me fc_v_sp_pre_me. a + m * fc_u_sp_pre_me = z + m * fc_v_sp_pre_me
intro i
intro a
intro z
intro hi
intro ha
intro hz
specialize hpw i
specialize hpw a
specialize hpw z
apply hpw
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
exact ha
exact hz
have hprefix : exists u v. x1 + m * u = x3 + m * v
specialize IH x1
specialize IH x3
apply IH
exact hpw_prefix
exact hnd_witness_witness_right_left
exact hqd_witness_witness_right_left
have hentry : exists u v. x + m * u = x2 + m * v
specialize hpw l
specialize hpw x
specialize hpw x2
apply hpw
specialize le_refl (S l)
exact le_refl
exact hnd_witness_witness_left
exact hqd_witness_witness_left
have hfold : exists u v. (x1 + x) + m * u = (x3 + x2) + m * v
specialize mod_eq_add m
specialize mod_eq_add x1
specialize mod_eq_add x3
specialize mod_eq_add x
specialize mod_eq_add x2
apply mod_eq_add
exact hprefix
exact hentry
rewrite hnd_witness_witness_right_right
rewrite hqd_witness_witness_right_right
exact hfold
qed

Occurrences

2,994

Distinct objects

1,079

Reused references

46

Depth

64

Cuts

84

Certificate SHA-256

3e8170b1fba3…

all_bits_zeroAll bits zero Quadratic reciprocity campaign checked

The empty decoded prefix consists only of bits, vacuously.

Prerequisites

add_eq_zero_right succ_ne_zero

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> exists ff_bit_empty. ((((exists ff_h_empty_decoded. ff_h_empty_decoded + S (ff_bit_empty) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (ff_bit_empty))) /\ (ff_bit_empty = 0 \/ ff_bit_empty = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> exists ff_bit_empty. ((((exists ff_h_empty_decoded. ff_h_empty_decoded + S (ff_bit_empty) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (ff_bit_empty))) /\ (ff_bit_empty = 0 \/ ff_bit_empty = 1)))
use add_eq_zero_right
use succ_ne_zero
intro b
intro c
intro l
intro hl
intro i
intro hi
rewrite hl at hi
exfalso
cases hi
have hsi : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi
qed

Occurrences

41

Distinct objects

41

Reused references

0

Depth

15

Cuts

2

Certificate SHA-256

1dff5495e030…

all_bits_prefix_succAll bits prefix succ Quadratic reciprocity campaign checked

Dropping the final entry preserves the all-bits invariant.

Prerequisites

le_succ

Used directly by

bit_count_succ_decompose

Exact expanded PA statement

Copy statement

forall b c l sl. sl = S l -> (forall ff_i_successor. (exists ff_lt_successor_bound. ff_lt_successor_bound + S ff_i_successor = sl) -> exists ff_bit_successor. ((((exists ff_h_successor_decoded. ff_h_successor_decoded + S (ff_bit_successor) = S ((S (ff_i_successor)) * c)) /\ exists ff_q_successor_decoded. b = ff_q_successor_decoded * S ((S (ff_i_successor)) * c) + (ff_bit_successor))) /\ (ff_bit_successor = 0 \/ ff_bit_successor = 1))) -> (forall ff_i_prefix. (exists ff_lt_prefix_bound. ff_lt_prefix_bound + S ff_i_prefix = l) -> exists ff_bit_prefix. ((((exists ff_h_prefix_decoded. ff_h_prefix_decoded + S (ff_bit_prefix) = S ((S (ff_i_prefix)) * c)) /\ exists ff_q_prefix_decoded. b = ff_q_prefix_decoded * S ((S (ff_i_prefix)) * c) + (ff_bit_prefix))) /\ (ff_bit_prefix = 0 \/ ff_bit_prefix = 1)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l sl. sl = S l -> (forall ff_i_successor. (exists ff_lt_successor_bound. ff_lt_successor_bound + S ff_i_successor = sl) -> exists ff_bit_successor. ((((exists ff_h_successor_decoded. ff_h_successor_decoded + S (ff_bit_successor) = S ((S (ff_i_successor)) * c)) /\ exists ff_q_successor_decoded. b = ff_q_successor_decoded * S ((S (ff_i_successor)) * c) + (ff_bit_successor))) /\ (ff_bit_successor = 0 \/ ff_bit_successor = 1))) -> (forall ff_i_prefix. (exists ff_lt_prefix_bound. ff_lt_prefix_bound + S ff_i_prefix = l) -> exists ff_bit_prefix. ((((exists ff_h_prefix_decoded. ff_h_prefix_decoded + S (ff_bit_prefix) = S ((S (ff_i_prefix)) * c)) /\ exists ff_q_prefix_decoded. b = ff_q_prefix_decoded * S ((S (ff_i_prefix)) * c) + (ff_bit_prefix))) /\ (ff_bit_prefix = 0 \/ ff_bit_prefix = 1)))
use le_succ
intro b
intro c
intro l
intro sl
intro hsl
intro hbits
rewrite hsl at hbits
intro i
intro hi
specialize hbits i
apply hbits
specialize le_succ (S i)
specialize le_succ l
apply le_succ
exact hi
qed

Occurrences

71

Distinct objects

69

Reused references

2

Depth

20

Cuts

2

Certificate SHA-256

ed2b35b3bbd8…

all_bits_last_succAll bits last succ Quadratic reciprocity campaign checked

The final entry of a nonempty all-bits prefix is zero or one.

Prerequisites

le_refl

Used directly by

bit_count_succ_decompose

Exact expanded PA statement

Copy statement

forall b c l sl. sl = S l -> (forall ff_i_successor. (exists ff_lt_successor_bound. ff_lt_successor_bound + S ff_i_successor = sl) -> exists ff_bit_successor. ((((exists ff_h_successor_decoded. ff_h_successor_decoded + S (ff_bit_successor) = S ((S (ff_i_successor)) * c)) /\ exists ff_q_successor_decoded. b = ff_q_successor_decoded * S ((S (ff_i_successor)) * c) + (ff_bit_successor))) /\ (ff_bit_successor = 0 \/ ff_bit_successor = 1))) -> exists a. ((((exists ff_h_last. ff_h_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_last. b = ff_q_last * S ((S (l)) * c) + (a))) /\ (a = 0 \/ a = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l sl. sl = S l -> (forall ff_i_successor. (exists ff_lt_successor_bound. ff_lt_successor_bound + S ff_i_successor = sl) -> exists ff_bit_successor. ((((exists ff_h_successor_decoded. ff_h_successor_decoded + S (ff_bit_successor) = S ((S (ff_i_successor)) * c)) /\ exists ff_q_successor_decoded. b = ff_q_successor_decoded * S ((S (ff_i_successor)) * c) + (ff_bit_successor))) /\ (ff_bit_successor = 0 \/ ff_bit_successor = 1))) -> exists a. ((((exists ff_h_last. ff_h_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_last. b = ff_q_last * S ((S (l)) * c) + (a))) /\ (a = 0 \/ a = 1))
use le_refl
intro b
intro c
intro l
intro sl
intro hsl
intro hbits
rewrite hsl at hbits
specialize hbits l
apply hbits
specialize le_refl (S l)
exact le_refl
qed

Occurrences

48

Distinct objects

48

Reused references

0

Depth

15

Cuts

2

Certificate SHA-256

11b88553710c…

bit_count_existsBit count exists Quadratic reciprocity campaign checked

Every all-bits prefix has a relational count of its ones.

Prerequisites

beta_sum_exists

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l. (forall ff_i_a. (exists ff_lt_a_bound. ff_lt_a_bound + S ff_i_a = l) -> exists ff_bit_a. ((((exists ff_h_a_decoded. ff_h_a_decoded + S (ff_bit_a) = S ((S (ff_i_a)) * c)) /\ exists ff_q_a_decoded. b = ff_q_a_decoded * S ((S (ff_i_a)) * c) + (ff_bit_a))) /\ (ff_bit_a = 0 \/ ff_bit_a = 1))) -> exists n. (((exists ff_u_b_sum ff_v_b_sum. ((((exists ff_h_b_sum_start. ff_h_b_sum_start + S (0) = S ((S (0)) * ff_v_b_sum)) /\ exists ff_q_b_sum_start. ff_u_b_sum = ff_q_b_sum_start * S ((S (0)) * ff_v_b_sum) + (0))) /\ ((((exists ff_h_b_sum_terminal. ff_h_b_sum_terminal + S (n) = S ((S (l)) * ff_v_b_sum)) /\ exists ff_q_b_sum_terminal. ff_u_b_sum = ff_q_b_sum_terminal * S ((S (l)) * ff_v_b_sum) + (n))) /\ forall ff_i_b_sum. (exists ff_lt_b_sum_bound. ff_lt_b_sum_bound + S ff_i_b_sum = l) -> exists ff_a_b_sum ff_r_b_sum ff_s_b_sum. ((((exists ff_h_b_sum_summand. ff_h_b_sum_summand + S (ff_a_b_sum) = S ((S (ff_i_b_sum)) * c)) /\ exists ff_q_b_sum_summand. b = ff_q_b_sum_summand * S ((S (ff_i_b_sum)) * c) + (ff_a_b_sum))) /\ ((((exists ff_h_b_sum_partial. ff_h_b_sum_partial + S (ff_r_b_sum) = S ((S (ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_partial. ff_u_b_sum = ff_q_b_sum_partial * S ((S (ff_i_b_sum)) * ff_v_b_sum) + (ff_r_b_sum))) /\ ((((exists ff_h_b_sum_successor. ff_h_b_sum_successor + S (ff_s_b_sum) = S ((S (S ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_successor. ff_u_b_sum = ff_q_b_sum_successor * S ((S (S ff_i_b_sum)) * ff_v_b_sum) + (ff_s_b_sum))) /\ ff_s_b_sum = ff_r_b_sum + ff_a_b_sum)))))) /\ (forall ff_i_b_bits. (exists ff_lt_b_bits_bound. ff_lt_b_bits_bound + S ff_i_b_bits = l) -> exists ff_bit_b_bits. ((((exists ff_h_b_bits_decoded. ff_h_b_bits_decoded + S (ff_bit_b_bits) = S ((S (ff_i_b_bits)) * c)) /\ exists ff_q_b_bits_decoded. b = ff_q_b_bits_decoded * S ((S (ff_i_b_bits)) * c) + (ff_bit_b_bits))) /\ (ff_bit_b_bits = 0 \/ ff_bit_b_bits = 1)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l. (forall ff_i_a. (exists ff_lt_a_bound. ff_lt_a_bound + S ff_i_a = l) -> exists ff_bit_a. ((((exists ff_h_a_decoded. ff_h_a_decoded + S (ff_bit_a) = S ((S (ff_i_a)) * c)) /\ exists ff_q_a_decoded. b = ff_q_a_decoded * S ((S (ff_i_a)) * c) + (ff_bit_a))) /\ (ff_bit_a = 0 \/ ff_bit_a = 1))) -> exists n. (((exists ff_u_b_sum ff_v_b_sum. ((((exists ff_h_b_sum_start. ff_h_b_sum_start + S (0) = S ((S (0)) * ff_v_b_sum)) /\ exists ff_q_b_sum_start. ff_u_b_sum = ff_q_b_sum_start * S ((S (0)) * ff_v_b_sum) + (0))) /\ ((((exists ff_h_b_sum_terminal. ff_h_b_sum_terminal + S (n) = S ((S (l)) * ff_v_b_sum)) /\ exists ff_q_b_sum_terminal. ff_u_b_sum = ff_q_b_sum_terminal * S ((S (l)) * ff_v_b_sum) + (n))) /\ forall ff_i_b_sum. (exists ff_lt_b_sum_bound. ff_lt_b_sum_bound + S ff_i_b_sum = l) -> exists ff_a_b_sum ff_r_b_sum ff_s_b_sum. ((((exists ff_h_b_sum_summand. ff_h_b_sum_summand + S (ff_a_b_sum) = S ((S (ff_i_b_sum)) * c)) /\ exists ff_q_b_sum_summand. b = ff_q_b_sum_summand * S ((S (ff_i_b_sum)) * c) + (ff_a_b_sum))) /\ ((((exists ff_h_b_sum_partial. ff_h_b_sum_partial + S (ff_r_b_sum) = S ((S (ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_partial. ff_u_b_sum = ff_q_b_sum_partial * S ((S (ff_i_b_sum)) * ff_v_b_sum) + (ff_r_b_sum))) /\ ((((exists ff_h_b_sum_successor. ff_h_b_sum_successor + S (ff_s_b_sum) = S ((S (S ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_successor. ff_u_b_sum = ff_q_b_sum_successor * S ((S (S ff_i_b_sum)) * ff_v_b_sum) + (ff_s_b_sum))) /\ ff_s_b_sum = ff_r_b_sum + ff_a_b_sum)))))) /\ (forall ff_i_b_bits. (exists ff_lt_b_bits_bound. ff_lt_b_bits_bound + S ff_i_b_bits = l) -> exists ff_bit_b_bits. ((((exists ff_h_b_bits_decoded. ff_h_b_bits_decoded + S (ff_bit_b_bits) = S ((S (ff_i_b_bits)) * c)) /\ exists ff_q_b_bits_decoded. b = ff_q_b_bits_decoded * S ((S (ff_i_b_bits)) * c) + (ff_bit_b_bits))) /\ (ff_bit_b_bits = 0 \/ ff_bit_b_bits = 1)))))
use beta_sum_exists
intro b
intro c
intro l
intro hbits
specialize beta_sum_exists b
specialize beta_sum_exists c
specialize beta_sum_exists l
cases beta_sum_exists
exists x
split
exact beta_sum_exists_witness
exact hbits
qed

Occurrences

30,514

Distinct objects

4,771

Reused references

236

Depth

87

Cuts

917

Certificate SHA-256

547a69863934…

bit_count_functionalBit count functional Quadratic reciprocity campaign checked

The relational count of a fixed all-bits prefix is unique.

Prerequisites

beta_sum_functional

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l n m. (((exists ff_u_l_sum ff_v_l_sum. ((((exists ff_h_l_sum_start. ff_h_l_sum_start + S (0) = S ((S (0)) * ff_v_l_sum)) /\ exists ff_q_l_sum_start. ff_u_l_sum = ff_q_l_sum_start * S ((S (0)) * ff_v_l_sum) + (0))) /\ ((((exists ff_h_l_sum_terminal. ff_h_l_sum_terminal + S (n) = S ((S (l)) * ff_v_l_sum)) /\ exists ff_q_l_sum_terminal. ff_u_l_sum = ff_q_l_sum_terminal * S ((S (l)) * ff_v_l_sum) + (n))) /\ forall ff_i_l_sum. (exists ff_lt_l_sum_bound. ff_lt_l_sum_bound + S ff_i_l_sum = l) -> exists ff_a_l_sum ff_r_l_sum ff_s_l_sum. ((((exists ff_h_l_sum_summand. ff_h_l_sum_summand + S (ff_a_l_sum) = S ((S (ff_i_l_sum)) * c)) /\ exists ff_q_l_sum_summand. b = ff_q_l_sum_summand * S ((S (ff_i_l_sum)) * c) + (ff_a_l_sum))) /\ ((((exists ff_h_l_sum_partial. ff_h_l_sum_partial + S (ff_r_l_sum) = S ((S (ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_partial. ff_u_l_sum = ff_q_l_sum_partial * S ((S (ff_i_l_sum)) * ff_v_l_sum) + (ff_r_l_sum))) /\ ((((exists ff_h_l_sum_successor. ff_h_l_sum_successor + S (ff_s_l_sum) = S ((S (S ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_successor. ff_u_l_sum = ff_q_l_sum_successor * S ((S (S ff_i_l_sum)) * ff_v_l_sum) + (ff_s_l_sum))) /\ ff_s_l_sum = ff_r_l_sum + ff_a_l_sum)))))) /\ (forall ff_i_l_bits. (exists ff_lt_l_bits_bound. ff_lt_l_bits_bound + S ff_i_l_bits = l) -> exists ff_bit_l_bits. ((((exists ff_h_l_bits_decoded. ff_h_l_bits_decoded + S (ff_bit_l_bits) = S ((S (ff_i_l_bits)) * c)) /\ exists ff_q_l_bits_decoded. b = ff_q_l_bits_decoded * S ((S (ff_i_l_bits)) * c) + (ff_bit_l_bits))) /\ (ff_bit_l_bits = 0 \/ ff_bit_l_bits = 1))))) -> (((exists ff_u_r_sum ff_v_r_sum. ((((exists ff_h_r_sum_start. ff_h_r_sum_start + S (0) = S ((S (0)) * ff_v_r_sum)) /\ exists ff_q_r_sum_start. ff_u_r_sum = ff_q_r_sum_start * S ((S (0)) * ff_v_r_sum) + (0))) /\ ((((exists ff_h_r_sum_terminal. ff_h_r_sum_terminal + S (m) = S ((S (l)) * ff_v_r_sum)) /\ exists ff_q_r_sum_terminal. ff_u_r_sum = ff_q_r_sum_terminal * S ((S (l)) * ff_v_r_sum) + (m))) /\ forall ff_i_r_sum. (exists ff_lt_r_sum_bound. ff_lt_r_sum_bound + S ff_i_r_sum = l) -> exists ff_a_r_sum ff_r_r_sum ff_s_r_sum. ((((exists ff_h_r_sum_summand. ff_h_r_sum_summand + S (ff_a_r_sum) = S ((S (ff_i_r_sum)) * c)) /\ exists ff_q_r_sum_summand. b = ff_q_r_sum_summand * S ((S (ff_i_r_sum)) * c) + (ff_a_r_sum))) /\ ((((exists ff_h_r_sum_partial. ff_h_r_sum_partial + S (ff_r_r_sum) = S ((S (ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_partial. ff_u_r_sum = ff_q_r_sum_partial * S ((S (ff_i_r_sum)) * ff_v_r_sum) + (ff_r_r_sum))) /\ ((((exists ff_h_r_sum_successor. ff_h_r_sum_successor + S (ff_s_r_sum) = S ((S (S ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_successor. ff_u_r_sum = ff_q_r_sum_successor * S ((S (S ff_i_r_sum)) * ff_v_r_sum) + (ff_s_r_sum))) /\ ff_s_r_sum = ff_r_r_sum + ff_a_r_sum)))))) /\ (forall ff_i_r_bits. (exists ff_lt_r_bits_bound. ff_lt_r_bits_bound + S ff_i_r_bits = l) -> exists ff_bit_r_bits. ((((exists ff_h_r_bits_decoded. ff_h_r_bits_decoded + S (ff_bit_r_bits) = S ((S (ff_i_r_bits)) * c)) /\ exists ff_q_r_bits_decoded. b = ff_q_r_bits_decoded * S ((S (ff_i_r_bits)) * c) + (ff_bit_r_bits))) /\ (ff_bit_r_bits = 0 \/ ff_bit_r_bits = 1))))) -> n = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n m. (((exists ff_u_l_sum ff_v_l_sum. ((((exists ff_h_l_sum_start. ff_h_l_sum_start + S (0) = S ((S (0)) * ff_v_l_sum)) /\ exists ff_q_l_sum_start. ff_u_l_sum = ff_q_l_sum_start * S ((S (0)) * ff_v_l_sum) + (0))) /\ ((((exists ff_h_l_sum_terminal. ff_h_l_sum_terminal + S (n) = S ((S (l)) * ff_v_l_sum)) /\ exists ff_q_l_sum_terminal. ff_u_l_sum = ff_q_l_sum_terminal * S ((S (l)) * ff_v_l_sum) + (n))) /\ forall ff_i_l_sum. (exists ff_lt_l_sum_bound. ff_lt_l_sum_bound + S ff_i_l_sum = l) -> exists ff_a_l_sum ff_r_l_sum ff_s_l_sum. ((((exists ff_h_l_sum_summand. ff_h_l_sum_summand + S (ff_a_l_sum) = S ((S (ff_i_l_sum)) * c)) /\ exists ff_q_l_sum_summand. b = ff_q_l_sum_summand * S ((S (ff_i_l_sum)) * c) + (ff_a_l_sum))) /\ ((((exists ff_h_l_sum_partial. ff_h_l_sum_partial + S (ff_r_l_sum) = S ((S (ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_partial. ff_u_l_sum = ff_q_l_sum_partial * S ((S (ff_i_l_sum)) * ff_v_l_sum) + (ff_r_l_sum))) /\ ((((exists ff_h_l_sum_successor. ff_h_l_sum_successor + S (ff_s_l_sum) = S ((S (S ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_successor. ff_u_l_sum = ff_q_l_sum_successor * S ((S (S ff_i_l_sum)) * ff_v_l_sum) + (ff_s_l_sum))) /\ ff_s_l_sum = ff_r_l_sum + ff_a_l_sum)))))) /\ (forall ff_i_l_bits. (exists ff_lt_l_bits_bound. ff_lt_l_bits_bound + S ff_i_l_bits = l) -> exists ff_bit_l_bits. ((((exists ff_h_l_bits_decoded. ff_h_l_bits_decoded + S (ff_bit_l_bits) = S ((S (ff_i_l_bits)) * c)) /\ exists ff_q_l_bits_decoded. b = ff_q_l_bits_decoded * S ((S (ff_i_l_bits)) * c) + (ff_bit_l_bits))) /\ (ff_bit_l_bits = 0 \/ ff_bit_l_bits = 1))))) -> (((exists ff_u_r_sum ff_v_r_sum. ((((exists ff_h_r_sum_start. ff_h_r_sum_start + S (0) = S ((S (0)) * ff_v_r_sum)) /\ exists ff_q_r_sum_start. ff_u_r_sum = ff_q_r_sum_start * S ((S (0)) * ff_v_r_sum) + (0))) /\ ((((exists ff_h_r_sum_terminal. ff_h_r_sum_terminal + S (m) = S ((S (l)) * ff_v_r_sum)) /\ exists ff_q_r_sum_terminal. ff_u_r_sum = ff_q_r_sum_terminal * S ((S (l)) * ff_v_r_sum) + (m))) /\ forall ff_i_r_sum. (exists ff_lt_r_sum_bound. ff_lt_r_sum_bound + S ff_i_r_sum = l) -> exists ff_a_r_sum ff_r_r_sum ff_s_r_sum. ((((exists ff_h_r_sum_summand. ff_h_r_sum_summand + S (ff_a_r_sum) = S ((S (ff_i_r_sum)) * c)) /\ exists ff_q_r_sum_summand. b = ff_q_r_sum_summand * S ((S (ff_i_r_sum)) * c) + (ff_a_r_sum))) /\ ((((exists ff_h_r_sum_partial. ff_h_r_sum_partial + S (ff_r_r_sum) = S ((S (ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_partial. ff_u_r_sum = ff_q_r_sum_partial * S ((S (ff_i_r_sum)) * ff_v_r_sum) + (ff_r_r_sum))) /\ ((((exists ff_h_r_sum_successor. ff_h_r_sum_successor + S (ff_s_r_sum) = S ((S (S ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_successor. ff_u_r_sum = ff_q_r_sum_successor * S ((S (S ff_i_r_sum)) * ff_v_r_sum) + (ff_s_r_sum))) /\ ff_s_r_sum = ff_r_r_sum + ff_a_r_sum)))))) /\ (forall ff_i_r_bits. (exists ff_lt_r_bits_bound. ff_lt_r_bits_bound + S ff_i_r_bits = l) -> exists ff_bit_r_bits. ((((exists ff_h_r_bits_decoded. ff_h_r_bits_decoded + S (ff_bit_r_bits) = S ((S (ff_i_r_bits)) * c)) /\ exists ff_q_r_bits_decoded. b = ff_q_r_bits_decoded * S ((S (ff_i_r_bits)) * c) + (ff_bit_r_bits))) /\ (ff_bit_r_bits = 0 \/ ff_bit_r_bits = 1))))) -> n = m
use beta_sum_functional
intro b
intro c
intro l
intro n
intro m
intro hn
intro hm
cases hn
cases hm
specialize beta_sum_functional b
specialize beta_sum_functional c
specialize beta_sum_functional l
specialize beta_sum_functional n
specialize beta_sum_functional m
apply beta_sum_functional
exact hn_left
exact hm_left
qed

Occurrences

1,488

Distinct objects

1,015

Reused references

39

Depth

62

Cuts

38

Certificate SHA-256

f92a1960a142…

bit_count_zeroBit count zero Quadratic reciprocity campaign checked

An empty bit prefix contains zero ones.

Prerequisites

beta_sum_zero

Used directly by

bit_count_bounded

Exact expanded PA statement

Copy statement

forall b c l n. l = 0 -> (((exists ff_u_zero_sum ff_v_zero_sum. ((((exists ff_h_zero_sum_start. ff_h_zero_sum_start + S (0) = S ((S (0)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_start. ff_u_zero_sum = ff_q_zero_sum_start * S ((S (0)) * ff_v_zero_sum) + (0))) /\ ((((exists ff_h_zero_sum_terminal. ff_h_zero_sum_terminal + S (n) = S ((S (l)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_terminal. ff_u_zero_sum = ff_q_zero_sum_terminal * S ((S (l)) * ff_v_zero_sum) + (n))) /\ forall ff_i_zero_sum. (exists ff_lt_zero_sum_bound. ff_lt_zero_sum_bound + S ff_i_zero_sum = l) -> exists ff_a_zero_sum ff_r_zero_sum ff_s_zero_sum. ((((exists ff_h_zero_sum_summand. ff_h_zero_sum_summand + S (ff_a_zero_sum) = S ((S (ff_i_zero_sum)) * c)) /\ exists ff_q_zero_sum_summand. b = ff_q_zero_sum_summand * S ((S (ff_i_zero_sum)) * c) + (ff_a_zero_sum))) /\ ((((exists ff_h_zero_sum_partial. ff_h_zero_sum_partial + S (ff_r_zero_sum) = S ((S (ff_i_zero_sum)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_partial. ff_u_zero_sum = ff_q_zero_sum_partial * S ((S (ff_i_zero_sum)) * ff_v_zero_sum) + (ff_r_zero_sum))) /\ ((((exists ff_h_zero_sum_successor. ff_h_zero_sum_successor + S (ff_s_zero_sum) = S ((S (S ff_i_zero_sum)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_successor. ff_u_zero_sum = ff_q_zero_sum_successor * S ((S (S ff_i_zero_sum)) * ff_v_zero_sum) + (ff_s_zero_sum))) /\ ff_s_zero_sum = ff_r_zero_sum + ff_a_zero_sum)))))) /\ (forall ff_i_zero_bits. (exists ff_lt_zero_bits_bound. ff_lt_zero_bits_bound + S ff_i_zero_bits = l) -> exists ff_bit_zero_bits. ((((exists ff_h_zero_bits_decoded. ff_h_zero_bits_decoded + S (ff_bit_zero_bits) = S ((S (ff_i_zero_bits)) * c)) /\ exists ff_q_zero_bits_decoded. b = ff_q_zero_bits_decoded * S ((S (ff_i_zero_bits)) * c) + (ff_bit_zero_bits))) /\ (ff_bit_zero_bits = 0 \/ ff_bit_zero_bits = 1))))) -> n = 0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. l = 0 -> (((exists ff_u_zero_sum ff_v_zero_sum. ((((exists ff_h_zero_sum_start. ff_h_zero_sum_start + S (0) = S ((S (0)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_start. ff_u_zero_sum = ff_q_zero_sum_start * S ((S (0)) * ff_v_zero_sum) + (0))) /\ ((((exists ff_h_zero_sum_terminal. ff_h_zero_sum_terminal + S (n) = S ((S (l)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_terminal. ff_u_zero_sum = ff_q_zero_sum_terminal * S ((S (l)) * ff_v_zero_sum) + (n))) /\ forall ff_i_zero_sum. (exists ff_lt_zero_sum_bound. ff_lt_zero_sum_bound + S ff_i_zero_sum = l) -> exists ff_a_zero_sum ff_r_zero_sum ff_s_zero_sum. ((((exists ff_h_zero_sum_summand. ff_h_zero_sum_summand + S (ff_a_zero_sum) = S ((S (ff_i_zero_sum)) * c)) /\ exists ff_q_zero_sum_summand. b = ff_q_zero_sum_summand * S ((S (ff_i_zero_sum)) * c) + (ff_a_zero_sum))) /\ ((((exists ff_h_zero_sum_partial. ff_h_zero_sum_partial + S (ff_r_zero_sum) = S ((S (ff_i_zero_sum)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_partial. ff_u_zero_sum = ff_q_zero_sum_partial * S ((S (ff_i_zero_sum)) * ff_v_zero_sum) + (ff_r_zero_sum))) /\ ((((exists ff_h_zero_sum_successor. ff_h_zero_sum_successor + S (ff_s_zero_sum) = S ((S (S ff_i_zero_sum)) * ff_v_zero_sum)) /\ exists ff_q_zero_sum_successor. ff_u_zero_sum = ff_q_zero_sum_successor * S ((S (S ff_i_zero_sum)) * ff_v_zero_sum) + (ff_s_zero_sum))) /\ ff_s_zero_sum = ff_r_zero_sum + ff_a_zero_sum)))))) /\ (forall ff_i_zero_bits. (exists ff_lt_zero_bits_bound. ff_lt_zero_bits_bound + S ff_i_zero_bits = l) -> exists ff_bit_zero_bits. ((((exists ff_h_zero_bits_decoded. ff_h_zero_bits_decoded + S (ff_bit_zero_bits) = S ((S (ff_i_zero_bits)) * c)) /\ exists ff_q_zero_bits_decoded. b = ff_q_zero_bits_decoded * S ((S (ff_i_zero_bits)) * c) + (ff_bit_zero_bits))) /\ (ff_bit_zero_bits = 0 \/ ff_bit_zero_bits = 1))))) -> n = 0
use beta_sum_zero
intro b
intro c
intro l
intro n
intro hl
intro hcount
rewrite hl at hcount
rewrite hl at hcount
rewrite hl at hcount
cases hcount
specialize beta_sum_zero b
specialize beta_sum_zero c
specialize beta_sum_zero n
apply beta_sum_zero
exact hcount_left
qed

Occurrences

1,216

Distinct objects

787

Reused references

37

Depth

61

Cuts

32

Certificate SHA-256

d7b3bef13355…

bit_count_succ_decomposeBit count succ decompose Quadratic reciprocity campaign checked

A successor count is its prefix count plus a final zero-or-one bit.

Prerequisites

beta_sum_succ_decompose all_bits_prefix_succ all_bits_last_succ beta_at_unique

Used directly by

bit_count_bounded

Exact expanded PA statement

Copy statement

forall b c l sl n. sl = S l -> (((exists ff_u_successor_sum ff_v_successor_sum. ((((exists ff_h_successor_sum_start. ff_h_successor_sum_start + S (0) = S ((S (0)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_start. ff_u_successor_sum = ff_q_successor_sum_start * S ((S (0)) * ff_v_successor_sum) + (0))) /\ ((((exists ff_h_successor_sum_terminal. ff_h_successor_sum_terminal + S (n) = S ((S (sl)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_terminal. ff_u_successor_sum = ff_q_successor_sum_terminal * S ((S (sl)) * ff_v_successor_sum) + (n))) /\ forall ff_i_successor_sum. (exists ff_lt_successor_sum_bound. ff_lt_successor_sum_bound + S ff_i_successor_sum = sl) -> exists ff_a_successor_sum ff_r_successor_sum ff_s_successor_sum. ((((exists ff_h_successor_sum_summand. ff_h_successor_sum_summand + S (ff_a_successor_sum) = S ((S (ff_i_successor_sum)) * c)) /\ exists ff_q_successor_sum_summand. b = ff_q_successor_sum_summand * S ((S (ff_i_successor_sum)) * c) + (ff_a_successor_sum))) /\ ((((exists ff_h_successor_sum_partial. ff_h_successor_sum_partial + S (ff_r_successor_sum) = S ((S (ff_i_successor_sum)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_partial. ff_u_successor_sum = ff_q_successor_sum_partial * S ((S (ff_i_successor_sum)) * ff_v_successor_sum) + (ff_r_successor_sum))) /\ ((((exists ff_h_successor_sum_successor. ff_h_successor_sum_successor + S (ff_s_successor_sum) = S ((S (S ff_i_successor_sum)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_successor. ff_u_successor_sum = ff_q_successor_sum_successor * S ((S (S ff_i_successor_sum)) * ff_v_successor_sum) + (ff_s_successor_sum))) /\ ff_s_successor_sum = ff_r_successor_sum + ff_a_successor_sum)))))) /\ (forall ff_i_successor_bits. (exists ff_lt_successor_bits_bound. ff_lt_successor_bits_bound + S ff_i_successor_bits = sl) -> exists ff_bit_successor_bits. ((((exists ff_h_successor_bits_decoded. ff_h_successor_bits_decoded + S (ff_bit_successor_bits) = S ((S (ff_i_successor_bits)) * c)) /\ exists ff_q_successor_bits_decoded. b = ff_q_successor_bits_decoded * S ((S (ff_i_successor_bits)) * c) + (ff_bit_successor_bits))) /\ (ff_bit_successor_bits = 0 \/ ff_bit_successor_bits = 1))))) -> exists a r. (((exists ff_h_last. ff_h_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_last. b = ff_q_last * S ((S (l)) * c) + (a))) /\ ((((exists ff_u_prefix_sum ff_v_prefix_sum. ((((exists ff_h_prefix_sum_start. ff_h_prefix_sum_start + S (0) = S ((S (0)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_start. ff_u_prefix_sum = ff_q_prefix_sum_start * S ((S (0)) * ff_v_prefix_sum) + (0))) /\ ((((exists ff_h_prefix_sum_terminal. ff_h_prefix_sum_terminal + S (r) = S ((S (l)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_terminal. ff_u_prefix_sum = ff_q_prefix_sum_terminal * S ((S (l)) * ff_v_prefix_sum) + (r))) /\ forall ff_i_prefix_sum. (exists ff_lt_prefix_sum_bound. ff_lt_prefix_sum_bound + S ff_i_prefix_sum = l) -> exists ff_a_prefix_sum ff_r_prefix_sum ff_s_prefix_sum. ((((exists ff_h_prefix_sum_summand. ff_h_prefix_sum_summand + S (ff_a_prefix_sum) = S ((S (ff_i_prefix_sum)) * c)) /\ exists ff_q_prefix_sum_summand. b = ff_q_prefix_sum_summand * S ((S (ff_i_prefix_sum)) * c) + (ff_a_prefix_sum))) /\ ((((exists ff_h_prefix_sum_partial. ff_h_prefix_sum_partial + S (ff_r_prefix_sum) = S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_partial. ff_u_prefix_sum = ff_q_prefix_sum_partial * S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_r_prefix_sum))) /\ ((((exists ff_h_prefix_sum_successor. ff_h_prefix_sum_successor + S (ff_s_prefix_sum) = S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_successor. ff_u_prefix_sum = ff_q_prefix_sum_successor * S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_s_prefix_sum))) /\ ff_s_prefix_sum = ff_r_prefix_sum + ff_a_prefix_sum)))))) /\ (forall ff_i_prefix_bits. (exists ff_lt_prefix_bits_bound. ff_lt_prefix_bits_bound + S ff_i_prefix_bits = l) -> exists ff_bit_prefix_bits. ((((exists ff_h_prefix_bits_decoded. ff_h_prefix_bits_decoded + S (ff_bit_prefix_bits) = S ((S (ff_i_prefix_bits)) * c)) /\ exists ff_q_prefix_bits_decoded. b = ff_q_prefix_bits_decoded * S ((S (ff_i_prefix_bits)) * c) + (ff_bit_prefix_bits))) /\ (ff_bit_prefix_bits = 0 \/ ff_bit_prefix_bits = 1))))) /\ ((a = 0 \/ a = 1) /\ n = r + a))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l sl n. sl = S l -> (((exists ff_u_successor_sum ff_v_successor_sum. ((((exists ff_h_successor_sum_start. ff_h_successor_sum_start + S (0) = S ((S (0)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_start. ff_u_successor_sum = ff_q_successor_sum_start * S ((S (0)) * ff_v_successor_sum) + (0))) /\ ((((exists ff_h_successor_sum_terminal. ff_h_successor_sum_terminal + S (n) = S ((S (sl)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_terminal. ff_u_successor_sum = ff_q_successor_sum_terminal * S ((S (sl)) * ff_v_successor_sum) + (n))) /\ forall ff_i_successor_sum. (exists ff_lt_successor_sum_bound. ff_lt_successor_sum_bound + S ff_i_successor_sum = sl) -> exists ff_a_successor_sum ff_r_successor_sum ff_s_successor_sum. ((((exists ff_h_successor_sum_summand. ff_h_successor_sum_summand + S (ff_a_successor_sum) = S ((S (ff_i_successor_sum)) * c)) /\ exists ff_q_successor_sum_summand. b = ff_q_successor_sum_summand * S ((S (ff_i_successor_sum)) * c) + (ff_a_successor_sum))) /\ ((((exists ff_h_successor_sum_partial. ff_h_successor_sum_partial + S (ff_r_successor_sum) = S ((S (ff_i_successor_sum)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_partial. ff_u_successor_sum = ff_q_successor_sum_partial * S ((S (ff_i_successor_sum)) * ff_v_successor_sum) + (ff_r_successor_sum))) /\ ((((exists ff_h_successor_sum_successor. ff_h_successor_sum_successor + S (ff_s_successor_sum) = S ((S (S ff_i_successor_sum)) * ff_v_successor_sum)) /\ exists ff_q_successor_sum_successor. ff_u_successor_sum = ff_q_successor_sum_successor * S ((S (S ff_i_successor_sum)) * ff_v_successor_sum) + (ff_s_successor_sum))) /\ ff_s_successor_sum = ff_r_successor_sum + ff_a_successor_sum)))))) /\ (forall ff_i_successor_bits. (exists ff_lt_successor_bits_bound. ff_lt_successor_bits_bound + S ff_i_successor_bits = sl) -> exists ff_bit_successor_bits. ((((exists ff_h_successor_bits_decoded. ff_h_successor_bits_decoded + S (ff_bit_successor_bits) = S ((S (ff_i_successor_bits)) * c)) /\ exists ff_q_successor_bits_decoded. b = ff_q_successor_bits_decoded * S ((S (ff_i_successor_bits)) * c) + (ff_bit_successor_bits))) /\ (ff_bit_successor_bits = 0 \/ ff_bit_successor_bits = 1))))) -> exists a r. (((exists ff_h_last. ff_h_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_last. b = ff_q_last * S ((S (l)) * c) + (a))) /\ ((((exists ff_u_prefix_sum ff_v_prefix_sum. ((((exists ff_h_prefix_sum_start. ff_h_prefix_sum_start + S (0) = S ((S (0)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_start. ff_u_prefix_sum = ff_q_prefix_sum_start * S ((S (0)) * ff_v_prefix_sum) + (0))) /\ ((((exists ff_h_prefix_sum_terminal. ff_h_prefix_sum_terminal + S (r) = S ((S (l)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_terminal. ff_u_prefix_sum = ff_q_prefix_sum_terminal * S ((S (l)) * ff_v_prefix_sum) + (r))) /\ forall ff_i_prefix_sum. (exists ff_lt_prefix_sum_bound. ff_lt_prefix_sum_bound + S ff_i_prefix_sum = l) -> exists ff_a_prefix_sum ff_r_prefix_sum ff_s_prefix_sum. ((((exists ff_h_prefix_sum_summand. ff_h_prefix_sum_summand + S (ff_a_prefix_sum) = S ((S (ff_i_prefix_sum)) * c)) /\ exists ff_q_prefix_sum_summand. b = ff_q_prefix_sum_summand * S ((S (ff_i_prefix_sum)) * c) + (ff_a_prefix_sum))) /\ ((((exists ff_h_prefix_sum_partial. ff_h_prefix_sum_partial + S (ff_r_prefix_sum) = S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_partial. ff_u_prefix_sum = ff_q_prefix_sum_partial * S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_r_prefix_sum))) /\ ((((exists ff_h_prefix_sum_successor. ff_h_prefix_sum_successor + S (ff_s_prefix_sum) = S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_successor. ff_u_prefix_sum = ff_q_prefix_sum_successor * S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_s_prefix_sum))) /\ ff_s_prefix_sum = ff_r_prefix_sum + ff_a_prefix_sum)))))) /\ (forall ff_i_prefix_bits. (exists ff_lt_prefix_bits_bound. ff_lt_prefix_bits_bound + S ff_i_prefix_bits = l) -> exists ff_bit_prefix_bits. ((((exists ff_h_prefix_bits_decoded. ff_h_prefix_bits_decoded + S (ff_bit_prefix_bits) = S ((S (ff_i_prefix_bits)) * c)) /\ exists ff_q_prefix_bits_decoded. b = ff_q_prefix_bits_decoded * S ((S (ff_i_prefix_bits)) * c) + (ff_bit_prefix_bits))) /\ (ff_bit_prefix_bits = 0 \/ ff_bit_prefix_bits = 1))))) /\ ((a = 0 \/ a = 1) /\ n = r + a))
use beta_sum_succ_decompose
use all_bits_prefix_succ
use all_bits_last_succ
use beta_at_unique
intro b
intro c
intro l
intro sl
intro n
intro hsl
intro hcount
rewrite hsl at hcount
rewrite hsl at hcount
rewrite hsl at hcount
rewrite hsl at hcount
cases hcount
have hsum : exists a r. (((exists ff_h_sum_last. ff_h_sum_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_sum_last. b = ff_q_sum_last * S ((S (l)) * c) + (a))) /\ ((exists ff_u_sum_prefix ff_v_sum_prefix. ((((exists ff_h_sum_prefix_start. ff_h_sum_prefix_start + S (0) = S ((S (0)) * ff_v_sum_prefix)) /\ exists ff_q_sum_prefix_start. ff_u_sum_prefix = ff_q_sum_prefix_start * S ((S (0)) * ff_v_sum_prefix) + (0))) /\ ((((exists ff_h_sum_prefix_terminal. ff_h_sum_prefix_terminal + S (r) = S ((S (l)) * ff_v_sum_prefix)) /\ exists ff_q_sum_prefix_terminal. ff_u_sum_prefix = ff_q_sum_prefix_terminal * S ((S (l)) * ff_v_sum_prefix) + (r))) /\ forall ff_i_sum_prefix. (exists ff_lt_sum_prefix_bound. ff_lt_sum_prefix_bound + S ff_i_sum_prefix = l) -> exists ff_a_sum_prefix ff_r_sum_prefix ff_s_sum_prefix. ((((exists ff_h_sum_prefix_summand. ff_h_sum_prefix_summand + S (ff_a_sum_prefix) = S ((S (ff_i_sum_prefix)) * c)) /\ exists ff_q_sum_prefix_summand. b = ff_q_sum_prefix_summand * S ((S (ff_i_sum_prefix)) * c) + (ff_a_sum_prefix))) /\ ((((exists ff_h_sum_prefix_partial. ff_h_sum_prefix_partial + S (ff_r_sum_prefix) = S ((S (ff_i_sum_prefix)) * ff_v_sum_prefix)) /\ exists ff_q_sum_prefix_partial. ff_u_sum_prefix = ff_q_sum_prefix_partial * S ((S (ff_i_sum_prefix)) * ff_v_sum_prefix) + (ff_r_sum_prefix))) /\ ((((exists ff_h_sum_prefix_successor. ff_h_sum_prefix_successor + S (ff_s_sum_prefix) = S ((S (S ff_i_sum_prefix)) * ff_v_sum_prefix)) /\ exists ff_q_sum_prefix_successor. ff_u_sum_prefix = ff_q_sum_prefix_successor * S ((S (S ff_i_sum_prefix)) * ff_v_sum_prefix) + (ff_s_sum_prefix))) /\ ff_s_sum_prefix = ff_r_sum_prefix + ff_a_sum_prefix)))))) /\ n = r + a)
specialize beta_sum_succ_decompose b
specialize beta_sum_succ_decompose c
specialize beta_sum_succ_decompose l
specialize beta_sum_succ_decompose n
apply beta_sum_succ_decompose
exact hcount_left
cases hsum
cases hsum_witness
cases hsum_witness_witness
cases hsum_witness_witness_right
have hlast : exists a. ((((exists ff_h_bits_last. ff_h_bits_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_bits_last. b = ff_q_bits_last * S ((S (l)) * c) + (a))) /\ (a = 0 \/ a = 1))
specialize all_bits_last_succ b
specialize all_bits_last_succ c
specialize all_bits_last_succ l
specialize all_bits_last_succ (S l)
apply all_bits_last_succ
refl
exact hcount_right
cases hlast
cases hlast_witness
have ha : x = x2
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique x2
apply beta_at_unique
exact hsum_witness_witness_left
exact hlast_witness_left
have hprefix : forall ff_i_kept_prefix. (exists ff_lt_kept_prefix_bound. ff_lt_kept_prefix_bound + S ff_i_kept_prefix = l) -> exists ff_bit_kept_prefix. ((((exists ff_h_kept_prefix_decoded. ff_h_kept_prefix_decoded + S (ff_bit_kept_prefix) = S ((S (ff_i_kept_prefix)) * c)) /\ exists ff_q_kept_prefix_decoded. b = ff_q_kept_prefix_decoded * S ((S (ff_i_kept_prefix)) * c) + (ff_bit_kept_prefix))) /\ (ff_bit_kept_prefix = 0 \/ ff_bit_kept_prefix = 1))
specialize all_bits_prefix_succ b
specialize all_bits_prefix_succ c
specialize all_bits_prefix_succ l
specialize all_bits_prefix_succ (S l)
apply all_bits_prefix_succ
refl
exact hcount_right
exists x
exists x1
split
exact hsum_witness_witness_left
split
split
exact hsum_witness_witness_right_left
exact hprefix
split
rewrite ha
rewrite ha
exact hlast_witness_right
exact hsum_witness_witness_right_right
qed

Occurrences

2,608

Distinct objects

949

Reused references

42

Depth

63

Cuts

73

Certificate SHA-256

e4dbc903d361…

bit_count_boundedBit count bounded Quadratic reciprocity campaign checked

A zero/one count never exceeds the length of its decoded prefix.

Prerequisites

bit_count_zero bit_count_succ_decompose add_succ_left add_assoc

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c l n. (((exists ff_u_b_sum ff_v_b_sum. ((((exists ff_h_b_sum_start. ff_h_b_sum_start + S (0) = S ((S (0)) * ff_v_b_sum)) /\ exists ff_q_b_sum_start. ff_u_b_sum = ff_q_b_sum_start * S ((S (0)) * ff_v_b_sum) + (0))) /\ ((((exists ff_h_b_sum_terminal. ff_h_b_sum_terminal + S (n) = S ((S (l)) * ff_v_b_sum)) /\ exists ff_q_b_sum_terminal. ff_u_b_sum = ff_q_b_sum_terminal * S ((S (l)) * ff_v_b_sum) + (n))) /\ forall ff_i_b_sum. (exists ff_lt_b_sum_bound. ff_lt_b_sum_bound + S ff_i_b_sum = l) -> exists ff_a_b_sum ff_r_b_sum ff_s_b_sum. ((((exists ff_h_b_sum_summand. ff_h_b_sum_summand + S (ff_a_b_sum) = S ((S (ff_i_b_sum)) * c)) /\ exists ff_q_b_sum_summand. b = ff_q_b_sum_summand * S ((S (ff_i_b_sum)) * c) + (ff_a_b_sum))) /\ ((((exists ff_h_b_sum_partial. ff_h_b_sum_partial + S (ff_r_b_sum) = S ((S (ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_partial. ff_u_b_sum = ff_q_b_sum_partial * S ((S (ff_i_b_sum)) * ff_v_b_sum) + (ff_r_b_sum))) /\ ((((exists ff_h_b_sum_successor. ff_h_b_sum_successor + S (ff_s_b_sum) = S ((S (S ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_successor. ff_u_b_sum = ff_q_b_sum_successor * S ((S (S ff_i_b_sum)) * ff_v_b_sum) + (ff_s_b_sum))) /\ ff_s_b_sum = ff_r_b_sum + ff_a_b_sum)))))) /\ (forall ff_i_b_bits. (exists ff_lt_b_bits_bound. ff_lt_b_bits_bound + S ff_i_b_bits = l) -> exists ff_bit_b_bits. ((((exists ff_h_b_bits_decoded. ff_h_b_bits_decoded + S (ff_bit_b_bits) = S ((S (ff_i_b_bits)) * c)) /\ exists ff_q_b_bits_decoded. b = ff_q_b_bits_decoded * S ((S (ff_i_b_bits)) * c) + (ff_bit_b_bits))) /\ (ff_bit_b_bits = 0 \/ ff_bit_b_bits = 1))))) -> exists h. h + n = l

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l n. (((exists ff_u_b_sum ff_v_b_sum. ((((exists ff_h_b_sum_start. ff_h_b_sum_start + S (0) = S ((S (0)) * ff_v_b_sum)) /\ exists ff_q_b_sum_start. ff_u_b_sum = ff_q_b_sum_start * S ((S (0)) * ff_v_b_sum) + (0))) /\ ((((exists ff_h_b_sum_terminal. ff_h_b_sum_terminal + S (n) = S ((S (l)) * ff_v_b_sum)) /\ exists ff_q_b_sum_terminal. ff_u_b_sum = ff_q_b_sum_terminal * S ((S (l)) * ff_v_b_sum) + (n))) /\ forall ff_i_b_sum. (exists ff_lt_b_sum_bound. ff_lt_b_sum_bound + S ff_i_b_sum = l) -> exists ff_a_b_sum ff_r_b_sum ff_s_b_sum. ((((exists ff_h_b_sum_summand. ff_h_b_sum_summand + S (ff_a_b_sum) = S ((S (ff_i_b_sum)) * c)) /\ exists ff_q_b_sum_summand. b = ff_q_b_sum_summand * S ((S (ff_i_b_sum)) * c) + (ff_a_b_sum))) /\ ((((exists ff_h_b_sum_partial. ff_h_b_sum_partial + S (ff_r_b_sum) = S ((S (ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_partial. ff_u_b_sum = ff_q_b_sum_partial * S ((S (ff_i_b_sum)) * ff_v_b_sum) + (ff_r_b_sum))) /\ ((((exists ff_h_b_sum_successor. ff_h_b_sum_successor + S (ff_s_b_sum) = S ((S (S ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_successor. ff_u_b_sum = ff_q_b_sum_successor * S ((S (S ff_i_b_sum)) * ff_v_b_sum) + (ff_s_b_sum))) /\ ff_s_b_sum = ff_r_b_sum + ff_a_b_sum)))))) /\ (forall ff_i_b_bits. (exists ff_lt_b_bits_bound. ff_lt_b_bits_bound + S ff_i_b_bits = l) -> exists ff_bit_b_bits. ((((exists ff_h_b_bits_decoded. ff_h_b_bits_decoded + S (ff_bit_b_bits) = S ((S (ff_i_b_bits)) * c)) /\ exists ff_q_b_bits_decoded. b = ff_q_b_bits_decoded * S ((S (ff_i_b_bits)) * c) + (ff_bit_b_bits))) /\ (ff_bit_b_bits = 0 \/ ff_bit_b_bits = 1))))) -> exists h. h + n = l
use bit_count_zero
use bit_count_succ_decompose
use add_succ_left
use add_assoc
intro b
intro c
induction l
intro n
intro hcount
have hn : n = 0
specialize bit_count_zero b
specialize bit_count_zero c
specialize bit_count_zero 0
specialize bit_count_zero n
apply bit_count_zero
refl
exact hcount
exists 0
rewrite hn
rewrite PA3
refl
intro n
intro hcount
have hdecomp : exists a r. (((exists ff_h_last. ff_h_last + S (a) = S ((S (l)) * c)) /\ exists ff_q_last. b = ff_q_last * S ((S (l)) * c) + (a))) /\ ((((exists ff_u_prefix_sum ff_v_prefix_sum. ((((exists ff_h_prefix_sum_start. ff_h_prefix_sum_start + S (0) = S ((S (0)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_start. ff_u_prefix_sum = ff_q_prefix_sum_start * S ((S (0)) * ff_v_prefix_sum) + (0))) /\ ((((exists ff_h_prefix_sum_terminal. ff_h_prefix_sum_terminal + S (r) = S ((S (l)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_terminal. ff_u_prefix_sum = ff_q_prefix_sum_terminal * S ((S (l)) * ff_v_prefix_sum) + (r))) /\ forall ff_i_prefix_sum. (exists ff_lt_prefix_sum_bound. ff_lt_prefix_sum_bound + S ff_i_prefix_sum = l) -> exists ff_a_prefix_sum ff_r_prefix_sum ff_s_prefix_sum. ((((exists ff_h_prefix_sum_summand. ff_h_prefix_sum_summand + S (ff_a_prefix_sum) = S ((S (ff_i_prefix_sum)) * c)) /\ exists ff_q_prefix_sum_summand. b = ff_q_prefix_sum_summand * S ((S (ff_i_prefix_sum)) * c) + (ff_a_prefix_sum))) /\ ((((exists ff_h_prefix_sum_partial. ff_h_prefix_sum_partial + S (ff_r_prefix_sum) = S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_partial. ff_u_prefix_sum = ff_q_prefix_sum_partial * S ((S (ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_r_prefix_sum))) /\ ((((exists ff_h_prefix_sum_successor. ff_h_prefix_sum_successor + S (ff_s_prefix_sum) = S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum)) /\ exists ff_q_prefix_sum_successor. ff_u_prefix_sum = ff_q_prefix_sum_successor * S ((S (S ff_i_prefix_sum)) * ff_v_prefix_sum) + (ff_s_prefix_sum))) /\ ff_s_prefix_sum = ff_r_prefix_sum + ff_a_prefix_sum)))))) /\ (forall ff_i_prefix_bits. (exists ff_lt_prefix_bits_bound. ff_lt_prefix_bits_bound + S ff_i_prefix_bits = l) -> exists ff_bit_prefix_bits. ((((exists ff_h_prefix_bits_decoded. ff_h_prefix_bits_decoded + S (ff_bit_prefix_bits) = S ((S (ff_i_prefix_bits)) * c)) /\ exists ff_q_prefix_bits_decoded. b = ff_q_prefix_bits_decoded * S ((S (ff_i_prefix_bits)) * c) + (ff_bit_prefix_bits))) /\ (ff_bit_prefix_bits = 0 \/ ff_bit_prefix_bits = 1))))) /\ ((a = 0 \/ a = 1) /\ n = r + a))
specialize bit_count_succ_decompose b
specialize bit_count_succ_decompose c
specialize bit_count_succ_decompose l
specialize bit_count_succ_decompose (S l)
specialize bit_count_succ_decompose n
apply bit_count_succ_decompose
refl
exact hcount
cases hdecomp
cases hdecomp_witness
cases hdecomp_witness_witness
cases hdecomp_witness_witness_right
cases hdecomp_witness_witness_right_right
have hr : exists h. h + x1 = l
specialize IH x1
apply IH
exact hdecomp_witness_witness_right_left
cases hr
cases hdecomp_witness_witness_right_right_left
exists S x2
rewrite hdecomp_witness_witness_right_right_right
rewrite hdecomp_witness_witness_right_right_left_left
rewrite PA3
specialize add_succ_left x2
specialize add_succ_left x1
rewrite add_succ_left
congr
exact hr_witness
exists x2
rewrite hdecomp_witness_witness_right_right_right
rewrite hdecomp_witness_witness_right_right_left_right
trans (x2 + x1) + 1
symm
apply add_assoc
rewrite PA4
rewrite PA3
congr
exact hr_witness
qed

Occurrences

3,987

Distinct objects

1,147

Reused references

45

Depth

65

Cuts

109

Certificate SHA-256

53aa30ac1659…

prime_coprime_or_dividesPrime coprime or divides Quadratic reciprocity campaign checked

A prime is constructively either coprime to a natural or divides it.

Prerequisites

gcd_exists_relational prime_divisor_eq_one_or_self is_gcd_one_to_coprime

Used directly by

prime_not_divides_coprime

Exact expanded PA statement

Copy statement

forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * k

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * k
use gcd_exists_relational
use prime_divisor_eq_one_or_self
use is_gcd_one_to_coprime
intro p
intro a
intro hp
have hg : exists g. (((exists x. p = g * x) /\ (exists y. a = g * y)) /\ forall c. (exists u. p = c * u) -> (exists v. a = c * v) -> exists w. g = c * w)
apply gcd_exists_relational
cases hg
have hgfull : (((exists u. p = x * u) /\ (exists v. a = x * v)) /\ forall c. (exists s. p = c * s) -> (exists t. a = c * t) -> exists w. x = c * w)
exact hg_witness
cases hg_witness
cases hg_witness_left
have hfactor : x = 1 \/ p = x
specialize prime_divisor_eq_one_or_self p
specialize prime_divisor_eq_one_or_self x
apply prime_divisor_eq_one_or_self
exact hp
exact hg_witness_left_left
cases hfactor
left
apply is_gcd_one_to_coprime
have hg1 : (((exists u. p = 1 * u) /\ (exists v. a = 1 * v)) /\ forall c. (exists s. p = c * s) -> (exists t. a = c * t) -> exists w. 1 = c * w)
rewrite <- hfactor_left
rewrite <- hfactor_left
rewrite <- hfactor_left
exact hgfull
exact hg1
right
cases hg_witness_left_right
exists x1
rewrite hfactor_right
exact hg_witness_left_right_witness
qed

Occurrences

1,572

Distinct objects

1,069

Reused references

66

Depth

47

Cuts

48

Certificate SHA-256

8d298ff62d8d…

prime_not_divides_coprimePrime not divides coprime Quadratic reciprocity campaign checked

A prime not dividing a natural is coprime to that natural.

Prerequisites

prime_coprime_or_divides

Used directly by

distinct_primes_coprime prime_mod_inverse prime_mod_cancel

Exact expanded PA statement

Copy statement

forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1
use prime_coprime_or_divides
intro p
intro a
intro hp
intro hnot
specialize prime_coprime_or_divides p
specialize prime_coprime_or_divides a
have hsplit : (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * k
apply prime_coprime_or_divides
exact hp
cases hsplit
exact hsplit_left
exfalso
apply hnot
exact hsplit_right
qed

Occurrences

1,588

Distinct objects

1,085

Reused references

66

Depth

48

Cuts

49

Certificate SHA-256

cc42611e7af1…

distinct_primes_coprimeDistinct primes coprime Quadratic reciprocity campaign checked

Distinct primes are coprime in the expanded common-divisor relation.

Prerequisites

prime_divisor_eq_one_or_self prime_not_divides_coprime

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p q. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> (~(q = 1) /\ forall c e. q = c * e -> c = 1 \/ e = 1) -> ~(p = q) -> forall d. (exists x. p = d * x) -> (exists y. q = d * y) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p q. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> (~(q = 1) /\ forall c e. q = c * e -> c = 1 \/ e = 1) -> ~(p = q) -> forall d. (exists x. p = d * x) -> (exists y. q = d * y) -> d = 1
use prime_divisor_eq_one_or_self
use prime_not_divides_coprime
intro p
intro q
intro hp
intro hq
intro hpq
have hnot : ~(exists k. q = p * k)
intro hdiv
specialize prime_divisor_eq_one_or_self q
specialize prime_divisor_eq_one_or_self p
have hfactor : p = 1 \/ q = p
apply prime_divisor_eq_one_or_self
exact hq
exact hdiv
cases hfactor
cases hp
apply hp_left
exact hfactor_left
apply hpq
symm
exact hfactor_right
specialize prime_not_divides_coprime p
specialize prime_not_divides_coprime q
apply prime_not_divides_coprime
exact hp
exact hnot
qed

Occurrences

1,675

Distinct objects

1,115

Reused references

67

Depth

50

Cuts

53

Certificate SHA-256

16d0eddcf30a…

coprime_balanced_mod_inverseCoprime balanced mod inverse Quadratic reciprocity campaign checked

Balanced Bezout coefficients give a subtraction-free modular inverse.

Prerequisites

coprime_balanced_bezout add_assoc

Used directly by

coprime_mod_inverse

Exact expanded PA statement

Copy statement

forall a m. (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists xp xn u v. a * xp + m * u = (1 + a * xn) + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a m. (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists xp xn u v. a * xp + m * u = (1 + a * xn) + m * v
use coprime_balanced_bezout
use add_assoc
intro a
intro m
intro hcop
have hbez : exists xp yp xn yn. a * xp + m * yp = 1 + (a * xn + m * yn)
specialize coprime_balanced_bezout a
specialize coprime_balanced_bezout m
apply coprime_balanced_bezout
exact hcop
cases hbez
cases hbez_witness
cases hbez_witness_witness
cases hbez_witness_witness_witness
exists x
exists x2
exists x1
exists x3
trans 1 + (a * x2 + m * x3)
exact hbez_witness_witness_witness_witness
symm
apply add_assoc
qed

Occurrences

2,365

Distinct objects

1,269

Reused references

82

Depth

49

Cuts

69

Certificate SHA-256

3b82b2213ae1…

coprime_mod_inverseCoprime mod inverse Quadratic reciprocity campaign checked

A nonzero modulus turns balanced Bezout data into a natural modular inverse.

Prerequisites

nonzero_is_succ coprime_balanced_mod_inverse mod_eq_refl mod_eq_add mod_eq_predecessor_cancel mod_eq_trans mul_add mul_assoc mul_comm

Used directly by

mod_eq_cancel_coprime prime_mod_inverse coprime_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall a m. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists z u v. a * z + m * u = 1 + m * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a m. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists z u v. a * z + m * u = 1 + m * v
use nonzero_is_succ
use coprime_balanced_mod_inverse
use mod_eq_refl
use mod_eq_add
use mod_eq_predecessor_cancel
use mod_eq_trans
use mul_add
use mul_assoc
use mul_comm
intro a
intro m
intro hm
intro hcop
have hms : exists k. m = S k
specialize nonzero_is_succ m
apply nonzero_is_succ
exact hm
have hbal : exists xp xn u v. a * xp + m * u = (1 + a * xn) + m * v
specialize coprime_balanced_mod_inverse a
specialize coprime_balanced_mod_inverse m
apply coprime_balanced_mod_inverse
exact hcop
cases hms
cases hbal
cases hbal_witness
cases hbal_witness_witness
cases hbal_witness_witness_witness
have hself : exists u v. (x * (a * x2)) + m * u = (x * (a * x2)) + m * v
specialize mod_eq_refl m
specialize mod_eq_refl (x * (a * x2))
apply mod_eq_refl
have hadd : exists u v. ((a * x1) + x * (a * x2)) + m * u = ((1 + a * x2) + x * (a * x2)) + m * v
specialize mod_eq_add m
specialize mod_eq_add (a * x1)
specialize mod_eq_add (1 + a * x2)
specialize mod_eq_add (x * (a * x2))
specialize mod_eq_add (x * (a * x2))
apply mod_eq_add
exists x3
exists x4
exact hbal_witness_witness_witness_witness
exact hself
have hcancel : exists u v. ((1 + a * x2) + x * (a * x2)) + m * u = 1 + m * v
specialize mod_eq_predecessor_cancel x
specialize mod_eq_predecessor_cancel 1
specialize mod_eq_predecessor_cancel (a * x2)
have hkcancel : exists u v. ((1 + a * x2) + x * (a * x2)) + S x * u = 1 + S x * v
apply mod_eq_predecessor_cancel
rewrite <- hms_witness at hkcancel
rewrite <- hms_witness at hkcancel
exact hkcancel
have hfinal : exists u v. ((a * x1) + x * (a * x2)) + m * u = 1 + m * v
specialize mod_eq_trans m
specialize mod_eq_trans ((a * x1) + x * (a * x2))
specialize mod_eq_trans ((1 + a * x2) + x * (a * x2))
specialize mod_eq_trans 1
apply mod_eq_trans
exact hadd
exact hcancel
have hnorm : a * (x1 + x * x2) = (a * x1) + x * (a * x2)
trans a * x1 + a * (x * x2)
apply mul_add
congr
refl
trans (a * x) * x2
symm
apply mul_assoc
trans (x * a) * x2
congr
apply mul_comm
refl
apply mul_assoc
exists x1 + x * x2
rewrite hnorm
exact hfinal
qed

Occurrences

3,820

Distinct objects

1,549

Reused references

94

Depth

51

Cuts

112

Certificate SHA-256

f60b0a6bf024…

mod_eq_cancel_coprimeMod eq cancel coprime Quadratic reciprocity campaign checked

A coprime factor cancels from balanced congruence at nonzero modulus.

Prerequisites

coprime_mod_inverse mod_eq_mul_right mod_eq_mul_left mod_eq_symm mod_eq_trans mul_assoc mul_comm mul_one

Used directly by

prime_mod_cancel

Exact expanded PA statement

Copy statement

forall m a x y. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> (exists u v. (a * x) + m * u = (a * y) + m * v) -> exists r s. x + m * r = y + m * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a x y. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> (exists u v. (a * x) + m * u = (a * y) + m * v) -> exists r s. x + m * r = y + m * s
use coprime_mod_inverse
use mod_eq_mul_right
use mod_eq_mul_left
use mod_eq_symm
use mod_eq_trans
use mul_assoc
use mul_comm
use mul_one
intro m
intro a
intro x
intro y
intro hm
intro hcop
intro hxy
have hinv : exists z u v. a * z + m * u = 1 + m * v
specialize coprime_mod_inverse a
specialize coprime_mod_inverse m
apply coprime_mod_inverse
exact hm
exact hcop
cases hinv
cases hinv_witness
cases hinv_witness_witness
have hzx : exists u v. (x * (a * x1)) + m * u = (x * 1) + m * v
specialize mod_eq_mul_left m
specialize mod_eq_mul_left (a * x1)
specialize mod_eq_mul_left 1
specialize mod_eq_mul_left x
apply mod_eq_mul_left
exists x2
exists x3
exact hinv_witness_witness_witness
have hnormx : x * (a * x1) = (a * x) * x1
trans (x * a) * x1
symm
apply mul_assoc
trans (a * x) * x1
congr
apply mul_comm
refl
refl
have honex : x * 1 = x
apply mul_one
have hxprod : exists u v. ((a * x) * x1) + m * u = x + m * v
cases hzx
cases hzx_witness
exists x4
exists x5
trans (x * (a * x1)) + m * x4
congr
symm
exact hnormx
refl
trans (x * 1) + m * x5
exact hzx_witness_witness
congr
exact honex
refl
have hxhprod : exists u v. x + m * u = ((a * x) * x1) + m * v
specialize mod_eq_symm m
specialize mod_eq_symm ((a * x) * x1)
specialize mod_eq_symm x
apply mod_eq_symm
exact hxprod
have hscaled : exists u v. ((a * x) * x1) + m * u = ((a * y) * x1) + m * v
specialize mod_eq_mul_right m
specialize mod_eq_mul_right (a * x)
specialize mod_eq_mul_right (a * y)
specialize mod_eq_mul_right x1
apply mod_eq_mul_right
exact hxy
have hzy : exists u v. (y * (a * x1)) + m * u = (y * 1) + m * v
specialize mod_eq_mul_left m
specialize mod_eq_mul_left (a * x1)
specialize mod_eq_mul_left 1
specialize mod_eq_mul_left y
apply mod_eq_mul_left
exists x2
exists x3
exact hinv_witness_witness_witness
have hnormy : y * (a * x1) = (a * y) * x1
trans (y * a) * x1
symm
apply mul_assoc
trans (a * y) * x1
congr
apply mul_comm
refl
refl
have honey : y * 1 = y
apply mul_one
have hyprod : exists u v. ((a * y) * x1) + m * u = y + m * v
cases hzy
cases hzy_witness
exists x4
exists x5
trans (y * (a * x1)) + m * x4
congr
symm
exact hnormy
refl
trans (y * 1) + m * x5
exact hzy_witness_witness
congr
exact honey
refl
have hmid : exists u v. x + m * u = ((a * y) * x1) + m * v
specialize mod_eq_trans m
specialize mod_eq_trans x
specialize mod_eq_trans ((a * x) * x1)
specialize mod_eq_trans ((a * y) * x1)
apply mod_eq_trans
exact hxhprod
exact hscaled
specialize mod_eq_trans m
specialize mod_eq_trans x
specialize mod_eq_trans ((a * y) * x1)
specialize mod_eq_trans y
apply mod_eq_trans
exact hmid
exact hyprod
qed

Occurrences

5,804

Distinct objects

1,762

Reused references

102

Depth

52

Cuts

169

Certificate SHA-256

d7a55457c6e7…

prime_mod_inversePrime mod inverse Quadratic reciprocity campaign checked

A nonzero residue modulo a prime has a natural modular inverse.

Prerequisites

prime_nonzero prime_not_divides_coprime coprime_mod_inverse

Used directly by

prime_bounded_nonzero_mod_inverse

Exact expanded PA statement

Copy statement

forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> exists z u v. a * z + p * u = 1 + p * v

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> exists z u v. a * z + p * u = 1 + p * v
use prime_nonzero
use prime_not_divides_coprime
use coprime_mod_inverse
intro p
intro a
intro hp
intro hnot
have hp0 : ~(p = 0)
intro hpzero
specialize prime_nonzero p
apply prime_nonzero
exact hp
exact hpzero
have hcop : forall d. (exists x. a = d * x) -> (exists y. p = d * y) -> d = 1
have hpacop : forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1
specialize prime_not_divides_coprime p
specialize prime_not_divides_coprime a
apply prime_not_divides_coprime
exact hp
exact hnot
intro d
intro hda
intro hdp
specialize hpacop d
apply hpacop
exact hdp
exact hda
specialize coprime_mod_inverse a
specialize coprime_mod_inverse p
apply coprime_mod_inverse
exact hp0
exact hcop
qed

Occurrences

5,491

Distinct objects

1,946

Reused references

118

Depth

54

Cuts

166

Certificate SHA-256

84f40e5a8c10…

prime_mod_cancelPrime mod cancel Quadratic reciprocity campaign checked

A nonzero residue factor cancels from congruence modulo a prime.

Prerequisites

prime_nonzero prime_not_divides_coprime coprime_symm mod_eq_cancel_coprime

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p a x y. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> (exists u v. (a * x) + p * u = (a * y) + p * v) -> exists r s. x + p * r = y + p * s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a x y. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> (exists u v. (a * x) + p * u = (a * y) + p * v) -> exists r s. x + p * r = y + p * s
use prime_nonzero
use prime_not_divides_coprime
use coprime_symm
use mod_eq_cancel_coprime
intro p
intro a
intro x
intro y
intro hp
intro hnot
intro hxy
have hp0 : ~(p = 0)
intro hpzero
specialize prime_nonzero p
apply prime_nonzero
exact hp
exact hpzero
have hpacop : forall d. (exists u. p = d * u) -> (exists v. a = d * v) -> d = 1
specialize prime_not_divides_coprime p
specialize prime_not_divides_coprime a
apply prime_not_divides_coprime
exact hp
exact hnot
have hapcop : forall d. (exists u. a = d * u) -> (exists v. p = d * v) -> d = 1
specialize coprime_symm p
specialize coprime_symm a
apply coprime_symm
exact hpacop
specialize mod_eq_cancel_coprime p
specialize mod_eq_cancel_coprime a
specialize mod_eq_cancel_coprime x
specialize mod_eq_cancel_coprime y
apply mod_eq_cancel_coprime
exact hp0
exact hapcop
exact hxy
qed

Occurrences

7,494

Distinct objects

2,178

Reused references

126

Depth

56

Cuts

224

Certificate SHA-256

e6fdc689812f…

factorial_existsFactorial exists Quadratic reciprocity campaign checked

Every natural has a beta-coded relational factorial value.

Prerequisites

beta_range_exists beta_product_exists

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n. exists z. (exists ff_b_exists ff_c_exists. ((forall ff_i_exists_range. (exists ff_lt_exists_range_bound. ff_lt_exists_range_bound + S ff_i_exists_range = n) -> (((exists ff_h_exists_range_decoded. ff_h_exists_range_decoded + S (1 + ff_i_exists_range) = S ((S (ff_i_exists_range)) * ff_c_exists)) /\ exists ff_q_exists_range_decoded. ff_b_exists = ff_q_exists_range_decoded * S ((S (ff_i_exists_range)) * ff_c_exists) + (1 + ff_i_exists_range)))) /\ (exists ff_u_exists_product ff_v_exists_product. ((((exists ff_h_exists_product_start. ff_h_exists_product_start + S (1) = S ((S (0)) * ff_v_exists_product)) /\ exists ff_q_exists_product_start. ff_u_exists_product = ff_q_exists_product_start * S ((S (0)) * ff_v_exists_product) + (1))) /\ ((((exists ff_h_exists_product_terminal. ff_h_exists_product_terminal + S (z) = S ((S (n)) * ff_v_exists_product)) /\ exists ff_q_exists_product_terminal. ff_u_exists_product = ff_q_exists_product_terminal * S ((S (n)) * ff_v_exists_product) + (z))) /\ forall ff_i_exists_product. (exists ff_lt_exists_product_bound. ff_lt_exists_product_bound + S ff_i_exists_product = n) -> exists ff_p_exists_product ff_r_exists_product ff_s_exists_product. ((((exists ff_h_exists_product_factor. ff_h_exists_product_factor + S (ff_p_exists_product) = S ((S (ff_i_exists_product)) * ff_c_exists)) /\ exists ff_q_exists_product_factor. ff_b_exists = ff_q_exists_product_factor * S ((S (ff_i_exists_product)) * ff_c_exists) + (ff_p_exists_product))) /\ ((((exists ff_h_exists_product_partial. ff_h_exists_product_partial + S (ff_r_exists_product) = S ((S (ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_partial. ff_u_exists_product = ff_q_exists_product_partial * S ((S (ff_i_exists_product)) * ff_v_exists_product) + (ff_r_exists_product))) /\ ((((exists ff_h_exists_product_successor. ff_h_exists_product_successor + S (ff_s_exists_product) = S ((S (S ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_successor. ff_u_exists_product = ff_q_exists_product_successor * S ((S (S ff_i_exists_product)) * ff_v_exists_product) + (ff_s_exists_product))) /\ ff_s_exists_product = ff_r_exists_product * ff_p_exists_product))))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. exists z. (exists ff_b_exists ff_c_exists. ((forall ff_i_exists_range. (exists ff_lt_exists_range_bound. ff_lt_exists_range_bound + S ff_i_exists_range = n) -> (((exists ff_h_exists_range_decoded. ff_h_exists_range_decoded + S (1 + ff_i_exists_range) = S ((S (ff_i_exists_range)) * ff_c_exists)) /\ exists ff_q_exists_range_decoded. ff_b_exists = ff_q_exists_range_decoded * S ((S (ff_i_exists_range)) * ff_c_exists) + (1 + ff_i_exists_range)))) /\ (exists ff_u_exists_product ff_v_exists_product. ((((exists ff_h_exists_product_start. ff_h_exists_product_start + S (1) = S ((S (0)) * ff_v_exists_product)) /\ exists ff_q_exists_product_start. ff_u_exists_product = ff_q_exists_product_start * S ((S (0)) * ff_v_exists_product) + (1))) /\ ((((exists ff_h_exists_product_terminal. ff_h_exists_product_terminal + S (z) = S ((S (n)) * ff_v_exists_product)) /\ exists ff_q_exists_product_terminal. ff_u_exists_product = ff_q_exists_product_terminal * S ((S (n)) * ff_v_exists_product) + (z))) /\ forall ff_i_exists_product. (exists ff_lt_exists_product_bound. ff_lt_exists_product_bound + S ff_i_exists_product = n) -> exists ff_p_exists_product ff_r_exists_product ff_s_exists_product. ((((exists ff_h_exists_product_factor. ff_h_exists_product_factor + S (ff_p_exists_product) = S ((S (ff_i_exists_product)) * ff_c_exists)) /\ exists ff_q_exists_product_factor. ff_b_exists = ff_q_exists_product_factor * S ((S (ff_i_exists_product)) * ff_c_exists) + (ff_p_exists_product))) /\ ((((exists ff_h_exists_product_partial. ff_h_exists_product_partial + S (ff_r_exists_product) = S ((S (ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_partial. ff_u_exists_product = ff_q_exists_product_partial * S ((S (ff_i_exists_product)) * ff_v_exists_product) + (ff_r_exists_product))) /\ ((((exists ff_h_exists_product_successor. ff_h_exists_product_successor + S (ff_s_exists_product) = S ((S (S ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_successor. ff_u_exists_product = ff_q_exists_product_successor * S ((S (S ff_i_exists_product)) * ff_v_exists_product) + (ff_s_exists_product))) /\ ff_s_exists_product = ff_r_exists_product * ff_p_exists_product))))))))
use beta_range_exists
use beta_product_exists
intro n
have hrange : exists b c. (forall i. (exists h. h + S i = n) -> ((exists h. h + S (1 + i) = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + (1 + i)))
specialize beta_range_exists 1
specialize beta_range_exists n
exact beta_range_exists
cases hrange
cases hrange_witness
specialize beta_product_exists x
specialize beta_product_exists x1
specialize beta_product_exists n
cases beta_product_exists
cases beta_product_exists_witness
cases beta_product_exists_witness_witness
exists x2
exists x
exists x1
split
exact hrange_witness_witness
exists x3
exists x4
exact beta_product_exists_witness_witness_witness
qed

Occurrences

59,841

Distinct objects

4,907

Reused references

241

Depth

88

Cuts

1,795

Certificate SHA-256

dd6a084e8622…

factorial_functionalFactorial functional Quadratic reciprocity campaign checked

The beta-coded relational factorial has a unique value.

