polyDegreeLt is closed under polyScale

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

Recorded description

If g has polyDegreeLt n, then for any Complex scale a and any index k>=n, a*g(k) is Complex.Equiv-zero -- i.e. polyScale a g again has polyDegreeLt n (via Complex.mul_zero applied to g's own vanishing coefficient), so the degree-bound predicate is closed under scaling by any constant, including zero.

Formal statement

theorem Complex.polyDegreeLt_polyScale : ((x0 : Complex) -> ((x1 : ((x1 : AxNat) -> Complex)) -> ((x2 : AxNat) -> ((x3 : Complex.polyDegreeLt x1 x2) -> ((x4 : AxNat) -> ((x5 : AxNat.le x2 x4) -> Complex.Equiv (Complex.mul x0 (x1 x4)) Complex.zero))))))

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Complex.polyDegreeLt_polyScale

Kind: kernel-term
Status: checked

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

Evidence notes

build_complex_prelude admits Complex.polyDegreeLt_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.polyDegreeLt_polyScale. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.polyDegreeLt_polyScale

Kind: exhaustive-enumeration
Status: checked

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