Int.fib_rec -- the Fibonacci recurrence at EVERY integer index

Identifier: F:int-fib-rec
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

fib (n + 2) = fib (n + 1) + fib n for every INTEGER n, negative ones included. Nat.fib_add_two is the natural-number half and says nothing below 0, and Int.fib's negSucc clause is a definition rather than a recurrence, so before this nothing in the development related fib(-k-1) to its neighbours.

Formal statement

theorem Int.fib_rec : ((x0 : Int) -> Eq.{1} Int (Int.fib (Int.add x0 (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))))) (Int.add (Int.fib (Int.add x0 Int.one)) (Int.fib x0)))

Dependencies

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

14 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Int.fib_rec

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

The pattern pins the entire rendered type; verified discriminating on this tree -- 1 against the real row, 0 against a row with the two summands transposed. Tab separators are matched with [[:space:]] and never with a backslash-t escape -- see this repository's documented GNU-grep/ugrep divergence, which silently made 68 checker commands match nothing. --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_rec

Kind: exhaustive-enumeration
Status: checked

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

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_rec-one-index-per-branch

Kind: witness-replay
Status: checked

Supports: The proof has three cases because the recurrence straddles zero, and the test reads one index from each and checks the closed arithmetic: n = 3 gives 5 = 3 + 2; n = -1 gives 1 = 0 + 1 (the subNatNat corner); n = -4 gives -1 = 2 + (-3) (the branch that does the sign algebra). Each is paired with a wrong right-hand side that must NOT be def_eq.

Checker command

test "$(cargo test -q -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::fib_rec_computes_the_recurrence_at_indices_of_both_signs 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. 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_rec)",
  "prior_art": [
    {
      "attribution": "statement-only; the Mathlib proof was not consulted. NOTE the overlap, deliberately recorded rather than hidden: the same proposition is already epistemic_status proved in this ledger as F:ml430-int-fib-add-two-739358dd, closed by a SEALED EXTERNAL Lean capsule under the autogenesis import route. That capsule lives outside this kernel's environment, so no int_prelude declaration can cite it; this row records the in-tree constructive declaration that Int.fib_add actually consumes. It is NOT an independent new mathematical result.",
      "what": "Int.fib_add_two, the same proposition with the summands transposed",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "who": "the Mathlib contributors",
      "year": 2026
    }
  ],
  "source": "constructed in this kernel; canonical type read from int_theorem_inventory."
}