The gcd is exactly the common divisors' upper bound

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

Recorded description

For all natural numbers k, m and n: k divides gcd(m, n) if and only if k divides m and k divides n. This is the universal property that pins down the greatest common divisor without appealing to an ordering of divisors.

Formal statement

theorem Nat.dvd_gcd_iff : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> Iff (AxNat.dvd x0 (AxNat.gcd x1 x2)) (And (AxNat.dvd x0 x1) (AxNat.dvd x0 x2)))))

Dependencies

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5 direct dependencies 1 direct dependents

Evidence

kernel-Nat.dvd_gcd_iff

Kind: kernel-term
Status: checked

Supports: For all natural numbers k, m and n: k divides gcd(m, n) if and only if k divides m and k divides n. This is the universal property that pins down the greatest common divisor without appealing to an ordering of divisors.

Checker command

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

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"
}