A multiset of primes carries exactly its recorded multiplicity

Identifier: F:nat-multiset-not-pow-succ-count-dvd-prod
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a multiset m whose every element is prime and a prime x: x raised to one more than the multiplicity of x in m does NOT divide the product of m.

Formal statement

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

Dependencies

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

Evidence

kernel-multiset-not-pow-succ-count-dvd-prod

Kind: kernel-term
Status: checked

Supports: Nat.Multiset.not_pow_succ_count_dvd_prod 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 -- Multiset.not_pow_succ_count_dvd_prod 2>/dev/null | grep -Ec '^Nat\.Multiset\.not_pow_succ_count_dvd_prod[[: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-multiset-not-pow-succ-count-dvd-prod

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)."
}