The rectangle partition at complementary predicates, with no hypothesis

Identifier: F:nat-countrectangle-partition-compl
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

`Nat.countRectangle_partition` specialised to `R := setCompl Q`, so no hypothesis remains. It exists for two reasons and the second is the load-bearing one. First, it is the form a consumer wants whenever the two halves are complementary everywhere. Second, and this is why it is registered separately: it PROVES THE GENERAL THEOREM'S HYPOTHESIS IS SATISFIABLE. A theorem whose hypothesis nothing can discharge is vacuous, and no axiom-footprint check, prelude build or inventory sweep can see that -- deriving this corollary constructs an actual inhabitant of that hypothesis inside the kernel, at a genuinely free `Q`. The per-point witness was not re-derived: `finite_set::compl_sum_eq` was already built there for `Nat.countRange_compl`'s induction step and was private; it is exported instead. WHAT THIS DOES NOT CLAIM: the ARGUMENT ORDER of the two `Nat` binders is not pinned by any numeric test -- swapping them consistently gives a true, admitted theorem and the partition totals `m*n` either way. That mutation SURVIVES, is recorded in ADR-1260's mutation table, and is caught only by reading the declared type.

Formal statement

theorem Nat.countRectangle_partition_compl : ((x0 : ((x0 : AxNat) -> ((x1 : AxNat) -> Bool))) -> ((x1 : AxNat) -> ((x2 : AxNat) -> Eq.{1} AxNat (AxNat.add (AxNat.sumRange (fun (x3 : AxNat) => AxNat.countRange (fun (x4 : AxNat) => x0 x3 x4) x2) x1) (AxNat.sumRange (fun (x3 : AxNat) => AxNat.countRange (AxNat.setCompl (fun (x4 : AxNat) => x0 x4 x3)) x1) x2)) (AxNat.mul x2 x1))))

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.countRectangle_partition_compl

Kind: kernel-term
Status: checked

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

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Nat.countRectangle_partition_compl 2>/dev/null | grep -cE '^Nat\.countRectangle_partition_compl[[:space:]]'

Evidence notes

Two independent failure modes, so the exit status depends on the finding rather than on the run completing: theorem_dependency_inventory exits non-zero when a NAMED filter matches nothing, and grep -c exits 1 printing 0 when the anchored line is absent. Run both ways for this declaration: the real name prints 1 and exits 0, a one-letter misspelling prints 0 and exits 1. Anchored with [[:space:]], never \t -- in a scripted (GNU) grep \t is a literal t. grep -c rather than grep -q, which would SIGPIPE the producer under pipefail. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack. Pass ONE name per invocation: this tool silently keeps only the FIRST name argument.

footprint-Nat.countRectangle_partition_compl

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty, which bounds Nat.countRectangle_partition_compl.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

--require-axiom-free exits non-zero when the named prelude's trusted surface (Axiom + Opaque + Quotient) is not empty, and errors rather than silently passing for a prelude the run never built. A declaration cannot depend on a trusted declaration the environment does not contain, so an empty nat surface bounds every declaration in it. This is a whole-prelude bound, not a per-declaration measurement; the per-declaration figure is the footprint column of kernel_declaration_projection, measured 0 for this row. Measured 2026-08-31: `ok: nat trusted surface = 0`.

Provenance

{
  "date": "2026-08-31",
  "curation": "curated",
  "established_by": "axeyum-lean-kernel build_nat_prelude (crates/axeyum-lean-kernel/src/nat_prelude/lattice_count.rs)",
  "source": "formal.statement, axiom_footprint and depends_on are taken verbatim from `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`, whose TSV rows are (prelude, kind, display name, axiom-footprint size, direct type declarations, direct declarations, direct theorems, Kernel::render_lean(declaration.ty())). depends_on is the direct-theorem column intersected with this ledger's registered facts. Nothing about the statement was hand transcribed; title, statement and the evidence notes were authored (lane quadratic-reciprocity, ADR-1260)."
}