AF0004

foundation_prime_factor_list_exists

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

G004: every positive natural has a genuinely constructed finite beta-coded prime-factor list and actual product trace; the existing sorted construction is used only to obtain witnesses, not required as a premise.

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. ~(n = 0) -> exists l b c. ((~(n = 0) /\ ((exists ff_u_fsat_exists_product ff_v_fsat_exists_product. ((((exists ff_h_fsat_exists_product_start. ff_h_fsat_exists_product_start + S (1) = S ((S (0)) * ff_v_fsat_exists_product)) /\ exists ff_q_fsat_exists_product_start. ff_u_fsat_exists_product = ff_q_fsat_exists_product_start * S ((S (0)) * ff_v_fsat_exists_product) + (1))) /\ ((((exists ff_h_fsat_exists_product_terminal. ff_h_fsat_exists_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_exists_product)) /\ exists ff_q_fsat_exists_product_terminal. ff_u_fsat_exists_product = ff_q_fsat_exists_product_terminal * S ((S (l)) * ff_v_fsat_exists_product) + (n))) /\ forall ff_i_fsat_exists_product. (exists ff_lt_fsat_exists_product_bound. ff_lt_fsat_exists_product_bound + S ff_i_fsat_exists_product = l) -> exists ff_p_fsat_exists_product ff_r_fsat_exists_product ff_s_fsat_exists_product. ((((exists ff_h_fsat_exists_product_factor. ff_h_fsat_exists_product_factor + S (ff_p_fsat_exists_product) = S ((S (ff_i_fsat_exists_product)) * c)) /\ exists ff_q_fsat_exists_product_factor. b = ff_q_fsat_exists_product_factor * S ((S (ff_i_fsat_exists_product)) * c) + (ff_p_fsat_exists_product))) /\ ((((exists ff_h_fsat_exists_product_partial. ff_h_fsat_exists_product_partial + S (ff_r_fsat_exists_product) = S ((S (ff_i_fsat_exists_product)) * ff_v_fsat_exists_product)) /\ exists ff_q_fsat_exists_product_partial. ff_u_fsat_exists_product = ff_q_fsat_exists_product_partial * S ((S (ff_i_fsat_exists_product)) * ff_v_fsat_exists_product) + (ff_r_fsat_exists_product))) /\ ((((exists ff_h_fsat_exists_product_successor. ff_h_fsat_exists_product_successor + S (ff_s_fsat_exists_product) = S ((S (S ff_i_fsat_exists_product)) * ff_v_fsat_exists_product)) /\ exists ff_q_fsat_exists_product_successor. ff_u_fsat_exists_product = ff_q_fsat_exists_product_successor * S ((S (S ff_i_fsat_exists_product)) * ff_v_fsat_exists_product) + (ff_s_fsat_exists_product))) /\ ff_s_fsat_exists_product = ff_r_fsat_exists_product * ff_p_fsat_exists_product)))))) /\ (forall ftsf_index_fsat_exists_primes. (exists ftsf_gap_fsat_exists_primes_bound. ftsf_gap_fsat_exists_primes_bound + S ftsf_index_fsat_exists_primes = (l)) -> exists ftsf_factor_fsat_exists_primes. ((((exists ff_h_ftsf_fsat_exists_primes_entry. ff_h_ftsf_fsat_exists_primes_entry + S (ftsf_factor_fsat_exists_primes) = S ((S (ftsf_index_fsat_exists_primes)) * c)) /\ exists ff_q_ftsf_fsat_exists_primes_entry. b = ff_q_ftsf_fsat_exists_primes_entry * S ((S (ftsf_index_fsat_exists_primes)) * c) + (ftsf_factor_fsat_exists_primes))) /\ ((~(ftsf_factor_fsat_exists_primes = 1) /\ forall frm_prime_left_ftsf_fsat_exists_primes_prime frm_prime_right_ftsf_fsat_exists_primes_prime. ftsf_factor_fsat_exists_primes = frm_prime_left_ftsf_fsat_exists_primes_prime * frm_prime_right_ftsf_fsat_exists_primes_prime -> frm_prime_left_ftsf_fsat_exists_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_exists_primes_prime = 1)))))))

Constructive proof overview

Generated structural guide

G004: every positive natural has a genuinely constructed finite beta-coded prime-factor list and actual product trace; the existing sorted construction is used only to obtain witnesses, not required as a premise.

The unchanged tactic script uses 1 declared prerequisite and contains 19 exact native proof lines.

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

Proof neighborhood

Direct dependencies

prime_factorization_existence Stable theorem; checked-use authorized

Direct dependents

AF001B prime_factorization_exists_unique_up_to_permutation

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

19 script commands · 9 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hn
02Use earlier factsL3–3

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

  1. L3
    specialize prime_factorization_existence n
03Establish hexistsL4–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime factorization existence.

  1. L4
    have hexists : ∃ l. ∃ b. ∃ c. Product(b,c,l,n) ∧ (AllPrime(b,c,l) ∧ (∀ x. Lt(S x,l) → ∃ y. ∃ z. BetaAt(b,c,x,y) ∧ (BetaAt(b,c,S x,z) ∧ y ≤ z)))Definitions: LtBetaAtProductAllPrime
  2. L5
    apply prime_factorization_existence
  3. L6
    exact hn
04Separate the logical casesL7–11

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

  1. L7
    cases hexists
  2. L8
    cases hexists_witness
  3. L9
    cases hexists_witness_witness
  4. L10
    cases hexists_witness_witness_witness
  5. L11
    cases hexists_witness_witness_witness_right
05Construct an explicit witnessL12–14

Supply the displayed value, then prove that it has the required property.

