A prime power passes through a factor the prime does not divide

Identifier: F:nat-prime-pow-dvd-of-dvd-mul-of-not-dvd
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a prime p, a natural b that p does not divide, an exponent c and any natural a: if p to the c divides a times b, then p to the c divides a.

Formal statement

theorem Nat.prime_pow_dvd_of_dvd_mul_of_not_dvd : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : And (AxNat.le (AxNat.succ (AxNat.succ AxNat.zero)) x0) (((x3 : AxNat) -> ((x4 : AxNat.dvd x3 x0) -> Or (Eq.{1} AxNat x3 (AxNat.succ AxNat.zero)) (Eq.{1} AxNat x3 x0))))) -> ((x4 : ((x4 : AxNat.dvd x0 x1) -> False)) -> ((x5 : AxNat) -> ((x6 : AxNat.dvd (AxNat.pow x0 x2) (AxNat.mul x5 x1)) -> AxNat.dvd (AxNat.pow x0 x2) x5)))))))

Dependencies

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

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

Evidence

kernel-prime-pow-dvd-of-dvd-mul-of-not-dvd

Kind: kernel-term
Status: checked

Supports: Nat.prime_pow_dvd_of_dvd_mul_of_not_dvd 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 -- prime_pow_dvd_of_dvd_mul_of_not_dvd 2>/dev/null | grep -Ec '^Nat\.prime_pow_dvd_of_dvd_mul_of_not_dvd[[: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.

footprint-prime-pow-dvd-of-dvd-mul-of-not-dvd

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 (Axiom, Opaque and Quotient), not `Declaration::Axiom` alone, and exits non-zero unless the surface is empty. The enumeration is per-environment rather than per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain. Read directly from `Kernel::axiom_footprint` for these ten `Nat.Multiset.*` theorems, `theorem_axiom_footprint` prints footprint size 0 and an empty axiom column for every one.

Provenance

{
  "date": "2026-09-02",
  "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; declared by `nat_prelude/multiset.rs` (lane nat-multiset, ADR-1520)."
}