Uniform continuity on an interval restricts to any sub-interval with the SAME modulus

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

Recorded description

If F is CReal.UniformlyContinuousOn [a,b], and a<=a'<=b'<=b, then F is CReal.UniformlyContinuousOn [a',b'] -- using the identical modulus witness, no new estimate. UniformlyContinuousOn's own spec only ever uses its range hypotheses (a<=x, x<=b, a<=y, y<=b) to state the closeness implication, and a<=a'<=x together with y<=b'<=b compose to exactly those via CReal.le_trans. Built directly against the witness constructor rather than through the recursor: the new spec is a lambda wrapping the ORIGINAL witness's own spec applied to the composed range proofs, so UniformlyContinuousOn's definition is never unfolded.

Formal statement

theorem CReal.uniformlyContinuousOn_restrict : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : CReal) -> ((x5 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x6 : CReal.le x1 x3) -> ((x7 : CReal.le x3 x4) -> ((x8 : CReal.le x4 x2) -> CReal.UniformlyContinuousOn x0 x3 x4)))))))))

Dependencies

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

2 direct dependencies 4 direct dependents

Evidence

kernel-CReal.uniformlyContinuousOn_restrict

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.uniformlyContinuousOn_restrict 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.uniformlyContinuousOn_restrict

Kind: exhaustive-enumeration
Status: checked

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

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.uniformlyContinuousOn_restrict, 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_continuity.rs, declare_uniformly_continuous_on_restrict)",
  "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."
}