The empty signed finite sum over the constructed integers is zero

Identifier: F:int-sumrange-zero
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

`Int.sumRange f 0 = 0`, the base equation of the signed finite sum. Closes by `Eq.refl`: `Int.sumRange` is a checked `Nat.rec` definition whose zero minor premise IS `Int.zero`, so the equation is a reduction, not a proof. Admitting it entails the definition it is stated about. `Kernel::axiom_footprint` is EMPTY. Admitted by the trusted gate on the first attempt.

Formal statement

theorem Int.sumRange_zero : ((x0 : ((x0 : AxNat) -> Int)) -> Eq.{1} Int (Int.sumRange x0 AxNat.zero) Int.zero)

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Int.sumRange_zero

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude's trusted surface is empty, which bounds Int.sumRange_zero.

Checker command

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

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 integer 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.

Provenance

{
  "date": "2026-08-31",
  "curation": "curated",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/sum.rs)",
  "source": "formal.statement and axiom_footprint 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, supplied by scripts/check-fact-depends-derived.py --fix. Nothing about the statement was hand transcribed; title, statement and the evidence notes were authored (lane int-sumrange)."
}