Mathlib v4.30 source proposition Nat.and_or_distrib_left

Identifier: F:ml430-nat-and-or-distrib-left-fe131f64
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (x y z : ℕ), x &&& (y ||| z) = x &&& y ||| x &&& z

Dependencies

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

Evidence

kernel-Nat.and_or_distrib_left

Kind: kernel-term
Status: checked

Supports: ∀ (x y z : ℕ), x &&& (y ||| z) = x &&& y ||| x &&& z

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.and_or_distrib_left` (`nat_prelude/and_or_distrib.rs`) through the trusted `Kernel::add_declaration` gate. `nat_theorem_inventory`'s rendered type is `(x0 x1 x2 : AxNat) -> Eq (land x0 (lor x1 x2)) (lor (land x0 x1) (land x0 x2))`, matching this fact's `formal.statement` up to variable naming. Anchored with `^Nat\.and_or_distrib_left[[:space:]]`; `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). Route: `Nat.eq_of_testBit_eq` extensionality (`xor_algebra.rs`) plus `Nat.testBit_land`/`Nat.testBit_lor` twice per side, reduced to a per-bit AND-distributes-over-OR identity closed by a nested `{0,1}` case split (8 leaves, each `refl`, `and_or_distrib.rs`'s `bit_and_or_distrib`). Mathlib's `&&&`/`|||` for `Nat` are `Nat.land`/`Nat.lor`, so reconciling this mirror to a NEW `Nat.and_or_distrib_left` theorem is honest under the mirror-flip criterion in CLAUDE.md.

compute-Nat.and_or_distrib_left

Kind: kernel-term
Status: checked

Supports: At (x,y,z) = (6,3,5): y|||z = 7, x&&&(y|||z) = 6&&&7 = 6; x&&&y = 2, x&&&z = 4, 2|||4 = 6 -- both sides 6, with two negative controls (inner-operator swap gives 0, outer-operator swap gives 7).

Checker command

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

footprint-Nat.and_or_distrib_left

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.and_or_distrib_left`'s footprint by [].

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.and_or_distrib_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.and_or_distrib_left`",
      "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"
    }
  ]
}