Mathlib v4.30 source proposition Int.fib_two_mul_add_one_pos

Identifier: F:ml430-int-fib-two-mul-add-one-pos-8977f65f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.fib_two_mul_add_one_pos` in the pinned Mathlib v4.30 source.

Formal statement

∀ {n : ℤ}, 0 < Int.fib (2 * n + 1)

Dependencies

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

Evidence

kernel-Int.fib_two_mul_add_one_pos

Kind: kernel-term
Status: checked

Supports: forall n : Int, 0 < Int.fib (2*n+1)

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example int_theorem_inventory -- fib_two_mul_add_one_pos 2>/dev/null | grep -xFc 'theorem	Int.fib_two_mul_add_one_pos		((x0 : Int) -> Int.lt Int.zero (Int.fib (Int.add (Int.mul (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) x0) (Int.ofNat (AxNat.succ AxNat.zero)))))')" -ge 1

Evidence notes

The kernel re-infers the theorem type from the constructed term. `Int.fib` did not exist in this kernel before this fact closed -- it is a new definition (`crates/axeyum-lean-kernel/src/int_prelude/fibonacci.rs::declare_fib`), the sign-extended `fib(-n) = (-1)^(n+1) fib(n)` built as one `Int.rec` case split with no new recursion device. The proof splits on `n`'s constructor: the `ofNat k` branch reduces `2k+1` and `fib`'s own case split PURELY (no lemma) down to a `Nat`-side `fib_pos_of_pos`/`zero_lt_succ` composition; the `negSucc j` branch needs one genuine non-structural fact, `(-1)^(2j) = 1` (`pow_neg_one_two_mul`, induction on `j` reusing `pow_neg_one_succ` from `fib_cassini`), then transports the resulting `Nat`-side positivity fact across `Eq Int (fib (negSucc (2j))) (ofNat (Nat.fib (2j+1)))`. `Int.fib`'s definition is independently checked by evaluation, not just by this theorem's type-check: `fib_computes_the_sign_extended_sequence` (`int_prelude_tests.rs`) reduces `fib` at six concrete indices (both signs) against the hand-computed sequence, with a negative control at `fib(-2)` guarding against a definition that dropped the sign.

footprint-Int.fib_two_mul_add_one_pos

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example theorem_axiom_footprint -- Int.fib_two_mul_add_one_pos 2>/dev/null | grep -xFc $'integer\tInt.fib_two_mul_add_one_pos\t0\t')" -ge 1

Evidence notes

The theorem-specific footprint command reports zero dependencies on axioms in the constructed Int environment (built over the axiom-free Nat prelude).

Provenance

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