Mathlib v4.30 source proposition Int.gcd_eq_gcd_ab

Identifier: F:ml430-int-gcd-eq-gcd-ab-63005aef
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (x y : ℤ), ↑(x.gcd y) = x * x.gcdA y + y * x.gcdB y

Dependencies

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4 direct dependencies 4 direct dependents

Evidence

kernel-Int.gcd_eq_gcd_ab_witnesses

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Int.gcd_eq_gcd_ab` in the pinned Mathlib v4.30 source, at the named computable witnesses.

Checker command

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

Evidence notes

`Int.gcd_eq_gcd_ab_witnesses` in `crates/axeyum-lean-kernel/src/int_prelude/bezout_witnesses.rs`. The name differs from Mathlib's `Int.gcd_eq_gcd_ab` only because THIS prelude already owned that name for the strictly weaker EXISTENTIAL form (`∀ a b, ∃ u v, ofNat (gcd a b) = a*u + b*v`, `int_prelude/gcd.rs`), which `crt.rs` and `modinv.rs` consume; the statement here is Mathlib's verbatim. The witnesses are genuine `Definition`s that return data: `Nat.xgcdAux` is the extended Euclidean recursion, structural on a FUEL argument rather than `WellFounded` (whose equation-compiler route carries `propext`/`Quot.sound`), with a trailing `Bool` selecting which coefficient to return so one recursion carries the pair. `Nat.xgcdAux_sound` is the induction on the fuel with both `m` and `n` generalized in the motive; `Int.gcdA`/`Int.gcdB` are Mathlib's signed definitions and the lift is a four-branch `Int.rec` case split.

footprint-Int.gcd_eq_gcd_ab_witnesses

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.gcd_eq_gcd_ab_witnesses 2>/dev/null | grep -xFc $'integer\tInt.gcd_eq_gcd_ab_witnesses\t0\t')" -ge 1

Evidence notes

The matched row is `integer<TAB>Int.gcd_eq_gcd_ab_witnesses<TAB>0<TAB>` with an EMPTY trailing footprint column, so a nonzero footprint or an absent declaration both give a count of 0 and fail. `grep -xF` with `$'...'` puts real tabs in the pattern before grep sees them -- do not rewrite this as `grep -E '\t'`, which GNU grep reads as a literal `t`. Run `--release`: in debug, building the full constructed environment stack-overflows, which is a resource limit and not a fact about this theorem.

compute-Int.gcdA-Int.gcdB

Kind: witness-replay
Status: checked

Supports: `Int.gcdA` and `Int.gcdB` are computable `Definition`s whose values satisfy the identity numerically, in all four sign branches.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::int_gcd_ab_compute_in_every_sign_branch 2>&1 | grep -c 'test result: ok. 1 passed')" -ge 1

Evidence notes

The theorem alone does NOT pin the algorithm down -- some pair of coefficients satisfies Bézout for any correct gcd, so a transposed selector or an off-by-one fuel would still type-check. This evaluates `Int.gcdA x y` and `Int.gcdB x y` to normal form against hand-computed answers at (3,2), (-3,2), (3,-2), (-3,-2), (0,5), (-4,6), then evaluates `x*gcdA + y*gcdB` and compares against the gcd. It is also the right shape of checker for a CONSTRUCTION: a theorem inventory reports zero rows for a `Definition` and cannot assert one exists. Matching `1 passed` rather than the exit status is deliberate -- a mistyped filter runs zero tests and exits 0.

Provenance

{
  "date": "2026-08-28",
  "established_by": "the `int-bezout-witnesses` lane, by constructing the extended Euclidean algorithm and re-deriving Bézout for it",
  "source": "statement-only extraction of `Int.gcd_eq_gcd_ab` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.gcd_eq_gcd_ab`",
      "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"
    }
  ]
}