AF0003

foundation_coprime_product_divisor

G003: coprimality and a witnessed divisor of the product construct an actual quotient of the other factor, with no positivity premise added.

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

∀ a. ∀ b. ∀ c. Coprime(a,b) ∧ Dvd(a,b · c) → Dvd(a,c)

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

Definition DAG

Dvd(d,n) · 2Coprime(a,b) · 1

Actual proof prerequisites

gauss_coprime_cancel · checked external prerequisite
Original expanded first-order statement 
forall a b c. ((forall frp_divisor_fsat_euclid. (exists frp_left_factor_fsat_euclid. a = frp_divisor_fsat_euclid * frp_left_factor_fsat_euclid) -> (exists frp_right_factor_fsat_euclid. b = frp_divisor_fsat_euclid * frp_right_factor_fsat_euclid) -> frp_divisor_fsat_euclid = 1) /\ (exists q. b * c = a * q)) -> exists q. c = a * q

Complete tactic proof in conservative notation

All 11 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

11 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro h
02Separate the logical casesL5–5

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

  1. L5
    cases h
03Use earlier factsL6–11

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

  1. L6
    specialize gauss_coprime_cancel (a)
  2. L7
    specialize gauss_coprime_cancel (b)
  3. L8
    specialize gauss_coprime_cancel (c)
  4. L9
    apply gauss_coprime_cancel
  5. L10
    exact h_left
  6. L11
    exact h_right

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006specialize gauss_coprime_cancel (a)
  7. 0007specialize gauss_coprime_cancel (b)
  8. 0008specialize gauss_coprime_cancel (c)
  9. 0009apply gauss_coprime_cancel
  10. 0010exact h_left
  11. 0011exact h_right