Every factor below the bound divides a bounded product

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

Recorded description

For a function f on the naturals and indices i < k: f applied to i divides the product of f over the range [0, k).

Formal statement

theorem Nat.dvd_prodRange_of_lt : ((x0 : ((x0 : AxNat) -> AxNat)) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.lt x1 x2) -> AxNat.dvd (x0 x1) (AxNat.prodRange x0 x2)))))

Dependencies

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7 direct dependencies 1 direct dependents

Evidence

kernel-dvd-prodrange-of-lt

Kind: kernel-term
Status: checked

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

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted kernel gate. The pattern is ANCHORED at the start of the line: `Nat.not_pow_succ_dvd_prodRange_of_lt` contains this theorem's name as a substring, so an unanchored match would pass on the wrong row.

footprint-dvd-prodrange-of-lt

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)."
}