polyEval: total Horner-style evaluation of a coefficient function over CReal

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

Recorded description

CReal.polyEval c n x := sumRange (fun i => c i * x^i) n -- evaluate the polynomial with coefficient function c at point x, truncated to degree < n. A total, axiom-free definition over the constructed reals; not a computed degree (see F:creal-polydegreelt's own notes for why no coefficient of a CReal-valued polynomial can be tested for zero).

Formal statement

def CReal.polyEval : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : AxNat) -> ((x2 : CReal) -> CReal)))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-CReal.polyEval

Kind: kernel-term
Status: checked

Supports: CReal.polyEval 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 CReal.polyEval --require-kind definition 2>/dev/null | grep -cE '^found[[:space:]]creal[[:space:]]definition[[:space:]]CReal\.polyEval[[:space:]]'

Evidence notes

CReal.polyEval is a Declaration::Definition. theorem_dependency_inventory / nat_theorem_inventory / prelude_theorem_inventory all filter to Declaration::Theorem and exclude Definitions/Inductives, so this fact uses the DIRECT 'kernel_declaration_projection --require-declaration' checker: 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\tcreal\tdefinition\tCReal.polyEval\t...' line. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.polyEval

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the creal prelude's trusted surface is empty, which bounds CReal.polyEval

Checker command

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

Evidence notes

Re-measured on this tree: creal: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: creal trusted surface = 0'. That bounds every declaration in the creal environment, including CReal.polyEval, 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_creal_prelude (crates/axeyum-lean-kernel/src/creal/polynomial.rs, declare_poly_eval)",
  "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); crates/ source was not touched to produce this batch."
}