A divisor of the modulus decides divisibility of a value from its remainder

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

Recorded description

For naturals a, b, c with a dividing b+1: a divides (c mod (b+1)) if and only if a divides c.

Formal statement

theorem Nat.dvd_mod_iff : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.dvd x0 (AxNat.succ x1)) -> Iff (AxNat.dvd x0 (AxNat.mod x2 (AxNat.succ x1))) (AxNat.dvd x0 x2)))))

Dependencies

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4 direct dependencies 4 direct dependents

Evidence

kernel-dvd_mod_iff

Kind: kernel-term
Status: checked

Supports: Nat.dvd_mod_iff 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 -- Nat.dvd_mod_iff 2>/dev/null | grep -Ec '^Nat\.dvd_mod_iff[[: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 both performs it and prints the admitted type, which is copied verbatim into formal.statement.

footprint-dvd_mod_iff

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 rather than Declaration::Axiom alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-25",
  "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."
}