Mathlib v4.30 source proposition Nat.even_add_one

Identifier: F:ml430-nat-even-add-one-15b5cb18
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Even (n + 1) ↔ ¬Even n

Dependencies

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3 direct dependencies 1 direct dependents

Evidence

kernel-Nat.even_add_one

Kind: kernel-term
Status: checked

Supports: `Nat.even_add_one` is admitted as a Theorem with EXACTLY the mirror's shape, pinned verbatim via the kernel's own renderer. Case-splits on `Nat.mod_two_eq_zero_or_one`: in each branch one side of the Iff is already fully decided independent of the other's hypothesis (n%2=0: Even n holds, so Even(n+1) and Not(Even n) are both false; n%2=1: Odd n holds, so Not(Even n) holds directly and Even(n+1) is built from the Odd witness via `Nat.succ_add`).

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory even_add_one 2>/dev/null | awk -F'\t' '$1 == "Nat.even_add_one" && $3 == "((x0 : AxNat) -> Iff (AxNat.Even (AxNat.add x0 (AxNat.succ AxNat.zero))) (Not (AxNat.Even x0)))"' | wc -l)" -ge 1

instance-Nat.even_add_one

Kind: kernel-term
Status: checked

Supports: Applies at a concrete odd instance (n := 3): mp(Even 4) type-checks against Not(Even 3), mpr(odd_not_even(3)) lands on a type defeq to Even 4, AND at a genuinely free `n` (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::even_add_one_applies_at_a_concrete_odd_instance_and_symbolically -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

footprint-Nat.even_add_one

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.even_add_one`. 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.even_add_one

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.even_add_one` 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 `Nat.mod_two_eq_zero_or_one`, `Nat.even_iff_mod_two_eq_zero`, `Nat.odd_iff_mod_two_eq_one`, `Nat.even_iff_odd_succ`, `Nat.odd_not_even` and `Nat.succ_add`. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat parity lemma about the SAME Nat.Even/Nat.add this kernel already has (no new definition introduced), so mirroring it under Mathlib's own name is honest under the mirror-flip criterion.",
  "source": "statement-only extraction of `Nat.even_add_one` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.even_add_one`",
      "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"
    }
  ]
}