The j-th power-series term, coefficients times x to the j, is a total constructed function

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

Recorded description

CReal.powerSeriesTerm : (Nat -> CReal) -> Nat -> CReal -> CReal := fun c j x => mul (c j) (pow x j) -- the j-th term of the power series with coefficient sequence c, evaluated at x. A bare Definition, asserting nothing beyond totality; the Ch.24 power-series machinery (Equiv-congruence, coefficient-boundedness domination, uniform convergence) is built on top of it as separate theorems, registered alongside this one in the same batch.

Formal statement

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

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-CReal.powerSeriesTerm

Kind: kernel-term
Status: checked

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

Evidence notes

CReal.powerSeriesTerm is a Declaration::Definition, so theorem_dependency_inventory / nat_theorem_inventory / prelude_theorem_inventory (which all filter to Declaration::Theorem) cannot see it. kernel_declaration_projection --require-declaration searches every constructed prelude's environment for an exact display-name match of the given kind and exits non-zero on no match. Verified directly on this tree: exit 0, count 1. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.powerSeriesTerm

Kind: exhaustive-enumeration
Status: checked

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

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.powerSeriesTerm, 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/power.rs, declare_power_series_term)",
  "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()). That output was piped to a scratchpad file and the exact row for this declaration's creal prelude row was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct theorem dependencies were cross-read from the same run's direct_theorems column (second-to-last field) and matched against the ledger's own registered facts (by parsing each candidate fact's formal.statement for its declared theorem/def name) to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}