AF000B

factor_permutation_unit_length_zero

The only prime factorization of one has empty length; this is an actual-product statement, not a convention imposed on a list.

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. PrimeFactorList(n,b,c,l) → n = 1 → l = 0

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

Definition DAG

PrimeFactorList(n,b,c,l) · 1

Actual proof prerequisites

beta_all_prime_product_one_iff_length_zero · checked external prerequisite
Original expanded first-order statement 
forall n b c l. ((~(n = 0) /\ ((exists ff_u_fsat_unit_product ff_v_fsat_unit_product. ((((exists ff_h_fsat_unit_product_start. ff_h_fsat_unit_product_start + S (1) = S ((S (0)) * ff_v_fsat_unit_product)) /\ exists ff_q_fsat_unit_product_start. ff_u_fsat_unit_product = ff_q_fsat_unit_product_start * S ((S (0)) * ff_v_fsat_unit_product) + (1))) /\ ((((exists ff_h_fsat_unit_product_terminal. ff_h_fsat_unit_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_unit_product)) /\ exists ff_q_fsat_unit_product_terminal. ff_u_fsat_unit_product = ff_q_fsat_unit_product_terminal * S ((S (l)) * ff_v_fsat_unit_product) + (n))) /\ forall ff_i_fsat_unit_product. (exists ff_lt_fsat_unit_product_bound. ff_lt_fsat_unit_product_bound + S ff_i_fsat_unit_product = l) -> exists ff_p_fsat_unit_product ff_r_fsat_unit_product ff_s_fsat_unit_product. ((((exists ff_h_fsat_unit_product_factor. ff_h_fsat_unit_product_factor + S (ff_p_fsat_unit_product) = S ((S (ff_i_fsat_unit_product)) * c)) /\ exists ff_q_fsat_unit_product_factor. b = ff_q_fsat_unit_product_factor * S ((S (ff_i_fsat_unit_product)) * c) + (ff_p_fsat_unit_product))) /\ ((((exists ff_h_fsat_unit_product_partial. ff_h_fsat_unit_product_partial + S (ff_r_fsat_unit_product) = S ((S (ff_i_fsat_unit_product)) * ff_v_fsat_unit_product)) /\ exists ff_q_fsat_unit_product_partial. ff_u_fsat_unit_product = ff_q_fsat_unit_product_partial * S ((S (ff_i_fsat_unit_product)) * ff_v_fsat_unit_product) + (ff_r_fsat_unit_product))) /\ ((((exists ff_h_fsat_unit_product_successor. ff_h_fsat_unit_product_successor + S (ff_s_fsat_unit_product) = S ((S (S ff_i_fsat_unit_product)) * ff_v_fsat_unit_product)) /\ exists ff_q_fsat_unit_product_successor. ff_u_fsat_unit_product = ff_q_fsat_unit_product_successor * S ((S (S ff_i_fsat_unit_product)) * ff_v_fsat_unit_product) + (ff_s_fsat_unit_product))) /\ ff_s_fsat_unit_product = ff_r_fsat_unit_product * ff_p_fsat_unit_product)))))) /\ (forall ftsf_index_fsat_unit_primes. (exists ftsf_gap_fsat_unit_primes_bound. ftsf_gap_fsat_unit_primes_bound + S ftsf_index_fsat_unit_primes = (l)) -> exists ftsf_factor_fsat_unit_primes. ((((exists ff_h_ftsf_fsat_unit_primes_entry. ff_h_ftsf_fsat_unit_primes_entry + S (ftsf_factor_fsat_unit_primes) = S ((S (ftsf_index_fsat_unit_primes)) * c)) /\ exists ff_q_ftsf_fsat_unit_primes_entry. b = ff_q_ftsf_fsat_unit_primes_entry * S ((S (ftsf_index_fsat_unit_primes)) * c) + (ftsf_factor_fsat_unit_primes))) /\ ((~(ftsf_factor_fsat_unit_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unit_primes_prime frm_prime_right_ftsf_fsat_unit_primes_prime. ftsf_factor_fsat_unit_primes = frm_prime_left_ftsf_fsat_unit_primes_prime * frm_prime_right_ftsf_fsat_unit_primes_prime -> frm_prime_left_ftsf_fsat_unit_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unit_primes_prime = 1))))))) -> n = 1 -> l = 0

Complete tactic proof in conservative notation

All 19 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

19 script commands · 5 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.

01Fix variables and assumptionsL1–6

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 hf
  6. L6
    intro hunit
02Separate the logical casesL7–8

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

  1. L7
    cases hf
  2. L8
    cases hf_right
03Establish hiffL9–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta all prime product one iff length zero.

  1. L9
    have hiff : (n = 1 -> l = 0) /\ (l = 0 -> n = 1)
  2. L10
    specialize beta_all_prime_product_one_iff_length_zero (b)
  3. L11
    specialize beta_all_prime_product_one_iff_length_zero (c)
  4. L12
    specialize beta_all_prime_product_one_iff_length_zero (l)
  5. L13
    specialize beta_all_prime_product_one_iff_length_zero (n)
  6. L14
    apply beta_all_prime_product_one_iff_length_zero
  7. L15
    exact hf_right_right
  8. L16
    exact hf_right_left
04Separate the logical casesL17–17

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

  1. L17
    cases hiff
05Use earlier factsL18–19

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

  1. L18
    apply hiff_left
  2. L19
    exact hunit

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hf
  6. 0006intro hunit
  7. 0007cases hf
  8. 0008cases hf_right
  9. 0009have hiff : (n = 1 -> l = 0) /\ (l = 0 -> n = 1)
  10. 0010specialize beta_all_prime_product_one_iff_length_zero (b)
  11. 0011specialize beta_all_prime_product_one_iff_length_zero (c)
  12. 0012specialize beta_all_prime_product_one_iff_length_zero (l)
  13. 0013specialize beta_all_prime_product_one_iff_length_zero (n)
  14. 0014apply beta_all_prime_product_one_iff_length_zero
  15. 0015exact hf_right_right
  16. 0016exact hf_right_left
  17. 0017cases hiff
  18. 0018apply hiff_left
  19. 0019exact hunit