Mathlib v4.30 source proposition Nat.and_mod_two_eq_one

Identifier: F:ml430-nat-and-mod-two-eq-one-3e873792
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℕ}, (a &&& b) % 2 = 1 ↔ a % 2 = 1 ∧ b % 2 = 1

Dependencies

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9 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.land_mod_two_eq_one

Kind: kernel-term
Status: checked

Supports: ∀ {a b : ℕ}, (a &&& b) % 2 = 1 ↔ a % 2 = 1 ∧ b % 2 = 1

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- land_mod_two_eq_one 2>/dev/null | grep -Ec '^Nat\.land_mod_two_eq_one[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.land_mod_two_eq_one` (`nat_prelude/land_low_bit.rs`) through the trusted `Kernel::add_declaration` gate. `nat_theorem_inventory`'s rendered type is `(x0 x1 : AxNat) -> Iff (Eq (mod (land x0 x1) 2) 1) (And (Eq (mod x0 2) 1) (Eq (mod x1 2) 1))`, matching this fact's `formal.statement` up to variable naming. Anchored with `^Nat\.land_mod_two_eq_one[[:space:]]` so a sibling name (e.g. `land_mod_two_eq_mul`) cannot satisfy this row; `grep -Ec` consumes the pipe so the tested count decides the exit status. Run `--release` -- the debug build of this inventory example SIGABRTs on stack depth (unrelated to this proof). Mathlib's `&&&` for `Nat` is the SAME function as `Nat.land`, so reconciling this mirror against a NEW `Nat.land_mod_two_eq_one` theorem is honest under the mirror-flip criterion in CLAUDE.md.

compute-Nat.land_mod_two_eq_one

Kind: kernel-term
Status: checked

Supports: land(3, 5) = 1 (0b011 & 0b101 = 0b001, both operands odd), so both sides of the Iff genuinely evaluate rather than vacuously matching.

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::land_mod_two_eq_one_applies_at_a_concrete_instance

footprint-Nat.land_mod_two_eq_one

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.land_mod_two_eq_one`'s footprint by [].

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane draw9-second-theorems",
  "source": "statement-only extraction of `Nat.and_mod_two_eq_one` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.and_mod_two_eq_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"
    }
  ]
}