Prerequisites

beta_range_transport_entry beta_product_transport_prefix beta_product_functional

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n z w. (exists ff_b_functional_l ff_c_functional_l. ((forall ff_i_functional_l_range. (exists ff_lt_functional_l_range_bound. ff_lt_functional_l_range_bound + S ff_i_functional_l_range = n) -> (((exists ff_h_functional_l_range_decoded. ff_h_functional_l_range_decoded + S (1 + ff_i_functional_l_range) = S ((S (ff_i_functional_l_range)) * ff_c_functional_l)) /\ exists ff_q_functional_l_range_decoded. ff_b_functional_l = ff_q_functional_l_range_decoded * S ((S (ff_i_functional_l_range)) * ff_c_functional_l) + (1 + ff_i_functional_l_range)))) /\ (exists ff_u_functional_l_product ff_v_functional_l_product. ((((exists ff_h_functional_l_product_start. ff_h_functional_l_product_start + S (1) = S ((S (0)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_start. ff_u_functional_l_product = ff_q_functional_l_product_start * S ((S (0)) * ff_v_functional_l_product) + (1))) /\ ((((exists ff_h_functional_l_product_terminal. ff_h_functional_l_product_terminal + S (z) = S ((S (n)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_terminal. ff_u_functional_l_product = ff_q_functional_l_product_terminal * S ((S (n)) * ff_v_functional_l_product) + (z))) /\ forall ff_i_functional_l_product. (exists ff_lt_functional_l_product_bound. ff_lt_functional_l_product_bound + S ff_i_functional_l_product = n) -> exists ff_p_functional_l_product ff_r_functional_l_product ff_s_functional_l_product. ((((exists ff_h_functional_l_product_factor. ff_h_functional_l_product_factor + S (ff_p_functional_l_product) = S ((S (ff_i_functional_l_product)) * ff_c_functional_l)) /\ exists ff_q_functional_l_product_factor. ff_b_functional_l = ff_q_functional_l_product_factor * S ((S (ff_i_functional_l_product)) * ff_c_functional_l) + (ff_p_functional_l_product))) /\ ((((exists ff_h_functional_l_product_partial. ff_h_functional_l_product_partial + S (ff_r_functional_l_product) = S ((S (ff_i_functional_l_product)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_partial. ff_u_functional_l_product = ff_q_functional_l_product_partial * S ((S (ff_i_functional_l_product)) * ff_v_functional_l_product) + (ff_r_functional_l_product))) /\ ((((exists ff_h_functional_l_product_successor. ff_h_functional_l_product_successor + S (ff_s_functional_l_product) = S ((S (S ff_i_functional_l_product)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_successor. ff_u_functional_l_product = ff_q_functional_l_product_successor * S ((S (S ff_i_functional_l_product)) * ff_v_functional_l_product) + (ff_s_functional_l_product))) /\ ff_s_functional_l_product = ff_r_functional_l_product * ff_p_functional_l_product)))))))) -> (exists ff_b_functional_r ff_c_functional_r. ((forall ff_i_functional_r_range. (exists ff_lt_functional_r_range_bound. ff_lt_functional_r_range_bound + S ff_i_functional_r_range = n) -> (((exists ff_h_functional_r_range_decoded. ff_h_functional_r_range_decoded + S (1 + ff_i_functional_r_range) = S ((S (ff_i_functional_r_range)) * ff_c_functional_r)) /\ exists ff_q_functional_r_range_decoded. ff_b_functional_r = ff_q_functional_r_range_decoded * S ((S (ff_i_functional_r_range)) * ff_c_functional_r) + (1 + ff_i_functional_r_range)))) /\ (exists ff_u_functional_r_product ff_v_functional_r_product. ((((exists ff_h_functional_r_product_start. ff_h_functional_r_product_start + S (1) = S ((S (0)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_start. ff_u_functional_r_product = ff_q_functional_r_product_start * S ((S (0)) * ff_v_functional_r_product) + (1))) /\ ((((exists ff_h_functional_r_product_terminal. ff_h_functional_r_product_terminal + S (w) = S ((S (n)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_terminal. ff_u_functional_r_product = ff_q_functional_r_product_terminal * S ((S (n)) * ff_v_functional_r_product) + (w))) /\ forall ff_i_functional_r_product. (exists ff_lt_functional_r_product_bound. ff_lt_functional_r_product_bound + S ff_i_functional_r_product = n) -> exists ff_p_functional_r_product ff_r_functional_r_product ff_s_functional_r_product. ((((exists ff_h_functional_r_product_factor. ff_h_functional_r_product_factor + S (ff_p_functional_r_product) = S ((S (ff_i_functional_r_product)) * ff_c_functional_r)) /\ exists ff_q_functional_r_product_factor. ff_b_functional_r = ff_q_functional_r_product_factor * S ((S (ff_i_functional_r_product)) * ff_c_functional_r) + (ff_p_functional_r_product))) /\ ((((exists ff_h_functional_r_product_partial. ff_h_functional_r_product_partial + S (ff_r_functional_r_product) = S ((S (ff_i_functional_r_product)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_partial. ff_u_functional_r_product = ff_q_functional_r_product_partial * S ((S (ff_i_functional_r_product)) * ff_v_functional_r_product) + (ff_r_functional_r_product))) /\ ((((exists ff_h_functional_r_product_successor. ff_h_functional_r_product_successor + S (ff_s_functional_r_product) = S ((S (S ff_i_functional_r_product)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_successor. ff_u_functional_r_product = ff_q_functional_r_product_successor * S ((S (S ff_i_functional_r_product)) * ff_v_functional_r_product) + (ff_s_functional_r_product))) /\ ff_s_functional_r_product = ff_r_functional_r_product * ff_p_functional_r_product)))))))) -> z = w

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n z w. (exists ff_b_functional_l ff_c_functional_l. ((forall ff_i_functional_l_range. (exists ff_lt_functional_l_range_bound. ff_lt_functional_l_range_bound + S ff_i_functional_l_range = n) -> (((exists ff_h_functional_l_range_decoded. ff_h_functional_l_range_decoded + S (1 + ff_i_functional_l_range) = S ((S (ff_i_functional_l_range)) * ff_c_functional_l)) /\ exists ff_q_functional_l_range_decoded. ff_b_functional_l = ff_q_functional_l_range_decoded * S ((S (ff_i_functional_l_range)) * ff_c_functional_l) + (1 + ff_i_functional_l_range)))) /\ (exists ff_u_functional_l_product ff_v_functional_l_product. ((((exists ff_h_functional_l_product_start. ff_h_functional_l_product_start + S (1) = S ((S (0)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_start. ff_u_functional_l_product = ff_q_functional_l_product_start * S ((S (0)) * ff_v_functional_l_product) + (1))) /\ ((((exists ff_h_functional_l_product_terminal. ff_h_functional_l_product_terminal + S (z) = S ((S (n)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_terminal. ff_u_functional_l_product = ff_q_functional_l_product_terminal * S ((S (n)) * ff_v_functional_l_product) + (z))) /\ forall ff_i_functional_l_product. (exists ff_lt_functional_l_product_bound. ff_lt_functional_l_product_bound + S ff_i_functional_l_product = n) -> exists ff_p_functional_l_product ff_r_functional_l_product ff_s_functional_l_product. ((((exists ff_h_functional_l_product_factor. ff_h_functional_l_product_factor + S (ff_p_functional_l_product) = S ((S (ff_i_functional_l_product)) * ff_c_functional_l)) /\ exists ff_q_functional_l_product_factor. ff_b_functional_l = ff_q_functional_l_product_factor * S ((S (ff_i_functional_l_product)) * ff_c_functional_l) + (ff_p_functional_l_product))) /\ ((((exists ff_h_functional_l_product_partial. ff_h_functional_l_product_partial + S (ff_r_functional_l_product) = S ((S (ff_i_functional_l_product)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_partial. ff_u_functional_l_product = ff_q_functional_l_product_partial * S ((S (ff_i_functional_l_product)) * ff_v_functional_l_product) + (ff_r_functional_l_product))) /\ ((((exists ff_h_functional_l_product_successor. ff_h_functional_l_product_successor + S (ff_s_functional_l_product) = S ((S (S ff_i_functional_l_product)) * ff_v_functional_l_product)) /\ exists ff_q_functional_l_product_successor. ff_u_functional_l_product = ff_q_functional_l_product_successor * S ((S (S ff_i_functional_l_product)) * ff_v_functional_l_product) + (ff_s_functional_l_product))) /\ ff_s_functional_l_product = ff_r_functional_l_product * ff_p_functional_l_product)))))))) -> (exists ff_b_functional_r ff_c_functional_r. ((forall ff_i_functional_r_range. (exists ff_lt_functional_r_range_bound. ff_lt_functional_r_range_bound + S ff_i_functional_r_range = n) -> (((exists ff_h_functional_r_range_decoded. ff_h_functional_r_range_decoded + S (1 + ff_i_functional_r_range) = S ((S (ff_i_functional_r_range)) * ff_c_functional_r)) /\ exists ff_q_functional_r_range_decoded. ff_b_functional_r = ff_q_functional_r_range_decoded * S ((S (ff_i_functional_r_range)) * ff_c_functional_r) + (1 + ff_i_functional_r_range)))) /\ (exists ff_u_functional_r_product ff_v_functional_r_product. ((((exists ff_h_functional_r_product_start. ff_h_functional_r_product_start + S (1) = S ((S (0)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_start. ff_u_functional_r_product = ff_q_functional_r_product_start * S ((S (0)) * ff_v_functional_r_product) + (1))) /\ ((((exists ff_h_functional_r_product_terminal. ff_h_functional_r_product_terminal + S (w) = S ((S (n)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_terminal. ff_u_functional_r_product = ff_q_functional_r_product_terminal * S ((S (n)) * ff_v_functional_r_product) + (w))) /\ forall ff_i_functional_r_product. (exists ff_lt_functional_r_product_bound. ff_lt_functional_r_product_bound + S ff_i_functional_r_product = n) -> exists ff_p_functional_r_product ff_r_functional_r_product ff_s_functional_r_product. ((((exists ff_h_functional_r_product_factor. ff_h_functional_r_product_factor + S (ff_p_functional_r_product) = S ((S (ff_i_functional_r_product)) * ff_c_functional_r)) /\ exists ff_q_functional_r_product_factor. ff_b_functional_r = ff_q_functional_r_product_factor * S ((S (ff_i_functional_r_product)) * ff_c_functional_r) + (ff_p_functional_r_product))) /\ ((((exists ff_h_functional_r_product_partial. ff_h_functional_r_product_partial + S (ff_r_functional_r_product) = S ((S (ff_i_functional_r_product)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_partial. ff_u_functional_r_product = ff_q_functional_r_product_partial * S ((S (ff_i_functional_r_product)) * ff_v_functional_r_product) + (ff_r_functional_r_product))) /\ ((((exists ff_h_functional_r_product_successor. ff_h_functional_r_product_successor + S (ff_s_functional_r_product) = S ((S (S ff_i_functional_r_product)) * ff_v_functional_r_product)) /\ exists ff_q_functional_r_product_successor. ff_u_functional_r_product = ff_q_functional_r_product_successor * S ((S (S ff_i_functional_r_product)) * ff_v_functional_r_product) + (ff_s_functional_r_product))) /\ ff_s_functional_r_product = ff_r_functional_r_product * ff_p_functional_r_product)))))))) -> z = w
use beta_range_transport_entry
use beta_product_transport_prefix
use beta_product_functional
intro n
intro z
intro w
intro hz
intro hw
cases hz
cases hz_witness
cases hz_witness_witness
cases hw
cases hw_witness
cases hw_witness_witness
have htransport : exists ff_u_factorial_transport ff_v_factorial_transport. ((((exists ff_h_factorial_transport_start. ff_h_factorial_transport_start + S (1) = S ((S (0)) * ff_v_factorial_transport)) /\ exists ff_q_factorial_transport_start. ff_u_factorial_transport = ff_q_factorial_transport_start * S ((S (0)) * ff_v_factorial_transport) + (1))) /\ ((((exists ff_h_factorial_transport_terminal. ff_h_factorial_transport_terminal + S (z) = S ((S (n)) * ff_v_factorial_transport)) /\ exists ff_q_factorial_transport_terminal. ff_u_factorial_transport = ff_q_factorial_transport_terminal * S ((S (n)) * ff_v_factorial_transport) + (z))) /\ forall ff_i_factorial_transport. (exists ff_lt_factorial_transport_bound. ff_lt_factorial_transport_bound + S ff_i_factorial_transport = n) -> exists ff_p_factorial_transport ff_r_factorial_transport ff_s_factorial_transport. ((((exists ff_h_factorial_transport_factor. ff_h_factorial_transport_factor + S (ff_p_factorial_transport) = S ((S (ff_i_factorial_transport)) * x3)) /\ exists ff_q_factorial_transport_factor. x2 = ff_q_factorial_transport_factor * S ((S (ff_i_factorial_transport)) * x3) + (ff_p_factorial_transport))) /\ ((((exists ff_h_factorial_transport_partial. ff_h_factorial_transport_partial + S (ff_r_factorial_transport) = S ((S (ff_i_factorial_transport)) * ff_v_factorial_transport)) /\ exists ff_q_factorial_transport_partial. ff_u_factorial_transport = ff_q_factorial_transport_partial * S ((S (ff_i_factorial_transport)) * ff_v_factorial_transport) + (ff_r_factorial_transport))) /\ ((((exists ff_h_factorial_transport_successor. ff_h_factorial_transport_successor + S (ff_s_factorial_transport) = S ((S (S ff_i_factorial_transport)) * ff_v_factorial_transport)) /\ exists ff_q_factorial_transport_successor. ff_u_factorial_transport = ff_q_factorial_transport_successor * S ((S (S ff_i_factorial_transport)) * ff_v_factorial_transport) + (ff_s_factorial_transport))) /\ ff_s_factorial_transport = ff_r_factorial_transport * ff_p_factorial_transport)))))
specialize beta_product_transport_prefix x
specialize beta_product_transport_prefix x1
specialize beta_product_transport_prefix x2
specialize beta_product_transport_prefix x3
specialize beta_product_transport_prefix n
specialize beta_product_transport_prefix z
apply beta_product_transport_prefix
exact hz_witness_witness_right
intro i
intro p
intro hi
intro hp
specialize beta_range_transport_entry x
specialize beta_range_transport_entry x1
specialize beta_range_transport_entry x2
specialize beta_range_transport_entry x3
specialize beta_range_transport_entry 1
specialize beta_range_transport_entry n
have hentries : forall i p. (exists h. h + S i = n) -> (((exists ff_h_factorial_transport_l. ff_h_factorial_transport_l + S (p) = S ((S (i)) * x1)) /\ exists ff_q_factorial_transport_l. x = ff_q_factorial_transport_l * S ((S (i)) * x1) + (p))) -> (((exists ff_h_factorial_transport_r. ff_h_factorial_transport_r + S (p) = S ((S (i)) * x3)) /\ exists ff_q_factorial_transport_r. x2 = ff_q_factorial_transport_r * S ((S (i)) * x3) + (p)))
apply beta_range_transport_entry
exact hz_witness_witness_left
exact hw_witness_witness_left
specialize hentries i
specialize hentries p
apply hentries
exact hi
exact hp
cases htransport
cases htransport_witness
cases hw_witness_witness_right
cases hw_witness_witness_right_witness
specialize beta_product_functional x2
specialize beta_product_functional x3
specialize beta_product_functional n
specialize beta_product_functional z
specialize beta_product_functional x4
specialize beta_product_functional x5
specialize beta_product_functional w
specialize beta_product_functional x6
specialize beta_product_functional x7
apply beta_product_functional
exact htransport_witness_witness
exact hw_witness_witness_right_witness_witness
qed

Occurrences

2,704

Distinct objects

1,110

Reused references

40

Depth

63

Cuts

71

Certificate SHA-256

53e13da0d0ce…

factorial_zeroFactorial zero Quadratic reciprocity campaign checked

The relational factorial of zero is one.

Prerequisites

beta_product_zero

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n z. n = 0 -> (exists ff_b_zero ff_c_zero. ((forall ff_i_zero_range. (exists ff_lt_zero_range_bound. ff_lt_zero_range_bound + S ff_i_zero_range = n) -> (((exists ff_h_zero_range_decoded. ff_h_zero_range_decoded + S (1 + ff_i_zero_range) = S ((S (ff_i_zero_range)) * ff_c_zero)) /\ exists ff_q_zero_range_decoded. ff_b_zero = ff_q_zero_range_decoded * S ((S (ff_i_zero_range)) * ff_c_zero) + (1 + ff_i_zero_range)))) /\ (exists ff_u_zero_product ff_v_zero_product. ((((exists ff_h_zero_product_start. ff_h_zero_product_start + S (1) = S ((S (0)) * ff_v_zero_product)) /\ exists ff_q_zero_product_start. ff_u_zero_product = ff_q_zero_product_start * S ((S (0)) * ff_v_zero_product) + (1))) /\ ((((exists ff_h_zero_product_terminal. ff_h_zero_product_terminal + S (z) = S ((S (n)) * ff_v_zero_product)) /\ exists ff_q_zero_product_terminal. ff_u_zero_product = ff_q_zero_product_terminal * S ((S (n)) * ff_v_zero_product) + (z))) /\ forall ff_i_zero_product. (exists ff_lt_zero_product_bound. ff_lt_zero_product_bound + S ff_i_zero_product = n) -> exists ff_p_zero_product ff_r_zero_product ff_s_zero_product. ((((exists ff_h_zero_product_factor. ff_h_zero_product_factor + S (ff_p_zero_product) = S ((S (ff_i_zero_product)) * ff_c_zero)) /\ exists ff_q_zero_product_factor. ff_b_zero = ff_q_zero_product_factor * S ((S (ff_i_zero_product)) * ff_c_zero) + (ff_p_zero_product))) /\ ((((exists ff_h_zero_product_partial. ff_h_zero_product_partial + S (ff_r_zero_product) = S ((S (ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_partial. ff_u_zero_product = ff_q_zero_product_partial * S ((S (ff_i_zero_product)) * ff_v_zero_product) + (ff_r_zero_product))) /\ ((((exists ff_h_zero_product_successor. ff_h_zero_product_successor + S (ff_s_zero_product) = S ((S (S ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_successor. ff_u_zero_product = ff_q_zero_product_successor * S ((S (S ff_i_zero_product)) * ff_v_zero_product) + (ff_s_zero_product))) /\ ff_s_zero_product = ff_r_zero_product * ff_p_zero_product)))))))) -> z = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n z. n = 0 -> (exists ff_b_zero ff_c_zero. ((forall ff_i_zero_range. (exists ff_lt_zero_range_bound. ff_lt_zero_range_bound + S ff_i_zero_range = n) -> (((exists ff_h_zero_range_decoded. ff_h_zero_range_decoded + S (1 + ff_i_zero_range) = S ((S (ff_i_zero_range)) * ff_c_zero)) /\ exists ff_q_zero_range_decoded. ff_b_zero = ff_q_zero_range_decoded * S ((S (ff_i_zero_range)) * ff_c_zero) + (1 + ff_i_zero_range)))) /\ (exists ff_u_zero_product ff_v_zero_product. ((((exists ff_h_zero_product_start. ff_h_zero_product_start + S (1) = S ((S (0)) * ff_v_zero_product)) /\ exists ff_q_zero_product_start. ff_u_zero_product = ff_q_zero_product_start * S ((S (0)) * ff_v_zero_product) + (1))) /\ ((((exists ff_h_zero_product_terminal. ff_h_zero_product_terminal + S (z) = S ((S (n)) * ff_v_zero_product)) /\ exists ff_q_zero_product_terminal. ff_u_zero_product = ff_q_zero_product_terminal * S ((S (n)) * ff_v_zero_product) + (z))) /\ forall ff_i_zero_product. (exists ff_lt_zero_product_bound. ff_lt_zero_product_bound + S ff_i_zero_product = n) -> exists ff_p_zero_product ff_r_zero_product ff_s_zero_product. ((((exists ff_h_zero_product_factor. ff_h_zero_product_factor + S (ff_p_zero_product) = S ((S (ff_i_zero_product)) * ff_c_zero)) /\ exists ff_q_zero_product_factor. ff_b_zero = ff_q_zero_product_factor * S ((S (ff_i_zero_product)) * ff_c_zero) + (ff_p_zero_product))) /\ ((((exists ff_h_zero_product_partial. ff_h_zero_product_partial + S (ff_r_zero_product) = S ((S (ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_partial. ff_u_zero_product = ff_q_zero_product_partial * S ((S (ff_i_zero_product)) * ff_v_zero_product) + (ff_r_zero_product))) /\ ((((exists ff_h_zero_product_successor. ff_h_zero_product_successor + S (ff_s_zero_product) = S ((S (S ff_i_zero_product)) * ff_v_zero_product)) /\ exists ff_q_zero_product_successor. ff_u_zero_product = ff_q_zero_product_successor * S ((S (S ff_i_zero_product)) * ff_v_zero_product) + (ff_s_zero_product))) /\ ff_s_zero_product = ff_r_zero_product * ff_p_zero_product)))))))) -> z = 1
use beta_product_zero
intro n
intro z
intro hn
intro hfactorial
rewrite hn at hfactorial
rewrite hn at hfactorial
rewrite hn at hfactorial
rewrite hn at hfactorial
cases hfactorial
cases hfactorial_witness
cases hfactorial_witness_witness
specialize beta_product_zero x
specialize beta_product_zero x1
specialize beta_product_zero z
apply beta_product_zero
exact hfactorial_witness_witness_right
qed

Occurrences

1,223

Distinct objects

794

Reused references

37

Depth

61

Cuts

32

Certificate SHA-256

34038cedbd37…

factorial_succ_decomposeFactorial succ decompose Quadratic reciprocity campaign checked

A successor factorial is its predecessor factorial times the successor.

Prerequisites

beta_product_succ_decompose beta_range_entry_eq le_refl le_succ add_succ_left zero_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n sn z. sn = S n -> (exists ff_b_successor ff_c_successor. ((forall ff_i_successor_range. (exists ff_lt_successor_range_bound. ff_lt_successor_range_bound + S ff_i_successor_range = sn) -> (((exists ff_h_successor_range_decoded. ff_h_successor_range_decoded + S (1 + ff_i_successor_range) = S ((S (ff_i_successor_range)) * ff_c_successor)) /\ exists ff_q_successor_range_decoded. ff_b_successor = ff_q_successor_range_decoded * S ((S (ff_i_successor_range)) * ff_c_successor) + (1 + ff_i_successor_range)))) /\ (exists ff_u_successor_product ff_v_successor_product. ((((exists ff_h_successor_product_start. ff_h_successor_product_start + S (1) = S ((S (0)) * ff_v_successor_product)) /\ exists ff_q_successor_product_start. ff_u_successor_product = ff_q_successor_product_start * S ((S (0)) * ff_v_successor_product) + (1))) /\ ((((exists ff_h_successor_product_terminal. ff_h_successor_product_terminal + S (z) = S ((S (sn)) * ff_v_successor_product)) /\ exists ff_q_successor_product_terminal. ff_u_successor_product = ff_q_successor_product_terminal * S ((S (sn)) * ff_v_successor_product) + (z))) /\ forall ff_i_successor_product. (exists ff_lt_successor_product_bound. ff_lt_successor_product_bound + S ff_i_successor_product = sn) -> exists ff_p_successor_product ff_r_successor_product ff_s_successor_product. ((((exists ff_h_successor_product_factor. ff_h_successor_product_factor + S (ff_p_successor_product) = S ((S (ff_i_successor_product)) * ff_c_successor)) /\ exists ff_q_successor_product_factor. ff_b_successor = ff_q_successor_product_factor * S ((S (ff_i_successor_product)) * ff_c_successor) + (ff_p_successor_product))) /\ ((((exists ff_h_successor_product_partial. ff_h_successor_product_partial + S (ff_r_successor_product) = S ((S (ff_i_successor_product)) * ff_v_successor_product)) /\ exists ff_q_successor_product_partial. ff_u_successor_product = ff_q_successor_product_partial * S ((S (ff_i_successor_product)) * ff_v_successor_product) + (ff_r_successor_product))) /\ ((((exists ff_h_successor_product_successor. ff_h_successor_product_successor + S (ff_s_successor_product) = S ((S (S ff_i_successor_product)) * ff_v_successor_product)) /\ exists ff_q_successor_product_successor. ff_u_successor_product = ff_q_successor_product_successor * S ((S (S ff_i_successor_product)) * ff_v_successor_product) + (ff_s_successor_product))) /\ ff_s_successor_product = ff_r_successor_product * ff_p_successor_product)))))))) -> exists r. (exists ff_b_predecessor ff_c_predecessor. ((forall ff_i_predecessor_range. (exists ff_lt_predecessor_range_bound. ff_lt_predecessor_range_bound + S ff_i_predecessor_range = n) -> (((exists ff_h_predecessor_range_decoded. ff_h_predecessor_range_decoded + S (1 + ff_i_predecessor_range) = S ((S (ff_i_predecessor_range)) * ff_c_predecessor)) /\ exists ff_q_predecessor_range_decoded. ff_b_predecessor = ff_q_predecessor_range_decoded * S ((S (ff_i_predecessor_range)) * ff_c_predecessor) + (1 + ff_i_predecessor_range)))) /\ (exists ff_u_predecessor_product ff_v_predecessor_product. ((((exists ff_h_predecessor_product_start. ff_h_predecessor_product_start + S (1) = S ((S (0)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_start. ff_u_predecessor_product = ff_q_predecessor_product_start * S ((S (0)) * ff_v_predecessor_product) + (1))) /\ ((((exists ff_h_predecessor_product_terminal. ff_h_predecessor_product_terminal + S (r) = S ((S (n)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_terminal. ff_u_predecessor_product = ff_q_predecessor_product_terminal * S ((S (n)) * ff_v_predecessor_product) + (r))) /\ forall ff_i_predecessor_product. (exists ff_lt_predecessor_product_bound. ff_lt_predecessor_product_bound + S ff_i_predecessor_product = n) -> exists ff_p_predecessor_product ff_r_predecessor_product ff_s_predecessor_product. ((((exists ff_h_predecessor_product_factor. ff_h_predecessor_product_factor + S (ff_p_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_c_predecessor)) /\ exists ff_q_predecessor_product_factor. ff_b_predecessor = ff_q_predecessor_product_factor * S ((S (ff_i_predecessor_product)) * ff_c_predecessor) + (ff_p_predecessor_product))) /\ ((((exists ff_h_predecessor_product_partial. ff_h_predecessor_product_partial + S (ff_r_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_partial. ff_u_predecessor_product = ff_q_predecessor_product_partial * S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_r_predecessor_product))) /\ ((((exists ff_h_predecessor_product_successor. ff_h_predecessor_product_successor + S (ff_s_predecessor_product) = S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_successor. ff_u_predecessor_product = ff_q_predecessor_product_successor * S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_s_predecessor_product))) /\ ff_s_predecessor_product = ff_r_predecessor_product * ff_p_predecessor_product)))))))) /\ z = r * S n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n sn z. sn = S n -> (exists ff_b_successor ff_c_successor. ((forall ff_i_successor_range. (exists ff_lt_successor_range_bound. ff_lt_successor_range_bound + S ff_i_successor_range = sn) -> (((exists ff_h_successor_range_decoded. ff_h_successor_range_decoded + S (1 + ff_i_successor_range) = S ((S (ff_i_successor_range)) * ff_c_successor)) /\ exists ff_q_successor_range_decoded. ff_b_successor = ff_q_successor_range_decoded * S ((S (ff_i_successor_range)) * ff_c_successor) + (1 + ff_i_successor_range)))) /\ (exists ff_u_successor_product ff_v_successor_product. ((((exists ff_h_successor_product_start. ff_h_successor_product_start + S (1) = S ((S (0)) * ff_v_successor_product)) /\ exists ff_q_successor_product_start. ff_u_successor_product = ff_q_successor_product_start * S ((S (0)) * ff_v_successor_product) + (1))) /\ ((((exists ff_h_successor_product_terminal. ff_h_successor_product_terminal + S (z) = S ((S (sn)) * ff_v_successor_product)) /\ exists ff_q_successor_product_terminal. ff_u_successor_product = ff_q_successor_product_terminal * S ((S (sn)) * ff_v_successor_product) + (z))) /\ forall ff_i_successor_product. (exists ff_lt_successor_product_bound. ff_lt_successor_product_bound + S ff_i_successor_product = sn) -> exists ff_p_successor_product ff_r_successor_product ff_s_successor_product. ((((exists ff_h_successor_product_factor. ff_h_successor_product_factor + S (ff_p_successor_product) = S ((S (ff_i_successor_product)) * ff_c_successor)) /\ exists ff_q_successor_product_factor. ff_b_successor = ff_q_successor_product_factor * S ((S (ff_i_successor_product)) * ff_c_successor) + (ff_p_successor_product))) /\ ((((exists ff_h_successor_product_partial. ff_h_successor_product_partial + S (ff_r_successor_product) = S ((S (ff_i_successor_product)) * ff_v_successor_product)) /\ exists ff_q_successor_product_partial. ff_u_successor_product = ff_q_successor_product_partial * S ((S (ff_i_successor_product)) * ff_v_successor_product) + (ff_r_successor_product))) /\ ((((exists ff_h_successor_product_successor. ff_h_successor_product_successor + S (ff_s_successor_product) = S ((S (S ff_i_successor_product)) * ff_v_successor_product)) /\ exists ff_q_successor_product_successor. ff_u_successor_product = ff_q_successor_product_successor * S ((S (S ff_i_successor_product)) * ff_v_successor_product) + (ff_s_successor_product))) /\ ff_s_successor_product = ff_r_successor_product * ff_p_successor_product)))))))) -> exists r. (exists ff_b_predecessor ff_c_predecessor. ((forall ff_i_predecessor_range. (exists ff_lt_predecessor_range_bound. ff_lt_predecessor_range_bound + S ff_i_predecessor_range = n) -> (((exists ff_h_predecessor_range_decoded. ff_h_predecessor_range_decoded + S (1 + ff_i_predecessor_range) = S ((S (ff_i_predecessor_range)) * ff_c_predecessor)) /\ exists ff_q_predecessor_range_decoded. ff_b_predecessor = ff_q_predecessor_range_decoded * S ((S (ff_i_predecessor_range)) * ff_c_predecessor) + (1 + ff_i_predecessor_range)))) /\ (exists ff_u_predecessor_product ff_v_predecessor_product. ((((exists ff_h_predecessor_product_start. ff_h_predecessor_product_start + S (1) = S ((S (0)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_start. ff_u_predecessor_product = ff_q_predecessor_product_start * S ((S (0)) * ff_v_predecessor_product) + (1))) /\ ((((exists ff_h_predecessor_product_terminal. ff_h_predecessor_product_terminal + S (r) = S ((S (n)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_terminal. ff_u_predecessor_product = ff_q_predecessor_product_terminal * S ((S (n)) * ff_v_predecessor_product) + (r))) /\ forall ff_i_predecessor_product. (exists ff_lt_predecessor_product_bound. ff_lt_predecessor_product_bound + S ff_i_predecessor_product = n) -> exists ff_p_predecessor_product ff_r_predecessor_product ff_s_predecessor_product. ((((exists ff_h_predecessor_product_factor. ff_h_predecessor_product_factor + S (ff_p_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_c_predecessor)) /\ exists ff_q_predecessor_product_factor. ff_b_predecessor = ff_q_predecessor_product_factor * S ((S (ff_i_predecessor_product)) * ff_c_predecessor) + (ff_p_predecessor_product))) /\ ((((exists ff_h_predecessor_product_partial. ff_h_predecessor_product_partial + S (ff_r_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_partial. ff_u_predecessor_product = ff_q_predecessor_product_partial * S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_r_predecessor_product))) /\ ((((exists ff_h_predecessor_product_successor. ff_h_predecessor_product_successor + S (ff_s_predecessor_product) = S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_successor. ff_u_predecessor_product = ff_q_predecessor_product_successor * S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_s_predecessor_product))) /\ ff_s_predecessor_product = ff_r_predecessor_product * ff_p_predecessor_product)))))))) /\ z = r * S n
use beta_product_succ_decompose
use beta_range_entry_eq
use le_refl
use le_succ
use add_succ_left
use zero_add
intro n
intro sn
intro z
intro hsn
intro hfactorial
rewrite hsn at hfactorial
rewrite hsn at hfactorial
rewrite hsn at hfactorial
rewrite hsn at hfactorial
cases hfactorial
cases hfactorial_witness
cases hfactorial_witness_witness
have hdecomp : exists p r. (((exists ff_h_factorial_succ_factor. ff_h_factorial_succ_factor + S (p) = S ((S (n)) * x1)) /\ exists ff_q_factorial_succ_factor. x = ff_q_factorial_succ_factor * S ((S (n)) * x1) + (p))) /\ ((exists ff_u_factorial_succ_prefix ff_v_factorial_succ_prefix. ((((exists ff_h_factorial_succ_prefix_start. ff_h_factorial_succ_prefix_start + S (1) = S ((S (0)) * ff_v_factorial_succ_prefix)) /\ exists ff_q_factorial_succ_prefix_start. ff_u_factorial_succ_prefix = ff_q_factorial_succ_prefix_start * S ((S (0)) * ff_v_factorial_succ_prefix) + (1))) /\ ((((exists ff_h_factorial_succ_prefix_terminal. ff_h_factorial_succ_prefix_terminal + S (r) = S ((S (n)) * ff_v_factorial_succ_prefix)) /\ exists ff_q_factorial_succ_prefix_terminal. ff_u_factorial_succ_prefix = ff_q_factorial_succ_prefix_terminal * S ((S (n)) * ff_v_factorial_succ_prefix) + (r))) /\ forall ff_i_factorial_succ_prefix. (exists ff_lt_factorial_succ_prefix_bound. ff_lt_factorial_succ_prefix_bound + S ff_i_factorial_succ_prefix = n) -> exists ff_p_factorial_succ_prefix ff_r_factorial_succ_prefix ff_s_factorial_succ_prefix. ((((exists ff_h_factorial_succ_prefix_factor. ff_h_factorial_succ_prefix_factor + S (ff_p_factorial_succ_prefix) = S ((S (ff_i_factorial_succ_prefix)) * x1)) /\ exists ff_q_factorial_succ_prefix_factor. x = ff_q_factorial_succ_prefix_factor * S ((S (ff_i_factorial_succ_prefix)) * x1) + (ff_p_factorial_succ_prefix))) /\ ((((exists ff_h_factorial_succ_prefix_partial. ff_h_factorial_succ_prefix_partial + S (ff_r_factorial_succ_prefix) = S ((S (ff_i_factorial_succ_prefix)) * ff_v_factorial_succ_prefix)) /\ exists ff_q_factorial_succ_prefix_partial. ff_u_factorial_succ_prefix = ff_q_factorial_succ_prefix_partial * S ((S (ff_i_factorial_succ_prefix)) * ff_v_factorial_succ_prefix) + (ff_r_factorial_succ_prefix))) /\ ((((exists ff_h_factorial_succ_prefix_successor. ff_h_factorial_succ_prefix_successor + S (ff_s_factorial_succ_prefix) = S ((S (S ff_i_factorial_succ_prefix)) * ff_v_factorial_succ_prefix)) /\ exists ff_q_factorial_succ_prefix_successor. ff_u_factorial_succ_prefix = ff_q_factorial_succ_prefix_successor * S ((S (S ff_i_factorial_succ_prefix)) * ff_v_factorial_succ_prefix) + (ff_s_factorial_succ_prefix))) /\ ff_s_factorial_succ_prefix = ff_r_factorial_succ_prefix * ff_p_factorial_succ_prefix)))))) /\ z = r * p)
specialize beta_product_succ_decompose x
specialize beta_product_succ_decompose x1
specialize beta_product_succ_decompose n
specialize beta_product_succ_decompose z
apply beta_product_succ_decompose
exact hfactorial_witness_witness_right
cases hdecomp
cases hdecomp_witness
cases hdecomp_witness_witness
cases hdecomp_witness_witness_right
have hp : x2 = 1 + n
specialize beta_range_entry_eq x
specialize beta_range_entry_eq x1
specialize beta_range_entry_eq 1
specialize beta_range_entry_eq (S n)
specialize beta_range_entry_eq n
specialize beta_range_entry_eq x2
apply beta_range_entry_eq
exact hfactorial_witness_witness_left
specialize le_refl (S n)
exact le_refl
exact hdecomp_witness_witness_left
exists x3
split
exists x
exists x1
split
intro i
intro hi
specialize hfactorial_witness_witness_left i
apply hfactorial_witness_witness_left
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
exact hdecomp_witness_witness_right_left
trans x3 * x2
exact hdecomp_witness_witness_right_right
rewrite hp
congr
refl
specialize add_succ_left 0
specialize add_succ_left n
trans S (0 + n)
exact add_succ_left
congr
apply zero_add
qed

Occurrences

2,594

Distinct objects

891

Reused references

44

Depth

63

Cuts

74

Certificate SHA-256

8e876f042ad4…

pow_one_from_zero_successorPow one from zero successor Quadratic reciprocity campaign checked

A successor of a zero exponent gives the relational first power.

Prerequisites

pow_successor_decompose pow_zero one_mul

Used directly by

pow_one

Exact expanded PA statement

Copy statement

forall a z e n. z = 0 -> e = S z -> (exists ff_b_one_carrier ff_c_one_carrier. ((forall ff_i_one_carrier_repeat. (exists ff_lt_one_carrier_repeat_bound. ff_lt_one_carrier_repeat_bound + S ff_i_one_carrier_repeat = e) -> (((exists ff_h_one_carrier_repeat_decoded. ff_h_one_carrier_repeat_decoded + S (a) = S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_repeat_decoded. ff_b_one_carrier = ff_q_one_carrier_repeat_decoded * S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier) + (a)))) /\ (exists ff_u_one_carrier_product ff_v_one_carrier_product. ((((exists ff_h_one_carrier_product_start. ff_h_one_carrier_product_start + S (1) = S ((S (0)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_start. ff_u_one_carrier_product = ff_q_one_carrier_product_start * S ((S (0)) * ff_v_one_carrier_product) + (1))) /\ ((((exists ff_h_one_carrier_product_terminal. ff_h_one_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_terminal. ff_u_one_carrier_product = ff_q_one_carrier_product_terminal * S ((S (e)) * ff_v_one_carrier_product) + (n))) /\ forall ff_i_one_carrier_product. (exists ff_lt_one_carrier_product_bound. ff_lt_one_carrier_product_bound + S ff_i_one_carrier_product = e) -> exists ff_p_one_carrier_product ff_r_one_carrier_product ff_s_one_carrier_product. ((((exists ff_h_one_carrier_product_factor. ff_h_one_carrier_product_factor + S (ff_p_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_product_factor. ff_b_one_carrier = ff_q_one_carrier_product_factor * S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier) + (ff_p_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_partial. ff_h_one_carrier_product_partial + S (ff_r_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_partial. ff_u_one_carrier_product = ff_q_one_carrier_product_partial * S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_r_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_successor. ff_h_one_carrier_product_successor + S (ff_s_one_carrier_product) = S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_successor. ff_u_one_carrier_product = ff_q_one_carrier_product_successor * S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_s_one_carrier_product))) /\ ff_s_one_carrier_product = ff_r_one_carrier_product * ff_p_one_carrier_product)))))))) -> n = a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a z e n. z = 0 -> e = S z -> (exists ff_b_one_carrier ff_c_one_carrier. ((forall ff_i_one_carrier_repeat. (exists ff_lt_one_carrier_repeat_bound. ff_lt_one_carrier_repeat_bound + S ff_i_one_carrier_repeat = e) -> (((exists ff_h_one_carrier_repeat_decoded. ff_h_one_carrier_repeat_decoded + S (a) = S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_repeat_decoded. ff_b_one_carrier = ff_q_one_carrier_repeat_decoded * S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier) + (a)))) /\ (exists ff_u_one_carrier_product ff_v_one_carrier_product. ((((exists ff_h_one_carrier_product_start. ff_h_one_carrier_product_start + S (1) = S ((S (0)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_start. ff_u_one_carrier_product = ff_q_one_carrier_product_start * S ((S (0)) * ff_v_one_carrier_product) + (1))) /\ ((((exists ff_h_one_carrier_product_terminal. ff_h_one_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_terminal. ff_u_one_carrier_product = ff_q_one_carrier_product_terminal * S ((S (e)) * ff_v_one_carrier_product) + (n))) /\ forall ff_i_one_carrier_product. (exists ff_lt_one_carrier_product_bound. ff_lt_one_carrier_product_bound + S ff_i_one_carrier_product = e) -> exists ff_p_one_carrier_product ff_r_one_carrier_product ff_s_one_carrier_product. ((((exists ff_h_one_carrier_product_factor. ff_h_one_carrier_product_factor + S (ff_p_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_product_factor. ff_b_one_carrier = ff_q_one_carrier_product_factor * S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier) + (ff_p_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_partial. ff_h_one_carrier_product_partial + S (ff_r_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_partial. ff_u_one_carrier_product = ff_q_one_carrier_product_partial * S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_r_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_successor. ff_h_one_carrier_product_successor + S (ff_s_one_carrier_product) = S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_successor. ff_u_one_carrier_product = ff_q_one_carrier_product_successor * S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_s_one_carrier_product))) /\ ff_s_one_carrier_product = ff_r_one_carrier_product * ff_p_one_carrier_product)))))))) -> n = a
use pow_successor_decompose
use pow_zero
use one_mul
intro a
intro z
intro e
intro n
intro hz
intro he
intro hpow
have hstep : exists r. (exists ff_b_one_predecessor ff_c_one_predecessor. ((forall ff_i_one_predecessor_repeat. (exists ff_lt_one_predecessor_repeat_bound. ff_lt_one_predecessor_repeat_bound + S ff_i_one_predecessor_repeat = z) -> (((exists ff_h_one_predecessor_repeat_decoded. ff_h_one_predecessor_repeat_decoded + S (a) = S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor)) /\ exists ff_q_one_predecessor_repeat_decoded. ff_b_one_predecessor = ff_q_one_predecessor_repeat_decoded * S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor) + (a)))) /\ (exists ff_u_one_predecessor_product ff_v_one_predecessor_product. ((((exists ff_h_one_predecessor_product_start. ff_h_one_predecessor_product_start + S (1) = S ((S (0)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_start. ff_u_one_predecessor_product = ff_q_one_predecessor_product_start * S ((S (0)) * ff_v_one_predecessor_product) + (1))) /\ ((((exists ff_h_one_predecessor_product_terminal. ff_h_one_predecessor_product_terminal + S (r) = S ((S (z)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_terminal. ff_u_one_predecessor_product = ff_q_one_predecessor_product_terminal * S ((S (z)) * ff_v_one_predecessor_product) + (r))) /\ forall ff_i_one_predecessor_product. (exists ff_lt_one_predecessor_product_bound. ff_lt_one_predecessor_product_bound + S ff_i_one_predecessor_product = z) -> exists ff_p_one_predecessor_product ff_r_one_predecessor_product ff_s_one_predecessor_product. ((((exists ff_h_one_predecessor_product_factor. ff_h_one_predecessor_product_factor + S (ff_p_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor)) /\ exists ff_q_one_predecessor_product_factor. ff_b_one_predecessor = ff_q_one_predecessor_product_factor * S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor) + (ff_p_one_predecessor_product))) /\ ((((exists ff_h_one_predecessor_product_partial. ff_h_one_predecessor_product_partial + S (ff_r_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_partial. ff_u_one_predecessor_product = ff_q_one_predecessor_product_partial * S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_r_one_predecessor_product))) /\ ((((exists ff_h_one_predecessor_product_successor. ff_h_one_predecessor_product_successor + S (ff_s_one_predecessor_product) = S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_successor. ff_u_one_predecessor_product = ff_q_one_predecessor_product_successor * S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_s_one_predecessor_product))) /\ ff_s_one_predecessor_product = ff_r_one_predecessor_product * ff_p_one_predecessor_product)))))))) /\ n = r * a
specialize pow_successor_decompose a
specialize pow_successor_decompose z
specialize pow_successor_decompose e
specialize pow_successor_decompose n
apply pow_successor_decompose
exact he
exact hpow
cases hstep
cases hstep_witness
have hr : x = 1
specialize pow_zero a
specialize pow_zero z
specialize pow_zero x
apply pow_zero
exact hz
exact hstep_witness_left
trans x * a
exact hstep_witness_right
rewrite hr
specialize one_mul a
exact one_mul
qed

Occurrences

3,827

Distinct objects

1,043

Reused references

47

Depth

64

Cuts

107

Certificate SHA-256

d709c0bb9f53…

pow_onePow one Quadratic reciprocity campaign checked

The relational first power of a natural is the natural itself.

Prerequisites

pow_one_from_zero_successor

Used directly by

pow_two_from_one_successor

Exact expanded PA statement

Copy statement

forall a e n. e = 1 -> (exists ff_b_one ff_c_one. ((forall ff_i_one_repeat. (exists ff_lt_one_repeat_bound. ff_lt_one_repeat_bound + S ff_i_one_repeat = e) -> (((exists ff_h_one_repeat_decoded. ff_h_one_repeat_decoded + S (a) = S ((S (ff_i_one_repeat)) * ff_c_one)) /\ exists ff_q_one_repeat_decoded. ff_b_one = ff_q_one_repeat_decoded * S ((S (ff_i_one_repeat)) * ff_c_one) + (a)))) /\ (exists ff_u_one_product ff_v_one_product. ((((exists ff_h_one_product_start. ff_h_one_product_start + S (1) = S ((S (0)) * ff_v_one_product)) /\ exists ff_q_one_product_start. ff_u_one_product = ff_q_one_product_start * S ((S (0)) * ff_v_one_product) + (1))) /\ ((((exists ff_h_one_product_terminal. ff_h_one_product_terminal + S (n) = S ((S (e)) * ff_v_one_product)) /\ exists ff_q_one_product_terminal. ff_u_one_product = ff_q_one_product_terminal * S ((S (e)) * ff_v_one_product) + (n))) /\ forall ff_i_one_product. (exists ff_lt_one_product_bound. ff_lt_one_product_bound + S ff_i_one_product = e) -> exists ff_p_one_product ff_r_one_product ff_s_one_product. ((((exists ff_h_one_product_factor. ff_h_one_product_factor + S (ff_p_one_product) = S ((S (ff_i_one_product)) * ff_c_one)) /\ exists ff_q_one_product_factor. ff_b_one = ff_q_one_product_factor * S ((S (ff_i_one_product)) * ff_c_one) + (ff_p_one_product))) /\ ((((exists ff_h_one_product_partial. ff_h_one_product_partial + S (ff_r_one_product) = S ((S (ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_partial. ff_u_one_product = ff_q_one_product_partial * S ((S (ff_i_one_product)) * ff_v_one_product) + (ff_r_one_product))) /\ ((((exists ff_h_one_product_successor. ff_h_one_product_successor + S (ff_s_one_product) = S ((S (S ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_successor. ff_u_one_product = ff_q_one_product_successor * S ((S (S ff_i_one_product)) * ff_v_one_product) + (ff_s_one_product))) /\ ff_s_one_product = ff_r_one_product * ff_p_one_product)))))))) -> n = a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e n. e = 1 -> (exists ff_b_one ff_c_one. ((forall ff_i_one_repeat. (exists ff_lt_one_repeat_bound. ff_lt_one_repeat_bound + S ff_i_one_repeat = e) -> (((exists ff_h_one_repeat_decoded. ff_h_one_repeat_decoded + S (a) = S ((S (ff_i_one_repeat)) * ff_c_one)) /\ exists ff_q_one_repeat_decoded. ff_b_one = ff_q_one_repeat_decoded * S ((S (ff_i_one_repeat)) * ff_c_one) + (a)))) /\ (exists ff_u_one_product ff_v_one_product. ((((exists ff_h_one_product_start. ff_h_one_product_start + S (1) = S ((S (0)) * ff_v_one_product)) /\ exists ff_q_one_product_start. ff_u_one_product = ff_q_one_product_start * S ((S (0)) * ff_v_one_product) + (1))) /\ ((((exists ff_h_one_product_terminal. ff_h_one_product_terminal + S (n) = S ((S (e)) * ff_v_one_product)) /\ exists ff_q_one_product_terminal. ff_u_one_product = ff_q_one_product_terminal * S ((S (e)) * ff_v_one_product) + (n))) /\ forall ff_i_one_product. (exists ff_lt_one_product_bound. ff_lt_one_product_bound + S ff_i_one_product = e) -> exists ff_p_one_product ff_r_one_product ff_s_one_product. ((((exists ff_h_one_product_factor. ff_h_one_product_factor + S (ff_p_one_product) = S ((S (ff_i_one_product)) * ff_c_one)) /\ exists ff_q_one_product_factor. ff_b_one = ff_q_one_product_factor * S ((S (ff_i_one_product)) * ff_c_one) + (ff_p_one_product))) /\ ((((exists ff_h_one_product_partial. ff_h_one_product_partial + S (ff_r_one_product) = S ((S (ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_partial. ff_u_one_product = ff_q_one_product_partial * S ((S (ff_i_one_product)) * ff_v_one_product) + (ff_r_one_product))) /\ ((((exists ff_h_one_product_successor. ff_h_one_product_successor + S (ff_s_one_product) = S ((S (S ff_i_one_product)) * ff_v_one_product)) /\ exists ff_q_one_product_successor. ff_u_one_product = ff_q_one_product_successor * S ((S (S ff_i_one_product)) * ff_v_one_product) + (ff_s_one_product))) /\ ff_s_one_product = ff_r_one_product * ff_p_one_product)))))))) -> n = a
use pow_one_from_zero_successor
intro a
intro e
intro n
intro he
intro hpow
specialize pow_one_from_zero_successor a
specialize pow_one_from_zero_successor 0
specialize pow_one_from_zero_successor e
specialize pow_one_from_zero_successor n
apply pow_one_from_zero_successor
refl
exact he
exact hpow
qed

Occurrences

3,856

Distinct objects

1,072

Reused references

47

Depth

65

Cuts

108

Certificate SHA-256

16491d90b740…

pow_successor_pair_mulPow successor pair mul Quadratic reciprocity campaign checked

A successor power paired with its predecessor equals predecessor times base.

Prerequisites

pow_successor_decompose pow_functional

Used directly by

pow_mod_congruent

Exact expanded PA statement

Copy statement

forall a e se r n. se = S e -> (exists ff_b_pair_predecessor ff_c_pair_predecessor. ((forall ff_i_pair_predecessor_repeat. (exists ff_lt_pair_predecessor_repeat_bound. ff_lt_pair_predecessor_repeat_bound + S ff_i_pair_predecessor_repeat = e) -> (((exists ff_h_pair_predecessor_repeat_decoded. ff_h_pair_predecessor_repeat_decoded + S (a) = S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor)) /\ exists ff_q_pair_predecessor_repeat_decoded. ff_b_pair_predecessor = ff_q_pair_predecessor_repeat_decoded * S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor) + (a)))) /\ (exists ff_u_pair_predecessor_product ff_v_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_start. ff_h_pair_predecessor_product_start + S (1) = S ((S (0)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_start. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_start * S ((S (0)) * ff_v_pair_predecessor_product) + (1))) /\ ((((exists ff_h_pair_predecessor_product_terminal. ff_h_pair_predecessor_product_terminal + S (r) = S ((S (e)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_terminal. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_terminal * S ((S (e)) * ff_v_pair_predecessor_product) + (r))) /\ forall ff_i_pair_predecessor_product. (exists ff_lt_pair_predecessor_product_bound. ff_lt_pair_predecessor_product_bound + S ff_i_pair_predecessor_product = e) -> exists ff_p_pair_predecessor_product ff_r_pair_predecessor_product ff_s_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_factor. ff_h_pair_predecessor_product_factor + S (ff_p_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor)) /\ exists ff_q_pair_predecessor_product_factor. ff_b_pair_predecessor = ff_q_pair_predecessor_product_factor * S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor) + (ff_p_pair_predecessor_product))) /\ ((((exists ff_h_pair_predecessor_product_partial. ff_h_pair_predecessor_product_partial + S (ff_r_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_partial. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_partial * S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_r_pair_predecessor_product))) /\ ((((exists ff_h_pair_predecessor_product_successor. ff_h_pair_predecessor_product_successor + S (ff_s_pair_predecessor_product) = S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_successor. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_successor * S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_s_pair_predecessor_product))) /\ ff_s_pair_predecessor_product = ff_r_pair_predecessor_product * ff_p_pair_predecessor_product)))))))) -> (exists ff_b_pair_successor ff_c_pair_successor. ((forall ff_i_pair_successor_repeat. (exists ff_lt_pair_successor_repeat_bound. ff_lt_pair_successor_repeat_bound + S ff_i_pair_successor_repeat = se) -> (((exists ff_h_pair_successor_repeat_decoded. ff_h_pair_successor_repeat_decoded + S (a) = S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor)) /\ exists ff_q_pair_successor_repeat_decoded. ff_b_pair_successor = ff_q_pair_successor_repeat_decoded * S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor) + (a)))) /\ (exists ff_u_pair_successor_product ff_v_pair_successor_product. ((((exists ff_h_pair_successor_product_start. ff_h_pair_successor_product_start + S (1) = S ((S (0)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_start. ff_u_pair_successor_product = ff_q_pair_successor_product_start * S ((S (0)) * ff_v_pair_successor_product) + (1))) /\ ((((exists ff_h_pair_successor_product_terminal. ff_h_pair_successor_product_terminal + S (n) = S ((S (se)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_terminal. ff_u_pair_successor_product = ff_q_pair_successor_product_terminal * S ((S (se)) * ff_v_pair_successor_product) + (n))) /\ forall ff_i_pair_successor_product. (exists ff_lt_pair_successor_product_bound. ff_lt_pair_successor_product_bound + S ff_i_pair_successor_product = se) -> exists ff_p_pair_successor_product ff_r_pair_successor_product ff_s_pair_successor_product. ((((exists ff_h_pair_successor_product_factor. ff_h_pair_successor_product_factor + S (ff_p_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor)) /\ exists ff_q_pair_successor_product_factor. ff_b_pair_successor = ff_q_pair_successor_product_factor * S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor) + (ff_p_pair_successor_product))) /\ ((((exists ff_h_pair_successor_product_partial. ff_h_pair_successor_product_partial + S (ff_r_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_partial. ff_u_pair_successor_product = ff_q_pair_successor_product_partial * S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_r_pair_successor_product))) /\ ((((exists ff_h_pair_successor_product_successor. ff_h_pair_successor_product_successor + S (ff_s_pair_successor_product) = S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_successor. ff_u_pair_successor_product = ff_q_pair_successor_product_successor * S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_s_pair_successor_product))) /\ ff_s_pair_successor_product = ff_r_pair_successor_product * ff_p_pair_successor_product)))))))) -> n = r * a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e se r n. se = S e -> (exists ff_b_pair_predecessor ff_c_pair_predecessor. ((forall ff_i_pair_predecessor_repeat. (exists ff_lt_pair_predecessor_repeat_bound. ff_lt_pair_predecessor_repeat_bound + S ff_i_pair_predecessor_repeat = e) -> (((exists ff_h_pair_predecessor_repeat_decoded. ff_h_pair_predecessor_repeat_decoded + S (a) = S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor)) /\ exists ff_q_pair_predecessor_repeat_decoded. ff_b_pair_predecessor = ff_q_pair_predecessor_repeat_decoded * S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor) + (a)))) /\ (exists ff_u_pair_predecessor_product ff_v_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_start. ff_h_pair_predecessor_product_start + S (1) = S ((S (0)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_start. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_start * S ((S (0)) * ff_v_pair_predecessor_product) + (1))) /\ ((((exists ff_h_pair_predecessor_product_terminal. ff_h_pair_predecessor_product_terminal + S (r) = S ((S (e)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_terminal. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_terminal * S ((S (e)) * ff_v_pair_predecessor_product) + (r))) /\ forall ff_i_pair_predecessor_product. (exists ff_lt_pair_predecessor_product_bound. ff_lt_pair_predecessor_product_bound + S ff_i_pair_predecessor_product = e) -> exists ff_p_pair_predecessor_product ff_r_pair_predecessor_product ff_s_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_factor. ff_h_pair_predecessor_product_factor + S (ff_p_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor)) /\ exists ff_q_pair_predecessor_product_factor. ff_b_pair_predecessor = ff_q_pair_predecessor_product_factor * S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor) + (ff_p_pair_predecessor_product))) /\ ((((exists ff_h_pair_predecessor_product_partial. ff_h_pair_predecessor_product_partial + S (ff_r_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_partial. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_partial * S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_r_pair_predecessor_product))) /\ ((((exists ff_h_pair_predecessor_product_successor. ff_h_pair_predecessor_product_successor + S (ff_s_pair_predecessor_product) = S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\ exists ff_q_pair_predecessor_product_successor. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_successor * S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_s_pair_predecessor_product))) /\ ff_s_pair_predecessor_product = ff_r_pair_predecessor_product * ff_p_pair_predecessor_product)))))))) -> (exists ff_b_pair_successor ff_c_pair_successor. ((forall ff_i_pair_successor_repeat. (exists ff_lt_pair_successor_repeat_bound. ff_lt_pair_successor_repeat_bound + S ff_i_pair_successor_repeat = se) -> (((exists ff_h_pair_successor_repeat_decoded. ff_h_pair_successor_repeat_decoded + S (a) = S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor)) /\ exists ff_q_pair_successor_repeat_decoded. ff_b_pair_successor = ff_q_pair_successor_repeat_decoded * S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor) + (a)))) /\ (exists ff_u_pair_successor_product ff_v_pair_successor_product. ((((exists ff_h_pair_successor_product_start. ff_h_pair_successor_product_start + S (1) = S ((S (0)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_start. ff_u_pair_successor_product = ff_q_pair_successor_product_start * S ((S (0)) * ff_v_pair_successor_product) + (1))) /\ ((((exists ff_h_pair_successor_product_terminal. ff_h_pair_successor_product_terminal + S (n) = S ((S (se)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_terminal. ff_u_pair_successor_product = ff_q_pair_successor_product_terminal * S ((S (se)) * ff_v_pair_successor_product) + (n))) /\ forall ff_i_pair_successor_product. (exists ff_lt_pair_successor_product_bound. ff_lt_pair_successor_product_bound + S ff_i_pair_successor_product = se) -> exists ff_p_pair_successor_product ff_r_pair_successor_product ff_s_pair_successor_product. ((((exists ff_h_pair_successor_product_factor. ff_h_pair_successor_product_factor + S (ff_p_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor)) /\ exists ff_q_pair_successor_product_factor. ff_b_pair_successor = ff_q_pair_successor_product_factor * S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor) + (ff_p_pair_successor_product))) /\ ((((exists ff_h_pair_successor_product_partial. ff_h_pair_successor_product_partial + S (ff_r_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_partial. ff_u_pair_successor_product = ff_q_pair_successor_product_partial * S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_r_pair_successor_product))) /\ ((((exists ff_h_pair_successor_product_successor. ff_h_pair_successor_product_successor + S (ff_s_pair_successor_product) = S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\ exists ff_q_pair_successor_product_successor. ff_u_pair_successor_product = ff_q_pair_successor_product_successor * S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_s_pair_successor_product))) /\ ff_s_pair_successor_product = ff_r_pair_successor_product * ff_p_pair_successor_product)))))))) -> n = r * a
use pow_successor_decompose
use pow_functional
intro a
intro e
intro se
intro r
intro n
intro hse
intro hprevious
intro hsuccessor
have hstep : exists z. (exists ff_b_pair_decomposed ff_c_pair_decomposed. ((forall ff_i_pair_decomposed_repeat. (exists ff_lt_pair_decomposed_repeat_bound. ff_lt_pair_decomposed_repeat_bound + S ff_i_pair_decomposed_repeat = e) -> (((exists ff_h_pair_decomposed_repeat_decoded. ff_h_pair_decomposed_repeat_decoded + S (a) = S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed)) /\ exists ff_q_pair_decomposed_repeat_decoded. ff_b_pair_decomposed = ff_q_pair_decomposed_repeat_decoded * S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed) + (a)))) /\ (exists ff_u_pair_decomposed_product ff_v_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_start. ff_h_pair_decomposed_product_start + S (1) = S ((S (0)) * ff_v_pair_decomposed_product)) /\ exists ff_q_pair_decomposed_product_start. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_start * S ((S (0)) * ff_v_pair_decomposed_product) + (1))) /\ ((((exists ff_h_pair_decomposed_product_terminal. ff_h_pair_decomposed_product_terminal + S (z) = S ((S (e)) * ff_v_pair_decomposed_product)) /\ exists ff_q_pair_decomposed_product_terminal. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_terminal * S ((S (e)) * ff_v_pair_decomposed_product) + (z))) /\ forall ff_i_pair_decomposed_product. (exists ff_lt_pair_decomposed_product_bound. ff_lt_pair_decomposed_product_bound + S ff_i_pair_decomposed_product = e) -> exists ff_p_pair_decomposed_product ff_r_pair_decomposed_product ff_s_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_factor. ff_h_pair_decomposed_product_factor + S (ff_p_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed)) /\ exists ff_q_pair_decomposed_product_factor. ff_b_pair_decomposed = ff_q_pair_decomposed_product_factor * S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed) + (ff_p_pair_decomposed_product))) /\ ((((exists ff_h_pair_decomposed_product_partial. ff_h_pair_decomposed_product_partial + S (ff_r_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\ exists ff_q_pair_decomposed_product_partial. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_partial * S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_r_pair_decomposed_product))) /\ ((((exists ff_h_pair_decomposed_product_successor. ff_h_pair_decomposed_product_successor + S (ff_s_pair_decomposed_product) = S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\ exists ff_q_pair_decomposed_product_successor. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_successor * S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_s_pair_decomposed_product))) /\ ff_s_pair_decomposed_product = ff_r_pair_decomposed_product * ff_p_pair_decomposed_product)))))))) /\ n = z * a
specialize pow_successor_decompose a
specialize pow_successor_decompose e
specialize pow_successor_decompose se
specialize pow_successor_decompose n
apply pow_successor_decompose
exact hse
exact hsuccessor
cases hstep
cases hstep_witness
have hz : x = r
specialize pow_functional a
specialize pow_functional e
specialize pow_functional x
specialize pow_functional r
apply pow_functional
exact hstep_witness_left
exact hprevious
trans x * a
exact hstep_witness_right
rewrite hz
refl
qed

Occurrences

5,282

Distinct objects

1,293

Reused references

46

Depth

65

Cuts

145

Certificate SHA-256

be60a3277b36…

pow_mod_congruentPow mod congruent Quadratic reciprocity campaign checked

Balanced-congruent bases have congruent relational powers at every exponent.

Prerequisites

pow_zero pow_successor_decompose pow_successor_pair_mul mod_eq_refl mod_eq_mul

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m a b e x y. (exists qr_u_base qr_v_base. a + m * qr_u_base = b + m * qr_v_base) -> (exists ff_b_left ff_c_left. ((forall ff_i_left_repeat. (exists ff_lt_left_repeat_bound. ff_lt_left_repeat_bound + S ff_i_left_repeat = e) -> (((exists ff_h_left_repeat_decoded. ff_h_left_repeat_decoded + S (a) = S ((S (ff_i_left_repeat)) * ff_c_left)) /\ exists ff_q_left_repeat_decoded. ff_b_left = ff_q_left_repeat_decoded * S ((S (ff_i_left_repeat)) * ff_c_left) + (a)))) /\ (exists ff_u_left_product ff_v_left_product. ((((exists ff_h_left_product_start. ff_h_left_product_start + S (1) = S ((S (0)) * ff_v_left_product)) /\ exists ff_q_left_product_start. ff_u_left_product = ff_q_left_product_start * S ((S (0)) * ff_v_left_product) + (1))) /\ ((((exists ff_h_left_product_terminal. ff_h_left_product_terminal + S (x) = S ((S (e)) * ff_v_left_product)) /\ exists ff_q_left_product_terminal. ff_u_left_product = ff_q_left_product_terminal * S ((S (e)) * ff_v_left_product) + (x))) /\ forall ff_i_left_product. (exists ff_lt_left_product_bound. ff_lt_left_product_bound + S ff_i_left_product = e) -> exists ff_p_left_product ff_r_left_product ff_s_left_product. ((((exists ff_h_left_product_factor. ff_h_left_product_factor + S (ff_p_left_product) = S ((S (ff_i_left_product)) * ff_c_left)) /\ exists ff_q_left_product_factor. ff_b_left = ff_q_left_product_factor * S ((S (ff_i_left_product)) * ff_c_left) + (ff_p_left_product))) /\ ((((exists ff_h_left_product_partial. ff_h_left_product_partial + S (ff_r_left_product) = S ((S (ff_i_left_product)) * ff_v_left_product)) /\ exists ff_q_left_product_partial. ff_u_left_product = ff_q_left_product_partial * S ((S (ff_i_left_product)) * ff_v_left_product) + (ff_r_left_product))) /\ ((((exists ff_h_left_product_successor. ff_h_left_product_successor + S (ff_s_left_product) = S ((S (S ff_i_left_product)) * ff_v_left_product)) /\ exists ff_q_left_product_successor. ff_u_left_product = ff_q_left_product_successor * S ((S (S ff_i_left_product)) * ff_v_left_product) + (ff_s_left_product))) /\ ff_s_left_product = ff_r_left_product * ff_p_left_product)))))))) -> (exists ff_b_right ff_c_right. ((forall ff_i_right_repeat. (exists ff_lt_right_repeat_bound. ff_lt_right_repeat_bound + S ff_i_right_repeat = e) -> (((exists ff_h_right_repeat_decoded. ff_h_right_repeat_decoded + S (b) = S ((S (ff_i_right_repeat)) * ff_c_right)) /\ exists ff_q_right_repeat_decoded. ff_b_right = ff_q_right_repeat_decoded * S ((S (ff_i_right_repeat)) * ff_c_right) + (b)))) /\ (exists ff_u_right_product ff_v_right_product. ((((exists ff_h_right_product_start. ff_h_right_product_start + S (1) = S ((S (0)) * ff_v_right_product)) /\ exists ff_q_right_product_start. ff_u_right_product = ff_q_right_product_start * S ((S (0)) * ff_v_right_product) + (1))) /\ ((((exists ff_h_right_product_terminal. ff_h_right_product_terminal + S (y) = S ((S (e)) * ff_v_right_product)) /\ exists ff_q_right_product_terminal. ff_u_right_product = ff_q_right_product_terminal * S ((S (e)) * ff_v_right_product) + (y))) /\ forall ff_i_right_product. (exists ff_lt_right_product_bound. ff_lt_right_product_bound + S ff_i_right_product = e) -> exists ff_p_right_product ff_r_right_product ff_s_right_product. ((((exists ff_h_right_product_factor. ff_h_right_product_factor + S (ff_p_right_product) = S ((S (ff_i_right_product)) * ff_c_right)) /\ exists ff_q_right_product_factor. ff_b_right = ff_q_right_product_factor * S ((S (ff_i_right_product)) * ff_c_right) + (ff_p_right_product))) /\ ((((exists ff_h_right_product_partial. ff_h_right_product_partial + S (ff_r_right_product) = S ((S (ff_i_right_product)) * ff_v_right_product)) /\ exists ff_q_right_product_partial. ff_u_right_product = ff_q_right_product_partial * S ((S (ff_i_right_product)) * ff_v_right_product) + (ff_r_right_product))) /\ ((((exists ff_h_right_product_successor. ff_h_right_product_successor + S (ff_s_right_product) = S ((S (S ff_i_right_product)) * ff_v_right_product)) /\ exists ff_q_right_product_successor. ff_u_right_product = ff_q_right_product_successor * S ((S (S ff_i_right_product)) * ff_v_right_product) + (ff_s_right_product))) /\ ff_s_right_product = ff_r_right_product * ff_p_right_product)))))))) -> (exists qr_u_result qr_v_result. x + m * qr_u_result = y + m * qr_v_result)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b e x y. (exists qr_u_base qr_v_base. a + m * qr_u_base = b + m * qr_v_base) -> (exists ff_b_left ff_c_left. ((forall ff_i_left_repeat. (exists ff_lt_left_repeat_bound. ff_lt_left_repeat_bound + S ff_i_left_repeat = e) -> (((exists ff_h_left_repeat_decoded. ff_h_left_repeat_decoded + S (a) = S ((S (ff_i_left_repeat)) * ff_c_left)) /\ exists ff_q_left_repeat_decoded. ff_b_left = ff_q_left_repeat_decoded * S ((S (ff_i_left_repeat)) * ff_c_left) + (a)))) /\ (exists ff_u_left_product ff_v_left_product. ((((exists ff_h_left_product_start. ff_h_left_product_start + S (1) = S ((S (0)) * ff_v_left_product)) /\ exists ff_q_left_product_start. ff_u_left_product = ff_q_left_product_start * S ((S (0)) * ff_v_left_product) + (1))) /\ ((((exists ff_h_left_product_terminal. ff_h_left_product_terminal + S (x) = S ((S (e)) * ff_v_left_product)) /\ exists ff_q_left_product_terminal. ff_u_left_product = ff_q_left_product_terminal * S ((S (e)) * ff_v_left_product) + (x))) /\ forall ff_i_left_product. (exists ff_lt_left_product_bound. ff_lt_left_product_bound + S ff_i_left_product = e) -> exists ff_p_left_product ff_r_left_product ff_s_left_product. ((((exists ff_h_left_product_factor. ff_h_left_product_factor + S (ff_p_left_product) = S ((S (ff_i_left_product)) * ff_c_left)) /\ exists ff_q_left_product_factor. ff_b_left = ff_q_left_product_factor * S ((S (ff_i_left_product)) * ff_c_left) + (ff_p_left_product))) /\ ((((exists ff_h_left_product_partial. ff_h_left_product_partial + S (ff_r_left_product) = S ((S (ff_i_left_product)) * ff_v_left_product)) /\ exists ff_q_left_product_partial. ff_u_left_product = ff_q_left_product_partial * S ((S (ff_i_left_product)) * ff_v_left_product) + (ff_r_left_product))) /\ ((((exists ff_h_left_product_successor. ff_h_left_product_successor + S (ff_s_left_product) = S ((S (S ff_i_left_product)) * ff_v_left_product)) /\ exists ff_q_left_product_successor. ff_u_left_product = ff_q_left_product_successor * S ((S (S ff_i_left_product)) * ff_v_left_product) + (ff_s_left_product))) /\ ff_s_left_product = ff_r_left_product * ff_p_left_product)))))))) -> (exists ff_b_right ff_c_right. ((forall ff_i_right_repeat. (exists ff_lt_right_repeat_bound. ff_lt_right_repeat_bound + S ff_i_right_repeat = e) -> (((exists ff_h_right_repeat_decoded. ff_h_right_repeat_decoded + S (b) = S ((S (ff_i_right_repeat)) * ff_c_right)) /\ exists ff_q_right_repeat_decoded. ff_b_right = ff_q_right_repeat_decoded * S ((S (ff_i_right_repeat)) * ff_c_right) + (b)))) /\ (exists ff_u_right_product ff_v_right_product. ((((exists ff_h_right_product_start. ff_h_right_product_start + S (1) = S ((S (0)) * ff_v_right_product)) /\ exists ff_q_right_product_start. ff_u_right_product = ff_q_right_product_start * S ((S (0)) * ff_v_right_product) + (1))) /\ ((((exists ff_h_right_product_terminal. ff_h_right_product_terminal + S (y) = S ((S (e)) * ff_v_right_product)) /\ exists ff_q_right_product_terminal. ff_u_right_product = ff_q_right_product_terminal * S ((S (e)) * ff_v_right_product) + (y))) /\ forall ff_i_right_product. (exists ff_lt_right_product_bound. ff_lt_right_product_bound + S ff_i_right_product = e) -> exists ff_p_right_product ff_r_right_product ff_s_right_product. ((((exists ff_h_right_product_factor. ff_h_right_product_factor + S (ff_p_right_product) = S ((S (ff_i_right_product)) * ff_c_right)) /\ exists ff_q_right_product_factor. ff_b_right = ff_q_right_product_factor * S ((S (ff_i_right_product)) * ff_c_right) + (ff_p_right_product))) /\ ((((exists ff_h_right_product_partial. ff_h_right_product_partial + S (ff_r_right_product) = S ((S (ff_i_right_product)) * ff_v_right_product)) /\ exists ff_q_right_product_partial. ff_u_right_product = ff_q_right_product_partial * S ((S (ff_i_right_product)) * ff_v_right_product) + (ff_r_right_product))) /\ ((((exists ff_h_right_product_successor. ff_h_right_product_successor + S (ff_s_right_product) = S ((S (S ff_i_right_product)) * ff_v_right_product)) /\ exists ff_q_right_product_successor. ff_u_right_product = ff_q_right_product_successor * S ((S (S ff_i_right_product)) * ff_v_right_product) + (ff_s_right_product))) /\ ff_s_right_product = ff_r_right_product * ff_p_right_product)))))))) -> (exists qr_u_result qr_v_result. x + m * qr_u_result = y + m * qr_v_result)
use pow_zero
use pow_successor_decompose
use pow_successor_pair_mul
use mod_eq_refl
use mod_eq_mul
intro m
intro a
intro b
induction e
intro x
intro y
intro hab
intro hx
intro hy
have hx1 : x = 1
specialize pow_zero a
specialize pow_zero 0
specialize pow_zero x
apply pow_zero
refl
exact hx
have hy1 : y = 1
specialize pow_zero b
specialize pow_zero 0
specialize pow_zero y
apply pow_zero
refl
exact hy
rewrite hx1
rewrite hy1
specialize mod_eq_refl m
specialize mod_eq_refl 1
exact mod_eq_refl
intro x
intro y
intro hab
intro hx
intro hy
have hleft : exists r. (exists ff_b_left_step ff_c_left_step. ((forall ff_i_left_step_repeat. (exists ff_lt_left_step_repeat_bound. ff_lt_left_step_repeat_bound + S ff_i_left_step_repeat = e) -> (((exists ff_h_left_step_repeat_decoded. ff_h_left_step_repeat_decoded + S (a) = S ((S (ff_i_left_step_repeat)) * ff_c_left_step)) /\ exists ff_q_left_step_repeat_decoded. ff_b_left_step = ff_q_left_step_repeat_decoded * S ((S (ff_i_left_step_repeat)) * ff_c_left_step) + (a)))) /\ (exists ff_u_left_step_product ff_v_left_step_product. ((((exists ff_h_left_step_product_start. ff_h_left_step_product_start + S (1) = S ((S (0)) * ff_v_left_step_product)) /\ exists ff_q_left_step_product_start. ff_u_left_step_product = ff_q_left_step_product_start * S ((S (0)) * ff_v_left_step_product) + (1))) /\ ((((exists ff_h_left_step_product_terminal. ff_h_left_step_product_terminal + S (r) = S ((S (e)) * ff_v_left_step_product)) /\ exists ff_q_left_step_product_terminal. ff_u_left_step_product = ff_q_left_step_product_terminal * S ((S (e)) * ff_v_left_step_product) + (r))) /\ forall ff_i_left_step_product. (exists ff_lt_left_step_product_bound. ff_lt_left_step_product_bound + S ff_i_left_step_product = e) -> exists ff_p_left_step_product ff_r_left_step_product ff_s_left_step_product. ((((exists ff_h_left_step_product_factor. ff_h_left_step_product_factor + S (ff_p_left_step_product) = S ((S (ff_i_left_step_product)) * ff_c_left_step)) /\ exists ff_q_left_step_product_factor. ff_b_left_step = ff_q_left_step_product_factor * S ((S (ff_i_left_step_product)) * ff_c_left_step) + (ff_p_left_step_product))) /\ ((((exists ff_h_left_step_product_partial. ff_h_left_step_product_partial + S (ff_r_left_step_product) = S ((S (ff_i_left_step_product)) * ff_v_left_step_product)) /\ exists ff_q_left_step_product_partial. ff_u_left_step_product = ff_q_left_step_product_partial * S ((S (ff_i_left_step_product)) * ff_v_left_step_product) + (ff_r_left_step_product))) /\ ((((exists ff_h_left_step_product_successor. ff_h_left_step_product_successor + S (ff_s_left_step_product) = S ((S (S ff_i_left_step_product)) * ff_v_left_step_product)) /\ exists ff_q_left_step_product_successor. ff_u_left_step_product = ff_q_left_step_product_successor * S ((S (S ff_i_left_step_product)) * ff_v_left_step_product) + (ff_s_left_step_product))) /\ ff_s_left_step_product = ff_r_left_step_product * ff_p_left_step_product)))))))) /\ x = r * a
specialize pow_successor_decompose a
specialize pow_successor_decompose e
specialize pow_successor_decompose (S e)
specialize pow_successor_decompose x
apply pow_successor_decompose
refl
exact hx
cases hleft
cases hleft_witness
have hright : exists r. (exists ff_b_right_step ff_c_right_step. ((forall ff_i_right_step_repeat. (exists ff_lt_right_step_repeat_bound. ff_lt_right_step_repeat_bound + S ff_i_right_step_repeat = e) -> (((exists ff_h_right_step_repeat_decoded. ff_h_right_step_repeat_decoded + S (b) = S ((S (ff_i_right_step_repeat)) * ff_c_right_step)) /\ exists ff_q_right_step_repeat_decoded. ff_b_right_step = ff_q_right_step_repeat_decoded * S ((S (ff_i_right_step_repeat)) * ff_c_right_step) + (b)))) /\ (exists ff_u_right_step_product ff_v_right_step_product. ((((exists ff_h_right_step_product_start. ff_h_right_step_product_start + S (1) = S ((S (0)) * ff_v_right_step_product)) /\ exists ff_q_right_step_product_start. ff_u_right_step_product = ff_q_right_step_product_start * S ((S (0)) * ff_v_right_step_product) + (1))) /\ ((((exists ff_h_right_step_product_terminal. ff_h_right_step_product_terminal + S (r) = S ((S (e)) * ff_v_right_step_product)) /\ exists ff_q_right_step_product_terminal. ff_u_right_step_product = ff_q_right_step_product_terminal * S ((S (e)) * ff_v_right_step_product) + (r))) /\ forall ff_i_right_step_product. (exists ff_lt_right_step_product_bound. ff_lt_right_step_product_bound + S ff_i_right_step_product = e) -> exists ff_p_right_step_product ff_r_right_step_product ff_s_right_step_product. ((((exists ff_h_right_step_product_factor. ff_h_right_step_product_factor + S (ff_p_right_step_product) = S ((S (ff_i_right_step_product)) * ff_c_right_step)) /\ exists ff_q_right_step_product_factor. ff_b_right_step = ff_q_right_step_product_factor * S ((S (ff_i_right_step_product)) * ff_c_right_step) + (ff_p_right_step_product))) /\ ((((exists ff_h_right_step_product_partial. ff_h_right_step_product_partial + S (ff_r_right_step_product) = S ((S (ff_i_right_step_product)) * ff_v_right_step_product)) /\ exists ff_q_right_step_product_partial. ff_u_right_step_product = ff_q_right_step_product_partial * S ((S (ff_i_right_step_product)) * ff_v_right_step_product) + (ff_r_right_step_product))) /\ ((((exists ff_h_right_step_product_successor. ff_h_right_step_product_successor + S (ff_s_right_step_product) = S ((S (S ff_i_right_step_product)) * ff_v_right_step_product)) /\ exists ff_q_right_step_product_successor. ff_u_right_step_product = ff_q_right_step_product_successor * S ((S (S ff_i_right_step_product)) * ff_v_right_step_product) + (ff_s_right_step_product))) /\ ff_s_right_step_product = ff_r_right_step_product * ff_p_right_step_product)))))))) /\ y = r * b
specialize pow_successor_decompose b
specialize pow_successor_decompose e
specialize pow_successor_decompose (S e)
specialize pow_successor_decompose y
apply pow_successor_decompose
refl
exact hy
cases hright
cases hright_witness
have hxmul : x = x1 * a
specialize pow_successor_pair_mul a
specialize pow_successor_pair_mul e
specialize pow_successor_pair_mul (S e)
specialize pow_successor_pair_mul x1
specialize pow_successor_pair_mul x
apply pow_successor_pair_mul
refl
exact hleft_witness_left
exact hx
have hymul : y = x2 * b
specialize pow_successor_pair_mul b
specialize pow_successor_pair_mul e
specialize pow_successor_pair_mul (S e)
specialize pow_successor_pair_mul x2
specialize pow_successor_pair_mul y
apply pow_successor_pair_mul
refl
exact hright_witness_left
exact hy
have hpre : exists qr_u_predecessors qr_v_predecessors. x1 + m * qr_u_predecessors = x2 + m * qr_v_predecessors
specialize IH x1
specialize IH x2
apply IH
exact hab
exact hleft_witness_left
exact hright_witness_left
have hmul : exists pc_u_products pc_v_products. (x1 * a) + m * pc_u_products = (x2 * b) + m * pc_v_products
specialize mod_eq_mul m
specialize mod_eq_mul x1
specialize mod_eq_mul x2
specialize mod_eq_mul a
specialize mod_eq_mul b
apply mod_eq_mul
exact hpre
exact hab
rewrite hxmul
rewrite hymul
exact hmul
qed

Occurrences

10,671

Distinct objects

1,748

Reused references

63

Depth

68

Cuts

297

Certificate SHA-256

5009e017166c…

lt_three_casesLt three cases Quadratic reciprocity campaign checked

Every natural strictly below three is zero, one, or two.

Prerequisites

le_of_succ_le_succ le_eq_or_lt le_zero

Used directly by

lt_five_cases bounded_square_mod3_classify

Exact expanded PA statement

Copy statement

forall x. (exists h. h + S x = 3) -> x = 0 \/ x = 1 \/ x = 2

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall x. (exists h. h + S x = 3) -> x = 0 \/ x = 1 \/ x = 2
use le_of_succ_le_succ
use le_eq_or_lt
use le_zero
intro x
intro hb
have hle2 : exists h. h + x = 2
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 2
apply le_of_succ_le_succ
exact hb
have hc2 : x = 2 \/ exists h. h + S x = 2
specialize le_eq_or_lt x
specialize le_eq_or_lt 2
apply le_eq_or_lt
exact hle2
cases hc2
right
exact hc2_left
left
have hle1 : exists h. h + x = 1
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 1
apply le_of_succ_le_succ
exact hc2_right
have hc1 : x = 1 \/ exists h. h + S x = 1
specialize le_eq_or_lt x
specialize le_eq_or_lt 1
apply le_eq_or_lt
exact hle1
cases hc1
right
exact hc1_left
left
have hle0 : exists h. h + x = 0
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 0
apply le_of_succ_le_succ
exact hc1_right
specialize le_zero x
apply le_zero
exact hle0
qed

Occurrences

182

Distinct objects

178

Reused references

4

Depth

21

Cuts

7

Certificate SHA-256

f61975330ce8…

lt_five_casesLt five cases Quadratic reciprocity campaign checked

Every natural strictly below five is one of its canonical values.

Prerequisites

le_of_succ_le_succ le_eq_or_lt lt_three_cases

Used directly by

lt_seven_cases bounded_square_mod5_classify

Exact expanded PA statement

Copy statement

forall x. (exists h. h + S x = 5) -> x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall x. (exists h. h + S x = 5) -> x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4
use le_of_succ_le_succ
use le_eq_or_lt
use lt_three_cases
intro x
intro hb
have hle4 : exists h. h + x = 4
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 4
apply le_of_succ_le_succ
exact hb
have hc4 : x = 4 \/ exists h. h + S x = 4
specialize le_eq_or_lt x
specialize le_eq_or_lt 4
apply le_eq_or_lt
exact hle4
cases hc4
right
exact hc4_left
left
have hle3 : exists h. h + x = 3
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 3
apply le_of_succ_le_succ
exact hc4_right
have hc3 : x = 3 \/ exists h. h + S x = 3
specialize le_eq_or_lt x
specialize le_eq_or_lt 3
apply le_eq_or_lt
exact hle3
cases hc3
right
exact hc3_left
left
specialize lt_three_cases x
apply lt_three_cases
exact hc3_right
qed

Occurrences

331

Distinct objects

213

Reused references

6

Depth

24

Cuts

13

Certificate SHA-256

f70f1a4e36bb…

lt_seven_casesLt seven cases Quadratic reciprocity campaign checked

Every natural strictly below seven is one of its canonical values.

Prerequisites

le_of_succ_le_succ le_eq_or_lt lt_five_cases

Used directly by

bounded_square_mod7_classify

Exact expanded PA statement

Copy statement

forall x. (exists h. h + S x = 7) -> x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4 \/ x = 5 \/ x = 6

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall x. (exists h. h + S x = 7) -> x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4 \/ x = 5 \/ x = 6
use le_of_succ_le_succ
use le_eq_or_lt
use lt_five_cases
intro x
intro hb
have hle6 : exists h. h + x = 6
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 6
apply le_of_succ_le_succ
exact hb
have hc6 : x = 6 \/ exists h. h + S x = 6
specialize le_eq_or_lt x
specialize le_eq_or_lt 6
apply le_eq_or_lt
exact hle6
cases hc6
right
exact hc6_left
left
have hle5 : exists h. h + x = 5
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ 5
apply le_of_succ_le_succ
exact hc6_right
have hc5 : x = 5 \/ exists h. h + S x = 5
specialize le_eq_or_lt x
specialize le_eq_or_lt 5
apply le_eq_or_lt
exact hle5
cases hc5
right
exact hc5_left
left
specialize lt_five_cases x
apply lt_five_cases
exact hc5_right
qed

Occurrences

480

Distinct objects

248

Reused references

8

Depth

27

Cuts

19

Certificate SHA-256

60a0e0ace6b6…

bounded_square_mod3_classifyBounded square mod3 classify Quadratic reciprocity campaign checked

A bounded square modulo 3 has exactly a canonical square residue.

