AF0003

foundation_coprime_product_divisor

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

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

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 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 11 exact native proof lines.

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

Proof neighborhood

Direct dependencies

gauss_coprime_cancel Stable theorem; checked-use authorized

Direct dependents

none

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

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.

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 exact 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