Count distributes additively over append

Identifier: F:list-count-append
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For any element a and lists l1, l2 over Nat, the number of times a occurs in the concatenation of l1 and l2 equals the sum of how many times a occurs in l1 and how many times it occurs in l2.

Formal statement

theorem List.count_append : ((x0 : AxNat) -> ((x1 : List.{0} AxNat) -> ((x2 : List.{0} AxNat) -> Eq.{1} AxNat (List.count x0 (List.append AxNat x1 x2)) (AxNat.add (List.count x0 x1) (List.count x0 x2)))))

Dependencies

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2 direct dependencies 2 direct dependents

Evidence

kernel-list-count-append

Kind: kernel-term
Status: checked

Supports: List.count_append is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example list_theorem_inventory -- List.count_append 2>/dev/null | grep -Fc '((x0 : AxNat) -> ((x1 : List.{0} AxNat) -> ((x2 : List.{0} AxNat) -> Eq.{1} AxNat (List.count x0 (List.append AxNat x1 x2)) (AxNat.add (List.count x0 x1) (List.count x0 x2)))))')" -ge 1

Evidence notes

The pattern is the FULL admitted type (rendered by Kernel::render_lean via list_theorem_inventory), not the theorem's name -- a drifted statement would fail this check.

footprint-list-count-append

Kind: instance-pin
Status: checked

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

Checker command

cargo test -q --release -p axeyum-lean-kernel --lib -- list_prelude::perm::perm_tests::the_perm_theorems_declare_no_axioms --test-threads=1

Evidence notes

Asserts Kernel::axiom_footprint(name).is_empty() for this declaration, read directly from the kernel rather than asserted from a list.

evaluation-list-count-append

Kind: instance-pin
Status: checked

Supports: The List/Nat operations 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 -- list_prelude:: --test-threads=4

Evidence notes

The list_prelude:: sweep (list_prelude_tests.rs, bridge_tests.rs, perm_tests.rs) reduces List/Nat.Multiset operations to concrete numerals or nil/cons skeletons at tiny discriminating arguments and pairs positives with negative controls.

Provenance

{
  "date": "2026-09-03",
  "established_by": "axeyum-lean-kernel build_list_perm",
  "source": "theorem name and canonical type read directly via list_theorem_inventory, which prints render_lean of the admitted type; declared by list_prelude/perm.rs (lane list-carrier-2, ADR-1583 amending ADR-1579)."
}