Prerequisites

lt_three_cases mod_eq_trans mod_eq_bounded_unique

Used directly by

qres_mod3_canonical_iff

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 3) -> (exists sm_x_bc3. (exists sm_h_bc3. sm_h_bc3 + S sm_x_bc3 = 3) /\ exists sm_u_bc3 sm_v_bc3. sm_x_bc3 * sm_x_bc3 + 3 * sm_u_bc3 = a + 3 * sm_v_bc3) -> (a = 0 \/ a = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 3) -> (exists sm_x_bc3. (exists sm_h_bc3. sm_h_bc3 + S sm_x_bc3 = 3) /\ exists sm_u_bc3 sm_v_bc3. sm_x_bc3 * sm_x_bc3 + 3 * sm_u_bc3 = a + 3 * sm_v_bc3) -> (a = 0 \/ a = 1)
use lt_three_cases
use mod_eq_trans
use mod_eq_bounded_unique
intro a
intro ha
intro hb
cases hb
cases hb_witness
have hroots : x = 0 \/ x = 1 \/ x = 2
specialize lt_three_cases x
apply lt_three_cases
exact hb_witness_left
cases hroots
cases hroots_left
have hsq : exists u v. 0 * 0 + 3 * u = a + 3 * v
rewrite hroots_left_left at hb_witness_right
rewrite hroots_left_left at hb_witness_right
exact hb_witness_right
have hc : exists u v. 0 + 3 * u = (0 * 0) + 3 * v
exists 0
exists 0
norm_num
have hra : exists u v. 0 + 3 * u = a + 3 * v
specialize mod_eq_trans 3
specialize mod_eq_trans 0
specialize mod_eq_trans (0 * 0)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 0 = 3
exists 2
norm_num
have heq : 0 = a
specialize mod_eq_bounded_unique 3
specialize mod_eq_bounded_unique 0
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
symm
exact heq
have hsq : exists u v. 1 * 1 + 3 * u = a + 3 * v
rewrite hroots_left_right at hb_witness_right
rewrite hroots_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 3 * u = (1 * 1) + 3 * v
exists 0
exists 0
norm_num
have hra : exists u v. 1 + 3 * u = a + 3 * v
specialize mod_eq_trans 3
specialize mod_eq_trans 1
specialize mod_eq_trans (1 * 1)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 3
exists 1
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 3
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
have hsq : exists u v. 2 * 2 + 3 * u = a + 3 * v
rewrite hroots_right at hb_witness_right
rewrite hroots_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 3 * u = (2 * 2) + 3 * v
exists 1
exists 0
norm_num
have hra : exists u v. 1 + 3 * u = a + 3 * v
specialize mod_eq_trans 3
specialize mod_eq_trans 1
specialize mod_eq_trans (2 * 2)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 3
exists 1
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 3
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
qed

Occurrences

1,767

Distinct objects

1,149

Reused references

29

Depth

62

Cuts

42

Certificate SHA-256

2a8ba2241f72…

bounded_square_mod5_classifyBounded square mod5 classify Quadratic reciprocity campaign checked

A bounded square modulo 5 has exactly a canonical square residue.

Prerequisites

lt_five_cases mod_eq_trans mod_eq_bounded_unique

Used directly by

qres_mod5_canonical_iff

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 5) -> (exists sm_x_bc5. (exists sm_h_bc5. sm_h_bc5 + S sm_x_bc5 = 5) /\ exists sm_u_bc5 sm_v_bc5. sm_x_bc5 * sm_x_bc5 + 5 * sm_u_bc5 = a + 5 * sm_v_bc5) -> (a = 0 \/ a = 1 \/ a = 4)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 5) -> (exists sm_x_bc5. (exists sm_h_bc5. sm_h_bc5 + S sm_x_bc5 = 5) /\ exists sm_u_bc5 sm_v_bc5. sm_x_bc5 * sm_x_bc5 + 5 * sm_u_bc5 = a + 5 * sm_v_bc5) -> (a = 0 \/ a = 1 \/ a = 4)
use lt_five_cases
use mod_eq_trans
use mod_eq_bounded_unique
intro a
intro ha
intro hb
cases hb
cases hb_witness
have hroots : x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4
specialize lt_five_cases x
apply lt_five_cases
exact hb_witness_left
cases hroots
cases hroots_left
cases hroots_left_left
cases hroots_left_left_left
have hsq : exists u v. 0 * 0 + 5 * u = a + 5 * v
rewrite hroots_left_left_left_left at hb_witness_right
rewrite hroots_left_left_left_left at hb_witness_right
exact hb_witness_right
have hc : exists u v. 0 + 5 * u = (0 * 0) + 5 * v
exists 0
exists 0
norm_num
have hra : exists u v. 0 + 5 * u = a + 5 * v
specialize mod_eq_trans 5
specialize mod_eq_trans 0
specialize mod_eq_trans (0 * 0)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 0 = 5
exists 4
norm_num
have heq : 0 = a
specialize mod_eq_bounded_unique 5
specialize mod_eq_bounded_unique 0
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
left
symm
exact heq
have hsq : exists u v. 1 * 1 + 5 * u = a + 5 * v
rewrite hroots_left_left_left_right at hb_witness_right
rewrite hroots_left_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 5 * u = (1 * 1) + 5 * v
exists 0
exists 0
norm_num
have hra : exists u v. 1 + 5 * u = a + 5 * v
specialize mod_eq_trans 5
specialize mod_eq_trans 1
specialize mod_eq_trans (1 * 1)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 5
exists 3
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 5
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
right
symm
exact heq
have hsq : exists u v. 2 * 2 + 5 * u = a + 5 * v
rewrite hroots_left_left_right at hb_witness_right
rewrite hroots_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 4 + 5 * u = (2 * 2) + 5 * v
exists 0
exists 0
norm_num
have hra : exists u v. 4 + 5 * u = a + 5 * v
specialize mod_eq_trans 5
specialize mod_eq_trans 4
specialize mod_eq_trans (2 * 2)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 4 = 5
exists 0
norm_num
have heq : 4 = a
specialize mod_eq_bounded_unique 5
specialize mod_eq_bounded_unique 4
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
have hsq : exists u v. 3 * 3 + 5 * u = a + 5 * v
rewrite hroots_left_right at hb_witness_right
rewrite hroots_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 4 + 5 * u = (3 * 3) + 5 * v
exists 1
exists 0
norm_num
have hra : exists u v. 4 + 5 * u = a + 5 * v
specialize mod_eq_trans 5
specialize mod_eq_trans 4
specialize mod_eq_trans (3 * 3)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 4 = 5
exists 0
norm_num
have heq : 4 = a
specialize mod_eq_bounded_unique 5
specialize mod_eq_bounded_unique 4
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
have hsq : exists u v. 4 * 4 + 5 * u = a + 5 * v
rewrite hroots_right at hb_witness_right
rewrite hroots_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 5 * u = (4 * 4) + 5 * v
exists 3
exists 0
norm_num
have hra : exists u v. 1 + 5 * u = a + 5 * v
specialize mod_eq_trans 5
specialize mod_eq_trans 1
specialize mod_eq_trans (4 * 4)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 5
exists 3
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 5
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
right
symm
exact heq
qed

Occurrences

2,567

Distinct objects

1,835

Reused references

31

Depth

62

Cuts

48

Certificate SHA-256

ae64803ac8fb…

bounded_square_mod7_classifyBounded square mod7 classify Quadratic reciprocity campaign checked

A bounded square modulo 7 has exactly a canonical square residue.

Prerequisites

lt_seven_cases mod_eq_trans mod_eq_bounded_unique

Used directly by

qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 7) -> (exists sm_x_bc7. (exists sm_h_bc7. sm_h_bc7 + S sm_x_bc7 = 7) /\ exists sm_u_bc7 sm_v_bc7. sm_x_bc7 * sm_x_bc7 + 7 * sm_u_bc7 = a + 7 * sm_v_bc7) -> (a = 0 \/ a = 1 \/ a = 2 \/ a = 4)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 7) -> (exists sm_x_bc7. (exists sm_h_bc7. sm_h_bc7 + S sm_x_bc7 = 7) /\ exists sm_u_bc7 sm_v_bc7. sm_x_bc7 * sm_x_bc7 + 7 * sm_u_bc7 = a + 7 * sm_v_bc7) -> (a = 0 \/ a = 1 \/ a = 2 \/ a = 4)
use lt_seven_cases
use mod_eq_trans
use mod_eq_bounded_unique
intro a
intro ha
intro hb
cases hb
cases hb_witness
have hroots : x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4 \/ x = 5 \/ x = 6
specialize lt_seven_cases x
apply lt_seven_cases
exact hb_witness_left
cases hroots
cases hroots_left
cases hroots_left_left
cases hroots_left_left_left
cases hroots_left_left_left_left
cases hroots_left_left_left_left_left
have hsq : exists u v. 0 * 0 + 7 * u = a + 7 * v
rewrite hroots_left_left_left_left_left_left at hb_witness_right
rewrite hroots_left_left_left_left_left_left at hb_witness_right
exact hb_witness_right
have hc : exists u v. 0 + 7 * u = (0 * 0) + 7 * v
exists 0
exists 0
norm_num
have hra : exists u v. 0 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 0
specialize mod_eq_trans (0 * 0)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 0 = 7
exists 6
norm_num
have heq : 0 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 0
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
left
left
symm
exact heq
have hsq : exists u v. 1 * 1 + 7 * u = a + 7 * v
rewrite hroots_left_left_left_left_left_right at hb_witness_right
rewrite hroots_left_left_left_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 7 * u = (1 * 1) + 7 * v
exists 0
exists 0
norm_num
have hra : exists u v. 1 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 1
specialize mod_eq_trans (1 * 1)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 7
exists 5
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
left
right
symm
exact heq
have hsq : exists u v. 2 * 2 + 7 * u = a + 7 * v
rewrite hroots_left_left_left_left_right at hb_witness_right
rewrite hroots_left_left_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 4 + 7 * u = (2 * 2) + 7 * v
exists 0
exists 0
norm_num
have hra : exists u v. 4 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 4
specialize mod_eq_trans (2 * 2)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 4 = 7
exists 2
norm_num
have heq : 4 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 4
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
have hsq : exists u v. 3 * 3 + 7 * u = a + 7 * v
rewrite hroots_left_left_left_right at hb_witness_right
rewrite hroots_left_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 2 + 7 * u = (3 * 3) + 7 * v
exists 1
exists 0
norm_num
have hra : exists u v. 2 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 2
specialize mod_eq_trans (3 * 3)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 2 = 7
exists 4
norm_num
have heq : 2 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 2
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
right
symm
exact heq
have hsq : exists u v. 4 * 4 + 7 * u = a + 7 * v
rewrite hroots_left_left_right at hb_witness_right
rewrite hroots_left_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 2 + 7 * u = (4 * 4) + 7 * v
exists 2
exists 0
norm_num
have hra : exists u v. 2 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 2
specialize mod_eq_trans (4 * 4)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 2 = 7
exists 4
norm_num
have heq : 2 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 2
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
right
symm
exact heq
have hsq : exists u v. 5 * 5 + 7 * u = a + 7 * v
rewrite hroots_left_right at hb_witness_right
rewrite hroots_left_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 4 + 7 * u = (5 * 5) + 7 * v
exists 3
exists 0
norm_num
have hra : exists u v. 4 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 4
specialize mod_eq_trans (5 * 5)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 4 = 7
exists 2
norm_num
have heq : 4 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 4
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
right
symm
exact heq
have hsq : exists u v. 6 * 6 + 7 * u = a + 7 * v
rewrite hroots_right at hb_witness_right
rewrite hroots_right at hb_witness_right
exact hb_witness_right
have hc : exists u v. 1 + 7 * u = (6 * 6) + 7 * v
exists 5
exists 0
norm_num
have hra : exists u v. 1 + 7 * u = a + 7 * v
specialize mod_eq_trans 7
specialize mod_eq_trans 1
specialize mod_eq_trans (6 * 6)
specialize mod_eq_trans a
apply mod_eq_trans
exact hc
exact hsq
have hrb : exists h. h + S 1 = 7
exists 5
norm_num
have heq : 1 = a
specialize mod_eq_bounded_unique 7
specialize mod_eq_bounded_unique 1
specialize mod_eq_bounded_unique a
apply mod_eq_bounded_unique
exact hrb
exact ha
exact hra
left
left
right
symm
exact heq
qed

Occurrences

4,033

Distinct objects

3,187

Reused references

33

Depth

91

Cuts

54

Certificate SHA-256

474234f13310…

qres_mod3_zeroQres mod3 zero Quadratic reciprocity campaign checked

The canonical value 0 is a quadratic residue modulo 3.

Prerequisites

none — a root theorem

Used directly by

qres_mod3_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p3_0. exists sm_u_p3_0 sm_v_p3_0. sm_x_p3_0 * sm_x_p3_0 + 3 * sm_u_p3_0 = 0 + 3 * sm_v_p3_0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p3_0. exists sm_u_p3_0 sm_v_p3_0. sm_x_p3_0 * sm_x_p3_0 + 3 * sm_u_p3_0 = 0 + 3 * sm_v_p3_0
exists 0
exists 0
exists 0
norm_num
qed

Occurrences

40

Distinct objects

40

Reused references

0

Depth

14

Cuts

0

Certificate SHA-256

28a067d3acd7…

qres_mod3_oneQres mod3 one Quadratic reciprocity campaign checked

The canonical value 1 is a quadratic residue modulo 3.

Prerequisites

none — a root theorem

Used directly by

qres_mod3_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p3_1. exists sm_u_p3_1 sm_v_p3_1. sm_x_p3_1 * sm_x_p3_1 + 3 * sm_u_p3_1 = 1 + 3 * sm_v_p3_1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p3_1. exists sm_u_p3_1 sm_v_p3_1. sm_x_p3_1 * sm_x_p3_1 + 3 * sm_u_p3_1 = 1 + 3 * sm_v_p3_1
exists 1
exists 0
exists 0
norm_num
qed

Occurrences

57

Distinct objects

57

Reused references

0

Depth

14

Cuts

0

Certificate SHA-256

01a0e94a6526…

qres_mod5_zeroQres mod5 zero Quadratic reciprocity campaign checked

The canonical value 0 is a quadratic residue modulo 5.

Prerequisites

none — a root theorem

Used directly by

qres_mod5_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p5_0. exists sm_u_p5_0 sm_v_p5_0. sm_x_p5_0 * sm_x_p5_0 + 5 * sm_u_p5_0 = 0 + 5 * sm_v_p5_0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p5_0. exists sm_u_p5_0 sm_v_p5_0. sm_x_p5_0 * sm_x_p5_0 + 5 * sm_u_p5_0 = 0 + 5 * sm_v_p5_0
exists 0
exists 0
exists 0
norm_num
qed

Occurrences

44

Distinct objects

44

Reused references

0

Depth

16

Cuts

0

Certificate SHA-256

c81ed7cded0d…

qres_mod5_oneQres mod5 one Quadratic reciprocity campaign checked

The canonical value 1 is a quadratic residue modulo 5.

Prerequisites

none — a root theorem

Used directly by

qres_mod5_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p5_1. exists sm_u_p5_1 sm_v_p5_1. sm_x_p5_1 * sm_x_p5_1 + 5 * sm_u_p5_1 = 1 + 5 * sm_v_p5_1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p5_1. exists sm_u_p5_1 sm_v_p5_1. sm_x_p5_1 * sm_x_p5_1 + 5 * sm_u_p5_1 = 1 + 5 * sm_v_p5_1
exists 1
exists 0
exists 0
norm_num
qed

Occurrences

61

Distinct objects

61

Reused references

0

Depth

16

Cuts

0

Certificate SHA-256

53992204c10b…

qres_mod5_fourQres mod5 four Quadratic reciprocity campaign checked

The canonical value 4 is a quadratic residue modulo 5.

Prerequisites

none — a root theorem

Used directly by

qres_mod5_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p5_4. exists sm_u_p5_4 sm_v_p5_4. sm_x_p5_4 * sm_x_p5_4 + 5 * sm_u_p5_4 = 4 + 5 * sm_v_p5_4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p5_4. exists sm_u_p5_4 sm_v_p5_4. sm_x_p5_4 * sm_x_p5_4 + 5 * sm_u_p5_4 = 4 + 5 * sm_v_p5_4
exists 2
exists 0
exists 0
norm_num
qed

Occurrences

90

Distinct objects

90

Reused references

0

Depth

18

Cuts

0

Certificate SHA-256

108c167d7c36…

qres_mod7_zeroQres mod7 zero Quadratic reciprocity campaign checked

The canonical value 0 is a quadratic residue modulo 7.

Prerequisites

none — a root theorem

Used directly by

qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p7_0. exists sm_u_p7_0 sm_v_p7_0. sm_x_p7_0 * sm_x_p7_0 + 7 * sm_u_p7_0 = 0 + 7 * sm_v_p7_0

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p7_0. exists sm_u_p7_0 sm_v_p7_0. sm_x_p7_0 * sm_x_p7_0 + 7 * sm_u_p7_0 = 0 + 7 * sm_v_p7_0
exists 0
exists 0
exists 0
norm_num
qed

Occurrences

48

Distinct objects

48

Reused references

0

Depth

18

Cuts

0

Certificate SHA-256

9c4e083aa20a…

qres_mod7_oneQres mod7 one Quadratic reciprocity campaign checked

The canonical value 1 is a quadratic residue modulo 7.

Prerequisites

none — a root theorem

Used directly by

qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p7_1. exists sm_u_p7_1 sm_v_p7_1. sm_x_p7_1 * sm_x_p7_1 + 7 * sm_u_p7_1 = 1 + 7 * sm_v_p7_1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p7_1. exists sm_u_p7_1 sm_v_p7_1. sm_x_p7_1 * sm_x_p7_1 + 7 * sm_u_p7_1 = 1 + 7 * sm_v_p7_1
exists 1
exists 0
exists 0
norm_num
qed

Occurrences

65

Distinct objects

65

Reused references

0

Depth

18

Cuts

0

Certificate SHA-256

dd095ced1763…

qres_mod7_twoQres mod7 two Quadratic reciprocity campaign checked

The canonical value 2 is a quadratic residue modulo 7.

Prerequisites

none — a root theorem

Used directly by

qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p7_2. exists sm_u_p7_2 sm_v_p7_2. sm_x_p7_2 * sm_x_p7_2 + 7 * sm_u_p7_2 = 2 + 7 * sm_v_p7_2

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p7_2. exists sm_u_p7_2 sm_v_p7_2. sm_x_p7_2 * sm_x_p7_2 + 7 * sm_u_p7_2 = 2 + 7 * sm_v_p7_2
exists 3
exists 0
exists 1
norm_num
qed

Occurrences

208

Distinct objects

208

Reused references

0

Depth

27

Cuts

0

Certificate SHA-256

09fb90ef9c88…

qres_mod7_fourQres mod7 four Quadratic reciprocity campaign checked

The canonical value 4 is a quadratic residue modulo 7.

Prerequisites

none — a root theorem

Used directly by

qres_mod7_canonical_iff

Exact expanded PA statement

Copy statement

exists sm_x_p7_4. exists sm_u_p7_4 sm_v_p7_4. sm_x_p7_4 * sm_x_p7_4 + 7 * sm_u_p7_4 = 4 + 7 * sm_v_p7_4

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove exists sm_x_p7_4. exists sm_u_p7_4 sm_v_p7_4. sm_x_p7_4 * sm_x_p7_4 + 7 * sm_u_p7_4 = 4 + 7 * sm_v_p7_4
exists 2
exists 0
exists 0
norm_num
qed

Occurrences

94

Distinct objects

94

Reused references

0

Depth

18

Cuts

0

Certificate SHA-256

799fce5f8cc7…

qres_mod3_canonical_iffQres mod3 canonical iff Quadratic reciprocity campaign checked

For canonical values, the displayed list is exactly QRes modulo 3.

Prerequisites

quadratic_residue_bounded_equiv succ_ne_zero bounded_square_mod3_classify qres_mod3_zero qres_mod3_one

Used directly by

not_qres_mod3_two

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 3) -> (((exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3) -> (a = 0 \/ a = 1)) /\ ((a = 0 \/ a = 1) -> (exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 3) -> (((exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3) -> (a = 0 \/ a = 1)) /\ ((a = 0 \/ a = 1) -> (exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3)))
use quadratic_residue_bounded_equiv
use succ_ne_zero
use bounded_square_mod3_classify
use qres_mod3_zero
use qres_mod3_one
intro a
intro ha
split
intro hq
have hequiv : ((exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3) -> (exists sm_x_iffb3. (exists sm_h_iffb3. sm_h_iffb3 + S sm_x_iffb3 = 3) /\ exists sm_u_iffb3 sm_v_iffb3. sm_x_iffb3 * sm_x_iffb3 + 3 * sm_u_iffb3 = a + 3 * sm_v_iffb3)) /\ ((exists sm_x_iffb3. (exists sm_h_iffb3. sm_h_iffb3 + S sm_x_iffb3 = 3) /\ exists sm_u_iffb3 sm_v_iffb3. sm_x_iffb3 * sm_x_iffb3 + 3 * sm_u_iffb3 = a + 3 * sm_v_iffb3) -> (exists sm_x_iffq3. exists sm_u_iffq3 sm_v_iffq3. sm_x_iffq3 * sm_x_iffq3 + 3 * sm_u_iffq3 = a + 3 * sm_v_iffq3))
specialize quadratic_residue_bounded_equiv 3
specialize quadratic_residue_bounded_equiv a
apply quadratic_residue_bounded_equiv
specialize succ_ne_zero 2
exact succ_ne_zero
cases hequiv
have hb : exists sm_x_iffb3. (exists sm_h_iffb3. sm_h_iffb3 + S sm_x_iffb3 = 3) /\ exists sm_u_iffb3 sm_v_iffb3. sm_x_iffb3 * sm_x_iffb3 + 3 * sm_u_iffb3 = a + 3 * sm_v_iffb3
apply hequiv_left
exact hq
specialize bounded_square_mod3_classify a
apply bounded_square_mod3_classify
exact ha
exact hb
intro hcases
cases hcases
rewrite hcases_left
exact qres_mod3_zero
rewrite hcases_right
exact qres_mod3_one
qed

Occurrences

4,050

Distinct objects

1,850

Reused references

81

Depth

65

Cuts

110

Certificate SHA-256

9149885aeda4…

qres_mod5_canonical_iffQres mod5 canonical iff Quadratic reciprocity campaign checked

For canonical values, the displayed list is exactly QRes modulo 5.

Prerequisites

quadratic_residue_bounded_equiv succ_ne_zero bounded_square_mod5_classify qres_mod5_zero qres_mod5_one qres_mod5_four

Used directly by

not_qres_mod5_two not_qres_mod5_three

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 5) -> (((exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5) -> (a = 0 \/ a = 1 \/ a = 4)) /\ ((a = 0 \/ a = 1 \/ a = 4) -> (exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 5) -> (((exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5) -> (a = 0 \/ a = 1 \/ a = 4)) /\ ((a = 0 \/ a = 1 \/ a = 4) -> (exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5)))
use quadratic_residue_bounded_equiv
use succ_ne_zero
use bounded_square_mod5_classify
use qres_mod5_zero
use qres_mod5_one
use qres_mod5_four
intro a
intro ha
split
intro hq
have hequiv : ((exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5) -> (exists sm_x_iffb5. (exists sm_h_iffb5. sm_h_iffb5 + S sm_x_iffb5 = 5) /\ exists sm_u_iffb5 sm_v_iffb5. sm_x_iffb5 * sm_x_iffb5 + 5 * sm_u_iffb5 = a + 5 * sm_v_iffb5)) /\ ((exists sm_x_iffb5. (exists sm_h_iffb5. sm_h_iffb5 + S sm_x_iffb5 = 5) /\ exists sm_u_iffb5 sm_v_iffb5. sm_x_iffb5 * sm_x_iffb5 + 5 * sm_u_iffb5 = a + 5 * sm_v_iffb5) -> (exists sm_x_iffq5. exists sm_u_iffq5 sm_v_iffq5. sm_x_iffq5 * sm_x_iffq5 + 5 * sm_u_iffq5 = a + 5 * sm_v_iffq5))
specialize quadratic_residue_bounded_equiv 5
specialize quadratic_residue_bounded_equiv a
apply quadratic_residue_bounded_equiv
specialize succ_ne_zero 4
exact succ_ne_zero
cases hequiv
have hb : exists sm_x_iffb5. (exists sm_h_iffb5. sm_h_iffb5 + S sm_x_iffb5 = 5) /\ exists sm_u_iffb5 sm_v_iffb5. sm_x_iffb5 * sm_x_iffb5 + 5 * sm_u_iffb5 = a + 5 * sm_v_iffb5
apply hequiv_left
exact hq
specialize bounded_square_mod5_classify a
apply bounded_square_mod5_classify
exact ha
exact hb
intro hcases
cases hcases
cases hcases_left
rewrite hcases_left_left
exact qres_mod5_zero
rewrite hcases_left_right
exact qres_mod5_one
rewrite hcases_right
exact qres_mod5_four
qed

Occurrences

4,955

Distinct objects

2,641

Reused references

83

Depth

65

Cuts

117

Certificate SHA-256

d169f89bd8a7…

qres_mod7_canonical_iffQres mod7 canonical iff Quadratic reciprocity campaign checked

For canonical values, the displayed list is exactly QRes modulo 7.

Prerequisites

quadratic_residue_bounded_equiv succ_ne_zero bounded_square_mod7_classify qres_mod7_zero qres_mod7_one qres_mod7_two qres_mod7_four

Used directly by

not_qres_mod7_three not_qres_mod7_five not_qres_mod7_six

Exact expanded PA statement

Copy statement

forall a. (exists h. h + S a = 7) -> (((exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7) -> (a = 0 \/ a = 1 \/ a = 2 \/ a = 4)) /\ ((a = 0 \/ a = 1 \/ a = 2 \/ a = 4) -> (exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. (exists h. h + S a = 7) -> (((exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7) -> (a = 0 \/ a = 1 \/ a = 2 \/ a = 4)) /\ ((a = 0 \/ a = 1 \/ a = 2 \/ a = 4) -> (exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7)))
use quadratic_residue_bounded_equiv
use succ_ne_zero
use bounded_square_mod7_classify
use qres_mod7_zero
use qres_mod7_one
use qres_mod7_two
use qres_mod7_four
intro a
intro ha
split
intro hq
have hequiv : ((exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7) -> (exists sm_x_iffb7. (exists sm_h_iffb7. sm_h_iffb7 + S sm_x_iffb7 = 7) /\ exists sm_u_iffb7 sm_v_iffb7. sm_x_iffb7 * sm_x_iffb7 + 7 * sm_u_iffb7 = a + 7 * sm_v_iffb7)) /\ ((exists sm_x_iffb7. (exists sm_h_iffb7. sm_h_iffb7 + S sm_x_iffb7 = 7) /\ exists sm_u_iffb7 sm_v_iffb7. sm_x_iffb7 * sm_x_iffb7 + 7 * sm_u_iffb7 = a + 7 * sm_v_iffb7) -> (exists sm_x_iffq7. exists sm_u_iffq7 sm_v_iffq7. sm_x_iffq7 * sm_x_iffq7 + 7 * sm_u_iffq7 = a + 7 * sm_v_iffq7))
specialize quadratic_residue_bounded_equiv 7
specialize quadratic_residue_bounded_equiv a
apply quadratic_residue_bounded_equiv
specialize succ_ne_zero 6
exact succ_ne_zero
cases hequiv
have hb : exists sm_x_iffb7. (exists sm_h_iffb7. sm_h_iffb7 + S sm_x_iffb7 = 7) /\ exists sm_u_iffb7 sm_v_iffb7. sm_x_iffb7 * sm_x_iffb7 + 7 * sm_u_iffb7 = a + 7 * sm_v_iffb7
apply hequiv_left
exact hq
specialize bounded_square_mod7_classify a
apply bounded_square_mod7_classify
exact ha
exact hb
intro hcases
cases hcases
cases hcases_left
cases hcases_left_left
rewrite hcases_left_left_left
exact qres_mod7_zero
rewrite hcases_left_left_right
exact qres_mod7_one
rewrite hcases_left_right
exact qres_mod7_two
rewrite hcases_right
exact qres_mod7_four
qed

Occurrences

6,648

Distinct objects

4,220

Reused references

85

Depth

94

Cuts

124

Certificate SHA-256

dfcdeffddc5d…

not_qres_mod3_twoNot qres mod3 two Quadratic reciprocity campaign checked

The canonical value 2 is not a quadratic residue modulo 3.

Prerequisites

qres_mod3_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n3_2. exists sm_u_n3_2 sm_v_n3_2. sm_x_n3_2 * sm_x_n3_2 + 3 * sm_u_n3_2 = 2 + 3 * sm_v_n3_2)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n3_2. exists sm_u_n3_2 sm_v_n3_2. sm_x_n3_2 * sm_x_n3_2 + 3 * sm_u_n3_2 = 2 + 3 * sm_v_n3_2)
use qres_mod3_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 2 = 3
exists 0
norm_num
have hiff : (((exists sm_x_ni3_2. exists sm_u_ni3_2 sm_v_ni3_2. sm_x_ni3_2 * sm_x_ni3_2 + 3 * sm_u_ni3_2 = 2 + 3 * sm_v_ni3_2) -> (2 = 0 \/ 2 = 1)) /\ ((2 = 0 \/ 2 = 1) -> (exists sm_x_ni3_2. exists sm_u_ni3_2 sm_v_ni3_2. sm_x_ni3_2 * sm_x_ni3_2 + 3 * sm_u_ni3_2 = 2 + 3 * sm_v_ni3_2)))
specialize qres_mod3_canonical_iff 2
apply qres_mod3_canonical_iff
exact ha
cases hiff
have hcases : 2 = 0 \/ 2 = 1
apply hiff_left
exact hq
cases hcases
apply PA1
exact hcases_left
have hne_3_2_1_0 : 1 = 0
specialize succ_injective 1
specialize succ_injective 0
apply succ_injective
exact hcases_right
apply PA1
exact hne_3_2_1_0
qed

Occurrences

4,103

Distinct objects

1,903

Reused references

81

Depth

66

Cuts

112

Certificate SHA-256

ca0da90b4e2e…

not_qres_mod5_twoNot qres mod5 two Quadratic reciprocity campaign checked

The canonical value 2 is not a quadratic residue modulo 5.

Prerequisites

qres_mod5_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n5_2. exists sm_u_n5_2 sm_v_n5_2. sm_x_n5_2 * sm_x_n5_2 + 5 * sm_u_n5_2 = 2 + 5 * sm_v_n5_2)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n5_2. exists sm_u_n5_2 sm_v_n5_2. sm_x_n5_2 * sm_x_n5_2 + 5 * sm_u_n5_2 = 2 + 5 * sm_v_n5_2)
use qres_mod5_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 2 = 5
exists 2
norm_num
have hiff : (((exists sm_x_ni5_2. exists sm_u_ni5_2 sm_v_ni5_2. sm_x_ni5_2 * sm_x_ni5_2 + 5 * sm_u_ni5_2 = 2 + 5 * sm_v_ni5_2) -> (2 = 0 \/ 2 = 1 \/ 2 = 4)) /\ ((2 = 0 \/ 2 = 1 \/ 2 = 4) -> (exists sm_x_ni5_2. exists sm_u_ni5_2 sm_v_ni5_2. sm_x_ni5_2 * sm_x_ni5_2 + 5 * sm_u_ni5_2 = 2 + 5 * sm_v_ni5_2)))
specialize qres_mod5_canonical_iff 2
apply qres_mod5_canonical_iff
exact ha
cases hiff
have hcases : 2 = 0 \/ 2 = 1 \/ 2 = 4
apply hiff_left
exact hq
cases hcases
cases hcases_left
apply PA1
exact hcases_left_left
have hne_5_2_1_0 : 1 = 0
specialize succ_injective 1
specialize succ_injective 0
apply succ_injective
exact hcases_left_right
apply PA1
exact hne_5_2_1_0
have hne_5_2_4_0 : 1 = 3
specialize succ_injective 1
specialize succ_injective 3
apply succ_injective
exact hcases_right
have hne_5_2_4_1 : 0 = 2
specialize succ_injective 0
specialize succ_injective 2
apply succ_injective
exact hne_5_2_4_0
apply PA1
symm
exact hne_5_2_4_1
qed

Occurrences

5,025

Distinct objects

2,711

Reused references

83

Depth

66

Cuts

119

Certificate SHA-256

68ff6335aa92…

not_qres_mod5_threeNot qres mod5 three Quadratic reciprocity campaign checked

The canonical value 3 is not a quadratic residue modulo 5.

Prerequisites

qres_mod5_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n5_3. exists sm_u_n5_3 sm_v_n5_3. sm_x_n5_3 * sm_x_n5_3 + 5 * sm_u_n5_3 = 3 + 5 * sm_v_n5_3)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n5_3. exists sm_u_n5_3 sm_v_n5_3. sm_x_n5_3 * sm_x_n5_3 + 5 * sm_u_n5_3 = 3 + 5 * sm_v_n5_3)
use qres_mod5_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 3 = 5
exists 1
norm_num
have hiff : (((exists sm_x_ni5_3. exists sm_u_ni5_3 sm_v_ni5_3. sm_x_ni5_3 * sm_x_ni5_3 + 5 * sm_u_ni5_3 = 3 + 5 * sm_v_ni5_3) -> (3 = 0 \/ 3 = 1 \/ 3 = 4)) /\ ((3 = 0 \/ 3 = 1 \/ 3 = 4) -> (exists sm_x_ni5_3. exists sm_u_ni5_3 sm_v_ni5_3. sm_x_ni5_3 * sm_x_ni5_3 + 5 * sm_u_ni5_3 = 3 + 5 * sm_v_ni5_3)))
specialize qres_mod5_canonical_iff 3
apply qres_mod5_canonical_iff
exact ha
cases hiff
have hcases : 3 = 0 \/ 3 = 1 \/ 3 = 4
apply hiff_left
exact hq
cases hcases
cases hcases_left
apply PA1
exact hcases_left_left
have hne_5_3_1_0 : 2 = 0
specialize succ_injective 2
specialize succ_injective 0
apply succ_injective
exact hcases_left_right
apply PA1
exact hne_5_3_1_0
have hne_5_3_4_0 : 2 = 3
specialize succ_injective 2
specialize succ_injective 3
apply succ_injective
exact hcases_right
have hne_5_3_4_1 : 1 = 2
specialize succ_injective 1
specialize succ_injective 2
apply succ_injective
exact hne_5_3_4_0
have hne_5_3_4_2 : 0 = 1
specialize succ_injective 0
specialize succ_injective 1
apply succ_injective
exact hne_5_3_4_1
apply PA1
symm
exact hne_5_3_4_2
qed

Occurrences

5,034

Distinct objects

2,720

Reused references

83

Depth

66

Cuts

119

Certificate SHA-256

18527b818202…

not_qres_mod7_threeNot qres mod7 three Quadratic reciprocity campaign checked

The canonical value 3 is not a quadratic residue modulo 7.

Prerequisites

qres_mod7_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n7_3. exists sm_u_n7_3 sm_v_n7_3. sm_x_n7_3 * sm_x_n7_3 + 7 * sm_u_n7_3 = 3 + 7 * sm_v_n7_3)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n7_3. exists sm_u_n7_3 sm_v_n7_3. sm_x_n7_3 * sm_x_n7_3 + 7 * sm_u_n7_3 = 3 + 7 * sm_v_n7_3)
use qres_mod7_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 3 = 7
exists 3
norm_num
have hiff : (((exists sm_x_ni7_3. exists sm_u_ni7_3 sm_v_ni7_3. sm_x_ni7_3 * sm_x_ni7_3 + 7 * sm_u_ni7_3 = 3 + 7 * sm_v_ni7_3) -> (3 = 0 \/ 3 = 1 \/ 3 = 2 \/ 3 = 4)) /\ ((3 = 0 \/ 3 = 1 \/ 3 = 2 \/ 3 = 4) -> (exists sm_x_ni7_3. exists sm_u_ni7_3 sm_v_ni7_3. sm_x_ni7_3 * sm_x_ni7_3 + 7 * sm_u_ni7_3 = 3 + 7 * sm_v_ni7_3)))
specialize qres_mod7_canonical_iff 3
apply qres_mod7_canonical_iff
exact ha
cases hiff
have hcases : 3 = 0 \/ 3 = 1 \/ 3 = 2 \/ 3 = 4
apply hiff_left
exact hq
cases hcases
cases hcases_left
cases hcases_left_left
apply PA1
exact hcases_left_left_left
have hne_7_3_1_0 : 2 = 0
specialize succ_injective 2
specialize succ_injective 0
apply succ_injective
exact hcases_left_left_right
apply PA1
exact hne_7_3_1_0
have hne_7_3_2_0 : 2 = 1
specialize succ_injective 2
specialize succ_injective 1
apply succ_injective
exact hcases_left_right
have hne_7_3_2_1 : 1 = 0
specialize succ_injective 1
specialize succ_injective 0
apply succ_injective
exact hne_7_3_2_0
apply PA1
exact hne_7_3_2_1
have hne_7_3_4_0 : 2 = 3
specialize succ_injective 2
specialize succ_injective 3
apply succ_injective
exact hcases_right
have hne_7_3_4_1 : 1 = 2
specialize succ_injective 1
specialize succ_injective 2
apply succ_injective
exact hne_7_3_4_0
have hne_7_3_4_2 : 0 = 1
specialize succ_injective 0
specialize succ_injective 1
apply succ_injective
exact hne_7_3_4_1
apply PA1
symm
exact hne_7_3_4_2
qed

Occurrences

6,743

Distinct objects

4,315

Reused references

85

Depth

95

Cuts

126

Certificate SHA-256

f745ebc50714…

not_qres_mod7_fiveNot qres mod7 five Quadratic reciprocity campaign checked

The canonical value 5 is not a quadratic residue modulo 7.

Prerequisites

qres_mod7_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n7_5. exists sm_u_n7_5 sm_v_n7_5. sm_x_n7_5 * sm_x_n7_5 + 7 * sm_u_n7_5 = 5 + 7 * sm_v_n7_5)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n7_5. exists sm_u_n7_5 sm_v_n7_5. sm_x_n7_5 * sm_x_n7_5 + 7 * sm_u_n7_5 = 5 + 7 * sm_v_n7_5)
use qres_mod7_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 5 = 7
exists 1
norm_num
have hiff : (((exists sm_x_ni7_5. exists sm_u_ni7_5 sm_v_ni7_5. sm_x_ni7_5 * sm_x_ni7_5 + 7 * sm_u_ni7_5 = 5 + 7 * sm_v_ni7_5) -> (5 = 0 \/ 5 = 1 \/ 5 = 2 \/ 5 = 4)) /\ ((5 = 0 \/ 5 = 1 \/ 5 = 2 \/ 5 = 4) -> (exists sm_x_ni7_5. exists sm_u_ni7_5 sm_v_ni7_5. sm_x_ni7_5 * sm_x_ni7_5 + 7 * sm_u_ni7_5 = 5 + 7 * sm_v_ni7_5)))
specialize qres_mod7_canonical_iff 5
apply qres_mod7_canonical_iff
exact ha
cases hiff
have hcases : 5 = 0 \/ 5 = 1 \/ 5 = 2 \/ 5 = 4
apply hiff_left
exact hq
cases hcases
cases hcases_left
cases hcases_left_left
apply PA1
exact hcases_left_left_left
have hne_7_5_1_0 : 4 = 0
specialize succ_injective 4
specialize succ_injective 0
apply succ_injective
exact hcases_left_left_right
apply PA1
exact hne_7_5_1_0
have hne_7_5_2_0 : 4 = 1
specialize succ_injective 4
specialize succ_injective 1
apply succ_injective
exact hcases_left_right
have hne_7_5_2_1 : 3 = 0
specialize succ_injective 3
specialize succ_injective 0
apply succ_injective
exact hne_7_5_2_0
apply PA1
exact hne_7_5_2_1
have hne_7_5_4_0 : 4 = 3
specialize succ_injective 4
specialize succ_injective 3
apply succ_injective
exact hcases_right
have hne_7_5_4_1 : 3 = 2
specialize succ_injective 3
specialize succ_injective 2
apply succ_injective
exact hne_7_5_4_0
have hne_7_5_4_2 : 2 = 1
specialize succ_injective 2
specialize succ_injective 1
apply succ_injective
exact hne_7_5_4_1
have hne_7_5_4_3 : 1 = 0
specialize succ_injective 1
specialize succ_injective 0
apply succ_injective
exact hne_7_5_4_2
apply PA1
exact hne_7_5_4_3
qed

Occurrences

6,756

Distinct objects

4,328

Reused references

85

Depth

95

Cuts

126

Certificate SHA-256

04d3f09358e5…

not_qres_mod7_sixNot qres mod7 six Quadratic reciprocity campaign checked

The canonical value 6 is not a quadratic residue modulo 7.

Prerequisites

qres_mod7_canonical_iff succ_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

~(exists sm_x_n7_6. exists sm_u_n7_6 sm_v_n7_6. sm_x_n7_6 * sm_x_n7_6 + 7 * sm_u_n7_6 = 6 + 7 * sm_v_n7_6)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove ~(exists sm_x_n7_6. exists sm_u_n7_6 sm_v_n7_6. sm_x_n7_6 * sm_x_n7_6 + 7 * sm_u_n7_6 = 6 + 7 * sm_v_n7_6)
use qres_mod7_canonical_iff
use succ_injective
intro hq
have ha : exists h. h + S 6 = 7
exists 0
norm_num
have hiff : (((exists sm_x_ni7_6. exists sm_u_ni7_6 sm_v_ni7_6. sm_x_ni7_6 * sm_x_ni7_6 + 7 * sm_u_ni7_6 = 6 + 7 * sm_v_ni7_6) -> (6 = 0 \/ 6 = 1 \/ 6 = 2 \/ 6 = 4)) /\ ((6 = 0 \/ 6 = 1 \/ 6 = 2 \/ 6 = 4) -> (exists sm_x_ni7_6. exists sm_u_ni7_6 sm_v_ni7_6. sm_x_ni7_6 * sm_x_ni7_6 + 7 * sm_u_ni7_6 = 6 + 7 * sm_v_ni7_6)))
specialize qres_mod7_canonical_iff 6
apply qres_mod7_canonical_iff
exact ha
cases hiff
have hcases : 6 = 0 \/ 6 = 1 \/ 6 = 2 \/ 6 = 4
apply hiff_left
exact hq
cases hcases
cases hcases_left
cases hcases_left_left
apply PA1
exact hcases_left_left_left
have hne_7_6_1_0 : 5 = 0
specialize succ_injective 5
specialize succ_injective 0
apply succ_injective
exact hcases_left_left_right
apply PA1
exact hne_7_6_1_0
have hne_7_6_2_0 : 5 = 1
specialize succ_injective 5
specialize succ_injective 1
apply succ_injective
exact hcases_left_right
have hne_7_6_2_1 : 4 = 0
specialize succ_injective 4
specialize succ_injective 0
apply succ_injective
exact hne_7_6_2_0
apply PA1
exact hne_7_6_2_1
have hne_7_6_4_0 : 5 = 3
specialize succ_injective 5
specialize succ_injective 3
apply succ_injective
exact hcases_right
have hne_7_6_4_1 : 4 = 2
specialize succ_injective 4
specialize succ_injective 2
apply succ_injective
exact hne_7_6_4_0
have hne_7_6_4_2 : 3 = 1
specialize succ_injective 3
specialize succ_injective 1
apply succ_injective
exact hne_7_6_4_1
have hne_7_6_4_3 : 2 = 0
specialize succ_injective 2
specialize succ_injective 0
apply succ_injective
exact hne_7_6_4_2
apply PA1
exact hne_7_6_4_3
qed

Occurrences

6,761

Distinct objects

4,333

Reused references

85

Depth

95

Cuts

126

Certificate SHA-256

2792d0f8a79a…

pow_two_from_one_successorPow two from one successor Quadratic reciprocity campaign checked

A successor of exponent one gives the relational square.

Prerequisites

pow_successor_decompose pow_one

Used directly by

pow_two

Exact expanded PA statement

Copy statement

forall a o e n. o = 1 -> e = S o -> (exists ff_b_two_carrier ff_c_two_carrier. ((forall ff_i_two_carrier_repeat. (exists ff_lt_two_carrier_repeat_bound. ff_lt_two_carrier_repeat_bound + S ff_i_two_carrier_repeat = e) -> (((exists ff_h_two_carrier_repeat_decoded. ff_h_two_carrier_repeat_decoded + S (a) = S ((S (ff_i_two_carrier_repeat)) * ff_c_two_carrier)) /\ exists ff_q_two_carrier_repeat_decoded. ff_b_two_carrier = ff_q_two_carrier_repeat_decoded * S ((S (ff_i_two_carrier_repeat)) * ff_c_two_carrier) + (a)))) /\ (exists ff_u_two_carrier_product ff_v_two_carrier_product. ((((exists ff_h_two_carrier_product_start. ff_h_two_carrier_product_start + S (1) = S ((S (0)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_start. ff_u_two_carrier_product = ff_q_two_carrier_product_start * S ((S (0)) * ff_v_two_carrier_product) + (1))) /\ ((((exists ff_h_two_carrier_product_terminal. ff_h_two_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_terminal. ff_u_two_carrier_product = ff_q_two_carrier_product_terminal * S ((S (e)) * ff_v_two_carrier_product) + (n))) /\ forall ff_i_two_carrier_product. (exists ff_lt_two_carrier_product_bound. ff_lt_two_carrier_product_bound + S ff_i_two_carrier_product = e) -> exists ff_p_two_carrier_product ff_r_two_carrier_product ff_s_two_carrier_product. ((((exists ff_h_two_carrier_product_factor. ff_h_two_carrier_product_factor + S (ff_p_two_carrier_product) = S ((S (ff_i_two_carrier_product)) * ff_c_two_carrier)) /\ exists ff_q_two_carrier_product_factor. ff_b_two_carrier = ff_q_two_carrier_product_factor * S ((S (ff_i_two_carrier_product)) * ff_c_two_carrier) + (ff_p_two_carrier_product))) /\ ((((exists ff_h_two_carrier_product_partial. ff_h_two_carrier_product_partial + S (ff_r_two_carrier_product) = S ((S (ff_i_two_carrier_product)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_partial. ff_u_two_carrier_product = ff_q_two_carrier_product_partial * S ((S (ff_i_two_carrier_product)) * ff_v_two_carrier_product) + (ff_r_two_carrier_product))) /\ ((((exists ff_h_two_carrier_product_successor. ff_h_two_carrier_product_successor + S (ff_s_two_carrier_product) = S ((S (S ff_i_two_carrier_product)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_successor. ff_u_two_carrier_product = ff_q_two_carrier_product_successor * S ((S (S ff_i_two_carrier_product)) * ff_v_two_carrier_product) + (ff_s_two_carrier_product))) /\ ff_s_two_carrier_product = ff_r_two_carrier_product * ff_p_two_carrier_product)))))))) -> n = a * a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a o e n. o = 1 -> e = S o -> (exists ff_b_two_carrier ff_c_two_carrier. ((forall ff_i_two_carrier_repeat. (exists ff_lt_two_carrier_repeat_bound. ff_lt_two_carrier_repeat_bound + S ff_i_two_carrier_repeat = e) -> (((exists ff_h_two_carrier_repeat_decoded. ff_h_two_carrier_repeat_decoded + S (a) = S ((S (ff_i_two_carrier_repeat)) * ff_c_two_carrier)) /\ exists ff_q_two_carrier_repeat_decoded. ff_b_two_carrier = ff_q_two_carrier_repeat_decoded * S ((S (ff_i_two_carrier_repeat)) * ff_c_two_carrier) + (a)))) /\ (exists ff_u_two_carrier_product ff_v_two_carrier_product. ((((exists ff_h_two_carrier_product_start. ff_h_two_carrier_product_start + S (1) = S ((S (0)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_start. ff_u_two_carrier_product = ff_q_two_carrier_product_start * S ((S (0)) * ff_v_two_carrier_product) + (1))) /\ ((((exists ff_h_two_carrier_product_terminal. ff_h_two_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_terminal. ff_u_two_carrier_product = ff_q_two_carrier_product_terminal * S ((S (e)) * ff_v_two_carrier_product) + (n))) /\ forall ff_i_two_carrier_product. (exists ff_lt_two_carrier_product_bound. ff_lt_two_carrier_product_bound + S ff_i_two_carrier_product = e) -> exists ff_p_two_carrier_product ff_r_two_carrier_product ff_s_two_carrier_product. ((((exists ff_h_two_carrier_product_factor. ff_h_two_carrier_product_factor + S (ff_p_two_carrier_product) = S ((S (ff_i_two_carrier_product)) * ff_c_two_carrier)) /\ exists ff_q_two_carrier_product_factor. ff_b_two_carrier = ff_q_two_carrier_product_factor * S ((S (ff_i_two_carrier_product)) * ff_c_two_carrier) + (ff_p_two_carrier_product))) /\ ((((exists ff_h_two_carrier_product_partial. ff_h_two_carrier_product_partial + S (ff_r_two_carrier_product) = S ((S (ff_i_two_carrier_product)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_partial. ff_u_two_carrier_product = ff_q_two_carrier_product_partial * S ((S (ff_i_two_carrier_product)) * ff_v_two_carrier_product) + (ff_r_two_carrier_product))) /\ ((((exists ff_h_two_carrier_product_successor. ff_h_two_carrier_product_successor + S (ff_s_two_carrier_product) = S ((S (S ff_i_two_carrier_product)) * ff_v_two_carrier_product)) /\ exists ff_q_two_carrier_product_successor. ff_u_two_carrier_product = ff_q_two_carrier_product_successor * S ((S (S ff_i_two_carrier_product)) * ff_v_two_carrier_product) + (ff_s_two_carrier_product))) /\ ff_s_two_carrier_product = ff_r_two_carrier_product * ff_p_two_carrier_product)))))))) -> n = a * a
use pow_successor_decompose
use pow_one
intro a
intro o
intro e
intro n
intro ho
intro he
intro hpow
have hstep : exists r. (exists ff_b_two_predecessor ff_c_two_predecessor. ((forall ff_i_two_predecessor_repeat. (exists ff_lt_two_predecessor_repeat_bound. ff_lt_two_predecessor_repeat_bound + S ff_i_two_predecessor_repeat = o) -> (((exists ff_h_two_predecessor_repeat_decoded. ff_h_two_predecessor_repeat_decoded + S (a) = S ((S (ff_i_two_predecessor_repeat)) * ff_c_two_predecessor)) /\ exists ff_q_two_predecessor_repeat_decoded. ff_b_two_predecessor = ff_q_two_predecessor_repeat_decoded * S ((S (ff_i_two_predecessor_repeat)) * ff_c_two_predecessor) + (a)))) /\ (exists ff_u_two_predecessor_product ff_v_two_predecessor_product. ((((exists ff_h_two_predecessor_product_start. ff_h_two_predecessor_product_start + S (1) = S ((S (0)) * ff_v_two_predecessor_product)) /\ exists ff_q_two_predecessor_product_start. ff_u_two_predecessor_product = ff_q_two_predecessor_product_start * S ((S (0)) * ff_v_two_predecessor_product) + (1))) /\ ((((exists ff_h_two_predecessor_product_terminal. ff_h_two_predecessor_product_terminal + S (r) = S ((S (o)) * ff_v_two_predecessor_product)) /\ exists ff_q_two_predecessor_product_terminal. ff_u_two_predecessor_product = ff_q_two_predecessor_product_terminal * S ((S (o)) * ff_v_two_predecessor_product) + (r))) /\ forall ff_i_two_predecessor_product. (exists ff_lt_two_predecessor_product_bound. ff_lt_two_predecessor_product_bound + S ff_i_two_predecessor_product = o) -> exists ff_p_two_predecessor_product ff_r_two_predecessor_product ff_s_two_predecessor_product. ((((exists ff_h_two_predecessor_product_factor. ff_h_two_predecessor_product_factor + S (ff_p_two_predecessor_product) = S ((S (ff_i_two_predecessor_product)) * ff_c_two_predecessor)) /\ exists ff_q_two_predecessor_product_factor. ff_b_two_predecessor = ff_q_two_predecessor_product_factor * S ((S (ff_i_two_predecessor_product)) * ff_c_two_predecessor) + (ff_p_two_predecessor_product))) /\ ((((exists ff_h_two_predecessor_product_partial. ff_h_two_predecessor_product_partial + S (ff_r_two_predecessor_product) = S ((S (ff_i_two_predecessor_product)) * ff_v_two_predecessor_product)) /\ exists ff_q_two_predecessor_product_partial. ff_u_two_predecessor_product = ff_q_two_predecessor_product_partial * S ((S (ff_i_two_predecessor_product)) * ff_v_two_predecessor_product) + (ff_r_two_predecessor_product))) /\ ((((exists ff_h_two_predecessor_product_successor. ff_h_two_predecessor_product_successor + S (ff_s_two_predecessor_product) = S ((S (S ff_i_two_predecessor_product)) * ff_v_two_predecessor_product)) /\ exists ff_q_two_predecessor_product_successor. ff_u_two_predecessor_product = ff_q_two_predecessor_product_successor * S ((S (S ff_i_two_predecessor_product)) * ff_v_two_predecessor_product) + (ff_s_two_predecessor_product))) /\ ff_s_two_predecessor_product = ff_r_two_predecessor_product * ff_p_two_predecessor_product)))))))) /\ n = r * a
specialize pow_successor_decompose a
specialize pow_successor_decompose o
specialize pow_successor_decompose e
specialize pow_successor_decompose n
apply pow_successor_decompose
exact he
exact hpow
cases hstep
cases hstep_witness
have hr : x = a
specialize pow_one a
specialize pow_one o
specialize pow_one x
apply pow_one
exact ho
exact hstep_witness_left
trans x * a
exact hstep_witness_right
rewrite hr
refl
qed

Occurrences

6,431

Distinct objects

1,106

Reused references

48

Depth

67

Cuts

182

Certificate SHA-256

ebb79888472a…

pow_twoPow two Quadratic reciprocity campaign checked

The relational second power is exactly the square.

Prerequisites

pow_two_from_one_successor

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a e n. e = 2 -> (exists ff_b_two ff_c_two. ((forall ff_i_two_repeat. (exists ff_lt_two_repeat_bound. ff_lt_two_repeat_bound + S ff_i_two_repeat = e) -> (((exists ff_h_two_repeat_decoded. ff_h_two_repeat_decoded + S (a) = S ((S (ff_i_two_repeat)) * ff_c_two)) /\ exists ff_q_two_repeat_decoded. ff_b_two = ff_q_two_repeat_decoded * S ((S (ff_i_two_repeat)) * ff_c_two) + (a)))) /\ (exists ff_u_two_product ff_v_two_product. ((((exists ff_h_two_product_start. ff_h_two_product_start + S (1) = S ((S (0)) * ff_v_two_product)) /\ exists ff_q_two_product_start. ff_u_two_product = ff_q_two_product_start * S ((S (0)) * ff_v_two_product) + (1))) /\ ((((exists ff_h_two_product_terminal. ff_h_two_product_terminal + S (n) = S ((S (e)) * ff_v_two_product)) /\ exists ff_q_two_product_terminal. ff_u_two_product = ff_q_two_product_terminal * S ((S (e)) * ff_v_two_product) + (n))) /\ forall ff_i_two_product. (exists ff_lt_two_product_bound. ff_lt_two_product_bound + S ff_i_two_product = e) -> exists ff_p_two_product ff_r_two_product ff_s_two_product. ((((exists ff_h_two_product_factor. ff_h_two_product_factor + S (ff_p_two_product) = S ((S (ff_i_two_product)) * ff_c_two)) /\ exists ff_q_two_product_factor. ff_b_two = ff_q_two_product_factor * S ((S (ff_i_two_product)) * ff_c_two) + (ff_p_two_product))) /\ ((((exists ff_h_two_product_partial. ff_h_two_product_partial + S (ff_r_two_product) = S ((S (ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_partial. ff_u_two_product = ff_q_two_product_partial * S ((S (ff_i_two_product)) * ff_v_two_product) + (ff_r_two_product))) /\ ((((exists ff_h_two_product_successor. ff_h_two_product_successor + S (ff_s_two_product) = S ((S (S ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_successor. ff_u_two_product = ff_q_two_product_successor * S ((S (S ff_i_two_product)) * ff_v_two_product) + (ff_s_two_product))) /\ ff_s_two_product = ff_r_two_product * ff_p_two_product)))))))) -> n = a * a

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e n. e = 2 -> (exists ff_b_two ff_c_two. ((forall ff_i_two_repeat. (exists ff_lt_two_repeat_bound. ff_lt_two_repeat_bound + S ff_i_two_repeat = e) -> (((exists ff_h_two_repeat_decoded. ff_h_two_repeat_decoded + S (a) = S ((S (ff_i_two_repeat)) * ff_c_two)) /\ exists ff_q_two_repeat_decoded. ff_b_two = ff_q_two_repeat_decoded * S ((S (ff_i_two_repeat)) * ff_c_two) + (a)))) /\ (exists ff_u_two_product ff_v_two_product. ((((exists ff_h_two_product_start. ff_h_two_product_start + S (1) = S ((S (0)) * ff_v_two_product)) /\ exists ff_q_two_product_start. ff_u_two_product = ff_q_two_product_start * S ((S (0)) * ff_v_two_product) + (1))) /\ ((((exists ff_h_two_product_terminal. ff_h_two_product_terminal + S (n) = S ((S (e)) * ff_v_two_product)) /\ exists ff_q_two_product_terminal. ff_u_two_product = ff_q_two_product_terminal * S ((S (e)) * ff_v_two_product) + (n))) /\ forall ff_i_two_product. (exists ff_lt_two_product_bound. ff_lt_two_product_bound + S ff_i_two_product = e) -> exists ff_p_two_product ff_r_two_product ff_s_two_product. ((((exists ff_h_two_product_factor. ff_h_two_product_factor + S (ff_p_two_product) = S ((S (ff_i_two_product)) * ff_c_two)) /\ exists ff_q_two_product_factor. ff_b_two = ff_q_two_product_factor * S ((S (ff_i_two_product)) * ff_c_two) + (ff_p_two_product))) /\ ((((exists ff_h_two_product_partial. ff_h_two_product_partial + S (ff_r_two_product) = S ((S (ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_partial. ff_u_two_product = ff_q_two_product_partial * S ((S (ff_i_two_product)) * ff_v_two_product) + (ff_r_two_product))) /\ ((((exists ff_h_two_product_successor. ff_h_two_product_successor + S (ff_s_two_product) = S ((S (S ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_successor. ff_u_two_product = ff_q_two_product_successor * S ((S (S ff_i_two_product)) * ff_v_two_product) + (ff_s_two_product))) /\ ff_s_two_product = ff_r_two_product * ff_p_two_product)))))))) -> n = a * a
use pow_two_from_one_successor
intro a
intro e
intro n
intro he
intro hpow
specialize pow_two_from_one_successor a
specialize pow_two_from_one_successor 1
specialize pow_two_from_one_successor e
specialize pow_two_from_one_successor n
apply pow_two_from_one_successor
refl
exact he
exact hpow
qed

Occurrences

6,460

Distinct objects

1,135

Reused references

48

Depth

68

Cuts

183

Certificate SHA-256

87df55f94fa5…

pow_addPow add Quadratic reciprocity campaign checked

Relational powers turn addition of exponents into multiplication.

Prerequisites

pow_zero pow_functional pow_successor_decompose mul_one mul_assoc

Used directly by

pow_mul_exp

Exact expanded PA statement

Copy statement

forall a e f s x y z. s = e + f -> (exists ff_b_add_left ff_c_add_left. ((forall ff_i_add_left_repeat. (exists ff_lt_add_left_repeat_bound. ff_lt_add_left_repeat_bound + S ff_i_add_left_repeat = e) -> (((exists ff_h_add_left_repeat_decoded. ff_h_add_left_repeat_decoded + S (a) = S ((S (ff_i_add_left_repeat)) * ff_c_add_left)) /\ exists ff_q_add_left_repeat_decoded. ff_b_add_left = ff_q_add_left_repeat_decoded * S ((S (ff_i_add_left_repeat)) * ff_c_add_left) + (a)))) /\ (exists ff_u_add_left_product ff_v_add_left_product. ((((exists ff_h_add_left_product_start. ff_h_add_left_product_start + S (1) = S ((S (0)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_start. ff_u_add_left_product = ff_q_add_left_product_start * S ((S (0)) * ff_v_add_left_product) + (1))) /\ ((((exists ff_h_add_left_product_terminal. ff_h_add_left_product_terminal + S (x) = S ((S (e)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_terminal. ff_u_add_left_product = ff_q_add_left_product_terminal * S ((S (e)) * ff_v_add_left_product) + (x))) /\ forall ff_i_add_left_product. (exists ff_lt_add_left_product_bound. ff_lt_add_left_product_bound + S ff_i_add_left_product = e) -> exists ff_p_add_left_product ff_r_add_left_product ff_s_add_left_product. ((((exists ff_h_add_left_product_factor. ff_h_add_left_product_factor + S (ff_p_add_left_product) = S ((S (ff_i_add_left_product)) * ff_c_add_left)) /\ exists ff_q_add_left_product_factor. ff_b_add_left = ff_q_add_left_product_factor * S ((S (ff_i_add_left_product)) * ff_c_add_left) + (ff_p_add_left_product))) /\ ((((exists ff_h_add_left_product_partial. ff_h_add_left_product_partial + S (ff_r_add_left_product) = S ((S (ff_i_add_left_product)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_partial. ff_u_add_left_product = ff_q_add_left_product_partial * S ((S (ff_i_add_left_product)) * ff_v_add_left_product) + (ff_r_add_left_product))) /\ ((((exists ff_h_add_left_product_successor. ff_h_add_left_product_successor + S (ff_s_add_left_product) = S ((S (S ff_i_add_left_product)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_successor. ff_u_add_left_product = ff_q_add_left_product_successor * S ((S (S ff_i_add_left_product)) * ff_v_add_left_product) + (ff_s_add_left_product))) /\ ff_s_add_left_product = ff_r_add_left_product * ff_p_add_left_product)))))))) -> (exists ff_b_add_right ff_c_add_right. ((forall ff_i_add_right_repeat. (exists ff_lt_add_right_repeat_bound. ff_lt_add_right_repeat_bound + S ff_i_add_right_repeat = f) -> (((exists ff_h_add_right_repeat_decoded. ff_h_add_right_repeat_decoded + S (a) = S ((S (ff_i_add_right_repeat)) * ff_c_add_right)) /\ exists ff_q_add_right_repeat_decoded. ff_b_add_right = ff_q_add_right_repeat_decoded * S ((S (ff_i_add_right_repeat)) * ff_c_add_right) + (a)))) /\ (exists ff_u_add_right_product ff_v_add_right_product. ((((exists ff_h_add_right_product_start. ff_h_add_right_product_start + S (1) = S ((S (0)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_start. ff_u_add_right_product = ff_q_add_right_product_start * S ((S (0)) * ff_v_add_right_product) + (1))) /\ ((((exists ff_h_add_right_product_terminal. ff_h_add_right_product_terminal + S (y) = S ((S (f)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_terminal. ff_u_add_right_product = ff_q_add_right_product_terminal * S ((S (f)) * ff_v_add_right_product) + (y))) /\ forall ff_i_add_right_product. (exists ff_lt_add_right_product_bound. ff_lt_add_right_product_bound + S ff_i_add_right_product = f) -> exists ff_p_add_right_product ff_r_add_right_product ff_s_add_right_product. ((((exists ff_h_add_right_product_factor. ff_h_add_right_product_factor + S (ff_p_add_right_product) = S ((S (ff_i_add_right_product)) * ff_c_add_right)) /\ exists ff_q_add_right_product_factor. ff_b_add_right = ff_q_add_right_product_factor * S ((S (ff_i_add_right_product)) * ff_c_add_right) + (ff_p_add_right_product))) /\ ((((exists ff_h_add_right_product_partial. ff_h_add_right_product_partial + S (ff_r_add_right_product) = S ((S (ff_i_add_right_product)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_partial. ff_u_add_right_product = ff_q_add_right_product_partial * S ((S (ff_i_add_right_product)) * ff_v_add_right_product) + (ff_r_add_right_product))) /\ ((((exists ff_h_add_right_product_successor. ff_h_add_right_product_successor + S (ff_s_add_right_product) = S ((S (S ff_i_add_right_product)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_successor. ff_u_add_right_product = ff_q_add_right_product_successor * S ((S (S ff_i_add_right_product)) * ff_v_add_right_product) + (ff_s_add_right_product))) /\ ff_s_add_right_product = ff_r_add_right_product * ff_p_add_right_product)))))))) -> (exists ff_b_add_total ff_c_add_total. ((forall ff_i_add_total_repeat. (exists ff_lt_add_total_repeat_bound. ff_lt_add_total_repeat_bound + S ff_i_add_total_repeat = s) -> (((exists ff_h_add_total_repeat_decoded. ff_h_add_total_repeat_decoded + S (a) = S ((S (ff_i_add_total_repeat)) * ff_c_add_total)) /\ exists ff_q_add_total_repeat_decoded. ff_b_add_total = ff_q_add_total_repeat_decoded * S ((S (ff_i_add_total_repeat)) * ff_c_add_total) + (a)))) /\ (exists ff_u_add_total_product ff_v_add_total_product. ((((exists ff_h_add_total_product_start. ff_h_add_total_product_start + S (1) = S ((S (0)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_start. ff_u_add_total_product = ff_q_add_total_product_start * S ((S (0)) * ff_v_add_total_product) + (1))) /\ ((((exists ff_h_add_total_product_terminal. ff_h_add_total_product_terminal + S (z) = S ((S (s)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_terminal. ff_u_add_total_product = ff_q_add_total_product_terminal * S ((S (s)) * ff_v_add_total_product) + (z))) /\ forall ff_i_add_total_product. (exists ff_lt_add_total_product_bound. ff_lt_add_total_product_bound + S ff_i_add_total_product = s) -> exists ff_p_add_total_product ff_r_add_total_product ff_s_add_total_product. ((((exists ff_h_add_total_product_factor. ff_h_add_total_product_factor + S (ff_p_add_total_product) = S ((S (ff_i_add_total_product)) * ff_c_add_total)) /\ exists ff_q_add_total_product_factor. ff_b_add_total = ff_q_add_total_product_factor * S ((S (ff_i_add_total_product)) * ff_c_add_total) + (ff_p_add_total_product))) /\ ((((exists ff_h_add_total_product_partial. ff_h_add_total_product_partial + S (ff_r_add_total_product) = S ((S (ff_i_add_total_product)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_partial. ff_u_add_total_product = ff_q_add_total_product_partial * S ((S (ff_i_add_total_product)) * ff_v_add_total_product) + (ff_r_add_total_product))) /\ ((((exists ff_h_add_total_product_successor. ff_h_add_total_product_successor + S (ff_s_add_total_product) = S ((S (S ff_i_add_total_product)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_successor. ff_u_add_total_product = ff_q_add_total_product_successor * S ((S (S ff_i_add_total_product)) * ff_v_add_total_product) + (ff_s_add_total_product))) /\ ff_s_add_total_product = ff_r_add_total_product * ff_p_add_total_product)))))))) -> z = x * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e f s x y z. s = e + f -> (exists ff_b_add_left ff_c_add_left. ((forall ff_i_add_left_repeat. (exists ff_lt_add_left_repeat_bound. ff_lt_add_left_repeat_bound + S ff_i_add_left_repeat = e) -> (((exists ff_h_add_left_repeat_decoded. ff_h_add_left_repeat_decoded + S (a) = S ((S (ff_i_add_left_repeat)) * ff_c_add_left)) /\ exists ff_q_add_left_repeat_decoded. ff_b_add_left = ff_q_add_left_repeat_decoded * S ((S (ff_i_add_left_repeat)) * ff_c_add_left) + (a)))) /\ (exists ff_u_add_left_product ff_v_add_left_product. ((((exists ff_h_add_left_product_start. ff_h_add_left_product_start + S (1) = S ((S (0)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_start. ff_u_add_left_product = ff_q_add_left_product_start * S ((S (0)) * ff_v_add_left_product) + (1))) /\ ((((exists ff_h_add_left_product_terminal. ff_h_add_left_product_terminal + S (x) = S ((S (e)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_terminal. ff_u_add_left_product = ff_q_add_left_product_terminal * S ((S (e)) * ff_v_add_left_product) + (x))) /\ forall ff_i_add_left_product. (exists ff_lt_add_left_product_bound. ff_lt_add_left_product_bound + S ff_i_add_left_product = e) -> exists ff_p_add_left_product ff_r_add_left_product ff_s_add_left_product. ((((exists ff_h_add_left_product_factor. ff_h_add_left_product_factor + S (ff_p_add_left_product) = S ((S (ff_i_add_left_product)) * ff_c_add_left)) /\ exists ff_q_add_left_product_factor. ff_b_add_left = ff_q_add_left_product_factor * S ((S (ff_i_add_left_product)) * ff_c_add_left) + (ff_p_add_left_product))) /\ ((((exists ff_h_add_left_product_partial. ff_h_add_left_product_partial + S (ff_r_add_left_product) = S ((S (ff_i_add_left_product)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_partial. ff_u_add_left_product = ff_q_add_left_product_partial * S ((S (ff_i_add_left_product)) * ff_v_add_left_product) + (ff_r_add_left_product))) /\ ((((exists ff_h_add_left_product_successor. ff_h_add_left_product_successor + S (ff_s_add_left_product) = S ((S (S ff_i_add_left_product)) * ff_v_add_left_product)) /\ exists ff_q_add_left_product_successor. ff_u_add_left_product = ff_q_add_left_product_successor * S ((S (S ff_i_add_left_product)) * ff_v_add_left_product) + (ff_s_add_left_product))) /\ ff_s_add_left_product = ff_r_add_left_product * ff_p_add_left_product)))))))) -> (exists ff_b_add_right ff_c_add_right. ((forall ff_i_add_right_repeat. (exists ff_lt_add_right_repeat_bound. ff_lt_add_right_repeat_bound + S ff_i_add_right_repeat = f) -> (((exists ff_h_add_right_repeat_decoded. ff_h_add_right_repeat_decoded + S (a) = S ((S (ff_i_add_right_repeat)) * ff_c_add_right)) /\ exists ff_q_add_right_repeat_decoded. ff_b_add_right = ff_q_add_right_repeat_decoded * S ((S (ff_i_add_right_repeat)) * ff_c_add_right) + (a)))) /\ (exists ff_u_add_right_product ff_v_add_right_product. ((((exists ff_h_add_right_product_start. ff_h_add_right_product_start + S (1) = S ((S (0)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_start. ff_u_add_right_product = ff_q_add_right_product_start * S ((S (0)) * ff_v_add_right_product) + (1))) /\ ((((exists ff_h_add_right_product_terminal. ff_h_add_right_product_terminal + S (y) = S ((S (f)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_terminal. ff_u_add_right_product = ff_q_add_right_product_terminal * S ((S (f)) * ff_v_add_right_product) + (y))) /\ forall ff_i_add_right_product. (exists ff_lt_add_right_product_bound. ff_lt_add_right_product_bound + S ff_i_add_right_product = f) -> exists ff_p_add_right_product ff_r_add_right_product ff_s_add_right_product. ((((exists ff_h_add_right_product_factor. ff_h_add_right_product_factor + S (ff_p_add_right_product) = S ((S (ff_i_add_right_product)) * ff_c_add_right)) /\ exists ff_q_add_right_product_factor. ff_b_add_right = ff_q_add_right_product_factor * S ((S (ff_i_add_right_product)) * ff_c_add_right) + (ff_p_add_right_product))) /\ ((((exists ff_h_add_right_product_partial. ff_h_add_right_product_partial + S (ff_r_add_right_product) = S ((S (ff_i_add_right_product)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_partial. ff_u_add_right_product = ff_q_add_right_product_partial * S ((S (ff_i_add_right_product)) * ff_v_add_right_product) + (ff_r_add_right_product))) /\ ((((exists ff_h_add_right_product_successor. ff_h_add_right_product_successor + S (ff_s_add_right_product) = S ((S (S ff_i_add_right_product)) * ff_v_add_right_product)) /\ exists ff_q_add_right_product_successor. ff_u_add_right_product = ff_q_add_right_product_successor * S ((S (S ff_i_add_right_product)) * ff_v_add_right_product) + (ff_s_add_right_product))) /\ ff_s_add_right_product = ff_r_add_right_product * ff_p_add_right_product)))))))) -> (exists ff_b_add_total ff_c_add_total. ((forall ff_i_add_total_repeat. (exists ff_lt_add_total_repeat_bound. ff_lt_add_total_repeat_bound + S ff_i_add_total_repeat = s) -> (((exists ff_h_add_total_repeat_decoded. ff_h_add_total_repeat_decoded + S (a) = S ((S (ff_i_add_total_repeat)) * ff_c_add_total)) /\ exists ff_q_add_total_repeat_decoded. ff_b_add_total = ff_q_add_total_repeat_decoded * S ((S (ff_i_add_total_repeat)) * ff_c_add_total) + (a)))) /\ (exists ff_u_add_total_product ff_v_add_total_product. ((((exists ff_h_add_total_product_start. ff_h_add_total_product_start + S (1) = S ((S (0)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_start. ff_u_add_total_product = ff_q_add_total_product_start * S ((S (0)) * ff_v_add_total_product) + (1))) /\ ((((exists ff_h_add_total_product_terminal. ff_h_add_total_product_terminal + S (z) = S ((S (s)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_terminal. ff_u_add_total_product = ff_q_add_total_product_terminal * S ((S (s)) * ff_v_add_total_product) + (z))) /\ forall ff_i_add_total_product. (exists ff_lt_add_total_product_bound. ff_lt_add_total_product_bound + S ff_i_add_total_product = s) -> exists ff_p_add_total_product ff_r_add_total_product ff_s_add_total_product. ((((exists ff_h_add_total_product_factor. ff_h_add_total_product_factor + S (ff_p_add_total_product) = S ((S (ff_i_add_total_product)) * ff_c_add_total)) /\ exists ff_q_add_total_product_factor. ff_b_add_total = ff_q_add_total_product_factor * S ((S (ff_i_add_total_product)) * ff_c_add_total) + (ff_p_add_total_product))) /\ ((((exists ff_h_add_total_product_partial. ff_h_add_total_product_partial + S (ff_r_add_total_product) = S ((S (ff_i_add_total_product)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_partial. ff_u_add_total_product = ff_q_add_total_product_partial * S ((S (ff_i_add_total_product)) * ff_v_add_total_product) + (ff_r_add_total_product))) /\ ((((exists ff_h_add_total_product_successor. ff_h_add_total_product_successor + S (ff_s_add_total_product) = S ((S (S ff_i_add_total_product)) * ff_v_add_total_product)) /\ exists ff_q_add_total_product_successor. ff_u_add_total_product = ff_q_add_total_product_successor * S ((S (S ff_i_add_total_product)) * ff_v_add_total_product) + (ff_s_add_total_product))) /\ ff_s_add_total_product = ff_r_add_total_product * ff_p_add_total_product)))))))) -> z = x * y
use pow_zero
use pow_functional
use pow_successor_decompose
use mul_one
use mul_assoc
intro a
intro e
induction f
intro s
intro x
intro y
intro z
intro hs
intro hx
intro hy
intro hz
rewrite PA3 at hs
rewrite hs at hz
rewrite hs at hz
rewrite hs at hz
rewrite hs at hz
have hzx : z = x
specialize pow_functional a
specialize pow_functional e
specialize pow_functional z
specialize pow_functional x
apply pow_functional
exact hz
exact hx
have hy1 : y = 1
specialize pow_zero a
specialize pow_zero 0
specialize pow_zero y
apply pow_zero
refl
exact hy
rewrite hzx
rewrite hy1
specialize mul_one x
symm
exact mul_one
intro s
intro x
intro y
intro z
intro hs
intro hx
intro hy
intro hz
have hy_step : exists r. (exists ff_b_add_y_prefix ff_c_add_y_prefix. ((forall ff_i_add_y_prefix_repeat. (exists ff_lt_add_y_prefix_repeat_bound. ff_lt_add_y_prefix_repeat_bound + S ff_i_add_y_prefix_repeat = f) -> (((exists ff_h_add_y_prefix_repeat_decoded. ff_h_add_y_prefix_repeat_decoded + S (a) = S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix)) /\ exists ff_q_add_y_prefix_repeat_decoded. ff_b_add_y_prefix = ff_q_add_y_prefix_repeat_decoded * S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix) + (a)))) /\ (exists ff_u_add_y_prefix_product ff_v_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_start. ff_h_add_y_prefix_product_start + S (1) = S ((S (0)) * ff_v_add_y_prefix_product)) /\ exists ff_q_add_y_prefix_product_start. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_start * S ((S (0)) * ff_v_add_y_prefix_product) + (1))) /\ ((((exists ff_h_add_y_prefix_product_terminal. ff_h_add_y_prefix_product_terminal + S (r) = S ((S (f)) * ff_v_add_y_prefix_product)) /\ exists ff_q_add_y_prefix_product_terminal. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_terminal * S ((S (f)) * ff_v_add_y_prefix_product) + (r))) /\ forall ff_i_add_y_prefix_product. (exists ff_lt_add_y_prefix_product_bound. ff_lt_add_y_prefix_product_bound + S ff_i_add_y_prefix_product = f) -> exists ff_p_add_y_prefix_product ff_r_add_y_prefix_product ff_s_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_factor. ff_h_add_y_prefix_product_factor + S (ff_p_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix)) /\ exists ff_q_add_y_prefix_product_factor. ff_b_add_y_prefix = ff_q_add_y_prefix_product_factor * S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix) + (ff_p_add_y_prefix_product))) /\ ((((exists ff_h_add_y_prefix_product_partial. ff_h_add_y_prefix_product_partial + S (ff_r_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\ exists ff_q_add_y_prefix_product_partial. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_partial * S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_r_add_y_prefix_product))) /\ ((((exists ff_h_add_y_prefix_product_successor. ff_h_add_y_prefix_product_successor + S (ff_s_add_y_prefix_product) = S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\ exists ff_q_add_y_prefix_product_successor. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_successor * S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_s_add_y_prefix_product))) /\ ff_s_add_y_prefix_product = ff_r_add_y_prefix_product * ff_p_add_y_prefix_product)))))))) /\ y = r * a
specialize pow_successor_decompose a
specialize pow_successor_decompose f
specialize pow_successor_decompose (S f)
specialize pow_successor_decompose y
apply pow_successor_decompose
refl
exact hy
cases hy_step
cases hy_step_witness
have hst : s = S (e + f)
trans e + S f
exact hs
apply PA4
have hz_step : exists r. (exists pa_b_add_z_prefix pa_c_add_z_prefix. ((forall pa_i_add_z_prefix_repeat. (exists pa_lt_add_z_prefix_repeat_bound. pa_lt_add_z_prefix_repeat_bound + S pa_i_add_z_prefix_repeat = e + f) -> (((exists pa_h_add_z_prefix_repeat_decoded. pa_h_add_z_prefix_repeat_decoded + S (a) = S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix)) /\ exists pa_q_add_z_prefix_repeat_decoded. pa_b_add_z_prefix = pa_q_add_z_prefix_repeat_decoded * S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix) + (a)))) /\ (exists pa_u_add_z_prefix_product pa_v_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_start. pa_h_add_z_prefix_product_start + S (1) = S ((S (0)) * pa_v_add_z_prefix_product)) /\ exists pa_q_add_z_prefix_product_start. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_start * S ((S (0)) * pa_v_add_z_prefix_product) + (1))) /\ ((((exists pa_h_add_z_prefix_product_terminal. pa_h_add_z_prefix_product_terminal + S (r) = S ((S (e + f)) * pa_v_add_z_prefix_product)) /\ exists pa_q_add_z_prefix_product_terminal. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_terminal * S ((S (e + f)) * pa_v_add_z_prefix_product) + (r))) /\ forall pa_i_add_z_prefix_product. (exists pa_lt_add_z_prefix_product_bound. pa_lt_add_z_prefix_product_bound + S pa_i_add_z_prefix_product = e + f) -> exists pa_p_add_z_prefix_product pa_r_add_z_prefix_product pa_s_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_factor. pa_h_add_z_prefix_product_factor + S (pa_p_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix)) /\ exists pa_q_add_z_prefix_product_factor. pa_b_add_z_prefix = pa_q_add_z_prefix_product_factor * S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix) + (pa_p_add_z_prefix_product))) /\ ((((exists pa_h_add_z_prefix_product_partial. pa_h_add_z_prefix_product_partial + S (pa_r_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\ exists pa_q_add_z_prefix_product_partial. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_partial * S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_r_add_z_prefix_product))) /\ ((((exists pa_h_add_z_prefix_product_successor. pa_h_add_z_prefix_product_successor + S (pa_s_add_z_prefix_product) = S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\ exists pa_q_add_z_prefix_product_successor. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_successor * S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_s_add_z_prefix_product))) /\ pa_s_add_z_prefix_product = pa_r_add_z_prefix_product * pa_p_add_z_prefix_product)))))))) /\ z = r * a
specialize pow_successor_decompose a
specialize pow_successor_decompose (e + f)
specialize pow_successor_decompose s
specialize pow_successor_decompose z
apply pow_successor_decompose
exact hst
exact hz
cases hz_step
cases hz_step_witness
have hprefix : x2 = x * x1
specialize IH (e + f)
specialize IH x
specialize IH x1
specialize IH x2
apply IH
refl
exact hx
exact hy_step_witness_left
exact hz_step_witness_left
trans x2 * a
exact hz_step_witness_right
trans (x * x1) * a
congr
exact hprefix
refl
trans x * (x1 * a)
apply mul_assoc
congr
refl
symm
exact hy_step_witness_right
qed

Occurrences

6,744

Distinct objects

1,530

Reused references

59

Depth

66

Cuts

183

Certificate SHA-256

e7260f892c41…

pow_mul_expPow mul exp Quadratic reciprocity campaign checked

Iterated relational powers multiply their exponents.

