The product of a multiset sum is the product of the parts

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

Recorded description

For any two multisets over the naturals, the product of their sum is the product of their products. The two sides fold over three different bounds -- the sum's bound is the sum of the two bounds -- so neither factor's fold is a subfold of the left-hand side, and the proof collapses each tail rather than splitting a range.

Formal statement

theorem Nat.Multiset.prod_add : ((x0 : AxNat.Multiset) -> ((x1 : AxNat.Multiset) -> Eq.{1} AxNat (AxNat.Multiset.prod (AxNat.Multiset.add x0 x1)) (AxNat.mul (AxNat.Multiset.prod x0) (AxNat.Multiset.prod x1))))

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

nat-multiset-prod-add-1

Kind: kernel-term
Status: checked

Supports: Nat.Multiset.prod_add is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- Multiset.prod_add 2>&1) && test "$(printf "%s\n" "$out" | grep -cE '^Nat\.Multiset\.prod_add.2.\(\(x0 : AxNat\.Multiset\) -> \(\(x1 : AxNat\.Multiset\) -> Eq\.\{1\} AxNat \(AxNat\.Multiset\.prod \(AxNat\.Multiset\.add x0 x1\)\) \(AxNat\.mul \(AxNat\.Multiset\.prod x0\) \(AxNat\.Multiset\.prod x1\)\)\)\)$')" = 1

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted kernel gate, so a successful build IS the type-check. The pattern pins the ARITY and the FULL rendered type, not the theorem's presence: `scripts/new-fact.py` rejected a name-only pattern as population-only, because it catches the row disappearing but not the statement changing inside a row that is still there.

footprint-nat-multiset-prod-add

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` via `theorem_axiom_footprint`, this theorem's row prints footprint size 0 and an empty axiom column, and all 904 `nat` rows do. (Count the tool's OWN summary line, not `grep -c '^nat'` -- the summary starts with `nat:` too, so the naive count is 905.)

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_prod.rs` (lane nat-factorization)."
}