BT002W

coprime_symm

Stable checked-use theorem · independently kernel verified

Coprimality in its expanded common-divisor form is symmetric.

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 PA statement

forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> forall c. (exists u. b = c * u) -> (exists v. a = c * v) -> c = 1

Structural proof guide

Coprimality in its expanded common-divisor form is symmetric.

Direct prerequisites: none. The authored body proceeds by direct introduction and elimination.

Proof neighborhood

Direct dependencies

none

Direct dependents

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

10 script commands · 2 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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
  4. L4
    intro c
  5. L5
    intro hb
  6. L6
    intro ha
02Use earlier factsL7–10

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

  1. L7
    specialize h c
  2. L8
    apply h
  3. L9
    exact ha
  4. L10
    exact hb

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro c
  5. 0005intro hb
  6. 0006intro ha
  7. 0007specialize h c
  8. 0008apply h
  9. 0009exact ha
  10. 0010exact hb