Mathlib v4.30 source proposition Int.gcd_eq_one_of_gcd_mul_right_eq_one_left

Identifier: F:ml430-int-gcd-eq-one-of-gcd-mul-right-eq-one-left-8533eb82
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a : ℤ} {m n : ℕ}, a.gcd (↑m * ↑n) = 1 → a.gcd ↑m = 1

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Int.gcd_eq_one_of_gcd_mul_right_eq_one_left

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_gcd_eq_one_of_gcd_mul_right_eq_one` (`crates/axeyum-lean-kernel/src/int_prelude/gcd.rs`): `Int.mul (ofNat m) (ofNat n)` reduces to `ofNat (m*n)` by iota-reduction, so both the hypothesis and the conclusion unfold to plain `Nat.gcd` statements with no cast lemma needed; `Nat.dvd_mul` gives `m | m*n` directly, and `Nat.coprime_of_dvd_right` closes it.

footprint-Int.gcd_eq_one_of_gcd_mul_right_eq_one_left

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 int_theorem_inventory -- gcd_eq_one_of_gcd_mul_right_eq_one_left 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.gcd_eq_one_of_gcd_mul_right_eq_one_left[[:space:]]+\(\(')" -ge 1

Evidence notes

The regex requires the tab-separated footprint column to be EMPTY. `int_theorem_inventory` must be run `--release`; in debug it stack-overflows building the full constructed environment (unrelated to this theorem).

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.gcd_eq_one_of_gcd_mul_right_eq_one_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.gcd_eq_one_of_gcd_mul_right_eq_one_left`",
      "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"
    }
  ]
}