Prerequisites

pow_zero pow_successor_decompose pow_exists pow_add

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a e f p x y z. p = e * f -> (exists ff_b_mul_base ff_c_mul_base. ((forall ff_i_mul_base_repeat. (exists ff_lt_mul_base_repeat_bound. ff_lt_mul_base_repeat_bound + S ff_i_mul_base_repeat = e) -> (((exists ff_h_mul_base_repeat_decoded. ff_h_mul_base_repeat_decoded + S (a) = S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base)) /\ exists ff_q_mul_base_repeat_decoded. ff_b_mul_base = ff_q_mul_base_repeat_decoded * S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base) + (a)))) /\ (exists ff_u_mul_base_product ff_v_mul_base_product. ((((exists ff_h_mul_base_product_start. ff_h_mul_base_product_start + S (1) = S ((S (0)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_start. ff_u_mul_base_product = ff_q_mul_base_product_start * S ((S (0)) * ff_v_mul_base_product) + (1))) /\ ((((exists ff_h_mul_base_product_terminal. ff_h_mul_base_product_terminal + S (x) = S ((S (e)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_terminal. ff_u_mul_base_product = ff_q_mul_base_product_terminal * S ((S (e)) * ff_v_mul_base_product) + (x))) /\ forall ff_i_mul_base_product. (exists ff_lt_mul_base_product_bound. ff_lt_mul_base_product_bound + S ff_i_mul_base_product = e) -> exists ff_p_mul_base_product ff_r_mul_base_product ff_s_mul_base_product. ((((exists ff_h_mul_base_product_factor. ff_h_mul_base_product_factor + S (ff_p_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_c_mul_base)) /\ exists ff_q_mul_base_product_factor. ff_b_mul_base = ff_q_mul_base_product_factor * S ((S (ff_i_mul_base_product)) * ff_c_mul_base) + (ff_p_mul_base_product))) /\ ((((exists ff_h_mul_base_product_partial. ff_h_mul_base_product_partial + S (ff_r_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_partial. ff_u_mul_base_product = ff_q_mul_base_product_partial * S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_r_mul_base_product))) /\ ((((exists ff_h_mul_base_product_successor. ff_h_mul_base_product_successor + S (ff_s_mul_base_product) = S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_successor. ff_u_mul_base_product = ff_q_mul_base_product_successor * S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_s_mul_base_product))) /\ ff_s_mul_base_product = ff_r_mul_base_product * ff_p_mul_base_product)))))))) -> (exists ff_b_mul_outer ff_c_mul_outer. ((forall ff_i_mul_outer_repeat. (exists ff_lt_mul_outer_repeat_bound. ff_lt_mul_outer_repeat_bound + S ff_i_mul_outer_repeat = f) -> (((exists ff_h_mul_outer_repeat_decoded. ff_h_mul_outer_repeat_decoded + S (x) = S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_repeat_decoded. ff_b_mul_outer = ff_q_mul_outer_repeat_decoded * S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer) + (x)))) /\ (exists ff_u_mul_outer_product ff_v_mul_outer_product. ((((exists ff_h_mul_outer_product_start. ff_h_mul_outer_product_start + S (1) = S ((S (0)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_start. ff_u_mul_outer_product = ff_q_mul_outer_product_start * S ((S (0)) * ff_v_mul_outer_product) + (1))) /\ ((((exists ff_h_mul_outer_product_terminal. ff_h_mul_outer_product_terminal + S (y) = S ((S (f)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_terminal. ff_u_mul_outer_product = ff_q_mul_outer_product_terminal * S ((S (f)) * ff_v_mul_outer_product) + (y))) /\ forall ff_i_mul_outer_product. (exists ff_lt_mul_outer_product_bound. ff_lt_mul_outer_product_bound + S ff_i_mul_outer_product = f) -> exists ff_p_mul_outer_product ff_r_mul_outer_product ff_s_mul_outer_product. ((((exists ff_h_mul_outer_product_factor. ff_h_mul_outer_product_factor + S (ff_p_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_product_factor. ff_b_mul_outer = ff_q_mul_outer_product_factor * S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer) + (ff_p_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_partial. ff_h_mul_outer_product_partial + S (ff_r_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_partial. ff_u_mul_outer_product = ff_q_mul_outer_product_partial * S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_r_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_successor. ff_h_mul_outer_product_successor + S (ff_s_mul_outer_product) = S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_successor. ff_u_mul_outer_product = ff_q_mul_outer_product_successor * S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_s_mul_outer_product))) /\ ff_s_mul_outer_product = ff_r_mul_outer_product * ff_p_mul_outer_product)))))))) -> (exists ff_b_mul_total ff_c_mul_total. ((forall ff_i_mul_total_repeat. (exists ff_lt_mul_total_repeat_bound. ff_lt_mul_total_repeat_bound + S ff_i_mul_total_repeat = p) -> (((exists ff_h_mul_total_repeat_decoded. ff_h_mul_total_repeat_decoded + S (a) = S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total)) /\ exists ff_q_mul_total_repeat_decoded. ff_b_mul_total = ff_q_mul_total_repeat_decoded * S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total) + (a)))) /\ (exists ff_u_mul_total_product ff_v_mul_total_product. ((((exists ff_h_mul_total_product_start. ff_h_mul_total_product_start + S (1) = S ((S (0)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_start. ff_u_mul_total_product = ff_q_mul_total_product_start * S ((S (0)) * ff_v_mul_total_product) + (1))) /\ ((((exists ff_h_mul_total_product_terminal. ff_h_mul_total_product_terminal + S (z) = S ((S (p)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_terminal. ff_u_mul_total_product = ff_q_mul_total_product_terminal * S ((S (p)) * ff_v_mul_total_product) + (z))) /\ forall ff_i_mul_total_product. (exists ff_lt_mul_total_product_bound. ff_lt_mul_total_product_bound + S ff_i_mul_total_product = p) -> exists ff_p_mul_total_product ff_r_mul_total_product ff_s_mul_total_product. ((((exists ff_h_mul_total_product_factor. ff_h_mul_total_product_factor + S (ff_p_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_c_mul_total)) /\ exists ff_q_mul_total_product_factor. ff_b_mul_total = ff_q_mul_total_product_factor * S ((S (ff_i_mul_total_product)) * ff_c_mul_total) + (ff_p_mul_total_product))) /\ ((((exists ff_h_mul_total_product_partial. ff_h_mul_total_product_partial + S (ff_r_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_partial. ff_u_mul_total_product = ff_q_mul_total_product_partial * S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_r_mul_total_product))) /\ ((((exists ff_h_mul_total_product_successor. ff_h_mul_total_product_successor + S (ff_s_mul_total_product) = S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_successor. ff_u_mul_total_product = ff_q_mul_total_product_successor * S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_s_mul_total_product))) /\ ff_s_mul_total_product = ff_r_mul_total_product * ff_p_mul_total_product)))))))) -> y = z

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a e f p x y z. p = e * f -> (exists ff_b_mul_base ff_c_mul_base. ((forall ff_i_mul_base_repeat. (exists ff_lt_mul_base_repeat_bound. ff_lt_mul_base_repeat_bound + S ff_i_mul_base_repeat = e) -> (((exists ff_h_mul_base_repeat_decoded. ff_h_mul_base_repeat_decoded + S (a) = S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base)) /\ exists ff_q_mul_base_repeat_decoded. ff_b_mul_base = ff_q_mul_base_repeat_decoded * S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base) + (a)))) /\ (exists ff_u_mul_base_product ff_v_mul_base_product. ((((exists ff_h_mul_base_product_start. ff_h_mul_base_product_start + S (1) = S ((S (0)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_start. ff_u_mul_base_product = ff_q_mul_base_product_start * S ((S (0)) * ff_v_mul_base_product) + (1))) /\ ((((exists ff_h_mul_base_product_terminal. ff_h_mul_base_product_terminal + S (x) = S ((S (e)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_terminal. ff_u_mul_base_product = ff_q_mul_base_product_terminal * S ((S (e)) * ff_v_mul_base_product) + (x))) /\ forall ff_i_mul_base_product. (exists ff_lt_mul_base_product_bound. ff_lt_mul_base_product_bound + S ff_i_mul_base_product = e) -> exists ff_p_mul_base_product ff_r_mul_base_product ff_s_mul_base_product. ((((exists ff_h_mul_base_product_factor. ff_h_mul_base_product_factor + S (ff_p_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_c_mul_base)) /\ exists ff_q_mul_base_product_factor. ff_b_mul_base = ff_q_mul_base_product_factor * S ((S (ff_i_mul_base_product)) * ff_c_mul_base) + (ff_p_mul_base_product))) /\ ((((exists ff_h_mul_base_product_partial. ff_h_mul_base_product_partial + S (ff_r_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_partial. ff_u_mul_base_product = ff_q_mul_base_product_partial * S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_r_mul_base_product))) /\ ((((exists ff_h_mul_base_product_successor. ff_h_mul_base_product_successor + S (ff_s_mul_base_product) = S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_successor. ff_u_mul_base_product = ff_q_mul_base_product_successor * S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_s_mul_base_product))) /\ ff_s_mul_base_product = ff_r_mul_base_product * ff_p_mul_base_product)))))))) -> (exists ff_b_mul_outer ff_c_mul_outer. ((forall ff_i_mul_outer_repeat. (exists ff_lt_mul_outer_repeat_bound. ff_lt_mul_outer_repeat_bound + S ff_i_mul_outer_repeat = f) -> (((exists ff_h_mul_outer_repeat_decoded. ff_h_mul_outer_repeat_decoded + S (x) = S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_repeat_decoded. ff_b_mul_outer = ff_q_mul_outer_repeat_decoded * S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer) + (x)))) /\ (exists ff_u_mul_outer_product ff_v_mul_outer_product. ((((exists ff_h_mul_outer_product_start. ff_h_mul_outer_product_start + S (1) = S ((S (0)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_start. ff_u_mul_outer_product = ff_q_mul_outer_product_start * S ((S (0)) * ff_v_mul_outer_product) + (1))) /\ ((((exists ff_h_mul_outer_product_terminal. ff_h_mul_outer_product_terminal + S (y) = S ((S (f)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_terminal. ff_u_mul_outer_product = ff_q_mul_outer_product_terminal * S ((S (f)) * ff_v_mul_outer_product) + (y))) /\ forall ff_i_mul_outer_product. (exists ff_lt_mul_outer_product_bound. ff_lt_mul_outer_product_bound + S ff_i_mul_outer_product = f) -> exists ff_p_mul_outer_product ff_r_mul_outer_product ff_s_mul_outer_product. ((((exists ff_h_mul_outer_product_factor. ff_h_mul_outer_product_factor + S (ff_p_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_product_factor. ff_b_mul_outer = ff_q_mul_outer_product_factor * S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer) + (ff_p_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_partial. ff_h_mul_outer_product_partial + S (ff_r_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_partial. ff_u_mul_outer_product = ff_q_mul_outer_product_partial * S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_r_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_successor. ff_h_mul_outer_product_successor + S (ff_s_mul_outer_product) = S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_successor. ff_u_mul_outer_product = ff_q_mul_outer_product_successor * S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_s_mul_outer_product))) /\ ff_s_mul_outer_product = ff_r_mul_outer_product * ff_p_mul_outer_product)))))))) -> (exists ff_b_mul_total ff_c_mul_total. ((forall ff_i_mul_total_repeat. (exists ff_lt_mul_total_repeat_bound. ff_lt_mul_total_repeat_bound + S ff_i_mul_total_repeat = p) -> (((exists ff_h_mul_total_repeat_decoded. ff_h_mul_total_repeat_decoded + S (a) = S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total)) /\ exists ff_q_mul_total_repeat_decoded. ff_b_mul_total = ff_q_mul_total_repeat_decoded * S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total) + (a)))) /\ (exists ff_u_mul_total_product ff_v_mul_total_product. ((((exists ff_h_mul_total_product_start. ff_h_mul_total_product_start + S (1) = S ((S (0)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_start. ff_u_mul_total_product = ff_q_mul_total_product_start * S ((S (0)) * ff_v_mul_total_product) + (1))) /\ ((((exists ff_h_mul_total_product_terminal. ff_h_mul_total_product_terminal + S (z) = S ((S (p)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_terminal. ff_u_mul_total_product = ff_q_mul_total_product_terminal * S ((S (p)) * ff_v_mul_total_product) + (z))) /\ forall ff_i_mul_total_product. (exists ff_lt_mul_total_product_bound. ff_lt_mul_total_product_bound + S ff_i_mul_total_product = p) -> exists ff_p_mul_total_product ff_r_mul_total_product ff_s_mul_total_product. ((((exists ff_h_mul_total_product_factor. ff_h_mul_total_product_factor + S (ff_p_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_c_mul_total)) /\ exists ff_q_mul_total_product_factor. ff_b_mul_total = ff_q_mul_total_product_factor * S ((S (ff_i_mul_total_product)) * ff_c_mul_total) + (ff_p_mul_total_product))) /\ ((((exists ff_h_mul_total_product_partial. ff_h_mul_total_product_partial + S (ff_r_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_partial. ff_u_mul_total_product = ff_q_mul_total_product_partial * S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_r_mul_total_product))) /\ ((((exists ff_h_mul_total_product_successor. ff_h_mul_total_product_successor + S (ff_s_mul_total_product) = S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_successor. ff_u_mul_total_product = ff_q_mul_total_product_successor * S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_s_mul_total_product))) /\ ff_s_mul_total_product = ff_r_mul_total_product * ff_p_mul_total_product)))))))) -> y = z
use pow_zero
use pow_successor_decompose
use pow_exists
use pow_add
intro a
intro e
induction f
intro p
intro x
intro y
intro z
intro hp
intro hx
intro hy
intro hz
rewrite PA5 at hp
rewrite hp at hz
rewrite hp at hz
rewrite hp at hz
rewrite hp at hz
have hy1 : y = 1
specialize pow_zero x
specialize pow_zero 0
specialize pow_zero y
apply pow_zero
refl
exact hy
have hz1 : z = 1
specialize pow_zero a
specialize pow_zero 0
specialize pow_zero z
apply pow_zero
refl
exact hz
trans 1
exact hy1
symm
exact hz1
intro p
intro x
intro y
intro z
intro hp
intro hx
intro hy
intro hz
have hy_step : exists r. (exists ff_b_mul_y_prefix ff_c_mul_y_prefix. ((forall ff_i_mul_y_prefix_repeat. (exists ff_lt_mul_y_prefix_repeat_bound. ff_lt_mul_y_prefix_repeat_bound + S ff_i_mul_y_prefix_repeat = f) -> (((exists ff_h_mul_y_prefix_repeat_decoded. ff_h_mul_y_prefix_repeat_decoded + S (x) = S ((S (ff_i_mul_y_prefix_repeat)) * ff_c_mul_y_prefix)) /\ exists ff_q_mul_y_prefix_repeat_decoded. ff_b_mul_y_prefix = ff_q_mul_y_prefix_repeat_decoded * S ((S (ff_i_mul_y_prefix_repeat)) * ff_c_mul_y_prefix) + (x)))) /\ (exists ff_u_mul_y_prefix_product ff_v_mul_y_prefix_product. ((((exists ff_h_mul_y_prefix_product_start. ff_h_mul_y_prefix_product_start + S (1) = S ((S (0)) * ff_v_mul_y_prefix_product)) /\ exists ff_q_mul_y_prefix_product_start. ff_u_mul_y_prefix_product = ff_q_mul_y_prefix_product_start * S ((S (0)) * ff_v_mul_y_prefix_product) + (1))) /\ ((((exists ff_h_mul_y_prefix_product_terminal. ff_h_mul_y_prefix_product_terminal + S (r) = S ((S (f)) * ff_v_mul_y_prefix_product)) /\ exists ff_q_mul_y_prefix_product_terminal. ff_u_mul_y_prefix_product = ff_q_mul_y_prefix_product_terminal * S ((S (f)) * ff_v_mul_y_prefix_product) + (r))) /\ forall ff_i_mul_y_prefix_product. (exists ff_lt_mul_y_prefix_product_bound. ff_lt_mul_y_prefix_product_bound + S ff_i_mul_y_prefix_product = f) -> exists ff_p_mul_y_prefix_product ff_r_mul_y_prefix_product ff_s_mul_y_prefix_product. ((((exists ff_h_mul_y_prefix_product_factor. ff_h_mul_y_prefix_product_factor + S (ff_p_mul_y_prefix_product) = S ((S (ff_i_mul_y_prefix_product)) * ff_c_mul_y_prefix)) /\ exists ff_q_mul_y_prefix_product_factor. ff_b_mul_y_prefix = ff_q_mul_y_prefix_product_factor * S ((S (ff_i_mul_y_prefix_product)) * ff_c_mul_y_prefix) + (ff_p_mul_y_prefix_product))) /\ ((((exists ff_h_mul_y_prefix_product_partial. ff_h_mul_y_prefix_product_partial + S (ff_r_mul_y_prefix_product) = S ((S (ff_i_mul_y_prefix_product)) * ff_v_mul_y_prefix_product)) /\ exists ff_q_mul_y_prefix_product_partial. ff_u_mul_y_prefix_product = ff_q_mul_y_prefix_product_partial * S ((S (ff_i_mul_y_prefix_product)) * ff_v_mul_y_prefix_product) + (ff_r_mul_y_prefix_product))) /\ ((((exists ff_h_mul_y_prefix_product_successor. ff_h_mul_y_prefix_product_successor + S (ff_s_mul_y_prefix_product) = S ((S (S ff_i_mul_y_prefix_product)) * ff_v_mul_y_prefix_product)) /\ exists ff_q_mul_y_prefix_product_successor. ff_u_mul_y_prefix_product = ff_q_mul_y_prefix_product_successor * S ((S (S ff_i_mul_y_prefix_product)) * ff_v_mul_y_prefix_product) + (ff_s_mul_y_prefix_product))) /\ ff_s_mul_y_prefix_product = ff_r_mul_y_prefix_product * ff_p_mul_y_prefix_product)))))))) /\ y = r * x
specialize pow_successor_decompose x
specialize pow_successor_decompose f
specialize pow_successor_decompose (S f)
specialize pow_successor_decompose y
apply pow_successor_decompose
refl
exact hy
cases hy_step
cases hy_step_witness
have hqpow : exists r. (exists pa_b_mul_total_prefix pa_c_mul_total_prefix. ((forall pa_i_mul_total_prefix_repeat. (exists pa_lt_mul_total_prefix_repeat_bound. pa_lt_mul_total_prefix_repeat_bound + S pa_i_mul_total_prefix_repeat = e * f) -> (((exists pa_h_mul_total_prefix_repeat_decoded. pa_h_mul_total_prefix_repeat_decoded + S (a) = S ((S (pa_i_mul_total_prefix_repeat)) * pa_c_mul_total_prefix)) /\ exists pa_q_mul_total_prefix_repeat_decoded. pa_b_mul_total_prefix = pa_q_mul_total_prefix_repeat_decoded * S ((S (pa_i_mul_total_prefix_repeat)) * pa_c_mul_total_prefix) + (a)))) /\ (exists pa_u_mul_total_prefix_product pa_v_mul_total_prefix_product. ((((exists pa_h_mul_total_prefix_product_start. pa_h_mul_total_prefix_product_start + S (1) = S ((S (0)) * pa_v_mul_total_prefix_product)) /\ exists pa_q_mul_total_prefix_product_start. pa_u_mul_total_prefix_product = pa_q_mul_total_prefix_product_start * S ((S (0)) * pa_v_mul_total_prefix_product) + (1))) /\ ((((exists pa_h_mul_total_prefix_product_terminal. pa_h_mul_total_prefix_product_terminal + S (r) = S ((S (e * f)) * pa_v_mul_total_prefix_product)) /\ exists pa_q_mul_total_prefix_product_terminal. pa_u_mul_total_prefix_product = pa_q_mul_total_prefix_product_terminal * S ((S (e * f)) * pa_v_mul_total_prefix_product) + (r))) /\ forall pa_i_mul_total_prefix_product. (exists pa_lt_mul_total_prefix_product_bound. pa_lt_mul_total_prefix_product_bound + S pa_i_mul_total_prefix_product = e * f) -> exists pa_p_mul_total_prefix_product pa_r_mul_total_prefix_product pa_s_mul_total_prefix_product. ((((exists pa_h_mul_total_prefix_product_factor. pa_h_mul_total_prefix_product_factor + S (pa_p_mul_total_prefix_product) = S ((S (pa_i_mul_total_prefix_product)) * pa_c_mul_total_prefix)) /\ exists pa_q_mul_total_prefix_product_factor. pa_b_mul_total_prefix = pa_q_mul_total_prefix_product_factor * S ((S (pa_i_mul_total_prefix_product)) * pa_c_mul_total_prefix) + (pa_p_mul_total_prefix_product))) /\ ((((exists pa_h_mul_total_prefix_product_partial. pa_h_mul_total_prefix_product_partial + S (pa_r_mul_total_prefix_product) = S ((S (pa_i_mul_total_prefix_product)) * pa_v_mul_total_prefix_product)) /\ exists pa_q_mul_total_prefix_product_partial. pa_u_mul_total_prefix_product = pa_q_mul_total_prefix_product_partial * S ((S (pa_i_mul_total_prefix_product)) * pa_v_mul_total_prefix_product) + (pa_r_mul_total_prefix_product))) /\ ((((exists pa_h_mul_total_prefix_product_successor. pa_h_mul_total_prefix_product_successor + S (pa_s_mul_total_prefix_product) = S ((S (S pa_i_mul_total_prefix_product)) * pa_v_mul_total_prefix_product)) /\ exists pa_q_mul_total_prefix_product_successor. pa_u_mul_total_prefix_product = pa_q_mul_total_prefix_product_successor * S ((S (S pa_i_mul_total_prefix_product)) * pa_v_mul_total_prefix_product) + (pa_s_mul_total_prefix_product))) /\ pa_s_mul_total_prefix_product = pa_r_mul_total_prefix_product * pa_p_mul_total_prefix_product))))))))
specialize pow_exists a
specialize pow_exists (e * f)
exact pow_exists
cases hqpow
have hprefix : x1 = x2
specialize IH (e * f)
specialize IH x
specialize IH x1
specialize IH x2
apply IH
refl
exact hx
exact hy_step_witness_left
exact hqpow_witness
have hpsum : p = (e * f) + e
trans e * S f
exact hp
apply PA6
have htotal : z = x2 * x
specialize pow_add a
specialize pow_add (e * f)
specialize pow_add e
specialize pow_add p
specialize pow_add x2
specialize pow_add x
specialize pow_add z
apply pow_add
exact hpsum
exact hqpow_witness
exact hx
exact hz
trans x1 * x
exact hy_step_witness_right
trans x2 * x
congr
exact hprefix
refl
symm
exact htotal
qed

Occurrences

70,463

Distinct objects

5,786

Reused references

262

Depth

91

Cuts

2,086

Certificate SHA-256

95115103d3c2…

predecessor_square_mod_onePredecessor square mod one Quadratic reciprocity campaign checked

The predecessor of a successor squares to one modulo that successor.

Prerequisites

mul_one mul_succ_left add_assoc add_comm zero_add add_succ_left

Used directly by

pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall p r. p = S r -> exists gs_u_square gs_v_square. (r * r) + p * gs_u_square = (1) + p * gs_v_square

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p r. p = S r -> exists gs_u_square gs_v_square. (r * r) + p * gs_u_square = (1) + p * gs_v_square
use mul_one
use mul_succ_left
use add_assoc
use add_comm
use zero_add
use add_succ_left
intro p
intro r
intro hp
exists 1
exists r
rewrite hp
rewrite hp
simp [mul_one, mul_succ_left, add_assoc, add_comm, zero_add, add_succ_left]
qed

Occurrences

426

Distinct objects

234

Reused references

30

Depth

27

Cuts

13

Certificate SHA-256

47f8ce741b54…

even_successor_to_oddEven successor to odd Quadratic reciprocity campaign checked

If a successor is even, its predecessor is odd.

Prerequisites

parity_cases successor_odd_of_even even_not_odd

Used directly by

pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall n. (exists a. S n = 2 * a) -> exists b. n = 2 * b + 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. S n = 2 * a) -> exists b. n = 2 * b + 1
use parity_cases
use successor_odd_of_even
use even_not_odd
intro n
intro hse
specialize parity_cases n
cases parity_cases
cases parity_cases_witness
exfalso
have hso : exists b. S n = 2 * b + 1
specialize successor_odd_of_even n
apply successor_odd_of_even
exists x
exact parity_cases_witness_left
specialize even_not_odd (S n)
apply even_not_odd
exact hse
exact hso
exists x
exact parity_cases_witness_right
qed

Occurrences

1,074

Distinct objects

751

Reused references

35

Depth

62

Cuts

27

Certificate SHA-256

6d559c9eca8b…

odd_successor_to_evenOdd successor to even Quadratic reciprocity campaign checked

If a successor is odd, its predecessor is even.

Prerequisites

parity_cases successor_even_of_odd odd_not_even

Used directly by

pow_predecessor_parity_mod

Exact expanded PA statement

Copy statement

forall n. (exists a. S n = 2 * a + 1) -> exists b. n = 2 * b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n. (exists a. S n = 2 * a + 1) -> exists b. n = 2 * b
use parity_cases
use successor_even_of_odd
use odd_not_even
intro n
intro hso
specialize parity_cases n
cases parity_cases
cases parity_cases_witness
exists x
exact parity_cases_witness_left
exfalso
have hse : exists b. S n = 2 * b
specialize successor_even_of_odd n
apply successor_even_of_odd
exists x
exact parity_cases_witness_right
specialize odd_not_even (S n)
apply odd_not_even
exact hso
exact hse
qed

Occurrences

1,102

Distinct objects

773

Reused references

41

Depth

62

Cuts

27

Certificate SHA-256

fe3e17a0fce8…

pow_predecessor_parity_modPow predecessor parity mod Quadratic reciprocity campaign checked

Powers of the predecessor of p alternate between one and the predecessor modulo p.

Prerequisites

pow_zero pow_successor_decompose odd_not_even even_successor_to_odd odd_successor_to_even predecessor_square_mod_one mod_eq_refl mod_eq_mul mod_eq_trans one_mul

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p r e z. p = S r -> (exists ff_b_main ff_c_main. ((forall ff_i_main_repeat. (exists ff_lt_main_repeat_bound. ff_lt_main_repeat_bound + S ff_i_main_repeat = e) -> (((exists ff_h_main_repeat_decoded. ff_h_main_repeat_decoded + S (r) = S ((S (ff_i_main_repeat)) * ff_c_main)) /\ exists ff_q_main_repeat_decoded. ff_b_main = ff_q_main_repeat_decoded * S ((S (ff_i_main_repeat)) * ff_c_main) + (r)))) /\ (exists ff_u_main_product ff_v_main_product. ((((exists ff_h_main_product_start. ff_h_main_product_start + S (1) = S ((S (0)) * ff_v_main_product)) /\ exists ff_q_main_product_start. ff_u_main_product = ff_q_main_product_start * S ((S (0)) * ff_v_main_product) + (1))) /\ ((((exists ff_h_main_product_terminal. ff_h_main_product_terminal + S (z) = S ((S (e)) * ff_v_main_product)) /\ exists ff_q_main_product_terminal. ff_u_main_product = ff_q_main_product_terminal * S ((S (e)) * ff_v_main_product) + (z))) /\ forall ff_i_main_product. (exists ff_lt_main_product_bound. ff_lt_main_product_bound + S ff_i_main_product = e) -> exists ff_p_main_product ff_r_main_product ff_s_main_product. ((((exists ff_h_main_product_factor. ff_h_main_product_factor + S (ff_p_main_product) = S ((S (ff_i_main_product)) * ff_c_main)) /\ exists ff_q_main_product_factor. ff_b_main = ff_q_main_product_factor * S ((S (ff_i_main_product)) * ff_c_main) + (ff_p_main_product))) /\ ((((exists ff_h_main_product_partial. ff_h_main_product_partial + S (ff_r_main_product) = S ((S (ff_i_main_product)) * ff_v_main_product)) /\ exists ff_q_main_product_partial. ff_u_main_product = ff_q_main_product_partial * S ((S (ff_i_main_product)) * ff_v_main_product) + (ff_r_main_product))) /\ ((((exists ff_h_main_product_successor. ff_h_main_product_successor + S (ff_s_main_product) = S ((S (S ff_i_main_product)) * ff_v_main_product)) /\ exists ff_q_main_product_successor. ff_u_main_product = ff_q_main_product_successor * S ((S (S ff_i_main_product)) * ff_v_main_product) + (ff_s_main_product))) /\ ff_s_main_product = ff_r_main_product * ff_p_main_product)))))))) -> (((exists gs_even_main. e = 2 * gs_even_main) -> (exists gs_u_result_even gs_v_result_even. (z) + p * gs_u_result_even = (1) + p * gs_v_result_even)) /\ ((exists gs_odd_main. e = 2 * gs_odd_main + 1) -> (exists gs_u_result_odd gs_v_result_odd. (z) + p * gs_u_result_odd = (r) + p * gs_v_result_odd)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p r e z. p = S r -> (exists ff_b_main ff_c_main. ((forall ff_i_main_repeat. (exists ff_lt_main_repeat_bound. ff_lt_main_repeat_bound + S ff_i_main_repeat = e) -> (((exists ff_h_main_repeat_decoded. ff_h_main_repeat_decoded + S (r) = S ((S (ff_i_main_repeat)) * ff_c_main)) /\ exists ff_q_main_repeat_decoded. ff_b_main = ff_q_main_repeat_decoded * S ((S (ff_i_main_repeat)) * ff_c_main) + (r)))) /\ (exists ff_u_main_product ff_v_main_product. ((((exists ff_h_main_product_start. ff_h_main_product_start + S (1) = S ((S (0)) * ff_v_main_product)) /\ exists ff_q_main_product_start. ff_u_main_product = ff_q_main_product_start * S ((S (0)) * ff_v_main_product) + (1))) /\ ((((exists ff_h_main_product_terminal. ff_h_main_product_terminal + S (z) = S ((S (e)) * ff_v_main_product)) /\ exists ff_q_main_product_terminal. ff_u_main_product = ff_q_main_product_terminal * S ((S (e)) * ff_v_main_product) + (z))) /\ forall ff_i_main_product. (exists ff_lt_main_product_bound. ff_lt_main_product_bound + S ff_i_main_product = e) -> exists ff_p_main_product ff_r_main_product ff_s_main_product. ((((exists ff_h_main_product_factor. ff_h_main_product_factor + S (ff_p_main_product) = S ((S (ff_i_main_product)) * ff_c_main)) /\ exists ff_q_main_product_factor. ff_b_main = ff_q_main_product_factor * S ((S (ff_i_main_product)) * ff_c_main) + (ff_p_main_product))) /\ ((((exists ff_h_main_product_partial. ff_h_main_product_partial + S (ff_r_main_product) = S ((S (ff_i_main_product)) * ff_v_main_product)) /\ exists ff_q_main_product_partial. ff_u_main_product = ff_q_main_product_partial * S ((S (ff_i_main_product)) * ff_v_main_product) + (ff_r_main_product))) /\ ((((exists ff_h_main_product_successor. ff_h_main_product_successor + S (ff_s_main_product) = S ((S (S ff_i_main_product)) * ff_v_main_product)) /\ exists ff_q_main_product_successor. ff_u_main_product = ff_q_main_product_successor * S ((S (S ff_i_main_product)) * ff_v_main_product) + (ff_s_main_product))) /\ ff_s_main_product = ff_r_main_product * ff_p_main_product)))))))) -> (((exists gs_even_main. e = 2 * gs_even_main) -> (exists gs_u_result_even gs_v_result_even. (z) + p * gs_u_result_even = (1) + p * gs_v_result_even)) /\ ((exists gs_odd_main. e = 2 * gs_odd_main + 1) -> (exists gs_u_result_odd gs_v_result_odd. (z) + p * gs_u_result_odd = (r) + p * gs_v_result_odd)))
use pow_zero
use pow_successor_decompose
use odd_not_even
use even_successor_to_odd
use odd_successor_to_even
use predecessor_square_mod_one
use mod_eq_refl
use mod_eq_mul
use mod_eq_trans
use one_mul
intro p
intro r
induction e
intro z
intro hp
intro hpow
split
intro he
have hz : z = 1
specialize pow_zero r
specialize pow_zero 0
specialize pow_zero z
apply pow_zero
refl
exact hpow
rewrite hz
specialize mod_eq_refl p
specialize mod_eq_refl 1
exact mod_eq_refl
intro ho
exfalso
specialize odd_not_even 0
apply odd_not_even
exact ho
exists 0
norm_num
intro z
intro hp
intro hpow
have hstep : exists w. (exists ff_b_predecessor ff_c_predecessor. ((forall ff_i_predecessor_repeat. (exists ff_lt_predecessor_repeat_bound. ff_lt_predecessor_repeat_bound + S ff_i_predecessor_repeat = e) -> (((exists ff_h_predecessor_repeat_decoded. ff_h_predecessor_repeat_decoded + S (r) = S ((S (ff_i_predecessor_repeat)) * ff_c_predecessor)) /\ exists ff_q_predecessor_repeat_decoded. ff_b_predecessor = ff_q_predecessor_repeat_decoded * S ((S (ff_i_predecessor_repeat)) * ff_c_predecessor) + (r)))) /\ (exists ff_u_predecessor_product ff_v_predecessor_product. ((((exists ff_h_predecessor_product_start. ff_h_predecessor_product_start + S (1) = S ((S (0)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_start. ff_u_predecessor_product = ff_q_predecessor_product_start * S ((S (0)) * ff_v_predecessor_product) + (1))) /\ ((((exists ff_h_predecessor_product_terminal. ff_h_predecessor_product_terminal + S (w) = S ((S (e)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_terminal. ff_u_predecessor_product = ff_q_predecessor_product_terminal * S ((S (e)) * ff_v_predecessor_product) + (w))) /\ forall ff_i_predecessor_product. (exists ff_lt_predecessor_product_bound. ff_lt_predecessor_product_bound + S ff_i_predecessor_product = e) -> exists ff_p_predecessor_product ff_r_predecessor_product ff_s_predecessor_product. ((((exists ff_h_predecessor_product_factor. ff_h_predecessor_product_factor + S (ff_p_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_c_predecessor)) /\ exists ff_q_predecessor_product_factor. ff_b_predecessor = ff_q_predecessor_product_factor * S ((S (ff_i_predecessor_product)) * ff_c_predecessor) + (ff_p_predecessor_product))) /\ ((((exists ff_h_predecessor_product_partial. ff_h_predecessor_product_partial + S (ff_r_predecessor_product) = S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_partial. ff_u_predecessor_product = ff_q_predecessor_product_partial * S ((S (ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_r_predecessor_product))) /\ ((((exists ff_h_predecessor_product_successor. ff_h_predecessor_product_successor + S (ff_s_predecessor_product) = S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product)) /\ exists ff_q_predecessor_product_successor. ff_u_predecessor_product = ff_q_predecessor_product_successor * S ((S (S ff_i_predecessor_product)) * ff_v_predecessor_product) + (ff_s_predecessor_product))) /\ ff_s_predecessor_product = ff_r_predecessor_product * ff_p_predecessor_product)))))))) /\ z = w * r
specialize pow_successor_decompose r
specialize pow_successor_decompose e
specialize pow_successor_decompose (S e)
specialize pow_successor_decompose z
apply pow_successor_decompose
refl
exact hpow
cases hstep
cases hstep_witness
have hinv : (((exists gs_even_ih. e = 2 * gs_even_ih) -> (exists gs_u_ih_even gs_v_ih_even. (x) + p * gs_u_ih_even = (1) + p * gs_v_ih_even)) /\ ((exists gs_odd_ih. e = 2 * gs_odd_ih + 1) -> (exists gs_u_ih_odd gs_v_ih_odd. (x) + p * gs_u_ih_odd = (r) + p * gs_v_ih_odd)))
specialize IH x
apply IH
exact hp
exact hstep_witness_left
cases hinv
split
intro hse
have heo : exists a. e = 2 * a + 1
specialize even_successor_to_odd e
apply even_successor_to_odd
exact hse
have hwr : exists u v. x + p * u = r + p * v
apply hinv_right
exact heo
have hrr : exists u v. r + p * u = r + p * v
specialize mod_eq_refl p
specialize mod_eq_refl r
exact mod_eq_refl
have hmul : exists u v. (x * r) + p * u = (r * r) + p * v
specialize mod_eq_mul p
specialize mod_eq_mul x
specialize mod_eq_mul r
specialize mod_eq_mul r
specialize mod_eq_mul r
apply mod_eq_mul
exact hwr
exact hrr
have hsq : exists u v. (r * r) + p * u = 1 + p * v
specialize predecessor_square_mod_one p
specialize predecessor_square_mod_one r
apply predecessor_square_mod_one
exact hp
have hfinal : exists u v. (x * r) + p * u = 1 + p * v
specialize mod_eq_trans p
specialize mod_eq_trans (x * r)
specialize mod_eq_trans (r * r)
specialize mod_eq_trans 1
apply mod_eq_trans
exact hmul
exact hsq
rewrite hstep_witness_right
exact hfinal
intro hso
have hee : exists a. e = 2 * a
specialize odd_successor_to_even e
apply odd_successor_to_even
exact hso
have hw1 : exists u v. x + p * u = 1 + p * v
apply hinv_left
exact hee
have hrr : exists u v. r + p * u = r + p * v
specialize mod_eq_refl p
specialize mod_eq_refl r
exact mod_eq_refl
have hmul : exists u v. (x * r) + p * u = (1 * r) + p * v
specialize mod_eq_mul p
specialize mod_eq_mul x
specialize mod_eq_mul 1
specialize mod_eq_mul r
specialize mod_eq_mul r
apply mod_eq_mul
exact hw1
exact hrr
specialize one_mul r
rewrite one_mul at hmul
rewrite hstep_witness_right
exact hmul
qed

Occurrences

9,249

Distinct objects

1,758

Reused references

102

Depth

67

Cuts

254

Certificate SHA-256

85d62b696a87…

beta_range_injectiveBeta range injective Quadratic reciprocity campaign checked

Equal decoded values in one consecutive range have equal indices.

Prerequisites

beta_range_entry_eq add_left_cancel

Used directly by

beta_half_range_mod_injective

Exact expanded PA statement

Copy statement

forall b c a l i j x y. (forall ff_i_generic. (exists ff_lt_generic_bound. ff_lt_generic_bound + S ff_i_generic = l) -> (((exists ff_h_generic_decoded. ff_h_generic_decoded + S (a + ff_i_generic) = S ((S (ff_i_generic)) * c)) /\ exists ff_q_generic_decoded. b = ff_q_generic_decoded * S ((S (ff_i_generic)) * c) + (a + ff_i_generic)))) -> (exists gh_lt_generic_i. gh_lt_generic_i + S i = l) -> (exists gh_lt_generic_j. gh_lt_generic_j + S j = l) -> (((exists ff_h_generic_i. ff_h_generic_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_generic_i. b = ff_q_generic_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_generic_j. ff_h_generic_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_generic_j. b = ff_q_generic_j * S ((S (j)) * c) + (y))) -> x = y -> i = j

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c a l i j x y. (forall ff_i_generic. (exists ff_lt_generic_bound. ff_lt_generic_bound + S ff_i_generic = l) -> (((exists ff_h_generic_decoded. ff_h_generic_decoded + S (a + ff_i_generic) = S ((S (ff_i_generic)) * c)) /\ exists ff_q_generic_decoded. b = ff_q_generic_decoded * S ((S (ff_i_generic)) * c) + (a + ff_i_generic)))) -> (exists gh_lt_generic_i. gh_lt_generic_i + S i = l) -> (exists gh_lt_generic_j. gh_lt_generic_j + S j = l) -> (((exists ff_h_generic_i. ff_h_generic_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_generic_i. b = ff_q_generic_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_generic_j. ff_h_generic_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_generic_j. b = ff_q_generic_j * S ((S (j)) * c) + (y))) -> x = y -> i = j
use beta_range_entry_eq
use add_left_cancel
intro b
intro c
intro a
intro l
intro i
intro j
intro x
intro y
intro hrange
intro hi
intro hj
intro hxi
intro hyj
intro hxy
have hxia : x = a + i
specialize beta_range_entry_eq b
specialize beta_range_entry_eq c
specialize beta_range_entry_eq a
specialize beta_range_entry_eq l
specialize beta_range_entry_eq i
specialize beta_range_entry_eq x
apply beta_range_entry_eq
exact hrange
exact hi
exact hxi
have hyja : y = a + j
specialize beta_range_entry_eq b
specialize beta_range_entry_eq c
specialize beta_range_entry_eq a
specialize beta_range_entry_eq l
specialize beta_range_entry_eq j
specialize beta_range_entry_eq y
apply beta_range_entry_eq
exact hrange
exact hj
exact hyj
have haij : a + i = a + j
trans x
symm
exact hxia
trans y
exact hxy
exact hyja
specialize add_left_cancel a
specialize add_left_cancel i
specialize add_left_cancel j
apply add_left_cancel
exact haij
qed

Occurrences

1,338

Distinct objects

766

Reused references

38

Depth

61

Cuts

37

Certificate SHA-256

b2fab09a4ed7…

beta_half_range_entry_boundsBeta half range entry bounds Quadratic reciprocity campaign checked

Entries 1 through h in an odd half-range are nonzero and below p.

Prerequisites

beta_range_entry_eq zero_add add_succ_left mul_succ_left mul_zero_left add_assoc lt_of_le_of_lt

Used directly by

beta_half_range_mod_eq_value

Exact expanded PA statement

Copy statement

forall p h b c i x. p = 2 * h + 1 -> (forall gh_i_half. (exists gh_lt_half_bound. gh_lt_half_bound + S gh_i_half = h) -> (((exists gh_h_half. gh_h_half + S (1 + gh_i_half) = S ((S gh_i_half) * c)) /\ exists gh_q_half. b = gh_q_half * S ((S gh_i_half) * c) + (1 + gh_i_half)))) -> (exists gh_lt_half_i. gh_lt_half_i + S i = h) -> (((exists ff_h_half_i. ff_h_half_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_half_i. b = ff_q_half_i * S ((S (i)) * c) + (x))) -> (~(x = 0) /\ (exists gh_lt_half_value. gh_lt_half_value + S x = p))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p h b c i x. p = 2 * h + 1 -> (forall gh_i_half. (exists gh_lt_half_bound. gh_lt_half_bound + S gh_i_half = h) -> (((exists gh_h_half. gh_h_half + S (1 + gh_i_half) = S ((S gh_i_half) * c)) /\ exists gh_q_half. b = gh_q_half * S ((S gh_i_half) * c) + (1 + gh_i_half)))) -> (exists gh_lt_half_i. gh_lt_half_i + S i = h) -> (((exists ff_h_half_i. ff_h_half_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_half_i. b = ff_q_half_i * S ((S (i)) * c) + (x))) -> (~(x = 0) /\ (exists gh_lt_half_value. gh_lt_half_value + S x = p))
use beta_range_entry_eq
use zero_add
use add_succ_left
use mul_succ_left
use mul_zero_left
use add_assoc
use lt_of_le_of_lt
intro p
intro h
intro b
intro c
intro i
intro x
intro hp
intro hrange
intro hi
intro hxi
have hx : x = 1 + i
specialize beta_range_entry_eq b
specialize beta_range_entry_eq c
specialize beta_range_entry_eq 1
specialize beta_range_entry_eq h
specialize beta_range_entry_eq i
specialize beta_range_entry_eq x
apply beta_range_entry_eq
exact hrange
exact hi
exact hxi
have hone : 1 + i = S i
trans S (0 + i)
specialize add_succ_left 0
specialize add_succ_left i
exact add_succ_left
congr
specialize zero_add i
exact zero_add
have hxsi : x = S i
trans 1 + i
exact hx
exact hone
split
intro hx0
apply PA1
trans x
symm
exact hxsi
exact hx0
have hxh : exists k. k + x = h
rewrite hxsi
exact hi
have hhp : exists k. k + S h = p
exists h
rewrite hp
simp [mul_succ_left, mul_zero_left, add_succ_left, zero_add, add_assoc]
specialize lt_of_le_of_lt x
specialize lt_of_le_of_lt h
specialize lt_of_le_of_lt p
apply lt_of_le_of_lt
exact hxh
exact hhp
qed

Occurrences

1,585

Distinct objects

850

Reused references

46

Depth

61

Cuts

43

Certificate SHA-256

3752530cd637…

beta_half_range_mod_eq_valueBeta half range mod eq value Quadratic reciprocity campaign checked

Congruent entries in the odd half-range are equal as bounded residues.

Prerequisites

beta_half_range_entry_bounds mod_eq_bounded_unique

Used directly by

beta_half_range_mod_injective

Exact expanded PA statement

Copy statement

forall p h b c i j x y. p = 2 * h + 1 -> (forall gh_i_pair. (exists gh_lt_pair_bound. gh_lt_pair_bound + S gh_i_pair = h) -> (((exists gh_h_pair. gh_h_pair + S (1 + gh_i_pair) = S ((S gh_i_pair) * c)) /\ exists gh_q_pair. b = gh_q_pair * S ((S gh_i_pair) * c) + (1 + gh_i_pair)))) -> (exists gh_lt_pair_i. gh_lt_pair_i + S i = h) -> (exists gh_lt_pair_j. gh_lt_pair_j + S j = h) -> (((exists ff_h_pair_i. ff_h_pair_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_pair_i. b = ff_q_pair_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_pair_j. ff_h_pair_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_pair_j. b = ff_q_pair_j * S ((S (j)) * c) + (y))) -> (exists gh_u_pair gh_v_pair. x + p * gh_u_pair = y + p * gh_v_pair) -> x = y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p h b c i j x y. p = 2 * h + 1 -> (forall gh_i_pair. (exists gh_lt_pair_bound. gh_lt_pair_bound + S gh_i_pair = h) -> (((exists gh_h_pair. gh_h_pair + S (1 + gh_i_pair) = S ((S gh_i_pair) * c)) /\ exists gh_q_pair. b = gh_q_pair * S ((S gh_i_pair) * c) + (1 + gh_i_pair)))) -> (exists gh_lt_pair_i. gh_lt_pair_i + S i = h) -> (exists gh_lt_pair_j. gh_lt_pair_j + S j = h) -> (((exists ff_h_pair_i. ff_h_pair_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_pair_i. b = ff_q_pair_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_pair_j. ff_h_pair_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_pair_j. b = ff_q_pair_j * S ((S (j)) * c) + (y))) -> (exists gh_u_pair gh_v_pair. x + p * gh_u_pair = y + p * gh_v_pair) -> x = y
use beta_half_range_entry_bounds
use mod_eq_bounded_unique
intro p
intro h
intro b
intro c
intro i
intro j
intro x
intro y
intro hp
intro hrange
intro hi
intro hj
intro hxi
intro hyj
intro hxy
have hxb : ~(x = 0) /\ exists k. k + S x = p
specialize beta_half_range_entry_bounds p
specialize beta_half_range_entry_bounds h
specialize beta_half_range_entry_bounds b
specialize beta_half_range_entry_bounds c
specialize beta_half_range_entry_bounds i
specialize beta_half_range_entry_bounds x
apply beta_half_range_entry_bounds
exact hp
exact hrange
exact hi
exact hxi
have hyb : ~(y = 0) /\ exists k. k + S y = p
specialize beta_half_range_entry_bounds p
specialize beta_half_range_entry_bounds h
specialize beta_half_range_entry_bounds b
specialize beta_half_range_entry_bounds c
specialize beta_half_range_entry_bounds j
specialize beta_half_range_entry_bounds y
apply beta_half_range_entry_bounds
exact hp
exact hrange
exact hj
exact hyj
cases hxb
cases hyb
specialize mod_eq_bounded_unique p
specialize mod_eq_bounded_unique x
specialize mod_eq_bounded_unique y
apply mod_eq_bounded_unique
exact hxb_right
exact hyb_right
exact hxy
qed

Occurrences

2,603

Distinct objects

941

Reused references

48

Depth

62

Cuts

71

Certificate SHA-256

a3c337b1d50f…

beta_half_range_mod_injectiveBeta half range mod injective Quadratic reciprocity campaign checked

The odd half-range is injective modulo p at the level of indices.

Prerequisites

beta_half_range_mod_eq_value beta_range_injective

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p h b c i j x y. p = 2 * h + 1 -> (forall gh_i_pair. (exists gh_lt_pair_bound. gh_lt_pair_bound + S gh_i_pair = h) -> (((exists gh_h_pair. gh_h_pair + S (1 + gh_i_pair) = S ((S gh_i_pair) * c)) /\ exists gh_q_pair. b = gh_q_pair * S ((S gh_i_pair) * c) + (1 + gh_i_pair)))) -> (exists gh_lt_pair_i. gh_lt_pair_i + S i = h) -> (exists gh_lt_pair_j. gh_lt_pair_j + S j = h) -> (((exists ff_h_pair_i. ff_h_pair_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_pair_i. b = ff_q_pair_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_pair_j. ff_h_pair_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_pair_j. b = ff_q_pair_j * S ((S (j)) * c) + (y))) -> (exists gh_u_pair gh_v_pair. x + p * gh_u_pair = y + p * gh_v_pair) -> i = j

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p h b c i j x y. p = 2 * h + 1 -> (forall gh_i_pair. (exists gh_lt_pair_bound. gh_lt_pair_bound + S gh_i_pair = h) -> (((exists gh_h_pair. gh_h_pair + S (1 + gh_i_pair) = S ((S gh_i_pair) * c)) /\ exists gh_q_pair. b = gh_q_pair * S ((S gh_i_pair) * c) + (1 + gh_i_pair)))) -> (exists gh_lt_pair_i. gh_lt_pair_i + S i = h) -> (exists gh_lt_pair_j. gh_lt_pair_j + S j = h) -> (((exists ff_h_pair_i. ff_h_pair_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_pair_i. b = ff_q_pair_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_pair_j. ff_h_pair_j + S (y) = S ((S (j)) * c)) /\ exists ff_q_pair_j. b = ff_q_pair_j * S ((S (j)) * c) + (y))) -> (exists gh_u_pair gh_v_pair. x + p * gh_u_pair = y + p * gh_v_pair) -> i = j
use beta_half_range_mod_eq_value
use beta_range_injective
intro p
intro h
intro b
intro c
intro i
intro j
intro x
intro y
intro hp
intro hrange
intro hi
intro hj
intro hxi
intro hyj
intro hxy
have hvalue : x = y
specialize beta_half_range_mod_eq_value p
specialize beta_half_range_mod_eq_value h
specialize beta_half_range_mod_eq_value b
specialize beta_half_range_mod_eq_value c
specialize beta_half_range_mod_eq_value i
specialize beta_half_range_mod_eq_value j
specialize beta_half_range_mod_eq_value x
specialize beta_half_range_mod_eq_value y
apply beta_half_range_mod_eq_value
exact hp
exact hrange
exact hi
exact hj
exact hxi
exact hyj
exact hxy
specialize beta_range_injective b
specialize beta_range_injective c
specialize beta_range_injective 1
specialize beta_range_injective h
specialize beta_range_injective i
specialize beta_range_injective j
specialize beta_range_injective x
specialize beta_range_injective y
apply beta_range_injective
exact hrange
exact hi
exact hj
exact hxi
exact hyj
exact hvalue
qed

Occurrences

4,001

Distinct objects

1,052

Reused references

50

Depth

63

Cuts

110

Certificate SHA-256

9497e9ce70f5…

finite_surjective_zeroFinite surjective zero Quadratic reciprocity campaign checked

The empty decoded prefix is surjective onto the empty interval.

Prerequisites

add_eq_zero_right succ_ne_zero

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n. n = 0 -> (forall fp_value_zero. (exists fp_gap_zero_value. fp_gap_zero_value + S fp_value_zero = n) -> exists fp_i_zero. ((exists fp_gap_zero_index. fp_gap_zero_index + S fp_i_zero = n) /\ (((exists ff_h_zero_entry. ff_h_zero_entry + S (fp_value_zero) = S ((S (fp_i_zero)) * c)) /\ exists ff_q_zero_entry. b = ff_q_zero_entry * S ((S (fp_i_zero)) * c) + (fp_value_zero)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n. n = 0 -> (forall fp_value_zero. (exists fp_gap_zero_value. fp_gap_zero_value + S fp_value_zero = n) -> exists fp_i_zero. ((exists fp_gap_zero_index. fp_gap_zero_index + S fp_i_zero = n) /\ (((exists ff_h_zero_entry. ff_h_zero_entry + S (fp_value_zero) = S ((S (fp_i_zero)) * c)) /\ exists ff_q_zero_entry. b = ff_q_zero_entry * S ((S (fp_i_zero)) * c) + (fp_value_zero)))))
use add_eq_zero_right
use succ_ne_zero
intro b
intro c
intro n
intro hn
intro y
intro hy
rewrite hn at hy
exfalso
cases hy
have hsy : S y = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S y)
apply add_eq_zero_right
exact hy_witness
specialize succ_ne_zero y
apply succ_ne_zero
exact hsy
qed

Occurrences

41

Distinct objects

41

Reused references

0

Depth

15

Cuts

2

Certificate SHA-256

5784eb55a356…

finite_injective_prefix_succFinite injective prefix succ Quadratic reciprocity campaign checked

Injectivity of a successor prefix restricts to its old prefix.

Prerequisites

le_succ

Used directly by

finite_no_top_successor_gate finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix)
use le_succ
intro b
intro c
intro n
intro sn
intro hsn
intro hinj
rewrite hsn at hinj
rewrite hsn at hinj
intro i
intro j
intro x
intro hi
intro hj
intro hxi
intro hxj
specialize hinj i
specialize hinj j
specialize hinj x
apply hinj
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
specialize le_succ (S j)
specialize le_succ n
apply le_succ
exact hj
exact hxi
exact hxj
qed

Occurrences

105

Distinct objects

103

Reused references

2

Depth

34

Cuts

2

Certificate SHA-256

6e9e1361005b…

finite_lt_succ_eq_or_ltFinite lt succ eq or lt Quadratic reciprocity campaign checked

A value below a successor is the predecessor or lies below it.

Prerequisites

le_of_succ_le_succ le_eq_or_lt

Used directly by

beta_prefix_replace_exists finite_contains_decidable finite_bounded_prefix_without_top finite_surjective_succ_intro finite_last_is_top_from_prefix_surjective beta_product_replace_balance

Exact expanded PA statement

Copy statement

forall n x. (exists h. h + S x = S n) -> x = n \/ exists h. h + S x = n

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n x. (exists h. h + S x = S n) -> x = n \/ exists h. h + S x = n
use le_of_succ_le_succ
use le_eq_or_lt
intro n
intro x
intro hlt
have hle : exists h. h + x = n
specialize le_of_succ_le_succ x
specialize le_of_succ_le_succ n
apply le_of_succ_le_succ
exact hlt
specialize le_eq_or_lt x
specialize le_eq_or_lt n
apply le_eq_or_lt
exact hle
qed

Occurrences

128

Distinct objects

124

Reused references

4

Depth

21

Cuts

5

Certificate SHA-256

5d2bc17aeec8…

finite_bounded_entry_ltFinite bounded entry lt Quadratic reciprocity campaign checked

Every explicitly decoded entry of a bounded prefix satisfies its value bound.

Prerequisites

beta_at_unique

Used directly by

finite_swap_last_bounded

Exact expanded PA statement

Copy statement

forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l
use beta_at_unique
intro b
intro c
intro l
intro i
intro x
intro hbounded
intro hi
intro hentry
specialize hbounded i
have hdecoded : exists a. (((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) /\ exists h. h + S a = l)
apply hbounded
exact hi
cases hdecoded
cases hdecoded_witness
have hxa : x = x1
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique i
specialize beta_at_unique x
specialize beta_at_unique x1
apply beta_at_unique
exact hentry
exact hdecoded_witness_left
rewrite hxa
exact hdecoded_witness_right
qed

Occurrences

1,150

Distinct objects

721

Reused references

37

Depth

60

Cuts

31

Certificate SHA-256

a17f66fcd878…

beta_prefix_replace_existsBeta prefix replace exists Quadratic reciprocity campaign checked

Recode a finite beta prefix while replacing one interior entry.

Prerequisites

add_eq_zero_right succ_ne_zero finite_lt_succ_eq_or_lt beta_prefix_extend beta_at_exists beta_at_unique

Used directly by

beta_prefix_swap_last_from_entries

Exact expanded PA statement

Copy statement

forall b c i s k. (exists h. h + S i = k) -> exists z d. ((((exists ff_h_replace_entry. ff_h_replace_entry + S (s) = S ((S (i)) * d)) /\ exists ff_q_replace_entry. z = ff_q_replace_entry * S ((S (i)) * d) + (s))) /\ forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_replace_old. ff_h_replace_old + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_old. b = ff_q_replace_old * S ((S (j)) * c) + (a))) -> (((exists ff_h_replace_new. ff_h_replace_new + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_new. z = ff_q_replace_new * S ((S (j)) * d) + (a))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c i s k. (exists h. h + S i = k) -> exists z d. ((((exists ff_h_replace_entry. ff_h_replace_entry + S (s) = S ((S (i)) * d)) /\ exists ff_q_replace_entry. z = ff_q_replace_entry * S ((S (i)) * d) + (s))) /\ forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_replace_old. ff_h_replace_old + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_old. b = ff_q_replace_old * S ((S (j)) * c) + (a))) -> (((exists ff_h_replace_new. ff_h_replace_new + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_new. z = ff_q_replace_new * S ((S (j)) * d) + (a))))
use add_eq_zero_right
use succ_ne_zero
use finite_lt_succ_eq_or_lt
use beta_prefix_extend
use beta_at_exists
use beta_at_unique
intro b
intro c
intro i
intro s
induction k
intro hi
exfalso
cases hi
have hsi : S i = 0
specialize add_eq_zero_right x
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi
intro hi
have hisplit : i = k \/ exists h. h + S i = k
specialize finite_lt_succ_eq_or_lt k
specialize finite_lt_succ_eq_or_lt i
apply finite_lt_succ_eq_or_lt
exact hi
cases hisplit
specialize beta_prefix_extend k
specialize beta_prefix_extend b
specialize beta_prefix_extend c
specialize beta_prefix_extend s
cases beta_prefix_extend
cases beta_prefix_extend_witness
cases beta_prefix_extend_witness_witness
exists x
exists x1
split
rewrite hisplit_left
rewrite hisplit_left
exact beta_prefix_extend_witness_witness_left
intro j
intro a
intro hj
intro hji
intro hold
have hjsplit : j = k \/ exists h. h + S j = k
specialize finite_lt_succ_eq_or_lt k
specialize finite_lt_succ_eq_or_lt j
apply finite_lt_succ_eq_or_lt
exact hj
cases hjsplit
exfalso
apply hji
trans k
exact hjsplit_left
symm
exact hisplit_left
specialize beta_prefix_extend_witness_witness_right j
specialize beta_prefix_extend_witness_witness_right a
apply beta_prefix_extend_witness_witness_right
exact hjsplit_right
exact hold
have hreplaced : exists z d. (((exists h. h + S s = S ((S i) * d)) /\ exists q. z = q * S ((S i) * d) + s) /\ forall j a. (exists h. h + S j = k) -> ~(j = i) -> ((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a))
apply IH
exact hisplit_right
cases hreplaced
cases hreplaced_witness
cases hreplaced_witness_witness
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists k
cases beta_at_exists
specialize beta_prefix_extend k
specialize beta_prefix_extend x
specialize beta_prefix_extend x1
specialize beta_prefix_extend x2
cases beta_prefix_extend
cases beta_prefix_extend_witness
cases beta_prefix_extend_witness_witness
exists x3
exists x4
split
specialize beta_prefix_extend_witness_witness_right i
specialize beta_prefix_extend_witness_witness_right s
apply beta_prefix_extend_witness_witness_right
exact hisplit_right
exact hreplaced_witness_witness_left
intro j
intro a
intro hj
intro hji
intro hold
have hjsplit : j = k \/ exists h. h + S j = k
specialize finite_lt_succ_eq_or_lt k
specialize finite_lt_succ_eq_or_lt j
apply finite_lt_succ_eq_or_lt
exact hj
cases hjsplit
have hax : a = x2
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique k
specialize beta_at_unique a
specialize beta_at_unique x2
apply beta_at_unique
rewrite hjsplit_left at hold
rewrite hjsplit_left at hold
exact hold
exact beta_at_exists_witness
rewrite hjsplit_left
rewrite hjsplit_left
rewrite hax
rewrite hax
exact beta_prefix_extend_witness_witness_left
have hmiddle : ((exists h. h + S a = S ((S j) * x1)) /\ exists q. x = q * S ((S j) * x1) + a)
specialize hreplaced_witness_witness_right j
specialize hreplaced_witness_witness_right a
apply hreplaced_witness_witness_right
exact hjsplit_right
exact hji
exact hold
specialize beta_prefix_extend_witness_witness_right j
specialize beta_prefix_extend_witness_witness_right a
apply beta_prefix_extend_witness_witness_right
exact hjsplit_right
exact hmiddle
qed

Occurrences

30,981

Distinct objects

4,698

Reused references

231

Depth

84

Cuts

923

Certificate SHA-256

0bce1fe4a844…

beta_prefix_swap_last_from_entriesBeta prefix swap last from entries Quadratic reciprocity campaign checked

Swap a chosen interior beta entry with the last entry, given both decoded values.

Prerequisites

beta_prefix_replace_exists le_succ le_refl lt_irrefl_expanded

Used directly by

beta_prefix_swap_last_exists finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n i x y. (exists h. h + S i = n) -> (((exists ff_h_swap_old_i. ff_h_swap_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_old_i. b = ff_q_swap_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_old_n. ff_h_swap_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_old_n. b = ff_q_swap_old_n * S ((S (n)) * c) + (y))) -> exists z d. ((((exists ff_h_swap_new_i. ff_h_swap_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_new_i. z = ff_q_swap_new_i * S ((S (i)) * d) + (y))) /\ ((((exists ff_h_swap_new_n. ff_h_swap_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_new_n. z = ff_q_swap_new_n * S ((S (n)) * d) + (x))) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_old_j. ff_h_swap_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_old_j. b = ff_q_swap_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_new_j. ff_h_swap_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_new_j. z = ff_q_swap_new_j * S ((S (j)) * d) + (a)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n i x y. (exists h. h + S i = n) -> (((exists ff_h_swap_old_i. ff_h_swap_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_old_i. b = ff_q_swap_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_old_n. ff_h_swap_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_old_n. b = ff_q_swap_old_n * S ((S (n)) * c) + (y))) -> exists z d. ((((exists ff_h_swap_new_i. ff_h_swap_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_new_i. z = ff_q_swap_new_i * S ((S (i)) * d) + (y))) /\ ((((exists ff_h_swap_new_n. ff_h_swap_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_new_n. z = ff_q_swap_new_n * S ((S (n)) * d) + (x))) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_old_j. ff_h_swap_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_old_j. b = ff_q_swap_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_new_j. ff_h_swap_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_new_j. z = ff_q_swap_new_j * S ((S (j)) * d) + (a)))))
use beta_prefix_replace_exists
use le_succ
use le_refl
use lt_irrefl_expanded
intro b
intro c
intro n
intro i
intro x
intro y
intro hi
intro hxi
intro hyn
have hisn : exists h. h + S i = S n
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hin : ~(i = n)
intro hin_eq
specialize lt_irrefl_expanded n
apply lt_irrefl_expanded
rewrite hin_eq at hi
exact hi
have hni : ~(n = i)
intro hni_eq
apply hin
symm
exact hni_eq
have hfirst : exists z d. (((exists h. h + S y = S ((S i) * d)) /\ exists q. z = q * S ((S i) * d) + y) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a))
specialize beta_prefix_replace_exists b
specialize beta_prefix_replace_exists c
specialize beta_prefix_replace_exists i
specialize beta_prefix_replace_exists y
specialize beta_prefix_replace_exists (S n)
apply beta_prefix_replace_exists
exact hisn
cases hfirst
cases hfirst_witness
cases hfirst_witness_witness
have hfirst_n : ((exists h. h + S y = S ((S n) * x2)) /\ exists q. x1 = q * S ((S n) * x2) + y)
specialize hfirst_witness_witness_right n
specialize hfirst_witness_witness_right y
apply hfirst_witness_witness_right
exact hnsn
exact hni
exact hyn
have hsecond : exists z d. (((exists h. h + S x = S ((S n) * d)) /\ exists q. z = q * S ((S n) * d) + x) /\ forall j a. (exists h. h + S j = S n) -> ~(j = n) -> ((exists h. h + S a = S ((S j) * x2)) /\ exists q. x1 = q * S ((S j) * x2) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a))
specialize beta_prefix_replace_exists x1
specialize beta_prefix_replace_exists x2
specialize beta_prefix_replace_exists n
specialize beta_prefix_replace_exists x
specialize beta_prefix_replace_exists (S n)
apply beta_prefix_replace_exists
exact hnsn
cases hsecond
cases hsecond_witness
cases hsecond_witness_witness
exists x3
exists x4
split
specialize hsecond_witness_witness_right i
specialize hsecond_witness_witness_right y
apply hsecond_witness_witness_right
exact hisn
exact hin
exact hfirst_witness_witness_left
split
exact hsecond_witness_witness_left
intro j
intro a
intro hj
intro hji
intro hjn
intro hold
have hmiddle : ((exists h. h + S a = S ((S j) * x2)) /\ exists q. x1 = q * S ((S j) * x2) + a)
specialize hfirst_witness_witness_right j
specialize hfirst_witness_witness_right a
apply hfirst_witness_witness_right
exact hj
exact hji
exact hold
specialize hsecond_witness_witness_right j
specialize hsecond_witness_witness_right a
apply hsecond_witness_witness_right
exact hj
exact hjn
exact hmiddle
qed

Occurrences

31,221

Distinct objects

4,790

Reused references

234

Depth

85

Cuts

931

Certificate SHA-256

c89cd33df597…

beta_prefix_swap_last_existsBeta prefix swap last exists Quadratic reciprocity campaign checked

Construct an extensional beta code with an interior entry swapped with the last.

Prerequisites

beta_at_exists beta_prefix_swap_last_from_entries

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c n i. (exists h. h + S i = n) -> exists z d x y. ((((exists ff_h_swap_exists_old_i. ff_h_swap_exists_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_exists_old_i. b = ff_q_swap_exists_old_i * S ((S (i)) * c) + (x))) /\ ((((exists ff_h_swap_exists_old_n. ff_h_swap_exists_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_exists_old_n. b = ff_q_swap_exists_old_n * S ((S (n)) * c) + (y))) /\ ((((exists ff_h_swap_exists_new_i. ff_h_swap_exists_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_exists_new_i. z = ff_q_swap_exists_new_i * S ((S (i)) * d) + (y))) /\ ((((exists ff_h_swap_exists_new_n. ff_h_swap_exists_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_exists_new_n. z = ff_q_swap_exists_new_n * S ((S (n)) * d) + (x))) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_exists_old_j. ff_h_swap_exists_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_exists_old_j. b = ff_q_swap_exists_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_exists_new_j. ff_h_swap_exists_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_exists_new_j. z = ff_q_swap_exists_new_j * S ((S (j)) * d) + (a)))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n i. (exists h. h + S i = n) -> exists z d x y. ((((exists ff_h_swap_exists_old_i. ff_h_swap_exists_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_exists_old_i. b = ff_q_swap_exists_old_i * S ((S (i)) * c) + (x))) /\ ((((exists ff_h_swap_exists_old_n. ff_h_swap_exists_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_exists_old_n. b = ff_q_swap_exists_old_n * S ((S (n)) * c) + (y))) /\ ((((exists ff_h_swap_exists_new_i. ff_h_swap_exists_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_exists_new_i. z = ff_q_swap_exists_new_i * S ((S (i)) * d) + (y))) /\ ((((exists ff_h_swap_exists_new_n. ff_h_swap_exists_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_exists_new_n. z = ff_q_swap_exists_new_n * S ((S (n)) * d) + (x))) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_exists_old_j. ff_h_swap_exists_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_exists_old_j. b = ff_q_swap_exists_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_exists_new_j. ff_h_swap_exists_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_exists_new_j. z = ff_q_swap_exists_new_j * S ((S (j)) * d) + (a)))))))
use beta_at_exists
use beta_prefix_swap_last_from_entries
intro b
intro c
intro n
intro i
intro hi
have hdecode_i : forall u v k. exists a. ((exists h. h + S a = S ((S k) * v)) /\ exists q. u = q * S ((S k) * v) + a)
exact beta_at_exists
have hdecode_n : forall u v k. exists a. ((exists h. h + S a = S ((S k) * v)) /\ exists q. u = q * S ((S k) * v) + a)
exact beta_at_exists
specialize hdecode_i b
specialize hdecode_i c
specialize hdecode_i i
cases hdecode_i
specialize hdecode_n b
specialize hdecode_n c
specialize hdecode_n n
cases hdecode_n
have hswap : exists z d. (((exists h. h + S x1 = S ((S i) * d)) /\ exists q. z = q * S ((S i) * d) + x1) /\ (((exists h. h + S x = S ((S n) * d)) /\ exists q. z = q * S ((S n) * d) + x) /\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> ((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a)))
specialize beta_prefix_swap_last_from_entries b
specialize beta_prefix_swap_last_from_entries c
specialize beta_prefix_swap_last_from_entries n
specialize beta_prefix_swap_last_from_entries i
specialize beta_prefix_swap_last_from_entries x
specialize beta_prefix_swap_last_from_entries x1
apply beta_prefix_swap_last_from_entries
exact hi
exact hdecode_i_witness
exact hdecode_n_witness
cases hswap
cases hswap_witness
exists x2
exists x3
exists x
exists x1
split
exact hdecode_i_witness
split
exact hdecode_n_witness
exact hswap_witness_witness
qed

Occurrences

31,742

Distinct objects

4,832

Reused references

235

Depth

87

Cuts

948

Certificate SHA-256

b1e5c90f043e…

beta_prefix_swap_last_reflectBeta prefix swap last reflect Quadratic reciprocity campaign checked

Every decoded swapped entry reflects to one of the two moved entries or the original index.

Prerequisites

eq_decidable beta_at_exists beta_at_unique

Used directly by

finite_swap_last_injective finite_swap_last_surjective_back beta_reindex_alignment_swap_last

Exact expanded PA statement

Copy statement

forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
use eq_decidable
use beta_at_exists
use beta_at_unique
intro b
intro c
intro z
intro d
intro n
intro i
intro x
intro y
intro hnew_i
intro hnew_n
intro hpreserve
intro j
intro a
intro hj
intro hnew
specialize eq_decidable j
specialize eq_decidable i
cases eq_decidable
left
split
exact eq_decidable_left
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique i
specialize beta_at_unique a
specialize beta_at_unique y
apply beta_at_unique
rewrite eq_decidable_left at hnew
rewrite eq_decidable_left at hnew
exact hnew
exact hnew_i
specialize eq_decidable_before2 n
cases eq_decidable_before2
right
left
split
exact eq_decidable_before2_left
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique n
specialize beta_at_unique a
specialize beta_at_unique x
apply beta_at_unique
rewrite eq_decidable_before2_left at hnew
rewrite eq_decidable_before2_left at hnew
exact hnew
exact hnew_n
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists j
cases beta_at_exists
have htransport : ((exists h. h + S x1 = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + x1)
specialize hpreserve j
specialize hpreserve x1
apply hpreserve
exact hj
exact eq_decidable_right
exact eq_decidable_before2_right
exact beta_at_exists_witness
have hav : a = x1
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique j
specialize beta_at_unique a
specialize beta_at_unique x1
apply beta_at_unique
exact hnew
exact htransport
right
right
split
exact eq_decidable_right
split
exact eq_decidable_before2_right
rewrite hav
rewrite hav
exact beta_at_exists_witness
qed

Occurrences

1,765

Distinct objects

1,041

Reused references

55

Depth

62

Cuts

48

Certificate SHA-256

da963c32a931…

finite_swap_last_boundedFinite swap last bounded Quadratic reciprocity campaign checked

A swap-last recoding preserves boundedness of the full successor prefix.

Prerequisites

finite_bounded_entry_lt eq_decidable le_succ le_refl

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_bound_old. (exists fp_gap_swap_bound_old_index. fp_gap_swap_bound_old_index + S fp_i_swap_bound_old = sn) -> exists fp_value_swap_bound_old. ((((exists ff_h_swap_bound_old_entry. ff_h_swap_bound_old_entry + S (fp_value_swap_bound_old) = S ((S (fp_i_swap_bound_old)) * c)) /\ exists ff_q_swap_bound_old_entry. b = ff_q_swap_bound_old_entry * S ((S (fp_i_swap_bound_old)) * c) + (fp_value_swap_bound_old))) /\ (exists fp_gap_swap_bound_old_value. fp_gap_swap_bound_old_value + S fp_value_swap_bound_old = sn))) -> (((exists ff_h_swap_bound_old_i. ff_h_swap_bound_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_bound_old_i. b = ff_q_swap_bound_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_bound_old_n. ff_h_swap_bound_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_bound_old_n. b = ff_q_swap_bound_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_bound_new_i. ff_h_swap_bound_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_bound_new_i. z = ff_q_swap_bound_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_bound_new_n. ff_h_swap_bound_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_bound_new_n. z = ff_q_swap_bound_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_bound_old_j. ff_h_swap_bound_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_bound_old_j. b = ff_q_swap_bound_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_bound_new_j. ff_h_swap_bound_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_bound_new_j. z = ff_q_swap_bound_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_bound_new. (exists fp_gap_swap_bound_new_index. fp_gap_swap_bound_new_index + S fp_i_swap_bound_new = sn) -> exists fp_value_swap_bound_new. ((((exists ff_h_swap_bound_new_entry. ff_h_swap_bound_new_entry + S (fp_value_swap_bound_new) = S ((S (fp_i_swap_bound_new)) * d)) /\ exists ff_q_swap_bound_new_entry. z = ff_q_swap_bound_new_entry * S ((S (fp_i_swap_bound_new)) * d) + (fp_value_swap_bound_new))) /\ (exists fp_gap_swap_bound_new_value. fp_gap_swap_bound_new_value + S fp_value_swap_bound_new = sn)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_bound_old. (exists fp_gap_swap_bound_old_index. fp_gap_swap_bound_old_index + S fp_i_swap_bound_old = sn) -> exists fp_value_swap_bound_old. ((((exists ff_h_swap_bound_old_entry. ff_h_swap_bound_old_entry + S (fp_value_swap_bound_old) = S ((S (fp_i_swap_bound_old)) * c)) /\ exists ff_q_swap_bound_old_entry. b = ff_q_swap_bound_old_entry * S ((S (fp_i_swap_bound_old)) * c) + (fp_value_swap_bound_old))) /\ (exists fp_gap_swap_bound_old_value. fp_gap_swap_bound_old_value + S fp_value_swap_bound_old = sn))) -> (((exists ff_h_swap_bound_old_i. ff_h_swap_bound_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_bound_old_i. b = ff_q_swap_bound_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_bound_old_n. ff_h_swap_bound_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_bound_old_n. b = ff_q_swap_bound_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_bound_new_i. ff_h_swap_bound_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_bound_new_i. z = ff_q_swap_bound_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_bound_new_n. ff_h_swap_bound_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_bound_new_n. z = ff_q_swap_bound_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_bound_old_j. ff_h_swap_bound_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_bound_old_j. b = ff_q_swap_bound_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_bound_new_j. ff_h_swap_bound_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_bound_new_j. z = ff_q_swap_bound_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_bound_new. (exists fp_gap_swap_bound_new_index. fp_gap_swap_bound_new_index + S fp_i_swap_bound_new = sn) -> exists fp_value_swap_bound_new. ((((exists ff_h_swap_bound_new_entry. ff_h_swap_bound_new_entry + S (fp_value_swap_bound_new) = S ((S (fp_i_swap_bound_new)) * d)) /\ exists ff_q_swap_bound_new_entry. z = ff_q_swap_bound_new_entry * S ((S (fp_i_swap_bound_new)) * d) + (fp_value_swap_bound_new))) /\ (exists fp_gap_swap_bound_new_value. fp_gap_swap_bound_new_value + S fp_value_swap_bound_new = sn)))
use finite_bounded_entry_lt
use eq_decidable
use le_succ
use le_refl
intro b
intro c
intro z
intro d
intro n
intro sn
intro i
intro x
intro y
intro hsn
intro hi
intro hbounded
intro hold_i
intro hold_n
intro hnew_i
intro hnew_n
intro hpreserve
rewrite hsn at hbounded
rewrite hsn at hbounded
have hisn : exists h. h + S i = S n
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hentry_bound_i : forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l
exact finite_bounded_entry_lt
have hentry_bound_n : forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l
exact finite_bounded_entry_lt
have hxb : exists h. h + S x = S n
specialize hentry_bound_i b
specialize hentry_bound_i c
specialize hentry_bound_i (S n)
specialize hentry_bound_i i
specialize hentry_bound_i x
apply hentry_bound_i
exact hbounded
exact hisn
exact hold_i
have hyb : exists h. h + S y = S n
specialize hentry_bound_n b
specialize hentry_bound_n c
specialize hentry_bound_n (S n)
specialize hentry_bound_n n
specialize hentry_bound_n y
apply hentry_bound_n
exact hbounded
exact hnsn
exact hold_n
have heq_i : forall u v. u = v \/ ~(u = v)
exact eq_decidable
have heq_n : forall u v. u = v \/ ~(u = v)
exact eq_decidable
rewrite hsn
rewrite hsn
intro j
intro hj
specialize heq_i j
specialize heq_i i
cases heq_i
exists y
split
rewrite heq_i_left
rewrite heq_i_left
exact hnew_i
exact hyb
specialize heq_n j
specialize heq_n n
cases heq_n
exists x
split
rewrite heq_n_left
rewrite heq_n_left
exact hnew_n
exact hxb
specialize hbounded j
have hold : exists a. (((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a) /\ exists h. h + S a = S n)
apply hbounded
exact hj
cases hold
cases hold_witness
exists x1
split
specialize hpreserve j
specialize hpreserve x1
apply hpreserve
exact hj
exact heq_i_right
exact heq_n_right
exact hold_witness_left
exact hold_witness_right
qed

Occurrences

1,380

Distinct objects

907

Reused references

39

Depth

61

Cuts

37

Certificate SHA-256

d379ada457c5…

finite_swap_last_injectiveFinite swap last injective Quadratic reciprocity campaign checked

A swap-last recoding preserves injectivity of the full successor prefix.

