polyMul is the total antidiagonal (Cauchy-product) convolution

Identifier: F:complex-polymul
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

Complex.polyMul c g := fun k => sumRange (fun i => c(i) * g(k-i)) (k+1) -- the finite antidiagonal (Cauchy-product) convolution of two coefficient functions, a first-class named operation. A total, axiom-free definition.

Formal statement

def Complex.polyMul : ((x0 : ((x0 : AxNat) -> Complex)) -> ((x1 : ((x1 : AxNat) -> Complex)) -> ((x2 : AxNat) -> Complex)))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Complex.polyMul

Kind: kernel-term
Status: checked

Supports: Complex.polyMul 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 kernel_declaration_projection -- --require-declaration Complex.polyMul --require-kind definition 2>/dev/null | grep -cE '^found[[:space:]]complex[[:space:]]definition[[:space:]]Complex\.polyMul[[:space:]]'

Evidence notes

Complex.polyMul is a Declaration::Definition. theorem_dependency_inventory / nat_theorem_inventory / prelude_theorem_inventory all filter to Declaration::Theorem and exclude Definitions, so this fact uses the DIRECT 'kernel_declaration_projection --require-declaration' checker (added earlier in this session's batch): it searches every constructed prelude's environment for an exact display-name match and exits non-zero on no match or kind mismatch. Verified on this tree: the command prints exactly one 'found\tcomplex\tdefinition\tComplex.polyMul\t...' line (a definition built this early in the dependency order also shows up, separately, under the downstream complex/cpoint preludes that rebuild Complex from scratch as their own Kernel instance -- this checker deliberately anchors on the complex row only, the prelude Complex.polyMul is native to). --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.polyMul

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the complex prelude's trusted surface is empty, which bounds Complex.polyMul

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free complex

Evidence notes

Re-measured on this tree: complex: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: complex trusted surface = 0'. That bounds every declaration in the complex environment, including Complex.polyMul, since a declaration cannot depend on a trusted declaration the environment does not contain. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_complex_prelude (crates/axeyum-lean-kernel/src/complex/poly.rs, declare_poly_mul)",
  "source": "canonical type read via kernel_declaration_projection's own UNFILTERED emit mode (cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection, no --require-declaration flag), which prints, per constructed prelude, one TSV row per declaration whose last field is kernel.render_lean(declaration.ty()) -- the same Kernel::render_lean canonical form nat_theorem_inventory prints, just not filtered to Declaration::Theorem. That output was piped to a scratchpad file and the exact row for this declaration's own prelude label was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct dependency edges (where applicable) were cross-read from the same run's direct_theorems column and independently confirmed against theorem_dependency_inventory's own output on this tree. No new probe binary was written for this batch, since the in-tree tool already emits the canonical type; crates/ source was not touched to produce this batch."
}