ascFactorial(n, 0) = 1

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

Recorded description

The rising (ascending) factorial of n with 0 factors equals 1.

Formal statement

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

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 3 direct dependents

Evidence

kernel-Nat.ascFactorial_zero

Kind: kernel-term
Status: checked

Supports: n.ascFactorial 0 = 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- ascFactorial_zero 2>/dev/null | grep -Ec '^Nat\.ascFactorial_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.ascFactorial` is declared by structural recursion on its SECOND argument via `NatOps::define_binary` (`nat_prelude/asc_factorial.rs`), mirroring `Nat.descFactorial` but climbing with `Nat.add` instead of descending with `Nat.sub`, so `ascFactorial 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.ascFactorial_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::asc_factorial_evaluates_correctly

Evidence notes

`Kernel::def_eq` reduces `3.ascFactorial 0` directly (definition unfolding, no boundary theorem invoked) and confirms it is `1`, with negative reduction controls (`3.ascFactorial 2` must NOT be def-eq to the DESCENDING product `3*2 = 6`, nor to `3.descFactorial 2`) so the assertion cannot pass on a vacuous `def_eq` and specifically catches an `add`/`sub` copy-paste that would still type-check.

footprint-Nat.ascFactorial_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.ascFactorial` needed no fuel device: it is structural recursion on its second argument, exactly like `Nat.descFactorial`/`Nat.factorial`/`Nat.mul`, 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.ascFactorial` and its three theorems, and `Nat.multichoose` and its three theorems, all declared.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-asc-multichoose",
  "source": "one of two definitional boundary equations `declare_asc_factorial` (`crates/axeyum-lean-kernel/src/nat_prelude/asc_factorial.rs`) admits alongside the `Nat.ascFactorial` definition"
}