  1. L12
    exists x
  2. L13
    exists x1
  3. L14
    exists x2
06Separate the logical casesL15–15

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

  1. L15
    split
07Use earlier factsL16–16

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

  1. L16
    exact hn
08Separate the logical casesL17–17

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

  1. L17
    split
09Use earlier factsL18–19

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

  1. L18
    exact hexists_witness_witness_witness_left
  2. L19
    exact hexists_witness_witness_witness_right_left

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003specialize prime_factorization_existence n
  4. 0004have hexists : exists l b c. ((exists ff_u_fsat_canonical_product ff_v_fsat_canonical_product. ((((exists ff_h_fsat_canonical_product_start. ff_h_fsat_canonical_product_start + S (1) = S ((S (0)) * ff_v_fsat_canonical_product)) /\ exists ff_q_fsat_canonical_product_start. ff_u_fsat_canonical_product = ff_q_fsat_canonical_product_start * S ((S (0)) * ff_v_fsat_canonical_product) + (1))) /\ ((((exists ff_h_fsat_canonical_product_terminal. ff_h_fsat_canonical_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_canonical_product)) /\ exists ff_q_fsat_canonical_product_terminal. ff_u_fsat_canonical_product = ff_q_fsat_canonical_product_terminal * S ((S (l)) * ff_v_fsat_canonical_product) + (n))) /\ forall ff_i_fsat_canonical_product. (exists ff_lt_fsat_canonical_product_bound. ff_lt_fsat_canonical_product_bound + S ff_i_fsat_canonical_product = l) -> exists ff_p_fsat_canonical_product ff_r_fsat_canonical_product ff_s_fsat_canonical_product. ((((exists ff_h_fsat_canonical_product_factor. ff_h_fsat_canonical_product_factor + S (ff_p_fsat_canonical_product) = S ((S (ff_i_fsat_canonical_product)) * c)) /\ exists ff_q_fsat_canonical_product_factor. b = ff_q_fsat_canonical_product_factor * S ((S (ff_i_fsat_canonical_product)) * c) + (ff_p_fsat_canonical_product))) /\ ((((exists ff_h_fsat_canonical_product_partial. ff_h_fsat_canonical_product_partial + S (ff_r_fsat_canonical_product) = S ((S (ff_i_fsat_canonical_product)) * ff_v_fsat_canonical_product)) /\ exists ff_q_fsat_canonical_product_partial. ff_u_fsat_canonical_product = ff_q_fsat_canonical_product_partial * S ((S (ff_i_fsat_canonical_product)) * ff_v_fsat_canonical_product) + (ff_r_fsat_canonical_product))) /\ ((((exists ff_h_fsat_canonical_product_successor. ff_h_fsat_canonical_product_successor + S (ff_s_fsat_canonical_product) = S ((S (S ff_i_fsat_canonical_product)) * ff_v_fsat_canonical_product)) /\ exists ff_q_fsat_canonical_product_successor. ff_u_fsat_canonical_product = ff_q_fsat_canonical_product_successor * S ((S (S ff_i_fsat_canonical_product)) * ff_v_fsat_canonical_product) + (ff_s_fsat_canonical_product))) /\ ff_s_fsat_canonical_product = ff_r_fsat_canonical_product * ff_p_fsat_canonical_product)))))) /\ ((forall ftsf_index_fsat_canonical_primes. (exists ftsf_gap_fsat_canonical_primes_bound. ftsf_gap_fsat_canonical_primes_bound + S ftsf_index_fsat_canonical_primes = (l)) -> exists ftsf_factor_fsat_canonical_primes. ((((exists ff_h_ftsf_fsat_canonical_primes_entry. ff_h_ftsf_fsat_canonical_primes_entry + S (ftsf_factor_fsat_canonical_primes) = S ((S (ftsf_index_fsat_canonical_primes)) * c)) /\ exists ff_q_ftsf_fsat_canonical_primes_entry. b = ff_q_ftsf_fsat_canonical_primes_entry * S ((S (ftsf_index_fsat_canonical_primes)) * c) + (ftsf_factor_fsat_canonical_primes))) /\ ((~(ftsf_factor_fsat_canonical_primes = 1) /\ forall frm_prime_left_ftsf_fsat_canonical_primes_prime frm_prime_right_ftsf_fsat_canonical_primes_prime. ftsf_factor_fsat_canonical_primes = frm_prime_left_ftsf_fsat_canonical_primes_prime * frm_prime_right_ftsf_fsat_canonical_primes_prime -> frm_prime_left_ftsf_fsat_canonical_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_canonical_primes_prime = 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))))))
  5. 0005apply prime_factorization_existence
  6. 0006exact hn
  7. 0007cases hexists
  8. 0008cases hexists_witness
  9. 0009cases hexists_witness_witness
  10. 0010cases hexists_witness_witness_witness
  11. 0011cases hexists_witness_witness_witness_right
  12. 0012exists x
  13. 0013exists x1
  14. 0014exists x2
  15. 0015split
  16. 0016exact hn
  17. 0017split
  18. 0018exact hexists_witness_witness_witness_left
  19. 0019exact hexists_witness_witness_witness_right_left