Mathlib v4.30 source proposition Int.gcd_div

Identifier: F:ml430-int-gcd-div-5e01872f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b c : ℤ}, c ∣ a → c ∣ b → (a / c).gcd (b / c) = a.gcd b / c.natAbs

Dependencies

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

Evidence

kernel-Int.gcd_div

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`Int.gcd_div` in `crates/axeyum-lean-kernel/src/int_prelude/gcd.rs` (`declare_gcd_div`). Mirrors Mathlib v4.30's `Int.gcd_div` (`alias gcd_div := gcd_ediv`, `Mathlib/Data/Int/GCD.lean`) exactly: Lean 4 core's `Int.gcd_ediv` (`Init.Data.Int.Gcd`) states the same proposition over `/`, and core's own `instance : Div Int` binds `/` to `Int.ediv` "for compatibility with SMT-LIB" (`Init.Data.Int.DivMod.Basic`) -- the SAME division this development's `Int.ediv` matches bit for bit (verified against the pinned toolchain source, not inferred). This is a same-definition mirror (honest flip), not a restatement of a different proposition, and it carries NO restriction on `c`'s sign or `c = 0`, matching Mathlib's own unrestricted hypotheses (`c ∣ a`, `c ∣ b`) exactly -- `c = 0` is proved here as a genuine degenerate case, not excluded. Proved by mutual divisibility (Bezout on `a,b` and on `a/c,b/c`, entirely through `natAbs` identities with no case split on `c`'s sign needed for the algebra itself), NOT via Lean core's own route through `Nat.gcd_div`/`Nat.gcd_mul_left` -- neither exists in this development and either would need a fresh strong-induction principle over `Nat.gcd`'s well-founded recursion. `int_prelude::` sweep: 45 -> 47 (one bridge lemma `Int.emod_eq_zero_iff_dvd_general` plus this theorem).

footprint-Int.gcd_div

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

Evidence notes

The matched row is `integer<TAB>Int.gcd_div<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. Uses `/usr/bin/grep` explicitly (not the interactive `ugrep` alias) and ANSI-C-quoted literal tabs, per this repository's own grep-dialect gotcha. 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": "the `int-gcd-div` lane, completing the `int-emod-negative` lane's two landed sign-general division-algorithm lemmas (`Int.emod_natAbs_bound`, `Int.ediv_emod_unique_general`) with a third bridge lemma (`Int.emod_eq_zero_iff_dvd_general`) and the mutual-divisibility argument for `Int.gcd_div` itself",
  "source": "statement-only extraction of `Int.gcd_div` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.gcd_div`",
      "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"
    }
  ]
}