bit(true, n) = 2n + 1

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

Recorded description

Appending the bit `true` to the little end of `n` equals `2 * n + 1`.

Formal statement

theorem Nat.bit_true : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.bit Bool.true x0) (AxNat.add (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x0) (AxNat.succ AxNat.zero)))

Dependencies

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0 direct dependencies 2 direct dependents

Evidence

kernel-Nat.bit_true

Kind: kernel-term
Status: checked

Supports: bit(true, n) = 2*n + 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- bit_true 2>/dev/null | grep -Ec '^Nat\.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. `Nat.bit b n` is stated `add (mul 2 n) (cond b 1 0)`; at `b = true`, `cond true 1 0` iota-reduces to `1 = succ zero`, and `Nat.add`'s successor case (`add x (succ j) ≡ succ (add x j)`) collapses the sum to `succ (mul 2 n)` -- the same normal form the RHS `add (mul 2 n) 1` itself reduces to. The proof is `Eq.refl`; the kernel closes it by delta+iota alone, no case-split combinator and no induction. `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. The anchor `^Nat\.bit_true[[:space:]]` matters here for the same reason as the sibling `bit_false` fact -- `bit_true` is not itself a prefix of another declared name in this build, but the anchor keeps the checker robust if one is added later.

compute-Nat.bit_true

Kind: kernel-term
Status: checked

Supports: the equation instantiates correctly and names the true branch's `2n+1` form

Checker command

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

Evidence notes

`Kernel::def_eq` reduces `bit true n` at n in {0, 1, 6} to {1, 3, 13} directly (definition unfolding, no boundary theorem invoked), including negative controls (`bit false 6` must NOT def_eq `13`, `bit true 6` must NOT def_eq `12`), and separately confirms `Nat.bit_true`'s ADMITTED TYPE (via `Kernel::infer`) is exactly `Eq (bit true 6) (add (mul 2 6) 1)`.

footprint-Nat.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. `Nat.bit` needed no fuel device -- it is a plain non-recursive lambda over `Nat.add`/`Nat.mul`/`Bool.rec` -- so this command is what makes the axiom-freedom claim falsifiable rather than asserted.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 207-nat-bitwise",
  "source": "one of two definitional boundary equations `declare_bit_all` (`crates/axeyum-lean-kernel/src/nat_prelude/bits.rs`) admits alongside the `Nat.bit` definition"
}