Mathlib v4.30 source proposition Nat.add_descFactorial_eq_ascFactorial

Identifier: F:ml430-nat-add-descfactorial-eq-ascfactorial-5faac784
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.add_descFactorial_eq_ascFactorial` in the pinned Mathlib v4.30 source.

Formal statement

∀ (n k : ℕ), (n + k).descFactorial k = (n + 1).ascFactorial k

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Nat.add_descFactorial_eq_ascFactorial

Kind: kernel-term
Status: checked

Supports: ∀ (n k : ℕ), (n + k).descFactorial k = (n + 1).ascFactorial k

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- add_descFactorial_eq_ascFactorial 2>/dev/null | /usr/bin/grep -Ec '^Nat\.add_descFactorial_eq_ascFactorial[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.add_descFactorial_eq_ascFactorial` through the trusted `Kernel::add_declaration` gate, declared in new module `nat_prelude/add_desc_factorial_asc_factorial.rs`. No induction: both sides already have a closed form through the same shared RHS `k! * choose (n+k) k` -- `Nat.descFactorial_eq_factorial_mul_choose` instantiated at `(n+k, k)` and `Nat.ascFactorial_succ_eq_factorial_mul_choose` instantiated at `(n, k)` -- chained through that shared term (`Kernel::trans`/`Kernel::symm`, no new induction principle). `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.descFactorial (AxNat.add x0 x1) x1) (AxNat.ascFactorial (AxNat.succ x0) x1)))`, matching this fact's `formal.statement` (`AxNat.succ x0` is `n+1`). The `-ge 1` count-based test (not `grep -q`) requires the admitted declaration to actually be printed; grepping a made-up name greps to `0`.

footprint-Nat.add_descFactorial_eq_ascFactorial

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: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly via `theorem_names` (this declaration's `p.add_desc_factorial_eq_asc_factorial` entry was added there in the same commit that declared it), and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked above. `nat_prelude/add_desc_factorial_asc_factorial_tests.rs` independently checks this declaration's `axiom_footprint` too, plus a concrete instantiation at n=2,k=3 (both sides confirmed to reduce to 60) with a genuinely-discriminating negative control ((2+3).descFactorial 2 = 20). Full `nat_prelude::` sweep: 252 passed, 0 failed.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.add_descFactorial_eq_ascFactorial` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_descFactorial_eq_ascFactorial`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}