Mathlib v4.30 source proposition Nat.add_factorial_le_factorial_add

Identifier: F:ml430-nat-add-factorial-le-factorial-add-b0400cf6
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (i : ℕ) {n : ℕ}, 1 ≤ n → i + n.factorial ≤ (i + n).factorial

Dependencies

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9 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.add_factorial_le_factorial_add

Kind: kernel-term
Status: checked

Supports: ∀ (i : ℕ) {n : ℕ}, 1 ≤ n → i + n.factorial ≤ (i + n).factorial

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.add_factorial_le_factorial_add` through the trusted `Kernel::add_declaration` gate, declared in new module `nat_prelude/add_factorial_le.rs`. Proved by induction on `i` (`n` and the `1 <= n` hypothesis held fixed): base case (`i=0`) needs `zero_add` on both occurrences (`Nat.add` recurses on its RIGHT argument, so `add(zero,x)` is stuck for symbolic `x`); the step case needs `succ_add` on both occurrences plus one arithmetic fact (`1 <= (j+n)*(j+n)!`, from `one_le_mul`/`one_le_factorial`/`n <= j+n` via `le_add_right`+`add_comm`+the hypothesis via `le_trans`) fed through `add_le_add_left` -- the `factorial(succ x) = mul(factorial x, succ x)` unfold costs no lemma at all, since `factorial_succ` is itself proved by `Eq.refl` (`defs.rs`) and `mul` recurses right same as `add`, so the kernel accepts the assembled term by defeq at the final check (the same technique `divisibility.rs`'s existing `factorial_lt_of_lt` uses). `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le (AxNat.succ AxNat.zero) x1) -> AxNat.le (AxNat.add x0 (AxNat.factorial x1)) (AxNat.factorial (AxNat.add x0 x1)))))`, matching this fact's `formal.statement`. 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_factorial_le_factorial_add

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_factorial_le_factorial_add` 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_factorial_le_tests.rs` independently checks this declaration's `axiom_footprint` too, plus a concrete instantiation at i=2,n=3 (2+3! = 8, confirmed <= (2+3)! = 120). Full `nat_prelude::` sweep: 255 passed, 0 failed.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.add_factorial_le_factorial_add` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_factorial_le_factorial_add`",
      "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"
    }
  ]
}