Addition on the naturals is commutative

Identifier: F:nat-add-comm
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For all natural numbers a and b, a + b = b + a.

Formal statement

theorem Nat.add_comm : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.add x0 x1) (AxNat.add x1 x0)))

Dependencies

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2 direct dependencies 114 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-add-comm

Kind: kernel-term
Status: checked

Supports: For all natural numbers a and b, a + b = b + a.

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted kernel gate, so a successful build IS the type-check; the command both performs it and prints the admitted type, which is copied verbatim into formal.statement. Nothing here is transcribed.

footprint-add-comm

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

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

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the FULL trusted surface rather than Declaration::Axiom alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain. A per-theorem footprint (Lean's `#print axioms`) does not exist here yet.

Provenance

{
  "date": "2026-08-14",
  "established_by": "axeyum-lean-kernel nat_prelude (lean-kernel lane)",
  "source": "hand-built foundational library"
}