Mathlib v4.30 source proposition Int.ne_zero_of_gcd

Identifier: F:ml430-int-ne-zero-of-gcd-f71f00df
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {x y : ℤ}, x.gcd y ≠ 0 → x ≠ 0 ∨ y ≠ 0

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Int.ne_zero_of_gcd

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_ne_zero_of_gcd` (`crates/axeyum-lean-kernel/src/int_prelude/gcd.rs`): `Int.eq_em x 0` case split; the `x != 0` branch is `Or.inl` directly, and the `x = 0` branch builds `y != 0` as a bare lambda -- assuming `y = 0` too, `Int.gcd_zero_right 0` (`gcd 0 0 = natAbs 0`, defeq `0`) transported along both equalities contradicts the hypothesis.

footprint-Int.ne_zero_of_gcd

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 -- ne_zero_of_gcd 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.ne_zero_of_gcd[[: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.ne_zero_of_gcd` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.ne_zero_of_gcd`",
      "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"
    }
  ]
}