Mathlib v4.30 source proposition Int.exists_gcd_one'

Identifier: F:ml430-int-exists-gcd-one-657db3e2
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.exists_gcd_one'` in the pinned Mathlib v4.30 source.

Formal statement

∀ {m n : ℤ}, 0 < m.gcd n → ∃ g m' n', 0 < g ∧ m'.gcd n' = 1 ∧ m = m' * ↑g ∧ n = n' * ↑g

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Int.exists_gcd_one_prime

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Int.exists_gcd_one'` in the pinned Mathlib v4.30 source.

Checker command

cargo test -p axeyum-lean-kernel --lib int_prelude::

Evidence notes

`declare_exists_gcd_one_prime` (`crates/axeyum-lean-kernel/src/int_prelude/exists_gcd_one.rs`): reuses `declare_exists_gcd_one`'s construction verbatim at `g := gcd m n`, with the fact's own hypothesis `h : Lt zero g` doubling as the `0 < g` conjunct -- no new arithmetic, one more `Exists.intro`/`And.intro` layer wrapping `g` as an existential witness.

footprint-Int.exists_gcd_one_prime

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Int.exists_gcd_one 2>/dev/null | grep -Ec 'integer[[:space:]]Int\.exists_gcd_one'\''[[:space:]]0[[:space:]]*$')" -ge 1

Evidence notes

The matched row's trailing footprint column is EMPTY, so a nonzero footprint or an absent declaration both give a count of 0 and fail. The tool substring-matches on the query name, so passing `Int.exists_gcd_one` (without the trailing `'`) as the query still surfaces this row; the anchor picks the primed one out specifically.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.exists_gcd_one'` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.exists_gcd_one'`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}