Mathlib v4.30 source proposition Nat.add_one_lt_of_even

Identifier: F:ml430-nat-add-one-lt-of-even-3464b374
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, Even n → Even m → n < m → n + 1 < m

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.add_one_lt_of_even

Kind: kernel-term
Status: checked

Supports: `Nat.add_one_lt_of_even` is admitted as a Theorem with EXACTLY the mirror's shape, pinned verbatim via the kernel's own renderer. Built via `Nat.lt_or_eq_of_le` at `(n+1, m)` -- legal because `Nat.lt n m := Nat.le (succ n) m` and `add n 1` reduces to `succ n` -- ruling out the equality branch with `Nat.even_iff_odd_succ`/`Nat.odd_not_even`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory add_one_lt_of_even 2>/dev/null | awk -F'\t' '$1 == "Nat.add_one_lt_of_even" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.Even x0) -> ((x3 : AxNat.Even x1) -> ((x4 : AxNat.lt x0 x1) -> AxNat.lt (AxNat.add x0 (AxNat.succ AxNat.zero)) x1)))))"' | wc -l)" -ge 1

instance-Nat.add_one_lt_of_even

Kind: kernel-term
Status: checked

Supports: Applies at a concrete instance (n := 2, m := 6, both Even, landing on Lt 3 6) AND at a genuinely free (n, m) 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::add_one_lt_of_even_applies_concretely_symbolically_and_rejects_an_odd_m_control -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

footprint-Nat.add_one_lt_of_even

Kind: kernel-term
Status: checked

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

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.add_one_lt_of_even` 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.lt_or_eq_of_le`, `Nat.even_iff_odd_succ` and `Nat.odd_not_even`. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat order/parity lemma about the SAME Nat.lt/Nat.Even 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.add_one_lt_of_even` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_one_lt_of_even`",
      "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"
    }
  ]
}