FS0017 · theorem body

four_square_cross_intersection

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Any two injective bounded equal-length decoded prefixes with combined length exceeding their codomain have an explicit actual common value.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ b. ∀ c. ∀ d. ∀ e. ∀ l. ∀ p. (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,p)) → (∀ x. Lt(x,l) → ∃ y. BetaAt(d,e,x,y) ∧ Lt(y,p)) → InjectivePrefix(b,c,l) → InjectivePrefix(d,e,l) → Lt(p,l + l) → ∃ x. ∃ y. ∃ z. Lt(x,l) ∧ (Lt(y,l) ∧ (BetaAt(b,c,x,z) ∧ BetaAt(d,e,y,z)))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0013 BetaAt PD0025 InjectivePrefix

In local proof propositions

PD0002 Lt PD0013 BetaAt
Exact expanded first-order statement
forall b c d e l p. (forall fscp_index_left. (exists fscp_gap_left_index. fscp_gap_left_index + S (fscp_index_left) = (l)) -> exists fscp_value_left. ((((exists ff_h_fscp_left_entry. ff_h_fscp_left_entry + S (fscp_value_left) = S ((S (fscp_index_left)) * c)) /\ exists ff_q_fscp_left_entry. b = ff_q_fscp_left_entry * S ((S (fscp_index_left)) * c) + (fscp_value_left))) /\ (exists fscp_gap_left_value. fscp_gap_left_value + S (fscp_value_left) = (p)))) -> (forall fscp_index_right. (exists fscp_gap_right_index. fscp_gap_right_index + S (fscp_index_right) = (l)) -> exists fscp_value_right. ((((exists ff_h_fscp_right_entry. ff_h_fscp_right_entry + S (fscp_value_right) = S ((S (fscp_index_right)) * e)) /\ exists ff_q_fscp_right_entry. d = ff_q_fscp_right_entry * S ((S (fscp_index_right)) * e) + (fscp_value_right))) /\ (exists fscp_gap_right_value. fscp_gap_right_value + S (fscp_value_right) = (p)))) -> (forall fp_i_fscp_left fp_j_fscp_left fp_value_fscp_left. (exists fp_gap_fscp_left_i. fp_gap_fscp_left_i + S fp_i_fscp_left = l) -> (exists fp_gap_fscp_left_j. fp_gap_fscp_left_j + S fp_j_fscp_left = l) -> (((exists ff_h_fscp_left_left. ff_h_fscp_left_left + S (fp_value_fscp_left) = S ((S (fp_i_fscp_left)) * c)) /\ exists ff_q_fscp_left_left. b = ff_q_fscp_left_left * S ((S (fp_i_fscp_left)) * c) + (fp_value_fscp_left))) -> (((exists ff_h_fscp_left_right. ff_h_fscp_left_right + S (fp_value_fscp_left) = S ((S (fp_j_fscp_left)) * c)) /\ exists ff_q_fscp_left_right. b = ff_q_fscp_left_right * S ((S (fp_j_fscp_left)) * c) + (fp_value_fscp_left))) -> fp_i_fscp_left = fp_j_fscp_left) -> (forall fp_i_fscp_right fp_j_fscp_right fp_value_fscp_right. (exists fp_gap_fscp_right_i. fp_gap_fscp_right_i + S fp_i_fscp_right = l) -> (exists fp_gap_fscp_right_j. fp_gap_fscp_right_j + S fp_j_fscp_right = l) -> (((exists ff_h_fscp_right_left. ff_h_fscp_right_left + S (fp_value_fscp_right) = S ((S (fp_i_fscp_right)) * e)) /\ exists ff_q_fscp_right_left. d = ff_q_fscp_right_left * S ((S (fp_i_fscp_right)) * e) + (fp_value_fscp_right))) -> (((exists ff_h_fscp_right_right. ff_h_fscp_right_right + S (fp_value_fscp_right) = S ((S (fp_j_fscp_right)) * e)) /\ exists ff_q_fscp_right_right. d = ff_q_fscp_right_right * S ((S (fp_j_fscp_right)) * e) + (fp_value_fscp_right))) -> fp_i_fscp_right = fp_j_fscp_right) -> (exists fscp_gap_intersection_overflow. fscp_gap_intersection_overflow + S (p) = (l + l)) -> (exists fscp_left_result fscp_right_result fscp_value_result. ((exists fscp_gap_result_left_bound. fscp_gap_result_left_bound + S (fscp_left_result) = (l)) /\ ((exists fscp_gap_result_right_bound. fscp_gap_result_right_bound + S (fscp_right_result) = (l)) /\ ((((exists ff_h_fscp_result_left. ff_h_fscp_result_left + S (fscp_value_result) = S ((S (fscp_left_result)) * c)) /\ exists ff_q_fscp_result_left. b = ff_q_fscp_result_left * S ((S (fscp_left_result)) * c) + (fscp_value_result))) /\ (((exists ff_h_fscp_result_right. ff_h_fscp_result_right + S (fscp_value_result) = S ((S (fscp_right_result)) * e)) /\ exists ff_q_fscp_result_right. d = ff_q_fscp_result_right * S ((S (fscp_right_result)) * e) + (fscp_value_result)))))))

