Mathlib v4.30 source proposition Int.Odd.of_mul_left

Identifier: F:ml430-int-odd-of-mul-left-b580971e
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℤ}, Odd (m * n) → Odd m

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Int.odd_of_mul_left

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`Int.odd_of_mul_left` in `crates/axeyum-lean-kernel/src/int_prelude/parity.rs` (`declare_odd_of_mul_left`). Mirrors Mathlib v4.30's `Odd.of_mul_left` (`Mathlib/Algebra/Ring/Int/Parity.lean`) verbatim. Routes entirely through `Int.natAbs` being multiplicative (`nat_abs_mul`, `gcd.rs`) -- no `Int.rec` needed, since sign plays no role. Contrapositive: `Nat.Even (natAbs m)` gives `Nat.Even (natAbs m * natAbs n)` (a new helper `nat_even_mul_of_even_left`, via `Nat.right_distrib`), hence `Not (Nat.Odd (natAbs m * natAbs n))` (`Nat.even_not_odd`), refuting the rewritten hypothesis; a new helper `nat_not_even_implies_odd` (contrapositive of `Nat.even_or_odd_exists`) closes it. The `Nat`-level distributivity content used here has no home in `nat_prelude` yet and is built module-locally in `int_prelude/parity.rs`, not added to that crate.

footprint-Int.odd_of_mul_left

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

Evidence notes

The matched row is `integer<TAB>Int.odd_of_mul_left<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": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.Odd.of_mul_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.Odd.of_mul_left`",
      "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"
    }
  ]
}