The alternating-series lower bracket, closed against the actual limit

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

Recorded description

For a nonnegative, non-increasing sequence a, if the alternating partial sums sumRange t converge to a limit L, then for every m, the even-count partial sum E(m) is <= L. This is CReal.alternatingBracket's per-index bracket CLOSED against the true limit rather than against a further partial sum: for an arbitrary shift n, Nat.even_or_odd's computed parity split (this batch) case-splits n = k+k (landing on the E side of alternatingBracket's paired conclusion) or n = succ(k+k) (landing on the O side), transported across the resulting index identity; converges_lower_bound_shift at shift s := m+m then closes E(m) <= L.

Formal statement

theorem CReal.alternatingLowerBound : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : ((x1 : AxNat) -> CReal.le CReal.zero (x0 x1))) -> ((x2 : ((x2 : AxNat) -> CReal.le (x0 (AxNat.succ x2)) (x0 x2))) -> ((x3 : CReal) -> ((x4 : CReal.Converges (CReal.sumRange (fun (x4 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x4) (x0 x4))) x3) -> ((x5 : AxNat) -> CReal.le (CReal.sumRange (fun (x6 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x6) (x0 x6)) (AxNat.add x5 x5)) x3))))))

Dependencies

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

9 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.alternatingLowerBound

Kind: kernel-term
Status: checked

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

Evidence notes

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

footprint-CReal.alternatingLowerBound

Kind: exhaustive-enumeration
Status: checked

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

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.alternatingLowerBound, 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/alternating.rs, declare_alternating_lower_bound)",
  "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()) -- the same Kernel::render_lean canonical form nat_theorem_inventory prints, just not filtered to Declaration::Theorem. That output was piped to a scratchpad file and the exact row for this declaration's own prelude label 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 (field 6) and matched against the ledger's own registered kernel_theorem/formal.statement names to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}