Proof neighborhood

Direct theorem prerequisites

FS0016 four_square_cross_interleaved_prefix_exists FS0015 four_square_cross_pigeonhole

Direct theorem dependents

FS000K four_square_odd_prime_half_coordinate_seed FS004K four_square_odd_prime_modular_seed

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

35 script commands · 6 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
four square cross interleaved prefix existsfour square cross pigeonhole
01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro l
  6. L6
    intro p
  7. L7
    intro hleft
  8. L8
    intro hright
  9. L9
    intro hleft_injective
  10. L10
    intro hright_injective
02Fix variables and assumptionsL11–11

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro hoverflow
03Establish hcodeL12–18

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hcode : ∃ z. ∃ t. ∀ x. ∀ y. Lt(x,l + l) → BetaAt(z,t,x,y) → (∃ n. Lt(n,l) ∧ (BetaAt(b,c,n,y) ∧ x = n + n)) ∨ (∃ n. Lt(n,l) ∧ (BetaAt(d,e,n,y) ∧ x = S (n + n)))Definitions: Lt(x,l + l)BetaAt(z,t,x,y)Lt(n,l)BetaAt(b,c,n,y)BetaAt(d,e,n,y)Original native command in the exact edition
  2. L13
    specialize four_square_cross_interleaved_prefix_exists b
  3. L14
    specialize four_square_cross_interleaved_prefix_exists c
  4. L15
    specialize four_square_cross_interleaved_prefix_exists d
  5. L16
    specialize four_square_cross_interleaved_prefix_exists e
  6. L17
    specialize four_square_cross_interleaved_prefix_exists l
  7. L18
    exact four_square_cross_interleaved_prefix_exists
04Separate the logical casesL19–20

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L19
    cases hcode
  2. L20
    cases hcode_witness
05Use earlier factsL21–30

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L21
    specialize four_square_cross_pigeonhole b
  2. L22
    specialize four_square_cross_pigeonhole c
  3. L23
    specialize four_square_cross_pigeonhole d
  4. L24
    specialize four_square_cross_pigeonhole e
  5. L25
    specialize four_square_cross_pigeonhole x
  6. L26
    specialize four_square_cross_pigeonhole x1
  7. L27
    specialize four_square_cross_pigeonhole l
  8. L28
    specialize four_square_cross_pigeonhole p
  9. L29
    apply four_square_cross_pigeonhole
  10. L30
    exact hleft
06Use earlier factsL31–35

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L31
    exact hright
  2. L32
    exact hleft_injective
  3. L33
    exact hright_injective
  4. L34
    exact hcode_witness_witness
  5. L35
    exact hoverflow

