A finite sum of a constant, over the constructed naturals

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

Recorded description

Summing the constant `c` over `[0, n)` gives `mul c n`. The ORIENTATION is forced by the kernel rather than chosen: `Nat.mul` recurses on its RIGHT argument, so `mul c (succ j)` reduces to `add (mul c j) c`, which is exactly what `sumRange (fun _ => c) (succ j)` reduces to. The mirrored `mul n c` is equally TRUE (`Nat.mul` commutes) and is NOT provable by this proof term -- a mutation stating it is refused by the trusted gate with a TypeMismatch. Proved by induction on `n`, one congruence per step, no lemma outside the two definitions' own reduction rules.

Formal statement

theorem Nat.sumRange_const : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.sumRange (fun (x2 : AxNat) => x0) x1) (AxNat.mul x0 x1)))

Dependencies

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0 direct dependencies 3 direct dependents

Evidence

kernel-Nat.sumRange_const

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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

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)."
}