descFactorial(n, 0) = 1

Identifier: F:nat-desc-factorial-zero
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The falling (descending) factorial of n with 0 factors equals 1.

Formal statement

theorem Nat.descFactorial_zero : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.descFactorial x0 AxNat.zero) (AxNat.succ AxNat.zero))

Dependencies

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0 direct dependencies 2 direct dependents

Evidence

kernel-Nat.descFactorial_zero

Kind: kernel-term
Status: checked

Supports: n.descFactorial 0 = 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- descFactorial_zero 2>/dev/null | grep -Ec '^Nat\.descFactorial_zero[[: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.descFactorial` is declared by structural recursion on its SECOND argument via `NatOps::define_binary` (`nat_prelude/desc_factorial.rs`), so `descFactorial n zero` iota-reduces to the base case, `1`, directly -- the proof is `Eq.refl`, no induction or case split. `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.

compute-Nat.descFactorial_zero

Kind: kernel-term
Status: checked

Supports: the base case computes the right value at a concrete instance, not merely an opaque application

Checker command

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

Evidence notes

`Kernel::def_eq` reduces `5.descFactorial 0` directly (definition unfolding, no boundary theorem invoked) and confirms it is `1`, with a negative reduction control (`5.descFactorial 2` must NOT be def-eq to `120`) so the assertion cannot pass on a vacuous `def_eq`.

footprint-Nat.descFactorial_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. `Nat.descFactorial` needed no fuel device: it is structural recursion on its second argument, exactly like `Nat.factorial`/`Nat.mul`/`Nat.sub`, so nothing here was ever at risk of needing `WellFounded`/`Quot.sound`/`propext`. Measured 2026-08-28: `nat: axiom=0 opaque=0 quotient=0 total_trusted=0` with `Nat.descFactorial` and its four theorems declared.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 212-nat-factorial-variants",
  "source": "one of two definitional boundary equations `declare_desc_factorial` (`crates/axeyum-lean-kernel/src/nat_prelude/desc_factorial.rs`) admits alongside the `Nat.descFactorial` definition"
}