The length of an append is the sum of the two lengths

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

Recorded description

For lists of naturals, the length of the concatenation of l1 and l2 equals the sum of their individual lengths.

Formal statement

theorem List.length_append : ((x0 : Sort (1)) -> ((x1 : List.{0} x0) -> ((x2 : List.{0} x0) -> Eq.{1} AxNat (List.length x0 (List.append x0 x1 x2)) (AxNat.add (List.length x0 x1) (List.length x0 x2)))))

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

kernel-list-length-append

Kind: kernel-term
Status: checked

Supports: List.length_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.length_append 2>/dev/null | grep -Fc '((x0 : Sort (1)) -> ((x1 : List.{0} x0) -> ((x2 : List.{0} x0) -> Eq.{1} AxNat (List.length x0 (List.append x0 x1 x2)) (AxNat.add (List.length x0 x1) (List.length 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. build_list_nat_bridge admits this theorem only through the trusted kernel gate, so a successful build IS the type-check.

footprint-list-length-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::bridge::bridge_tests::the_bridge_theorems_declare_no_axioms --test-threads=1

Evidence notes

list_prelude::bridge::bridge_tests::the_bridge_theorems_declare_no_axioms asserts Kernel::axiom_footprint(name).is_empty() for this declaration (among others in the same coverage sweep), read directly from the kernel rather than asserted from a list.

evaluation-list-length-append

Kind: instance-pin
Status: checked

Supports: The List 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

list_prelude_tests.rs and bridge_tests.rs reduce List.length/append/map/foldr/reverse/sum/toMultiset/count to a concrete numeral or nil/cons skeleton at tiny discriminating arguments and pair every positive with the specific wrong formula its negative control rules out.

Provenance

{
  "date": "2026-09-03",
  "established_by": "axeyum-lean-kernel build_list_nat_bridge",
  "source": "theorem name and canonical type read directly via list_theorem_inventory, which prints render_lean of the admitted type; declared by list_prelude.rs/theorems.rs/bridge.rs (lane list-carrier-1, ADR-1579)."
}