Mathlib v4.30 source proposition Int.exists_gcd_one

Identifier: F:ml430-int-exists-gcd-one-d8820780
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 → ∃ m' n', m'.gcd n' = 1 ∧ m = m' * ↑(m.gcd n) ∧ n = n' * ↑(m.gcd n)

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Int.exists_gcd_one

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` (`crates/axeyum-lean-kernel/src/int_prelude/exists_gcd_one.rs`): witnesses `m' := m.ediv (ofNat (gcd m n))`, `n' := n.ediv (ofNat (gcd m n))`. Coprimality is `gcd.rs`'s already-checked `gcd_div_gcd_div_gcd` directly; `m = m'*c`/`n = n'*c` reuse that theorem's private `exact` closure (rebuilt locally) via `emod_eq_zero_iff_dvd` + `ediv_add_emod`, commuted with `mul_comm` to match the stated multiplication order.

footprint-Int.exists_gcd_one

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, so `Int.exists_gcd_one'` also appears in its output; the anchored `[[:space:]]0[[:space:]]*$` (whitespace, not `'`, right after the name) is what keeps this pattern from matching that row too. Pass exactly one name per invocation -- this tool keeps only the first.

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"
    }
  ]
}