Mathlib v4.30 source proposition Nat.add_factorial_lt_factorial_add

Identifier: F:ml430-nat-add-factorial-lt-factorial-add-7501a8c8
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {i n : ℕ}, 2 ≤ i → 1 ≤ n → i + n.factorial < (i + n).factorial

Dependencies

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

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

Evidence

kernel-Nat.add_factorial_lt_factorial_add

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Nat.add_factorial_lt_factorial_add` in the pinned Mathlib v4.30 source.

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::

Evidence notes

`declare_add_factorial_lt_factorial_add` (`crates/axeyum-lean-kernel/src/nat_prelude/add_factorial_lt.rs`): `le_dest`+`Exists.rec` peel `Le 2 i` into a witness `k` with `i = add 2 k`, then a `k`-indexed strict induction on `Lt (add (add 2 k) n!) (factorial (add (add 2 k) n))` with `n`/`Le 1 n` held fixed. The base case (`k=0`, reducing via `add 2 0 ≡ 2` by pure δ/ι) is proved directly via `factorial_lt_of_lt` + `factorial_le` (at `factorial 2 ≡ 2`, also pure δ/ι) + `mul_le_mul_left`; the step case is `add_factorial_le_factorial_add`'s own step function applied to an IH that is already one `succ` ahead (since `Lt a b` unfolds definitionally to `Le (succ a) b`), landing on the required shape with no extra slack needed.

footprint-Nat.add_factorial_lt_factorial_add

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.add_factorial_lt_factorial_add 2>/dev/null | grep -Ec 'nat[[:space:]]Nat\.add_factorial_lt_factorial_add[[:space:]]0[[:space:]]*$')" -ge 1

Evidence notes

The matched row's trailing footprint column is EMPTY, so a nonzero footprint or an absent declaration both give a count of 0 and fail. Pass exactly one name per invocation -- this tool keeps only the first.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.add_factorial_lt_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_lt_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"
    }
  ]
}