Prerequisites

beta_prefix_swap_last_reflect le_succ le_refl

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_inj_old fp_j_swap_inj_old fp_value_swap_inj_old. (exists fp_gap_swap_inj_old_i. fp_gap_swap_inj_old_i + S fp_i_swap_inj_old = sn) -> (exists fp_gap_swap_inj_old_j. fp_gap_swap_inj_old_j + S fp_j_swap_inj_old = sn) -> (((exists ff_h_swap_inj_old_left. ff_h_swap_inj_old_left + S (fp_value_swap_inj_old) = S ((S (fp_i_swap_inj_old)) * c)) /\ exists ff_q_swap_inj_old_left. b = ff_q_swap_inj_old_left * S ((S (fp_i_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> (((exists ff_h_swap_inj_old_right. ff_h_swap_inj_old_right + S (fp_value_swap_inj_old) = S ((S (fp_j_swap_inj_old)) * c)) /\ exists ff_q_swap_inj_old_right. b = ff_q_swap_inj_old_right * S ((S (fp_j_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> fp_i_swap_inj_old = fp_j_swap_inj_old) -> (((exists ff_h_swap_inj_old_i. ff_h_swap_inj_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_inj_old_i. b = ff_q_swap_inj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_inj_old_n. ff_h_swap_inj_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_inj_old_n. b = ff_q_swap_inj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_inj_new_i. ff_h_swap_inj_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_inj_new_i. z = ff_q_swap_inj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_inj_new_n. ff_h_swap_inj_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_inj_new_n. z = ff_q_swap_inj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_inj_old_j. ff_h_swap_inj_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_inj_old_j. b = ff_q_swap_inj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_inj_new_j. ff_h_swap_inj_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_inj_new_j. z = ff_q_swap_inj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_inj_new fp_j_swap_inj_new fp_value_swap_inj_new. (exists fp_gap_swap_inj_new_i. fp_gap_swap_inj_new_i + S fp_i_swap_inj_new = sn) -> (exists fp_gap_swap_inj_new_j. fp_gap_swap_inj_new_j + S fp_j_swap_inj_new = sn) -> (((exists ff_h_swap_inj_new_left. ff_h_swap_inj_new_left + S (fp_value_swap_inj_new) = S ((S (fp_i_swap_inj_new)) * d)) /\ exists ff_q_swap_inj_new_left. z = ff_q_swap_inj_new_left * S ((S (fp_i_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> (((exists ff_h_swap_inj_new_right. ff_h_swap_inj_new_right + S (fp_value_swap_inj_new) = S ((S (fp_j_swap_inj_new)) * d)) /\ exists ff_q_swap_inj_new_right. z = ff_q_swap_inj_new_right * S ((S (fp_j_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> fp_i_swap_inj_new = fp_j_swap_inj_new)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_inj_old fp_j_swap_inj_old fp_value_swap_inj_old. (exists fp_gap_swap_inj_old_i. fp_gap_swap_inj_old_i + S fp_i_swap_inj_old = sn) -> (exists fp_gap_swap_inj_old_j. fp_gap_swap_inj_old_j + S fp_j_swap_inj_old = sn) -> (((exists ff_h_swap_inj_old_left. ff_h_swap_inj_old_left + S (fp_value_swap_inj_old) = S ((S (fp_i_swap_inj_old)) * c)) /\ exists ff_q_swap_inj_old_left. b = ff_q_swap_inj_old_left * S ((S (fp_i_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> (((exists ff_h_swap_inj_old_right. ff_h_swap_inj_old_right + S (fp_value_swap_inj_old) = S ((S (fp_j_swap_inj_old)) * c)) /\ exists ff_q_swap_inj_old_right. b = ff_q_swap_inj_old_right * S ((S (fp_j_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> fp_i_swap_inj_old = fp_j_swap_inj_old) -> (((exists ff_h_swap_inj_old_i. ff_h_swap_inj_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_inj_old_i. b = ff_q_swap_inj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_inj_old_n. ff_h_swap_inj_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_inj_old_n. b = ff_q_swap_inj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_inj_new_i. ff_h_swap_inj_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_inj_new_i. z = ff_q_swap_inj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_inj_new_n. ff_h_swap_inj_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_inj_new_n. z = ff_q_swap_inj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_inj_old_j. ff_h_swap_inj_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_inj_old_j. b = ff_q_swap_inj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_inj_new_j. ff_h_swap_inj_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_inj_new_j. z = ff_q_swap_inj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_inj_new fp_j_swap_inj_new fp_value_swap_inj_new. (exists fp_gap_swap_inj_new_i. fp_gap_swap_inj_new_i + S fp_i_swap_inj_new = sn) -> (exists fp_gap_swap_inj_new_j. fp_gap_swap_inj_new_j + S fp_j_swap_inj_new = sn) -> (((exists ff_h_swap_inj_new_left. ff_h_swap_inj_new_left + S (fp_value_swap_inj_new) = S ((S (fp_i_swap_inj_new)) * d)) /\ exists ff_q_swap_inj_new_left. z = ff_q_swap_inj_new_left * S ((S (fp_i_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> (((exists ff_h_swap_inj_new_right. ff_h_swap_inj_new_right + S (fp_value_swap_inj_new) = S ((S (fp_j_swap_inj_new)) * d)) /\ exists ff_q_swap_inj_new_right. z = ff_q_swap_inj_new_right * S ((S (fp_j_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> fp_i_swap_inj_new = fp_j_swap_inj_new)
use beta_prefix_swap_last_reflect
use le_succ
use le_refl
intro b
intro c
intro z
intro d
intro n
intro sn
intro i
intro x
intro y
intro hsn
intro hi
intro hinjective
intro hold_i
intro hold_n
intro hnew_i
intro hnew_n
intro hpreserve
rewrite hsn at hinjective
rewrite hsn at hinjective
have hisn : exists h. h + S i = S n
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hreflect_j : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
exact beta_prefix_swap_last_reflect
have hreflect_k : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
exact beta_prefix_swap_last_reflect
rewrite hsn
rewrite hsn
intro j
intro k
intro a
intro hj
intro hk
intro hnew_j
intro hnew_k
specialize hreflect_j b
specialize hreflect_j c
specialize hreflect_j z
specialize hreflect_j d
specialize hreflect_j n
specialize hreflect_j i
specialize hreflect_j x
specialize hreflect_j y
have hreflect_entries_j : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
apply hreflect_j
exact hnew_i
exact hnew_n
exact hpreserve
specialize hreflect_entries_j j
specialize hreflect_entries_j a
have hclass_j : ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ ((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a)))))
apply hreflect_entries_j
exact hj
exact hnew_j
specialize hreflect_k b
specialize hreflect_k c
specialize hreflect_k z
specialize hreflect_k d
specialize hreflect_k n
specialize hreflect_k i
specialize hreflect_k x
specialize hreflect_k y
have hreflect_entries_k : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
apply hreflect_k
exact hnew_i
exact hnew_n
exact hpreserve
specialize hreflect_entries_k k
specialize hreflect_entries_k a
have hclass_k : ((k = i /\ a = y) \/ ((k = n /\ a = x) \/ (~(k = i) /\ (~(k = n) /\ ((exists h. h + S a = S ((S k) * c)) /\ exists q. b = q * S ((S k) * c) + a)))))
apply hreflect_entries_k
exact hk
exact hnew_k
cases hclass_j
cases hclass_j_left
cases hclass_k
cases hclass_k_left
trans i
exact hclass_j_left_left
symm
exact hclass_k_left_left
cases hclass_k_right
cases hclass_k_right_left
have hxy : x = y
trans a
symm
exact hclass_k_right_left_right
exact hclass_j_left_right
have hin : i = n
specialize hinjective i
specialize hinjective n
specialize hinjective x
apply hinjective
exact hisn
exact hnsn
exact hold_i
rewrite hxy
rewrite hxy
exact hold_n
trans i
exact hclass_j_left_left
trans n
exact hin
symm
exact hclass_k_right_left_left
cases hclass_k_right_right
cases hclass_k_right_right_right
have hnk : n = k
specialize hinjective n
specialize hinjective k
specialize hinjective y
apply hinjective
exact hnsn
exact hk
exact hold_n
rewrite <- hclass_j_left_right
rewrite <- hclass_j_left_right
exact hclass_k_right_right_right_right
exfalso
apply hclass_k_right_right_right_left
symm
exact hnk
cases hclass_j_right
cases hclass_j_right_left
cases hclass_k
cases hclass_k_left
have hxy2 : x = y
trans a
symm
exact hclass_j_right_left_right
exact hclass_k_left_right
have hin2 : n = i
specialize hinjective n
specialize hinjective i
specialize hinjective y
apply hinjective
exact hnsn
exact hisn
exact hold_n
rewrite <- hxy2
rewrite <- hxy2
exact hold_i
trans n
exact hclass_j_right_left_left
trans i
exact hin2
symm
exact hclass_k_left_left
cases hclass_k_right
cases hclass_k_right_left
trans n
exact hclass_j_right_left_left
symm
exact hclass_k_right_left_left
cases hclass_k_right_right
cases hclass_k_right_right_right
have hik : i = k
specialize hinjective i
specialize hinjective k
specialize hinjective x
apply hinjective
exact hisn
exact hk
exact hold_i
rewrite <- hclass_j_right_left_right
rewrite <- hclass_j_right_left_right
exact hclass_k_right_right_right_right
exfalso
apply hclass_k_right_right_left
symm
exact hik
cases hclass_j_right_right
cases hclass_j_right_right_right
cases hclass_k
cases hclass_k_left
have hjn : j = n
specialize hinjective j
specialize hinjective n
specialize hinjective y
apply hinjective
exact hj
exact hnsn
rewrite <- hclass_k_left_right
rewrite <- hclass_k_left_right
exact hclass_j_right_right_right_right
exact hold_n
exfalso
apply hclass_j_right_right_right_left
exact hjn
cases hclass_k_right
cases hclass_k_right_left
have hji : j = i
specialize hinjective j
specialize hinjective i
specialize hinjective x
apply hinjective
exact hj
exact hisn
rewrite <- hclass_k_right_left_right
rewrite <- hclass_k_right_left_right
exact hclass_j_right_right_right_right
exact hold_i
exfalso
apply hclass_j_right_right_left
exact hji
cases hclass_k_right_right
cases hclass_k_right_right_right
specialize hinjective j
specialize hinjective k
specialize hinjective a
apply hinjective
exact hj
exact hk
exact hclass_j_right_right_right_right
exact hclass_k_right_right_right_right
qed

Occurrences

2,203

Distinct objects

1,435

Reused references

57

Depth

63

Cuts

53

Certificate SHA-256

b9a49da8fb88…

finite_swap_last_surjective_backFinite swap last surjective back Quadratic reciprocity campaign checked

Surjectivity of a swapped successor prefix transports back to the original code.

Prerequisites

beta_prefix_swap_last_reflect le_succ le_refl

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (((exists ff_h_swap_surj_old_i. ff_h_swap_surj_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_surj_old_i. b = ff_q_swap_surj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_surj_old_n. ff_h_swap_surj_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_surj_old_n. b = ff_q_swap_surj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_surj_new_i. ff_h_swap_surj_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_surj_new_i. z = ff_q_swap_surj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_surj_new_n. ff_h_swap_surj_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_surj_new_n. z = ff_q_swap_surj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_surj_old_j. ff_h_swap_surj_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_surj_old_j. b = ff_q_swap_surj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_surj_new_j. ff_h_swap_surj_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_surj_new_j. z = ff_q_swap_surj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_value_swap_surj_new. (exists fp_gap_swap_surj_new_value. fp_gap_swap_surj_new_value + S fp_value_swap_surj_new = sn) -> exists fp_i_swap_surj_new. ((exists fp_gap_swap_surj_new_index. fp_gap_swap_surj_new_index + S fp_i_swap_surj_new = sn) /\ (((exists ff_h_swap_surj_new_entry. ff_h_swap_surj_new_entry + S (fp_value_swap_surj_new) = S ((S (fp_i_swap_surj_new)) * d)) /\ exists ff_q_swap_surj_new_entry. z = ff_q_swap_surj_new_entry * S ((S (fp_i_swap_surj_new)) * d) + (fp_value_swap_surj_new))))) -> (forall fp_value_swap_surj_old. (exists fp_gap_swap_surj_old_value. fp_gap_swap_surj_old_value + S fp_value_swap_surj_old = sn) -> exists fp_i_swap_surj_old. ((exists fp_gap_swap_surj_old_index. fp_gap_swap_surj_old_index + S fp_i_swap_surj_old = sn) /\ (((exists ff_h_swap_surj_old_entry. ff_h_swap_surj_old_entry + S (fp_value_swap_surj_old) = S ((S (fp_i_swap_surj_old)) * c)) /\ exists ff_q_swap_surj_old_entry. b = ff_q_swap_surj_old_entry * S ((S (fp_i_swap_surj_old)) * c) + (fp_value_swap_surj_old)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (((exists ff_h_swap_surj_old_i. ff_h_swap_surj_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_swap_surj_old_i. b = ff_q_swap_surj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_surj_old_n. ff_h_swap_surj_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_swap_surj_old_n. b = ff_q_swap_surj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_surj_new_i. ff_h_swap_surj_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_swap_surj_new_i. z = ff_q_swap_surj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_surj_new_n. ff_h_swap_surj_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_swap_surj_new_n. z = ff_q_swap_surj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_surj_old_j. ff_h_swap_surj_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_swap_surj_old_j. b = ff_q_swap_surj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_surj_new_j. ff_h_swap_surj_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_swap_surj_new_j. z = ff_q_swap_surj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_value_swap_surj_new. (exists fp_gap_swap_surj_new_value. fp_gap_swap_surj_new_value + S fp_value_swap_surj_new = sn) -> exists fp_i_swap_surj_new. ((exists fp_gap_swap_surj_new_index. fp_gap_swap_surj_new_index + S fp_i_swap_surj_new = sn) /\ (((exists ff_h_swap_surj_new_entry. ff_h_swap_surj_new_entry + S (fp_value_swap_surj_new) = S ((S (fp_i_swap_surj_new)) * d)) /\ exists ff_q_swap_surj_new_entry. z = ff_q_swap_surj_new_entry * S ((S (fp_i_swap_surj_new)) * d) + (fp_value_swap_surj_new))))) -> (forall fp_value_swap_surj_old. (exists fp_gap_swap_surj_old_value. fp_gap_swap_surj_old_value + S fp_value_swap_surj_old = sn) -> exists fp_i_swap_surj_old. ((exists fp_gap_swap_surj_old_index. fp_gap_swap_surj_old_index + S fp_i_swap_surj_old = sn) /\ (((exists ff_h_swap_surj_old_entry. ff_h_swap_surj_old_entry + S (fp_value_swap_surj_old) = S ((S (fp_i_swap_surj_old)) * c)) /\ exists ff_q_swap_surj_old_entry. b = ff_q_swap_surj_old_entry * S ((S (fp_i_swap_surj_old)) * c) + (fp_value_swap_surj_old)))))
use beta_prefix_swap_last_reflect
use le_succ
use le_refl
intro b
intro c
intro z
intro d
intro n
intro sn
intro i
intro x
intro y
intro hsn
intro hi
intro hold_i
intro hold_n
intro hnew_i
intro hnew_n
intro hpreserve
intro hsurjective
rewrite hsn at hsurjective
rewrite hsn at hsurjective
have hisn : exists h. h + S i = S n
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hreflect : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))
exact beta_prefix_swap_last_reflect
specialize hreflect b
specialize hreflect c
specialize hreflect z
specialize hreflect d
specialize hreflect n
specialize hreflect i
specialize hreflect x
specialize hreflect y
have hreflect_entries : forall j a. (exists h. h + S j = S n) -> ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a) -> ((j = i /\ a = y) \/ ((j = n /\ a = x) \/ (~(j = i) /\ (~(j = n) /\ ((exists h. h + S a = S ((S j) * c)) /\ exists q. b = q * S ((S j) * c) + a)))))
apply hreflect
exact hnew_i
exact hnew_n
exact hpreserve
rewrite hsn
rewrite hsn
intro a
intro ha
specialize hsurjective a
have hoccurs : exists j. ((exists h. h + S j = S n) /\ ((exists h. h + S a = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + a))
apply hsurjective
exact ha
cases hoccurs
cases hoccurs_witness
specialize hreflect_entries x1
specialize hreflect_entries a
have hsource : ((x1 = i /\ a = y) \/ ((x1 = n /\ a = x) \/ (~(x1 = i) /\ (~(x1 = n) /\ ((exists h. h + S a = S ((S x1) * c)) /\ exists q. b = q * S ((S x1) * c) + a)))))
apply hreflect_entries
exact hoccurs_witness_left
exact hoccurs_witness_right
cases hsource
cases hsource_left
exists n
split
exact hnsn
rewrite hsource_left_right
rewrite hsource_left_right
exact hold_n
cases hsource_right
cases hsource_right_left
exists i
split
exact hisn
rewrite hsource_right_left_right
rewrite hsource_right_left_right
exact hold_i
cases hsource_right_right
cases hsource_right_right_right
exists x1
split
exact hoccurs_witness_left
exact hsource_right_right_right_right
qed

Occurrences

1,929

Distinct objects

1,161

Reused references

57

Depth

63

Cuts

53

Certificate SHA-256

689f32aef19d…

finite_contains_decidableFinite contains decidable Quadratic reciprocity campaign checked

Occurrence of a value in a nonempty decoded prefix is constructively decidable.

Prerequisites

add_eq_zero_right succ_ne_zero finite_lt_succ_eq_or_lt beta_at_exists beta_at_unique eq_decidable le_refl le_succ

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c l y. ((exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))) \/ ~(exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c l y. ((exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))) \/ ~(exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))))
use add_eq_zero_right
use succ_ne_zero
use finite_lt_succ_eq_or_lt
use beta_at_exists
use beta_at_unique
use eq_decidable
use le_refl
use le_succ
intro b
intro c
induction l
intro y
right
intro hcontains
cases hcontains
cases hcontains_witness
cases hcontains_witness_left
have hsi : S x = 0
specialize add_eq_zero_right x1
specialize add_eq_zero_right (S x)
apply add_eq_zero_right
exact hcontains_witness_left_witness
specialize succ_ne_zero x
apply succ_ne_zero
exact hsi
intro y
have hpresent : (exists i. ((exists h. h + S i = l) /\ ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y))) \/ ~(exists i. ((exists h. h + S i = l) /\ ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y)))
specialize IH y
exact IH
cases hpresent
left
cases hpresent_left
cases hpresent_left_witness
exists x
split
specialize le_succ (S x)
specialize le_succ l
apply le_succ
exact hpresent_left_witness_left
exact hpresent_left_witness_right
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists l
cases beta_at_exists
specialize eq_decidable x
specialize eq_decidable y
cases eq_decidable
left
exists l
split
specialize le_refl (S l)
exact le_refl
rewrite eq_decidable_left at beta_at_exists_witness
rewrite eq_decidable_left at beta_at_exists_witness
exact beta_at_exists_witness
right
intro hfull
cases hfull
cases hfull_witness
have hindex : x1 = l \/ exists h. h + S x1 = l
specialize finite_lt_succ_eq_or_lt l
specialize finite_lt_succ_eq_or_lt x1
apply finite_lt_succ_eq_or_lt
exact hfull_witness_left
cases hindex
have hentry : ((exists h. h + S y = S ((S l) * c)) /\ exists q. b = q * S ((S l) * c) + y)
rewrite hindex_left at hfull_witness_right
rewrite hindex_left at hfull_witness_right
exact hfull_witness_right
have hxy : x = y
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique l
specialize beta_at_unique x
specialize beta_at_unique y
apply beta_at_unique
exact beta_at_exists_witness
exact hentry
apply eq_decidable_right
exact hxy
apply hpresent_right
exists x1
split
exact hindex_right
exact hfull_witness_right
qed

Occurrences

1,961

Distinct objects

1,140

Reused references

61

Depth

64

Cuts

60

Certificate SHA-256

264b86a40d0a…

finite_bounded_prefix_without_topFinite bounded prefix without top Quadratic reciprocity campaign checked

If a successor prefix omits its top value, its old prefix is bounded by the predecessor.

Prerequisites

le_succ finite_lt_succ_eq_or_lt

Used directly by

finite_no_top_successor_gate finite_bounded_injective_surjective finite_fixed_last_prefix_bounded

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall i. (exists h. h + S i = n) -> ~(((exists ff_h_top_i. ff_h_top_i + S (n) = S ((S (i)) * c)) /\ exists ff_q_top_i. b = ff_q_top_i * S ((S (i)) * c) + (n)))) -> (forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall i. (exists h. h + S i = n) -> ~(((exists ff_h_top_i. ff_h_top_i + S (n) = S ((S (i)) * c)) /\ exists ff_q_top_i. b = ff_q_top_i * S ((S (i)) * c) + (n)))) -> (forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n)))
use le_succ
use finite_lt_succ_eq_or_lt
intro b
intro c
intro n
intro sn
intro hsn
intro hbounded
intro hnotop
rewrite hsn at hbounded
rewrite hsn at hbounded
intro i
intro hi
specialize hbounded i
have hfull : exists x. (((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) /\ exists h. h + S x = S n)
apply hbounded
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
cases hfull
cases hfull_witness
have hsplit : x = n \/ exists h. h + S x = n
specialize finite_lt_succ_eq_or_lt n
specialize finite_lt_succ_eq_or_lt x
apply finite_lt_succ_eq_or_lt
exact hfull_witness_right
cases hsplit
exfalso
specialize hnotop i
apply hnotop
exact hi
rewrite <- hsplit_left
rewrite <- hsplit_left
exact hfull_witness_left
exists x
split
exact hfull_witness_left
exact hsplit_right
qed

Occurrences

216

Distinct objects

185

Reused references

5

Depth

23

Cuts

8

Certificate SHA-256

994ca043b11b…

finite_bounded_last_succFinite bounded last succ Quadratic reciprocity campaign checked

A bounded successor prefix exposes a bounded final decoded value.

Prerequisites

le_refl

Used directly by

finite_last_is_top_from_prefix_surjective finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> exists x. ((((exists ff_h_last_x. ff_h_last_x + S (x) = S ((S (n)) * c)) /\ exists ff_q_last_x. b = ff_q_last_x * S ((S (n)) * c) + (x))) /\ exists h. h + S x = S n)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> exists x. ((((exists ff_h_last_x. ff_h_last_x + S (x) = S ((S (n)) * c)) /\ exists ff_q_last_x. b = ff_q_last_x * S ((S (n)) * c) + (x))) /\ exists h. h + S x = S n)
use le_refl
intro b
intro c
intro n
intro sn
intro hsn
intro hbounded
rewrite hsn at hbounded
rewrite hsn at hbounded
specialize hbounded n
apply hbounded
specialize le_refl (S n)
exact le_refl
qed

Occurrences

53

Distinct objects

53

Reused references

0

Depth

17

Cuts

2

Certificate SHA-256

ac7f1147afe2…

finite_surjective_succ_introFinite surjective succ intro Quadratic reciprocity campaign checked

A surjective prefix plus its new top value is surjective at successor length.

Prerequisites

finite_lt_succ_eq_or_lt le_refl le_succ

Used directly by

finite_surjective_succ_from_prefix

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))
use finite_lt_succ_eq_or_lt
use le_refl
use le_succ
intro b
intro c
intro n
intro sn
intro hsn
intro hsurj
intro hlast
rewrite hsn
rewrite hsn
intro y
intro hy
have hsplit : y = n \/ exists h. h + S y = n
specialize finite_lt_succ_eq_or_lt n
specialize finite_lt_succ_eq_or_lt y
apply finite_lt_succ_eq_or_lt
exact hy
cases hsplit
exists n
split
specialize le_refl (S n)
exact le_refl
rewrite hsplit_left
rewrite hsplit_left
exact hlast
specialize hsurj y
have hpre : exists i. ((exists h. h + S i = n) /\ ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y))
apply hsurj
exact hsplit_right
cases hpre
cases hpre_witness
exists x
split
specialize le_succ (S x)
specialize le_succ n
apply le_succ
exact hpre_witness_left
exact hpre_witness_right
qed

Occurrences

243

Distinct objects

195

Reused references

6

Depth

22

Cuts

10

Certificate SHA-256

c2ee0348b812…

finite_last_is_top_from_prefix_surjectiveFinite last is top from prefix surjective Quadratic reciprocity campaign checked

A bounded injective successor sequence must place the new value last once its prefix is surjective.

Prerequisites

finite_bounded_last_succ finite_lt_succ_eq_or_lt le_refl le_succ lt_irrefl_expanded

Used directly by

finite_surjective_succ_from_prefix

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n)))
use finite_bounded_last_succ
use finite_lt_succ_eq_or_lt
use le_refl
use le_succ
use lt_irrefl_expanded
intro b
intro c
intro n
intro sn
intro hsn
intro hbounded
intro hinj
intro hsurj
rewrite hsn at hinj
rewrite hsn at hinj
have hlast : exists x. (((exists h. h + S x = S ((S n) * c)) /\ exists q. b = q * S ((S n) * c) + x) /\ exists h. h + S x = S n)
specialize finite_bounded_last_succ b
specialize finite_bounded_last_succ c
specialize finite_bounded_last_succ n
specialize finite_bounded_last_succ sn
apply finite_bounded_last_succ
exact hsn
exact hbounded
cases hlast
cases hlast_witness
have hsplit : x = n \/ exists h. h + S x = n
specialize finite_lt_succ_eq_or_lt n
specialize finite_lt_succ_eq_or_lt x
apply finite_lt_succ_eq_or_lt
exact hlast_witness_right
cases hsplit
rewrite hsplit_left at hlast_witness_left
rewrite hsplit_left at hlast_witness_left
exact hlast_witness_left
specialize hsurj x
have hpre : exists i. ((exists h. h + S i = n) /\ ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x))
apply hsurj
exact hsplit_right
cases hpre
cases hpre_witness
have hni : n = x1
specialize hinj n
specialize hinj x1
specialize hinj x
apply hinj
specialize le_refl (S n)
exact le_refl
specialize le_succ (S x1)
specialize le_succ n
apply le_succ
exact hpre_witness_left
exact hlast_witness_left
exact hpre_witness_right
exfalso
specialize lt_irrefl_expanded n
apply lt_irrefl_expanded
rewrite hni
exact hpre_witness_left
qed

Occurrences

402

Distinct objects

302

Reused references

8

Depth

30

Cuts

16

Certificate SHA-256

0a7e3280cd99…

finite_surjective_succ_from_prefixFinite surjective succ from prefix Quadratic reciprocity campaign checked

The available successor branch extends prefix surjectivity to the full prefix.

Prerequisites

finite_last_is_top_from_prefix_surjective finite_surjective_succ_intro

Used directly by

finite_no_top_successor_gate finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))
use finite_last_is_top_from_prefix_surjective
use finite_surjective_succ_intro
intro b
intro c
intro n
intro sn
intro hsn
intro hbounded
intro hinj
intro hsurj
have hlast : ((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n))
specialize finite_last_is_top_from_prefix_surjective b
specialize finite_last_is_top_from_prefix_surjective c
specialize finite_last_is_top_from_prefix_surjective n
specialize finite_last_is_top_from_prefix_surjective sn
apply finite_last_is_top_from_prefix_surjective
exact hsn
exact hbounded
exact hinj
exact hsurj
specialize finite_surjective_succ_intro b
specialize finite_surjective_succ_intro c
specialize finite_surjective_succ_intro n
specialize finite_surjective_succ_intro sn
apply finite_surjective_succ_intro
exact hsn
exact hsurj
exact hlast
qed

Occurrences

678

Distinct objects

385

Reused references

11

Depth

31

Cuts

28

Certificate SHA-256

78bdcadaa47a…

finite_no_top_successor_gateFinite no top successor gate Quadratic reciprocity campaign checked

The no-top branch of the constructive successor induction is complete.

Prerequisites

finite_bounded_prefix_without_top finite_injective_prefix_succ finite_surjective_succ_from_prefix

Used directly by

finite_bounded_injective_surjective

Exact expanded PA statement

Copy statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) -> ((forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n)))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) -> ((forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n)))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))
use finite_bounded_prefix_without_top
use finite_injective_prefix_succ
use finite_surjective_succ_from_prefix
intro b
intro c
intro n
intro sn
intro hsn
intro hbounded
intro hinj
intro hmissing
intro hinduction
have hnotop : forall i. (exists h. h + S i = n) -> ~((exists h. h + S n = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + n)
intro i
intro hi
intro hentry
apply hmissing
exists i
split
exact hi
exact hentry
have hprefix_bounded : forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))
specialize finite_bounded_prefix_without_top b
specialize finite_bounded_prefix_without_top c
specialize finite_bounded_prefix_without_top n
specialize finite_bounded_prefix_without_top sn
apply finite_bounded_prefix_without_top
exact hsn
exact hbounded
exact hnotop
have hprefix_injective : forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix
specialize finite_injective_prefix_succ b
specialize finite_injective_prefix_succ c
specialize finite_injective_prefix_succ n
specialize finite_injective_prefix_succ sn
apply finite_injective_prefix_succ
exact hsn
exact hinj
have hprefix_surjective : forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))
apply hinduction
exact hprefix_bounded
exact hprefix_injective
specialize finite_surjective_succ_from_prefix b
specialize finite_surjective_succ_from_prefix c
specialize finite_surjective_succ_from_prefix n
specialize finite_surjective_succ_from_prefix sn
apply finite_surjective_succ_from_prefix
exact hsn
exact hbounded
exact hinj
exact hprefix_surjective
qed

Occurrences

1,054

Distinct objects

553

Reused references

14

Depth

36

Cuts

41

Certificate SHA-256

2d6d4891922b…

finite_bounded_injective_surjectiveFinite bounded injective surjective Quadratic reciprocity campaign checked

Every bounded injective beta-coded prefix is surjective onto its finite interval.

Prerequisites

finite_surjective_zero finite_contains_decidable finite_bounded_last_succ beta_prefix_swap_last_from_entries finite_swap_last_bounded finite_swap_last_injective finite_bounded_prefix_without_top finite_injective_prefix_succ finite_surjective_succ_from_prefix finite_swap_last_surjective_back finite_no_top_successor_gate beta_at_unique le_succ le_refl lt_irrefl_expanded

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall n b c. (forall fp_i_pigeon_bounded. (exists fp_gap_pigeon_bounded_index. fp_gap_pigeon_bounded_index + S fp_i_pigeon_bounded = n) -> exists fp_value_pigeon_bounded. ((((exists ff_h_pigeon_bounded_entry. ff_h_pigeon_bounded_entry + S (fp_value_pigeon_bounded) = S ((S (fp_i_pigeon_bounded)) * c)) /\ exists ff_q_pigeon_bounded_entry. b = ff_q_pigeon_bounded_entry * S ((S (fp_i_pigeon_bounded)) * c) + (fp_value_pigeon_bounded))) /\ (exists fp_gap_pigeon_bounded_value. fp_gap_pigeon_bounded_value + S fp_value_pigeon_bounded = n))) -> (forall fp_i_pigeon_injective fp_j_pigeon_injective fp_value_pigeon_injective. (exists fp_gap_pigeon_injective_i. fp_gap_pigeon_injective_i + S fp_i_pigeon_injective = n) -> (exists fp_gap_pigeon_injective_j. fp_gap_pigeon_injective_j + S fp_j_pigeon_injective = n) -> (((exists ff_h_pigeon_injective_left. ff_h_pigeon_injective_left + S (fp_value_pigeon_injective) = S ((S (fp_i_pigeon_injective)) * c)) /\ exists ff_q_pigeon_injective_left. b = ff_q_pigeon_injective_left * S ((S (fp_i_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> (((exists ff_h_pigeon_injective_right. ff_h_pigeon_injective_right + S (fp_value_pigeon_injective) = S ((S (fp_j_pigeon_injective)) * c)) /\ exists ff_q_pigeon_injective_right. b = ff_q_pigeon_injective_right * S ((S (fp_j_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> fp_i_pigeon_injective = fp_j_pigeon_injective) -> (forall fp_value_pigeon_surjective. (exists fp_gap_pigeon_surjective_value. fp_gap_pigeon_surjective_value + S fp_value_pigeon_surjective = n) -> exists fp_i_pigeon_surjective. ((exists fp_gap_pigeon_surjective_index. fp_gap_pigeon_surjective_index + S fp_i_pigeon_surjective = n) /\ (((exists ff_h_pigeon_surjective_entry. ff_h_pigeon_surjective_entry + S (fp_value_pigeon_surjective) = S ((S (fp_i_pigeon_surjective)) * c)) /\ exists ff_q_pigeon_surjective_entry. b = ff_q_pigeon_surjective_entry * S ((S (fp_i_pigeon_surjective)) * c) + (fp_value_pigeon_surjective)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall n b c. (forall fp_i_pigeon_bounded. (exists fp_gap_pigeon_bounded_index. fp_gap_pigeon_bounded_index + S fp_i_pigeon_bounded = n) -> exists fp_value_pigeon_bounded. ((((exists ff_h_pigeon_bounded_entry. ff_h_pigeon_bounded_entry + S (fp_value_pigeon_bounded) = S ((S (fp_i_pigeon_bounded)) * c)) /\ exists ff_q_pigeon_bounded_entry. b = ff_q_pigeon_bounded_entry * S ((S (fp_i_pigeon_bounded)) * c) + (fp_value_pigeon_bounded))) /\ (exists fp_gap_pigeon_bounded_value. fp_gap_pigeon_bounded_value + S fp_value_pigeon_bounded = n))) -> (forall fp_i_pigeon_injective fp_j_pigeon_injective fp_value_pigeon_injective. (exists fp_gap_pigeon_injective_i. fp_gap_pigeon_injective_i + S fp_i_pigeon_injective = n) -> (exists fp_gap_pigeon_injective_j. fp_gap_pigeon_injective_j + S fp_j_pigeon_injective = n) -> (((exists ff_h_pigeon_injective_left. ff_h_pigeon_injective_left + S (fp_value_pigeon_injective) = S ((S (fp_i_pigeon_injective)) * c)) /\ exists ff_q_pigeon_injective_left. b = ff_q_pigeon_injective_left * S ((S (fp_i_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> (((exists ff_h_pigeon_injective_right. ff_h_pigeon_injective_right + S (fp_value_pigeon_injective) = S ((S (fp_j_pigeon_injective)) * c)) /\ exists ff_q_pigeon_injective_right. b = ff_q_pigeon_injective_right * S ((S (fp_j_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> fp_i_pigeon_injective = fp_j_pigeon_injective) -> (forall fp_value_pigeon_surjective. (exists fp_gap_pigeon_surjective_value. fp_gap_pigeon_surjective_value + S fp_value_pigeon_surjective = n) -> exists fp_i_pigeon_surjective. ((exists fp_gap_pigeon_surjective_index. fp_gap_pigeon_surjective_index + S fp_i_pigeon_surjective = n) /\ (((exists ff_h_pigeon_surjective_entry. ff_h_pigeon_surjective_entry + S (fp_value_pigeon_surjective) = S ((S (fp_i_pigeon_surjective)) * c)) /\ exists ff_q_pigeon_surjective_entry. b = ff_q_pigeon_surjective_entry * S ((S (fp_i_pigeon_surjective)) * c) + (fp_value_pigeon_surjective)))))
use finite_surjective_zero
use finite_contains_decidable
use finite_bounded_last_succ
use beta_prefix_swap_last_from_entries
use finite_swap_last_bounded
use finite_swap_last_injective
use finite_bounded_prefix_without_top
use finite_injective_prefix_succ
use finite_surjective_succ_from_prefix
use finite_swap_last_surjective_back
use finite_no_top_successor_gate
use beta_at_unique
use le_succ
use le_refl
use lt_irrefl_expanded
induction n
intro b
intro c
intro hbounded
intro hinjective
specialize finite_surjective_zero b
specialize finite_surjective_zero c
specialize finite_surjective_zero 0
apply finite_surjective_zero
refl
intro b
intro c
intro hbounded
intro hinjective
have hcontains : (exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) \/ ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n)))))
specialize finite_contains_decidable b
specialize finite_contains_decidable c
specialize finite_contains_decidable n
specialize finite_contains_decidable n
exact finite_contains_decidable
cases hcontains
cases hcontains_left
cases hcontains_left_witness
have hlast : exists y. (((exists h. h + S y = S ((S n) * c)) /\ exists q. b = q * S ((S n) * c) + y) /\ exists h. h + S y = S n)
specialize finite_bounded_last_succ b
specialize finite_bounded_last_succ c
specialize finite_bounded_last_succ n
specialize finite_bounded_last_succ (S n)
apply finite_bounded_last_succ
refl
exact hbounded
cases hlast
cases hlast_witness
have hswap : exists z d. ((((exists ff_h_pigeon_swap_new_i. ff_h_pigeon_swap_new_i + S (x1) = S ((S (x)) * d)) /\ exists ff_q_pigeon_swap_new_i. z = ff_q_pigeon_swap_new_i * S ((S (x)) * d) + (x1))) /\ ((((exists ff_h_pigeon_swap_new_n. ff_h_pigeon_swap_new_n + S (n) = S ((S (n)) * d)) /\ exists ff_q_pigeon_swap_new_n. z = ff_q_pigeon_swap_new_n * S ((S (n)) * d) + (n))) /\ forall j a. (exists h. h + S j = S n) -> ~(j = x) -> ~(j = n) -> (((exists ff_h_pigeon_swap_old_other. ff_h_pigeon_swap_old_other + S (a) = S ((S (j)) * c)) /\ exists ff_q_pigeon_swap_old_other. b = ff_q_pigeon_swap_old_other * S ((S (j)) * c) + (a))) -> (((exists ff_h_pigeon_swap_new_other. ff_h_pigeon_swap_new_other + S (a) = S ((S (j)) * d)) /\ exists ff_q_pigeon_swap_new_other. z = ff_q_pigeon_swap_new_other * S ((S (j)) * d) + (a)))))
specialize beta_prefix_swap_last_from_entries b
specialize beta_prefix_swap_last_from_entries c
specialize beta_prefix_swap_last_from_entries n
specialize beta_prefix_swap_last_from_entries x
specialize beta_prefix_swap_last_from_entries n
specialize beta_prefix_swap_last_from_entries x1
apply beta_prefix_swap_last_from_entries
exact hcontains_left_witness_left
exact hcontains_left_witness_right
exact hlast_witness_left
cases hswap
cases hswap_witness
cases hswap_witness_witness
cases hswap_witness_witness_right
have hswap_bounded : forall fp_i_pigeon_swapped_bounded. (exists fp_gap_pigeon_swapped_bounded_index. fp_gap_pigeon_swapped_bounded_index + S fp_i_pigeon_swapped_bounded = S n) -> exists fp_value_pigeon_swapped_bounded. ((((exists ff_h_pigeon_swapped_bounded_entry. ff_h_pigeon_swapped_bounded_entry + S (fp_value_pigeon_swapped_bounded) = S ((S (fp_i_pigeon_swapped_bounded)) * x3)) /\ exists ff_q_pigeon_swapped_bounded_entry. x2 = ff_q_pigeon_swapped_bounded_entry * S ((S (fp_i_pigeon_swapped_bounded)) * x3) + (fp_value_pigeon_swapped_bounded))) /\ (exists fp_gap_pigeon_swapped_bounded_value. fp_gap_pigeon_swapped_bounded_value + S fp_value_pigeon_swapped_bounded = S n))
specialize finite_swap_last_bounded b
specialize finite_swap_last_bounded c
specialize finite_swap_last_bounded x2
specialize finite_swap_last_bounded x3
specialize finite_swap_last_bounded n
specialize finite_swap_last_bounded (S n)
specialize finite_swap_last_bounded x
specialize finite_swap_last_bounded n
specialize finite_swap_last_bounded x1
apply finite_swap_last_bounded
refl
exact hcontains_left_witness_left
exact hbounded
exact hcontains_left_witness_right
exact hlast_witness_left
exact hswap_witness_witness_left
exact hswap_witness_witness_right_left
exact hswap_witness_witness_right_right
have hswap_injective : forall fp_i_pigeon_swapped_injective fp_j_pigeon_swapped_injective fp_value_pigeon_swapped_injective. (exists fp_gap_pigeon_swapped_injective_i. fp_gap_pigeon_swapped_injective_i + S fp_i_pigeon_swapped_injective = S n) -> (exists fp_gap_pigeon_swapped_injective_j. fp_gap_pigeon_swapped_injective_j + S fp_j_pigeon_swapped_injective = S n) -> (((exists ff_h_pigeon_swapped_injective_left. ff_h_pigeon_swapped_injective_left + S (fp_value_pigeon_swapped_injective) = S ((S (fp_i_pigeon_swapped_injective)) * x3)) /\ exists ff_q_pigeon_swapped_injective_left. x2 = ff_q_pigeon_swapped_injective_left * S ((S (fp_i_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> (((exists ff_h_pigeon_swapped_injective_right. ff_h_pigeon_swapped_injective_right + S (fp_value_pigeon_swapped_injective) = S ((S (fp_j_pigeon_swapped_injective)) * x3)) /\ exists ff_q_pigeon_swapped_injective_right. x2 = ff_q_pigeon_swapped_injective_right * S ((S (fp_j_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> fp_i_pigeon_swapped_injective = fp_j_pigeon_swapped_injective
specialize finite_swap_last_injective b
specialize finite_swap_last_injective c
specialize finite_swap_last_injective x2
specialize finite_swap_last_injective x3
specialize finite_swap_last_injective n
specialize finite_swap_last_injective (S n)
specialize finite_swap_last_injective x
specialize finite_swap_last_injective n
specialize finite_swap_last_injective x1
apply finite_swap_last_injective
refl
exact hcontains_left_witness_left
exact hinjective
exact hcontains_left_witness_right
exact hlast_witness_left
exact hswap_witness_witness_left
exact hswap_witness_witness_right_left
exact hswap_witness_witness_right_right
have hnotop : forall j. (exists h. h + S j = n) -> ~(((exists ff_h_pigeon_top_j. ff_h_pigeon_top_j + S (n) = S ((S (j)) * x3)) /\ exists ff_q_pigeon_top_j. x2 = ff_q_pigeon_top_j * S ((S (j)) * x3) + (n)))
intro j
intro hj
intro htop
have hjsn : exists h. h + S j = S n
specialize le_succ (S j)
specialize le_succ n
apply le_succ
exact hj
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hjneq : j = n
specialize hswap_injective j
specialize hswap_injective n
specialize hswap_injective n
apply hswap_injective
exact hjsn
exact hnsn
exact htop
exact hswap_witness_witness_right_left
specialize lt_irrefl_expanded n
apply lt_irrefl_expanded
rewrite hjneq at hj
exact hj
have hprefix_bounded : forall fp_i_pigeon_swapped_prefix_bounded. (exists fp_gap_pigeon_swapped_prefix_bounded_index. fp_gap_pigeon_swapped_prefix_bounded_index + S fp_i_pigeon_swapped_prefix_bounded = n) -> exists fp_value_pigeon_swapped_prefix_bounded. ((((exists ff_h_pigeon_swapped_prefix_bounded_entry. ff_h_pigeon_swapped_prefix_bounded_entry + S (fp_value_pigeon_swapped_prefix_bounded) = S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3)) /\ exists ff_q_pigeon_swapped_prefix_bounded_entry. x2 = ff_q_pigeon_swapped_prefix_bounded_entry * S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3) + (fp_value_pigeon_swapped_prefix_bounded))) /\ (exists fp_gap_pigeon_swapped_prefix_bounded_value. fp_gap_pigeon_swapped_prefix_bounded_value + S fp_value_pigeon_swapped_prefix_bounded = n))
specialize finite_bounded_prefix_without_top x2
specialize finite_bounded_prefix_without_top x3
specialize finite_bounded_prefix_without_top n
specialize finite_bounded_prefix_without_top (S n)
apply finite_bounded_prefix_without_top
refl
exact hswap_bounded
exact hnotop
have hprefix_injective : forall fp_i_pigeon_swapped_prefix_injective fp_j_pigeon_swapped_prefix_injective fp_value_pigeon_swapped_prefix_injective. (exists fp_gap_pigeon_swapped_prefix_injective_i. fp_gap_pigeon_swapped_prefix_injective_i + S fp_i_pigeon_swapped_prefix_injective = n) -> (exists fp_gap_pigeon_swapped_prefix_injective_j. fp_gap_pigeon_swapped_prefix_injective_j + S fp_j_pigeon_swapped_prefix_injective = n) -> (((exists ff_h_pigeon_swapped_prefix_injective_left. ff_h_pigeon_swapped_prefix_injective_left + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3)) /\ exists ff_q_pigeon_swapped_prefix_injective_left. x2 = ff_q_pigeon_swapped_prefix_injective_left * S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> (((exists ff_h_pigeon_swapped_prefix_injective_right. ff_h_pigeon_swapped_prefix_injective_right + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3)) /\ exists ff_q_pigeon_swapped_prefix_injective_right. x2 = ff_q_pigeon_swapped_prefix_injective_right * S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> fp_i_pigeon_swapped_prefix_injective = fp_j_pigeon_swapped_prefix_injective
specialize finite_injective_prefix_succ x2
specialize finite_injective_prefix_succ x3
specialize finite_injective_prefix_succ n
specialize finite_injective_prefix_succ (S n)
apply finite_injective_prefix_succ
refl
exact hswap_injective
have hprefix_surjective : forall fp_value_pigeon_swapped_prefix_surjective. (exists fp_gap_pigeon_swapped_prefix_surjective_value. fp_gap_pigeon_swapped_prefix_surjective_value + S fp_value_pigeon_swapped_prefix_surjective = n) -> exists fp_i_pigeon_swapped_prefix_surjective. ((exists fp_gap_pigeon_swapped_prefix_surjective_index. fp_gap_pigeon_swapped_prefix_surjective_index + S fp_i_pigeon_swapped_prefix_surjective = n) /\ (((exists ff_h_pigeon_swapped_prefix_surjective_entry. ff_h_pigeon_swapped_prefix_surjective_entry + S (fp_value_pigeon_swapped_prefix_surjective) = S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3)) /\ exists ff_q_pigeon_swapped_prefix_surjective_entry. x2 = ff_q_pigeon_swapped_prefix_surjective_entry * S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3) + (fp_value_pigeon_swapped_prefix_surjective))))
specialize IH x2
specialize IH x3
apply IH
exact hprefix_bounded
exact hprefix_injective
have hswap_surjective : forall fp_value_pigeon_swapped_surjective. (exists fp_gap_pigeon_swapped_surjective_value. fp_gap_pigeon_swapped_surjective_value + S fp_value_pigeon_swapped_surjective = S n) -> exists fp_i_pigeon_swapped_surjective. ((exists fp_gap_pigeon_swapped_surjective_index. fp_gap_pigeon_swapped_surjective_index + S fp_i_pigeon_swapped_surjective = S n) /\ (((exists ff_h_pigeon_swapped_surjective_entry. ff_h_pigeon_swapped_surjective_entry + S (fp_value_pigeon_swapped_surjective) = S ((S (fp_i_pigeon_swapped_surjective)) * x3)) /\ exists ff_q_pigeon_swapped_surjective_entry. x2 = ff_q_pigeon_swapped_surjective_entry * S ((S (fp_i_pigeon_swapped_surjective)) * x3) + (fp_value_pigeon_swapped_surjective))))
specialize finite_surjective_succ_from_prefix x2
specialize finite_surjective_succ_from_prefix x3
specialize finite_surjective_succ_from_prefix n
specialize finite_surjective_succ_from_prefix (S n)
apply finite_surjective_succ_from_prefix
refl
exact hswap_bounded
exact hswap_injective
exact hprefix_surjective
specialize finite_swap_last_surjective_back b
specialize finite_swap_last_surjective_back c
specialize finite_swap_last_surjective_back x2
specialize finite_swap_last_surjective_back x3
specialize finite_swap_last_surjective_back n
specialize finite_swap_last_surjective_back (S n)
specialize finite_swap_last_surjective_back x
specialize finite_swap_last_surjective_back n
specialize finite_swap_last_surjective_back x1
apply finite_swap_last_surjective_back
refl
exact hcontains_left_witness_left
exact hcontains_left_witness_right
exact hlast_witness_left
exact hswap_witness_witness_left
exact hswap_witness_witness_right_left
exact hswap_witness_witness_right_right
exact hswap_surjective
specialize finite_no_top_successor_gate b
specialize finite_no_top_successor_gate c
specialize finite_no_top_successor_gate n
specialize finite_no_top_successor_gate (S n)
apply finite_no_top_successor_gate
refl
exact hbounded
exact hinjective
exact hcontains_right
intro hprefix_bounded
intro hprefix_injective
specialize IH b
specialize IH c
apply IH
exact hprefix_bounded
exact hprefix_injective
qed

Occurrences

42,463

Distinct objects

6,399

Reused references

274

Depth

89

Cuts

1,266

Certificate SHA-256

3e4a5f2303e7…

beta_prefix_replace_reflectBeta prefix replace reflect Quadratic reciprocity campaign checked

A decoded entry of a one-position replacement is either the replacement or the original entry.

Prerequisites

eq_decidable beta_at_exists beta_at_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c z d k i y. (exists h. h + S i = k) -> (((exists ff_h_replace_reflect_new_i. ff_h_replace_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_replace_reflect_new_i. z = ff_q_replace_reflect_new_i * S ((S (i)) * d) + (y))) -> (forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_replace_reflect_old_j. ff_h_replace_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_reflect_old_j. b = ff_q_replace_reflect_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_replace_reflect_new_j. ff_h_replace_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_reflect_new_j. z = ff_q_replace_reflect_new_j * S ((S (j)) * d) + (a)))) -> forall j a. (exists h. h + S j = k) -> (((exists ff_h_replace_reflect_new_j. ff_h_replace_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_reflect_new_j. z = ff_q_replace_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ (~(j = i) /\ (((exists ff_h_replace_reflect_old_j. ff_h_replace_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_reflect_old_j. b = ff_q_replace_reflect_old_j * S ((S (j)) * c) + (a)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d k i y. (exists h. h + S i = k) -> (((exists ff_h_replace_reflect_new_i. ff_h_replace_reflect_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_replace_reflect_new_i. z = ff_q_replace_reflect_new_i * S ((S (i)) * d) + (y))) -> (forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_replace_reflect_old_j. ff_h_replace_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_reflect_old_j. b = ff_q_replace_reflect_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_replace_reflect_new_j. ff_h_replace_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_reflect_new_j. z = ff_q_replace_reflect_new_j * S ((S (j)) * d) + (a)))) -> forall j a. (exists h. h + S j = k) -> (((exists ff_h_replace_reflect_new_j. ff_h_replace_reflect_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_replace_reflect_new_j. z = ff_q_replace_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\ a = y) \/ (~(j = i) /\ (((exists ff_h_replace_reflect_old_j. ff_h_replace_reflect_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_replace_reflect_old_j. b = ff_q_replace_reflect_old_j * S ((S (j)) * c) + (a)))))
use eq_decidable
use beta_at_exists
use beta_at_unique
intro b
intro c
intro z
intro d
intro k
intro i
intro y
intro hi
intro hnew_i
intro hpreserve
intro j
intro a
intro hj
intro hnew
specialize eq_decidable j
specialize eq_decidable i
cases eq_decidable
left
split
exact eq_decidable_left
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique i
specialize beta_at_unique a
specialize beta_at_unique y
apply beta_at_unique
rewrite eq_decidable_left at hnew
rewrite eq_decidable_left at hnew
exact hnew
exact hnew_i
specialize beta_at_exists b
specialize beta_at_exists c
specialize beta_at_exists j
cases beta_at_exists
have htransport : ((exists h. h + S x = S ((S j) * d)) /\ exists q. z = q * S ((S j) * d) + x)
specialize hpreserve j
specialize hpreserve x
apply hpreserve
exact hj
exact eq_decidable_right
exact beta_at_exists_witness
have hax : a = x
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique j
specialize beta_at_unique a
specialize beta_at_unique x
apply beta_at_unique
exact hnew
exact htransport
right
split
exact eq_decidable_right
rewrite hax
rewrite hax
exact beta_at_exists_witness
qed

Occurrences

1,735

Distinct objects

1,011

Reused references

55

Depth

62

Cuts

48

Certificate SHA-256

3e50d9582fa7…

beta_product_replace_balanceBeta product replace balance Quadratic reciprocity campaign checked

Replacing one factor balances the old and new finite products by the exchanged values.

Prerequisites

add_eq_zero_right succ_ne_zero finite_lt_succ_eq_or_lt beta_product_succ_decompose beta_product_transport_prefix beta_product_functional beta_at_unique mul_assoc mul_comm le_succ le_refl lt_irrefl_expanded

Used directly by

beta_product_swap_last_invariant

Exact expanded PA statement

Copy statement

forall k b c z d i x y p q. (exists h. h + S i = k) -> (((exists ff_h_balance_old_i. ff_h_balance_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_balance_old_i. b = ff_q_balance_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_balance_new_i. ff_h_balance_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_balance_new_i. z = ff_q_balance_new_i * S ((S (i)) * d) + (y))) -> (forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_balance_old_j. ff_h_balance_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_balance_old_j. b = ff_q_balance_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_balance_new_j. ff_h_balance_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_balance_new_j. z = ff_q_balance_new_j * S ((S (j)) * d) + (a)))) -> (exists ff_u_balance_old ff_v_balance_old. ((((exists ff_h_balance_old_start. ff_h_balance_old_start + S (1) = S ((S (0)) * ff_v_balance_old)) /\ exists ff_q_balance_old_start. ff_u_balance_old = ff_q_balance_old_start * S ((S (0)) * ff_v_balance_old) + (1))) /\ ((((exists ff_h_balance_old_terminal. ff_h_balance_old_terminal + S (p) = S ((S (k)) * ff_v_balance_old)) /\ exists ff_q_balance_old_terminal. ff_u_balance_old = ff_q_balance_old_terminal * S ((S (k)) * ff_v_balance_old) + (p))) /\ forall ff_i_balance_old. (exists ff_lt_balance_old_bound. ff_lt_balance_old_bound + S ff_i_balance_old = k) -> exists ff_p_balance_old ff_r_balance_old ff_s_balance_old. ((((exists ff_h_balance_old_factor. ff_h_balance_old_factor + S (ff_p_balance_old) = S ((S (ff_i_balance_old)) * c)) /\ exists ff_q_balance_old_factor. b = ff_q_balance_old_factor * S ((S (ff_i_balance_old)) * c) + (ff_p_balance_old))) /\ ((((exists ff_h_balance_old_partial. ff_h_balance_old_partial + S (ff_r_balance_old) = S ((S (ff_i_balance_old)) * ff_v_balance_old)) /\ exists ff_q_balance_old_partial. ff_u_balance_old = ff_q_balance_old_partial * S ((S (ff_i_balance_old)) * ff_v_balance_old) + (ff_r_balance_old))) /\ ((((exists ff_h_balance_old_successor. ff_h_balance_old_successor + S (ff_s_balance_old) = S ((S (S ff_i_balance_old)) * ff_v_balance_old)) /\ exists ff_q_balance_old_successor. ff_u_balance_old = ff_q_balance_old_successor * S ((S (S ff_i_balance_old)) * ff_v_balance_old) + (ff_s_balance_old))) /\ ff_s_balance_old = ff_r_balance_old * ff_p_balance_old)))))) -> (exists ff_u_balance_new ff_v_balance_new. ((((exists ff_h_balance_new_start. ff_h_balance_new_start + S (1) = S ((S (0)) * ff_v_balance_new)) /\ exists ff_q_balance_new_start. ff_u_balance_new = ff_q_balance_new_start * S ((S (0)) * ff_v_balance_new) + (1))) /\ ((((exists ff_h_balance_new_terminal. ff_h_balance_new_terminal + S (q) = S ((S (k)) * ff_v_balance_new)) /\ exists ff_q_balance_new_terminal. ff_u_balance_new = ff_q_balance_new_terminal * S ((S (k)) * ff_v_balance_new) + (q))) /\ forall ff_i_balance_new. (exists ff_lt_balance_new_bound. ff_lt_balance_new_bound + S ff_i_balance_new = k) -> exists ff_p_balance_new ff_r_balance_new ff_s_balance_new. ((((exists ff_h_balance_new_factor. ff_h_balance_new_factor + S (ff_p_balance_new) = S ((S (ff_i_balance_new)) * d)) /\ exists ff_q_balance_new_factor. z = ff_q_balance_new_factor * S ((S (ff_i_balance_new)) * d) + (ff_p_balance_new))) /\ ((((exists ff_h_balance_new_partial. ff_h_balance_new_partial + S (ff_r_balance_new) = S ((S (ff_i_balance_new)) * ff_v_balance_new)) /\ exists ff_q_balance_new_partial. ff_u_balance_new = ff_q_balance_new_partial * S ((S (ff_i_balance_new)) * ff_v_balance_new) + (ff_r_balance_new))) /\ ((((exists ff_h_balance_new_successor. ff_h_balance_new_successor + S (ff_s_balance_new) = S ((S (S ff_i_balance_new)) * ff_v_balance_new)) /\ exists ff_q_balance_new_successor. ff_u_balance_new = ff_q_balance_new_successor * S ((S (S ff_i_balance_new)) * ff_v_balance_new) + (ff_s_balance_new))) /\ ff_s_balance_new = ff_r_balance_new * ff_p_balance_new)))))) -> q * x = p * y

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k b c z d i x y p q. (exists h. h + S i = k) -> (((exists ff_h_balance_old_i. ff_h_balance_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_balance_old_i. b = ff_q_balance_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_balance_new_i. ff_h_balance_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_balance_new_i. z = ff_q_balance_new_i * S ((S (i)) * d) + (y))) -> (forall j a. (exists h. h + S j = k) -> ~(j = i) -> (((exists ff_h_balance_old_j. ff_h_balance_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_balance_old_j. b = ff_q_balance_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_balance_new_j. ff_h_balance_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_balance_new_j. z = ff_q_balance_new_j * S ((S (j)) * d) + (a)))) -> (exists ff_u_balance_old ff_v_balance_old. ((((exists ff_h_balance_old_start. ff_h_balance_old_start + S (1) = S ((S (0)) * ff_v_balance_old)) /\ exists ff_q_balance_old_start. ff_u_balance_old = ff_q_balance_old_start * S ((S (0)) * ff_v_balance_old) + (1))) /\ ((((exists ff_h_balance_old_terminal. ff_h_balance_old_terminal + S (p) = S ((S (k)) * ff_v_balance_old)) /\ exists ff_q_balance_old_terminal. ff_u_balance_old = ff_q_balance_old_terminal * S ((S (k)) * ff_v_balance_old) + (p))) /\ forall ff_i_balance_old. (exists ff_lt_balance_old_bound. ff_lt_balance_old_bound + S ff_i_balance_old = k) -> exists ff_p_balance_old ff_r_balance_old ff_s_balance_old. ((((exists ff_h_balance_old_factor. ff_h_balance_old_factor + S (ff_p_balance_old) = S ((S (ff_i_balance_old)) * c)) /\ exists ff_q_balance_old_factor. b = ff_q_balance_old_factor * S ((S (ff_i_balance_old)) * c) + (ff_p_balance_old))) /\ ((((exists ff_h_balance_old_partial. ff_h_balance_old_partial + S (ff_r_balance_old) = S ((S (ff_i_balance_old)) * ff_v_balance_old)) /\ exists ff_q_balance_old_partial. ff_u_balance_old = ff_q_balance_old_partial * S ((S (ff_i_balance_old)) * ff_v_balance_old) + (ff_r_balance_old))) /\ ((((exists ff_h_balance_old_successor. ff_h_balance_old_successor + S (ff_s_balance_old) = S ((S (S ff_i_balance_old)) * ff_v_balance_old)) /\ exists ff_q_balance_old_successor. ff_u_balance_old = ff_q_balance_old_successor * S ((S (S ff_i_balance_old)) * ff_v_balance_old) + (ff_s_balance_old))) /\ ff_s_balance_old = ff_r_balance_old * ff_p_balance_old)))))) -> (exists ff_u_balance_new ff_v_balance_new. ((((exists ff_h_balance_new_start. ff_h_balance_new_start + S (1) = S ((S (0)) * ff_v_balance_new)) /\ exists ff_q_balance_new_start. ff_u_balance_new = ff_q_balance_new_start * S ((S (0)) * ff_v_balance_new) + (1))) /\ ((((exists ff_h_balance_new_terminal. ff_h_balance_new_terminal + S (q) = S ((S (k)) * ff_v_balance_new)) /\ exists ff_q_balance_new_terminal. ff_u_balance_new = ff_q_balance_new_terminal * S ((S (k)) * ff_v_balance_new) + (q))) /\ forall ff_i_balance_new. (exists ff_lt_balance_new_bound. ff_lt_balance_new_bound + S ff_i_balance_new = k) -> exists ff_p_balance_new ff_r_balance_new ff_s_balance_new. ((((exists ff_h_balance_new_factor. ff_h_balance_new_factor + S (ff_p_balance_new) = S ((S (ff_i_balance_new)) * d)) /\ exists ff_q_balance_new_factor. z = ff_q_balance_new_factor * S ((S (ff_i_balance_new)) * d) + (ff_p_balance_new))) /\ ((((exists ff_h_balance_new_partial. ff_h_balance_new_partial + S (ff_r_balance_new) = S ((S (ff_i_balance_new)) * ff_v_balance_new)) /\ exists ff_q_balance_new_partial. ff_u_balance_new = ff_q_balance_new_partial * S ((S (ff_i_balance_new)) * ff_v_balance_new) + (ff_r_balance_new))) /\ ((((exists ff_h_balance_new_successor. ff_h_balance_new_successor + S (ff_s_balance_new) = S ((S (S ff_i_balance_new)) * ff_v_balance_new)) /\ exists ff_q_balance_new_successor. ff_u_balance_new = ff_q_balance_new_successor * S ((S (S ff_i_balance_new)) * ff_v_balance_new) + (ff_s_balance_new))) /\ ff_s_balance_new = ff_r_balance_new * ff_p_balance_new)))))) -> q * x = p * y
use add_eq_zero_right
use succ_ne_zero
use finite_lt_succ_eq_or_lt
use beta_product_succ_decompose
use beta_product_transport_prefix
use beta_product_functional
use beta_at_unique
use mul_assoc
use mul_comm
use le_succ
use le_refl
use lt_irrefl_expanded
induction k
intro b
intro c
intro z
intro d
intro i
intro x
intro y
intro p
intro q
intro hi
exfalso
cases hi
have hsi : S i = 0
specialize add_eq_zero_right x1
specialize add_eq_zero_right (S i)
apply add_eq_zero_right
exact hi_witness
specialize succ_ne_zero i
apply succ_ne_zero
exact hsi
intro b
intro c
intro z
intro d
intro i
intro x
intro y
intro p
intro q
intro hi
intro hold_i
intro hnew_i
intro hpreserve
intro hproduct_old
intro hproduct_new
have hisplit : i = k \/ exists h. h + S i = k
specialize finite_lt_succ_eq_or_lt k
specialize finite_lt_succ_eq_or_lt i
apply finite_lt_succ_eq_or_lt
exact hi
have hold_decomp : exists a r. (((exists ff_h_balance_old_last. ff_h_balance_old_last + S (a) = S ((S (k)) * c)) /\ exists ff_q_balance_old_last. b = ff_q_balance_old_last * S ((S (k)) * c) + (a))) /\ ((exists ff_u_balance_old_prefix ff_v_balance_old_prefix. ((((exists ff_h_balance_old_prefix_start. ff_h_balance_old_prefix_start + S (1) = S ((S (0)) * ff_v_balance_old_prefix)) /\ exists ff_q_balance_old_prefix_start. ff_u_balance_old_prefix = ff_q_balance_old_prefix_start * S ((S (0)) * ff_v_balance_old_prefix) + (1))) /\ ((((exists ff_h_balance_old_prefix_terminal. ff_h_balance_old_prefix_terminal + S (r) = S ((S (k)) * ff_v_balance_old_prefix)) /\ exists ff_q_balance_old_prefix_terminal. ff_u_balance_old_prefix = ff_q_balance_old_prefix_terminal * S ((S (k)) * ff_v_balance_old_prefix) + (r))) /\ forall ff_i_balance_old_prefix. (exists ff_lt_balance_old_prefix_bound. ff_lt_balance_old_prefix_bound + S ff_i_balance_old_prefix = k) -> exists ff_p_balance_old_prefix ff_r_balance_old_prefix ff_s_balance_old_prefix. ((((exists ff_h_balance_old_prefix_factor. ff_h_balance_old_prefix_factor + S (ff_p_balance_old_prefix) = S ((S (ff_i_balance_old_prefix)) * c)) /\ exists ff_q_balance_old_prefix_factor. b = ff_q_balance_old_prefix_factor * S ((S (ff_i_balance_old_prefix)) * c) + (ff_p_balance_old_prefix))) /\ ((((exists ff_h_balance_old_prefix_partial. ff_h_balance_old_prefix_partial + S (ff_r_balance_old_prefix) = S ((S (ff_i_balance_old_prefix)) * ff_v_balance_old_prefix)) /\ exists ff_q_balance_old_prefix_partial. ff_u_balance_old_prefix = ff_q_balance_old_prefix_partial * S ((S (ff_i_balance_old_prefix)) * ff_v_balance_old_prefix) + (ff_r_balance_old_prefix))) /\ ((((exists ff_h_balance_old_prefix_successor. ff_h_balance_old_prefix_successor + S (ff_s_balance_old_prefix) = S ((S (S ff_i_balance_old_prefix)) * ff_v_balance_old_prefix)) /\ exists ff_q_balance_old_prefix_successor. ff_u_balance_old_prefix = ff_q_balance_old_prefix_successor * S ((S (S ff_i_balance_old_prefix)) * ff_v_balance_old_prefix) + (ff_s_balance_old_prefix))) /\ ff_s_balance_old_prefix = ff_r_balance_old_prefix * ff_p_balance_old_prefix)))))) /\ p = r * a)
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose k
specialize beta_product_succ_decompose p
apply beta_product_succ_decompose
exact hproduct_old
have hnew_decomp : exists a r. (((exists ff_h_balance_new_last. ff_h_balance_new_last + S (a) = S ((S (k)) * d)) /\ exists ff_q_balance_new_last. z = ff_q_balance_new_last * S ((S (k)) * d) + (a))) /\ ((exists ff_u_balance_new_prefix ff_v_balance_new_prefix. ((((exists ff_h_balance_new_prefix_start. ff_h_balance_new_prefix_start + S (1) = S ((S (0)) * ff_v_balance_new_prefix)) /\ exists ff_q_balance_new_prefix_start. ff_u_balance_new_prefix = ff_q_balance_new_prefix_start * S ((S (0)) * ff_v_balance_new_prefix) + (1))) /\ ((((exists ff_h_balance_new_prefix_terminal. ff_h_balance_new_prefix_terminal + S (r) = S ((S (k)) * ff_v_balance_new_prefix)) /\ exists ff_q_balance_new_prefix_terminal. ff_u_balance_new_prefix = ff_q_balance_new_prefix_terminal * S ((S (k)) * ff_v_balance_new_prefix) + (r))) /\ forall ff_i_balance_new_prefix. (exists ff_lt_balance_new_prefix_bound. ff_lt_balance_new_prefix_bound + S ff_i_balance_new_prefix = k) -> exists ff_p_balance_new_prefix ff_r_balance_new_prefix ff_s_balance_new_prefix. ((((exists ff_h_balance_new_prefix_factor. ff_h_balance_new_prefix_factor + S (ff_p_balance_new_prefix) = S ((S (ff_i_balance_new_prefix)) * d)) /\ exists ff_q_balance_new_prefix_factor. z = ff_q_balance_new_prefix_factor * S ((S (ff_i_balance_new_prefix)) * d) + (ff_p_balance_new_prefix))) /\ ((((exists ff_h_balance_new_prefix_partial. ff_h_balance_new_prefix_partial + S (ff_r_balance_new_prefix) = S ((S (ff_i_balance_new_prefix)) * ff_v_balance_new_prefix)) /\ exists ff_q_balance_new_prefix_partial. ff_u_balance_new_prefix = ff_q_balance_new_prefix_partial * S ((S (ff_i_balance_new_prefix)) * ff_v_balance_new_prefix) + (ff_r_balance_new_prefix))) /\ ((((exists ff_h_balance_new_prefix_successor. ff_h_balance_new_prefix_successor + S (ff_s_balance_new_prefix) = S ((S (S ff_i_balance_new_prefix)) * ff_v_balance_new_prefix)) /\ exists ff_q_balance_new_prefix_successor. ff_u_balance_new_prefix = ff_q_balance_new_prefix_successor * S ((S (S ff_i_balance_new_prefix)) * ff_v_balance_new_prefix) + (ff_s_balance_new_prefix))) /\ ff_s_balance_new_prefix = ff_r_balance_new_prefix * ff_p_balance_new_prefix)))))) /\ q = r * a)
specialize beta_product_succ_decompose z
specialize beta_product_succ_decompose d
specialize beta_product_succ_decompose k
specialize beta_product_succ_decompose q
apply beta_product_succ_decompose
exact hproduct_new
cases hold_decomp
cases hold_decomp_witness
cases hold_decomp_witness_witness
cases hold_decomp_witness_witness_right
cases hnew_decomp
cases hnew_decomp_witness
cases hnew_decomp_witness_witness
cases hnew_decomp_witness_witness_right
cases hisplit
have hax : x1 = x
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique k
specialize beta_at_unique x1
specialize beta_at_unique x
apply beta_at_unique
exact hold_decomp_witness_witness_left
rewrite hisplit_left at hold_i
rewrite hisplit_left at hold_i
exact hold_i
have hby : x3 = y
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique k
specialize beta_at_unique x3
specialize beta_at_unique y
apply beta_at_unique
exact hnew_decomp_witness_witness_left
rewrite hisplit_left at hnew_i
rewrite hisplit_left at hnew_i
exact hnew_i
have hprefix_transport : exists ff_u_balance_transported_prefix ff_v_balance_transported_prefix. ((((exists ff_h_balance_transported_prefix_start. ff_h_balance_transported_prefix_start + S (1) = S ((S (0)) * ff_v_balance_transported_prefix)) /\ exists ff_q_balance_transported_prefix_start. ff_u_balance_transported_prefix = ff_q_balance_transported_prefix_start * S ((S (0)) * ff_v_balance_transported_prefix) + (1))) /\ ((((exists ff_h_balance_transported_prefix_terminal. ff_h_balance_transported_prefix_terminal + S (x2) = S ((S (k)) * ff_v_balance_transported_prefix)) /\ exists ff_q_balance_transported_prefix_terminal. ff_u_balance_transported_prefix = ff_q_balance_transported_prefix_terminal * S ((S (k)) * ff_v_balance_transported_prefix) + (x2))) /\ forall ff_i_balance_transported_prefix. (exists ff_lt_balance_transported_prefix_bound. ff_lt_balance_transported_prefix_bound + S ff_i_balance_transported_prefix = k) -> exists ff_p_balance_transported_prefix ff_r_balance_transported_prefix ff_s_balance_transported_prefix. ((((exists ff_h_balance_transported_prefix_factor. ff_h_balance_transported_prefix_factor + S (ff_p_balance_transported_prefix) = S ((S (ff_i_balance_transported_prefix)) * d)) /\ exists ff_q_balance_transported_prefix_factor. z = ff_q_balance_transported_prefix_factor * S ((S (ff_i_balance_transported_prefix)) * d) + (ff_p_balance_transported_prefix))) /\ ((((exists ff_h_balance_transported_prefix_partial. ff_h_balance_transported_prefix_partial + S (ff_r_balance_transported_prefix) = S ((S (ff_i_balance_transported_prefix)) * ff_v_balance_transported_prefix)) /\ exists ff_q_balance_transported_prefix_partial. ff_u_balance_transported_prefix = ff_q_balance_transported_prefix_partial * S ((S (ff_i_balance_transported_prefix)) * ff_v_balance_transported_prefix) + (ff_r_balance_transported_prefix))) /\ ((((exists ff_h_balance_transported_prefix_successor. ff_h_balance_transported_prefix_successor + S (ff_s_balance_transported_prefix) = S ((S (S ff_i_balance_transported_prefix)) * ff_v_balance_transported_prefix)) /\ exists ff_q_balance_transported_prefix_successor. ff_u_balance_transported_prefix = ff_q_balance_transported_prefix_successor * S ((S (S ff_i_balance_transported_prefix)) * ff_v_balance_transported_prefix) + (ff_s_balance_transported_prefix))) /\ ff_s_balance_transported_prefix = ff_r_balance_transported_prefix * ff_p_balance_transported_prefix)))))
specialize beta_product_transport_prefix b
specialize beta_product_transport_prefix c
specialize beta_product_transport_prefix z
specialize beta_product_transport_prefix d
specialize beta_product_transport_prefix k
specialize beta_product_transport_prefix x2
apply beta_product_transport_prefix
exact hold_decomp_witness_witness_right_left
intro j
intro a
intro hj
intro hold
specialize hpreserve j
specialize hpreserve a
apply hpreserve
specialize le_succ (S j)
specialize le_succ k
apply le_succ
exact hj
intro hjk
specialize lt_irrefl_expanded k
apply lt_irrefl_expanded
rewrite hjk at hj
rewrite hisplit_left at hj
exact hj
exact hold
cases hprefix_transport
cases hprefix_transport_witness
cases hnew_decomp_witness_witness_right_left
cases hnew_decomp_witness_witness_right_left_witness
rewrite hold_decomp_witness_witness_right_right
rewrite hnew_decomp_witness_witness_right_right
rewrite hax
rewrite hby
trans (x4 * x) * y
simp [mul_assoc, mul_comm]
congr
congr
symm
specialize beta_product_functional z
specialize beta_product_functional d
specialize beta_product_functional k
specialize beta_product_functional x2
specialize beta_product_functional x5
specialize beta_product_functional x6
specialize beta_product_functional x4
specialize beta_product_functional x7
specialize beta_product_functional x8
apply beta_product_functional
exact hprefix_transport_witness_witness
exact hnew_decomp_witness_witness_right_left_witness_witness
refl
refl
have hki : ~(k = i)
intro hki_eq
specialize lt_irrefl_expanded k
apply lt_irrefl_expanded
rewrite <- hki_eq at hisplit_right
exact hisplit_right
have hlast_new : ((exists h. h + S x1 = S ((S k) * d)) /\ exists w. z = w * S ((S k) * d) + x1)
specialize hpreserve k
specialize hpreserve x1
apply hpreserve
specialize le_refl (S k)
exact le_refl
exact hki
exact hold_decomp_witness_witness_left
have hlast_eq : x3 = x1
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique k
specialize beta_at_unique x3
specialize beta_at_unique x1
apply beta_at_unique
exact hnew_decomp_witness_witness_left
exact hlast_new
have hprefix_preserve : forall j a. (exists h. h + S j = k) -> ~(j = i) -> ((exists h. h + S a = S ((S j) * c)) /\ exists w. b = w * S ((S j) * c) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists w. z = w * S ((S j) * d) + a)
intro j
intro a
intro hj
intro hji
intro hold
specialize hpreserve j
specialize hpreserve a
apply hpreserve
specialize le_succ (S j)
specialize le_succ k
apply le_succ
exact hj
exact hji
exact hold
have hbalance : x4 * x = x2 * y
specialize IH b
specialize IH c
specialize IH z
specialize IH d
specialize IH i
specialize IH x
specialize IH y
specialize IH x2
specialize IH x4
apply IH
exact hisplit_right
exact hold_i
exact hnew_i
exact hprefix_preserve
exact hold_decomp_witness_witness_right_left
exact hnew_decomp_witness_witness_right_left
rewrite hold_decomp_witness_witness_right_right
rewrite hnew_decomp_witness_witness_right_right
rewrite hlast_eq
trans (x4 * x) * x1
simp [mul_assoc, mul_comm]
rewrite hbalance
simp [mul_assoc, mul_comm]
qed

Occurrences

4,780

Distinct objects

1,552

Reused references

56

Depth

66

Cuts

130

Certificate SHA-256

e4f1bc0e72bf…

beta_product_swap_last_invariantBeta product swap last invariant Quadratic reciprocity campaign checked

Swapping an interior beta-coded factor with the last factor preserves the exact finite product.

