Subtraction undoes addition when the order holds

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

Recorded description

For naturals a and b with a <= b: (b - a) + a = b.

Formal statement

theorem Nat.sub_add_cancel : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le x0 x1) -> Eq.{1} AxNat (AxNat.add (AxNat.sub x1 x0) x0) x1)))

Dependencies

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4 direct dependencies 39 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-sub_add_cancel

Kind: kernel-term
Status: checked

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

Checker command

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

footprint-sub_add_cancel

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.

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_nat_prelude",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}