A multiset's product is divisible by each element to its multiplicity

Identifier: F:nat-multiset-pow-count-dvd-prod
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a multiset m of naturals and any natural x: x raised to the multiplicity of x in m divides the product of m.

Formal statement

theorem Nat.Multiset.pow_count_dvd_prod : ((x0 : AxNat.Multiset) -> ((x1 : AxNat) -> AxNat.dvd (AxNat.pow x1 (AxNat.Multiset.count x0 x1)) (AxNat.Multiset.prod x0)))

Dependencies

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4 direct dependencies 1 direct dependents

Evidence

kernel-multiset-pow-count-dvd-prod

Kind: kernel-term
Status: checked

Supports: Nat.Multiset.pow_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.pow_count_dvd_prod 2>/dev/null | grep -Ec '^Nat\.Multiset\.pow_count_dvd_prod[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted kernel gate. `crates/axeyum-lean-kernel/src/nat_prelude/multiset_tests.rs` additionally INSTANTIATES it at concrete arguments and checks the inferred statement is the arithmetic fact it should be (`4 | 12` at the multiset {2,2,3} and the point 2, with `8 | 12` as the negative control) -- the trusted gate cannot tell that `count` and `prod` unfold to the right numbers.

footprint-multiset-pow-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)."
}