A power-series term is dominated by M*r^j whenever the coefficients are bounded by M and 0<=x<=r

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

Recorded description

For a coefficient sequence c and bound M with |c j| <= M for every j, and points x, r with 0 <= x <= r: for every j, |powerSeriesTerm c j x| <= M * r^j. Proved by combining base-monotonicity of CReal.pow (0<=x<=r gives x^j<=r^j since 0<=x) with a two-sided abs_le argument and CReal.abs_mul_le_of_bounds -- no induction on j is needed, everything is a direct combination for an arbitrary j. This is the coefficient-boundedness domination package the Weierstrass M-test needs to prove uniform convergence of a power series (see CReal.powerSeriesUniformConvergesOn, registered in the same batch).

Formal statement

theorem CReal.powerSeriesTerm_abs_le : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : CReal) -> ((x2 : ((x2 : AxNat) -> CReal.le (CReal.abs (x0 x2)) x1)) -> ((x3 : CReal) -> ((x4 : CReal) -> ((x5 : CReal.le CReal.zero x3) -> ((x6 : CReal.le x3 x4) -> ((x7 : AxNat) -> CReal.le (CReal.abs (CReal.powerSeriesTerm x0 x7 x3)) (CReal.mul x1 (CReal.pow x4 x7))))))))))

Dependencies

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

13 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.powerSeriesTerm_abs_le

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.powerSeriesTerm_abs_le through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c asserts the exact tab-anchored line. Verified with /usr/bin/grep directly (not the ugrep-backed interactive `grep` function) against a freshly built --release binary on this tree, returning count 1. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.powerSeriesTerm_abs_le

Kind: exhaustive-enumeration
Status: checked

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

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_abs_le, 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_abs_le)",
  "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."
}