polyEval is scalar-linear: evaluating a scaled polynomial equals scaling the evaluation

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

Recorded description

CReal.Equiv (polyEval (polyScale s c) n x) (s * polyEval c n x) -- polyEval commutes with scalar multiplication, for every scalar s, coefficient function c, degree bound n, and point x.

Formal statement

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

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-CReal.polyEval_polyScale

Kind: kernel-term
Status: checked

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

Evidence notes

The producing prelude admits CReal.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/renamed theorem cannot read as a re-derived one), and grep -c (never -q) both avoids a SIGPIPE-under-pipefail false negative and independently asserts the exact tab-anchored line is present. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.polyEval_polyScale

Kind: exhaustive-enumeration
Status: checked

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

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_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_creal_prelude (crates/axeyum-lean-kernel/src/creal/polynomial.rs, declare_poly_eval_poly_scale)",
  "source": "theorem name, canonical type and direct dependency edges read via theorem_dependency_inventory's own per-name output (cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- <Name>), which prints '<Name>\\t<comma-separated direct deps>'; the canonical type itself was cross-extracted from kernel_declaration_projection's unfiltered TSV dump the same way as the Definition facts in this batch, and confirmed byte-for-byte identical between the two tools' output for every name in this batch. crates/ source was not touched to produce this batch."
}