Mathlib v4.30 source proposition Nat.Odd.of_mul_left

Identifier: F:ml430-nat-odd-of-mul-left-2c6c2553
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, Odd (m * n) → Odd m

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Nat.odd_of_mul_left

Kind: kernel-term
Status: checked

Supports: `Nat.odd_of_mul_left` (this kernel's name for Mathlib's `Nat.Odd.of_mul_left`, a dot-notation extension) is admitted as a Theorem with EXACTLY the mirror's shape, pinned verbatim via the kernel's own renderer. Case-splits `m` via `Nat.even_or_odd_exists`: the `Even m` branch contradicts `Odd (m*n)` through the private helper `Nat.even_mul_of_even_left` plus `Nat.odd_not_even`; the `Odd m` branch is the goal already.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory odd_of_mul_left 2>/dev/null | awk -F'\t' '$1 == "Nat.odd_of_mul_left" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.Odd (AxNat.mul x0 x1)) -> AxNat.Odd x0)))"' | wc -l)" -ge 1

instance-Nat.odd_of_mul_left

Kind: kernel-term
Status: checked

Supports: Applies at a concrete discriminating instance (m := 3, n := 5, mul m n := 15, Odd, landing on Odd 3) with a transposed negative control (must NOT land on Odd 5), AND at a genuinely free (m, n) pair (a symbolic restatement built via the same NatOps closure machinery `d.theorem` uses).

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::odd_of_mul_left_and_right_apply_at_a_concrete_instance_and_symbolically -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

footprint-Nat.odd_of_mul_left

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.odd_of_mul_left` and `Nat.even_mul_of_even_left`. The `nat` prelude's trusted surface stays 0: no Axiom, no Opaque, no Quotient.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free nat

coverage-Nat.odd_of_mul_left

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.odd_of_mul_left` derived from `kernel.environment()` directly, not from an inventory list -- a Nat-namespace declaration missing from `theorem_names` fails this test by name.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane nat-parity-div (2026-08-30): built directly as a kernel declaration in the new file nat_prelude/parity_div.rs, over a new private helper `Nat.even_mul_of_even_left` (Even m -> Even (m*n), via `Nat.right_distrib` on the `k+k` witness), `Nat.even_or_odd_exists` and `Nat.odd_not_even`. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat parity/multiplication lemma about the SAME Nat.mul/Nat.Odd this kernel already has (no new definition introduced), so mirroring it is honest under the mirror-flip criterion.",
  "source": "statement-only extraction of `Nat.Odd.of_mul_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.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"
    }
  ]
}