bit(false, n) <= bit(true, n)

Identifier: F:nat-bit-false-le-bit-true
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every n, appending bit `false` to n's little end is at most appending bit `true`.

Formal statement

theorem Nat.bit_false_le_bit_true : ((x0 : AxNat) -> AxNat.le (AxNat.bit Bool.false x0) (AxNat.bit Bool.true x0))

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.bit_false_le_bit_true

Kind: kernel-term
Status: checked

Supports: bit(false, n) <= bit(true, n), for every n

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. The proof term is `le_succ (mul 2 n)` -- the already-proved generic lemma `m <= succ m` instantiated at `m := mul 2 n` -- accepted directly by the kernel's definitional-equality check, because `bit false n` unfolds to `mul 2 n` and `bit true n` unfolds to `succ (mul 2 n)` (delta+iota, per `F:nat-bit-false`/`F:nat-bit-true`). No case-split combinator and no induction were needed for this theorem specifically. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the count is tested, so the exit status depends on what the run found.

instantiation-Nat.bit_false_le_bit_true

Kind: kernel-term
Status: checked

Supports: the statement instantiates to the promised inequality, in the promised direction, at a concrete witness

Checker command

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

Evidence notes

Applies `Nat.bit_false_le_bit_true` at `n = 6` and confirms (via `Kernel::infer` + `Kernel::def_eq`) the inferred conclusion is exactly `Le (bit false 6) (bit true 6)`, with a REVERSED negative control (`Le (bit true 6) (bit false 6)`, i.e. `Le 13 12`) that must NOT def_eq the inferred type -- the direction is genuinely checked, not merely present.

footprint-Nat.bit_false_le_bit_true

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. `bit_false_le_bit_true`'s proof only ever composes `Nat.le_refl` and `Nat.le_step` (via the `le_succ` theorem) and `Nat.rec`, none of them asserted, so this row stays empty.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 207-nat-bitwise",
  "source": "`declare_bit_all` (`crates/axeyum-lean-kernel/src/nat_prelude/bits.rs`), a defeq corollary of `le_succ`"
}