Mathlib v4.30 source proposition Nat.bit_ne_zero

Identifier: F:ml430-nat-bit-ne-zero-181bd65c
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (b : Bool) {n : ℕ}, n ≠ 0 → Nat.bit b n ≠ 0

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.bit_ne_zero

Kind: kernel-term
Status: checked

Supports: ∀ (b : Bool) {n : ℕ}, n ≠ 0 → Nat.bit b n ≠ 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.bit_ne_zero` through the trusted `Kernel::add_declaration` gate, declared in `nat_prelude/bit_extra.rs` (a new module, kept separate from `bits.rs`'s own dense theorem set). `nat_theorem_inventory`'s rendered type is `((x0 : Bool) -> ((x1 : AxNat) -> ((x2 : ((x2 : Eq.{1} AxNat x1 AxNat.zero) -> False)) -> ((x3 : Eq.{1} AxNat (AxNat.bit x0 x1) AxNat.zero) -> False))))`, matching this fact's `formal.statement`. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Verified both ways: the real name greps to a count `-ge 1`; grepping a made-up name (`Nat.bit_ne_zero_bogus`) greps to `0`.

footprint-Nat.bit_ne_zero

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q -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 every individual theorem's footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly via `theorem_names` (this declaration's `p.bit_ne_zero` entry was added there in the same session), and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked above. `nat_prelude/bit_extra_tests.rs` independently checks this declaration's `axiom_footprint` too, plus a concrete instantiation with a discriminating negative control.

Provenance

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