AF000C

factor_permutation_prime_member

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_prime_divisor_product_member Stable theorem; checked-use authorized

Direct dependents

AF0019 prime_factor_lists_matching_by_length

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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