Mathlib v4.30 source proposition Nat.fib_le_fib_succ

Identifier: F:ml430-nat-fib-le-fib-succ-d1ef4a3d
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Nat.fib n ≤ Nat.fib (n + 1)

Dependencies

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4 direct dependencies 2 direct dependents

Evidence

kernel-Nat.fib_le_succ

Kind: kernel-term
Status: checked

Supports: ∀ {n : ℕ}, Nat.fib n ≤ Nat.fib (n + 1)

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.fib_le_succ : forall n, fib n <= fib (succ n)` through the trusted `Kernel::add_declaration` gate, which re-derives the type from the proof term. This is literally the pinned Mathlib v4.30 proposition `Nat.fib_le_fib_succ` (`n + 1` renders as `succ n`) -- found by comparing this fact's formal.statement against every candidate `nat_theorem_inventory --release` row a crude name match against the 1,117-theorem kernel inventory surfaced, not assumed from the name match alone. `--release` is required: the debug build of this binary can SIGABRT on stack depth for the full constructed-environment inventories (not this one specifically, but the convention is followed uniformly). Demonstrated to discriminate: substituting a single wrong token (e.g. flipping the `<=` operands) into the expected line makes this exact command exit 1 against the same kernel build.

footprint-Nat.fib_le_succ

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 --release --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory` reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, and `Nat.fib_le_succ` is declared inside `build_nat_prelude`, so it cannot depend on a trusted declaration the Nat environment does not contain. Demonstrated to discriminate: `--require-axiom-free axreal` against the SAME kernel build exits 1 (`axreal trusted surface = 30, expected 0`), so the flag is not satisfied unconditionally.

Provenance

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