Uniform convergence is closed under termwise addition

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

Recorded description

If F converges uniformly to G on [a,b] and H converges uniformly to K on [a,b], then the termwise sum (fun n x => F n x + H n x) converges uniformly on [a,b] to the pointwise sum (fun x => G x + K x). The proof rearranges a four-term sum (two convergence gaps) via add4_comm so the two gap terms land together, fuses the two ofRat-embedded rational gaps into one via CReal.ofRat_add, rebuilds a single close_within bound via CReal.abs_le_of_two_sided, and widens that bound by an exact Rat equality (Rat.natDivSucc_add) to the stated combined rate.

Formal statement

theorem CReal.uniform_converges_add : ((x0 : ((x0 : AxNat) -> ((x1 : CReal) -> CReal))) -> ((x1 : ((x1 : AxNat) -> ((x2 : CReal) -> CReal))) -> ((x2 : ((x2 : CReal) -> CReal)) -> ((x3 : ((x3 : CReal) -> CReal)) -> ((x4 : CReal) -> ((x5 : CReal) -> ((x6 : CReal.UniformConvergesOn x0 x2 x4 x5) -> ((x7 : CReal.UniformConvergesOn x1 x3 x4 x5) -> CReal.UniformConvergesOn (fun (x8 : AxNat) => fun (x9 : CReal) => CReal.add (x0 x8 x9) (x1 x8 x9)) (fun (x8 : CReal) => CReal.add (x2 x8) (x3 x8)) x4 x5))))))))

Dependencies

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

Evidence

kernel-CReal.uniform_converges_add

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.uniform_converges_add 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.uniform_converges_add

Kind: exhaustive-enumeration
Status: checked

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

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_add, 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_add)",
  "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."
}