Mathlib v4.30 source proposition Nat.le_fib_add_one

Identifier: F:ml430-nat-le-fib-add-one-5284f0bf
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n : ℕ), n ≤ Nat.fib n + 1

Dependencies

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

10 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.le_fib_add_one

Kind: kernel-term
Status: checked

Supports: n <= Nat.fib n + 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example nat_theorem_inventory -- le_fib_add_one 2>/dev/null | grep -xFc 'Nat.le_fib_add_one	1	((x0 : AxNat) -> AxNat.le x0 (AxNat.add (AxNat.fib x0) (AxNat.succ AxNat.zero)))')" -ge 1

Evidence notes

The kernel re-infers the theorem type from the constructed term, matching Mathlib's unconditional `n <= fib n + 1`. Proved by splitting at `Nat.lt_or_ge n 5` (ops::cases_lt_or_ge): the `Le 5 n` side chains le_fib_self with le_add_right + le_trans; the `Lt n 5` side is a genuine 5-way case split to concrete n in {0,1,2,3,4} (ops::cases_lt_bound, a new finite-cases eliminator), each branch closed by le_add_right (or zero_le at n=0) at a hand-picked slack defeq to fib(n)+1 for that tiny literal n. The bound is TIGHT (equality) at n=2,3,4, which is why a bare pair-induction cannot prove this (docs/plan/status/228-fib-2.md's algebraic analysis).

footprint-Nat.le_fib_add_one

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example theorem_axiom_footprint -- Nat.le_fib_add_one 2>/dev/null | grep -xFc $'nat\tNat.le_fib_add_one\t0\t')" -ge 1

Evidence notes

The theorem-specific footprint command reports zero dependencies on axioms in the constructed Nat environment.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.le_fib_add_one` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.le_fib_add_one`",
      "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"
    }
  ]
}