Prerequisites

beta_product_replace_balance beta_product_succ_decompose beta_at_unique le_succ le_refl lt_irrefl_expanded

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall b c z d n i x y p q. (exists h. h + S i = n) -> (((exists ff_h_product_swap_old_i. ff_h_product_swap_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_product_swap_old_i. b = ff_q_product_swap_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_product_swap_old_n. ff_h_product_swap_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_product_swap_old_n. b = ff_q_product_swap_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_product_swap_new_i. ff_h_product_swap_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_product_swap_new_i. z = ff_q_product_swap_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_product_swap_new_n. ff_h_product_swap_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_product_swap_new_n. z = ff_q_product_swap_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_product_swap_old_j. ff_h_product_swap_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_product_swap_old_j. b = ff_q_product_swap_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_product_swap_new_j. ff_h_product_swap_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_product_swap_new_j. z = ff_q_product_swap_new_j * S ((S (j)) * d) + (a)))) -> (exists ff_u_product_swap_old ff_v_product_swap_old. ((((exists ff_h_product_swap_old_start. ff_h_product_swap_old_start + S (1) = S ((S (0)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_start. ff_u_product_swap_old = ff_q_product_swap_old_start * S ((S (0)) * ff_v_product_swap_old) + (1))) /\ ((((exists ff_h_product_swap_old_terminal. ff_h_product_swap_old_terminal + S (p) = S ((S (S n)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_terminal. ff_u_product_swap_old = ff_q_product_swap_old_terminal * S ((S (S n)) * ff_v_product_swap_old) + (p))) /\ forall ff_i_product_swap_old. (exists ff_lt_product_swap_old_bound. ff_lt_product_swap_old_bound + S ff_i_product_swap_old = S n) -> exists ff_p_product_swap_old ff_r_product_swap_old ff_s_product_swap_old. ((((exists ff_h_product_swap_old_factor. ff_h_product_swap_old_factor + S (ff_p_product_swap_old) = S ((S (ff_i_product_swap_old)) * c)) /\ exists ff_q_product_swap_old_factor. b = ff_q_product_swap_old_factor * S ((S (ff_i_product_swap_old)) * c) + (ff_p_product_swap_old))) /\ ((((exists ff_h_product_swap_old_partial. ff_h_product_swap_old_partial + S (ff_r_product_swap_old) = S ((S (ff_i_product_swap_old)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_partial. ff_u_product_swap_old = ff_q_product_swap_old_partial * S ((S (ff_i_product_swap_old)) * ff_v_product_swap_old) + (ff_r_product_swap_old))) /\ ((((exists ff_h_product_swap_old_successor. ff_h_product_swap_old_successor + S (ff_s_product_swap_old) = S ((S (S ff_i_product_swap_old)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_successor. ff_u_product_swap_old = ff_q_product_swap_old_successor * S ((S (S ff_i_product_swap_old)) * ff_v_product_swap_old) + (ff_s_product_swap_old))) /\ ff_s_product_swap_old = ff_r_product_swap_old * ff_p_product_swap_old)))))) -> (exists ff_u_product_swap_new ff_v_product_swap_new. ((((exists ff_h_product_swap_new_start. ff_h_product_swap_new_start + S (1) = S ((S (0)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_start. ff_u_product_swap_new = ff_q_product_swap_new_start * S ((S (0)) * ff_v_product_swap_new) + (1))) /\ ((((exists ff_h_product_swap_new_terminal. ff_h_product_swap_new_terminal + S (q) = S ((S (S n)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_terminal. ff_u_product_swap_new = ff_q_product_swap_new_terminal * S ((S (S n)) * ff_v_product_swap_new) + (q))) /\ forall ff_i_product_swap_new. (exists ff_lt_product_swap_new_bound. ff_lt_product_swap_new_bound + S ff_i_product_swap_new = S n) -> exists ff_p_product_swap_new ff_r_product_swap_new ff_s_product_swap_new. ((((exists ff_h_product_swap_new_factor. ff_h_product_swap_new_factor + S (ff_p_product_swap_new) = S ((S (ff_i_product_swap_new)) * d)) /\ exists ff_q_product_swap_new_factor. z = ff_q_product_swap_new_factor * S ((S (ff_i_product_swap_new)) * d) + (ff_p_product_swap_new))) /\ ((((exists ff_h_product_swap_new_partial. ff_h_product_swap_new_partial + S (ff_r_product_swap_new) = S ((S (ff_i_product_swap_new)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_partial. ff_u_product_swap_new = ff_q_product_swap_new_partial * S ((S (ff_i_product_swap_new)) * ff_v_product_swap_new) + (ff_r_product_swap_new))) /\ ((((exists ff_h_product_swap_new_successor. ff_h_product_swap_new_successor + S (ff_s_product_swap_new) = S ((S (S ff_i_product_swap_new)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_successor. ff_u_product_swap_new = ff_q_product_swap_new_successor * S ((S (S ff_i_product_swap_new)) * ff_v_product_swap_new) + (ff_s_product_swap_new))) /\ ff_s_product_swap_new = ff_r_product_swap_new * ff_p_product_swap_new)))))) -> p = q

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b c z d n i x y p q. (exists h. h + S i = n) -> (((exists ff_h_product_swap_old_i. ff_h_product_swap_old_i + S (x) = S ((S (i)) * c)) /\ exists ff_q_product_swap_old_i. b = ff_q_product_swap_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_product_swap_old_n. ff_h_product_swap_old_n + S (y) = S ((S (n)) * c)) /\ exists ff_q_product_swap_old_n. b = ff_q_product_swap_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_product_swap_new_i. ff_h_product_swap_new_i + S (y) = S ((S (i)) * d)) /\ exists ff_q_product_swap_new_i. z = ff_q_product_swap_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_product_swap_new_n. ff_h_product_swap_new_n + S (x) = S ((S (n)) * d)) /\ exists ff_q_product_swap_new_n. z = ff_q_product_swap_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_product_swap_old_j. ff_h_product_swap_old_j + S (a) = S ((S (j)) * c)) /\ exists ff_q_product_swap_old_j. b = ff_q_product_swap_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_product_swap_new_j. ff_h_product_swap_new_j + S (a) = S ((S (j)) * d)) /\ exists ff_q_product_swap_new_j. z = ff_q_product_swap_new_j * S ((S (j)) * d) + (a)))) -> (exists ff_u_product_swap_old ff_v_product_swap_old. ((((exists ff_h_product_swap_old_start. ff_h_product_swap_old_start + S (1) = S ((S (0)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_start. ff_u_product_swap_old = ff_q_product_swap_old_start * S ((S (0)) * ff_v_product_swap_old) + (1))) /\ ((((exists ff_h_product_swap_old_terminal. ff_h_product_swap_old_terminal + S (p) = S ((S (S n)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_terminal. ff_u_product_swap_old = ff_q_product_swap_old_terminal * S ((S (S n)) * ff_v_product_swap_old) + (p))) /\ forall ff_i_product_swap_old. (exists ff_lt_product_swap_old_bound. ff_lt_product_swap_old_bound + S ff_i_product_swap_old = S n) -> exists ff_p_product_swap_old ff_r_product_swap_old ff_s_product_swap_old. ((((exists ff_h_product_swap_old_factor. ff_h_product_swap_old_factor + S (ff_p_product_swap_old) = S ((S (ff_i_product_swap_old)) * c)) /\ exists ff_q_product_swap_old_factor. b = ff_q_product_swap_old_factor * S ((S (ff_i_product_swap_old)) * c) + (ff_p_product_swap_old))) /\ ((((exists ff_h_product_swap_old_partial. ff_h_product_swap_old_partial + S (ff_r_product_swap_old) = S ((S (ff_i_product_swap_old)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_partial. ff_u_product_swap_old = ff_q_product_swap_old_partial * S ((S (ff_i_product_swap_old)) * ff_v_product_swap_old) + (ff_r_product_swap_old))) /\ ((((exists ff_h_product_swap_old_successor. ff_h_product_swap_old_successor + S (ff_s_product_swap_old) = S ((S (S ff_i_product_swap_old)) * ff_v_product_swap_old)) /\ exists ff_q_product_swap_old_successor. ff_u_product_swap_old = ff_q_product_swap_old_successor * S ((S (S ff_i_product_swap_old)) * ff_v_product_swap_old) + (ff_s_product_swap_old))) /\ ff_s_product_swap_old = ff_r_product_swap_old * ff_p_product_swap_old)))))) -> (exists ff_u_product_swap_new ff_v_product_swap_new. ((((exists ff_h_product_swap_new_start. ff_h_product_swap_new_start + S (1) = S ((S (0)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_start. ff_u_product_swap_new = ff_q_product_swap_new_start * S ((S (0)) * ff_v_product_swap_new) + (1))) /\ ((((exists ff_h_product_swap_new_terminal. ff_h_product_swap_new_terminal + S (q) = S ((S (S n)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_terminal. ff_u_product_swap_new = ff_q_product_swap_new_terminal * S ((S (S n)) * ff_v_product_swap_new) + (q))) /\ forall ff_i_product_swap_new. (exists ff_lt_product_swap_new_bound. ff_lt_product_swap_new_bound + S ff_i_product_swap_new = S n) -> exists ff_p_product_swap_new ff_r_product_swap_new ff_s_product_swap_new. ((((exists ff_h_product_swap_new_factor. ff_h_product_swap_new_factor + S (ff_p_product_swap_new) = S ((S (ff_i_product_swap_new)) * d)) /\ exists ff_q_product_swap_new_factor. z = ff_q_product_swap_new_factor * S ((S (ff_i_product_swap_new)) * d) + (ff_p_product_swap_new))) /\ ((((exists ff_h_product_swap_new_partial. ff_h_product_swap_new_partial + S (ff_r_product_swap_new) = S ((S (ff_i_product_swap_new)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_partial. ff_u_product_swap_new = ff_q_product_swap_new_partial * S ((S (ff_i_product_swap_new)) * ff_v_product_swap_new) + (ff_r_product_swap_new))) /\ ((((exists ff_h_product_swap_new_successor. ff_h_product_swap_new_successor + S (ff_s_product_swap_new) = S ((S (S ff_i_product_swap_new)) * ff_v_product_swap_new)) /\ exists ff_q_product_swap_new_successor. ff_u_product_swap_new = ff_q_product_swap_new_successor * S ((S (S ff_i_product_swap_new)) * ff_v_product_swap_new) + (ff_s_product_swap_new))) /\ ff_s_product_swap_new = ff_r_product_swap_new * ff_p_product_swap_new)))))) -> p = q
use beta_product_replace_balance
use beta_product_succ_decompose
use beta_at_unique
use le_succ
use le_refl
use lt_irrefl_expanded
intro b
intro c
intro z
intro d
intro n
intro i
intro x
intro y
intro p
intro q
intro hi
intro hold_i
intro hold_n
intro hnew_i
intro hnew_n
intro hpreserve
intro hproduct_old
intro hproduct_new
have hold_decomp : exists a r. (((exists ff_h_swap_old_last. ff_h_swap_old_last + S (a) = S ((S (n)) * c)) /\ exists ff_q_swap_old_last. b = ff_q_swap_old_last * S ((S (n)) * c) + (a))) /\ ((exists ff_u_swap_old_prefix ff_v_swap_old_prefix. ((((exists ff_h_swap_old_prefix_start. ff_h_swap_old_prefix_start + S (1) = S ((S (0)) * ff_v_swap_old_prefix)) /\ exists ff_q_swap_old_prefix_start. ff_u_swap_old_prefix = ff_q_swap_old_prefix_start * S ((S (0)) * ff_v_swap_old_prefix) + (1))) /\ ((((exists ff_h_swap_old_prefix_terminal. ff_h_swap_old_prefix_terminal + S (r) = S ((S (n)) * ff_v_swap_old_prefix)) /\ exists ff_q_swap_old_prefix_terminal. ff_u_swap_old_prefix = ff_q_swap_old_prefix_terminal * S ((S (n)) * ff_v_swap_old_prefix) + (r))) /\ forall ff_i_swap_old_prefix. (exists ff_lt_swap_old_prefix_bound. ff_lt_swap_old_prefix_bound + S ff_i_swap_old_prefix = n) -> exists ff_p_swap_old_prefix ff_r_swap_old_prefix ff_s_swap_old_prefix. ((((exists ff_h_swap_old_prefix_factor. ff_h_swap_old_prefix_factor + S (ff_p_swap_old_prefix) = S ((S (ff_i_swap_old_prefix)) * c)) /\ exists ff_q_swap_old_prefix_factor. b = ff_q_swap_old_prefix_factor * S ((S (ff_i_swap_old_prefix)) * c) + (ff_p_swap_old_prefix))) /\ ((((exists ff_h_swap_old_prefix_partial. ff_h_swap_old_prefix_partial + S (ff_r_swap_old_prefix) = S ((S (ff_i_swap_old_prefix)) * ff_v_swap_old_prefix)) /\ exists ff_q_swap_old_prefix_partial. ff_u_swap_old_prefix = ff_q_swap_old_prefix_partial * S ((S (ff_i_swap_old_prefix)) * ff_v_swap_old_prefix) + (ff_r_swap_old_prefix))) /\ ((((exists ff_h_swap_old_prefix_successor. ff_h_swap_old_prefix_successor + S (ff_s_swap_old_prefix) = S ((S (S ff_i_swap_old_prefix)) * ff_v_swap_old_prefix)) /\ exists ff_q_swap_old_prefix_successor. ff_u_swap_old_prefix = ff_q_swap_old_prefix_successor * S ((S (S ff_i_swap_old_prefix)) * ff_v_swap_old_prefix) + (ff_s_swap_old_prefix))) /\ ff_s_swap_old_prefix = ff_r_swap_old_prefix * ff_p_swap_old_prefix)))))) /\ p = r * a)
specialize beta_product_succ_decompose b
specialize beta_product_succ_decompose c
specialize beta_product_succ_decompose n
specialize beta_product_succ_decompose p
apply beta_product_succ_decompose
exact hproduct_old
have hnew_decomp : exists a r. (((exists ff_h_swap_new_last. ff_h_swap_new_last + S (a) = S ((S (n)) * d)) /\ exists ff_q_swap_new_last. z = ff_q_swap_new_last * S ((S (n)) * d) + (a))) /\ ((exists ff_u_swap_new_prefix ff_v_swap_new_prefix. ((((exists ff_h_swap_new_prefix_start. ff_h_swap_new_prefix_start + S (1) = S ((S (0)) * ff_v_swap_new_prefix)) /\ exists ff_q_swap_new_prefix_start. ff_u_swap_new_prefix = ff_q_swap_new_prefix_start * S ((S (0)) * ff_v_swap_new_prefix) + (1))) /\ ((((exists ff_h_swap_new_prefix_terminal. ff_h_swap_new_prefix_terminal + S (r) = S ((S (n)) * ff_v_swap_new_prefix)) /\ exists ff_q_swap_new_prefix_terminal. ff_u_swap_new_prefix = ff_q_swap_new_prefix_terminal * S ((S (n)) * ff_v_swap_new_prefix) + (r))) /\ forall ff_i_swap_new_prefix. (exists ff_lt_swap_new_prefix_bound. ff_lt_swap_new_prefix_bound + S ff_i_swap_new_prefix = n) -> exists ff_p_swap_new_prefix ff_r_swap_new_prefix ff_s_swap_new_prefix. ((((exists ff_h_swap_new_prefix_factor. ff_h_swap_new_prefix_factor + S (ff_p_swap_new_prefix) = S ((S (ff_i_swap_new_prefix)) * d)) /\ exists ff_q_swap_new_prefix_factor. z = ff_q_swap_new_prefix_factor * S ((S (ff_i_swap_new_prefix)) * d) + (ff_p_swap_new_prefix))) /\ ((((exists ff_h_swap_new_prefix_partial. ff_h_swap_new_prefix_partial + S (ff_r_swap_new_prefix) = S ((S (ff_i_swap_new_prefix)) * ff_v_swap_new_prefix)) /\ exists ff_q_swap_new_prefix_partial. ff_u_swap_new_prefix = ff_q_swap_new_prefix_partial * S ((S (ff_i_swap_new_prefix)) * ff_v_swap_new_prefix) + (ff_r_swap_new_prefix))) /\ ((((exists ff_h_swap_new_prefix_successor. ff_h_swap_new_prefix_successor + S (ff_s_swap_new_prefix) = S ((S (S ff_i_swap_new_prefix)) * ff_v_swap_new_prefix)) /\ exists ff_q_swap_new_prefix_successor. ff_u_swap_new_prefix = ff_q_swap_new_prefix_successor * S ((S (S ff_i_swap_new_prefix)) * ff_v_swap_new_prefix) + (ff_s_swap_new_prefix))) /\ ff_s_swap_new_prefix = ff_r_swap_new_prefix * ff_p_swap_new_prefix)))))) /\ q = r * a)
specialize beta_product_succ_decompose z
specialize beta_product_succ_decompose d
specialize beta_product_succ_decompose n
specialize beta_product_succ_decompose q
apply beta_product_succ_decompose
exact hproduct_new
cases hold_decomp
cases hold_decomp_witness
cases hold_decomp_witness_witness
cases hold_decomp_witness_witness_right
cases hnew_decomp
cases hnew_decomp_witness
cases hnew_decomp_witness_witness
cases hnew_decomp_witness_witness_right
have hold_last : x1 = y
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique n
specialize beta_at_unique x1
specialize beta_at_unique y
apply beta_at_unique
exact hold_decomp_witness_witness_left
exact hold_n
have hnew_last : x3 = x
specialize beta_at_unique z
specialize beta_at_unique d
specialize beta_at_unique n
specialize beta_at_unique x3
specialize beta_at_unique x
apply beta_at_unique
exact hnew_decomp_witness_witness_left
exact hnew_n
have hprefix_preserve : forall j a. (exists h. h + S j = n) -> ~(j = i) -> ((exists h. h + S a = S ((S j) * c)) /\ exists w. b = w * S ((S j) * c) + a) -> ((exists h. h + S a = S ((S j) * d)) /\ exists w. z = w * S ((S j) * d) + a)
intro j
intro a
intro hj
intro hji
intro hold
specialize hpreserve j
specialize hpreserve a
apply hpreserve
specialize le_succ (S j)
specialize le_succ n
apply le_succ
exact hj
exact hji
intro hjn
specialize lt_irrefl_expanded n
apply lt_irrefl_expanded
rewrite hjn at hj
exact hj
exact hold
have hbalance : x4 * x = x2 * y
specialize beta_product_replace_balance n
specialize beta_product_replace_balance b
specialize beta_product_replace_balance c
specialize beta_product_replace_balance z
specialize beta_product_replace_balance d
specialize beta_product_replace_balance i
specialize beta_product_replace_balance x
specialize beta_product_replace_balance y
specialize beta_product_replace_balance x2
specialize beta_product_replace_balance x4
apply beta_product_replace_balance
exact hi
exact hold_i
exact hnew_i
exact hprefix_preserve
exact hold_decomp_witness_witness_right_left
exact hnew_decomp_witness_witness_right_left
rewrite hold_decomp_witness_witness_right_right
rewrite hnew_decomp_witness_witness_right_right
rewrite hold_last
rewrite hnew_last
symm
exact hbalance
qed

Occurrences

7,439

Distinct objects

1,685

Reused references

61

Depth

67

Cuts

205

Certificate SHA-256

ee7ff4aa1c45…

finite_fixed_last_prefix_boundedFinite fixed last prefix bounded Quadratic reciprocity campaign checked

A bounded injective successor reindexing fixed at its last position is bounded on the old prefix.

Prerequisites

finite_bounded_prefix_without_top le_succ le_refl lt_irrefl_expanded

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall r s n. (forall fp_i_fixed_last_bounded_succ. (exists fp_gap_fixed_last_bounded_succ_index. fp_gap_fixed_last_bounded_succ_index + S fp_i_fixed_last_bounded_succ = S n) -> exists fp_value_fixed_last_bounded_succ. ((((exists ff_h_fixed_last_bounded_succ_entry. ff_h_fixed_last_bounded_succ_entry + S (fp_value_fixed_last_bounded_succ) = S ((S (fp_i_fixed_last_bounded_succ)) * s)) /\ exists ff_q_fixed_last_bounded_succ_entry. r = ff_q_fixed_last_bounded_succ_entry * S ((S (fp_i_fixed_last_bounded_succ)) * s) + (fp_value_fixed_last_bounded_succ))) /\ (exists fp_gap_fixed_last_bounded_succ_value. fp_gap_fixed_last_bounded_succ_value + S fp_value_fixed_last_bounded_succ = S n))) -> (forall fp_i_fixed_last_injective_succ fp_j_fixed_last_injective_succ fp_value_fixed_last_injective_succ. (exists fp_gap_fixed_last_injective_succ_i. fp_gap_fixed_last_injective_succ_i + S fp_i_fixed_last_injective_succ = S n) -> (exists fp_gap_fixed_last_injective_succ_j. fp_gap_fixed_last_injective_succ_j + S fp_j_fixed_last_injective_succ = S n) -> (((exists ff_h_fixed_last_injective_succ_left. ff_h_fixed_last_injective_succ_left + S (fp_value_fixed_last_injective_succ) = S ((S (fp_i_fixed_last_injective_succ)) * s)) /\ exists ff_q_fixed_last_injective_succ_left. r = ff_q_fixed_last_injective_succ_left * S ((S (fp_i_fixed_last_injective_succ)) * s) + (fp_value_fixed_last_injective_succ))) -> (((exists ff_h_fixed_last_injective_succ_right. ff_h_fixed_last_injective_succ_right + S (fp_value_fixed_last_injective_succ) = S ((S (fp_j_fixed_last_injective_succ)) * s)) /\ exists ff_q_fixed_last_injective_succ_right. r = ff_q_fixed_last_injective_succ_right * S ((S (fp_j_fixed_last_injective_succ)) * s) + (fp_value_fixed_last_injective_succ))) -> fp_i_fixed_last_injective_succ = fp_j_fixed_last_injective_succ) -> (((exists ff_h_fixed_last_entry. ff_h_fixed_last_entry + S (n) = S ((S (n)) * s)) /\ exists ff_q_fixed_last_entry. r = ff_q_fixed_last_entry * S ((S (n)) * s) + (n))) -> (forall fp_i_fixed_last_bounded_prefix. (exists fp_gap_fixed_last_bounded_prefix_index. fp_gap_fixed_last_bounded_prefix_index + S fp_i_fixed_last_bounded_prefix = n) -> exists fp_value_fixed_last_bounded_prefix. ((((exists ff_h_fixed_last_bounded_prefix_entry. ff_h_fixed_last_bounded_prefix_entry + S (fp_value_fixed_last_bounded_prefix) = S ((S (fp_i_fixed_last_bounded_prefix)) * s)) /\ exists ff_q_fixed_last_bounded_prefix_entry. r = ff_q_fixed_last_bounded_prefix_entry * S ((S (fp_i_fixed_last_bounded_prefix)) * s) + (fp_value_fixed_last_bounded_prefix))) /\ (exists fp_gap_fixed_last_bounded_prefix_value. fp_gap_fixed_last_bounded_prefix_value + S fp_value_fixed_last_bounded_prefix = n)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall r s n. (forall fp_i_fixed_last_bounded_succ. (exists fp_gap_fixed_last_bounded_succ_index. fp_gap_fixed_last_bounded_succ_index + S fp_i_fixed_last_bounded_succ = S n) -> exists fp_value_fixed_last_bounded_succ. ((((exists ff_h_fixed_last_bounded_succ_entry. ff_h_fixed_last_bounded_succ_entry + S (fp_value_fixed_last_bounded_succ) = S ((S (fp_i_fixed_last_bounded_succ)) * s)) /\ exists ff_q_fixed_last_bounded_succ_entry. r = ff_q_fixed_last_bounded_succ_entry * S ((S (fp_i_fixed_last_bounded_succ)) * s) + (fp_value_fixed_last_bounded_succ))) /\ (exists fp_gap_fixed_last_bounded_succ_value. fp_gap_fixed_last_bounded_succ_value + S fp_value_fixed_last_bounded_succ = S n))) -> (forall fp_i_fixed_last_injective_succ fp_j_fixed_last_injective_succ fp_value_fixed_last_injective_succ. (exists fp_gap_fixed_last_injective_succ_i. fp_gap_fixed_last_injective_succ_i + S fp_i_fixed_last_injective_succ = S n) -> (exists fp_gap_fixed_last_injective_succ_j. fp_gap_fixed_last_injective_succ_j + S fp_j_fixed_last_injective_succ = S n) -> (((exists ff_h_fixed_last_injective_succ_left. ff_h_fixed_last_injective_succ_left + S (fp_value_fixed_last_injective_succ) = S ((S (fp_i_fixed_last_injective_succ)) * s)) /\ exists ff_q_fixed_last_injective_succ_left. r = ff_q_fixed_last_injective_succ_left * S ((S (fp_i_fixed_last_injective_succ)) * s) + (fp_value_fixed_last_injective_succ))) -> (((exists ff_h_fixed_last_injective_succ_right. ff_h_fixed_last_injective_succ_right + S (fp_value_fixed_last_injective_succ) = S ((S (fp_j_fixed_last_injective_succ)) * s)) /\ exists ff_q_fixed_last_injective_succ_right. r = ff_q_fixed_last_injective_succ_right * S ((S (fp_j_fixed_last_injective_succ)) * s) + (fp_value_fixed_last_injective_succ))) -> fp_i_fixed_last_injective_succ = fp_j_fixed_last_injective_succ) -> (((exists ff_h_fixed_last_entry. ff_h_fixed_last_entry + S (n) = S ((S (n)) * s)) /\ exists ff_q_fixed_last_entry. r = ff_q_fixed_last_entry * S ((S (n)) * s) + (n))) -> (forall fp_i_fixed_last_bounded_prefix. (exists fp_gap_fixed_last_bounded_prefix_index. fp_gap_fixed_last_bounded_prefix_index + S fp_i_fixed_last_bounded_prefix = n) -> exists fp_value_fixed_last_bounded_prefix. ((((exists ff_h_fixed_last_bounded_prefix_entry. ff_h_fixed_last_bounded_prefix_entry + S (fp_value_fixed_last_bounded_prefix) = S ((S (fp_i_fixed_last_bounded_prefix)) * s)) /\ exists ff_q_fixed_last_bounded_prefix_entry. r = ff_q_fixed_last_bounded_prefix_entry * S ((S (fp_i_fixed_last_bounded_prefix)) * s) + (fp_value_fixed_last_bounded_prefix))) /\ (exists fp_gap_fixed_last_bounded_prefix_value. fp_gap_fixed_last_bounded_prefix_value + S fp_value_fixed_last_bounded_prefix = n)))
use finite_bounded_prefix_without_top
use le_succ
use le_refl
use lt_irrefl_expanded
intro r
intro s
intro n
intro hbounded
intro hinjective
intro hlast
have hnotop : forall i. (exists h. h + S i = n) -> ~((exists h. h + S n = S ((S i) * s)) /\ exists q. r = q * S ((S i) * s) + n)
intro i
intro hi
intro htop
have hisn : exists h. h + S i = S n
specialize le_succ (S i)
specialize le_succ n
apply le_succ
exact hi
have hnsn : exists h. h + S n = S n
specialize le_refl (S n)
exact le_refl
have hin : i = n
specialize hinjective i
specialize hinjective n
specialize hinjective n
apply hinjective
exact hisn
exact hnsn
exact htop
exact hlast
specialize lt_irrefl_expanded n
apply lt_irrefl_expanded
rewrite hin at hi
exact hi
specialize finite_bounded_prefix_without_top r
specialize finite_bounded_prefix_without_top s
specialize finite_bounded_prefix_without_top n
specialize finite_bounded_prefix_without_top (S n)
apply finite_bounded_prefix_without_top
refl
exact hbounded
exact hnotop
qed

Occurrences

409

Distinct objects

294

Reused references

8

Depth

24

Cuts

16

Certificate SHA-256

f17e5ed17a10…

beta_reindex_alignment_swap_lastBeta reindex alignment swap last Quadratic reciprocity campaign checked

Simultaneous interior/final swaps of an index code and target factors preserve alignment.

Prerequisites

beta_prefix_swap_last_reflect beta_at_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall r s u v b c z d w e n i m x y. (((exists ff_h_align_swap_map_i. ff_h_align_swap_map_i + S (m) = S ((S (i)) * v)) /\ exists ff_q_align_swap_map_i. u = ff_q_align_swap_map_i * S ((S (i)) * v) + (m))) -> (((exists ff_h_align_swap_map_n. ff_h_align_swap_map_n + S (n) = S ((S (n)) * v)) /\ exists ff_q_align_swap_map_n. u = ff_q_align_swap_map_n * S ((S (n)) * v) + (n))) -> (forall k j. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_align_swap_map_old. ff_h_align_swap_map_old + S (j) = S ((S (k)) * s)) /\ exists ff_q_align_swap_map_old. r = ff_q_align_swap_map_old * S ((S (k)) * s) + (j))) -> (((exists ff_h_align_swap_map_new. ff_h_align_swap_map_new + S (j) = S ((S (k)) * v)) /\ exists ff_q_align_swap_map_new. u = ff_q_align_swap_map_new * S ((S (k)) * v) + (j)))) -> (((exists ff_h_align_swap_source_m. ff_h_align_swap_source_m + S (y) = S ((S (m)) * c)) /\ exists ff_q_align_swap_source_m. b = ff_q_align_swap_source_m * S ((S (m)) * c) + (y))) -> (((exists ff_h_align_swap_source_n. ff_h_align_swap_source_n + S (x) = S ((S (n)) * c)) /\ exists ff_q_align_swap_source_n. b = ff_q_align_swap_source_n * S ((S (n)) * c) + (x))) -> (((exists ff_h_align_swap_target_i. ff_h_align_swap_target_i + S (y) = S ((S (i)) * e)) /\ exists ff_q_align_swap_target_i. w = ff_q_align_swap_target_i * S ((S (i)) * e) + (y))) -> (((exists ff_h_align_swap_target_n. ff_h_align_swap_target_n + S (x) = S ((S (n)) * e)) /\ exists ff_q_align_swap_target_n. w = ff_q_align_swap_target_n * S ((S (n)) * e) + (x))) -> (forall k a. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_align_swap_target_old. ff_h_align_swap_target_old + S (a) = S ((S (k)) * d)) /\ exists ff_q_align_swap_target_old. z = ff_q_align_swap_target_old * S ((S (k)) * d) + (a))) -> (((exists ff_h_align_swap_target_new. ff_h_align_swap_target_new + S (a) = S ((S (k)) * e)) /\ exists ff_q_align_swap_target_new. w = ff_q_align_swap_target_new * S ((S (k)) * e) + (a)))) -> (forall fpr_i_align_swap_old fpr_j_align_swap_old fpr_x_align_swap_old. (exists fpr_h_align_swap_old. fpr_h_align_swap_old + S fpr_i_align_swap_old = S n) -> (((exists ff_h_align_swap_old_map. ff_h_align_swap_old_map + S (fpr_j_align_swap_old) = S ((S (fpr_i_align_swap_old)) * s)) /\ exists ff_q_align_swap_old_map. r = ff_q_align_swap_old_map * S ((S (fpr_i_align_swap_old)) * s) + (fpr_j_align_swap_old))) -> (((exists ff_h_align_swap_old_source. ff_h_align_swap_old_source + S (fpr_x_align_swap_old) = S ((S (fpr_j_align_swap_old)) * c)) /\ exists ff_q_align_swap_old_source. b = ff_q_align_swap_old_source * S ((S (fpr_j_align_swap_old)) * c) + (fpr_x_align_swap_old))) -> (((exists ff_h_align_swap_old_target. ff_h_align_swap_old_target + S (fpr_x_align_swap_old) = S ((S (fpr_i_align_swap_old)) * d)) /\ exists ff_q_align_swap_old_target. z = ff_q_align_swap_old_target * S ((S (fpr_i_align_swap_old)) * d) + (fpr_x_align_swap_old)))) -> (forall fpr_i_align_swap_new fpr_j_align_swap_new fpr_x_align_swap_new. (exists fpr_h_align_swap_new. fpr_h_align_swap_new + S fpr_i_align_swap_new = S n) -> (((exists ff_h_align_swap_new_map. ff_h_align_swap_new_map + S (fpr_j_align_swap_new) = S ((S (fpr_i_align_swap_new)) * v)) /\ exists ff_q_align_swap_new_map. u = ff_q_align_swap_new_map * S ((S (fpr_i_align_swap_new)) * v) + (fpr_j_align_swap_new))) -> (((exists ff_h_align_swap_new_source. ff_h_align_swap_new_source + S (fpr_x_align_swap_new) = S ((S (fpr_j_align_swap_new)) * c)) /\ exists ff_q_align_swap_new_source. b = ff_q_align_swap_new_source * S ((S (fpr_j_align_swap_new)) * c) + (fpr_x_align_swap_new))) -> (((exists ff_h_align_swap_new_target. ff_h_align_swap_new_target + S (fpr_x_align_swap_new) = S ((S (fpr_i_align_swap_new)) * e)) /\ exists ff_q_align_swap_new_target. w = ff_q_align_swap_new_target * S ((S (fpr_i_align_swap_new)) * e) + (fpr_x_align_swap_new))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall r s u v b c z d w e n i m x y. (((exists ff_h_align_swap_map_i. ff_h_align_swap_map_i + S (m) = S ((S (i)) * v)) /\ exists ff_q_align_swap_map_i. u = ff_q_align_swap_map_i * S ((S (i)) * v) + (m))) -> (((exists ff_h_align_swap_map_n. ff_h_align_swap_map_n + S (n) = S ((S (n)) * v)) /\ exists ff_q_align_swap_map_n. u = ff_q_align_swap_map_n * S ((S (n)) * v) + (n))) -> (forall k j. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_align_swap_map_old. ff_h_align_swap_map_old + S (j) = S ((S (k)) * s)) /\ exists ff_q_align_swap_map_old. r = ff_q_align_swap_map_old * S ((S (k)) * s) + (j))) -> (((exists ff_h_align_swap_map_new. ff_h_align_swap_map_new + S (j) = S ((S (k)) * v)) /\ exists ff_q_align_swap_map_new. u = ff_q_align_swap_map_new * S ((S (k)) * v) + (j)))) -> (((exists ff_h_align_swap_source_m. ff_h_align_swap_source_m + S (y) = S ((S (m)) * c)) /\ exists ff_q_align_swap_source_m. b = ff_q_align_swap_source_m * S ((S (m)) * c) + (y))) -> (((exists ff_h_align_swap_source_n. ff_h_align_swap_source_n + S (x) = S ((S (n)) * c)) /\ exists ff_q_align_swap_source_n. b = ff_q_align_swap_source_n * S ((S (n)) * c) + (x))) -> (((exists ff_h_align_swap_target_i. ff_h_align_swap_target_i + S (y) = S ((S (i)) * e)) /\ exists ff_q_align_swap_target_i. w = ff_q_align_swap_target_i * S ((S (i)) * e) + (y))) -> (((exists ff_h_align_swap_target_n. ff_h_align_swap_target_n + S (x) = S ((S (n)) * e)) /\ exists ff_q_align_swap_target_n. w = ff_q_align_swap_target_n * S ((S (n)) * e) + (x))) -> (forall k a. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_align_swap_target_old. ff_h_align_swap_target_old + S (a) = S ((S (k)) * d)) /\ exists ff_q_align_swap_target_old. z = ff_q_align_swap_target_old * S ((S (k)) * d) + (a))) -> (((exists ff_h_align_swap_target_new. ff_h_align_swap_target_new + S (a) = S ((S (k)) * e)) /\ exists ff_q_align_swap_target_new. w = ff_q_align_swap_target_new * S ((S (k)) * e) + (a)))) -> (forall fpr_i_align_swap_old fpr_j_align_swap_old fpr_x_align_swap_old. (exists fpr_h_align_swap_old. fpr_h_align_swap_old + S fpr_i_align_swap_old = S n) -> (((exists ff_h_align_swap_old_map. ff_h_align_swap_old_map + S (fpr_j_align_swap_old) = S ((S (fpr_i_align_swap_old)) * s)) /\ exists ff_q_align_swap_old_map. r = ff_q_align_swap_old_map * S ((S (fpr_i_align_swap_old)) * s) + (fpr_j_align_swap_old))) -> (((exists ff_h_align_swap_old_source. ff_h_align_swap_old_source + S (fpr_x_align_swap_old) = S ((S (fpr_j_align_swap_old)) * c)) /\ exists ff_q_align_swap_old_source. b = ff_q_align_swap_old_source * S ((S (fpr_j_align_swap_old)) * c) + (fpr_x_align_swap_old))) -> (((exists ff_h_align_swap_old_target. ff_h_align_swap_old_target + S (fpr_x_align_swap_old) = S ((S (fpr_i_align_swap_old)) * d)) /\ exists ff_q_align_swap_old_target. z = ff_q_align_swap_old_target * S ((S (fpr_i_align_swap_old)) * d) + (fpr_x_align_swap_old)))) -> (forall fpr_i_align_swap_new fpr_j_align_swap_new fpr_x_align_swap_new. (exists fpr_h_align_swap_new. fpr_h_align_swap_new + S fpr_i_align_swap_new = S n) -> (((exists ff_h_align_swap_new_map. ff_h_align_swap_new_map + S (fpr_j_align_swap_new) = S ((S (fpr_i_align_swap_new)) * v)) /\ exists ff_q_align_swap_new_map. u = ff_q_align_swap_new_map * S ((S (fpr_i_align_swap_new)) * v) + (fpr_j_align_swap_new))) -> (((exists ff_h_align_swap_new_source. ff_h_align_swap_new_source + S (fpr_x_align_swap_new) = S ((S (fpr_j_align_swap_new)) * c)) /\ exists ff_q_align_swap_new_source. b = ff_q_align_swap_new_source * S ((S (fpr_j_align_swap_new)) * c) + (fpr_x_align_swap_new))) -> (((exists ff_h_align_swap_new_target. ff_h_align_swap_new_target + S (fpr_x_align_swap_new) = S ((S (fpr_i_align_swap_new)) * e)) /\ exists ff_q_align_swap_new_target. w = ff_q_align_swap_new_target * S ((S (fpr_i_align_swap_new)) * e) + (fpr_x_align_swap_new))))
use beta_prefix_swap_last_reflect
use beta_at_unique
intro r
intro s
intro u
intro v
intro b
intro c
intro z
intro d
intro w
intro e
intro n
intro i
intro m
intro x
intro y
intro hmap_i
intro hmap_n
intro hmap_preserve
intro hsource_m
intro hsource_n
intro htarget_i
intro htarget_n
intro htarget_preserve
intro haligned
have hreflect : forall k j. (exists h. h + S k = S n) -> ((exists h. h + S j = S ((S k) * v)) /\ exists q. u = q * S ((S k) * v) + j) -> (k = i /\ j = m) \/ ((k = n /\ j = n) \/ (~(k = i) /\ (~(k = n) /\ ((exists h. h + S j = S ((S k) * s)) /\ exists q. r = q * S ((S k) * s) + j))))
specialize beta_prefix_swap_last_reflect r
specialize beta_prefix_swap_last_reflect s
specialize beta_prefix_swap_last_reflect u
specialize beta_prefix_swap_last_reflect v
specialize beta_prefix_swap_last_reflect n
specialize beta_prefix_swap_last_reflect i
specialize beta_prefix_swap_last_reflect n
specialize beta_prefix_swap_last_reflect m
apply beta_prefix_swap_last_reflect
exact hmap_i
exact hmap_n
exact hmap_preserve
intro k
intro j
intro a
intro hk
intro hmap
intro hsource
specialize hreflect k
specialize hreflect j
have hcases : (k = i /\ j = m) \/ ((k = n /\ j = n) \/ (~(k = i) /\ (~(k = n) /\ ((exists h. h + S j = S ((S k) * s)) /\ exists q. r = q * S ((S k) * s) + j))))
apply hreflect
exact hk
exact hmap
cases hcases
cases hcases_left
have hay : a = y
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique m
specialize beta_at_unique a
specialize beta_at_unique y
apply beta_at_unique
rewrite hcases_left_right at hsource
rewrite hcases_left_right at hsource
exact hsource
exact hsource_m
rewrite hcases_left_left
rewrite hcases_left_left
rewrite hay
rewrite hay
exact htarget_i
cases hcases_right
cases hcases_right_left
have hax : a = x
specialize beta_at_unique b
specialize beta_at_unique c
specialize beta_at_unique n
specialize beta_at_unique a
specialize beta_at_unique x
apply beta_at_unique
rewrite hcases_right_left_right at hsource
rewrite hcases_right_left_right at hsource
exact hsource
exact hsource_n
rewrite hcases_right_left_left
rewrite hcases_right_left_left
rewrite hax
rewrite hax
exact htarget_n
cases hcases_right_right
cases hcases_right_right_right
have hold_target : ((exists h. h + S a = S ((S k) * d)) /\ exists q. z = q * S ((S k) * d) + a)
specialize haligned k
specialize haligned j
specialize haligned a
apply haligned
exact hk
exact hcases_right_right_right_right
exact hsource
specialize htarget_preserve k
specialize htarget_preserve a
apply htarget_preserve
exact hk
exact hcases_right_right_left
exact hcases_right_right_right_left
exact hold_target
qed

Occurrences

3,057

Distinct objects

1,212

Reused references

56

Depth

63

Cuts

80

Certificate SHA-256

04543723c864…

prime_is_succ_succPrime is succ succ Quadratic reciprocity campaign checked

Every prime natural is the second successor of a natural.

Prerequisites

prime_nonzero nonzero_is_succ

Used directly by

prime_bounded_nonzero_mod_inverse

Exact expanded PA statement

Copy statement

forall p. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> exists k. p = S (S k)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> exists k. p = S (S k)
use prime_nonzero
use nonzero_is_succ
intro p
intro hp
have hp0 : ~(p = 0)
intro hpzero
specialize prime_nonzero p
apply prime_nonzero
exact hp
exact hpzero
have hps : exists k. p = S k
specialize nonzero_is_succ p
apply nonzero_is_succ
exact hp0
cases hps
have hx0 : ~(x = 0)
intro hx0
cases hp
apply hp_left
rewrite hps_witness
rewrite hx0
refl
have hxs : exists k. x = S k
specialize nonzero_is_succ x
apply nonzero_is_succ
exact hx0
cases hxs
exists x1
rewrite hps_witness
rewrite hxs_witness
refl
qed

Occurrences

98

Distinct objects

98

Reused references

0

Depth

13

Cuts

4

Certificate SHA-256

e0cd8df0205c…

prime_bounded_nonzero_mod_inversePrime bounded nonzero mod inverse Quadratic reciprocity campaign checked

A nonzero residue below a prime has a nonzero bounded inverse.

Prerequisites

prime_is_succ_succ prime_nonzero divisor_le_nonzero lt_not_le prime_mod_inverse division_remainder_exists mul_comm remainder_decomposition_to_mod_eq mod_eq_mul_left mod_eq_symm mod_eq_trans mod_eq_bounded_unique succ_ne_zero

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall p a. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> ~(a = 0) -> (exists qrbu_gap_a_lt_p. qrbu_gap_a_lt_p + S a = p) -> (exists qrbu_inverse_bounded_inverse. (~(qrbu_inverse_bounded_inverse = 0) /\ ((exists qrbu_gap_bounded_inverse_bound. qrbu_gap_bounded_inverse_bound + S qrbu_inverse_bounded_inverse = p) /\ (exists qrbu_mod_left_bounded_inverse_mod qrbu_mod_right_bounded_inverse_mod. a * qrbu_inverse_bounded_inverse + p * qrbu_mod_left_bounded_inverse_mod = 1 + p * qrbu_mod_right_bounded_inverse_mod))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p a. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> ~(a = 0) -> (exists qrbu_gap_a_lt_p. qrbu_gap_a_lt_p + S a = p) -> (exists qrbu_inverse_bounded_inverse. (~(qrbu_inverse_bounded_inverse = 0) /\ ((exists qrbu_gap_bounded_inverse_bound. qrbu_gap_bounded_inverse_bound + S qrbu_inverse_bounded_inverse = p) /\ (exists qrbu_mod_left_bounded_inverse_mod qrbu_mod_right_bounded_inverse_mod. a * qrbu_inverse_bounded_inverse + p * qrbu_mod_left_bounded_inverse_mod = 1 + p * qrbu_mod_right_bounded_inverse_mod))))
use prime_is_succ_succ
use prime_nonzero
use divisor_le_nonzero
use lt_not_le
use prime_mod_inverse
use division_remainder_exists
use mul_comm
use remainder_decomposition_to_mod_eq
use mod_eq_mul_left
use mod_eq_symm
use mod_eq_trans
use mod_eq_bounded_unique
use succ_ne_zero
intro p
intro a
intro hp
intro ha0
intro hap
have hp0 : ~(p = 0)
intro hpzero
specialize prime_nonzero p
apply prime_nonzero
exact hp
exact hpzero
have hnotdiv : ~(exists k. a = p * k)
intro hdiv
have hpa : exists t. t + p = a
specialize divisor_le_nonzero p
specialize divisor_le_nonzero a
apply divisor_le_nonzero
exact ha0
exact hdiv
specialize lt_not_le a
specialize lt_not_le p
apply lt_not_le
exact hap
exact hpa
have hinv : exists z u v. a * z + p * u = 1 + p * v
specialize prime_mod_inverse p
specialize prime_mod_inverse a
apply prime_mod_inverse
exact hp
exact hnotdiv
cases hinv
cases hinv_witness
cases hinv_witness_witness
have hdivz : exists q r. x = p * q + r /\ exists h. h + S r = p
specialize division_remainder_exists p
specialize division_remainder_exists x
apply division_remainder_exists
exact hp0
cases hdivz
cases hdivz_witness
cases hdivz_witness_witness
have hzdecomp : x = x3 * p + x4
trans p * x3 + x4
exact hdivz_witness_witness_left
congr
apply mul_comm
refl
have hzr : exists u v. x + p * u = x4 + p * v
specialize remainder_decomposition_to_mod_eq p
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq x3
specialize remainder_decomposition_to_mod_eq x4
apply remainder_decomposition_to_mod_eq
exact hzdecomp
have hscaled : exists u v. (a * x) + p * u = (a * x4) + p * v
specialize mod_eq_mul_left p
specialize mod_eq_mul_left x
specialize mod_eq_mul_left x4
specialize mod_eq_mul_left a
apply mod_eq_mul_left
exact hzr
have hrz : exists u v. (a * x4) + p * u = (a * x) + p * v
specialize mod_eq_symm p
specialize mod_eq_symm (a * x)
specialize mod_eq_symm (a * x4)
apply mod_eq_symm
exact hscaled
have hfinal : exists u v. (a * x4) + p * u = 1 + p * v
specialize mod_eq_trans p
specialize mod_eq_trans (a * x4)
specialize mod_eq_trans (a * x)
specialize mod_eq_trans 1
apply mod_eq_trans
exact hrz
exists x1
exists x2
exact hinv_witness_witness_witness
have hp2 : exists k. p = S (S k)
specialize prime_is_succ_succ p
apply prime_is_succ_succ
exact hp
cases hp2
have h0bound : exists h. h + S 0 = p
exists S x5
rewrite hp2_witness
simp
have h1bound : exists h. h + S 1 = p
exists x5
rewrite hp2_witness
simp
have hr0 : ~(x4 = 0)
intro hrzero
have hzeroone : exists u v. 0 + p * u = 1 + p * v
cases hfinal
cases hfinal_witness
exists x6
exists x7
trans (a * x4) + p * x6
congr
symm
trans a * 0
congr
refl
exact hrzero
apply PA5
refl
exact hfinal_witness_witness
have hzeroeqone : 0 = 1
specialize mod_eq_bounded_unique p
specialize mod_eq_bounded_unique 0
specialize mod_eq_bounded_unique 1
apply mod_eq_bounded_unique
exact h0bound
exact h1bound
exact hzeroone
specialize succ_ne_zero 0
apply succ_ne_zero
symm
exact hzeroeqone
exists x4
split
exact hr0
split
exact hdivz_witness_witness_right
exact hfinal
qed

Occurrences

8,684

Distinct objects

2,801

Reused references

151

Depth

71

Cuts

260

Certificate SHA-256

a954ec4e9c70…

canonical_remainder_existsCanonical remainder existence Division checked

Every dividend has a canonical remainder for each nonzero modulus.

Prerequisites

division_remainder_exists

Used directly by

canonical_remainder_exists_unique coprime_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall m n. ~(m = 0) -> exists r. (((exists hcr_quotient_result. n = m * hcr_quotient_result + r) /\ exists hcr_gap_result. hcr_gap_result + S r = m))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. ~(m = 0) -> exists r. (((exists hcr_quotient_result. n = m * hcr_quotient_result + r) /\ exists hcr_gap_result. hcr_gap_result + S r = m))
use division_remainder_exists
intro m
intro n
intro hm
have hdivision : exists q r. n = m * q + r /\ exists gap. gap + S r = m
specialize division_remainder_exists m
specialize division_remainder_exists n
apply division_remainder_exists
exact hm
cases hdivision
cases hdivision_witness
cases hdivision_witness_witness
exists x1
split
exists x
exact hdivision_witness_witness_left
exact hdivision_witness_witness_right
qed

Occurrences

238

Distinct objects

215

Reused references

16

Depth

29

Cuts

6

Certificate SHA-256

02da660e4be1…

canonical_remainder_functionalCanonical remainder functionality Division checked

Canonical remainders for a fixed modulus are functional whenever they exist.

Prerequisites

division_remainder_unique

Used directly by

canonical_remainder_exists_unique

Exact expanded PA statement

Copy statement

forall m n r s. (((exists hcr_quotient_result. n = m * hcr_quotient_result + r) /\ exists hcr_gap_result. hcr_gap_result + S r = m)) -> (((exists hcr_quotient_comparison. n = m * hcr_quotient_comparison + s) /\ exists hcr_gap_comparison. hcr_gap_comparison + S s = m)) -> r = s

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n r s. (((exists hcr_quotient_result. n = m * hcr_quotient_result + r) /\ exists hcr_gap_result. hcr_gap_result + S r = m)) -> (((exists hcr_quotient_comparison. n = m * hcr_quotient_comparison + s) /\ exists hcr_gap_comparison. hcr_gap_comparison + S s = m)) -> r = s
use division_remainder_unique
intro m
intro n
intro r
intro s
intro hr
intro hs
cases hr
cases hs
cases hr_left
cases hs_left
have hunique : x = x1 /\ r = s
specialize division_remainder_unique m
specialize division_remainder_unique n
specialize division_remainder_unique x
specialize division_remainder_unique r
specialize division_remainder_unique x1
specialize division_remainder_unique s
apply division_remainder_unique
exact hr_left_witness
exact hr_right
exact hs_left_witness
exact hs_right
cases hunique
exact hunique_right
qed

Occurrences

885

Distinct objects

575

Reused references

22

Depth

58

Cuts

23

Certificate SHA-256

075a5bc6bceb…

canonical_remainder_zero_impossibleCanonical remainder excludes modulus zero Division checked

The canonical-remainder relation has no inhabitant at modulus zero.

Prerequisites

succ_ne_zero

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n r. m = 0 -> ~(((exists hcr_quotient_zero_impossible. n = m * hcr_quotient_zero_impossible + r) /\ exists hcr_gap_zero_impossible. hcr_gap_zero_impossible + S r = m))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n r. m = 0 -> ~(((exists hcr_quotient_zero_impossible. n = m * hcr_quotient_zero_impossible + r) /\ exists hcr_gap_zero_impossible. hcr_gap_zero_impossible + S r = m))
use succ_ne_zero
intro m
intro n
intro r
intro hm
intro hrem
cases hrem
rewrite hm at hrem_right
cases hrem_right
have hsucc : S (x + r) = 0
trans x + S r
symm
apply PA4
exact hrem_right_witness
specialize succ_ne_zero (x + r)
apply succ_ne_zero
exact hsucc
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

13

Cuts

1

Certificate SHA-256

ba801ceeb223…

canonical_remainder_exists_uniqueUnique canonical remainder Division checked

For every nonzero modulus, the canonical remainder exists uniquely; the comparison remainder is proved equal to the chosen remainder.

Prerequisites

canonical_remainder_exists canonical_remainder_functional

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n. ~(m = 0) -> exists r. ((((exists hcr_quotient_unique_result. n = m * hcr_quotient_unique_result + r) /\ exists hcr_gap_unique_result. hcr_gap_unique_result + S r = m)) /\ forall s. (((exists hcr_quotient_unique_comparison. n = m * hcr_quotient_unique_comparison + s) /\ exists hcr_gap_unique_comparison. hcr_gap_unique_comparison + S s = m)) -> s = r)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n. ~(m = 0) -> exists r. ((((exists hcr_quotient_unique_result. n = m * hcr_quotient_unique_result + r) /\ exists hcr_gap_unique_result. hcr_gap_unique_result + S r = m)) /\ forall s. (((exists hcr_quotient_unique_comparison. n = m * hcr_quotient_unique_comparison + s) /\ exists hcr_gap_unique_comparison. hcr_gap_unique_comparison + S s = m)) -> s = r)
use canonical_remainder_exists
use canonical_remainder_functional
intro m
intro n
intro hm
have hexists : exists r. (((exists hcr_quotient_unique_result. n = m * hcr_quotient_unique_result + r) /\ exists hcr_gap_unique_result. hcr_gap_unique_result + S r = m))
specialize canonical_remainder_exists m
specialize canonical_remainder_exists n
apply canonical_remainder_exists
exact hm
cases hexists
exists x
split
exact hexists_witness
intro s
intro hs
specialize canonical_remainder_functional m
specialize canonical_remainder_functional n
specialize canonical_remainder_functional s
specialize canonical_remainder_functional x
apply canonical_remainder_functional
exact hs
exact hexists_witness
qed

Occurrences

1,148

Distinct objects

765

Reused references

39

Depth

60

Cuts

31

Certificate SHA-256

40bd3cda0f0b…

canonical_remainders_characterize_mod_eqCanonical remainders characterize congruence Congruence & CRT checked

Two canonical remainders are equal exactly when their dividends are congruent modulo the shared modulus.

Prerequisites

mul_comm remainder_decomposition_to_mod_eq mod_eq_symm mod_eq_trans mod_eq_bounded_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m a b r s. (((exists hcr_quotient_left. a = m * hcr_quotient_left + r) /\ exists hcr_gap_left. hcr_gap_left + S r = m)) -> (((exists hcr_quotient_right. b = m * hcr_quotient_right + s) /\ exists hcr_gap_right. hcr_gap_right + S s = m)) -> (((exists hcc_mod_left_source hcc_mod_right_source. a + m * hcc_mod_left_source = b + m * hcc_mod_right_source) -> r = s) /\ (r = s -> (exists hcc_mod_left_result hcc_mod_right_result. a + m * hcc_mod_left_result = b + m * hcc_mod_right_result)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m a b r s. (((exists hcr_quotient_left. a = m * hcr_quotient_left + r) /\ exists hcr_gap_left. hcr_gap_left + S r = m)) -> (((exists hcr_quotient_right. b = m * hcr_quotient_right + s) /\ exists hcr_gap_right. hcr_gap_right + S s = m)) -> (((exists hcc_mod_left_source hcc_mod_right_source. a + m * hcc_mod_left_source = b + m * hcc_mod_right_source) -> r = s) /\ (r = s -> (exists hcc_mod_left_result hcc_mod_right_result. a + m * hcc_mod_left_result = b + m * hcc_mod_right_result)))
use mul_comm
use remainder_decomposition_to_mod_eq
use mod_eq_symm
use mod_eq_trans
use mod_eq_bounded_unique
intro m
intro a
intro b
intro r
intro s
intro hr
intro hs
cases hr
cases hs
cases hr_left
cases hs_left
have hadecomp : a = x * m + r
trans m * x + r
exact hr_left_witness
congr
apply mul_comm
refl
have hbdecomp : b = x1 * m + s
trans m * x1 + s
exact hs_left_witness
congr
apply mul_comm
refl
have har : exists hcc_mod_left_a_r hcc_mod_right_a_r. a + m * hcc_mod_left_a_r = r + m * hcc_mod_right_a_r
specialize remainder_decomposition_to_mod_eq m
specialize remainder_decomposition_to_mod_eq a
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq r
apply remainder_decomposition_to_mod_eq
exact hadecomp
have hbs : exists hcc_mod_left_b_s hcc_mod_right_b_s. b + m * hcc_mod_left_b_s = s + m * hcc_mod_right_b_s
specialize remainder_decomposition_to_mod_eq m
specialize remainder_decomposition_to_mod_eq b
specialize remainder_decomposition_to_mod_eq x1
specialize remainder_decomposition_to_mod_eq s
apply remainder_decomposition_to_mod_eq
exact hbdecomp
split
intro hab
have hra : exists hcc_mod_left_r_a hcc_mod_right_r_a. r + m * hcc_mod_left_r_a = a + m * hcc_mod_right_r_a
specialize mod_eq_symm m
specialize mod_eq_symm a
specialize mod_eq_symm r
apply mod_eq_symm
exact har
have hrb : exists hcc_mod_left_r_b hcc_mod_right_r_b. r + m * hcc_mod_left_r_b = b + m * hcc_mod_right_r_b
specialize mod_eq_trans m
specialize mod_eq_trans r
specialize mod_eq_trans a
specialize mod_eq_trans b
apply mod_eq_trans
exact hra
exact hab
have hrs : exists hcc_mod_left_r_s hcc_mod_right_r_s. r + m * hcc_mod_left_r_s = s + m * hcc_mod_right_r_s
specialize mod_eq_trans m
specialize mod_eq_trans r
specialize mod_eq_trans b
specialize mod_eq_trans s
apply mod_eq_trans
exact hrb
exact hbs
specialize mod_eq_bounded_unique m
specialize mod_eq_bounded_unique r
specialize mod_eq_bounded_unique s
apply mod_eq_bounded_unique
exact hr_right
exact hs_right
exact hrs
intro hrs_equal
rewrite hrs_equal at har
have hsb : exists hcc_mod_left_s_b hcc_mod_right_s_b. s + m * hcc_mod_left_s_b = b + m * hcc_mod_right_s_b
specialize mod_eq_symm m
specialize mod_eq_symm b
specialize mod_eq_symm s
apply mod_eq_symm
exact hbs
specialize mod_eq_trans m
specialize mod_eq_trans a
specialize mod_eq_trans s
specialize mod_eq_trans b
apply mod_eq_trans
exact har
exact hsb
qed

Occurrences

1,888

Distinct objects

908

Reused references

43

Depth

64

Cuts

53

Certificate SHA-256

f15b6a2109e7…

bounded_mod_inverse_uniqueUniqueness of bounded modular inverses GCD, Bézout & coprimality checked

Two bounded inverses of the same residue are equal.

Prerequisites

mod_eq_symm mod_eq_mul_left mod_eq_mul_right mul_assoc mul_comm mul_one one_mul mod_eq_trans mod_eq_bounded_unique

Used directly by

coprime_iff_unique_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall p x y z. (exists wip_strict_gap_unique_y_bound. wip_strict_gap_unique_y_bound + S y = p) -> (exists wip_strict_gap_unique_z_bound. wip_strict_gap_unique_z_bound + S z = p) -> (exists wip_mod_left_unique_xy wip_mod_right_unique_xy. x * y + p * wip_mod_left_unique_xy = 1 + p * wip_mod_right_unique_xy) -> (exists wip_mod_left_unique_xz wip_mod_right_unique_xz. x * z + p * wip_mod_left_unique_xz = 1 + p * wip_mod_right_unique_xz) -> y = z

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall p x y z. (exists wip_strict_gap_unique_y_bound. wip_strict_gap_unique_y_bound + S y = p) -> (exists wip_strict_gap_unique_z_bound. wip_strict_gap_unique_z_bound + S z = p) -> (exists wip_mod_left_unique_xy wip_mod_right_unique_xy. x * y + p * wip_mod_left_unique_xy = 1 + p * wip_mod_right_unique_xy) -> (exists wip_mod_left_unique_xz wip_mod_right_unique_xz. x * z + p * wip_mod_left_unique_xz = 1 + p * wip_mod_right_unique_xz) -> y = z
use mod_eq_symm
use mod_eq_mul_left
use mod_eq_mul_right
use mul_assoc
use mul_comm
use mul_one
use one_mul
use mod_eq_trans
use mod_eq_bounded_unique
intro p
intro x
intro y
intro z
intro hy
intro hz
intro hxy
intro hxz
have hreverse : exists wip_mod_left_unique_reverse_xz wip_mod_right_unique_reverse_xz. 1 + p * wip_mod_left_unique_reverse_xz = x * z + p * wip_mod_right_unique_reverse_xz
specialize mod_eq_symm p
specialize mod_eq_symm (x * z)
specialize mod_eq_symm 1
apply mod_eq_symm
exact hxz
have hleftscaled : exists wip_mod_left_unique_left_scaled wip_mod_right_unique_left_scaled. y * 1 + p * wip_mod_left_unique_left_scaled = y * (x * z) + p * wip_mod_right_unique_left_scaled
specialize mod_eq_mul_left p
specialize mod_eq_mul_left 1
specialize mod_eq_mul_left (x * z)
specialize mod_eq_mul_left y
apply mod_eq_mul_left
exact hreverse
have hrightscaled : exists wip_mod_left_unique_right_scaled wip_mod_right_unique_right_scaled. (x * y) * z + p * wip_mod_left_unique_right_scaled = 1 * z + p * wip_mod_right_unique_right_scaled
specialize mod_eq_mul_right p
specialize mod_eq_mul_right (x * y)
specialize mod_eq_mul_right 1
specialize mod_eq_mul_right z
apply mod_eq_mul_right
exact hxy
have hleft : y * 1 = y
specialize mul_one y
exact mul_one
have hmiddle : y * (x * z) = (x * y) * z
trans (y * x) * z
symm
specialize mul_assoc y
specialize mul_assoc x
specialize mul_assoc z
apply mul_assoc
congr
apply mul_comm
refl
have hright : 1 * z = z
specialize one_mul z
exact one_mul
rewrite hleft at hleftscaled
rewrite hmiddle at hleftscaled
rewrite hright at hrightscaled
have hyz : exists wip_mod_left_unique_result wip_mod_right_unique_result. y + p * wip_mod_left_unique_result = z + p * wip_mod_right_unique_result
specialize mod_eq_trans p
specialize mod_eq_trans y
specialize mod_eq_trans ((x * y) * z)
specialize mod_eq_trans z
apply mod_eq_trans
exact hleftscaled
exact hrightscaled
specialize mod_eq_bounded_unique p
specialize mod_eq_bounded_unique y
specialize mod_eq_bounded_unique z
apply mod_eq_bounded_unique
exact hy
exact hz
exact hyz
qed

Occurrences

2,914

Distinct objects

1,001

Reused references

61

Depth

68

Cuts

84

Certificate SHA-256

fd4a80c79f23…

coprime_bounded_mod_inverseCoprime residues have bounded modular inverses GCD, Bézout & coprimality checked

Every coprime residue has an inverse in the canonical interval below a nonzero modulus.

Prerequisites

canonical_remainder_exists coprime_mod_inverse mul_comm remainder_decomposition_to_mod_eq mod_eq_mul_left mod_eq_symm mod_eq_trans

Used directly by

coprime_iff_unique_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall a m. ~(m = 0) -> (forall hmi_divisor_assumption. (exists hmi_left_factor_assumption. a = hmi_divisor_assumption * hmi_left_factor_assumption) -> (exists hmi_right_factor_assumption. m = hmi_divisor_assumption * hmi_right_factor_assumption) -> hmi_divisor_assumption = 1) -> exists r. (exists hmi_gap_result_bound. hmi_gap_result_bound + S r = m) /\ (exists hmi_left_offset_result_inverse hmi_right_offset_result_inverse. a * r + m * hmi_left_offset_result_inverse = 1 + m * hmi_right_offset_result_inverse)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a m. ~(m = 0) -> (forall hmi_divisor_assumption. (exists hmi_left_factor_assumption. a = hmi_divisor_assumption * hmi_left_factor_assumption) -> (exists hmi_right_factor_assumption. m = hmi_divisor_assumption * hmi_right_factor_assumption) -> hmi_divisor_assumption = 1) -> exists r. (exists hmi_gap_result_bound. hmi_gap_result_bound + S r = m) /\ (exists hmi_left_offset_result_inverse hmi_right_offset_result_inverse. a * r + m * hmi_left_offset_result_inverse = 1 + m * hmi_right_offset_result_inverse)
use canonical_remainder_exists
use coprime_mod_inverse
use mul_comm
use remainder_decomposition_to_mod_eq
use mod_eq_mul_left
use mod_eq_symm
use mod_eq_trans
intro a
intro m
intro hm
intro hcop
have hinv : exists z u v. a * z + m * u = 1 + m * v
specialize coprime_mod_inverse a
specialize coprime_mod_inverse m
apply coprime_mod_inverse
exact hm
exact hcop
cases hinv
cases hinv_witness
cases hinv_witness_witness
have hrem : exists r. (((exists hcr_quotient_inverse_witness. x = m * hcr_quotient_inverse_witness + r) /\ exists hcr_gap_inverse_witness. hcr_gap_inverse_witness + S r = m))
specialize canonical_remainder_exists m
specialize canonical_remainder_exists x
apply canonical_remainder_exists
exact hm
cases hrem
cases hrem_witness
cases hrem_witness_left
have hzdecomp : x = x4 * m + x3
trans m * x4 + x3
exact hrem_witness_left_witness
congr
apply mul_comm
refl
have hzr : exists u v. x + m * u = x3 + m * v
specialize remainder_decomposition_to_mod_eq m
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq x4
specialize remainder_decomposition_to_mod_eq x3
apply remainder_decomposition_to_mod_eq
exact hzdecomp
have hscaled : exists u v. (a * x) + m * u = (a * x3) + m * v
specialize mod_eq_mul_left m
specialize mod_eq_mul_left x
specialize mod_eq_mul_left x3
specialize mod_eq_mul_left a
apply mod_eq_mul_left
exact hzr
have hreverse : exists u v. (a * x3) + m * u = (a * x) + m * v
specialize mod_eq_symm m
specialize mod_eq_symm (a * x)
specialize mod_eq_symm (a * x3)
apply mod_eq_symm
exact hscaled
have hfinal : exists u v. (a * x3) + m * u = 1 + m * v
specialize mod_eq_trans m
specialize mod_eq_trans (a * x3)
specialize mod_eq_trans (a * x)
specialize mod_eq_trans 1
apply mod_eq_trans
exact hreverse
exists x1
exists x2
exact hinv_witness_witness_witness
exists x3
split
exact hrem_witness_right
exact hfinal
qed

Occurrences

5,675

Distinct objects

1,752

Reused references

102

Depth

53

Cuts

168

Certificate SHA-256

00e30d9df307…

mod_inverse_implies_coprimeA modular inverse implies coprimality GCD, Bézout & coprimality checked

Any natural modular inverse forces coprimality, without a nonzero-modulus side condition.

Prerequisites

common_divisor_divides_balanced_result zero_add divisor_one

Used directly by

coprime_iff_unique_bounded_mod_inverse

Exact expanded PA statement

Copy statement

forall a m z. (exists hmi_left_offset_converse_assumption hmi_right_offset_converse_assumption. a * z + m * hmi_left_offset_converse_assumption = 1 + m * hmi_right_offset_converse_assumption) -> (forall hmi_divisor_converse_result. (exists hmi_left_factor_converse_result. a = hmi_divisor_converse_result * hmi_left_factor_converse_result) -> (exists hmi_right_factor_converse_result. m = hmi_divisor_converse_result * hmi_right_factor_converse_result) -> hmi_divisor_converse_result = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a m z. (exists hmi_left_offset_converse_assumption hmi_right_offset_converse_assumption. a * z + m * hmi_left_offset_converse_assumption = 1 + m * hmi_right_offset_converse_assumption) -> (forall hmi_divisor_converse_result. (exists hmi_left_factor_converse_result. a = hmi_divisor_converse_result * hmi_left_factor_converse_result) -> (exists hmi_right_factor_converse_result. m = hmi_divisor_converse_result * hmi_right_factor_converse_result) -> hmi_divisor_converse_result = 1)
use common_divisor_divides_balanced_result
use zero_add
use divisor_one
intro a
intro m
intro z
intro hinv
intro d
intro hda
intro hdm
cases hinv
cases hinv_witness
have hbez : a * z + m * x = 1 + (a * 0 + m * x1)
trans 1 + m * x1
exact hinv_witness_witness
congr
refl
symm
trans 0 + m * x1
congr
apply PA5
refl
apply zero_add
have hdivone : exists w. 1 = d * w
specialize common_divisor_divides_balanced_result d
specialize common_divisor_divides_balanced_result a
specialize common_divisor_divides_balanced_result m
specialize common_divisor_divides_balanced_result 1
specialize common_divisor_divides_balanced_result z
specialize common_divisor_divides_balanced_result x
specialize common_divisor_divides_balanced_result 0
specialize common_divisor_divides_balanced_result x1
apply common_divisor_divides_balanced_result
exact hda
exact hdm
exact hbez
specialize divisor_one d
apply divisor_one
exact hdivone
qed

Occurrences

874

Distinct objects

602

Reused references

42

Depth

40

Cuts

23

Certificate SHA-256

64b57d743e89…

coprime_iff_unique_bounded_mod_inverseCoprimality iff unique bounded modular inverse GCD, Bézout & coprimality checked

For a nonzero modulus, coprimality is equivalent to unique existence of a modular inverse in the canonical interval.

Prerequisites

coprime_bounded_mod_inverse bounded_mod_inverse_unique mod_inverse_implies_coprime

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a m. ~(m = 0) -> ((forall hmi_divisor_package_coprime. (exists hmi_left_factor_package_coprime. a = hmi_divisor_package_coprime * hmi_left_factor_package_coprime) -> (exists hmi_right_factor_package_coprime. m = hmi_divisor_package_coprime * hmi_right_factor_package_coprime) -> hmi_divisor_package_coprime = 1) -> (exists hmi_solution_package_result. ((((exists hmi_gap_package_result_chosen_bound. hmi_gap_package_result_chosen_bound + S hmi_solution_package_result = m) /\ (exists hmi_left_offset_package_result_chosen_inverse hmi_right_offset_package_result_chosen_inverse. a * hmi_solution_package_result + m * hmi_left_offset_package_result_chosen_inverse = 1 + m * hmi_right_offset_package_result_chosen_inverse))) /\ forall hmi_comparison_package_result. (((exists hmi_gap_package_result_compared_bound. hmi_gap_package_result_compared_bound + S hmi_comparison_package_result = m) /\ (exists hmi_left_offset_package_result_compared_inverse hmi_right_offset_package_result_compared_inverse. a * hmi_comparison_package_result + m * hmi_left_offset_package_result_compared_inverse = 1 + m * hmi_right_offset_package_result_compared_inverse))) -> hmi_comparison_package_result = hmi_solution_package_result))) /\ ((exists hmi_solution_package_result. ((((exists hmi_gap_package_result_chosen_bound. hmi_gap_package_result_chosen_bound + S hmi_solution_package_result = m) /\ (exists hmi_left_offset_package_result_chosen_inverse hmi_right_offset_package_result_chosen_inverse. a * hmi_solution_package_result + m * hmi_left_offset_package_result_chosen_inverse = 1 + m * hmi_right_offset_package_result_chosen_inverse))) /\ forall hmi_comparison_package_result. (((exists hmi_gap_package_result_compared_bound. hmi_gap_package_result_compared_bound + S hmi_comparison_package_result = m) /\ (exists hmi_left_offset_package_result_compared_inverse hmi_right_offset_package_result_compared_inverse. a * hmi_comparison_package_result + m * hmi_left_offset_package_result_compared_inverse = 1 + m * hmi_right_offset_package_result_compared_inverse))) -> hmi_comparison_package_result = hmi_solution_package_result)) -> (forall hmi_divisor_package_coprime. (exists hmi_left_factor_package_coprime. a = hmi_divisor_package_coprime * hmi_left_factor_package_coprime) -> (exists hmi_right_factor_package_coprime. m = hmi_divisor_package_coprime * hmi_right_factor_package_coprime) -> hmi_divisor_package_coprime = 1))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a m. ~(m = 0) -> ((forall hmi_divisor_package_coprime. (exists hmi_left_factor_package_coprime. a = hmi_divisor_package_coprime * hmi_left_factor_package_coprime) -> (exists hmi_right_factor_package_coprime. m = hmi_divisor_package_coprime * hmi_right_factor_package_coprime) -> hmi_divisor_package_coprime = 1) -> (exists hmi_solution_package_result. ((((exists hmi_gap_package_result_chosen_bound. hmi_gap_package_result_chosen_bound + S hmi_solution_package_result = m) /\ (exists hmi_left_offset_package_result_chosen_inverse hmi_right_offset_package_result_chosen_inverse. a * hmi_solution_package_result + m * hmi_left_offset_package_result_chosen_inverse = 1 + m * hmi_right_offset_package_result_chosen_inverse))) /\ forall hmi_comparison_package_result. (((exists hmi_gap_package_result_compared_bound. hmi_gap_package_result_compared_bound + S hmi_comparison_package_result = m) /\ (exists hmi_left_offset_package_result_compared_inverse hmi_right_offset_package_result_compared_inverse. a * hmi_comparison_package_result + m * hmi_left_offset_package_result_compared_inverse = 1 + m * hmi_right_offset_package_result_compared_inverse))) -> hmi_comparison_package_result = hmi_solution_package_result))) /\ ((exists hmi_solution_package_result. ((((exists hmi_gap_package_result_chosen_bound. hmi_gap_package_result_chosen_bound + S hmi_solution_package_result = m) /\ (exists hmi_left_offset_package_result_chosen_inverse hmi_right_offset_package_result_chosen_inverse. a * hmi_solution_package_result + m * hmi_left_offset_package_result_chosen_inverse = 1 + m * hmi_right_offset_package_result_chosen_inverse))) /\ forall hmi_comparison_package_result. (((exists hmi_gap_package_result_compared_bound. hmi_gap_package_result_compared_bound + S hmi_comparison_package_result = m) /\ (exists hmi_left_offset_package_result_compared_inverse hmi_right_offset_package_result_compared_inverse. a * hmi_comparison_package_result + m * hmi_left_offset_package_result_compared_inverse = 1 + m * hmi_right_offset_package_result_compared_inverse))) -> hmi_comparison_package_result = hmi_solution_package_result)) -> (forall hmi_divisor_package_coprime. (exists hmi_left_factor_package_coprime. a = hmi_divisor_package_coprime * hmi_left_factor_package_coprime) -> (exists hmi_right_factor_package_coprime. m = hmi_divisor_package_coprime * hmi_right_factor_package_coprime) -> hmi_divisor_package_coprime = 1))
use coprime_bounded_mod_inverse
use bounded_mod_inverse_unique
use mod_inverse_implies_coprime
intro a
intro m
intro hm
split
intro hcop
have hexists : exists u. (((exists hmi_gap_package_exists_bound. hmi_gap_package_exists_bound + S u = m) /\ (exists hmi_left_offset_package_exists_inverse hmi_right_offset_package_exists_inverse. a * u + m * hmi_left_offset_package_exists_inverse = 1 + m * hmi_right_offset_package_exists_inverse)))
specialize coprime_bounded_mod_inverse a
specialize coprime_bounded_mod_inverse m
apply coprime_bounded_mod_inverse
exact hm
exact hcop
cases hexists
exists x
split
exact hexists_witness
intro v
intro hv
cases hv
cases hexists_witness
specialize bounded_mod_inverse_unique m
specialize bounded_mod_inverse_unique a
specialize bounded_mod_inverse_unique v
specialize bounded_mod_inverse_unique x
apply bounded_mod_inverse_unique
exact hv_left
exact hexists_witness_left
exact hv_right
exact hexists_witness_right
intro hunique
cases hunique
cases hunique_witness
cases hunique_witness_left
specialize mod_inverse_implies_coprime a
specialize mod_inverse_implies_coprime m
specialize mod_inverse_implies_coprime x
apply mod_inverse_implies_coprime
exact hunique_witness_left_right
qed

Occurrences

9,512

Distinct objects

2,538

Reused references

142

Depth

70

Cuts

278

Certificate SHA-256

4cce94405a13…

is_lcm_multiple_leftA relational LCM is a multiple of its left input GCD, Bézout & coprimality checked

A relational lcm is a multiple of its left input.

Prerequisites

none — a root theorem

Used directly by

mod_eq_lcm_iff_pair

Exact expanded PA statement

Copy statement

forall l a b. ((((exists hlcm_left_factor_projection_left. l = a * hlcm_left_factor_projection_left) /\ (exists hlcm_right_factor_projection_left. l = b * hlcm_right_factor_projection_left)) /\ forall hlcm_common_projection_left. (exists hlcm_left_common_projection_left. hlcm_common_projection_left = a * hlcm_left_common_projection_left) -> (exists hlcm_right_common_projection_left. hlcm_common_projection_left = b * hlcm_right_common_projection_left) -> exists hlcm_least_factor_projection_left. hlcm_common_projection_left = l * hlcm_least_factor_projection_left)) -> exists x. l = a * x

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l a b. ((((exists hlcm_left_factor_projection_left. l = a * hlcm_left_factor_projection_left) /\ (exists hlcm_right_factor_projection_left. l = b * hlcm_right_factor_projection_left)) /\ forall hlcm_common_projection_left. (exists hlcm_left_common_projection_left. hlcm_common_projection_left = a * hlcm_left_common_projection_left) -> (exists hlcm_right_common_projection_left. hlcm_common_projection_left = b * hlcm_right_common_projection_left) -> exists hlcm_least_factor_projection_left. hlcm_common_projection_left = l * hlcm_least_factor_projection_left)) -> exists x. l = a * x
intro l
intro a
intro b
intro h
cases h
cases h_left
exact h_left_left
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

13

Cuts

0

Certificate SHA-256

848a8df1aab9…

is_lcm_multiple_rightA relational LCM is a multiple of its right input GCD, Bézout & coprimality checked

A relational lcm is a multiple of its right input.

Prerequisites

none — a root theorem

Used directly by

mod_eq_lcm_iff_pair

Exact expanded PA statement

Copy statement

forall l a b. ((((exists hlcm_left_factor_projection_right. l = a * hlcm_left_factor_projection_right) /\ (exists hlcm_right_factor_projection_right. l = b * hlcm_right_factor_projection_right)) /\ forall hlcm_common_projection_right. (exists hlcm_left_common_projection_right. hlcm_common_projection_right = a * hlcm_left_common_projection_right) -> (exists hlcm_right_common_projection_right. hlcm_common_projection_right = b * hlcm_right_common_projection_right) -> exists hlcm_least_factor_projection_right. hlcm_common_projection_right = l * hlcm_least_factor_projection_right)) -> exists x. l = b * x

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l a b. ((((exists hlcm_left_factor_projection_right. l = a * hlcm_left_factor_projection_right) /\ (exists hlcm_right_factor_projection_right. l = b * hlcm_right_factor_projection_right)) /\ forall hlcm_common_projection_right. (exists hlcm_left_common_projection_right. hlcm_common_projection_right = a * hlcm_left_common_projection_right) -> (exists hlcm_right_common_projection_right. hlcm_common_projection_right = b * hlcm_right_common_projection_right) -> exists hlcm_least_factor_projection_right. hlcm_common_projection_right = l * hlcm_least_factor_projection_right)) -> exists x. l = b * x
intro l
intro a
intro b
intro h
cases h
cases h_left
exact h_left_right
qed

Occurrences

21

Distinct objects

21

Reused references

0

Depth

13

Cuts

0

Certificate SHA-256

bdbcd5dadc77…

is_lcm_leastA relational LCM divides every common multiple GCD, Bézout & coprimality checked

A relational lcm divides every common multiple.

Prerequisites

none — a root theorem

Used directly by

mod_eq_lcm_merge

Exact expanded PA statement

Copy statement

forall l a b c. ((((exists hlcm_left_factor_least. l = a * hlcm_left_factor_least) /\ (exists hlcm_right_factor_least. l = b * hlcm_right_factor_least)) /\ forall hlcm_common_least. (exists hlcm_left_common_least. hlcm_common_least = a * hlcm_left_common_least) -> (exists hlcm_right_common_least. hlcm_common_least = b * hlcm_right_common_least) -> exists hlcm_least_factor_least. hlcm_common_least = l * hlcm_least_factor_least)) -> (exists x. c = a * x) -> (exists y. c = b * y) -> exists z. c = l * z

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l a b c. ((((exists hlcm_left_factor_least. l = a * hlcm_left_factor_least) /\ (exists hlcm_right_factor_least. l = b * hlcm_right_factor_least)) /\ forall hlcm_common_least. (exists hlcm_left_common_least. hlcm_common_least = a * hlcm_left_common_least) -> (exists hlcm_right_common_least. hlcm_common_least = b * hlcm_right_common_least) -> exists hlcm_least_factor_least. hlcm_common_least = l * hlcm_least_factor_least)) -> (exists x. c = a * x) -> (exists y. c = b * y) -> exists z. c = l * z
intro l
intro a
intro b
intro c
intro h
intro ha
intro hb
cases h
specialize h_right c
apply h_right
exact ha
exact hb
qed

Occurrences

24

Distinct objects

24

Reused references

0

Depth

16

Cuts

0

Certificate SHA-256

90f411bbe254…

is_lcm_symmSymmetry of relational LCM GCD, Bézout & coprimality checked

The expanded relational lcm specification is symmetric.

Prerequisites

none — a root theorem

Used directly by

is_lcm_zero_left

Exact expanded PA statement

Copy statement

forall l a b. ((((exists hlcm_left_factor_symmetry_source. l = a * hlcm_left_factor_symmetry_source) /\ (exists hlcm_right_factor_symmetry_source. l = b * hlcm_right_factor_symmetry_source)) /\ forall hlcm_common_symmetry_source. (exists hlcm_left_common_symmetry_source. hlcm_common_symmetry_source = a * hlcm_left_common_symmetry_source) -> (exists hlcm_right_common_symmetry_source. hlcm_common_symmetry_source = b * hlcm_right_common_symmetry_source) -> exists hlcm_least_factor_symmetry_source. hlcm_common_symmetry_source = l * hlcm_least_factor_symmetry_source)) -> ((((exists hlcm_left_factor_symmetry_target. l = b * hlcm_left_factor_symmetry_target) /\ (exists hlcm_right_factor_symmetry_target. l = a * hlcm_right_factor_symmetry_target)) /\ forall hlcm_common_symmetry_target. (exists hlcm_left_common_symmetry_target. hlcm_common_symmetry_target = b * hlcm_left_common_symmetry_target) -> (exists hlcm_right_common_symmetry_target. hlcm_common_symmetry_target = a * hlcm_right_common_symmetry_target) -> exists hlcm_least_factor_symmetry_target. hlcm_common_symmetry_target = l * hlcm_least_factor_symmetry_target))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l a b. ((((exists hlcm_left_factor_symmetry_source. l = a * hlcm_left_factor_symmetry_source) /\ (exists hlcm_right_factor_symmetry_source. l = b * hlcm_right_factor_symmetry_source)) /\ forall hlcm_common_symmetry_source. (exists hlcm_left_common_symmetry_source. hlcm_common_symmetry_source = a * hlcm_left_common_symmetry_source) -> (exists hlcm_right_common_symmetry_source. hlcm_common_symmetry_source = b * hlcm_right_common_symmetry_source) -> exists hlcm_least_factor_symmetry_source. hlcm_common_symmetry_source = l * hlcm_least_factor_symmetry_source)) -> ((((exists hlcm_left_factor_symmetry_target. l = b * hlcm_left_factor_symmetry_target) /\ (exists hlcm_right_factor_symmetry_target. l = a * hlcm_right_factor_symmetry_target)) /\ forall hlcm_common_symmetry_target. (exists hlcm_left_common_symmetry_target. hlcm_common_symmetry_target = b * hlcm_left_common_symmetry_target) -> (exists hlcm_right_common_symmetry_target. hlcm_common_symmetry_target = a * hlcm_right_common_symmetry_target) -> exists hlcm_least_factor_symmetry_target. hlcm_common_symmetry_target = l * hlcm_least_factor_symmetry_target))
intro l
intro a
intro b
intro h
cases h
cases h_left
split
split
exact h_left_right
exact h_left_left
intro c
intro hbc
intro hac
specialize h_right c
apply h_right
exact hac
exact hbc
qed

Occurrences

36

Distinct objects

36

Reused references

0

Depth

21

Cuts

0

Certificate SHA-256

bf63a0e526c3…

is_lcm_uniqueUniqueness of relational LCM GCD, Bézout & coprimality checked

The expanded relational lcm specification is single-valued.

Prerequisites

multiple_antisymm

Used directly by

canonical_lcm_exists_unique gcd_lcm_product

Exact expanded PA statement

Copy statement

forall l m a b. ((((exists hlcm_left_factor_unique_left. l = a * hlcm_left_factor_unique_left) /\ (exists hlcm_right_factor_unique_left. l = b * hlcm_right_factor_unique_left)) /\ forall hlcm_common_unique_left. (exists hlcm_left_common_unique_left. hlcm_common_unique_left = a * hlcm_left_common_unique_left) -> (exists hlcm_right_common_unique_left. hlcm_common_unique_left = b * hlcm_right_common_unique_left) -> exists hlcm_least_factor_unique_left. hlcm_common_unique_left = l * hlcm_least_factor_unique_left)) -> ((((exists hlcm_left_factor_unique_right. m = a * hlcm_left_factor_unique_right) /\ (exists hlcm_right_factor_unique_right. m = b * hlcm_right_factor_unique_right)) /\ forall hlcm_common_unique_right. (exists hlcm_left_common_unique_right. hlcm_common_unique_right = a * hlcm_left_common_unique_right) -> (exists hlcm_right_common_unique_right. hlcm_common_unique_right = b * hlcm_right_common_unique_right) -> exists hlcm_least_factor_unique_right. hlcm_common_unique_right = m * hlcm_least_factor_unique_right)) -> l = m

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m a b. ((((exists hlcm_left_factor_unique_left. l = a * hlcm_left_factor_unique_left) /\ (exists hlcm_right_factor_unique_left. l = b * hlcm_right_factor_unique_left)) /\ forall hlcm_common_unique_left. (exists hlcm_left_common_unique_left. hlcm_common_unique_left = a * hlcm_left_common_unique_left) -> (exists hlcm_right_common_unique_left. hlcm_common_unique_left = b * hlcm_right_common_unique_left) -> exists hlcm_least_factor_unique_left. hlcm_common_unique_left = l * hlcm_least_factor_unique_left)) -> ((((exists hlcm_left_factor_unique_right. m = a * hlcm_left_factor_unique_right) /\ (exists hlcm_right_factor_unique_right. m = b * hlcm_right_factor_unique_right)) /\ forall hlcm_common_unique_right. (exists hlcm_left_common_unique_right. hlcm_common_unique_right = a * hlcm_left_common_unique_right) -> (exists hlcm_right_common_unique_right. hlcm_common_unique_right = b * hlcm_right_common_unique_right) -> exists hlcm_least_factor_unique_right. hlcm_common_unique_right = m * hlcm_least_factor_unique_right)) -> l = m
use multiple_antisymm
intro l
intro m
intro a
intro b
intro hl
intro hm
cases hl
cases hl_left
cases hm
cases hm_left
have hlm : exists q. m = l * q
specialize hl_right m
apply hl_right
exact hm_left_left
exact hm_left_right
have hml : exists q. l = m * q
specialize hm_right l
apply hm_right
exact hl_left_left
exact hl_left_right
specialize multiple_antisymm l
specialize multiple_antisymm m
apply multiple_antisymm
exact hlm
exact hml
qed

Occurrences

680

Distinct objects

561

Reused references

35

Depth

34

Cuts

20

Certificate SHA-256

d39024dce998…

is_lcm_zero_rightLCM with zero on the right GCD, Bézout & coprimality checked

Zero is an lcm of every natural and zero.

Prerequisites

multiple_zero

Used directly by

is_lcm_zero_left

Exact expanded PA statement

Copy statement

forall a. ((((exists hlcm_left_factor_zero_right. 0 = a * hlcm_left_factor_zero_right) /\ (exists hlcm_right_factor_zero_right. 0 = 0 * hlcm_right_factor_zero_right)) /\ forall hlcm_common_zero_right. (exists hlcm_left_common_zero_right. hlcm_common_zero_right = a * hlcm_left_common_zero_right) -> (exists hlcm_right_common_zero_right. hlcm_common_zero_right = 0 * hlcm_right_common_zero_right) -> exists hlcm_least_factor_zero_right. hlcm_common_zero_right = 0 * hlcm_least_factor_zero_right))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a. ((((exists hlcm_left_factor_zero_right. 0 = a * hlcm_left_factor_zero_right) /\ (exists hlcm_right_factor_zero_right. 0 = 0 * hlcm_right_factor_zero_right)) /\ forall hlcm_common_zero_right. (exists hlcm_left_common_zero_right. hlcm_common_zero_right = a * hlcm_left_common_zero_right) -> (exists hlcm_right_common_zero_right. hlcm_common_zero_right = 0 * hlcm_right_common_zero_right) -> exists hlcm_least_factor_zero_right. hlcm_common_zero_right = 0 * hlcm_least_factor_zero_right))
use multiple_zero
intro a
split
split
specialize multiple_zero a
exact multiple_zero
specialize multiple_zero 0
exact multiple_zero
intro c
intro hca
intro hc0
exact hc0
qed

Occurrences

25

Distinct objects

25

Reused references

0

Depth

7

Cuts

1

Certificate SHA-256

166243d22548…

is_lcm_zero_leftLCM with zero on the left GCD, Bézout & coprimality checked

Zero is an lcm of zero and every natural.

