polyEval is linear over polyScale, up to Complex.Equiv

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

Recorded description

For any a, g, n, x: polyEval (polyScale a g) n x is Complex.Equiv to a * polyEval g n x -- polyEval is linear (homogeneous) over polyScale, up to Complex.Equiv.

Formal statement

theorem Complex.polyEval_polyScale : ((x0 : Complex) -> ((x1 : ((x1 : AxNat) -> Complex)) -> ((x2 : AxNat) -> ((x3 : Complex) -> Complex.Equiv (Complex.polyEval (Complex.polyScale x0 x1) x2 x3) (Complex.mul x0 (Complex.polyEval x1 x2 x3))))))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

19 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Complex.polyEval_polyScale

Kind: kernel-term
Status: checked

Supports: Complex.polyEval_polyScale 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 -- Complex.polyEval_polyScale 2>/dev/null | grep -cE '^Complex\.polyEval_polyScale[[:space:]]'

Evidence notes

build_complex_prelude admits Complex.polyEval_polyScale through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted theorem cannot read as a re-derived one), and grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command prints exactly one matching line for Complex.polyEval_polyScale. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.polyEval_polyScale

Kind: exhaustive-enumeration
Status: checked

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

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.polyEval_polyScale, 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_eval_poly_scale)",
  "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."
}