Bezout's identity holds for the natural gcd

Identifier: F:nat-gcd-bezout
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For all natural numbers m and n there exist natural numbers mp, mn, np, nn with gcd(m, n) + m*mn + n*nn = m*mp + n*np. This is Bezout's identity in balanced form: the usual `gcd(m,n) = m*x + n*y` with integer x, y, rearranged so both sides are natural numbers.

Formal statement

theorem Nat.gcd_bezout : ((x0 : AxNat) -> ((x1 : AxNat) -> AxNat.bezout x0 x1 (AxNat.gcd x0 x1)))

Dependencies

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16 direct dependencies 7 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.gcd_bezout

Kind: kernel-term
Status: checked

Supports: For all natural numbers m and n there exist natural numbers mp, mn, np, nn with gcd(m, n) + m*mn + n*nn = m*mp + n*np. This is Bezout's identity in balanced form: the usual `gcd(m,n) = m*x + n*y` with integer x, y, rearranged so both sides are natural numbers.

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing the row at all is a machine-checked proof. TIGHTENED 2026-08-16: the command was `cargo test -p axeyum-lean-kernel --lib nat_prelude`, a whole-suite run that passes or fails identically for every fact citing it and would stay green if THIS theorem were deleted. It now names its own subject twice over -- `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -q` requires the admitted declaration to be printed.

footprint-Nat.gcd_bezout

Kind: exhaustive-enumeration
Status: checked

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

Checker command

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

Evidence notes

`nat_axiom_inventory` enumerates the built Nat environment and reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0` (re-run 2026-08-14 by this lane). That is the whole environment, not this theorem, and that is what makes it sound as a footprint: a theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual footprint by []. Counting `Declaration::Axiom` alone would NOT have sufficed -- `Opaque` has no proof body and `Quotient` admits `Quot.sound`, one of Lean's own axioms -- so the enumeration covers all three.

Provenance

{
  "date": "2026-08-14",
  "established_by": "axeyum-lean-kernel build_nat_prelude, re-measured by lane facts-number",
  "source": "proposition extracted from the S:number / S:arithmetic-structure strands of the math-education concept graph; statement authored here, nothing copied"
}