The exact mesh-point split identity discharged from a uniform-continuity witness alone

Identifier: F:creal-riemannsum-split-exact-of-uc
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The same exact split identity as CReal.riemannSum_split_exact -- riemannSum F a b (succ m_ac + m_cb) Equiv riemannSum F a c m_ac + riemannSum F c b m_cb at the mesh-point split c -- but with the hand-supplied Equiv-congruence hypothesis on F replaced by a CReal.UniformlyContinuousOn F a b witness together with a Equiv a<=b proof: uniform continuity on [a,b] plus both endpoints of the resample being inside [a,b] (via CReal.riemannSum_sample_in_bounds) is enough to derive the congruence CReal.congrOfUniformlyContinuous needs at exactly the sample points this split's own proof uses. This removes the free-standing global-congruence hypothesis riemannSum_split_exact required, replacing it with the same uniform-continuity witness CReal.integral itself is built from.

Formal statement

theorem CReal.riemannSum_split_exact_of_uc : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : AxNat) -> ((x4 : AxNat) -> ((x5 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x6 : CReal.le x1 x2) -> CReal.Equiv (CReal.riemannSum x0 x1 x2 (AxNat.add (AxNat.succ x3) x4)) (CReal.add (CReal.riemannSum x0 x1 (CReal.add x1 (CReal.mul (CReal.ofNat (AxNat.succ x3)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.succ x3) x4)))))) x3) (CReal.riemannSum x0 (CReal.add x1 (CReal.mul (CReal.ofNat (AxNat.succ x3)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.succ x3) x4)))))) x2 x4)))))))))

Dependencies

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31 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.riemannSum_split_exact_of_uc

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.riemannSum_split_exact_of_uc 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.riemannSum_split_exact_of_uc

Kind: exhaustive-enumeration
Status: checked

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

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.riemannSum_split_exact_of_uc, 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_riemann_sum_split_exact_of_uc)",
  "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."
}