Absolute convergence implies convergence, Cauchy form

Identifier: F:creal-sumrange-cauchy-of-abs-cauchy
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

forall f, Cauchy (sumRange (fun k => abs (f k))) -> Cauchy (sumRange f) -- absolute convergence implies convergence for a SIGNED series, in Cauchy form. A direct corollary of sumRange_cauchy_of_dominated at g := fun k => abs (f k): the pointwise hypothesis forall k, le (abs (f k)) (g k) is le_refl (abs (f k)) after one beta reduction on g k, so no new real-analysis content is needed. This is what makes the comparison test usable on a series whose terms need not be nonnegative.

Formal statement

theorem CReal.sumRange_cauchy_of_abs_cauchy : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : CReal.Cauchy (CReal.sumRange (fun (x1 : AxNat) => CReal.abs (x0 x1)))) -> CReal.Cauchy (CReal.sumRange x0)))

Dependencies

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

2 direct dependencies 1 direct dependents

Evidence

kernel-CReal.sumRange_cauchy_of_abs_cauchy

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.sumRange_cauchy_of_abs_cauchy through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command currently prints exactly one matching line for CReal.sumRange_cauchy_of_abs_cauchy. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.sumRange_cauchy_of_abs_cauchy

Kind: exhaustive-enumeration
Status: checked

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

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. --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/series.rs, declare_sum_range_cauchy_of_abs_cauchy)",
  "source": "theorem name and canonical type read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint(), theorem_dependencies()) -- no in-tree example prints this theorem's own type directly. Direct dependency edges cross-checked against theorem_dependency_inventory's own output on the same tree. The probe was built and run in the session scratchpad and deleted after use; crates/ source was not touched to produce this batch."
}