CReal.supOn exists as a total, axiom-free CReal function -- the supremum value of a uniformly continuous function on [a, b]

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

Recorded description

There is a total function CReal.supOn taking a function F, endpoints a <= b, and a witness that F is uniformly continuous on [a,b], and returning a CReal -- the limit of the sequence of finite mesh maxima (CReal.supSeq), built as a CReal.mk with a regularity proof. Admitted by the trusted kernel gate with an empty axiom footprint. This is the supremum VALUE only: CReal.supOn_ub and CReal.supOn_approx_lub (registered separately) are what CHARACTERIZE it as a supremum, and CReal.evt_attained_max_decides_sign proves the exact form (an attaining maximiser) is unavailable from these hypotheses alone.

Formal statement

def CReal.supOn : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal.le x1 x2) -> ((x4 : CReal.UniformlyContinuousOn x0 x1 x2) -> CReal)))))

Dependencies

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0 direct dependencies 3 direct dependents

Evidence

kernel-CReal.supOn

Kind: kernel-term
Status: checked

Supports: CReal.supOn 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 kernel_declaration_projection -- --require-declaration CReal.supOn --require-kind definition 2>/dev/null | grep -cE '^found[[:space:]]creal[[:space:]]definition[[:space:]]CReal\.supOn[[:space:]]'

Evidence notes

CReal.supOn is a Declaration::Definition, not a Theorem, and every in-tree theorem inventory (theorem_dependency_inventory, prelude_theorem_inventory, nat_theorem_inventory) deliberately excludes Definitions. kernel_declaration_projection's '--require-declaration <name> --require-kind <kind>' searches every constructed prelude's environment for an EXACT display-name match of the given kind and exits non-zero on no match, so a deleted or renamed CReal.supOn cannot read as present. This is the same route ADR-0692 used to independently re-derive that CReal.supOn exists (as opposed to CReal.supOn_upper_bound, a name that never existed) -- see that ADR and ADR-0895 for the history of why this fact was absent from the ledger until now despite the declaration itself having landed earlier. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.supOn

Kind: exhaustive-enumeration
Status: checked

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

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.supOn, 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. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-30",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/supremum.rs, declare_sup_on)",
  "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) on this lane's own worktree build, filtered to the creal-prelude row for this declaration."
}