Mathlib v4.30 source proposition Int.natAbs_emod_two

Identifier: F:ml430-int-natabs-emod-two-18514063
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (i : ℤ), i.natAbs % 2 = (i % 2).natAbs

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Int.nat_abs_emod_two

Kind: kernel-term
Status: checked

Supports: `Int.nat_abs_emod_two` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : Int) -> Eq.{1} AxNat (AxNat.mod (Int.natAbs x0) (AxNat.succ (AxNat.succ AxNat.zero))) (Int.natAbs (Int.emod x0 (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))))))`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "theorem" && $2 == "Int.nat_abs_emod_two" && $3 == "" && $4 == "((x0 : Int) -> Eq.{1} AxNat (AxNat.mod (Int.natAbs x0) (AxNat.succ (AxNat.succ AxNat.zero))) (Int.natAbs (Int.emod x0 (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))))))"' | wc -l)" -ge 1

footprint-Int.nat_abs_emod_two

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Int.nat_abs_emod_two`. The `integer` prelude's trusted surface is empty (no Axiom, no Opaque, no Quotient), which bounds every declaration in it, including this one.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free integer

Provenance

{
  "date": "2026-08-29",
  "established_by": "lane score-the-blind-population (2026-09-01), proving the held-out `integer-absolute-value` family under the protocol preregistered at commit 067d675a3; declared in crates/axeyum-lean-kernel/src/int_prelude/nat_abs_mirrors.rs",
  "source": "statement-only extraction of `Int.natAbs_emod_two` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.natAbs_emod_two`",
      "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"
    }
  ]
}