Counting a predicate IS summing its selector, definitionally

Identifier: F:nat-countrange-eq-sumrange
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

`countRange f n` equals `sumRange (fun k => bool_select_nat (f k) 1 0) n` by `Eq.refl`. This is not a coincidence to be proved but a fact about the two definitions: `Nat.countRange` (`nat_prelude/totient.rs`) and `Nat.sumRange` (`nat_prelude/defs.rs`) are the SAME `Nat.rec` -- motive `fun _ => Nat`, base `zero`, step `fun j ih => add ih (g j)` -- differing only in what `g` is. WHAT THIS IS FOR: it lets a proof move between the counting world and the summing world, which is what the rectangle partition needs in order to apply Fubini to a `countRange`. WHAT IT DOES NOT CLAIM: being `Eq.refl`, it cannot be wrong the way a proved theorem can; its value is that it PINS the two definitions staying the same recursor. If someone changed `countRange`'s step to accumulate on the left, this declaration would stop being admitted.

Formal statement

theorem Nat.countRange_eq_sumRange : ((x0 : ((x0 : AxNat) -> Bool)) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.countRange x0 x1) (AxNat.sumRange (fun (x2 : AxNat) => Bool.rec.{1} (fun (x3 : Bool) => AxNat) AxNat.zero (AxNat.succ AxNat.zero) (x0 x2)) x1)))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 2 direct dependents

Evidence

kernel-Nat.countRange_eq_sumRange

Kind: kernel-term
Status: checked

Supports: Nat.countRange_eq_sumRange 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.countRange_eq_sumRange 2>/dev/null | grep -cE '^Nat\.countRange_eq_sumRange[[: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.countRange_eq_sumRange

Kind: exhaustive-enumeration
Status: checked

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

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)."
}