AF0008

factor_permutation_product_exists

Construct an actual product value and trace for every finite beta prefix, including empty prefixes.

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

∀ b. ∀ c. ∀ l. ∃ n. Product(b,c,l,n)

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

Definition DAG

Product(b,c,l,z) · 1

Actual proof prerequisites

beta_product_exists_unique · checked external prerequisite
Original expanded first-order statement 
forall b c l. exists n. (exists ff_u_fsat_product_exists ff_v_fsat_product_exists. ((((exists ff_h_fsat_product_exists_start. ff_h_fsat_product_exists_start + S (1) = S ((S (0)) * ff_v_fsat_product_exists)) /\ exists ff_q_fsat_product_exists_start. ff_u_fsat_product_exists = ff_q_fsat_product_exists_start * S ((S (0)) * ff_v_fsat_product_exists) + (1))) /\ ((((exists ff_h_fsat_product_exists_terminal. ff_h_fsat_product_exists_terminal + S (n) = S ((S (l)) * ff_v_fsat_product_exists)) /\ exists ff_q_fsat_product_exists_terminal. ff_u_fsat_product_exists = ff_q_fsat_product_exists_terminal * S ((S (l)) * ff_v_fsat_product_exists) + (n))) /\ forall ff_i_fsat_product_exists. (exists ff_lt_fsat_product_exists_bound. ff_lt_fsat_product_exists_bound + S ff_i_fsat_product_exists = l) -> exists ff_p_fsat_product_exists ff_r_fsat_product_exists ff_s_fsat_product_exists. ((((exists ff_h_fsat_product_exists_factor. ff_h_fsat_product_exists_factor + S (ff_p_fsat_product_exists) = S ((S (ff_i_fsat_product_exists)) * c)) /\ exists ff_q_fsat_product_exists_factor. b = ff_q_fsat_product_exists_factor * S ((S (ff_i_fsat_product_exists)) * c) + (ff_p_fsat_product_exists))) /\ ((((exists ff_h_fsat_product_exists_partial. ff_h_fsat_product_exists_partial + S (ff_r_fsat_product_exists) = S ((S (ff_i_fsat_product_exists)) * ff_v_fsat_product_exists)) /\ exists ff_q_fsat_product_exists_partial. ff_u_fsat_product_exists = ff_q_fsat_product_exists_partial * S ((S (ff_i_fsat_product_exists)) * ff_v_fsat_product_exists) + (ff_r_fsat_product_exists))) /\ ((((exists ff_h_fsat_product_exists_successor. ff_h_fsat_product_exists_successor + S (ff_s_fsat_product_exists) = S ((S (S ff_i_fsat_product_exists)) * ff_v_fsat_product_exists)) /\ exists ff_q_fsat_product_exists_successor. ff_u_fsat_product_exists = ff_q_fsat_product_exists_successor * S ((S (S ff_i_fsat_product_exists)) * ff_v_fsat_product_exists) + (ff_s_fsat_product_exists))) /\ ff_s_fsat_product_exists = ff_r_fsat_product_exists * ff_p_fsat_product_exists))))))

Complete tactic proof in conservative notation

All 10 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

10 script commands · 5 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 assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Use earlier factsL4–6

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

  1. L4
    specialize beta_product_exists_unique b
  2. L5
    specialize beta_product_exists_unique c
  3. L6
    specialize beta_product_exists_unique l
03Separate the logical casesL7–8

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

  1. L7
    cases beta_product_exists_unique
  2. L8
    cases beta_product_exists_unique_witness
04Construct an explicit witnessL9–9

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

  1. L9
    exists x
05Use earlier factsL10–10

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

  1. L10
    exact beta_product_exists_unique_witness_left

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004specialize beta_product_exists_unique b
  5. 0005specialize beta_product_exists_unique c
  6. 0006specialize beta_product_exists_unique l
  7. 0007cases beta_product_exists_unique
  8. 0008cases beta_product_exists_unique_witness
  9. 0009exists x
  10. 0010exact beta_product_exists_unique_witness_left