Mathlib v4.30 source proposition Int.fib_two_mul

Identifier: F:ml430-int-fib-two-mul-0e70f3dd
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n : ℤ), Int.fib (2 * n) = Int.fib n * (2 * Int.fib (n + 1) - Int.fib n)

Dependencies

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

Evidence

kernel-Int.fib_two_mul

Kind: kernel-term
Status: checked

Supports: Int.fib_two_mul is admitted by the trusted kernel gate, and its canonical type is the pinned Mathlib statement symbol for symbol: ((x0 : Int) -> Eq.{1} Int (Int.fib (Int.mul (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) x0)) (Int.mul (Int.fib x0) (Int.sub (Int.mul (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (Int.fib (Int.add x0 Int.one))) (Int.fib x0)))).

Checker command

cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory -- fib_two_mul 2>/dev/null | grep -cE '^theorem[[:space:]]+Int\.fib_two_mul[[:space:]]+\(\(x0[[:space:]]+:[[:space:]]+Int\)[[:space:]]+->[[:space:]]+Eq\.\{1\}[[:space:]]+Int[[:space:]]+\(Int\.fib[[:space:]]+\(Int\.mul[[:space:]]+\(Int\.ofNat[[:space:]]+\(AxNat\.succ[[:space:]]+\(AxNat\.succ[[:space:]]+AxNat\.zero\)\)\)[[:space:]]+x0\)\)[[:space:]]+\(Int\.mul[[:space:]]+\(Int\.fib[[:space:]]+x0\)[[:space:]]+\(Int\.sub[[:space:]]+\(Int\.mul[[:space:]]+\(Int\.ofNat[[:space:]]+\(AxNat\.succ[[:space:]]+\(AxNat\.succ[[:space:]]+AxNat\.zero\)\)\)[[:space:]]+\(Int\.fib[[:space:]]+\(Int\.add[[:space:]]+x0[[:space:]]+Int\.one\)\)\)[[:space:]]+\(Int\.fib[[:space:]]+x0\)\)\)\)$'

Evidence notes

The pattern pins the ENTIRE rendered type, not just the name, so a transposed factor (`fib n` and `fib(n+1)` swapped inside the `sub`, or `sub` folded to `add`) cannot pass. Verified discriminating on this tree: 1 against the real row, 0 against a row with `Int.sub` rewritten to `Int.add` and 0 against a row with the `2` literal (`AxNat.succ (AxNat.succ AxNat.zero)`) collapsed to `AxNat.zero`. Tab separators matched with `[[:space:]]`, never `\t` -- see this repository's documented GNU-grep/ugrep divergence. `--release` is MANDATORY: this binary also builds creal/complex/cpoint, which overflow the default debug thread stack, and a debug run's empty output would read as ABSENT.

footprint-Int.fib_two_mul

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude's trusted surface is empty, which bounds Int.fib_two_mul

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free integer

Evidence notes

Re-measured on this tree: integer trusted surface = 0. --require-axiom-free exits nonzero when the count is not zero, so the exit status depends on the finding. --release is MANDATORY: this binary also builds creal/complex/cpoint, which overflow the default debug thread stack, and a debug run's empty output would read as ABSENT.

evaluation-Int.fib_two_mul-each-sign

Kind: witness-replay
Status: checked

Supports: fib_two_mul instantiated at n=5 (fib(10)=55=fib(5)*(2*fib(6)-fib(5))=5*11) and n=-3 (fib(-6)=-8=fib(-3)*(2*fib(-2)-fib(-3))=2*(-4)), each paired with the value the swapped-coefficient formula fib(n)*(2*fib(n)-fib(n+1)) gives (10 at both indices), which must NOT be def_eq. n=5 was chosen so a transposed factor changes the numeric result.

Checker command

test "$(cargo test -q -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::fib_two_mul_computes_at_a_concrete_index_of_each_sign 2>/dev/null | grep -cE '^test result: ok\. 1 passed')" -eq 1

Evidence notes

crates/axeyum-lean-kernel/src/int_prelude/int_prelude_tests.rs. The command asserts EXACTLY ONE test passed, so a filter matching nothing fails rather than exiting 0.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/fibonacci.rs, declare_fib_two_mul)",
  "prior_art": [
    {
      "attribution": "the proposition statement was read from the pinned statement-only inventory; the proof PRODUCED here is this kernel's own construction (Int.fib_add plus a purpose-built subtraction bridge and ring algebra), not a transcription of Mathlib's tactic proof, which was never consulted.",
      "what": "the theorem declaration `Int.fib_two_mul`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "who": "the Mathlib contributors",
      "year": 2026
    }
  ],
  "source": "proved in-tree over the CONSTRUCTED integers. The canonical type was read from int_theorem_inventory's own output and compared symbol for symbol against the pinned Mathlib statement already recorded in formal.statement, which is unchanged."
}