A range sum splits into its selected and unselected parts

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

Recorded description

`sumRangeIf p f n + sumRangeIf (setCompl p) f n = sumRange f n` -- a range sum splits into the part its predicate selects and the part it does not. WHY THERE IS NO HYPOTHESIS: this kernel has no `Bool.not` (measured -- `shape_search --name-like bnot` returns ABSENT), and `Nat.setCompl p := fun k => if p k then false else true` (`finite_set.rs`) IS that missing function, already declared and already carrying its own involutivity and De Morgan laws. So the split needs no complementarity side condition at all, unlike `Nat.countRectangle_partition`, which takes one. HOW IT IS PROVED: the additive twin of `Nat.countRange_compl`, and its proof is that proof with `1`/`0` selection replaced by `f j`/`0` -- induction on `n`, one four-term regroup, and one pointwise `Bool.rec` whose two branches are NOT symmetric (`v + 0` reduces by `Nat.add`'s own right-argument rule, `0 + v` does not and needs `Nat.zero_add`). WHY IT DID NOT EXIST: two prior lanes (ADR-1540, ADR-1544) both stopped at the same wall and both MEASURED it rather than guessing -- `examples/shape_search --name-like sumRangeIf` returns ABSENT in every prelude, against a `prodRangeIf` positive control returning 12 declarations. This lane re-ran that measurement on a freshly built binary (declarations=2092, so not a stale artifact reporting a false absence) and got the same two verdicts. `Kernel::axiom_footprint` is EMPTY. Admitted on the first attempt.

Formal statement

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

Dependencies

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

4 direct dependencies 0 direct dependents

Evidence

kernel-Nat.sumRangeIf_compl

Kind: kernel-term
Status: checked

Supports: Nat.sumRangeIf_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.sumRangeIf_compl 2>/dev/null | grep -cE '^Nat\.sumRangeIf_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 WITH A NEGATIVE CONTROL and it was: the real name prints 1 with both pipeline stages at 0, a one-character typo prints 0 with both stages at 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. Pass ONE name per invocation: this tool silently keeps only the FIRST name argument.

footprint-Nat.sumRangeIf_compl

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the natural-number prelude's trusted surface is empty, which bounds Nat.sumRangeIf_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 measured 0 by the axiom-footprint assertion in subset_sum_tests.rs.

numeric-Nat.sumRangeIf_compl

Kind: exhaustive-enumeration
Status: checked

Supports: ADR-1552's check script sweeps this statement's arithmetic over 8450 (modulus, multiplier, bound) instances for the hypothesis-free rows, 519 coprime instances for the counting identity, and 399 coprime odd pairs plus all 240 ordered pairs of distinct odd primes below 60 for Eisenstein's lemma; the control table refutes eight wrong readings at named witnesses.

Checker command

python3 docs/research/09-decisions/adr-1552-eisenstein-checks.py

Evidence notes

Exit status depends on the finding: a claim that fails, or a control that behaves other than as recorded, exits 1. Verified by mutating the script itself in a scratch copy with one uniquely-named file per mutant (so the stale-__pycache__ trap cannot report the previous mutant's result): 16 of 16 mutations exit 1. Two recorded SURVIVORS, M9 and M10, are the argument order of the congruence and of `Even (F + N)` -- invisible to every numeric check and guarded only by the character-for-character type pins in subset_sum_tests.rs.

Provenance

{
  "date": "2026-09-02",
  "curation": "curated",
  "established_by": "axeyum-lean-kernel build_nat_prelude (crates/axeyum-lean-kernel/src/nat_prelude/subset_sum.rs)",
  "source": "formal.statement is taken verbatim from the kernel's own rendering (Kernel::render_lean of the admitted declaration type; the same string is pinned character for character in subset_sum_tests.rs). depends_on is the direct-theorem column of theorem_dependency_inventory intersected with this ledger's registered facts, so it is an INTERSECTION and not the full dependency set -- direct dependencies with no fact row are omitted. Nothing about the statement was hand transcribed; title, statement and the evidence notes were authored (lane eisenstein-3)."
}