AF000C

factor_permutation_prime_member

An actual prime divisor is found at an actual occurrence of every unordered prime factorization of the product.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

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.

The original division, gcd, cancellation, and factor-existence foundations are exposed through checked wrappers. Unordered uniqueness adds an actual bounded, injective, surjective index map matching repeated prime occurrences. The empty factor list represents one, not zero.

Exact theorem in conservative defined notation

∀ n. ∀ b. ∀ c. ∀ l. ∀ p. PrimeFactorList(n,b,c,l) → Prime(p) → Dvd(p,n) → ∃ x. Lt(x,l) ∧ BetaAt(b,c,x,p)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

PrimeFactorList(n,b,c,l) · 1Lt(a,b) · 1Dvd(d,n) · 1Prime(p) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

beta_prime_divisor_product_member · checked external prerequisite
Original expanded first-order statement 
forall n b c l p. ((~(n = 0) /\ ((exists ff_u_fsat_member_factorization_product ff_v_fsat_member_factorization_product. ((((exists ff_h_fsat_member_factorization_product_start. ff_h_fsat_member_factorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_member_factorization_product)) /\ exists ff_q_fsat_member_factorization_product_start. ff_u_fsat_member_factorization_product = ff_q_fsat_member_factorization_product_start * S ((S (0)) * ff_v_fsat_member_factorization_product) + (1))) /\ ((((exists ff_h_fsat_member_factorization_product_terminal. ff_h_fsat_member_factorization_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_member_factorization_product)) /\ exists ff_q_fsat_member_factorization_product_terminal. ff_u_fsat_member_factorization_product = ff_q_fsat_member_factorization_product_terminal * S ((S (l)) * ff_v_fsat_member_factorization_product) + (n))) /\ forall ff_i_fsat_member_factorization_product. (exists ff_lt_fsat_member_factorization_product_bound. ff_lt_fsat_member_factorization_product_bound + S ff_i_fsat_member_factorization_product = l) -> exists ff_p_fsat_member_factorization_product ff_r_fsat_member_factorization_product ff_s_fsat_member_factorization_product. ((((exists ff_h_fsat_member_factorization_product_factor. ff_h_fsat_member_factorization_product_factor + S (ff_p_fsat_member_factorization_product) = S ((S (ff_i_fsat_member_factorization_product)) * c)) /\ exists ff_q_fsat_member_factorization_product_factor. b = ff_q_fsat_member_factorization_product_factor * S ((S (ff_i_fsat_member_factorization_product)) * c) + (ff_p_fsat_member_factorization_product))) /\ ((((exists ff_h_fsat_member_factorization_product_partial. ff_h_fsat_member_factorization_product_partial + S (ff_r_fsat_member_factorization_product) = S ((S (ff_i_fsat_member_factorization_product)) * ff_v_fsat_member_factorization_product)) /\ exists ff_q_fsat_member_factorization_product_partial. ff_u_fsat_member_factorization_product = ff_q_fsat_member_factorization_product_partial * S ((S (ff_i_fsat_member_factorization_product)) * ff_v_fsat_member_factorization_product) + (ff_r_fsat_member_factorization_product))) /\ ((((exists ff_h_fsat_member_factorization_product_successor. ff_h_fsat_member_factorization_product_successor + S (ff_s_fsat_member_factorization_product) = S ((S (S ff_i_fsat_member_factorization_product)) * ff_v_fsat_member_factorization_product)) /\ exists ff_q_fsat_member_factorization_product_successor. ff_u_fsat_member_factorization_product = ff_q_fsat_member_factorization_product_successor * S ((S (S ff_i_fsat_member_factorization_product)) * ff_v_fsat_member_factorization_product) + (ff_s_fsat_member_factorization_product))) /\ ff_s_fsat_member_factorization_product = ff_r_fsat_member_factorization_product * ff_p_fsat_member_factorization_product)))))) /\ (forall ftsf_index_fsat_member_factorization_primes. (exists ftsf_gap_fsat_member_factorization_primes_bound. ftsf_gap_fsat_member_factorization_primes_bound + S ftsf_index_fsat_member_factorization_primes = (l)) -> exists ftsf_factor_fsat_member_factorization_primes. ((((exists ff_h_ftsf_fsat_member_factorization_primes_entry. ff_h_ftsf_fsat_member_factorization_primes_entry + S (ftsf_factor_fsat_member_factorization_primes) = S ((S (ftsf_index_fsat_member_factorization_primes)) * c)) /\ exists ff_q_ftsf_fsat_member_factorization_primes_entry. b = ff_q_ftsf_fsat_member_factorization_primes_entry * S ((S (ftsf_index_fsat_member_factorization_primes)) * c) + (ftsf_factor_fsat_member_factorization_primes))) /\ ((~(ftsf_factor_fsat_member_factorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_member_factorization_primes_prime frm_prime_right_ftsf_fsat_member_factorization_primes_prime. ftsf_factor_fsat_member_factorization_primes = frm_prime_left_ftsf_fsat_member_factorization_primes_prime * frm_prime_right_ftsf_fsat_member_factorization_primes_prime -> frm_prime_left_ftsf_fsat_member_factorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_member_factorization_primes_prime = 1))))))) -> ((~(p = 1) /\ forall frm_prime_left_pfp_member_prime frm_prime_right_pfp_member_prime. p = frm_prime_left_pfp_member_prime * frm_prime_right_pfp_member_prime -> frm_prime_left_pfp_member_prime = 1 \/ frm_prime_right_pfp_member_prime = 1)) -> (exists q. n = p * q) -> exists i. (exists pfp_gap_member_bound. pfp_gap_member_bound + S (i) = (l)) /\ (((exists ff_h_pfp_member. ff_h_pfp_member + S (p) = S ((S (i)) * c)) /\ exists ff_q_pfp_member. b = ff_q_pfp_member * S ((S (i)) * c) + (p)))

Complete tactic proof in conservative notation

All 20 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

20 script commands · 3 reading checkpoints · 0 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.

01Fix variables and assumptionsL1–8

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

  1. L1
    intro n
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro p
  6. L6
    intro hf
  7. L7
    intro hp
  8. L8
    intro hdiv
02Separate the logical casesL9–10

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

  1. L9
    cases hf
  2. L10
    cases hf_right
03Use earlier factsL11–20

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

  1. L11
    specialize beta_prime_divisor_product_member (b)
  2. L12
    specialize beta_prime_divisor_product_member (c)
  3. L13
    specialize beta_prime_divisor_product_member (l)
  4. L14
    specialize beta_prime_divisor_product_member (n)
  5. L15
    specialize beta_prime_divisor_product_member (p)
  6. L16
    apply beta_prime_divisor_product_member
  7. L17
    exact hp
  8. L18
    exact hf_right_right
  9. L19
    exact hf_right_left
  10. L20
    exact hdiv

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro p
  6. 0006intro hf
  7. 0007intro hp
  8. 0008intro hdiv
  9. 0009cases hf
  10. 0010cases hf_right
  11. 0011specialize beta_prime_divisor_product_member (b)
  12. 0012specialize beta_prime_divisor_product_member (c)
  13. 0013specialize beta_prime_divisor_product_member (l)
  14. 0014specialize beta_prime_divisor_product_member (n)
  15. 0015specialize beta_prime_divisor_product_member (p)
  16. 0016apply beta_prime_divisor_product_member
  17. 0017exact hp
  18. 0018exact hf_right_right
  19. 0019exact hf_right_left
  20. 0020exact hdiv