ascFactorial(n, k+1) = (n+k) * ascFactorial(n, k)

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

Recorded description

The rising factorial of n with k+1 factors equals (n + k) times the rising factorial of n with k factors.

Formal statement

theorem Nat.ascFactorial_succ : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.ascFactorial x0 (AxNat.succ x1)) (AxNat.mul (AxNat.add x0 x1) (AxNat.ascFactorial x0 x1))))

Dependencies

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1 direct dependencies 1 direct dependents

Evidence

kernel-Nat.ascFactorial_succ

Kind: kernel-term
Status: checked

Supports: n.ascFactorial (k+1) = (n + k) * n.ascFactorial k

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- ascFactorial_succ 2>/dev/null | grep -Ec '^Nat\.ascFactorial_succ[[: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. `Nat.ascFactorial n (succ k)` iota-reduces (via `NatOps::define_binary`'s recursor application) directly to `(n + k) * ascFactorial n k` -- the proof is `Eq.refl`, no rewrite. `n + k` is Lean's own `Nat.add`, matching Mathlib's `Nat.ascFactorial (n : ℕ) : ℕ → ℕ | 0 => 1 | k + 1 => (n + k) * ascFactorial n k` argument order exactly. `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_succ

Kind: kernel-term
Status: checked

Supports: the successor equation composes correctly across multiple steps, at a concrete instance

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 2` (two applications of the successor equation) to `12` (`3 * 4`), with negative reduction controls forbidding the vacuous case AND forbidding the DESCENDING value `3*2 = 6` at the same arguments, so this cannot pass without the successor equation actually unrolling the ASCENDING product.

footprint-Nat.ascFactorial_succ

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. 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": "the second of two definitional boundary equations `declare_asc_factorial` (`crates/axeyum-lean-kernel/src/nat_prelude/asc_factorial.rs`) admits alongside the `Nat.ascFactorial` definition"
}