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro l
  6. 0006intro p
  7. 0007intro hleft
  8. 0008intro hright
  9. 0009intro hleft_injective
  10. 0010intro hright_injective
  11. 0011intro hoverflow
  12. 0012have hcode : ∃ z. ∃ t. ∀ x. ∀ y. Lt(x,l + l) → BetaAt(z,t,x,y) → (∃ n. Lt(n,l) ∧ (BetaAt(b,c,n,y) ∧ x = n + n)) ∨ (∃ n. Lt(n,l) ∧ (BetaAt(d,e,n,y) ∧ x = S (n + n)))
    Exact native replay linehave hcode : exists z t. (forall fscp_index_intersection_code fscp_value_intersection_code. (exists fscp_gap_intersection_code_index. fscp_gap_intersection_code_index + S (fscp_index_intersection_code) = (l + l)) -> (((exists ff_h_fscp_intersection_code_source. ff_h_fscp_intersection_code_source + S (fscp_value_intersection_code) = S ((S (fscp_index_intersection_code)) * t)) /\ exists ff_q_fscp_intersection_code_source. z = ff_q_fscp_intersection_code_source * S ((S (fscp_index_intersection_code)) * t) + (fscp_value_intersection_code))) -> (((exists fscp_left_intersection_code. ((exists fscp_gap_intersection_code_left_bound. fscp_gap_intersection_code_left_bound + S (fscp_left_intersection_code) = (l)) /\ ((((exists ff_h_fscp_intersection_code_left. ff_h_fscp_intersection_code_left + S (fscp_value_intersection_code) = S ((S (fscp_left_intersection_code)) * c)) /\ exists ff_q_fscp_intersection_code_left. b = ff_q_fscp_intersection_code_left * S ((S (fscp_left_intersection_code)) * c) + (fscp_value_intersection_code))) /\ fscp_index_intersection_code = fscp_left_intersection_code + fscp_left_intersection_code))) \/ (exists fscp_right_intersection_code. ((exists fscp_gap_intersection_code_right_bound. fscp_gap_intersection_code_right_bound + S (fscp_right_intersection_code) = (l)) /\ ((((exists ff_h_fscp_intersection_code_right. ff_h_fscp_intersection_code_right + S (fscp_value_intersection_code) = S ((S (fscp_right_intersection_code)) * e)) /\ exists ff_q_fscp_intersection_code_right. d = ff_q_fscp_intersection_code_right * S ((S (fscp_right_intersection_code)) * e) + (fscp_value_intersection_code))) /\ fscp_index_intersection_code = S (fscp_right_intersection_code + fscp_right_intersection_code)))))))
  13. 0013specialize four_square_cross_interleaved_prefix_exists b
  14. 0014specialize four_square_cross_interleaved_prefix_exists c
  15. 0015specialize four_square_cross_interleaved_prefix_exists d
  16. 0016specialize four_square_cross_interleaved_prefix_exists e
  17. 0017specialize four_square_cross_interleaved_prefix_exists l
  18. 0018exact four_square_cross_interleaved_prefix_exists
  19. 0019cases hcode
  20. 0020cases hcode_witness
  21. 0021specialize four_square_cross_pigeonhole b
  22. 0022specialize four_square_cross_pigeonhole c
  23. 0023specialize four_square_cross_pigeonhole d
  24. 0024specialize four_square_cross_pigeonhole e
  25. 0025specialize four_square_cross_pigeonhole x
  26. 0026specialize four_square_cross_pigeonhole x1
  27. 0027specialize four_square_cross_pigeonhole l
  28. 0028specialize four_square_cross_pigeonhole p
  29. 0029apply four_square_cross_pigeonhole
  30. 0030exact hleft
  31. 0031exact hright
  32. 0032exact hleft_injective
  33. 0033exact hright_injective
  34. 0034exact hcode_witness_witness
  35. 0035exact hoverflow