two naturals with the same bits at every index are equal

Identifier: F:nat-eq-of-testbit-eq
Proof route: kernel-lean
External status: Not recorded
Axiom footprint: Empty

Recorded description

For every natural m, n: if testBit(m, i) equals testBit(n, i) for every i, then m equals n.

Formal statement

theorem Nat.eq_of_testBit_eq : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : ((x2 : AxNat) -> Eq.{1} AxNat (AxNat.testBit x0 x2) (AxNat.testBit x1 x2))) -> Eq.{1} AxNat x0 x1)))

Dependencies

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15 direct dependencies 5 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.eq_of_testBit_eq

Kind: kernel-term
Status: checked

Supports: Nat.eq_of_testBit_eq is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- eq_of_testBit_eq 2>/dev/null | grep -Ec '^Nat\.eq_of_testBit_eq[[:space:]]'

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe so the tested count (expected >= 1) decides the exit status, never `grep -q` under pipefail. Anchored with `[[:space:]]`, never a literal tab, per this ledger's own grep-dialect gotcha. `--release` is mandatory: this tool builds `creal`/`complex`/`cpoint`, which overflow the default debug thread stack.

compute-Nat.eq_of_testBit_eq

Kind: kernel-term
Status: checked

Supports: At a concrete REFLEXIVE instance (m = n = 6, bits trivially equal via refl at every i), the theorem's conclusion def-eqs to Eq 6 6 and NOT to the negative control Eq 6 7; and the theorem re-derives against a genuinely free (m, n) pair with an assumed (not derivable) bit-equality hypothesis, not merely a concrete instance.

Checker command

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

Evidence notes

A concrete instantiation alone can hide a defect a symbolic build exposes (and vice versa), so both are checked. A genuinely DISCRIMINATING concrete instance is not available for this statement without first proving what it proves (two distinct naturals cannot honestly be given the same bits without deriving m = n), so the concrete leg instead uses the REFLEXIVE case (m = n) with a negative control on the CONCLUSION (Eq 6 6, not Eq 6 7), and the symbolic leg is the one that exercises the theorem's real content -- it type-checks against a genuinely free (m, n) pair carrying only an ASSUMED bit-equality hypothesis, never derived from m = n.

footprint-Nat.eq_of_testBit_eq

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

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

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the FULL trusted surface rather than `Declaration::Axiom` alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-xor-algebra",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}