0 < bit(true, n)

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

Recorded description

Appending the bit `true` to the little end of any `n` produces a strictly positive result.

Formal statement

theorem Nat.bit_true_pos : ((x0 : AxNat) -> AxNat.lt AxNat.zero (AxNat.bit Bool.true x0))

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.bit_true_pos

Kind: kernel-term
Status: checked

Supports: 0 < bit(true, n), for every n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- bit_true_pos 2>/dev/null | grep -Ec '^Nat\.bit_true_pos[[: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 `zero_lt_succ (mul 2 n)` -- the already-proved generic lemma `0 < succ m` instantiated at `m := mul 2 n` -- accepted directly by the kernel's definitional-equality check, because `bit true n` unfolds (delta on `Nat.bit`, iota on the inner `Bool.rec`, iota again on `Nat.add`'s successor case) to exactly `succ (mul 2 n)`. No case-split combinator and no induction were needed for this theorem specifically; the case split is already paid for by `Nat.bit`'s own construction. `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_true_pos

Kind: kernel-term
Status: checked

Supports: the statement instantiates to the promised strict inequality 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_true_pos` at `n = 6` and confirms (via `Kernel::infer` + `Kernel::def_eq`) the inferred conclusion is exactly `Lt 0 (bit true 6)`, i.e. `Lt 0 13` after full reduction -- a positive statement at a concrete, nontrivial instance rather than a vacuous universal.

footprint-Nat.bit_true_pos

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_true_pos`'s proof only ever composes `Nat.zero_le`, `Nat.le_succ_succ` (via the `zero_lt_succ` term-builder), 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 `zero_lt_succ`"
}