Gauss's-lemma per-term congruence: a*k = (sign) * gaussFold(pp,a,k) mod pp

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

Recorded description

The per-term congruence underneath Gauss's lemma: for any modulus pp > 0 and any a, k, the product a*k is congruent mod pp to eps * gaussFold(pp,a,k), where eps is -1 when `AxNat.gaussSignNeg pp a k` holds (the least residue of a*k exceeds pp/2) and +1 otherwise, and gaussFold folds a 'negative' residue r back to pp - r. Stated over the constructed integers because eps is; `Kernel::axiom_footprint` is EMPTY. WHAT THIS DOES NOT CLAIM. Nothing about the product over k, and nothing about primality: the only hypothesis is 0 < pp, so this holds at composite moduli and at a not coprime to pp -- it is Gauss's lemma's termwise input, not a criterion of its own.

Formal statement

theorem Int.gaussTermModEq : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.lt AxNat.zero x0) -> Int.ModEq (Int.ofNat x0) (Int.mul (Int.ofNat x1) (Int.ofNat x2)) (Int.mul (Bool.rec.{1} (fun (x4 : Bool) => Int) Int.one (Int.neg Int.one) (AxNat.gaussSignNeg x0 x1 x2)) (Int.ofNat (AxNat.gaussFold x0 x1 x2)))))))

Dependencies

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7 direct dependencies 1 direct dependents

Evidence

kernel-Int.gaussTermModEq

Kind: kernel-term
Status: checked

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

footprint-Int.gaussTermModEq

Kind: exhaustive-enumeration
Status: checked

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

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, including Int.gaussTermModEq. 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",
  "generated_by": "scripts/gen-kernel-facts.py",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/)",
  "source": "Derived mechanically from the unfiltered emit of `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`, which prints one TSV row per declaration whose fields are (prelude, kind, display name, axiom-footprint size, direct type declarations, direct declarations, direct theorems, Kernel::render_lean(declaration.ty())). formal.statement is that last field verbatim; depends_on is the direct-theorem column intersected with this ledger's registered facts; axiom_footprint is the footprint-size column, cross-checked by the whole-prelude nat_axiom_inventory run recorded in the second evidence row. No field was hand-transcribed and no prose was authored. `title`, `statement` and `notes` were then REPLACED by hand (lane gauss-final, ADR-1130); every other field is the generator's output unmodified."
}