A uniform limit of uniformly continuous functions is uniformly continuous

Identifier: F:creal-uniform-limit-uniformly-continuous
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For F : Nat -> CReal -> CReal, G : CReal -> CReal, a b : CReal: if F converges uniformly to G on [a,b] (CReal.UniformConvergesOn F G a b) and every F n is uniformly continuous on [a,b], then G is uniformly continuous on [a,b]. Ch.24's headline theorem. Proving it requires CONSTRUCTING a UniformlyContinuousOn witness for G, whose modulus field is built from literal Nat data derived from the convergence rate -- exactly the construction UniformConvergesOn's Type-valued (not Prop-valued) design exists to make possible, since Exists.rec cannot eliminate a Prop witness into Type-valued data.

Formal statement

theorem CReal.uniform_limit_uniformly_continuous : ((x0 : ((x0 : AxNat) -> ((x1 : CReal) -> CReal))) -> ((x1 : ((x1 : CReal) -> CReal)) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : CReal.UniformConvergesOn x0 x1 x2 x3) -> ((x5 : ((x5 : AxNat) -> CReal.UniformlyContinuousOn (x0 x5) x2 x3)) -> CReal.UniformlyContinuousOn x1 x2 x3))))))

Dependencies

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

Evidence

kernel-CReal.uniform_limit_uniformly_continuous

Kind: kernel-term
Status: checked

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

Evidence notes

The producing prelude admits CReal.uniform_limit_uniformly_continuous 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_limit_uniformly_continuous

Kind: exhaustive-enumeration
Status: checked

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

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_limit_uniformly_continuous, 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_limit_uniformly_continuous)",
  "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."
}