x*(1/2)^n converges uniformly to the zero function on [0,1]

Identifier: F:creal-uniform-converges-geom-half
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

CReal.UniformConvergesOn (fun n x => mul x (pow (ofRat 1/2) n)) (fun _ => zero) zero one -- the sequence F n x := x*(1/2)^n converges uniformly to the zero function on [0,1]. A fully CONCRETE instance (a := zero, b := one are baked into the statement, unlike uniform_converges_id's arbitrary a, b), and a SECOND entry point admitting CReal.UniformConvergesOn's constructor, dispatched separately from declare_uniform_converges_id because the two witnesses are genuinely different constructions, not the same proof instantiated at different arguments.

Formal statement

theorem CReal.uniform_converges_geom_half : CReal.UniformConvergesOn (fun (x0 : AxNat) => fun (x1 : CReal) => CReal.mul x1 (CReal.pow (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero))) x0)) (fun (x0 : CReal) => CReal.zero) CReal.zero CReal.one

Dependencies

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

20 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.uniform_converges_geom_half

Kind: kernel-term
Status: checked

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

Evidence notes

The producing prelude admits CReal.uniform_converges_geom_half through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted/renamed theorem cannot read as a re-derived one), and grep -c (never -q) both avoids a SIGPIPE-under-pipefail false negative and independently asserts the exact tab-anchored line is present. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.uniform_converges_geom_half

Kind: exhaustive-enumeration
Status: checked

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

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.uniform_converges_geom_half, 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/uniform_convergence.rs, declare_uniform_converges_geom_half)",
  "source": "theorem name, canonical type and direct dependency edges read via theorem_dependency_inventory's own per-name output (cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- <Name>), which prints '<Name>\\t<comma-separated direct deps>'; the canonical type itself was cross-extracted from kernel_declaration_projection's unfiltered TSV dump the same way as the Definition facts in this batch, and confirmed byte-for-byte identical between the two tools' output for every name in this batch. crates/ source was not touched to produce this batch."
}