A sum over a disjoint union splits

Identifier: F:nat-finset-sum-union-disjoint
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

If two finite sets do not meet -- their memberships' pointwise intersection is false at every index -- then the sum of any summand over their union is the sum over the first plus the sum over the second.

Formal statement

theorem Nat.Finset.sum_union_disjoint : ((x0 : AxNat.Finset) -> ((x1 : AxNat.Finset) -> ((x2 : ((x2 : AxNat) -> AxNat)) -> ((x3 : ((x3 : AxNat) -> Eq.{1} Bool (AxNat.setInter (AxNat.Finset.memB x0) (AxNat.Finset.memB x1) x3) Bool.false)) -> Eq.{1} AxNat (AxNat.Finset.sum (AxNat.Finset.union x0 x1) x2) (AxNat.add (AxNat.Finset.sum x0 x2) (AxNat.Finset.sum x1 x2))))))

Dependencies

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8 direct dependencies 1 direct dependents

Evidence

kernel-nat-finset-sum-union-disjoint

Kind: kernel-term
Status: checked

Supports: Nat.Finset.sum_union_disjoint 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.Finset.sum_union_disjoint 2>/dev/null | grep -Fc 'Eq.{1} AxNat (AxNat.Finset.sum (AxNat.Finset.union x0 x1) x2) (AxNat.add (AxNat.Finset.sum x0 x2) (AxNat.Finset.sum x1 x2))')" -ge 1

Evidence notes

The pattern is a DISTINGUISHING substring of the admitted type, not the theorem's name: an existence check would still pass if the statement drifted. Verified to fail on a perturbed pattern before this row was written. `build_nat_prelude` admits this theorem only through the trusted kernel gate, so a successful build IS the type-check.

footprint-nat-finset-sum-union-disjoint

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- read from Kernel::axiom_footprint for this declaration, not from a maintained list

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Finset 2>/dev/null | awk -F'\t' '$1 == "nat" && $2 == "Nat.Finset.sum_union_disjoint" && $3 == "0"' | wc -l)" -ge 1

Evidence notes

`theorem_axiom_footprint` prints, per declaration, the size and contents of `Kernel::axiom_footprint`. The pattern pins BOTH the declaration name and the size 0 with an empty axiom column, so a proof that reached for a trusted declaration would fail the check rather than be reported as axiom-free. The `nat` prelude's whole trusted surface is empty, which bounds this independently.

evaluation-nat-finset-sum-union-disjoint

Kind: instance-pin
Status: checked

Supports: The definitions this statement is about COMPUTE the intended values, with negative controls -- the kernel cannot tell a Definition is wrong

Checker command

cargo test -q --release -p axeyum-lean-kernel --lib -- nat_prelude::finset_tests --test-threads=4

Evidence notes

`Nat.Finset`'s operations are admitted on their TYPE; a `card` that computed the wrong number would have the right type, an empty axiom footprint, and would pass every other check in this ledger. `finset_tests.rs` reduces each operation to a numeral or a Bool at tiny discriminating arguments and pairs every positive with the specific wrong formula its negative control rules out. It caught one wrong hand-computed expectation while being written.

Provenance

{
  "date": "2026-09-03",
  "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/finset.rs` (lane finset-role, ADR-1577)."
}