Prerequisites

is_lcm_zero_right is_lcm_symm

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall b. ((((exists hlcm_left_factor_zero_left. 0 = 0 * hlcm_left_factor_zero_left) /\ (exists hlcm_right_factor_zero_left. 0 = b * hlcm_right_factor_zero_left)) /\ forall hlcm_common_zero_left. (exists hlcm_left_common_zero_left. hlcm_common_zero_left = 0 * hlcm_left_common_zero_left) -> (exists hlcm_right_common_zero_left. hlcm_common_zero_left = b * hlcm_right_common_zero_left) -> exists hlcm_least_factor_zero_left. hlcm_common_zero_left = 0 * hlcm_least_factor_zero_left))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall b. ((((exists hlcm_left_factor_zero_left. 0 = 0 * hlcm_left_factor_zero_left) /\ (exists hlcm_right_factor_zero_left. 0 = b * hlcm_right_factor_zero_left)) /\ forall hlcm_common_zero_left. (exists hlcm_left_common_zero_left. hlcm_common_zero_left = 0 * hlcm_left_common_zero_left) -> (exists hlcm_right_common_zero_left. hlcm_common_zero_left = b * hlcm_right_common_zero_left) -> exists hlcm_least_factor_zero_left. hlcm_common_zero_left = 0 * hlcm_least_factor_zero_left))
use is_lcm_zero_right
use is_lcm_symm
intro b
have h : ((((exists hlcm_left_factor_zero_left_middle. 0 = b * hlcm_left_factor_zero_left_middle) /\ (exists hlcm_right_factor_zero_left_middle. 0 = 0 * hlcm_right_factor_zero_left_middle)) /\ forall hlcm_common_zero_left_middle. (exists hlcm_left_common_zero_left_middle. hlcm_common_zero_left_middle = b * hlcm_left_common_zero_left_middle) -> (exists hlcm_right_common_zero_left_middle. hlcm_common_zero_left_middle = 0 * hlcm_right_common_zero_left_middle) -> exists hlcm_least_factor_zero_left_middle. hlcm_common_zero_left_middle = 0 * hlcm_least_factor_zero_left_middle))
specialize is_lcm_zero_right b
exact is_lcm_zero_right
specialize is_lcm_symm 0
specialize is_lcm_symm b
specialize is_lcm_symm 0
apply is_lcm_symm
exact h
qed

Occurrences

71

Distinct objects

71

Reused references

0

Depth

23

Cuts

3

Certificate SHA-256

a4202018cabb…

balanced_bezout_one_implies_coprimeBalanced Bezout result one implies coprimality GCD, Bézout & coprimality checked

A balanced natural Bezout equation with result one forces the two inputs to be coprime.

Prerequisites

common_divisor_divides_balanced_result divisor_one

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall a b xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn) -> forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> d = 1

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn) -> forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> d = 1
use common_divisor_divides_balanced_result
use divisor_one
intro a
intro b
intro xp
intro yp
intro xn
intro yn
intro hbez
intro d
intro ha
intro hb
specialize divisor_one d
apply divisor_one
specialize common_divisor_divides_balanced_result d
specialize common_divisor_divides_balanced_result a
specialize common_divisor_divides_balanced_result b
specialize common_divisor_divides_balanced_result 1
specialize common_divisor_divides_balanced_result xp
specialize common_divisor_divides_balanced_result yp
specialize common_divisor_divides_balanced_result xn
specialize common_divisor_divides_balanced_result yn
apply common_divisor_divides_balanced_result
exact ha
exact hb
exact hbez
qed

Occurrences

871

Distinct objects

616

Reused references

41

Depth

40

Cuts

22

Certificate SHA-256

50e944a45f51…

coprime_product_is_lcmThe product of coprime naturals is an LCM GCD, Bézout & coprimality checked

The product of coprime naturals satisfies the universal relational LCM specification.

Prerequisites

mul_comm gauss_coprime_cancel mul_assoc

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall a b. (forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> d = 1) -> ((((exists hlcm_left_factor_coprime_product. a * b = a * hlcm_left_factor_coprime_product) /\ (exists hlcm_right_factor_coprime_product. a * b = b * hlcm_right_factor_coprime_product)) /\ forall hlcm_common_coprime_product. (exists hlcm_left_common_coprime_product. hlcm_common_coprime_product = a * hlcm_left_common_coprime_product) -> (exists hlcm_right_common_coprime_product. hlcm_common_coprime_product = b * hlcm_right_common_coprime_product) -> exists hlcm_least_factor_coprime_product. hlcm_common_coprime_product = a * b * hlcm_least_factor_coprime_product))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> d = 1) -> ((((exists hlcm_left_factor_coprime_product. a * b = a * hlcm_left_factor_coprime_product) /\ (exists hlcm_right_factor_coprime_product. a * b = b * hlcm_right_factor_coprime_product)) /\ forall hlcm_common_coprime_product. (exists hlcm_left_common_coprime_product. hlcm_common_coprime_product = a * hlcm_left_common_coprime_product) -> (exists hlcm_right_common_coprime_product. hlcm_common_coprime_product = b * hlcm_right_common_coprime_product) -> exists hlcm_least_factor_coprime_product. hlcm_common_coprime_product = a * b * hlcm_least_factor_coprime_product))
use mul_comm
use gauss_coprime_cancel
use mul_assoc
intro a
intro b
intro hcop
split
split
exists b
refl
exists a
apply mul_comm
intro c
intro ha
intro hb
cases ha
cases hb
have hdiv : exists q. b * x1 = a * q
exists x
trans c
symm
exact hb_witness
exact ha_witness
have hfactor : exists w. x1 = a * w
specialize gauss_coprime_cancel a
specialize gauss_coprime_cancel b
specialize gauss_coprime_cancel x1
apply gauss_coprime_cancel
exact hcop
exact hdiv
cases hfactor
exists x2
trans b * x1
exact hb_witness
trans b * (a * x2)
rewrite hfactor_witness
refl
trans (b * a) * x2
symm
apply mul_assoc
congr
apply mul_comm
refl
qed

Occurrences

4,191

Distinct objects

1,552

Reused references

95

Depth

53

Cuts

121

Certificate SHA-256

1bb51a1a4ac2…

is_lcm_scale_nonzeroNonzero scaling preserves relational LCM GCD, Bézout & coprimality checked

A nonzero common left scale preserves the universal relational LCM specification.

Prerequisites

mul_assoc mul_left_cancel_nonzero

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall k l a b. ~(k = 0) -> ((((exists hscale_left_factor_source. l = a * hscale_left_factor_source) /\ (exists hscale_right_factor_source. l = b * hscale_right_factor_source)) /\ forall hscale_common_source. (exists hscale_left_common_source. hscale_common_source = a * hscale_left_common_source) -> (exists hscale_right_common_source. hscale_common_source = b * hscale_right_common_source) -> exists hscale_least_factor_source. hscale_common_source = l * hscale_least_factor_source)) -> ((((exists hscale_left_factor_target. (k * l) = (k * a) * hscale_left_factor_target) /\ (exists hscale_right_factor_target. (k * l) = (k * b) * hscale_right_factor_target)) /\ forall hscale_common_target. (exists hscale_left_common_target. hscale_common_target = (k * a) * hscale_left_common_target) -> (exists hscale_right_common_target. hscale_common_target = (k * b) * hscale_right_common_target) -> exists hscale_least_factor_target. hscale_common_target = (k * l) * hscale_least_factor_target))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k l a b. ~(k = 0) -> ((((exists hscale_left_factor_source. l = a * hscale_left_factor_source) /\ (exists hscale_right_factor_source. l = b * hscale_right_factor_source)) /\ forall hscale_common_source. (exists hscale_left_common_source. hscale_common_source = a * hscale_left_common_source) -> (exists hscale_right_common_source. hscale_common_source = b * hscale_right_common_source) -> exists hscale_least_factor_source. hscale_common_source = l * hscale_least_factor_source)) -> ((((exists hscale_left_factor_target. (k * l) = (k * a) * hscale_left_factor_target) /\ (exists hscale_right_factor_target. (k * l) = (k * b) * hscale_right_factor_target)) /\ forall hscale_common_target. (exists hscale_left_common_target. hscale_common_target = (k * a) * hscale_left_common_target) -> (exists hscale_right_common_target. hscale_common_target = (k * b) * hscale_right_common_target) -> exists hscale_least_factor_target. hscale_common_target = (k * l) * hscale_least_factor_target))
use mul_assoc
use mul_left_cancel_nonzero
intro k
intro l
intro a
intro b
intro hk
intro h
cases h
cases h_left
cases h_left_left
cases h_left_right
split
split
exists x
rewrite h_left_left_witness
symm
apply mul_assoc
exists x1
rewrite h_left_right_witness
symm
apply mul_assoc
intro c
intro hca
intro hcb
cases hca
cases hcb
have hca_norm : c = k * (a * x2)
trans (k * a) * x2
exact hca_witness
apply mul_assoc
have hcb_norm : c = k * (b * x3)
trans (k * b) * x3
exact hcb_witness
apply mul_assoc
have hab : a * x2 = b * x3
specialize mul_left_cancel_nonzero k
specialize mul_left_cancel_nonzero (a * x2)
specialize mul_left_cancel_nonzero (b * x3)
apply mul_left_cancel_nonzero
exact hk
trans c
symm
exact hca_norm
exact hcb_norm
have hleast : exists w. a * x2 = l * w
specialize h_right (a * x2)
apply h_right
exists x2
refl
exists x3
exact hab
cases hleast
exists x4
trans k * (a * x2)
exact hca_norm
trans k * (l * x4)
congr
refl
exact hleast_witness
symm
apply mul_assoc
qed

Occurrences

430

Distinct objects

371

Reused references

13

Depth

27

Cuts

11

Certificate SHA-256

e53557b877b4…

balanced_bezout_cancel_gcdCancel a nonzero gcd from balanced Bezout GCD, Bézout & coprimality checked

Cancel a nonzero common gcd factor from a balanced Bezout equation.

Prerequisites

mul_left_cancel_nonzero mul_add mul_assoc mul_one

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall g a b A B xp yp xn yn. ~(g = 0) -> a = g * A -> b = g * B -> a * xp + b * yp = g + (a * xn + b * yn) -> A * xp + B * yp = 1 + (A * xn + B * yn)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b A B xp yp xn yn. ~(g = 0) -> a = g * A -> b = g * B -> a * xp + b * yp = g + (a * xn + b * yn) -> A * xp + B * yp = 1 + (A * xn + B * yn)
use mul_left_cancel_nonzero
use mul_add
use mul_assoc
use mul_one
intro g
intro a
intro b
intro A
intro B
intro xp
intro yp
intro xn
intro yn
intro hg
intro ha
intro hb
intro hbez
specialize mul_left_cancel_nonzero g
specialize mul_left_cancel_nonzero (A * xp + B * yp)
specialize mul_left_cancel_nonzero (1 + (A * xn + B * yn))
apply mul_left_cancel_nonzero
exact hg
trans g * (A * xp) + g * (B * yp)
apply mul_add
trans (g * A) * xp + (g * B) * yp
congr
symm
apply mul_assoc
symm
apply mul_assoc
trans a * xp + b * yp
rewrite ha
rewrite hb
refl
trans g + (a * xn + b * yn)
exact hbez
trans g + ((g * A) * xn + (g * B) * yn)
rewrite ha
rewrite hb
refl
trans g * 1 + ((g * A) * xn + (g * B) * yn)
congr
symm
apply mul_one
refl
trans g * 1 + (g * (A * xn) + g * (B * yn))
congr
refl
congr
apply mul_assoc
apply mul_assoc
trans g * 1 + g * (A * xn + B * yn)
congr
refl
symm
apply mul_add
symm
apply mul_add
qed

Occurrences

549

Distinct objects

409

Reused references

18

Depth

38

Cuts

15

Certificate SHA-256

2f0613e9f78c…

gcd_zero_inputsA zero gcd forces zero inputs GCD, Bézout & coprimality checked

A zero relational gcd can divide only the zero input pair.

Prerequisites

mul_zero_left

Used directly by

gcd_lcm_compatible_exists

Exact expanded PA statement

Copy statement

forall g a b. g = 0 -> ((((exists hag_left_factor_zero_inputs. a = g * hag_left_factor_zero_inputs) /\ (exists hag_right_factor_zero_inputs. b = g * hag_right_factor_zero_inputs)) /\ forall hag_divisor_zero_inputs. (exists hag_common_left_zero_inputs. a = hag_divisor_zero_inputs * hag_common_left_zero_inputs) -> (exists hag_common_right_zero_inputs. b = hag_divisor_zero_inputs * hag_common_right_zero_inputs) -> exists hag_greatest_factor_zero_inputs. g = hag_divisor_zero_inputs * hag_greatest_factor_zero_inputs)) -> (a = 0 /\ b = 0)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b. g = 0 -> ((((exists hag_left_factor_zero_inputs. a = g * hag_left_factor_zero_inputs) /\ (exists hag_right_factor_zero_inputs. b = g * hag_right_factor_zero_inputs)) /\ forall hag_divisor_zero_inputs. (exists hag_common_left_zero_inputs. a = hag_divisor_zero_inputs * hag_common_left_zero_inputs) -> (exists hag_common_right_zero_inputs. b = hag_divisor_zero_inputs * hag_common_right_zero_inputs) -> exists hag_greatest_factor_zero_inputs. g = hag_divisor_zero_inputs * hag_greatest_factor_zero_inputs)) -> (a = 0 /\ b = 0)
use mul_zero_left
intro g
intro a
intro b
intro hg
intro h
cases h
cases h_left
cases h_left_left
cases h_left_right
split
trans g * x
exact h_left_left_witness
rewrite hg
apply mul_zero_left
trans g * x1
exact h_left_right_witness
rewrite hg
apply mul_zero_left
qed

Occurrences

62

Distinct objects

62

Reused references

0

Depth

21

Cuts

1

Certificate SHA-256

0cfd18ba4f88…

gcd_lcm_compatible_existsCompatible gcd and LCM witnesses exist GCD, Bézout & coprimality checked

Every pair has compatible relational gcd and lcm witnesses whose product is the product of the inputs, including zero inputs.

Prerequisites

gcd_balanced_bezout_exists eq_decidable gcd_zero_inputs is_lcm_zero_left balanced_bezout_cancel_gcd balanced_bezout_one_implies_coprime coprime_product_is_lcm is_lcm_scale_nonzero mul_assoc mul_comm

Used directly by

lcm_exists_relational gcd_lcm_product

Exact expanded PA statement

Copy statement

forall a b. exists g l. ((((((exists hag_left_factor_compatible. a = g * hag_left_factor_compatible) /\ (exists hag_right_factor_compatible. b = g * hag_right_factor_compatible)) /\ forall hag_divisor_compatible. (exists hag_common_left_compatible. a = hag_divisor_compatible * hag_common_left_compatible) -> (exists hag_common_right_compatible. b = hag_divisor_compatible * hag_common_right_compatible) -> exists hag_greatest_factor_compatible. g = hag_divisor_compatible * hag_greatest_factor_compatible)) /\ ((((exists hlcm_left_factor_compatible. l = a * hlcm_left_factor_compatible) /\ (exists hlcm_right_factor_compatible. l = b * hlcm_right_factor_compatible)) /\ forall hlcm_common_compatible. (exists hlcm_left_common_compatible. hlcm_common_compatible = a * hlcm_left_common_compatible) -> (exists hlcm_right_common_compatible. hlcm_common_compatible = b * hlcm_right_common_compatible) -> exists hlcm_least_factor_compatible. hlcm_common_compatible = l * hlcm_least_factor_compatible))) /\ g * l = a * b)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists g l. ((((((exists hag_left_factor_compatible. a = g * hag_left_factor_compatible) /\ (exists hag_right_factor_compatible. b = g * hag_right_factor_compatible)) /\ forall hag_divisor_compatible. (exists hag_common_left_compatible. a = hag_divisor_compatible * hag_common_left_compatible) -> (exists hag_common_right_compatible. b = hag_divisor_compatible * hag_common_right_compatible) -> exists hag_greatest_factor_compatible. g = hag_divisor_compatible * hag_greatest_factor_compatible)) /\ ((((exists hlcm_left_factor_compatible. l = a * hlcm_left_factor_compatible) /\ (exists hlcm_right_factor_compatible. l = b * hlcm_right_factor_compatible)) /\ forall hlcm_common_compatible. (exists hlcm_left_common_compatible. hlcm_common_compatible = a * hlcm_left_common_compatible) -> (exists hlcm_right_common_compatible. hlcm_common_compatible = b * hlcm_right_common_compatible) -> exists hlcm_least_factor_compatible. hlcm_common_compatible = l * hlcm_least_factor_compatible))) /\ g * l = a * b)
use gcd_balanced_bezout_exists
use eq_decidable
use gcd_zero_inputs
use is_lcm_zero_left
use balanced_bezout_cancel_gcd
use balanced_bezout_one_implies_coprime
use coprime_product_is_lcm
use is_lcm_scale_nonzero
use mul_assoc
use mul_comm
intro a
intro b
have hgb : exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))
specialize gcd_balanced_bezout_exists a
specialize gcd_balanced_bezout_exists b
apply gcd_balanced_bezout_exists
cases hgb
cases hgb_witness
have hzero : x = 0 \/ ~(x = 0)
specialize eq_decidable x
specialize eq_decidable 0
exact eq_decidable
cases hzero
have habzero : a = 0 /\ b = 0
specialize gcd_zero_inputs x
specialize gcd_zero_inputs a
specialize gcd_zero_inputs b
apply gcd_zero_inputs
exact hzero_left
exact hgb_witness_left
cases habzero
have hlcmzero : ((((exists hlcm_left_factor_compatible_zero. 0 = 0 * hlcm_left_factor_compatible_zero) /\ (exists hlcm_right_factor_compatible_zero. 0 = 0 * hlcm_right_factor_compatible_zero)) /\ forall hlcm_common_compatible_zero. (exists hlcm_left_common_compatible_zero. hlcm_common_compatible_zero = 0 * hlcm_left_common_compatible_zero) -> (exists hlcm_right_common_compatible_zero. hlcm_common_compatible_zero = 0 * hlcm_right_common_compatible_zero) -> exists hlcm_least_factor_compatible_zero. hlcm_common_compatible_zero = 0 * hlcm_least_factor_compatible_zero))
specialize is_lcm_zero_left 0
exact is_lcm_zero_left
exists x
exists 0
split
split
exact hgb_witness_left
rewrite habzero_left
rewrite habzero_left
rewrite habzero_right
rewrite habzero_right
exact hlcmzero
rewrite hzero_left
rewrite habzero_left
rewrite habzero_right
refl
cases hgb_witness_left
cases hgb_witness_left_left
cases hgb_witness_left_left_left
cases hgb_witness_left_left_right
cases hgb_witness_right
cases hgb_witness_right_witness
cases hgb_witness_right_witness_witness
cases hgb_witness_right_witness_witness_witness
have hquotbez : x1 * x3 + x2 * x4 = 1 + (x1 * x5 + x2 * x6)
specialize balanced_bezout_cancel_gcd x
specialize balanced_bezout_cancel_gcd a
specialize balanced_bezout_cancel_gcd b
specialize balanced_bezout_cancel_gcd x1
specialize balanced_bezout_cancel_gcd x2
specialize balanced_bezout_cancel_gcd x3
specialize balanced_bezout_cancel_gcd x4
specialize balanced_bezout_cancel_gcd x5
specialize balanced_bezout_cancel_gcd x6
apply balanced_bezout_cancel_gcd
exact hzero_right
exact hgb_witness_left_left_left_witness
exact hgb_witness_left_left_right_witness
exact hgb_witness_right_witness_witness_witness_witness
have hcop : forall d. (exists u. x1 = d * u) -> (exists v. x2 = d * v) -> d = 1
specialize balanced_bezout_one_implies_coprime x1
specialize balanced_bezout_one_implies_coprime x2
specialize balanced_bezout_one_implies_coprime x3
specialize balanced_bezout_one_implies_coprime x4
specialize balanced_bezout_one_implies_coprime x5
specialize balanced_bezout_one_implies_coprime x6
apply balanced_bezout_one_implies_coprime
exact hquotbez
have hbase : ((((exists hlcm_left_factor_compatible_base. x1 * x2 = x1 * hlcm_left_factor_compatible_base) /\ (exists hlcm_right_factor_compatible_base. x1 * x2 = x2 * hlcm_right_factor_compatible_base)) /\ forall hlcm_common_compatible_base. (exists hlcm_left_common_compatible_base. hlcm_common_compatible_base = x1 * hlcm_left_common_compatible_base) -> (exists hlcm_right_common_compatible_base. hlcm_common_compatible_base = x2 * hlcm_right_common_compatible_base) -> exists hlcm_least_factor_compatible_base. hlcm_common_compatible_base = x1 * x2 * hlcm_least_factor_compatible_base))
specialize coprime_product_is_lcm x1
specialize coprime_product_is_lcm x2
apply coprime_product_is_lcm
exact hcop
have hscaled : ((((exists hscale_left_factor_compatible_scaled. (x * (x1 * x2)) = (x * x1) * hscale_left_factor_compatible_scaled) /\ (exists hscale_right_factor_compatible_scaled. (x * (x1 * x2)) = (x * x2) * hscale_right_factor_compatible_scaled)) /\ forall hscale_common_compatible_scaled. (exists hscale_left_common_compatible_scaled. hscale_common_compatible_scaled = (x * x1) * hscale_left_common_compatible_scaled) -> (exists hscale_right_common_compatible_scaled. hscale_common_compatible_scaled = (x * x2) * hscale_right_common_compatible_scaled) -> exists hscale_least_factor_compatible_scaled. hscale_common_compatible_scaled = (x * (x1 * x2)) * hscale_least_factor_compatible_scaled))
specialize is_lcm_scale_nonzero x
specialize is_lcm_scale_nonzero (x1 * x2)
specialize is_lcm_scale_nonzero x1
specialize is_lcm_scale_nonzero x2
apply is_lcm_scale_nonzero
exact hzero_right
exact hbase
exists x
exists x * (x1 * x2)
split
split
exact hgb_witness_left
rewrite hgb_witness_left_left_left_witness
rewrite hgb_witness_left_left_left_witness
rewrite hgb_witness_left_left_right_witness
rewrite hgb_witness_left_left_right_witness
exact hscaled
rewrite hgb_witness_left_left_left_witness
rewrite hgb_witness_left_left_right_witness
trans x * (x1 * (x * x2))
congr
refl
trans (x * x1) * x2
symm
apply mul_assoc
trans (x1 * x) * x2
congr
apply mul_comm
refl
apply mul_assoc
symm
apply mul_assoc
qed

Occurrences

9,038

Distinct objects

2,390

Reused references

121

Depth

60

Cuts

257

Certificate SHA-256

7f93d3556a9f…

lcm_exists_relationalRelational LCM exists GCD, Bézout & coprimality checked

Every pair of naturals has a relational LCM; this is the direct LCM projection of the compatible pair theorem.

Prerequisites

gcd_lcm_compatible_exists

Used directly by

canonical_lcm_exists_unique

Exact expanded PA statement

Copy statement

forall a b. exists l. ((((exists hlcm_left_factor_existence. l = a * hlcm_left_factor_existence) /\ (exists hlcm_right_factor_existence. l = b * hlcm_right_factor_existence)) /\ forall hlcm_common_existence. (exists hlcm_left_common_existence. hlcm_common_existence = a * hlcm_left_common_existence) -> (exists hlcm_right_common_existence. hlcm_common_existence = b * hlcm_right_common_existence) -> exists hlcm_least_factor_existence. hlcm_common_existence = l * hlcm_least_factor_existence))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists l. ((((exists hlcm_left_factor_existence. l = a * hlcm_left_factor_existence) /\ (exists hlcm_right_factor_existence. l = b * hlcm_right_factor_existence)) /\ forall hlcm_common_existence. (exists hlcm_left_common_existence. hlcm_common_existence = a * hlcm_left_common_existence) -> (exists hlcm_right_common_existence. hlcm_common_existence = b * hlcm_right_common_existence) -> exists hlcm_least_factor_existence. hlcm_common_existence = l * hlcm_least_factor_existence))
use gcd_lcm_compatible_exists
intro a
intro b
specialize gcd_lcm_compatible_exists a
specialize gcd_lcm_compatible_exists b
cases gcd_lcm_compatible_exists
cases gcd_lcm_compatible_exists_witness
cases gcd_lcm_compatible_exists_witness_witness
cases gcd_lcm_compatible_exists_witness_witness_left
exists x1
exact gcd_lcm_compatible_exists_witness_witness_left_right
qed

Occurrences

9,071

Distinct objects

2,423

Reused references

121

Depth

61

Cuts

258

Certificate SHA-256

804dd563fab9…

canonical_lcm_exists_uniqueCanonical LCM exists uniquely GCD, Bézout & coprimality checked

The relational LCM value exists uniquely; divisibility witnesses themselves are not claimed to be unique.

Prerequisites

lcm_exists_relational is_lcm_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall a b. exists l. (((((exists hlcm_left_factor_unique_chosen. l = a * hlcm_left_factor_unique_chosen) /\ (exists hlcm_right_factor_unique_chosen. l = b * hlcm_right_factor_unique_chosen)) /\ forall hlcm_common_unique_chosen. (exists hlcm_left_common_unique_chosen. hlcm_common_unique_chosen = a * hlcm_left_common_unique_chosen) -> (exists hlcm_right_common_unique_chosen. hlcm_common_unique_chosen = b * hlcm_right_common_unique_chosen) -> exists hlcm_least_factor_unique_chosen. hlcm_common_unique_chosen = l * hlcm_least_factor_unique_chosen)) /\ forall m. ((((exists hlcm_left_factor_unique_compared. m = a * hlcm_left_factor_unique_compared) /\ (exists hlcm_right_factor_unique_compared. m = b * hlcm_right_factor_unique_compared)) /\ forall hlcm_common_unique_compared. (exists hlcm_left_common_unique_compared. hlcm_common_unique_compared = a * hlcm_left_common_unique_compared) -> (exists hlcm_right_common_unique_compared. hlcm_common_unique_compared = b * hlcm_right_common_unique_compared) -> exists hlcm_least_factor_unique_compared. hlcm_common_unique_compared = m * hlcm_least_factor_unique_compared)) -> m = l)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. exists l. (((((exists hlcm_left_factor_unique_chosen. l = a * hlcm_left_factor_unique_chosen) /\ (exists hlcm_right_factor_unique_chosen. l = b * hlcm_right_factor_unique_chosen)) /\ forall hlcm_common_unique_chosen. (exists hlcm_left_common_unique_chosen. hlcm_common_unique_chosen = a * hlcm_left_common_unique_chosen) -> (exists hlcm_right_common_unique_chosen. hlcm_common_unique_chosen = b * hlcm_right_common_unique_chosen) -> exists hlcm_least_factor_unique_chosen. hlcm_common_unique_chosen = l * hlcm_least_factor_unique_chosen)) /\ forall m. ((((exists hlcm_left_factor_unique_compared. m = a * hlcm_left_factor_unique_compared) /\ (exists hlcm_right_factor_unique_compared. m = b * hlcm_right_factor_unique_compared)) /\ forall hlcm_common_unique_compared. (exists hlcm_left_common_unique_compared. hlcm_common_unique_compared = a * hlcm_left_common_unique_compared) -> (exists hlcm_right_common_unique_compared. hlcm_common_unique_compared = b * hlcm_right_common_unique_compared) -> exists hlcm_least_factor_unique_compared. hlcm_common_unique_compared = m * hlcm_least_factor_unique_compared)) -> m = l)
use lcm_exists_relational
use is_lcm_unique
intro a
intro b
specialize lcm_exists_relational a
specialize lcm_exists_relational b
cases lcm_exists_relational
exists x
split
exact lcm_exists_relational_witness
intro m
intro hm
specialize is_lcm_unique m
specialize is_lcm_unique x
specialize is_lcm_unique a
specialize is_lcm_unique b
apply is_lcm_unique
exact hm
exact lcm_exists_relational_witness
qed

Occurrences

9,791

Distinct objects

2,565

Reused references

127

Depth

62

Cuts

280

Certificate SHA-256

e40bc5728296…

gcd_lcm_productThe gcd-LCM product identity GCD, Bézout & coprimality checked

Any relational gcd and LCM pair satisfies the gcd--LCM product identity, by uniqueness from a compatible pair.

Prerequisites

gcd_lcm_compatible_exists is_gcd_unique is_lcm_unique

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall g l a b. ((((exists hag_left_factor_product_gcd_assumption. a = g * hag_left_factor_product_gcd_assumption) /\ (exists hag_right_factor_product_gcd_assumption. b = g * hag_right_factor_product_gcd_assumption)) /\ forall hag_divisor_product_gcd_assumption. (exists hag_common_left_product_gcd_assumption. a = hag_divisor_product_gcd_assumption * hag_common_left_product_gcd_assumption) -> (exists hag_common_right_product_gcd_assumption. b = hag_divisor_product_gcd_assumption * hag_common_right_product_gcd_assumption) -> exists hag_greatest_factor_product_gcd_assumption. g = hag_divisor_product_gcd_assumption * hag_greatest_factor_product_gcd_assumption)) -> ((((exists hlcm_left_factor_product_lcm_assumption. l = a * hlcm_left_factor_product_lcm_assumption) /\ (exists hlcm_right_factor_product_lcm_assumption. l = b * hlcm_right_factor_product_lcm_assumption)) /\ forall hlcm_common_product_lcm_assumption. (exists hlcm_left_common_product_lcm_assumption. hlcm_common_product_lcm_assumption = a * hlcm_left_common_product_lcm_assumption) -> (exists hlcm_right_common_product_lcm_assumption. hlcm_common_product_lcm_assumption = b * hlcm_right_common_product_lcm_assumption) -> exists hlcm_least_factor_product_lcm_assumption. hlcm_common_product_lcm_assumption = l * hlcm_least_factor_product_lcm_assumption)) -> g * l = a * b

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g l a b. ((((exists hag_left_factor_product_gcd_assumption. a = g * hag_left_factor_product_gcd_assumption) /\ (exists hag_right_factor_product_gcd_assumption. b = g * hag_right_factor_product_gcd_assumption)) /\ forall hag_divisor_product_gcd_assumption. (exists hag_common_left_product_gcd_assumption. a = hag_divisor_product_gcd_assumption * hag_common_left_product_gcd_assumption) -> (exists hag_common_right_product_gcd_assumption. b = hag_divisor_product_gcd_assumption * hag_common_right_product_gcd_assumption) -> exists hag_greatest_factor_product_gcd_assumption. g = hag_divisor_product_gcd_assumption * hag_greatest_factor_product_gcd_assumption)) -> ((((exists hlcm_left_factor_product_lcm_assumption. l = a * hlcm_left_factor_product_lcm_assumption) /\ (exists hlcm_right_factor_product_lcm_assumption. l = b * hlcm_right_factor_product_lcm_assumption)) /\ forall hlcm_common_product_lcm_assumption. (exists hlcm_left_common_product_lcm_assumption. hlcm_common_product_lcm_assumption = a * hlcm_left_common_product_lcm_assumption) -> (exists hlcm_right_common_product_lcm_assumption. hlcm_common_product_lcm_assumption = b * hlcm_right_common_product_lcm_assumption) -> exists hlcm_least_factor_product_lcm_assumption. hlcm_common_product_lcm_assumption = l * hlcm_least_factor_product_lcm_assumption)) -> g * l = a * b
use gcd_lcm_compatible_exists
use is_gcd_unique
use is_lcm_unique
intro g
intro l
intro a
intro b
intro hg
intro hl
specialize gcd_lcm_compatible_exists a
specialize gcd_lcm_compatible_exists b
cases gcd_lcm_compatible_exists
cases gcd_lcm_compatible_exists_witness
cases gcd_lcm_compatible_exists_witness_witness
cases gcd_lcm_compatible_exists_witness_witness_left
have hgeq : g = x
specialize is_gcd_unique g
specialize is_gcd_unique x
specialize is_gcd_unique a
specialize is_gcd_unique b
apply is_gcd_unique
exact hg
exact gcd_lcm_compatible_exists_witness_witness_left_left
have hleq : l = x1
specialize is_lcm_unique l
specialize is_lcm_unique x1
specialize is_lcm_unique a
specialize is_lcm_unique b
apply is_lcm_unique
exact hl
exact gcd_lcm_compatible_exists_witness_witness_left_right
rewrite hgeq
rewrite hleq
exact gcd_lcm_compatible_exists_witness_witness_right
qed

Occurrences

10,441

Distinct objects

2,569

Reused references

128

Depth

61

Cuts

300

Certificate SHA-256

8f7608a596be…

mod_eq_zero_iff_eqMod eq zero iff eq Congruence & CRT checked

Balanced congruence modulo zero is exactly equality.

Prerequisites

mul_zero_left

Used directly by

crt_solution_unique_lcm_zero mod_eq_decidable

Exact expanded PA statement

Copy statement

forall a b. (((exists hgcrt_mod_left_zero_source hgcrt_mod_right_zero_source. a + 0 * hgcrt_mod_left_zero_source = b + 0 * hgcrt_mod_right_zero_source) -> a = b) /\ (a = b -> (exists hgcrt_mod_left_zero_result hgcrt_mod_right_zero_result. a + 0 * hgcrt_mod_left_zero_result = b + 0 * hgcrt_mod_right_zero_result)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall a b. (((exists hgcrt_mod_left_zero_source hgcrt_mod_right_zero_source. a + 0 * hgcrt_mod_left_zero_source = b + 0 * hgcrt_mod_right_zero_source) -> a = b) /\ (a = b -> (exists hgcrt_mod_left_zero_result hgcrt_mod_right_zero_result. a + 0 * hgcrt_mod_left_zero_result = b + 0 * hgcrt_mod_right_zero_result)))
use mul_zero_left
intro a
intro b
split
intro h
cases h
cases h_witness
trans a + 0 * x
symm
trans a + 0
congr
refl
apply mul_zero_left
apply PA3
trans b + 0 * x1
exact h_witness_witness
trans b + 0
congr
refl
apply mul_zero_left
apply PA3
intro h
exists 0
exists 0
rewrite h
refl
qed

Occurrences

55

Distinct objects

55

Reused references

0

Depth

13

Cuts

1

Certificate SHA-256

af253fd988c1…

mod_eq_scaleMod eq scale Congruence & CRT checked

Scaling values and their modulus preserves balanced congruence.

Prerequisites

mul_add mul_assoc

Used directly by

crt_scaled_common_remainder_lift

Exact expanded PA statement

Copy statement

forall k m a b. (exists hgcrt_mod_left_scale_source hgcrt_mod_right_scale_source. a + m * hgcrt_mod_left_scale_source = b + m * hgcrt_mod_right_scale_source) -> (exists hgcrt_mod_left_scale_result hgcrt_mod_right_scale_result. (k * a) + (k * m) * hgcrt_mod_left_scale_result = (k * b) + (k * m) * hgcrt_mod_right_scale_result)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall k m a b. (exists hgcrt_mod_left_scale_source hgcrt_mod_right_scale_source. a + m * hgcrt_mod_left_scale_source = b + m * hgcrt_mod_right_scale_source) -> (exists hgcrt_mod_left_scale_result hgcrt_mod_right_scale_result. (k * a) + (k * m) * hgcrt_mod_left_scale_result = (k * b) + (k * m) * hgcrt_mod_right_scale_result)
use mul_add
use mul_assoc
intro k
intro m
intro a
intro b
intro h
cases h
cases h_witness
exists x
exists x1
trans k * a + k * (m * x)
congr
refl
apply mul_assoc
trans k * (a + m * x)
symm
apply mul_add
trans k * (b + m * x1)
congr
refl
exact h_witness_witness
trans k * b + k * (m * x1)
apply mul_add
congr
refl
symm
apply mul_assoc
qed

Occurrences

235

Distinct objects

146

Reused references

13

Depth

21

Cuts

5

Certificate SHA-256

d9c5bcb8145d…

crt_solution_pair_congruentCrt solution pair congruent Congruence & CRT checked

Any two solutions of one binary CRT problem agree modulo each modulus.

Prerequisites

mod_eq_symm mod_eq_trans

Used directly by

crt_solution_class_iff_lcm

Exact expanded PA statement

Copy statement

forall m n a b x y. (((exists hgcrt_mod_left_pair_x_left hgcrt_mod_right_pair_x_left. x + m * hgcrt_mod_left_pair_x_left = a + m * hgcrt_mod_right_pair_x_left) /\ (exists hgcrt_mod_left_pair_x_right hgcrt_mod_right_pair_x_right. x + n * hgcrt_mod_left_pair_x_right = b + n * hgcrt_mod_right_pair_x_right))) -> (((exists hgcrt_mod_left_pair_y_left hgcrt_mod_right_pair_y_left. y + m * hgcrt_mod_left_pair_y_left = a + m * hgcrt_mod_right_pair_y_left) /\ (exists hgcrt_mod_left_pair_y_right hgcrt_mod_right_pair_y_right. y + n * hgcrt_mod_left_pair_y_right = b + n * hgcrt_mod_right_pair_y_right))) -> ((exists hgcrt_mod_left_pair_mod_m hgcrt_mod_right_pair_mod_m. x + m * hgcrt_mod_left_pair_mod_m = y + m * hgcrt_mod_right_pair_mod_m) /\ (exists hgcrt_mod_left_pair_mod_n hgcrt_mod_right_pair_mod_n. x + n * hgcrt_mod_left_pair_mod_n = y + n * hgcrt_mod_right_pair_mod_n))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n a b x y. (((exists hgcrt_mod_left_pair_x_left hgcrt_mod_right_pair_x_left. x + m * hgcrt_mod_left_pair_x_left = a + m * hgcrt_mod_right_pair_x_left) /\ (exists hgcrt_mod_left_pair_x_right hgcrt_mod_right_pair_x_right. x + n * hgcrt_mod_left_pair_x_right = b + n * hgcrt_mod_right_pair_x_right))) -> (((exists hgcrt_mod_left_pair_y_left hgcrt_mod_right_pair_y_left. y + m * hgcrt_mod_left_pair_y_left = a + m * hgcrt_mod_right_pair_y_left) /\ (exists hgcrt_mod_left_pair_y_right hgcrt_mod_right_pair_y_right. y + n * hgcrt_mod_left_pair_y_right = b + n * hgcrt_mod_right_pair_y_right))) -> ((exists hgcrt_mod_left_pair_mod_m hgcrt_mod_right_pair_mod_m. x + m * hgcrt_mod_left_pair_mod_m = y + m * hgcrt_mod_right_pair_mod_m) /\ (exists hgcrt_mod_left_pair_mod_n hgcrt_mod_right_pair_mod_n. x + n * hgcrt_mod_left_pair_mod_n = y + n * hgcrt_mod_right_pair_mod_n))
use mod_eq_symm
use mod_eq_trans
intro m
intro n
intro a
intro b
intro x
intro y
intro hx
intro hy
cases hx
cases hy
split
have hay : exists hgcrt_mod_left_pair_a_y hgcrt_mod_right_pair_a_y. a + m * hgcrt_mod_left_pair_a_y = y + m * hgcrt_mod_right_pair_a_y
specialize mod_eq_symm m
specialize mod_eq_symm y
specialize mod_eq_symm a
apply mod_eq_symm
exact hy_left
specialize mod_eq_trans m
specialize mod_eq_trans x
specialize mod_eq_trans a
specialize mod_eq_trans y
apply mod_eq_trans
exact hx_left
exact hay
have hby : exists hgcrt_mod_left_pair_b_y hgcrt_mod_right_pair_b_y. b + n * hgcrt_mod_left_pair_b_y = y + n * hgcrt_mod_right_pair_b_y
specialize mod_eq_symm n
specialize mod_eq_symm y
specialize mod_eq_symm b
apply mod_eq_symm
exact hy_right
specialize mod_eq_trans n
specialize mod_eq_trans x
specialize mod_eq_trans b
specialize mod_eq_trans y
apply mod_eq_trans
exact hx_right
exact hby
qed

Occurrences

307

Distinct objects

259

Reused references

16

Depth

31

Cuts

8

Certificate SHA-256

a3425a60d00e…

crt_common_solution_implies_gcd_compatibleCrt common solution implies gcd compatible Congruence & CRT checked

A common CRT solution forces the residues to be congruent modulo every relational gcd.

Prerequisites

is_gcd_dvd_left is_gcd_dvd_right mod_eq_of_mod_eq_multiple mod_eq_symm mod_eq_trans

Used directly by

crt_incompatibility_obstructs_solution generalized_binary_crt_solvable_iff

Exact expanded PA statement

Copy statement

forall g m n a b x. ((((exists hag_left_factor_crt_necessity. m = g * hag_left_factor_crt_necessity) /\ (exists hag_right_factor_crt_necessity. n = g * hag_right_factor_crt_necessity)) /\ forall hag_divisor_crt_necessity. (exists hag_common_left_crt_necessity. m = hag_divisor_crt_necessity * hag_common_left_crt_necessity) -> (exists hag_common_right_crt_necessity. n = hag_divisor_crt_necessity * hag_common_right_crt_necessity) -> exists hag_greatest_factor_crt_necessity. g = hag_divisor_crt_necessity * hag_greatest_factor_crt_necessity)) -> (((exists hgcrt_mod_left_necessity_solution_left hgcrt_mod_right_necessity_solution_left. x + m * hgcrt_mod_left_necessity_solution_left = a + m * hgcrt_mod_right_necessity_solution_left) /\ (exists hgcrt_mod_left_necessity_solution_right hgcrt_mod_right_necessity_solution_right. x + n * hgcrt_mod_left_necessity_solution_right = b + n * hgcrt_mod_right_necessity_solution_right))) -> (exists hgcrt_mod_left_necessity_compatibility hgcrt_mod_right_necessity_compatibility. a + g * hgcrt_mod_left_necessity_compatibility = b + g * hgcrt_mod_right_necessity_compatibility)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b x. ((((exists hag_left_factor_crt_necessity. m = g * hag_left_factor_crt_necessity) /\ (exists hag_right_factor_crt_necessity. n = g * hag_right_factor_crt_necessity)) /\ forall hag_divisor_crt_necessity. (exists hag_common_left_crt_necessity. m = hag_divisor_crt_necessity * hag_common_left_crt_necessity) -> (exists hag_common_right_crt_necessity. n = hag_divisor_crt_necessity * hag_common_right_crt_necessity) -> exists hag_greatest_factor_crt_necessity. g = hag_divisor_crt_necessity * hag_greatest_factor_crt_necessity)) -> (((exists hgcrt_mod_left_necessity_solution_left hgcrt_mod_right_necessity_solution_left. x + m * hgcrt_mod_left_necessity_solution_left = a + m * hgcrt_mod_right_necessity_solution_left) /\ (exists hgcrt_mod_left_necessity_solution_right hgcrt_mod_right_necessity_solution_right. x + n * hgcrt_mod_left_necessity_solution_right = b + n * hgcrt_mod_right_necessity_solution_right))) -> (exists hgcrt_mod_left_necessity_compatibility hgcrt_mod_right_necessity_compatibility. a + g * hgcrt_mod_left_necessity_compatibility = b + g * hgcrt_mod_right_necessity_compatibility)
use is_gcd_dvd_left
use is_gcd_dvd_right
use mod_eq_of_mod_eq_multiple
use mod_eq_symm
use mod_eq_trans
intro g
intro m
intro n
intro a
intro b
intro x
intro hg
intro hx
cases hx
have hgm : exists q. m = g * q
specialize is_gcd_dvd_left g
specialize is_gcd_dvd_left m
specialize is_gcd_dvd_left n
apply is_gcd_dvd_left
exact hg
have hgn : exists q. n = g * q
specialize is_gcd_dvd_right g
specialize is_gcd_dvd_right m
specialize is_gcd_dvd_right n
apply is_gcd_dvd_right
exact hg
have hxa : exists hgcrt_mod_left_necessity_x_a hgcrt_mod_right_necessity_x_a. x + g * hgcrt_mod_left_necessity_x_a = a + g * hgcrt_mod_right_necessity_x_a
specialize mod_eq_of_mod_eq_multiple g
specialize mod_eq_of_mod_eq_multiple m
specialize mod_eq_of_mod_eq_multiple x
specialize mod_eq_of_mod_eq_multiple a
apply mod_eq_of_mod_eq_multiple
exact hgm
exact hx_left
have hxb : exists hgcrt_mod_left_necessity_x_b hgcrt_mod_right_necessity_x_b. x + g * hgcrt_mod_left_necessity_x_b = b + g * hgcrt_mod_right_necessity_x_b
specialize mod_eq_of_mod_eq_multiple g
specialize mod_eq_of_mod_eq_multiple n
specialize mod_eq_of_mod_eq_multiple x
specialize mod_eq_of_mod_eq_multiple b
apply mod_eq_of_mod_eq_multiple
exact hgn
exact hx_right
have hax : exists hgcrt_mod_left_necessity_a_x hgcrt_mod_right_necessity_a_x. a + g * hgcrt_mod_left_necessity_a_x = x + g * hgcrt_mod_right_necessity_a_x
specialize mod_eq_symm g
specialize mod_eq_symm x
specialize mod_eq_symm a
apply mod_eq_symm
exact hxa
specialize mod_eq_trans g
specialize mod_eq_trans a
specialize mod_eq_trans x
specialize mod_eq_trans b
apply mod_eq_trans
exact hax
exact hxb
qed

Occurrences

518

Distinct objects

388

Reused references

22

Depth

34

Cuts

14

Certificate SHA-256

aa5633a5e767…

crt_incompatibility_obstructs_solutionCrt incompatibility obstructs solution Congruence & CRT checked

Failure of gcd compatibility constructively refutes every common CRT solution.

Prerequisites

crt_common_solution_implies_gcd_compatible

Used directly by

generalized_binary_crt_solution_or_obstruction

Exact expanded PA statement

Copy statement

forall g m n a b. ((((exists hag_left_factor_crt_obstruction. m = g * hag_left_factor_crt_obstruction) /\ (exists hag_right_factor_crt_obstruction. n = g * hag_right_factor_crt_obstruction)) /\ forall hag_divisor_crt_obstruction. (exists hag_common_left_crt_obstruction. m = hag_divisor_crt_obstruction * hag_common_left_crt_obstruction) -> (exists hag_common_right_crt_obstruction. n = hag_divisor_crt_obstruction * hag_common_right_crt_obstruction) -> exists hag_greatest_factor_crt_obstruction. g = hag_divisor_crt_obstruction * hag_greatest_factor_crt_obstruction)) -> ~(exists hgcrt_mod_left_obstruction_compatibility hgcrt_mod_right_obstruction_compatibility. a + g * hgcrt_mod_left_obstruction_compatibility = b + g * hgcrt_mod_right_obstruction_compatibility) -> ~(exists x. (((exists hgcrt_mod_left_obstruction_solution_left hgcrt_mod_right_obstruction_solution_left. x + m * hgcrt_mod_left_obstruction_solution_left = a + m * hgcrt_mod_right_obstruction_solution_left) /\ (exists hgcrt_mod_left_obstruction_solution_right hgcrt_mod_right_obstruction_solution_right. x + n * hgcrt_mod_left_obstruction_solution_right = b + n * hgcrt_mod_right_obstruction_solution_right))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b. ((((exists hag_left_factor_crt_obstruction. m = g * hag_left_factor_crt_obstruction) /\ (exists hag_right_factor_crt_obstruction. n = g * hag_right_factor_crt_obstruction)) /\ forall hag_divisor_crt_obstruction. (exists hag_common_left_crt_obstruction. m = hag_divisor_crt_obstruction * hag_common_left_crt_obstruction) -> (exists hag_common_right_crt_obstruction. n = hag_divisor_crt_obstruction * hag_common_right_crt_obstruction) -> exists hag_greatest_factor_crt_obstruction. g = hag_divisor_crt_obstruction * hag_greatest_factor_crt_obstruction)) -> ~(exists hgcrt_mod_left_obstruction_compatibility hgcrt_mod_right_obstruction_compatibility. a + g * hgcrt_mod_left_obstruction_compatibility = b + g * hgcrt_mod_right_obstruction_compatibility) -> ~(exists x. (((exists hgcrt_mod_left_obstruction_solution_left hgcrt_mod_right_obstruction_solution_left. x + m * hgcrt_mod_left_obstruction_solution_left = a + m * hgcrt_mod_right_obstruction_solution_left) /\ (exists hgcrt_mod_left_obstruction_solution_right hgcrt_mod_right_obstruction_solution_right. x + n * hgcrt_mod_left_obstruction_solution_right = b + n * hgcrt_mod_right_obstruction_solution_right))))
use crt_common_solution_implies_gcd_compatible
intro g
intro m
intro n
intro a
intro b
intro hg
intro hnot
intro hsolution
cases hsolution
apply hnot
specialize crt_common_solution_implies_gcd_compatible g
specialize crt_common_solution_implies_gcd_compatible m
specialize crt_common_solution_implies_gcd_compatible n
specialize crt_common_solution_implies_gcd_compatible a
specialize crt_common_solution_implies_gcd_compatible b
specialize crt_common_solution_implies_gcd_compatible x
apply crt_common_solution_implies_gcd_compatible
exact hg
exact hsolution_witness
qed

Occurrences

560

Distinct objects

430

Reused references

22

Depth

35

Cuts

15

Certificate SHA-256

3e304725fcdf…

is_gcd_quotients_coprime_nonzeroIs gcd quotients coprime nonzero Congruence & CRT checked

Nonzero gcd cofactors are coprime by the greatest-divisor property.

Prerequisites

is_gcd_greatest mul_assoc mul_one mul_left_cancel_nonzero divisor_one

Used directly by

generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall g m n M N. ((((exists hag_left_factor_quotient_assumption. m = g * hag_left_factor_quotient_assumption) /\ (exists hag_right_factor_quotient_assumption. n = g * hag_right_factor_quotient_assumption)) /\ forall hag_divisor_quotient_assumption. (exists hag_common_left_quotient_assumption. m = hag_divisor_quotient_assumption * hag_common_left_quotient_assumption) -> (exists hag_common_right_quotient_assumption. n = hag_divisor_quotient_assumption * hag_common_right_quotient_assumption) -> exists hag_greatest_factor_quotient_assumption. g = hag_divisor_quotient_assumption * hag_greatest_factor_quotient_assumption)) -> ~(g = 0) -> m = g * M -> n = g * N -> (forall hmi_divisor_quotient_result. (exists hmi_left_factor_quotient_result. M = hmi_divisor_quotient_result * hmi_left_factor_quotient_result) -> (exists hmi_right_factor_quotient_result. N = hmi_divisor_quotient_result * hmi_right_factor_quotient_result) -> hmi_divisor_quotient_result = 1)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n M N. ((((exists hag_left_factor_quotient_assumption. m = g * hag_left_factor_quotient_assumption) /\ (exists hag_right_factor_quotient_assumption. n = g * hag_right_factor_quotient_assumption)) /\ forall hag_divisor_quotient_assumption. (exists hag_common_left_quotient_assumption. m = hag_divisor_quotient_assumption * hag_common_left_quotient_assumption) -> (exists hag_common_right_quotient_assumption. n = hag_divisor_quotient_assumption * hag_common_right_quotient_assumption) -> exists hag_greatest_factor_quotient_assumption. g = hag_divisor_quotient_assumption * hag_greatest_factor_quotient_assumption)) -> ~(g = 0) -> m = g * M -> n = g * N -> (forall hmi_divisor_quotient_result. (exists hmi_left_factor_quotient_result. M = hmi_divisor_quotient_result * hmi_left_factor_quotient_result) -> (exists hmi_right_factor_quotient_result. N = hmi_divisor_quotient_result * hmi_right_factor_quotient_result) -> hmi_divisor_quotient_result = 1)
use is_gcd_greatest
use mul_assoc
use mul_one
use mul_left_cancel_nonzero
use divisor_one
intro g
intro m
intro n
intro M
intro N
intro hg
intro hg0
intro hm
intro hn
intro d
intro hdM
intro hdN
cases hdM
cases hdN
have hdm : exists u. m = (g * d) * u
exists x
trans g * M
exact hm
trans g * (d * x)
congr
refl
exact hdM_witness
symm
apply mul_assoc
have hdn : exists v. n = (g * d) * v
exists x1
trans g * N
exact hn
trans g * (d * x1)
congr
refl
exact hdN_witness
symm
apply mul_assoc
have hdg : exists w. g = (g * d) * w
specialize is_gcd_greatest g
specialize is_gcd_greatest m
specialize is_gcd_greatest n
specialize is_gcd_greatest (g * d)
apply is_gcd_greatest
exact hg
exact hdm
exact hdn
cases hdg
have hnorm : g = g * (d * x2)
trans (g * d) * x2
exact hdg_witness
apply mul_assoc
have hone : 1 = d * x2
specialize mul_left_cancel_nonzero g
specialize mul_left_cancel_nonzero 1
specialize mul_left_cancel_nonzero (d * x2)
apply mul_left_cancel_nonzero
exact hg0
trans g
apply mul_one
exact hnorm
specialize divisor_one d
apply divisor_one
exists x2
exact hone
qed

Occurrences

660

Distinct objects

562

Reused references

34

Depth

33

Cuts

19

Certificate SHA-256

7c3914650910…

mod_eq_common_remainder_decompositionMod eq common remainder decomposition Congruence & CRT checked

Compatible residues share one bounded remainder modulo their gcd.

Prerequisites

division_remainder_exists remainder_decomposition_to_mod_eq mod_eq_symm mod_eq_trans mod_eq_to_remainder_decomposition mul_comm

Used directly by

generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall g a b. ~(g = 0) -> (exists hgcrt_mod_left_common_remainder_assumption hgcrt_mod_right_common_remainder_assumption. a + g * hgcrt_mod_left_common_remainder_assumption = b + g * hgcrt_mod_right_common_remainder_assumption) -> exists A B r. ((a = g * A + r /\ b = g * B + r) /\ (exists hmi_gap_common_remainder_bound. hmi_gap_common_remainder_bound + S r = g))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g a b. ~(g = 0) -> (exists hgcrt_mod_left_common_remainder_assumption hgcrt_mod_right_common_remainder_assumption. a + g * hgcrt_mod_left_common_remainder_assumption = b + g * hgcrt_mod_right_common_remainder_assumption) -> exists A B r. ((a = g * A + r /\ b = g * B + r) /\ (exists hmi_gap_common_remainder_bound. hmi_gap_common_remainder_bound + S r = g))
use division_remainder_exists
use remainder_decomposition_to_mod_eq
use mod_eq_symm
use mod_eq_trans
use mod_eq_to_remainder_decomposition
use mul_comm
intro g
intro a
intro b
intro hg0
intro hab
have hdiva : exists q r. a = g * q + r /\ exists h. h + S r = g
specialize division_remainder_exists g
specialize division_remainder_exists a
apply division_remainder_exists
exact hg0
cases hdiva
cases hdiva_witness
cases hdiva_witness_witness
have haqr : a = x * g + x1
trans g * x + x1
exact hdiva_witness_witness_left
congr
apply mul_comm
refl
have har : exists hgcrt_mod_left_common_remainder_a_r hgcrt_mod_right_common_remainder_a_r. a + g * hgcrt_mod_left_common_remainder_a_r = x1 + g * hgcrt_mod_right_common_remainder_a_r
specialize remainder_decomposition_to_mod_eq g
specialize remainder_decomposition_to_mod_eq a
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq x1
apply remainder_decomposition_to_mod_eq
exact haqr
have hba : exists hgcrt_mod_left_common_remainder_b_a hgcrt_mod_right_common_remainder_b_a. b + g * hgcrt_mod_left_common_remainder_b_a = a + g * hgcrt_mod_right_common_remainder_b_a
specialize mod_eq_symm g
specialize mod_eq_symm a
specialize mod_eq_symm b
apply mod_eq_symm
exact hab
have hbr : exists hgcrt_mod_left_common_remainder_b_r hgcrt_mod_right_common_remainder_b_r. b + g * hgcrt_mod_left_common_remainder_b_r = x1 + g * hgcrt_mod_right_common_remainder_b_r
specialize mod_eq_trans g
specialize mod_eq_trans b
specialize mod_eq_trans a
specialize mod_eq_trans x1
apply mod_eq_trans
exact hba
exact har
have hqb : exists q. b = q * g + x1
specialize mod_eq_to_remainder_decomposition g
specialize mod_eq_to_remainder_decomposition b
specialize mod_eq_to_remainder_decomposition x1
apply mod_eq_to_remainder_decomposition
exact hg0
exact hdiva_witness_witness_right
exact hbr
cases hqb
exists x
exists x2
exists x1
split
split
exact hdiva_witness_witness_left
trans x2 * g + x1
exact hqb_witness
congr
apply mul_comm
refl
exact hdiva_witness_witness_right
qed

Occurrences

2,894

Distinct objects

1,075

Reused references

64

Depth

69

Cuts

83

Certificate SHA-256

fadb87563251…

crt_scaled_common_remainder_liftCrt scaled common remainder lift Congruence & CRT checked

A coprime cofactor CRT solution scales and lifts through a common remainder.

Prerequisites

binary_crt mod_eq_scale mod_eq_refl mod_eq_add

Used directly by

generalized_binary_crt_sufficient_nonzero

Exact expanded PA statement

Copy statement

forall g m n a b M N A B r. m = g * M -> n = g * N -> a = g * A + r -> b = g * B + r -> ~(M = 0) -> ~(N = 0) -> (forall hmi_divisor_lift_assumption. (exists hmi_left_factor_lift_assumption. M = hmi_divisor_lift_assumption * hmi_left_factor_lift_assumption) -> (exists hmi_right_factor_lift_assumption. N = hmi_divisor_lift_assumption * hmi_right_factor_lift_assumption) -> hmi_divisor_lift_assumption = 1) -> exists x. (((exists hgcrt_mod_left_lift_result_left hgcrt_mod_right_lift_result_left. x + m * hgcrt_mod_left_lift_result_left = a + m * hgcrt_mod_right_lift_result_left) /\ (exists hgcrt_mod_left_lift_result_right hgcrt_mod_right_lift_result_right. x + n * hgcrt_mod_left_lift_result_right = b + n * hgcrt_mod_right_lift_result_right)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b M N A B r. m = g * M -> n = g * N -> a = g * A + r -> b = g * B + r -> ~(M = 0) -> ~(N = 0) -> (forall hmi_divisor_lift_assumption. (exists hmi_left_factor_lift_assumption. M = hmi_divisor_lift_assumption * hmi_left_factor_lift_assumption) -> (exists hmi_right_factor_lift_assumption. N = hmi_divisor_lift_assumption * hmi_right_factor_lift_assumption) -> hmi_divisor_lift_assumption = 1) -> exists x. (((exists hgcrt_mod_left_lift_result_left hgcrt_mod_right_lift_result_left. x + m * hgcrt_mod_left_lift_result_left = a + m * hgcrt_mod_right_lift_result_left) /\ (exists hgcrt_mod_left_lift_result_right hgcrt_mod_right_lift_result_right. x + n * hgcrt_mod_left_lift_result_right = b + n * hgcrt_mod_right_lift_result_right)))
use binary_crt
use mod_eq_scale
use mod_eq_refl
use mod_eq_add
intro g
intro m
intro n
intro a
intro b
intro M
intro N
intro A
intro B
intro r
intro hm
intro hn
intro ha
intro hb
intro hM
intro hN
intro hcop
have hcrt : exists x. (((exists hgcrt_mod_left_lift_reduced_left hgcrt_mod_right_lift_reduced_left. x + M * hgcrt_mod_left_lift_reduced_left = A + M * hgcrt_mod_right_lift_reduced_left) /\ (exists hgcrt_mod_left_lift_reduced_right hgcrt_mod_right_lift_reduced_right. x + N * hgcrt_mod_left_lift_reduced_right = B + N * hgcrt_mod_right_lift_reduced_right)))
specialize binary_crt M
specialize binary_crt N
specialize binary_crt A
specialize binary_crt B
apply binary_crt
exact hM
exact hN
exact hcop
cases hcrt
cases hcrt_witness
have hls : exists hgcrt_mod_left_lift_ls hgcrt_mod_right_lift_ls. (g * x) + (g * M) * hgcrt_mod_left_lift_ls = (g * A) + (g * M) * hgcrt_mod_right_lift_ls
specialize mod_eq_scale g
specialize mod_eq_scale M
specialize mod_eq_scale x
specialize mod_eq_scale A
apply mod_eq_scale
exact hcrt_witness_left
have hrs : exists hgcrt_mod_left_lift_rs hgcrt_mod_right_lift_rs. (g * x) + (g * N) * hgcrt_mod_left_lift_rs = (g * B) + (g * N) * hgcrt_mod_right_lift_rs
specialize mod_eq_scale g
specialize mod_eq_scale N
specialize mod_eq_scale x
specialize mod_eq_scale B
apply mod_eq_scale
exact hcrt_witness_right
have hlr : exists hgcrt_mod_left_lift_lr hgcrt_mod_right_lift_lr. r + (g * M) * hgcrt_mod_left_lift_lr = r + (g * M) * hgcrt_mod_right_lift_lr
specialize mod_eq_refl (g * M)
specialize mod_eq_refl r
apply mod_eq_refl
have hrr : exists hgcrt_mod_left_lift_rr hgcrt_mod_right_lift_rr. r + (g * N) * hgcrt_mod_left_lift_rr = r + (g * N) * hgcrt_mod_right_lift_rr
specialize mod_eq_refl (g * N)
specialize mod_eq_refl r
apply mod_eq_refl
have hll : exists hgcrt_mod_left_lift_ll hgcrt_mod_right_lift_ll. (g * x + r) + (g * M) * hgcrt_mod_left_lift_ll = (g * A + r) + (g * M) * hgcrt_mod_right_lift_ll
specialize mod_eq_add (g * M)
specialize mod_eq_add (g * x)
specialize mod_eq_add (g * A)
specialize mod_eq_add r
specialize mod_eq_add r
apply mod_eq_add
exact hls
exact hlr
have hrl : exists hgcrt_mod_left_lift_rl hgcrt_mod_right_lift_rl. (g * x + r) + (g * N) * hgcrt_mod_left_lift_rl = (g * B + r) + (g * N) * hgcrt_mod_right_lift_rl
specialize mod_eq_add (g * N)
specialize mod_eq_add (g * x)
specialize mod_eq_add (g * B)
specialize mod_eq_add r
specialize mod_eq_add r
apply mod_eq_add
exact hrs
exact hrr
exists g * x + r
split
rewrite <- hm at hll
rewrite <- hm at hll
rewrite <- ha at hll
exact hll
rewrite <- hn at hrl
rewrite <- hn at hrl
rewrite <- hb at hrl
exact hrl
qed

Occurrences

5,745

Distinct objects

2,062

Reused references

113

Depth

52

Cuts

163

Certificate SHA-256

983f84cbd94a…

generalized_binary_crt_sufficient_nonzeroGeneralized binary crt sufficient nonzero Congruence & CRT checked

Gcd compatibility is sufficient for a common solution when both moduli are nonzero.

Prerequisites

is_gcd_dvd_left is_gcd_dvd_right mul_zero_left is_gcd_quotients_coprime_nonzero mod_eq_common_remainder_decomposition crt_scaled_common_remainder_lift

Used directly by

generalized_binary_crt_sufficient

Exact expanded PA statement

Copy statement

forall g m n a b. ~(m = 0) -> ~(n = 0) -> ((((exists hag_left_factor_sufficient_assumption. m = g * hag_left_factor_sufficient_assumption) /\ (exists hag_right_factor_sufficient_assumption. n = g * hag_right_factor_sufficient_assumption)) /\ forall hag_divisor_sufficient_assumption. (exists hag_common_left_sufficient_assumption. m = hag_divisor_sufficient_assumption * hag_common_left_sufficient_assumption) -> (exists hag_common_right_sufficient_assumption. n = hag_divisor_sufficient_assumption * hag_common_right_sufficient_assumption) -> exists hag_greatest_factor_sufficient_assumption. g = hag_divisor_sufficient_assumption * hag_greatest_factor_sufficient_assumption)) -> (exists hgcrt_mod_left_sufficient_compatibility hgcrt_mod_right_sufficient_compatibility. a + g * hgcrt_mod_left_sufficient_compatibility = b + g * hgcrt_mod_right_sufficient_compatibility) -> exists x. (((exists hgcrt_mod_left_sufficient_result_left hgcrt_mod_right_sufficient_result_left. x + m * hgcrt_mod_left_sufficient_result_left = a + m * hgcrt_mod_right_sufficient_result_left) /\ (exists hgcrt_mod_left_sufficient_result_right hgcrt_mod_right_sufficient_result_right. x + n * hgcrt_mod_left_sufficient_result_right = b + n * hgcrt_mod_right_sufficient_result_right)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b. ~(m = 0) -> ~(n = 0) -> ((((exists hag_left_factor_sufficient_assumption. m = g * hag_left_factor_sufficient_assumption) /\ (exists hag_right_factor_sufficient_assumption. n = g * hag_right_factor_sufficient_assumption)) /\ forall hag_divisor_sufficient_assumption. (exists hag_common_left_sufficient_assumption. m = hag_divisor_sufficient_assumption * hag_common_left_sufficient_assumption) -> (exists hag_common_right_sufficient_assumption. n = hag_divisor_sufficient_assumption * hag_common_right_sufficient_assumption) -> exists hag_greatest_factor_sufficient_assumption. g = hag_divisor_sufficient_assumption * hag_greatest_factor_sufficient_assumption)) -> (exists hgcrt_mod_left_sufficient_compatibility hgcrt_mod_right_sufficient_compatibility. a + g * hgcrt_mod_left_sufficient_compatibility = b + g * hgcrt_mod_right_sufficient_compatibility) -> exists x. (((exists hgcrt_mod_left_sufficient_result_left hgcrt_mod_right_sufficient_result_left. x + m * hgcrt_mod_left_sufficient_result_left = a + m * hgcrt_mod_right_sufficient_result_left) /\ (exists hgcrt_mod_left_sufficient_result_right hgcrt_mod_right_sufficient_result_right. x + n * hgcrt_mod_left_sufficient_result_right = b + n * hgcrt_mod_right_sufficient_result_right)))
use is_gcd_dvd_left
use is_gcd_dvd_right
use mul_zero_left
use is_gcd_quotients_coprime_nonzero
use mod_eq_common_remainder_decomposition
use crt_scaled_common_remainder_lift
intro g
intro m
intro n
intro a
intro b
intro hm0
intro hn0
intro hgcd
intro hcompat
have hmfactor : exists M. m = g * M
specialize is_gcd_dvd_left g
specialize is_gcd_dvd_left m
specialize is_gcd_dvd_left n
apply is_gcd_dvd_left
exact hgcd
have hnfactor : exists N. n = g * N
specialize is_gcd_dvd_right g
specialize is_gcd_dvd_right m
specialize is_gcd_dvd_right n
apply is_gcd_dvd_right
exact hgcd
cases hmfactor
cases hnfactor
have hg0 : ~(g = 0)
intro hgz
apply hm0
trans g * x
exact hmfactor_witness
rewrite hgz
apply mul_zero_left
have hM0 : ~(x = 0)
intro hx0
apply hm0
trans g * x
exact hmfactor_witness
rewrite hx0
apply PA5
have hN0 : ~(x1 = 0)
intro hx10
apply hn0
trans g * x1
exact hnfactor_witness
rewrite hx10
apply PA5
have hcop : forall hmi_divisor_main_cofactors. (exists hmi_left_factor_main_cofactors. x = hmi_divisor_main_cofactors * hmi_left_factor_main_cofactors) -> (exists hmi_right_factor_main_cofactors. x1 = hmi_divisor_main_cofactors * hmi_right_factor_main_cofactors) -> hmi_divisor_main_cofactors = 1
specialize is_gcd_quotients_coprime_nonzero g
specialize is_gcd_quotients_coprime_nonzero m
specialize is_gcd_quotients_coprime_nonzero n
specialize is_gcd_quotients_coprime_nonzero x
specialize is_gcd_quotients_coprime_nonzero x1
apply is_gcd_quotients_coprime_nonzero
exact hgcd
exact hg0
exact hmfactor_witness
exact hnfactor_witness
have hrem : exists A B r. ((a = g * A + r /\ b = g * B + r) /\ exists h. h + S r = g)
specialize mod_eq_common_remainder_decomposition g
specialize mod_eq_common_remainder_decomposition a
specialize mod_eq_common_remainder_decomposition b
apply mod_eq_common_remainder_decomposition
exact hg0
exact hcompat
cases hrem
cases hrem_witness
cases hrem_witness_witness
cases hrem_witness_witness_witness
cases hrem_witness_witness_witness_left
specialize crt_scaled_common_remainder_lift g
specialize crt_scaled_common_remainder_lift m
specialize crt_scaled_common_remainder_lift n
specialize crt_scaled_common_remainder_lift a
specialize crt_scaled_common_remainder_lift b
specialize crt_scaled_common_remainder_lift x
specialize crt_scaled_common_remainder_lift x1
specialize crt_scaled_common_remainder_lift x2
specialize crt_scaled_common_remainder_lift x3
specialize crt_scaled_common_remainder_lift x4
apply crt_scaled_common_remainder_lift
exact hmfactor_witness
exact hnfactor_witness
exact hrem_witness_witness_witness_left_left
exact hrem_witness_witness_witness_left_right
exact hM0
exact hN0
exact hcop
qed

Occurrences

9,482

Distinct objects

3,147

Reused references

156

Depth

74

Cuts

271

Certificate SHA-256

632ca3c58954…

generalized_binary_crt_sufficient_zero_leftGeneralized binary crt sufficient zero left Congruence & CRT checked

If the left modulus is zero, gcd compatibility constructs a solution by choosing the left residue.

Prerequisites

is_gcd_symm is_gcd_zero_right is_gcd_unique mod_eq_refl

Used directly by

generalized_binary_crt_sufficient

Exact expanded PA statement

Copy statement

forall g n a b. ((((exists hage_left_factor_zero_left_gcd. 0 = g * hage_left_factor_zero_left_gcd) /\ (exists hage_right_factor_zero_left_gcd. n = g * hage_right_factor_zero_left_gcd)) /\ forall hage_divisor_zero_left_gcd. (exists hage_common_left_zero_left_gcd. 0 = hage_divisor_zero_left_gcd * hage_common_left_zero_left_gcd) -> (exists hage_common_right_zero_left_gcd. n = hage_divisor_zero_left_gcd * hage_common_right_zero_left_gcd) -> exists hage_greatest_factor_zero_left_gcd. g = hage_divisor_zero_left_gcd * hage_greatest_factor_zero_left_gcd)) -> (exists hgcrt_mod_left_zero_left_compatibility hgcrt_mod_right_zero_left_compatibility. a + g * hgcrt_mod_left_zero_left_compatibility = b + g * hgcrt_mod_right_zero_left_compatibility) -> exists x. (((exists hgcrt_mod_left_zero_left_solution_left hgcrt_mod_right_zero_left_solution_left. x + 0 * hgcrt_mod_left_zero_left_solution_left = a + 0 * hgcrt_mod_right_zero_left_solution_left) /\ (exists hgcrt_mod_left_zero_left_solution_right hgcrt_mod_right_zero_left_solution_right. x + n * hgcrt_mod_left_zero_left_solution_right = b + n * hgcrt_mod_right_zero_left_solution_right)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g n a b. ((((exists hage_left_factor_zero_left_gcd. 0 = g * hage_left_factor_zero_left_gcd) /\ (exists hage_right_factor_zero_left_gcd. n = g * hage_right_factor_zero_left_gcd)) /\ forall hage_divisor_zero_left_gcd. (exists hage_common_left_zero_left_gcd. 0 = hage_divisor_zero_left_gcd * hage_common_left_zero_left_gcd) -> (exists hage_common_right_zero_left_gcd. n = hage_divisor_zero_left_gcd * hage_common_right_zero_left_gcd) -> exists hage_greatest_factor_zero_left_gcd. g = hage_divisor_zero_left_gcd * hage_greatest_factor_zero_left_gcd)) -> (exists hgcrt_mod_left_zero_left_compatibility hgcrt_mod_right_zero_left_compatibility. a + g * hgcrt_mod_left_zero_left_compatibility = b + g * hgcrt_mod_right_zero_left_compatibility) -> exists x. (((exists hgcrt_mod_left_zero_left_solution_left hgcrt_mod_right_zero_left_solution_left. x + 0 * hgcrt_mod_left_zero_left_solution_left = a + 0 * hgcrt_mod_right_zero_left_solution_left) /\ (exists hgcrt_mod_left_zero_left_solution_right hgcrt_mod_right_zero_left_solution_right. x + n * hgcrt_mod_left_zero_left_solution_right = b + n * hgcrt_mod_right_zero_left_solution_right)))
use is_gcd_symm
use is_gcd_zero_right
use is_gcd_unique
use mod_eq_refl
intro g
intro n
intro a
intro b
intro hg
intro hab
have hsym : (((exists hage_left_factor_zero_left_gcd_swapped. n = g * hage_left_factor_zero_left_gcd_swapped) /\ (exists hage_right_factor_zero_left_gcd_swapped. 0 = g * hage_right_factor_zero_left_gcd_swapped)) /\ forall hage_divisor_zero_left_gcd_swapped. (exists hage_common_left_zero_left_gcd_swapped. n = hage_divisor_zero_left_gcd_swapped * hage_common_left_zero_left_gcd_swapped) -> (exists hage_common_right_zero_left_gcd_swapped. 0 = hage_divisor_zero_left_gcd_swapped * hage_common_right_zero_left_gcd_swapped) -> exists hage_greatest_factor_zero_left_gcd_swapped. g = hage_divisor_zero_left_gcd_swapped * hage_greatest_factor_zero_left_gcd_swapped)
specialize is_gcd_symm g
specialize is_gcd_symm 0
specialize is_gcd_symm n
apply is_gcd_symm
exact hg
have hn : (((exists hage_left_factor_zero_left_gcd_base. n = n * hage_left_factor_zero_left_gcd_base) /\ (exists hage_right_factor_zero_left_gcd_base. 0 = n * hage_right_factor_zero_left_gcd_base)) /\ forall hage_divisor_zero_left_gcd_base. (exists hage_common_left_zero_left_gcd_base. n = hage_divisor_zero_left_gcd_base * hage_common_left_zero_left_gcd_base) -> (exists hage_common_right_zero_left_gcd_base. 0 = hage_divisor_zero_left_gcd_base * hage_common_right_zero_left_gcd_base) -> exists hage_greatest_factor_zero_left_gcd_base. n = hage_divisor_zero_left_gcd_base * hage_greatest_factor_zero_left_gcd_base)
specialize is_gcd_zero_right n
exact is_gcd_zero_right
have hgn : g = n
specialize is_gcd_unique g
specialize is_gcd_unique n
specialize is_gcd_unique n
specialize is_gcd_unique 0
apply is_gcd_unique
exact hsym
exact hn
rewrite hgn at hab
rewrite hgn at hab
exists a
split
specialize mod_eq_refl 0
specialize mod_eq_refl a
exact mod_eq_refl
exact hab
qed

Occurrences

834

Distinct objects

682

Reused references

36

Depth

37

Cuts

28

Certificate SHA-256

a45dc68d2381…

generalized_binary_crt_sufficient_zero_rightGeneralized binary crt sufficient zero right Congruence & CRT checked

If the right modulus is zero, gcd compatibility constructs a solution by choosing the right residue.

Prerequisites

is_gcd_zero_right is_gcd_unique mod_eq_symm mod_eq_refl

Used directly by

generalized_binary_crt_sufficient

Exact expanded PA statement

Copy statement

forall g m a b. ((((exists hage_left_factor_zero_right_gcd. m = g * hage_left_factor_zero_right_gcd) /\ (exists hage_right_factor_zero_right_gcd. 0 = g * hage_right_factor_zero_right_gcd)) /\ forall hage_divisor_zero_right_gcd. (exists hage_common_left_zero_right_gcd. m = hage_divisor_zero_right_gcd * hage_common_left_zero_right_gcd) -> (exists hage_common_right_zero_right_gcd. 0 = hage_divisor_zero_right_gcd * hage_common_right_zero_right_gcd) -> exists hage_greatest_factor_zero_right_gcd. g = hage_divisor_zero_right_gcd * hage_greatest_factor_zero_right_gcd)) -> (exists hgcrt_mod_left_zero_right_compatibility hgcrt_mod_right_zero_right_compatibility. a + g * hgcrt_mod_left_zero_right_compatibility = b + g * hgcrt_mod_right_zero_right_compatibility) -> exists x. (((exists hgcrt_mod_left_zero_right_solution_left hgcrt_mod_right_zero_right_solution_left. x + m * hgcrt_mod_left_zero_right_solution_left = a + m * hgcrt_mod_right_zero_right_solution_left) /\ (exists hgcrt_mod_left_zero_right_solution_right hgcrt_mod_right_zero_right_solution_right. x + 0 * hgcrt_mod_left_zero_right_solution_right = b + 0 * hgcrt_mod_right_zero_right_solution_right)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m a b. ((((exists hage_left_factor_zero_right_gcd. m = g * hage_left_factor_zero_right_gcd) /\ (exists hage_right_factor_zero_right_gcd. 0 = g * hage_right_factor_zero_right_gcd)) /\ forall hage_divisor_zero_right_gcd. (exists hage_common_left_zero_right_gcd. m = hage_divisor_zero_right_gcd * hage_common_left_zero_right_gcd) -> (exists hage_common_right_zero_right_gcd. 0 = hage_divisor_zero_right_gcd * hage_common_right_zero_right_gcd) -> exists hage_greatest_factor_zero_right_gcd. g = hage_divisor_zero_right_gcd * hage_greatest_factor_zero_right_gcd)) -> (exists hgcrt_mod_left_zero_right_compatibility hgcrt_mod_right_zero_right_compatibility. a + g * hgcrt_mod_left_zero_right_compatibility = b + g * hgcrt_mod_right_zero_right_compatibility) -> exists x. (((exists hgcrt_mod_left_zero_right_solution_left hgcrt_mod_right_zero_right_solution_left. x + m * hgcrt_mod_left_zero_right_solution_left = a + m * hgcrt_mod_right_zero_right_solution_left) /\ (exists hgcrt_mod_left_zero_right_solution_right hgcrt_mod_right_zero_right_solution_right. x + 0 * hgcrt_mod_left_zero_right_solution_right = b + 0 * hgcrt_mod_right_zero_right_solution_right)))
use is_gcd_zero_right
use is_gcd_unique
use mod_eq_symm
use mod_eq_refl
intro g
intro m
intro a
intro b
intro hg
intro hab
have hm : (((exists hage_left_factor_zero_right_gcd_base. m = m * hage_left_factor_zero_right_gcd_base) /\ (exists hage_right_factor_zero_right_gcd_base. 0 = m * hage_right_factor_zero_right_gcd_base)) /\ forall hage_divisor_zero_right_gcd_base. (exists hage_common_left_zero_right_gcd_base. m = hage_divisor_zero_right_gcd_base * hage_common_left_zero_right_gcd_base) -> (exists hage_common_right_zero_right_gcd_base. 0 = hage_divisor_zero_right_gcd_base * hage_common_right_zero_right_gcd_base) -> exists hage_greatest_factor_zero_right_gcd_base. m = hage_divisor_zero_right_gcd_base * hage_greatest_factor_zero_right_gcd_base)
specialize is_gcd_zero_right m
exact is_gcd_zero_right
have hgm : g = m
specialize is_gcd_unique g
specialize is_gcd_unique m
specialize is_gcd_unique m
specialize is_gcd_unique 0
apply is_gcd_unique
exact hg
exact hm
rewrite hgm at hab
rewrite hgm at hab
exists b
split
specialize mod_eq_symm m
specialize mod_eq_symm a
specialize mod_eq_symm b
apply mod_eq_symm
exact hab
specialize mod_eq_refl 0
specialize mod_eq_refl b
exact mod_eq_refl
qed

Occurrences

805

Distinct objects

653

Reused references

36

Depth

36

Cuts

28

Certificate SHA-256

015740ead687…

generalized_binary_crt_sufficientGeneralized binary crt sufficient Congruence & CRT checked

Gcd compatibility constructs a common solution for arbitrary natural moduli, including zero.

