PA0037

is_gcd_one_to_coprime

Stable checked-use theorem · independently closed

A relational gcd witness one implies expanded coprimality.

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

Structural proof guide

Generated structural guide

A relational gcd witness one implies expanded coprimality.

Use the direct prerequisites divisor_one as previously established PA formulas.

The proof proceeds by case analysis (1), intermediate claims (1).

Referenced ingredients

Proof neighborhood

Direct dependencies

Direct dependents

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

15 script commands · 4 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.

Named ingredients (1)
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 hg
  4. L4
    intro c
  5. L5
    intro ha
  6. L6
    intro hb
02Separate the logical casesL7–7

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

  1. L7
    cases hg
03Use earlier factsL8–8

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

  1. L8
    specialize hg_right c
04Establish hdL9–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hg right.

  1. L9
    have hd : exists w. 1 = c * w
  2. L10
    apply hg_right
  3. L11
    exact ha
  4. L12
    exact hb
  5. L13
    specialize divisor_one c
  6. L14
    apply divisor_one
  7. L15
    exact hd

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hg
  4. 0004intro c
  5. 0005intro ha
  6. 0006intro hb
  7. 0007cases hg
  8. 0008specialize hg_right c
  9. 0009have hd : exists w. 1 = c * w
  10. 0010apply hg_right
  11. 0011exact ha
  12. 0012exact hb
  13. 0013specialize divisor_one c
  14. 0014apply divisor_one
  15. 0015exact hd