Mathlib v4.30 source proposition Int.gcd_greatest

Identifier: F:ml430-int-gcd-greatest-5b31c5fe
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b d : ℤ}, 0 ≤ d → d ∣ a → d ∣ b → (∀ (e : ℤ), e ∣ a → e ∣ b → e ∣ d) → d = ↑(a.gcd b)

Dependencies

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

11 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Int.gcd_greatest

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_gcd_greatest` (`crates/axeyum-lean-kernel/src/int_prelude/gcd.rs`): both `Int.dvd` directions come from the universal property already proved -- `d ∣ ofNat (gcd a b)` is `Int.dvd_gcd` fed `d`'s own two hypotheses directly (`c := d`), and `ofNat (gcd a b) ∣ d` is the fact's own universal hypothesis applied at `e := ofNat (gcd a b)` fed `Int.gcd_dvd_left`/`Int.gcd_dvd_right`. Mutual `Int.dvd` transports to mutual `Nat.dvd` between `natAbs d` and `gcd a b` via `Int.nat_abs_dvd_nat_abs_of_dvd` (`natAbs (ofNat n)` reduces to `n` by `rfl`), closed by the private `nat_dvd_antisymm` engine `gcd_comm`/`gcd_zero_right` already use, and `0 ≤ d` lifts the resulting `Eq Nat (natAbs d) (gcd a b)` back up to `Eq Int d (ofNat (gcd a b))` via `Int.of_nat_nat_abs_of_nonneg`.

footprint-Int.gcd_greatest

Kind: exhaustive-enumeration
Status: checked

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

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Int.gcd_greatest 2>/dev/null | grep -Ec 'integer[[:space:]]Int\.gcd_greatest[[:space:]]0[[:space:]]*$')" -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.gcd_greatest` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.gcd_greatest`",
      "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"
    }
  ]
}