descFactorial(n, k+1) = (n-k) * descFactorial(n, k)

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

Recorded description

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

Formal statement

theorem Nat.descFactorial_succ : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.descFactorial x0 (AxNat.succ x1)) (AxNat.mul (AxNat.sub x0 x1) (AxNat.descFactorial x0 x1))))

Dependencies

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1 direct dependencies 3 direct dependents

Evidence

kernel-Nat.descFactorial_succ

Kind: kernel-term
Status: checked

Supports: n.descFactorial (k+1) = (n - k) * n.descFactorial k

Checker command

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

Kind: kernel-term
Status: checked

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

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 2` (two applications of the successor equation) to `20` (`5 * 4`) and `5.descFactorial 5` to `120` (`5!`), each with a negative reduction control forbidding the vacuous case, so this cannot pass without the successor equation actually unrolling the product.

footprint-Nat.descFactorial_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.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": "the second of two definitional boundary equations `declare_desc_factorial` (`crates/axeyum-lean-kernel/src/nat_prelude/desc_factorial.rs`) admits alongside the `Nat.descFactorial` definition"
}