gcd is commutative on the integers

Identifier: F:int-gcd-comm
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For all integers a and b, gcd(a, b) = gcd(b, a). (The integer gcd returns a natural number.)

Formal statement

theorem Int.gcd_comm : ((x0 : Int) -> ((x1 : Int) -> Eq.{1} AxNat (Int.gcd x0 x1) (Int.gcd x1 x0)))

Dependencies

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

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

Evidence

kernel-Int.gcd_comm

Kind: kernel-term
Status: checked

Supports: For all integers a and b, gcd(a, b) = gcd(b, a). (The integer gcd returns a natural number.)

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example int_theorem_inventory -- gcd_comm 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.gcd_comm[[:space:]]')" -ge 1

Evidence notes

`build_int_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `int_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -q` requires the admitted `theorem`-kind row to be printed.

footprint-Int.gcd_comm

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Int prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory --require-axiom-free integer` enumerates the built Int environment (which nests the Nat development) and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_int_prelude, lane rado-claim-ledger",
  "source": "one of the theorems build_int_prelude admits; registered to close the flywheel's fact-ledger arrow for recently landed Int theorems"
}