The base-1/2 geometric partial sums form a Cauchy sequence

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

Recorded description

Cauchy (sumRange (fun n => pow half n)) -- the geometric series' convergence, Spivak Ch.22-23's standard geometric-decay estimate at the concrete ratio 1/2. Eliminates the Nat.le_total disjunction (never branching on the undecidable CReal.le itself) and calls the ordered-pair bound geomCauchyOrderedHalf at whichever of (m,n)/(n,m) satisfies its own a<=b side condition, with the single fixed witness K := 7. This is CLAUDE.md's named CReal.geomCauchy, and it is distinct from the not-yet-registered general-base CReal.geomCauchyOfLt (on an unmerged sibling branch, out of scope for this batch).

Formal statement

theorem CReal.geomCauchy : CReal.Cauchy (CReal.sumRange (fun (x0 : AxNat) => CReal.pow (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero))) x0))

Dependencies

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5 direct dependencies 1 direct dependents

Evidence

kernel-CReal.geomCauchy

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.geomCauchy 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.geomCauchy. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.geomCauchy

Kind: exhaustive-enumeration
Status: checked

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

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/exponential.rs, declare_geom_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."
}