A power divides a higher power of the same base

Identifier: F:nat-pow-dvd-pow-of-le
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For naturals a, i and j with i at most j: a to the i divides a to the j.

Formal statement

theorem Nat.pow_dvd_pow_of_le : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.le x1 x2) -> AxNat.dvd (AxNat.pow x0 x1) (AxNat.pow x0 x2)))))

Dependencies

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3 direct dependencies 1 direct dependents

Evidence

kernel-pow-dvd-pow-of-le

Kind: kernel-term
Status: checked

Supports: Nat.pow_dvd_pow_of_le 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 -- pow_dvd_pow_of_le 2>/dev/null | grep -Ec '^Nat\.pow_dvd_pow_of_le[[: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; the command prints the admitted type, copied verbatim into formal.statement, and exits non-zero when the name is absent.

footprint-pow-dvd-pow-of-le

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)."
}