Mathlib v4.30 source proposition Int.fib_two_mul_add_one_eq_natFib_natAbs

Identifier: F:ml430-int-fib-two-mul-add-one-eq-natfib-natabs-61a8342b
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℤ}, Int.fib (2 * n + 1) = ↑(Nat.fib (2 * n + 1).natAbs)

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Int.fib_two_mul_add_one_eq_natfib_natabs

Kind: kernel-term
Status: checked

Supports: Int.fib_two_mul_add_one_eq_natfib_natabs is admitted by the trusted kernel gate with the type recorded in formal.kernel_statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory 2>/dev/null | grep -cE '^theorem[[:space:]]Int\.fib_two_mul_add_one_eq_natfib_natabs[[:space:]]'

Evidence notes

int_theorem_inventory lists every Declaration::Theorem in the integer prelude with its canonical type; the grep asserts the exact tab-anchored name. --release is MANDATORY: this binary also builds creal/complex/cpoint, which overflow the default debug thread stack. Confirmed to exit 1 on a deliberately wrong name (…_WRONG) and 0 on the real one.

footprint-Int.fib_two_mul_add_one_eq_natfib_natabs

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude's trusted surface is empty, which bounds Int.fib_two_mul_add_one_eq_natfib_natabs and its stepping-stone Int.odd_two_mul_add_one.

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. --release is MANDATORY here.

evaluation-Int.fib_two_mul_add_one_eq_natfib_natabs-both-signs

Kind: witness-replay
Status: checked

Supports: The composite theorem instantiated at n := 1 (index 2*1+1=3, fib 3 = ofNat (Nat.fib 3) = ofNat 2) and n := negSucc 1 (n = -2, index 2*(-2)+1 = -3 = negSucc 2, fib (-3) = ofNat (Nat.fib 3) = ofNat 2) -- checked by Kernel::infer + Kernel::def_eq against the closed-form equation, exercising odd_two_mul_add_one composed with fib_of_odd end to end, not either lemma in isolation.

Checker command

cargo test -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::fib_two_mul_add_one_eq_natfib_natabs_applies_at_a_concrete_index_of_each_sign 2>&1 | grep -c '^test result: ok\. 1 passed'

Evidence notes

crates/axeyum-lean-kernel/src/int_prelude/int_prelude_tests.rs. Must be run with an explicit test-name filter and a NONZERO pass count confirmed -- confirmed to exit 1 (count 0) against a deliberately misspelled test name.

coverage-Int.fib_two_mul_add_one_eq_natfib_natabs

Kind: witness-replay
Status: checked

Supports: int_prelude_tests.rs's every_int_declaration_is_checked_and_axiom_free asserts, against the ENVIRONMENT rather than a hand-maintained list, that every live Int.* Definition/Theorem (both new declarations included) is listed in derived_laws/derived_lemmas/asserted_laws/definition_names and carries an empty axiom footprint.

Checker command

cargo test -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::every_int_declaration_is_checked_and_axiom_free 2>&1 | grep -c '^test result: ok\. 1 passed'

Evidence notes

This test failed (panicked, naming both new declarations by name) before Int.odd_two_mul_add_one and Int.fib_two_mul_add_one_eq_natfib_natabs were added to derived_laws; it passes now.

Provenance

{
  "date": "2026-09-01",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/fibonacci.rs, declare_odd_two_mul_add_one + declare_fib_two_mul_add_one_eq_natfib_natabs)",
  "source": "canonical type read via int_theorem_inventory's own output (cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory), which prints one row per Declaration::Theorem in the integer prelude with kernel.render_lean(declaration.ty()) as the last field. That row was grepped and copied verbatim into formal.kernel_statement.",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.fib_two_mul_add_one_eq_natFib_natAbs`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0), Mathlib/Data/Int/Fib/Basic.lean:45",
      "year": 2026,
      "attribution": "Mathlib's own proof is one line, `fib_of_odd <| odd_two_mul_add_one n`, and this kernel's route is exactly that shape once `Int.Odd` is stated via `natAbs` (F:int-odd) -- `fib_of_odd` was already proved (F:ml430-int-fib-of-odd-66560495); the new work is `Int.odd_two_mul_add_one`, proved by a case split on `n` reducing `2*n+1` to a Nat magnitude of the form `succ (mul 2 _)` in both signs, closed by a witness built from the ALREADY-DECLARED `second_supplementary.rs::two_mul_eq_add_self`/`odd_predicate` helpers (built for an unrelated quadratic-reciprocity modulus, reused unchanged here)."
    }
  ]
}