Mathlib v4.30 source proposition Int.even_add

Identifier: F:ml430-int-even-add-3c4536e3
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℤ}, Even (m + n) ↔ (Even m ↔ Even n)

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Int.even_add

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`Int.even_add` in `crates/axeyum-lean-kernel/src/int_prelude/parity.rs` (`declare_even_add`). Mirrors Mathlib v4.30's `Int.even_add` verbatim. Proved via the additive compatibility law for `emod` (`modeq_add`, built from `Int.ModEq`'s already-general `mod_eq_add_right`/`mod_eq_add_left`/`mod_eq_trans` in `modeq.rs` -- no fresh `Int.rec` case split on `Int.add`'s branch table was needed) plus the converses of the already-landed `even_implies_emod_zero`/`odd_implies_emod_one` (`emod_zero_implies_even`/`emod_one_implies_odd`, new in this lane). A four-way case split on `m`'s and `n`'s parity (`emod_two_eq_zero_or_one`) closes each combination via a generic `Iff`-combinator (`iff_fact`).

footprint-Int.even_add

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

Evidence notes

The matched row is `integer<TAB>Int.even_add<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. NOTE: passing `Int.even_add` as the CLI argument to this tool ALSO returns `Int.even_add'`/`Int.even_add_one` rows (a prefix-match, not an exact filter), so the `-xF` exact-line grep is load-bearing, not cosmetic -- verified both to pass for real (count 1) and to fail on a mutated footprint (`...\t1\t`, count 0) and a nonexistent name (`Int.even_add_bogus`, count 0). 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.even_add` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.even_add`",
      "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"
    }
  ]
}