Mathlib v4.30 source proposition Int.emod_two_ne_zero

Identifier: F:ml430-int-emod-two-ne-zero-d07d008f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℤ}, ¬n % 2 = 0 ↔ n % 2 = 1

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Int.emod_two_ne_zero

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`Int.emod_two_ne_zero` in `crates/axeyum-lean-kernel/src/int_prelude/parity.rs` (`declare_emod_two_ne_zero`). Mirrors Mathlib v4.30's `Int.emod_two_ne_zero` (`Mathlib/Algebra/Group/Int/Even.lean`) verbatim -- a pure `n % 2` fact, no `Even`/`Odd` involved. Proved from a new internal helper `Int.emod_two_eq_zero_or_one` (`Int.rec` on `n` plus `Nat.mod_two_eq_zero_or_one` on the bound `Nat` field of each branch, landed the same day) by `Or.elim` plus a small `0 ≠ 1 : Int` refutation (`zero_lt_one` rewritten to `Lt one one`, refuted by `lt_irrefl`).

footprint-Int.emod_two_ne_zero

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

Evidence notes

The matched row is `integer<TAB>Int.emod_two_ne_zero<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.emod_two_ne_zero` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.emod_two_ne_zero`",
      "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"
    }
  ]
}