Uniform continuity on [a,b] gives Equiv-congruence, but ONLY inside [a,b]

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

Recorded description

If F is CReal.UniformlyContinuousOn [a,b], and x, y both lie in [a,b] (a<=x<=b, a<=y<=b) and x Equiv y, then F x Equiv F y. This is the DOMAIN-RESTRICTED half of the Equiv-congruence hypothesis CReal.riemannSum_split_exact needs. IT CANNOT BE STRENGTHENED TO THE GLOBAL STATEMENT (forall x y, Equiv x y -> Equiv (F x) (F y)) riemannSum_split_exact's own general form assumes, for a structural reason rather than a missing lemma: CReal.UniformlyContinuousOn's spec says nothing about F's behaviour outside [a,b] at all, so nothing constrains F's values there, and a genuinely global congruence claim is simply FALSE for an arbitrary function that is merely uniformly continuous on [a,b]. Using this lemma to discharge riemannSum_split_exact's own hcong hypothesis at a concrete uniformly-continuous F therefore ALSO needs each sample point the split's proof applies hcong to be shown to lie inside [a,b] first (via CReal.riemannSum_sample_in_bounds-style reasoning) -- which is exactly what CReal.riemannSum_split_exact_of_uc (registered separately in this batch) assembles.

Formal statement

theorem CReal.congrOfUniformlyContinuous : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x4 : CReal) -> ((x5 : CReal) -> ((x6 : CReal.le x1 x4) -> ((x7 : CReal.le x4 x2) -> ((x8 : CReal.le x1 x5) -> ((x9 : CReal.le x5 x2) -> ((x10 : CReal.Equiv x4 x5) -> CReal.Equiv (x0 x4) (x0 x5))))))))))))

Dependencies

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

11 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.congrOfUniformlyContinuous

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.congrOfUniformlyContinuous 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. Verified with /usr/bin/grep directly (not the ugrep-backed interactive `grep` function) against a freshly built --release binary on this tree, returning count 1. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.congrOfUniformlyContinuous

Kind: exhaustive-enumeration
Status: checked

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

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.congrOfUniformlyContinuous, 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/integral.rs, declare_congr_of_uniformly_continuous)",
  "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()). That output was piped to a scratchpad file and the exact row for this declaration's creal prelude row 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 (second-to-last field) and matched against the ledger's own registered facts (by parsing each candidate fact's formal.statement for its declared theorem/def name) to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}