Mathlib v4.30 source proposition Int.dvd_of_dvd_mul_left_of_gcd_one

Identifier: F:ml430-int-dvd-of-dvd-mul-left-of-gcd-one-649e349b
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b c : ℤ}, a ∣ b * c → a.gcd c = 1 → a ∣ b

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

3 direct dependencies 1 direct dependents

Evidence

kernel-Int.dvd_of_dvd_mul_left_of_gcd_one

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_dvd_of_dvd_mul_left_of_gcd_one` (`crates/axeyum-lean-kernel/src/int_prelude/gcd.rs`): `Int.gauss_lemma` applied at `(a, c, b)` needs `a ∣ (c*b)`, not the given `a ∣ (b*c)`; the two witnesses differ by `Int.mul_comm b c`, so the hypothesis is eliminated, its equation rewritten, and the divisibility re-introduced before feeding `gauss_lemma` (same pattern `declare_dvd_mul_left` uses for `a ∣ (b*a)` from `a ∣ (a*b)`).

footprint-Int.dvd_of_dvd_mul_left_of_gcd_one

Kind: exhaustive-enumeration
Status: checked

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

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Int.dvd_of_dvd_mul_left_of_gcd_one 2>/dev/null | grep -Ec 'integer[[:space:]]Int\.dvd_of_dvd_mul_left_of_gcd_one[[:space:]]0[[:space:]]*$')" -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. Run `--release`: in debug, building the full constructed environment stack-overflows, which is a resource limit and not a fact about this theorem.

Provenance

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