The alternating-series bracket, dual form: every partial sum sits at or below the odd bracket

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

Recorded description

For a nonnegative, non-increasing sequence a (∀k, 0 <= a k; ∀k, a(k+1) <= a k), and every m and i: the even-count partial sum E(m+i) and the odd-count partial sum O(m+i) both lie at or BELOW O(m). This is the dual of CReal.alternatingBracket (which brackets from BELOW, at E(m)): same induction shape on i, using the odd bracket's own one-step decrease (private o_step_le, mirroring e_step_le's one-step increase) instead. Needed because this development has no converges_upper_bound_shift primitive: closing an upper bound L <= O(m) on the true limit routes through converges_lower_bound_shift applied to the NEGATED sequence, which needs this dual bracket's per-index upper bound, not alternatingBracket's lower one.

Formal statement

theorem CReal.alternatingBracketUpper : ((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 : AxNat) -> ((x4 : AxNat) -> And (CReal.le (CReal.sumRange (fun (x5 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x5) (x0 x5)) (AxNat.add (AxNat.add x3 x4) (AxNat.add x3 x4))) (CReal.sumRange (fun (x5 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x5) (x0 x5)) (AxNat.succ (AxNat.add x3 x3)))) (CReal.le (CReal.sumRange (fun (x5 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x5) (x0 x5)) (AxNat.succ (AxNat.add (AxNat.add x3 x4) (AxNat.add x3 x4)))) (CReal.sumRange (fun (x5 : AxNat) => CReal.mul (CReal.pow (CReal.neg CReal.one) x5) (x0 x5)) (AxNat.succ (AxNat.add x3 x3)))))))))

Dependencies

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

Evidence

kernel-CReal.alternatingBracketUpper

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.alternatingBracketUpper 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.alternatingBracketUpper

Kind: exhaustive-enumeration
Status: checked

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

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.alternatingBracketUpper, 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_bracket_upper)",
  "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."
}