Mathlib v4.30 source proposition Nat.add_factorial_succ_lt_factorial_add_succ

Identifier: F:ml430-nat-add-factorial-succ-lt-factorial-add-succ-ec0fa8d3
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.add_factorial_succ_lt_factorial_add_succ

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_add_factorial_succ_lt_factorial_add_succ` (`crates/axeyum-lean-kernel/src/nat_prelude/add_factorial_lt.rs`): immediate corollary of `Nat.add_factorial_lt_factorial_add` at `n := succ n`, discharging its `Le 1 (succ n)` hypothesis with `NatOps::zero_lt_succ`; `(n+1)!` and `(i+n+1)!` both reduce to `factorial (succ n)`/`factorial (succ (add i n))` by pure δ/ι (the literal `1` sits on the RIGHT of `add`'s right-recursion, so it is free), so no extra rewrite is needed beyond the parent theorem's application.

footprint-Nat.add_factorial_succ_lt_factorial_add_succ

Kind: exhaustive-enumeration
Status: checked

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

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.add_factorial_succ_lt_factorial_add_succ 2>/dev/null | grep -Ec 'nat[[:space:]]Nat\.add_factorial_succ_lt_factorial_add_succ[[:space:]]0[[:space:]]*$')" -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_succ_lt_factorial_add_succ` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_factorial_succ_lt_factorial_add_succ`",
      "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"
    }
  ]
}