A signed finite sum reduces modulo n term by term, with no positivity hypothesis

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

Recorded description

`(forall k, f k = g k [MOD n]) -> Int.sumRange f m = Int.sumRange g m [MOD n]`. WHAT IS DELIBERATELY ABSENT: any `0 < n` premise. `Int.modEq_prodRange`, the product analogue, carries one, and transporting it here would have been wrong rather than merely cautious -- the product's step goes through `Int.ModEq.mul`, whose statement in this prelude is positivity-scoped, while the sum's step goes through `Int.ModEq.add_right` and `Int.ModEq.add_left`, both proved UNCONDITIONALLY in the modulus. With the hypothesis every consumer of the mod-2 reader Eisenstein's lemma needs would have had to discharge `0 < 2`. A test asserts the difference in BOTH directions, so the absence is measured rather than an artefact of how the check reads types. `Kernel::axiom_footprint` is EMPTY. Admitted by the trusted gate on the first attempt.

Formal statement

theorem Int.modEq_sumRange : ((x0 : Int) -> ((x1 : ((x1 : AxNat) -> Int)) -> ((x2 : ((x2 : AxNat) -> Int)) -> ((x3 : AxNat) -> ((x4 : ((x4 : AxNat) -> Int.ModEq x0 (x1 x4) (x2 x4))) -> Int.ModEq x0 (Int.sumRange x1 x3) (Int.sumRange x2 x3))))))

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Int.modEq_sumRange

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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

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)."
}