Prerequisites

eq_decidable generalized_binary_crt_sufficient_zero_left generalized_binary_crt_sufficient_zero_right generalized_binary_crt_sufficient_nonzero

Used directly by

generalized_binary_crt_solvable_iff generalized_binary_crt_canonical_boundary generalized_binary_crt_solution_or_obstruction

Exact expanded PA statement

Copy statement

forall g m n a b. ((((exists hag_left_factor_total_sufficiency_gcd. m = g * hag_left_factor_total_sufficiency_gcd) /\ (exists hag_right_factor_total_sufficiency_gcd. n = g * hag_right_factor_total_sufficiency_gcd)) /\ forall hag_divisor_total_sufficiency_gcd. (exists hag_common_left_total_sufficiency_gcd. m = hag_divisor_total_sufficiency_gcd * hag_common_left_total_sufficiency_gcd) -> (exists hag_common_right_total_sufficiency_gcd. n = hag_divisor_total_sufficiency_gcd * hag_common_right_total_sufficiency_gcd) -> exists hag_greatest_factor_total_sufficiency_gcd. g = hag_divisor_total_sufficiency_gcd * hag_greatest_factor_total_sufficiency_gcd)) -> (exists hgcrt_mod_left_total_sufficiency_compatibility hgcrt_mod_right_total_sufficiency_compatibility. a + g * hgcrt_mod_left_total_sufficiency_compatibility = b + g * hgcrt_mod_right_total_sufficiency_compatibility) -> exists x. (((exists hgcrt_mod_left_total_sufficiency_solution_left hgcrt_mod_right_total_sufficiency_solution_left. x + m * hgcrt_mod_left_total_sufficiency_solution_left = a + m * hgcrt_mod_right_total_sufficiency_solution_left) /\ (exists hgcrt_mod_left_total_sufficiency_solution_right hgcrt_mod_right_total_sufficiency_solution_right. x + n * hgcrt_mod_left_total_sufficiency_solution_right = b + n * hgcrt_mod_right_total_sufficiency_solution_right)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b. ((((exists hag_left_factor_total_sufficiency_gcd. m = g * hag_left_factor_total_sufficiency_gcd) /\ (exists hag_right_factor_total_sufficiency_gcd. n = g * hag_right_factor_total_sufficiency_gcd)) /\ forall hag_divisor_total_sufficiency_gcd. (exists hag_common_left_total_sufficiency_gcd. m = hag_divisor_total_sufficiency_gcd * hag_common_left_total_sufficiency_gcd) -> (exists hag_common_right_total_sufficiency_gcd. n = hag_divisor_total_sufficiency_gcd * hag_common_right_total_sufficiency_gcd) -> exists hag_greatest_factor_total_sufficiency_gcd. g = hag_divisor_total_sufficiency_gcd * hag_greatest_factor_total_sufficiency_gcd)) -> (exists hgcrt_mod_left_total_sufficiency_compatibility hgcrt_mod_right_total_sufficiency_compatibility. a + g * hgcrt_mod_left_total_sufficiency_compatibility = b + g * hgcrt_mod_right_total_sufficiency_compatibility) -> exists x. (((exists hgcrt_mod_left_total_sufficiency_solution_left hgcrt_mod_right_total_sufficiency_solution_left. x + m * hgcrt_mod_left_total_sufficiency_solution_left = a + m * hgcrt_mod_right_total_sufficiency_solution_left) /\ (exists hgcrt_mod_left_total_sufficiency_solution_right hgcrt_mod_right_total_sufficiency_solution_right. x + n * hgcrt_mod_left_total_sufficiency_solution_right = b + n * hgcrt_mod_right_total_sufficiency_solution_right)))
use eq_decidable
use generalized_binary_crt_sufficient_zero_left
use generalized_binary_crt_sufficient_zero_right
use generalized_binary_crt_sufficient_nonzero
intro g
intro m
intro n
intro a
intro b
intro hg
intro hab
have hmzero : m = 0 \/ ~(m = 0)
specialize eq_decidable m
specialize eq_decidable 0
exact eq_decidable
cases hmzero
rewrite hmzero_left at hg
rewrite hmzero_left at hg
rewrite hmzero_left
rewrite hmzero_left
specialize generalized_binary_crt_sufficient_zero_left g
specialize generalized_binary_crt_sufficient_zero_left n
specialize generalized_binary_crt_sufficient_zero_left a
specialize generalized_binary_crt_sufficient_zero_left b
apply generalized_binary_crt_sufficient_zero_left
exact hg
exact hab
have hnzero : n = 0 \/ ~(n = 0)
specialize eq_decidable n
specialize eq_decidable 0
exact eq_decidable
cases hnzero
rewrite hnzero_left at hg
rewrite hnzero_left at hg
rewrite hnzero_left
rewrite hnzero_left
specialize generalized_binary_crt_sufficient_zero_right g
specialize generalized_binary_crt_sufficient_zero_right m
specialize generalized_binary_crt_sufficient_zero_right a
specialize generalized_binary_crt_sufficient_zero_right b
apply generalized_binary_crt_sufficient_zero_right
exact hg
exact hab
specialize generalized_binary_crt_sufficient_nonzero g
specialize generalized_binary_crt_sufficient_nonzero m
specialize generalized_binary_crt_sufficient_nonzero n
specialize generalized_binary_crt_sufficient_nonzero a
specialize generalized_binary_crt_sufficient_nonzero b
apply generalized_binary_crt_sufficient_nonzero
exact hmzero_right
exact hnzero_right
exact hg
exact hab
qed

Occurrences

11,240

Distinct objects

3,495

Reused references

168

Depth

78

Cuts

331

Certificate SHA-256

f22596043fc0…

generalized_binary_crt_solvable_iffGeneralized binary crt solvable iff Congruence & CRT checked

For arbitrary natural moduli, a binary CRT system is solvable exactly when its residues are congruent modulo a relational gcd.

Prerequisites

crt_common_solution_implies_gcd_compatible generalized_binary_crt_sufficient

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall g m n a b. ((((exists hag_left_factor_total_iff_gcd. m = g * hag_left_factor_total_iff_gcd) /\ (exists hag_right_factor_total_iff_gcd. n = g * hag_right_factor_total_iff_gcd)) /\ forall hag_divisor_total_iff_gcd. (exists hag_common_left_total_iff_gcd. m = hag_divisor_total_iff_gcd * hag_common_left_total_iff_gcd) -> (exists hag_common_right_total_iff_gcd. n = hag_divisor_total_iff_gcd * hag_common_right_total_iff_gcd) -> exists hag_greatest_factor_total_iff_gcd. g = hag_divisor_total_iff_gcd * hag_greatest_factor_total_iff_gcd)) -> (((exists x. (((exists hgcrt_mod_left_total_iff_forward_solution_left hgcrt_mod_right_total_iff_forward_solution_left. x + m * hgcrt_mod_left_total_iff_forward_solution_left = a + m * hgcrt_mod_right_total_iff_forward_solution_left) /\ (exists hgcrt_mod_left_total_iff_forward_solution_right hgcrt_mod_right_total_iff_forward_solution_right. x + n * hgcrt_mod_left_total_iff_forward_solution_right = b + n * hgcrt_mod_right_total_iff_forward_solution_right)))) -> (exists hgcrt_mod_left_total_iff_forward_compatibility hgcrt_mod_right_total_iff_forward_compatibility. a + g * hgcrt_mod_left_total_iff_forward_compatibility = b + g * hgcrt_mod_right_total_iff_forward_compatibility)) /\ ((exists hgcrt_mod_left_total_iff_reverse_compatibility hgcrt_mod_right_total_iff_reverse_compatibility. a + g * hgcrt_mod_left_total_iff_reverse_compatibility = b + g * hgcrt_mod_right_total_iff_reverse_compatibility) -> exists x. (((exists hgcrt_mod_left_total_iff_reverse_solution_left hgcrt_mod_right_total_iff_reverse_solution_left. x + m * hgcrt_mod_left_total_iff_reverse_solution_left = a + m * hgcrt_mod_right_total_iff_reverse_solution_left) /\ (exists hgcrt_mod_left_total_iff_reverse_solution_right hgcrt_mod_right_total_iff_reverse_solution_right. x + n * hgcrt_mod_left_total_iff_reverse_solution_right = b + n * hgcrt_mod_right_total_iff_reverse_solution_right)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b. ((((exists hag_left_factor_total_iff_gcd. m = g * hag_left_factor_total_iff_gcd) /\ (exists hag_right_factor_total_iff_gcd. n = g * hag_right_factor_total_iff_gcd)) /\ forall hag_divisor_total_iff_gcd. (exists hag_common_left_total_iff_gcd. m = hag_divisor_total_iff_gcd * hag_common_left_total_iff_gcd) -> (exists hag_common_right_total_iff_gcd. n = hag_divisor_total_iff_gcd * hag_common_right_total_iff_gcd) -> exists hag_greatest_factor_total_iff_gcd. g = hag_divisor_total_iff_gcd * hag_greatest_factor_total_iff_gcd)) -> (((exists x. (((exists hgcrt_mod_left_total_iff_forward_solution_left hgcrt_mod_right_total_iff_forward_solution_left. x + m * hgcrt_mod_left_total_iff_forward_solution_left = a + m * hgcrt_mod_right_total_iff_forward_solution_left) /\ (exists hgcrt_mod_left_total_iff_forward_solution_right hgcrt_mod_right_total_iff_forward_solution_right. x + n * hgcrt_mod_left_total_iff_forward_solution_right = b + n * hgcrt_mod_right_total_iff_forward_solution_right)))) -> (exists hgcrt_mod_left_total_iff_forward_compatibility hgcrt_mod_right_total_iff_forward_compatibility. a + g * hgcrt_mod_left_total_iff_forward_compatibility = b + g * hgcrt_mod_right_total_iff_forward_compatibility)) /\ ((exists hgcrt_mod_left_total_iff_reverse_compatibility hgcrt_mod_right_total_iff_reverse_compatibility. a + g * hgcrt_mod_left_total_iff_reverse_compatibility = b + g * hgcrt_mod_right_total_iff_reverse_compatibility) -> exists x. (((exists hgcrt_mod_left_total_iff_reverse_solution_left hgcrt_mod_right_total_iff_reverse_solution_left. x + m * hgcrt_mod_left_total_iff_reverse_solution_left = a + m * hgcrt_mod_right_total_iff_reverse_solution_left) /\ (exists hgcrt_mod_left_total_iff_reverse_solution_right hgcrt_mod_right_total_iff_reverse_solution_right. x + n * hgcrt_mod_left_total_iff_reverse_solution_right = b + n * hgcrt_mod_right_total_iff_reverse_solution_right)))))
use crt_common_solution_implies_gcd_compatible
use generalized_binary_crt_sufficient
intro g
intro m
intro n
intro a
intro b
intro hg
split
intro hsolution
cases hsolution
specialize crt_common_solution_implies_gcd_compatible g
specialize crt_common_solution_implies_gcd_compatible m
specialize crt_common_solution_implies_gcd_compatible n
specialize crt_common_solution_implies_gcd_compatible a
specialize crt_common_solution_implies_gcd_compatible b
specialize crt_common_solution_implies_gcd_compatible x
apply crt_common_solution_implies_gcd_compatible
exact hg
exact hsolution_witness
intro hcompat
specialize generalized_binary_crt_sufficient g
specialize generalized_binary_crt_sufficient m
specialize generalized_binary_crt_sufficient n
specialize generalized_binary_crt_sufficient a
specialize generalized_binary_crt_sufficient b
apply generalized_binary_crt_sufficient
exact hg
exact hcompat
qed

Occurrences

11,825

Distinct objects

3,658

Reused references

173

Depth

80

Cuts

347

Certificate SHA-256

f3767085f631…

mod_eq_ordered_gap_multipleMod eq ordered gap multiple Congruence & CRT checked

The directed gap between two congruent naturals is a multiple of the modulus.

Prerequisites

add_comm add_assoc add_left_cancel factor_difference

Used directly by

mod_eq_lcm_merge

Exact expanded PA statement

Copy statement

forall d k x y. k + x = y -> (exists hgcrt_mod_left_ordered_gap_assumption hgcrt_mod_right_ordered_gap_assumption. x + d * hgcrt_mod_left_ordered_gap_assumption = y + d * hgcrt_mod_right_ordered_gap_assumption) -> (exists hgcrt_divides_factor_ordered_gap_result. k = d * hgcrt_divides_factor_ordered_gap_result)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d k x y. k + x = y -> (exists hgcrt_mod_left_ordered_gap_assumption hgcrt_mod_right_ordered_gap_assumption. x + d * hgcrt_mod_left_ordered_gap_assumption = y + d * hgcrt_mod_right_ordered_gap_assumption) -> (exists hgcrt_divides_factor_ordered_gap_result. k = d * hgcrt_divides_factor_ordered_gap_result)
use add_comm
use add_assoc
use add_left_cancel
use factor_difference
intro d
intro k
intro x
intro y
intro hgap
intro hmod
cases hmod
cases hmod_witness
rewrite <- hgap at hmod_witness_witness
have hcancel : d * x1 = k + d * x2
specialize add_left_cancel x
specialize add_left_cancel (d * x1)
specialize add_left_cancel (k + d * x2)
apply add_left_cancel
trans (k + x) + d * x2
exact hmod_witness_witness
trans (x + k) + d * x2
congr
apply add_comm
refl
apply add_assoc
have hfactor : d * x1 = d * x2 + k
trans k + d * x2
exact hcancel
apply add_comm
specialize factor_difference d
specialize factor_difference x1
specialize factor_difference x2
specialize factor_difference k
apply factor_difference
exact hfactor
qed

Occurrences

558

Distinct objects

310

Reused references

16

Depth

30

Cuts

17

Certificate SHA-256

49758875c6ef…

mod_eq_lcm_mergeMod eq lcm merge Congruence & CRT checked

Congruence modulo both inputs merges to congruence modulo a relational lcm.

Prerequisites

le_total mod_eq_symm mod_eq_ordered_gap_multiple is_lcm_least mul_comm remainder_decomposition_to_mod_eq

Used directly by

mod_eq_lcm_iff_pair

Exact expanded PA statement

Copy statement

forall l m n x y. ((((exists hlcm_left_factor_merge. l = m * hlcm_left_factor_merge) /\ (exists hlcm_right_factor_merge. l = n * hlcm_right_factor_merge)) /\ forall hlcm_common_merge. (exists hlcm_left_common_merge. hlcm_common_merge = m * hlcm_left_common_merge) -> (exists hlcm_right_common_merge. hlcm_common_merge = n * hlcm_right_common_merge) -> exists hlcm_least_factor_merge. hlcm_common_merge = l * hlcm_least_factor_merge)) -> (exists hgcrt_mod_left_merge_m hgcrt_mod_right_merge_m. x + m * hgcrt_mod_left_merge_m = y + m * hgcrt_mod_right_merge_m) -> (exists hgcrt_mod_left_merge_n hgcrt_mod_right_merge_n. x + n * hgcrt_mod_left_merge_n = y + n * hgcrt_mod_right_merge_n) -> (exists hgcrt_mod_left_merge_l hgcrt_mod_right_merge_l. x + l * hgcrt_mod_left_merge_l = y + l * hgcrt_mod_right_merge_l)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m n x y. ((((exists hlcm_left_factor_merge. l = m * hlcm_left_factor_merge) /\ (exists hlcm_right_factor_merge. l = n * hlcm_right_factor_merge)) /\ forall hlcm_common_merge. (exists hlcm_left_common_merge. hlcm_common_merge = m * hlcm_left_common_merge) -> (exists hlcm_right_common_merge. hlcm_common_merge = n * hlcm_right_common_merge) -> exists hlcm_least_factor_merge. hlcm_common_merge = l * hlcm_least_factor_merge)) -> (exists hgcrt_mod_left_merge_m hgcrt_mod_right_merge_m. x + m * hgcrt_mod_left_merge_m = y + m * hgcrt_mod_right_merge_m) -> (exists hgcrt_mod_left_merge_n hgcrt_mod_right_merge_n. x + n * hgcrt_mod_left_merge_n = y + n * hgcrt_mod_right_merge_n) -> (exists hgcrt_mod_left_merge_l hgcrt_mod_right_merge_l. x + l * hgcrt_mod_left_merge_l = y + l * hgcrt_mod_right_merge_l)
use le_total
use mod_eq_symm
use mod_eq_ordered_gap_multiple
use is_lcm_least
use mul_comm
use remainder_decomposition_to_mod_eq
intro l
intro m
intro n
intro x
intro y
intro hl
intro hm
intro hn
have horder : x <= y \/ y <= x
specialize le_total x
specialize le_total y
exact le_total
cases horder
cases horder_left
have hmk : exists q. x1 = m * q
specialize mod_eq_ordered_gap_multiple m
specialize mod_eq_ordered_gap_multiple x1
specialize mod_eq_ordered_gap_multiple x
specialize mod_eq_ordered_gap_multiple y
apply mod_eq_ordered_gap_multiple
exact horder_left_witness
exact hm
have hnk : exists q. x1 = n * q
specialize mod_eq_ordered_gap_multiple n
specialize mod_eq_ordered_gap_multiple x1
specialize mod_eq_ordered_gap_multiple x
specialize mod_eq_ordered_gap_multiple y
apply mod_eq_ordered_gap_multiple
exact horder_left_witness
exact hn
have hlk : exists q. x1 = l * q
specialize is_lcm_least l
specialize is_lcm_least m
specialize is_lcm_least n
specialize is_lcm_least x1
apply is_lcm_least
exact hl
exact hmk
exact hnk
cases hlk
have hdecomp : y = x2 * l + x
trans x1 + x
symm
exact horder_left_witness
rewrite hlk_witness
congr
apply mul_comm
refl
have hyx : exists hgcrt_mod_left_merge_l_reverse hgcrt_mod_right_merge_l_reverse. y + l * hgcrt_mod_left_merge_l_reverse = x + l * hgcrt_mod_right_merge_l_reverse
specialize remainder_decomposition_to_mod_eq l
specialize remainder_decomposition_to_mod_eq y
specialize remainder_decomposition_to_mod_eq x2
specialize remainder_decomposition_to_mod_eq x
apply remainder_decomposition_to_mod_eq
exact hdecomp
specialize mod_eq_symm l
specialize mod_eq_symm y
specialize mod_eq_symm x
apply mod_eq_symm
exact hyx
cases horder_right
have hmyx : exists hgcrt_mod_left_merge_m_reverse hgcrt_mod_right_merge_m_reverse. y + m * hgcrt_mod_left_merge_m_reverse = x + m * hgcrt_mod_right_merge_m_reverse
specialize mod_eq_symm m
specialize mod_eq_symm x
specialize mod_eq_symm y
apply mod_eq_symm
exact hm
have hnyx : exists hgcrt_mod_left_merge_n_reverse hgcrt_mod_right_merge_n_reverse. y + n * hgcrt_mod_left_merge_n_reverse = x + n * hgcrt_mod_right_merge_n_reverse
specialize mod_eq_symm n
specialize mod_eq_symm x
specialize mod_eq_symm y
apply mod_eq_symm
exact hn
have hmk : exists q. x1 = m * q
specialize mod_eq_ordered_gap_multiple m
specialize mod_eq_ordered_gap_multiple x1
specialize mod_eq_ordered_gap_multiple y
specialize mod_eq_ordered_gap_multiple x
apply mod_eq_ordered_gap_multiple
exact horder_right_witness
exact hmyx
have hnk : exists q. x1 = n * q
specialize mod_eq_ordered_gap_multiple n
specialize mod_eq_ordered_gap_multiple x1
specialize mod_eq_ordered_gap_multiple y
specialize mod_eq_ordered_gap_multiple x
apply mod_eq_ordered_gap_multiple
exact horder_right_witness
exact hnyx
have hlk : exists q. x1 = l * q
specialize is_lcm_least l
specialize is_lcm_least m
specialize is_lcm_least n
specialize is_lcm_least x1
apply is_lcm_least
exact hl
exact hmk
exact hnk
cases hlk
have hdecomp : x = x2 * l + y
trans x1 + y
symm
exact horder_right_witness
rewrite hlk_witness
congr
apply mul_comm
refl
specialize remainder_decomposition_to_mod_eq l
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq x2
specialize remainder_decomposition_to_mod_eq y
apply remainder_decomposition_to_mod_eq
exact hdecomp
qed

Occurrences

1,315

Distinct objects

653

Reused references

33

Depth

33

Cuts

39

Certificate SHA-256

437802e4814c…

mod_eq_lcm_iff_pairMod eq lcm iff pair Congruence & CRT checked

Congruence modulo a relational lcm is equivalent to congruence modulo both inputs.

Prerequisites

is_lcm_multiple_left is_lcm_multiple_right mod_eq_of_mod_eq_multiple mod_eq_lcm_merge

Used directly by

crt_solution_class_iff_lcm

Exact expanded PA statement

Copy statement

forall l m n x y. ((((exists hlcm_left_factor_iff_pair. l = m * hlcm_left_factor_iff_pair) /\ (exists hlcm_right_factor_iff_pair. l = n * hlcm_right_factor_iff_pair)) /\ forall hlcm_common_iff_pair. (exists hlcm_left_common_iff_pair. hlcm_common_iff_pair = m * hlcm_left_common_iff_pair) -> (exists hlcm_right_common_iff_pair. hlcm_common_iff_pair = n * hlcm_right_common_iff_pair) -> exists hlcm_least_factor_iff_pair. hlcm_common_iff_pair = l * hlcm_least_factor_iff_pair)) -> (((exists hgcrt_mod_left_iff_l_forward hgcrt_mod_right_iff_l_forward. x + l * hgcrt_mod_left_iff_l_forward = y + l * hgcrt_mod_right_iff_l_forward) -> ((exists hgcrt_mod_left_iff_m_forward hgcrt_mod_right_iff_m_forward. x + m * hgcrt_mod_left_iff_m_forward = y + m * hgcrt_mod_right_iff_m_forward) /\ (exists hgcrt_mod_left_iff_n_forward hgcrt_mod_right_iff_n_forward. x + n * hgcrt_mod_left_iff_n_forward = y + n * hgcrt_mod_right_iff_n_forward))) /\ (((exists hgcrt_mod_left_iff_m_reverse hgcrt_mod_right_iff_m_reverse. x + m * hgcrt_mod_left_iff_m_reverse = y + m * hgcrt_mod_right_iff_m_reverse) /\ (exists hgcrt_mod_left_iff_n_reverse hgcrt_mod_right_iff_n_reverse. x + n * hgcrt_mod_left_iff_n_reverse = y + n * hgcrt_mod_right_iff_n_reverse)) -> (exists hgcrt_mod_left_iff_l_reverse hgcrt_mod_right_iff_l_reverse. x + l * hgcrt_mod_left_iff_l_reverse = y + l * hgcrt_mod_right_iff_l_reverse)))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m n x y. ((((exists hlcm_left_factor_iff_pair. l = m * hlcm_left_factor_iff_pair) /\ (exists hlcm_right_factor_iff_pair. l = n * hlcm_right_factor_iff_pair)) /\ forall hlcm_common_iff_pair. (exists hlcm_left_common_iff_pair. hlcm_common_iff_pair = m * hlcm_left_common_iff_pair) -> (exists hlcm_right_common_iff_pair. hlcm_common_iff_pair = n * hlcm_right_common_iff_pair) -> exists hlcm_least_factor_iff_pair. hlcm_common_iff_pair = l * hlcm_least_factor_iff_pair)) -> (((exists hgcrt_mod_left_iff_l_forward hgcrt_mod_right_iff_l_forward. x + l * hgcrt_mod_left_iff_l_forward = y + l * hgcrt_mod_right_iff_l_forward) -> ((exists hgcrt_mod_left_iff_m_forward hgcrt_mod_right_iff_m_forward. x + m * hgcrt_mod_left_iff_m_forward = y + m * hgcrt_mod_right_iff_m_forward) /\ (exists hgcrt_mod_left_iff_n_forward hgcrt_mod_right_iff_n_forward. x + n * hgcrt_mod_left_iff_n_forward = y + n * hgcrt_mod_right_iff_n_forward))) /\ (((exists hgcrt_mod_left_iff_m_reverse hgcrt_mod_right_iff_m_reverse. x + m * hgcrt_mod_left_iff_m_reverse = y + m * hgcrt_mod_right_iff_m_reverse) /\ (exists hgcrt_mod_left_iff_n_reverse hgcrt_mod_right_iff_n_reverse. x + n * hgcrt_mod_left_iff_n_reverse = y + n * hgcrt_mod_right_iff_n_reverse)) -> (exists hgcrt_mod_left_iff_l_reverse hgcrt_mod_right_iff_l_reverse. x + l * hgcrt_mod_left_iff_l_reverse = y + l * hgcrt_mod_right_iff_l_reverse)))
use is_lcm_multiple_left
use is_lcm_multiple_right
use mod_eq_of_mod_eq_multiple
use mod_eq_lcm_merge
intro l
intro m
intro n
intro x
intro y
intro hl
split
intro hxy
have hml : exists q. l = m * q
specialize is_lcm_multiple_left l
specialize is_lcm_multiple_left m
specialize is_lcm_multiple_left n
apply is_lcm_multiple_left
exact hl
have hnl : exists q. l = n * q
specialize is_lcm_multiple_right l
specialize is_lcm_multiple_right m
specialize is_lcm_multiple_right n
apply is_lcm_multiple_right
exact hl
split
specialize mod_eq_of_mod_eq_multiple m
specialize mod_eq_of_mod_eq_multiple l
specialize mod_eq_of_mod_eq_multiple x
specialize mod_eq_of_mod_eq_multiple y
apply mod_eq_of_mod_eq_multiple
exact hml
exact hxy
specialize mod_eq_of_mod_eq_multiple n
specialize mod_eq_of_mod_eq_multiple l
specialize mod_eq_of_mod_eq_multiple x
specialize mod_eq_of_mod_eq_multiple y
apply mod_eq_of_mod_eq_multiple
exact hnl
exact hxy
intro hpair
cases hpair
specialize mod_eq_lcm_merge l
specialize mod_eq_lcm_merge m
specialize mod_eq_lcm_merge n
specialize mod_eq_lcm_merge x
specialize mod_eq_lcm_merge y
apply mod_eq_lcm_merge
exact hl
exact hpair_left
exact hpair_right
qed

Occurrences

1,570

Distinct objects

864

Reused references

45

Depth

37

Cuts

46

Certificate SHA-256

8e89d2f01135…

crt_solution_class_iff_lcmCrt solution class iff lcm Congruence & CRT checked

Relative to one fixed common solution, all solutions are exactly its relational-lcm congruence class.

Prerequisites

crt_solution_pair_congruent mod_eq_lcm_iff_pair mod_eq_trans

Used directly by

crt_solution_unique_lcm_zero crt_solution_canonical_remainder_nonzero

Exact expanded PA statement

Copy statement

forall l m n a b x y. ((((exists hlcm_left_factor_solution_class. l = m * hlcm_left_factor_solution_class) /\ (exists hlcm_right_factor_solution_class. l = n * hlcm_right_factor_solution_class)) /\ forall hlcm_common_solution_class. (exists hlcm_left_common_solution_class. hlcm_common_solution_class = m * hlcm_left_common_solution_class) -> (exists hlcm_right_common_solution_class. hlcm_common_solution_class = n * hlcm_right_common_solution_class) -> exists hlcm_least_factor_solution_class. hlcm_common_solution_class = l * hlcm_least_factor_solution_class)) -> (((exists hgcrt_mod_left_class_fixed_left hgcrt_mod_right_class_fixed_left. x + m * hgcrt_mod_left_class_fixed_left = a + m * hgcrt_mod_right_class_fixed_left) /\ (exists hgcrt_mod_left_class_fixed_right hgcrt_mod_right_class_fixed_right. x + n * hgcrt_mod_left_class_fixed_right = b + n * hgcrt_mod_right_class_fixed_right))) -> (((((exists hgcrt_mod_left_class_candidate_forward_left hgcrt_mod_right_class_candidate_forward_left. y + m * hgcrt_mod_left_class_candidate_forward_left = a + m * hgcrt_mod_right_class_candidate_forward_left) /\ (exists hgcrt_mod_left_class_candidate_forward_right hgcrt_mod_right_class_candidate_forward_right. y + n * hgcrt_mod_left_class_candidate_forward_right = b + n * hgcrt_mod_right_class_candidate_forward_right))) -> (exists hgcrt_mod_left_class_l_forward hgcrt_mod_right_class_l_forward. y + l * hgcrt_mod_left_class_l_forward = x + l * hgcrt_mod_right_class_l_forward)) /\ ((exists hgcrt_mod_left_class_l_reverse hgcrt_mod_right_class_l_reverse. y + l * hgcrt_mod_left_class_l_reverse = x + l * hgcrt_mod_right_class_l_reverse) -> (((exists hgcrt_mod_left_class_candidate_reverse_left hgcrt_mod_right_class_candidate_reverse_left. y + m * hgcrt_mod_left_class_candidate_reverse_left = a + m * hgcrt_mod_right_class_candidate_reverse_left) /\ (exists hgcrt_mod_left_class_candidate_reverse_right hgcrt_mod_right_class_candidate_reverse_right. y + n * hgcrt_mod_left_class_candidate_reverse_right = b + n * hgcrt_mod_right_class_candidate_reverse_right)))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m n a b x y. ((((exists hlcm_left_factor_solution_class. l = m * hlcm_left_factor_solution_class) /\ (exists hlcm_right_factor_solution_class. l = n * hlcm_right_factor_solution_class)) /\ forall hlcm_common_solution_class. (exists hlcm_left_common_solution_class. hlcm_common_solution_class = m * hlcm_left_common_solution_class) -> (exists hlcm_right_common_solution_class. hlcm_common_solution_class = n * hlcm_right_common_solution_class) -> exists hlcm_least_factor_solution_class. hlcm_common_solution_class = l * hlcm_least_factor_solution_class)) -> (((exists hgcrt_mod_left_class_fixed_left hgcrt_mod_right_class_fixed_left. x + m * hgcrt_mod_left_class_fixed_left = a + m * hgcrt_mod_right_class_fixed_left) /\ (exists hgcrt_mod_left_class_fixed_right hgcrt_mod_right_class_fixed_right. x + n * hgcrt_mod_left_class_fixed_right = b + n * hgcrt_mod_right_class_fixed_right))) -> (((((exists hgcrt_mod_left_class_candidate_forward_left hgcrt_mod_right_class_candidate_forward_left. y + m * hgcrt_mod_left_class_candidate_forward_left = a + m * hgcrt_mod_right_class_candidate_forward_left) /\ (exists hgcrt_mod_left_class_candidate_forward_right hgcrt_mod_right_class_candidate_forward_right. y + n * hgcrt_mod_left_class_candidate_forward_right = b + n * hgcrt_mod_right_class_candidate_forward_right))) -> (exists hgcrt_mod_left_class_l_forward hgcrt_mod_right_class_l_forward. y + l * hgcrt_mod_left_class_l_forward = x + l * hgcrt_mod_right_class_l_forward)) /\ ((exists hgcrt_mod_left_class_l_reverse hgcrt_mod_right_class_l_reverse. y + l * hgcrt_mod_left_class_l_reverse = x + l * hgcrt_mod_right_class_l_reverse) -> (((exists hgcrt_mod_left_class_candidate_reverse_left hgcrt_mod_right_class_candidate_reverse_left. y + m * hgcrt_mod_left_class_candidate_reverse_left = a + m * hgcrt_mod_right_class_candidate_reverse_left) /\ (exists hgcrt_mod_left_class_candidate_reverse_right hgcrt_mod_right_class_candidate_reverse_right. y + n * hgcrt_mod_left_class_candidate_reverse_right = b + n * hgcrt_mod_right_class_candidate_reverse_right)))))
use crt_solution_pair_congruent
use mod_eq_lcm_iff_pair
use mod_eq_trans
intro l
intro m
intro n
intro a
intro b
intro x
intro y
intro hl
intro hx
split
intro hy
have hpair : ((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n))
specialize crt_solution_pair_congruent m
specialize crt_solution_pair_congruent n
specialize crt_solution_pair_congruent a
specialize crt_solution_pair_congruent b
specialize crt_solution_pair_congruent y
specialize crt_solution_pair_congruent x
apply crt_solution_pair_congruent
exact hy
exact hx
have hiff : (((exists hgcrt_mod_left_class_l_aux hgcrt_mod_right_class_l_aux. y + l * hgcrt_mod_left_class_l_aux = x + l * hgcrt_mod_right_class_l_aux) -> ((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n))) /\ (((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n)) -> (exists hgcrt_mod_left_class_l_aux hgcrt_mod_right_class_l_aux. y + l * hgcrt_mod_left_class_l_aux = x + l * hgcrt_mod_right_class_l_aux)))
specialize mod_eq_lcm_iff_pair l
specialize mod_eq_lcm_iff_pair m
specialize mod_eq_lcm_iff_pair n
specialize mod_eq_lcm_iff_pair y
specialize mod_eq_lcm_iff_pair x
apply mod_eq_lcm_iff_pair
exact hl
cases hiff
apply hiff_right
exact hpair
intro hlyx
have hiff2 : (((exists hgcrt_mod_left_class_l_aux hgcrt_mod_right_class_l_aux. y + l * hgcrt_mod_left_class_l_aux = x + l * hgcrt_mod_right_class_l_aux) -> ((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n))) /\ (((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n)) -> (exists hgcrt_mod_left_class_l_aux hgcrt_mod_right_class_l_aux. y + l * hgcrt_mod_left_class_l_aux = x + l * hgcrt_mod_right_class_l_aux)))
specialize mod_eq_lcm_iff_pair l
specialize mod_eq_lcm_iff_pair m
specialize mod_eq_lcm_iff_pair n
specialize mod_eq_lcm_iff_pair y
specialize mod_eq_lcm_iff_pair x
apply mod_eq_lcm_iff_pair
exact hl
cases hiff2
have hpair2 : ((exists hgcrt_mod_left_class_pair_m hgcrt_mod_right_class_pair_m. y + m * hgcrt_mod_left_class_pair_m = x + m * hgcrt_mod_right_class_pair_m) /\ (exists hgcrt_mod_left_class_pair_n hgcrt_mod_right_class_pair_n. y + n * hgcrt_mod_left_class_pair_n = x + n * hgcrt_mod_right_class_pair_n))
apply hiff2_left
exact hlyx
cases hpair2
cases hx
split
specialize mod_eq_trans m
specialize mod_eq_trans y
specialize mod_eq_trans x
specialize mod_eq_trans a
apply mod_eq_trans
exact hpair2_left
exact hx_left
specialize mod_eq_trans n
specialize mod_eq_trans y
specialize mod_eq_trans x
specialize mod_eq_trans b
apply mod_eq_trans
exact hpair2_right
exact hx_right
qed

Occurrences

2,208

Distinct objects

1,055

Reused references

50

Depth

39

Cuts

63

Certificate SHA-256

7917de5c171d…

crt_solution_unique_lcm_zeroCrt solution unique lcm zero Congruence & CRT checked

At relational lcm zero, every common solution equals a fixed common solution.

Prerequisites

crt_solution_class_iff_lcm mod_eq_zero_iff_eq

Used directly by

generalized_binary_crt_canonical_boundary

Exact expanded PA statement

Copy statement

forall l m n a b x y. l = 0 -> ((((exists hlcm_left_factor_unique_zero. l = m * hlcm_left_factor_unique_zero) /\ (exists hlcm_right_factor_unique_zero. l = n * hlcm_right_factor_unique_zero)) /\ forall hlcm_common_unique_zero. (exists hlcm_left_common_unique_zero. hlcm_common_unique_zero = m * hlcm_left_common_unique_zero) -> (exists hlcm_right_common_unique_zero. hlcm_common_unique_zero = n * hlcm_right_common_unique_zero) -> exists hlcm_least_factor_unique_zero. hlcm_common_unique_zero = l * hlcm_least_factor_unique_zero)) -> (((exists hgcrt_mod_left_unique_zero_fixed_left hgcrt_mod_right_unique_zero_fixed_left. x + m * hgcrt_mod_left_unique_zero_fixed_left = a + m * hgcrt_mod_right_unique_zero_fixed_left) /\ (exists hgcrt_mod_left_unique_zero_fixed_right hgcrt_mod_right_unique_zero_fixed_right. x + n * hgcrt_mod_left_unique_zero_fixed_right = b + n * hgcrt_mod_right_unique_zero_fixed_right))) -> (((exists hgcrt_mod_left_unique_zero_candidate_left hgcrt_mod_right_unique_zero_candidate_left. y + m * hgcrt_mod_left_unique_zero_candidate_left = a + m * hgcrt_mod_right_unique_zero_candidate_left) /\ (exists hgcrt_mod_left_unique_zero_candidate_right hgcrt_mod_right_unique_zero_candidate_right. y + n * hgcrt_mod_left_unique_zero_candidate_right = b + n * hgcrt_mod_right_unique_zero_candidate_right))) -> y = x

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m n a b x y. l = 0 -> ((((exists hlcm_left_factor_unique_zero. l = m * hlcm_left_factor_unique_zero) /\ (exists hlcm_right_factor_unique_zero. l = n * hlcm_right_factor_unique_zero)) /\ forall hlcm_common_unique_zero. (exists hlcm_left_common_unique_zero. hlcm_common_unique_zero = m * hlcm_left_common_unique_zero) -> (exists hlcm_right_common_unique_zero. hlcm_common_unique_zero = n * hlcm_right_common_unique_zero) -> exists hlcm_least_factor_unique_zero. hlcm_common_unique_zero = l * hlcm_least_factor_unique_zero)) -> (((exists hgcrt_mod_left_unique_zero_fixed_left hgcrt_mod_right_unique_zero_fixed_left. x + m * hgcrt_mod_left_unique_zero_fixed_left = a + m * hgcrt_mod_right_unique_zero_fixed_left) /\ (exists hgcrt_mod_left_unique_zero_fixed_right hgcrt_mod_right_unique_zero_fixed_right. x + n * hgcrt_mod_left_unique_zero_fixed_right = b + n * hgcrt_mod_right_unique_zero_fixed_right))) -> (((exists hgcrt_mod_left_unique_zero_candidate_left hgcrt_mod_right_unique_zero_candidate_left. y + m * hgcrt_mod_left_unique_zero_candidate_left = a + m * hgcrt_mod_right_unique_zero_candidate_left) /\ (exists hgcrt_mod_left_unique_zero_candidate_right hgcrt_mod_right_unique_zero_candidate_right. y + n * hgcrt_mod_left_unique_zero_candidate_right = b + n * hgcrt_mod_right_unique_zero_candidate_right))) -> y = x
use crt_solution_class_iff_lcm
use mod_eq_zero_iff_eq
intro l
intro m
intro n
intro a
intro b
intro x
intro y
intro hl
intro hlcm
intro hx
intro hy
have hiff : (((((exists hgcrt_mod_left_unique_zero_iff_forward_left hgcrt_mod_right_unique_zero_iff_forward_left. y + m * hgcrt_mod_left_unique_zero_iff_forward_left = a + m * hgcrt_mod_right_unique_zero_iff_forward_left) /\ (exists hgcrt_mod_left_unique_zero_iff_forward_right hgcrt_mod_right_unique_zero_iff_forward_right. y + n * hgcrt_mod_left_unique_zero_iff_forward_right = b + n * hgcrt_mod_right_unique_zero_iff_forward_right))) -> (exists hgcrt_mod_left_unique_zero_mod_forward hgcrt_mod_right_unique_zero_mod_forward. y + l * hgcrt_mod_left_unique_zero_mod_forward = x + l * hgcrt_mod_right_unique_zero_mod_forward)) /\ ((exists hgcrt_mod_left_unique_zero_mod_reverse hgcrt_mod_right_unique_zero_mod_reverse. y + l * hgcrt_mod_left_unique_zero_mod_reverse = x + l * hgcrt_mod_right_unique_zero_mod_reverse) -> (((exists hgcrt_mod_left_unique_zero_iff_reverse_left hgcrt_mod_right_unique_zero_iff_reverse_left. y + m * hgcrt_mod_left_unique_zero_iff_reverse_left = a + m * hgcrt_mod_right_unique_zero_iff_reverse_left) /\ (exists hgcrt_mod_left_unique_zero_iff_reverse_right hgcrt_mod_right_unique_zero_iff_reverse_right. y + n * hgcrt_mod_left_unique_zero_iff_reverse_right = b + n * hgcrt_mod_right_unique_zero_iff_reverse_right)))))
specialize crt_solution_class_iff_lcm l
specialize crt_solution_class_iff_lcm m
specialize crt_solution_class_iff_lcm n
specialize crt_solution_class_iff_lcm a
specialize crt_solution_class_iff_lcm b
specialize crt_solution_class_iff_lcm x
specialize crt_solution_class_iff_lcm y
apply crt_solution_class_iff_lcm
exact hlcm
exact hx
cases hiff
have hmod : exists hgcrt_mod_left_unique_zero_mod_result hgcrt_mod_right_unique_zero_mod_result. y + l * hgcrt_mod_left_unique_zero_mod_result = x + l * hgcrt_mod_right_unique_zero_mod_result
apply hiff_left
exact hy
rewrite hl at hmod
rewrite hl at hmod
specialize mod_eq_zero_iff_eq y
specialize mod_eq_zero_iff_eq x
cases mod_eq_zero_iff_eq
apply mod_eq_zero_iff_eq_left
exact hmod
qed

Occurrences

2,300

Distinct objects

1,126

Reused references

51

Depth

40

Cuts

66

Certificate SHA-256

05a457c953d5…

crt_solution_canonical_remainder_nonzeroCrt solution canonical remainder nonzero Congruence & CRT checked

At nonzero relational lcm, every solvable binary CRT system has a unique solution below the lcm.

Prerequisites

division_remainder_exists mul_comm remainder_decomposition_to_mod_eq mod_eq_symm crt_solution_class_iff_lcm mod_eq_bounded_unique

Used directly by

generalized_binary_crt_canonical_boundary

Exact expanded PA statement

Copy statement

forall l m n a b x. ~(l = 0) -> ((((exists hlcm_left_factor_canonical_nonzero. l = m * hlcm_left_factor_canonical_nonzero) /\ (exists hlcm_right_factor_canonical_nonzero. l = n * hlcm_right_factor_canonical_nonzero)) /\ forall hlcm_common_canonical_nonzero. (exists hlcm_left_common_canonical_nonzero. hlcm_common_canonical_nonzero = m * hlcm_left_common_canonical_nonzero) -> (exists hlcm_right_common_canonical_nonzero. hlcm_common_canonical_nonzero = n * hlcm_right_common_canonical_nonzero) -> exists hlcm_least_factor_canonical_nonzero. hlcm_common_canonical_nonzero = l * hlcm_least_factor_canonical_nonzero)) -> (((exists hgcrt_mod_left_canonical_nonzero_fixed_left hgcrt_mod_right_canonical_nonzero_fixed_left. x + m * hgcrt_mod_left_canonical_nonzero_fixed_left = a + m * hgcrt_mod_right_canonical_nonzero_fixed_left) /\ (exists hgcrt_mod_left_canonical_nonzero_fixed_right hgcrt_mod_right_canonical_nonzero_fixed_right. x + n * hgcrt_mod_left_canonical_nonzero_fixed_right = b + n * hgcrt_mod_right_canonical_nonzero_fixed_right))) -> exists r. (((exists hgcrt_below_gap_canonical_nonzero_result. hgcrt_below_gap_canonical_nonzero_result + S r = l) /\ (((exists hgcrt_mod_left_canonical_nonzero_result_left hgcrt_mod_right_canonical_nonzero_result_left. r + m * hgcrt_mod_left_canonical_nonzero_result_left = a + m * hgcrt_mod_right_canonical_nonzero_result_left) /\ (exists hgcrt_mod_left_canonical_nonzero_result_right hgcrt_mod_right_canonical_nonzero_result_right. r + n * hgcrt_mod_left_canonical_nonzero_result_right = b + n * hgcrt_mod_right_canonical_nonzero_result_right)))) /\ ((exists hgcrt_mod_left_canonical_nonzero_result hgcrt_mod_right_canonical_nonzero_result. r + l * hgcrt_mod_left_canonical_nonzero_result = x + l * hgcrt_mod_right_canonical_nonzero_result) /\ forall s. (exists hgcrt_below_gap_canonical_nonzero_comparison. hgcrt_below_gap_canonical_nonzero_comparison + S s = l) -> (((exists hgcrt_mod_left_canonical_nonzero_comparison_left hgcrt_mod_right_canonical_nonzero_comparison_left. s + m * hgcrt_mod_left_canonical_nonzero_comparison_left = a + m * hgcrt_mod_right_canonical_nonzero_comparison_left) /\ (exists hgcrt_mod_left_canonical_nonzero_comparison_right hgcrt_mod_right_canonical_nonzero_comparison_right. s + n * hgcrt_mod_left_canonical_nonzero_comparison_right = b + n * hgcrt_mod_right_canonical_nonzero_comparison_right))) -> s = r))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall l m n a b x. ~(l = 0) -> ((((exists hlcm_left_factor_canonical_nonzero. l = m * hlcm_left_factor_canonical_nonzero) /\ (exists hlcm_right_factor_canonical_nonzero. l = n * hlcm_right_factor_canonical_nonzero)) /\ forall hlcm_common_canonical_nonzero. (exists hlcm_left_common_canonical_nonzero. hlcm_common_canonical_nonzero = m * hlcm_left_common_canonical_nonzero) -> (exists hlcm_right_common_canonical_nonzero. hlcm_common_canonical_nonzero = n * hlcm_right_common_canonical_nonzero) -> exists hlcm_least_factor_canonical_nonzero. hlcm_common_canonical_nonzero = l * hlcm_least_factor_canonical_nonzero)) -> (((exists hgcrt_mod_left_canonical_nonzero_fixed_left hgcrt_mod_right_canonical_nonzero_fixed_left. x + m * hgcrt_mod_left_canonical_nonzero_fixed_left = a + m * hgcrt_mod_right_canonical_nonzero_fixed_left) /\ (exists hgcrt_mod_left_canonical_nonzero_fixed_right hgcrt_mod_right_canonical_nonzero_fixed_right. x + n * hgcrt_mod_left_canonical_nonzero_fixed_right = b + n * hgcrt_mod_right_canonical_nonzero_fixed_right))) -> exists r. (((exists hgcrt_below_gap_canonical_nonzero_result. hgcrt_below_gap_canonical_nonzero_result + S r = l) /\ (((exists hgcrt_mod_left_canonical_nonzero_result_left hgcrt_mod_right_canonical_nonzero_result_left. r + m * hgcrt_mod_left_canonical_nonzero_result_left = a + m * hgcrt_mod_right_canonical_nonzero_result_left) /\ (exists hgcrt_mod_left_canonical_nonzero_result_right hgcrt_mod_right_canonical_nonzero_result_right. r + n * hgcrt_mod_left_canonical_nonzero_result_right = b + n * hgcrt_mod_right_canonical_nonzero_result_right)))) /\ ((exists hgcrt_mod_left_canonical_nonzero_result hgcrt_mod_right_canonical_nonzero_result. r + l * hgcrt_mod_left_canonical_nonzero_result = x + l * hgcrt_mod_right_canonical_nonzero_result) /\ forall s. (exists hgcrt_below_gap_canonical_nonzero_comparison. hgcrt_below_gap_canonical_nonzero_comparison + S s = l) -> (((exists hgcrt_mod_left_canonical_nonzero_comparison_left hgcrt_mod_right_canonical_nonzero_comparison_left. s + m * hgcrt_mod_left_canonical_nonzero_comparison_left = a + m * hgcrt_mod_right_canonical_nonzero_comparison_left) /\ (exists hgcrt_mod_left_canonical_nonzero_comparison_right hgcrt_mod_right_canonical_nonzero_comparison_right. s + n * hgcrt_mod_left_canonical_nonzero_comparison_right = b + n * hgcrt_mod_right_canonical_nonzero_comparison_right))) -> s = r))
use division_remainder_exists
use mul_comm
use remainder_decomposition_to_mod_eq
use mod_eq_symm
use crt_solution_class_iff_lcm
use mod_eq_bounded_unique
intro l
intro m
intro n
intro a
intro b
intro x
intro hl
intro hlcm
intro hx
have hdivision : exists q r. x = l * q + r /\ exists gap. gap + S r = l
specialize division_remainder_exists l
specialize division_remainder_exists x
apply division_remainder_exists
exact hl
cases hdivision
cases hdivision_witness
cases hdivision_witness_witness
have hxdecomp : x = x1 * l + x2
trans l * x1 + x2
exact hdivision_witness_witness_left
congr
apply mul_comm
refl
have hxx2 : exists hgcrt_mod_left_canonical_runtime_x_x2 hgcrt_mod_right_canonical_runtime_x_x2. x + l * hgcrt_mod_left_canonical_runtime_x_x2 = x2 + l * hgcrt_mod_right_canonical_runtime_x_x2
specialize remainder_decomposition_to_mod_eq l
specialize remainder_decomposition_to_mod_eq x
specialize remainder_decomposition_to_mod_eq x1
specialize remainder_decomposition_to_mod_eq x2
apply remainder_decomposition_to_mod_eq
exact hxdecomp
have hx2x : exists hgcrt_mod_left_canonical_runtime_x2_x hgcrt_mod_right_canonical_runtime_x2_x. x2 + l * hgcrt_mod_left_canonical_runtime_x2_x = x + l * hgcrt_mod_right_canonical_runtime_x2_x
specialize mod_eq_symm l
specialize mod_eq_symm x
specialize mod_eq_symm x2
apply mod_eq_symm
exact hxx2
have hclass : (((((exists hgcrt_mod_left_canonical_runtime_class_forward_left hgcrt_mod_right_canonical_runtime_class_forward_left. x2 + m * hgcrt_mod_left_canonical_runtime_class_forward_left = a + m * hgcrt_mod_right_canonical_runtime_class_forward_left) /\ (exists hgcrt_mod_left_canonical_runtime_class_forward_right hgcrt_mod_right_canonical_runtime_class_forward_right. x2 + n * hgcrt_mod_left_canonical_runtime_class_forward_right = b + n * hgcrt_mod_right_canonical_runtime_class_forward_right))) -> (exists hgcrt_mod_left_canonical_runtime_class_forward hgcrt_mod_right_canonical_runtime_class_forward. x2 + l * hgcrt_mod_left_canonical_runtime_class_forward = x + l * hgcrt_mod_right_canonical_runtime_class_forward)) /\ ((exists hgcrt_mod_left_canonical_runtime_class_reverse hgcrt_mod_right_canonical_runtime_class_reverse. x2 + l * hgcrt_mod_left_canonical_runtime_class_reverse = x + l * hgcrt_mod_right_canonical_runtime_class_reverse) -> (((exists hgcrt_mod_left_canonical_runtime_class_reverse_left hgcrt_mod_right_canonical_runtime_class_reverse_left. x2 + m * hgcrt_mod_left_canonical_runtime_class_reverse_left = a + m * hgcrt_mod_right_canonical_runtime_class_reverse_left) /\ (exists hgcrt_mod_left_canonical_runtime_class_reverse_right hgcrt_mod_right_canonical_runtime_class_reverse_right. x2 + n * hgcrt_mod_left_canonical_runtime_class_reverse_right = b + n * hgcrt_mod_right_canonical_runtime_class_reverse_right)))))
specialize crt_solution_class_iff_lcm l
specialize crt_solution_class_iff_lcm m
specialize crt_solution_class_iff_lcm n
specialize crt_solution_class_iff_lcm a
specialize crt_solution_class_iff_lcm b
specialize crt_solution_class_iff_lcm x
specialize crt_solution_class_iff_lcm x2
apply crt_solution_class_iff_lcm
exact hlcm
exact hx
cases hclass
have hrsol : ((exists hgcrt_mod_left_canonical_runtime_result_left hgcrt_mod_right_canonical_runtime_result_left. x2 + m * hgcrt_mod_left_canonical_runtime_result_left = a + m * hgcrt_mod_right_canonical_runtime_result_left) /\ (exists hgcrt_mod_left_canonical_runtime_result_right hgcrt_mod_right_canonical_runtime_result_right. x2 + n * hgcrt_mod_left_canonical_runtime_result_right = b + n * hgcrt_mod_right_canonical_runtime_result_right))
apply hclass_right
exact hx2x
exists x2
split
split
exact hdivision_witness_witness_right
exact hrsol
split
exact hx2x
intro s
intro hsbound
intro hssol
have hclasss : (((((exists hgcrt_mod_left_canonical_runtime_comparison_forward_left hgcrt_mod_right_canonical_runtime_comparison_forward_left. s + m * hgcrt_mod_left_canonical_runtime_comparison_forward_left = a + m * hgcrt_mod_right_canonical_runtime_comparison_forward_left) /\ (exists hgcrt_mod_left_canonical_runtime_comparison_forward_right hgcrt_mod_right_canonical_runtime_comparison_forward_right. s + n * hgcrt_mod_left_canonical_runtime_comparison_forward_right = b + n * hgcrt_mod_right_canonical_runtime_comparison_forward_right))) -> (exists hgcrt_mod_left_canonical_runtime_comparison_forward hgcrt_mod_right_canonical_runtime_comparison_forward. s + l * hgcrt_mod_left_canonical_runtime_comparison_forward = x2 + l * hgcrt_mod_right_canonical_runtime_comparison_forward)) /\ ((exists hgcrt_mod_left_canonical_runtime_comparison_reverse hgcrt_mod_right_canonical_runtime_comparison_reverse. s + l * hgcrt_mod_left_canonical_runtime_comparison_reverse = x2 + l * hgcrt_mod_right_canonical_runtime_comparison_reverse) -> (((exists hgcrt_mod_left_canonical_runtime_comparison_reverse_left hgcrt_mod_right_canonical_runtime_comparison_reverse_left. s + m * hgcrt_mod_left_canonical_runtime_comparison_reverse_left = a + m * hgcrt_mod_right_canonical_runtime_comparison_reverse_left) /\ (exists hgcrt_mod_left_canonical_runtime_comparison_reverse_right hgcrt_mod_right_canonical_runtime_comparison_reverse_right. s + n * hgcrt_mod_left_canonical_runtime_comparison_reverse_right = b + n * hgcrt_mod_right_canonical_runtime_comparison_reverse_right)))))
specialize crt_solution_class_iff_lcm l
specialize crt_solution_class_iff_lcm m
specialize crt_solution_class_iff_lcm n
specialize crt_solution_class_iff_lcm a
specialize crt_solution_class_iff_lcm b
specialize crt_solution_class_iff_lcm x2
specialize crt_solution_class_iff_lcm s
apply crt_solution_class_iff_lcm
exact hlcm
exact hrsol
cases hclasss
have hsmod : exists hgcrt_mod_left_canonical_runtime_comparison_result hgcrt_mod_right_canonical_runtime_comparison_result. s + l * hgcrt_mod_left_canonical_runtime_comparison_result = x2 + l * hgcrt_mod_right_canonical_runtime_comparison_result
apply hclasss_left
exact hssol
specialize mod_eq_bounded_unique l
specialize mod_eq_bounded_unique s
specialize mod_eq_bounded_unique x2
apply mod_eq_bounded_unique
exact hsbound
exact hdivision_witness_witness_right
exact hsmod
qed

Occurrences

4,086

Distinct objects

1,668

Reused references

79

Depth

65

Cuts

116

Certificate SHA-256

c3c9f2d9d2cb…

generalized_binary_crt_canonical_boundaryGeneralized binary crt canonical boundary Congruence & CRT checked

Every compatible binary CRT system has the correct canonical boundary: exact uniqueness at lcm zero or one unique bounded representative at nonzero lcm.

Prerequisites

eq_decidable generalized_binary_crt_sufficient crt_solution_unique_lcm_zero crt_solution_canonical_remainder_nonzero

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall g l m n a b. ((((exists hag_left_factor_canonical_boundary. m = g * hag_left_factor_canonical_boundary) /\ (exists hag_right_factor_canonical_boundary. n = g * hag_right_factor_canonical_boundary)) /\ forall hag_divisor_canonical_boundary. (exists hag_common_left_canonical_boundary. m = hag_divisor_canonical_boundary * hag_common_left_canonical_boundary) -> (exists hag_common_right_canonical_boundary. n = hag_divisor_canonical_boundary * hag_common_right_canonical_boundary) -> exists hag_greatest_factor_canonical_boundary. g = hag_divisor_canonical_boundary * hag_greatest_factor_canonical_boundary)) -> ((((exists hlcm_left_factor_canonical_boundary. l = m * hlcm_left_factor_canonical_boundary) /\ (exists hlcm_right_factor_canonical_boundary. l = n * hlcm_right_factor_canonical_boundary)) /\ forall hlcm_common_canonical_boundary. (exists hlcm_left_common_canonical_boundary. hlcm_common_canonical_boundary = m * hlcm_left_common_canonical_boundary) -> (exists hlcm_right_common_canonical_boundary. hlcm_common_canonical_boundary = n * hlcm_right_common_canonical_boundary) -> exists hlcm_least_factor_canonical_boundary. hlcm_common_canonical_boundary = l * hlcm_least_factor_canonical_boundary)) -> (exists hgcrt_mod_left_canonical_boundary_compatibility hgcrt_mod_right_canonical_boundary_compatibility. a + g * hgcrt_mod_left_canonical_boundary_compatibility = b + g * hgcrt_mod_right_canonical_boundary_compatibility) -> (((l = 0 /\ exists x. ((((exists hgcrt_mod_left_canonical_boundary_fixed_left hgcrt_mod_right_canonical_boundary_fixed_left. x + m * hgcrt_mod_left_canonical_boundary_fixed_left = a + m * hgcrt_mod_right_canonical_boundary_fixed_left) /\ (exists hgcrt_mod_left_canonical_boundary_fixed_right hgcrt_mod_right_canonical_boundary_fixed_right. x + n * hgcrt_mod_left_canonical_boundary_fixed_right = b + n * hgcrt_mod_right_canonical_boundary_fixed_right))) /\ forall y. (((exists hgcrt_mod_left_canonical_boundary_zero_candidate_left hgcrt_mod_right_canonical_boundary_zero_candidate_left. y + m * hgcrt_mod_left_canonical_boundary_zero_candidate_left = a + m * hgcrt_mod_right_canonical_boundary_zero_candidate_left) /\ (exists hgcrt_mod_left_canonical_boundary_zero_candidate_right hgcrt_mod_right_canonical_boundary_zero_candidate_right. y + n * hgcrt_mod_left_canonical_boundary_zero_candidate_right = b + n * hgcrt_mod_right_canonical_boundary_zero_candidate_right))) -> y = x))) \/ ((~(l = 0) /\ exists r. (((exists hgcrt_below_gap_canonical_boundary_result. hgcrt_below_gap_canonical_boundary_result + S r = l) /\ (((exists hgcrt_mod_left_canonical_boundary_result_left hgcrt_mod_right_canonical_boundary_result_left. r + m * hgcrt_mod_left_canonical_boundary_result_left = a + m * hgcrt_mod_right_canonical_boundary_result_left) /\ (exists hgcrt_mod_left_canonical_boundary_result_right hgcrt_mod_right_canonical_boundary_result_right. r + n * hgcrt_mod_left_canonical_boundary_result_right = b + n * hgcrt_mod_right_canonical_boundary_result_right)))) /\ forall s. (exists hgcrt_below_gap_canonical_boundary_comparison. hgcrt_below_gap_canonical_boundary_comparison + S s = l) -> (((exists hgcrt_mod_left_canonical_boundary_comparison_left hgcrt_mod_right_canonical_boundary_comparison_left. s + m * hgcrt_mod_left_canonical_boundary_comparison_left = a + m * hgcrt_mod_right_canonical_boundary_comparison_left) /\ (exists hgcrt_mod_left_canonical_boundary_comparison_right hgcrt_mod_right_canonical_boundary_comparison_right. s + n * hgcrt_mod_left_canonical_boundary_comparison_right = b + n * hgcrt_mod_right_canonical_boundary_comparison_right))) -> s = r))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g l m n a b. ((((exists hag_left_factor_canonical_boundary. m = g * hag_left_factor_canonical_boundary) /\ (exists hag_right_factor_canonical_boundary. n = g * hag_right_factor_canonical_boundary)) /\ forall hag_divisor_canonical_boundary. (exists hag_common_left_canonical_boundary. m = hag_divisor_canonical_boundary * hag_common_left_canonical_boundary) -> (exists hag_common_right_canonical_boundary. n = hag_divisor_canonical_boundary * hag_common_right_canonical_boundary) -> exists hag_greatest_factor_canonical_boundary. g = hag_divisor_canonical_boundary * hag_greatest_factor_canonical_boundary)) -> ((((exists hlcm_left_factor_canonical_boundary. l = m * hlcm_left_factor_canonical_boundary) /\ (exists hlcm_right_factor_canonical_boundary. l = n * hlcm_right_factor_canonical_boundary)) /\ forall hlcm_common_canonical_boundary. (exists hlcm_left_common_canonical_boundary. hlcm_common_canonical_boundary = m * hlcm_left_common_canonical_boundary) -> (exists hlcm_right_common_canonical_boundary. hlcm_common_canonical_boundary = n * hlcm_right_common_canonical_boundary) -> exists hlcm_least_factor_canonical_boundary. hlcm_common_canonical_boundary = l * hlcm_least_factor_canonical_boundary)) -> (exists hgcrt_mod_left_canonical_boundary_compatibility hgcrt_mod_right_canonical_boundary_compatibility. a + g * hgcrt_mod_left_canonical_boundary_compatibility = b + g * hgcrt_mod_right_canonical_boundary_compatibility) -> (((l = 0 /\ exists x. ((((exists hgcrt_mod_left_canonical_boundary_fixed_left hgcrt_mod_right_canonical_boundary_fixed_left. x + m * hgcrt_mod_left_canonical_boundary_fixed_left = a + m * hgcrt_mod_right_canonical_boundary_fixed_left) /\ (exists hgcrt_mod_left_canonical_boundary_fixed_right hgcrt_mod_right_canonical_boundary_fixed_right. x + n * hgcrt_mod_left_canonical_boundary_fixed_right = b + n * hgcrt_mod_right_canonical_boundary_fixed_right))) /\ forall y. (((exists hgcrt_mod_left_canonical_boundary_zero_candidate_left hgcrt_mod_right_canonical_boundary_zero_candidate_left. y + m * hgcrt_mod_left_canonical_boundary_zero_candidate_left = a + m * hgcrt_mod_right_canonical_boundary_zero_candidate_left) /\ (exists hgcrt_mod_left_canonical_boundary_zero_candidate_right hgcrt_mod_right_canonical_boundary_zero_candidate_right. y + n * hgcrt_mod_left_canonical_boundary_zero_candidate_right = b + n * hgcrt_mod_right_canonical_boundary_zero_candidate_right))) -> y = x))) \/ ((~(l = 0) /\ exists r. (((exists hgcrt_below_gap_canonical_boundary_result. hgcrt_below_gap_canonical_boundary_result + S r = l) /\ (((exists hgcrt_mod_left_canonical_boundary_result_left hgcrt_mod_right_canonical_boundary_result_left. r + m * hgcrt_mod_left_canonical_boundary_result_left = a + m * hgcrt_mod_right_canonical_boundary_result_left) /\ (exists hgcrt_mod_left_canonical_boundary_result_right hgcrt_mod_right_canonical_boundary_result_right. r + n * hgcrt_mod_left_canonical_boundary_result_right = b + n * hgcrt_mod_right_canonical_boundary_result_right)))) /\ forall s. (exists hgcrt_below_gap_canonical_boundary_comparison. hgcrt_below_gap_canonical_boundary_comparison + S s = l) -> (((exists hgcrt_mod_left_canonical_boundary_comparison_left hgcrt_mod_right_canonical_boundary_comparison_left. s + m * hgcrt_mod_left_canonical_boundary_comparison_left = a + m * hgcrt_mod_right_canonical_boundary_comparison_left) /\ (exists hgcrt_mod_left_canonical_boundary_comparison_right hgcrt_mod_right_canonical_boundary_comparison_right. s + n * hgcrt_mod_left_canonical_boundary_comparison_right = b + n * hgcrt_mod_right_canonical_boundary_comparison_right))) -> s = r))))
use eq_decidable
use generalized_binary_crt_sufficient
use crt_solution_unique_lcm_zero
use crt_solution_canonical_remainder_nonzero
intro g
intro l
intro m
intro n
intro a
intro b
intro hg
intro hl
intro hab
have hexists : exists x. (((exists hgcrt_mod_left_canonical_boundary_fixed_left hgcrt_mod_right_canonical_boundary_fixed_left. x + m * hgcrt_mod_left_canonical_boundary_fixed_left = a + m * hgcrt_mod_right_canonical_boundary_fixed_left) /\ (exists hgcrt_mod_left_canonical_boundary_fixed_right hgcrt_mod_right_canonical_boundary_fixed_right. x + n * hgcrt_mod_left_canonical_boundary_fixed_right = b + n * hgcrt_mod_right_canonical_boundary_fixed_right)))
specialize generalized_binary_crt_sufficient g
specialize generalized_binary_crt_sufficient m
specialize generalized_binary_crt_sufficient n
specialize generalized_binary_crt_sufficient a
specialize generalized_binary_crt_sufficient b
apply generalized_binary_crt_sufficient
exact hg
exact hab
cases hexists
have hlzero : l = 0 \/ ~(l = 0)
specialize eq_decidable l
specialize eq_decidable 0
exact eq_decidable
cases hlzero
left
split
exact hlzero_left
exists x
split
exact hexists_witness
intro y
intro hy
specialize crt_solution_unique_lcm_zero l
specialize crt_solution_unique_lcm_zero m
specialize crt_solution_unique_lcm_zero n
specialize crt_solution_unique_lcm_zero a
specialize crt_solution_unique_lcm_zero b
specialize crt_solution_unique_lcm_zero x
specialize crt_solution_unique_lcm_zero y
apply crt_solution_unique_lcm_zero
exact hlzero_left
exact hl
exact hexists_witness
exact hy
right
split
exact hlzero_right
have hcanonical : exists r. (((exists hgcrt_below_gap_canonical_boundary_runtime_result. hgcrt_below_gap_canonical_boundary_runtime_result + S r = l) /\ (((exists hgcrt_mod_left_canonical_boundary_runtime_result_left hgcrt_mod_right_canonical_boundary_runtime_result_left. r + m * hgcrt_mod_left_canonical_boundary_runtime_result_left = a + m * hgcrt_mod_right_canonical_boundary_runtime_result_left) /\ (exists hgcrt_mod_left_canonical_boundary_runtime_result_right hgcrt_mod_right_canonical_boundary_runtime_result_right. r + n * hgcrt_mod_left_canonical_boundary_runtime_result_right = b + n * hgcrt_mod_right_canonical_boundary_runtime_result_right)))) /\ ((exists hgcrt_mod_left_canonical_boundary_runtime_mod hgcrt_mod_right_canonical_boundary_runtime_mod. r + l * hgcrt_mod_left_canonical_boundary_runtime_mod = x + l * hgcrt_mod_right_canonical_boundary_runtime_mod) /\ forall s. (exists hgcrt_below_gap_canonical_boundary_runtime_comparison. hgcrt_below_gap_canonical_boundary_runtime_comparison + S s = l) -> (((exists hgcrt_mod_left_canonical_boundary_runtime_comparison_left hgcrt_mod_right_canonical_boundary_runtime_comparison_left. s + m * hgcrt_mod_left_canonical_boundary_runtime_comparison_left = a + m * hgcrt_mod_right_canonical_boundary_runtime_comparison_left) /\ (exists hgcrt_mod_left_canonical_boundary_runtime_comparison_right hgcrt_mod_right_canonical_boundary_runtime_comparison_right. s + n * hgcrt_mod_left_canonical_boundary_runtime_comparison_right = b + n * hgcrt_mod_right_canonical_boundary_runtime_comparison_right))) -> s = r))
specialize crt_solution_canonical_remainder_nonzero l
specialize crt_solution_canonical_remainder_nonzero m
specialize crt_solution_canonical_remainder_nonzero n
specialize crt_solution_canonical_remainder_nonzero a
specialize crt_solution_canonical_remainder_nonzero b
specialize crt_solution_canonical_remainder_nonzero x
apply crt_solution_canonical_remainder_nonzero
exact hlzero_right
exact hl
exact hexists_witness
cases hcanonical
exists x1
split
cases hcanonical_witness
exact hcanonical_witness_left
cases hcanonical_witness
cases hcanonical_witness_right
exact hcanonical_witness_right_right
qed

Occurrences

17,750

Distinct objects

4,239

Reused references

188

Depth

80

Cuts

517

Certificate SHA-256

59bc1b3f5a5b…

mod_eq_decidableMod eq decidable Congruence & CRT checked

Balanced congruence is constructively decidable for every natural modulus.

Prerequisites

eq_decidable mod_eq_zero_iff_eq mod_eq_decidable_nonzero

Used directly by

generalized_binary_crt_solution_or_obstruction

Exact expanded PA statement

Copy statement

forall d a b. (exists hgcrt_mod_left_decision_yes hgcrt_mod_right_decision_yes. a + d * hgcrt_mod_left_decision_yes = b + d * hgcrt_mod_right_decision_yes) \/ ~(exists hgcrt_mod_left_decision_no hgcrt_mod_right_decision_no. a + d * hgcrt_mod_left_decision_no = b + d * hgcrt_mod_right_decision_no)

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall d a b. (exists hgcrt_mod_left_decision_yes hgcrt_mod_right_decision_yes. a + d * hgcrt_mod_left_decision_yes = b + d * hgcrt_mod_right_decision_yes) \/ ~(exists hgcrt_mod_left_decision_no hgcrt_mod_right_decision_no. a + d * hgcrt_mod_left_decision_no = b + d * hgcrt_mod_right_decision_no)
use eq_decidable
use mod_eq_zero_iff_eq
use mod_eq_decidable_nonzero
intro d
intro a
intro b
have hd : d = 0 \/ ~(d = 0)
specialize eq_decidable d
specialize eq_decidable 0
exact eq_decidable
cases hd
have hzero : (((exists hgcrt_mod_left_decision_zero_forward hgcrt_mod_right_decision_zero_forward. a + 0 * hgcrt_mod_left_decision_zero_forward = b + 0 * hgcrt_mod_right_decision_zero_forward) -> a = b) /\ (a = b -> (exists hgcrt_mod_left_decision_zero_reverse hgcrt_mod_right_decision_zero_reverse. a + 0 * hgcrt_mod_left_decision_zero_reverse = b + 0 * hgcrt_mod_right_decision_zero_reverse)))
specialize mod_eq_zero_iff_eq a
specialize mod_eq_zero_iff_eq b
exact mod_eq_zero_iff_eq
cases hzero
have hab : a = b \/ ~(a = b)
specialize eq_decidable a
specialize eq_decidable b
exact eq_decidable
cases hab
left
rewrite hd_left
rewrite hd_left
apply hzero_right
exact hab_left
right
intro hmod
rewrite hd_left at hmod
rewrite hd_left at hmod
apply hab_right
apply hzero_left
exact hmod
specialize mod_eq_decidable_nonzero d
specialize mod_eq_decidable_nonzero a
specialize mod_eq_decidable_nonzero b
apply mod_eq_decidable_nonzero
exact hd_right
qed

Occurrences

2,339

Distinct objects

1,217

Reused references

62

Depth

70

Cuts

65

Certificate SHA-256

7ffa64c2d06d…

generalized_binary_crt_solution_or_obstructionGeneralized binary crt solution or obstruction Congruence & CRT checked

Every generalized binary CRT instance constructively returns a solution or a certified incompatibility obstruction.

Prerequisites

mod_eq_decidable generalized_binary_crt_sufficient crt_incompatibility_obstructs_solution

Used directly by

generalized_binary_crt_total_decision

Exact expanded PA statement

Copy statement

forall g m n a b. ((((exists hag_left_factor_decision_boundary. m = g * hag_left_factor_decision_boundary) /\ (exists hag_right_factor_decision_boundary. n = g * hag_right_factor_decision_boundary)) /\ forall hag_divisor_decision_boundary. (exists hag_common_left_decision_boundary. m = hag_divisor_decision_boundary * hag_common_left_decision_boundary) -> (exists hag_common_right_decision_boundary. n = hag_divisor_decision_boundary * hag_common_right_decision_boundary) -> exists hag_greatest_factor_decision_boundary. g = hag_divisor_decision_boundary * hag_greatest_factor_decision_boundary)) -> (((exists hgcrt_mod_left_decision_boundary_compatible hgcrt_mod_right_decision_boundary_compatible. a + g * hgcrt_mod_left_decision_boundary_compatible = b + g * hgcrt_mod_right_decision_boundary_compatible) /\ exists x. (((exists hgcrt_mod_left_decision_boundary_positive_left hgcrt_mod_right_decision_boundary_positive_left. x + m * hgcrt_mod_left_decision_boundary_positive_left = a + m * hgcrt_mod_right_decision_boundary_positive_left) /\ (exists hgcrt_mod_left_decision_boundary_positive_right hgcrt_mod_right_decision_boundary_positive_right. x + n * hgcrt_mod_left_decision_boundary_positive_right = b + n * hgcrt_mod_right_decision_boundary_positive_right)))) \/ (~(exists hgcrt_mod_left_decision_boundary_incompatible hgcrt_mod_right_decision_boundary_incompatible. a + g * hgcrt_mod_left_decision_boundary_incompatible = b + g * hgcrt_mod_right_decision_boundary_incompatible) /\ ~(exists x. (((exists hgcrt_mod_left_decision_boundary_negative_left hgcrt_mod_right_decision_boundary_negative_left. x + m * hgcrt_mod_left_decision_boundary_negative_left = a + m * hgcrt_mod_right_decision_boundary_negative_left) /\ (exists hgcrt_mod_left_decision_boundary_negative_right hgcrt_mod_right_decision_boundary_negative_right. x + n * hgcrt_mod_left_decision_boundary_negative_right = b + n * hgcrt_mod_right_decision_boundary_negative_right))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall g m n a b. ((((exists hag_left_factor_decision_boundary. m = g * hag_left_factor_decision_boundary) /\ (exists hag_right_factor_decision_boundary. n = g * hag_right_factor_decision_boundary)) /\ forall hag_divisor_decision_boundary. (exists hag_common_left_decision_boundary. m = hag_divisor_decision_boundary * hag_common_left_decision_boundary) -> (exists hag_common_right_decision_boundary. n = hag_divisor_decision_boundary * hag_common_right_decision_boundary) -> exists hag_greatest_factor_decision_boundary. g = hag_divisor_decision_boundary * hag_greatest_factor_decision_boundary)) -> (((exists hgcrt_mod_left_decision_boundary_compatible hgcrt_mod_right_decision_boundary_compatible. a + g * hgcrt_mod_left_decision_boundary_compatible = b + g * hgcrt_mod_right_decision_boundary_compatible) /\ exists x. (((exists hgcrt_mod_left_decision_boundary_positive_left hgcrt_mod_right_decision_boundary_positive_left. x + m * hgcrt_mod_left_decision_boundary_positive_left = a + m * hgcrt_mod_right_decision_boundary_positive_left) /\ (exists hgcrt_mod_left_decision_boundary_positive_right hgcrt_mod_right_decision_boundary_positive_right. x + n * hgcrt_mod_left_decision_boundary_positive_right = b + n * hgcrt_mod_right_decision_boundary_positive_right)))) \/ (~(exists hgcrt_mod_left_decision_boundary_incompatible hgcrt_mod_right_decision_boundary_incompatible. a + g * hgcrt_mod_left_decision_boundary_incompatible = b + g * hgcrt_mod_right_decision_boundary_incompatible) /\ ~(exists x. (((exists hgcrt_mod_left_decision_boundary_negative_left hgcrt_mod_right_decision_boundary_negative_left. x + m * hgcrt_mod_left_decision_boundary_negative_left = a + m * hgcrt_mod_right_decision_boundary_negative_left) /\ (exists hgcrt_mod_left_decision_boundary_negative_right hgcrt_mod_right_decision_boundary_negative_right. x + n * hgcrt_mod_left_decision_boundary_negative_right = b + n * hgcrt_mod_right_decision_boundary_negative_right))))))
use mod_eq_decidable
use generalized_binary_crt_sufficient
use crt_incompatibility_obstructs_solution
intro g
intro m
intro n
intro a
intro b
intro hgcd
have hcompat : (exists hgcrt_mod_left_decision_runtime_compatible hgcrt_mod_right_decision_runtime_compatible. a + g * hgcrt_mod_left_decision_runtime_compatible = b + g * hgcrt_mod_right_decision_runtime_compatible) \/ ~(exists hgcrt_mod_left_decision_runtime_incompatible hgcrt_mod_right_decision_runtime_incompatible. a + g * hgcrt_mod_left_decision_runtime_incompatible = b + g * hgcrt_mod_right_decision_runtime_incompatible)
specialize mod_eq_decidable g
specialize mod_eq_decidable a
specialize mod_eq_decidable b
apply mod_eq_decidable
cases hcompat
left
split
exact hcompat_left
specialize generalized_binary_crt_sufficient g
specialize generalized_binary_crt_sufficient m
specialize generalized_binary_crt_sufficient n
specialize generalized_binary_crt_sufficient a
specialize generalized_binary_crt_sufficient b
apply generalized_binary_crt_sufficient
exact hgcd
exact hcompat_left
right
split
exact hcompat_right
intro hsolution
specialize crt_incompatibility_obstructs_solution g
specialize crt_incompatibility_obstructs_solution m
specialize crt_incompatibility_obstructs_solution n
specialize crt_incompatibility_obstructs_solution a
specialize crt_incompatibility_obstructs_solution b
apply crt_incompatibility_obstructs_solution
exact hgcd
exact hcompat_right
exact hsolution
qed

Occurrences

14,182

Distinct objects

3,909

Reused references

182

Depth

80

Cuts

414

Certificate SHA-256

3eda3a762221…

generalized_binary_crt_total_decisionGeneralized binary crt total decision Congruence & CRT checked

Every raw binary CRT input returns a relational gcd and either a solution or a certified incompatibility obstruction.

Prerequisites

gcd_exists_relational generalized_binary_crt_solution_or_obstruction

Used directly by

no direct dependent

Exact expanded PA statement

Copy statement

forall m n a b. exists g. (((((exists hag_left_factor_total_decision. m = g * hag_left_factor_total_decision) /\ (exists hag_right_factor_total_decision. n = g * hag_right_factor_total_decision)) /\ forall hag_divisor_total_decision. (exists hag_common_left_total_decision. m = hag_divisor_total_decision * hag_common_left_total_decision) -> (exists hag_common_right_total_decision. n = hag_divisor_total_decision * hag_common_right_total_decision) -> exists hag_greatest_factor_total_decision. g = hag_divisor_total_decision * hag_greatest_factor_total_decision)) /\ (((exists hgcrt_mod_left_total_decision_compatible hgcrt_mod_right_total_decision_compatible. a + g * hgcrt_mod_left_total_decision_compatible = b + g * hgcrt_mod_right_total_decision_compatible) /\ exists x. (((exists hgcrt_mod_left_total_decision_positive_left hgcrt_mod_right_total_decision_positive_left. x + m * hgcrt_mod_left_total_decision_positive_left = a + m * hgcrt_mod_right_total_decision_positive_left) /\ (exists hgcrt_mod_left_total_decision_positive_right hgcrt_mod_right_total_decision_positive_right. x + n * hgcrt_mod_left_total_decision_positive_right = b + n * hgcrt_mod_right_total_decision_positive_right)))) \/ (~(exists hgcrt_mod_left_total_decision_incompatible hgcrt_mod_right_total_decision_incompatible. a + g * hgcrt_mod_left_total_decision_incompatible = b + g * hgcrt_mod_right_total_decision_incompatible) /\ ~(exists x. (((exists hgcrt_mod_left_total_decision_negative_left hgcrt_mod_right_total_decision_negative_left. x + m * hgcrt_mod_left_total_decision_negative_left = a + m * hgcrt_mod_right_total_decision_negative_left) /\ (exists hgcrt_mod_left_total_decision_negative_right hgcrt_mod_right_total_decision_negative_right. x + n * hgcrt_mod_left_total_decision_negative_right = b + n * hgcrt_mod_right_total_decision_negative_right)))))))

Complete replay recipe: dependency imports + authored proof

Copy proof

pa prove forall m n a b. exists g. (((((exists hag_left_factor_total_decision. m = g * hag_left_factor_total_decision) /\ (exists hag_right_factor_total_decision. n = g * hag_right_factor_total_decision)) /\ forall hag_divisor_total_decision. (exists hag_common_left_total_decision. m = hag_divisor_total_decision * hag_common_left_total_decision) -> (exists hag_common_right_total_decision. n = hag_divisor_total_decision * hag_common_right_total_decision) -> exists hag_greatest_factor_total_decision. g = hag_divisor_total_decision * hag_greatest_factor_total_decision)) /\ (((exists hgcrt_mod_left_total_decision_compatible hgcrt_mod_right_total_decision_compatible. a + g * hgcrt_mod_left_total_decision_compatible = b + g * hgcrt_mod_right_total_decision_compatible) /\ exists x. (((exists hgcrt_mod_left_total_decision_positive_left hgcrt_mod_right_total_decision_positive_left. x + m * hgcrt_mod_left_total_decision_positive_left = a + m * hgcrt_mod_right_total_decision_positive_left) /\ (exists hgcrt_mod_left_total_decision_positive_right hgcrt_mod_right_total_decision_positive_right. x + n * hgcrt_mod_left_total_decision_positive_right = b + n * hgcrt_mod_right_total_decision_positive_right)))) \/ (~(exists hgcrt_mod_left_total_decision_incompatible hgcrt_mod_right_total_decision_incompatible. a + g * hgcrt_mod_left_total_decision_incompatible = b + g * hgcrt_mod_right_total_decision_incompatible) /\ ~(exists x. (((exists hgcrt_mod_left_total_decision_negative_left hgcrt_mod_right_total_decision_negative_left. x + m * hgcrt_mod_left_total_decision_negative_left = a + m * hgcrt_mod_right_total_decision_negative_left) /\ (exists hgcrt_mod_left_total_decision_negative_right hgcrt_mod_right_total_decision_negative_right. x + n * hgcrt_mod_left_total_decision_negative_right = b + n * hgcrt_mod_right_total_decision_negative_right)))))))
use gcd_exists_relational
use generalized_binary_crt_solution_or_obstruction
intro m
intro n
intro a
intro b
specialize gcd_exists_relational m
specialize gcd_exists_relational n
cases gcd_exists_relational
exists x
split
exact gcd_exists_relational_witness
specialize generalized_binary_crt_solution_or_obstruction x
specialize generalized_binary_crt_solution_or_obstruction m
specialize generalized_binary_crt_solution_or_obstruction n
specialize generalized_binary_crt_solution_or_obstruction a
specialize generalized_binary_crt_solution_or_obstruction b
apply generalized_binary_crt_solution_or_obstruction
exact gcd_exists_relational_witness
qed

Occurrences

15,492

Distinct objects

4,052

Reused references

189

Depth

82

Cuts

454

Certificate SHA-256

20ff304fface…

bezout_integer_coefficientsBézout identity with integer coefficients GCD, Bézout & coprimality representation boundary

The conventional reusable theorem quantifies over signed coefficients.

Peano Lab quantifies only over naturals and has no integer encoding or subtraction. A balanced natural reformulation is possible, but it is a different interface and has now been designed explicitly and checked under a separate name.

Depends conceptually on

gcd_exists_relational

Related representational choices: a reviewed integer-pair encoding; or a separately named balanced-natural Bézout relation.

This card deliberately contains no native proof script or certificate. The separately named balanced four-natural Bézout theorem is checked.

How to use this atlas

• Start with the guided route if the exact formulas are unfamiliar.

• Select division_remainder_exists, euclid_prime_dvd_product, beta_prefix_product_trace_exists, prime_factorization_existence, or fundamental_theorem_of_arithmetic in the focused navigator to see the main dependency spine.

• A theorem card’s prerequisite and dependent chips are bidirectional links. Browser Back and Forward therefore become mathematical navigation controls.

• The pa lib NAME label identifies a theorem in the Stable checked-use registry. “Native source” and “Vault note” point to immutable proof material; hosted-browser deployment is tracked separately.

• The single boundary card is intentionally not presented as proved. It keeps the conventional integer-coefficient Bézout interface distinct from the checked balanced-natural theorem.

This page is regenerated with:

python3 scripts/build_arithmetic_book_atlas.py
python3 scripts/build_arithmetic_book_atlas.py --check