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Artifact · proved

[generated] kernel theorem CReal.riemannSum_sharedAccuracyClose (creal prelude, axiom-free, prose not curated)

This page records one proposition and the evidence attached to it. Text from the source ledger is shown as record data.

Identifier
F:creal-riemannsum-sharedaccuracyclose
Proof route
kernel-lean
External status
Not recorded
Axiom footprint
Empty

Recorded description

MECHANICALLY GENERATED, UNREVIEWED PROSE -- this sentence deliberately makes NO mathematical characterisation of the theorem. What is asserted, and all that is asserted, is this: the kernel declaration `CReal.riemannSum_sharedAccuracyClose` is a `Declaration::Theorem` admitted into the environment by `build_creal_prelude` through the trusted `Kernel::add_declaration` gate, which re-derives its type from its proof term; its type is recorded verbatim in `formal.statement`; and its axiom footprint, as computed by `Kernel::axiom_footprint`, is empty. The authoritative content of this fact is `formal.statement`. A human-readable characterisation of what CReal.riemannSum_sharedAccuracyClose SAYS has not been supplied, because a generator cannot supply one honestly -- see `notes`.

Formal statement 
theorem CReal.riemannSum_sharedAccuracyClose : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : AxNat) -> ((x4 : AxNat) -> ((x5 : AxNat) -> ((x6 : CReal.le x1 x2) -> ((x7 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x8 : AxNat) -> ((x9 : AxNat) -> ((x10 : AxNat) -> ((x11 : AxNat) -> CReal.Within (Rat.sub (CReal.seq (CReal.riemannSum x0 x1 x2 (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4)) x8) (CReal.seq (CReal.riemannSum x0 x1 x2 (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5)) x9)) (Rat.add (Rat.add (Rat.add (Rat.add (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.add (AxNat.mul (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4)) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5)) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4))) (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x10)))) ((fun (x12 : AxNat) => Rat.add (CReal.seq (CReal.mul (CReal.ofNat (AxNat.succ (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4))) (CReal.mul (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x3)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4)))))) x12) (Rat.natDivSucc (AxNat.succ (AxNat.succ AxNat.zero)) x12)) x10)) (Rat.add (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x10))) (Rat.natDivSucc (AxNat.succ AxNat.zero) x8))) (Rat.add (Rat.add (Rat.add (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.add (AxNat.mul (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4)) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5)) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x4))) (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x11)))) ((fun (x12 : AxNat) => Rat.add (CReal.seq (CReal.mul (CReal.ofNat (AxNat.succ (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5))) (CReal.mul (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x3)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.add (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x7 x3)) (CReal.bound (CReal.add x2 (CReal.neg x1)))) x5)))))) x12) (Rat.natDivSucc (AxNat.succ (AxNat.succ AxNat.zero)) x12)) x11)) (Rat.add (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x11))) (Rat.natDivSucc (AxNat.succ AxNat.zero) x9))))))))))))))))

Dependencies

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

Direct dependencies appear to the left. The current fact is in the center. Facts that depend directly on it appear to the right. Riemann sums over a fixed inter Closeness at an arbitrary share Addition on the naturals is ass Addition on the naturals is com Multiplication on the naturals A two-sided bound survives rati Negating a two-sided bound keep Negating a rational difference Current fact [generated] kernel theorem CRea [generated] kernel theorem CRea A sufficiently fine fixed-mesh
11 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.riemannSum_sharedAccuracyClose

Kind
kernel-term
Status
checked

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

Two independent failure modes, so the exit status depends on the finding rather than on the run completing: theorem_dependency_inventory exits non-zero when a NAMED filter matches nothing, and grep -c exits 1 printing 0 when the anchored line is absent. Anchored with [[:space:]], never \t -- in a scripted (GNU) grep \t is a literal t. grep -c rather than grep -q, which would SIGPIPE the producer under pipefail. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.riemannSum_sharedAccuracyClose

Kind
exhaustive-enumeration
Status
checked

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

Checker command 
cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free creal
Evidence notes

--require-axiom-free exits non-zero when the named prelude's trusted surface (Axiom + Opaque + Quotient) is not empty, and errors rather than silently passing for a prelude the run never built. A declaration cannot depend on a trusted declaration the environment does not contain, so an empty creal surface bounds every declaration in it, including CReal.riemannSum_sharedAccuracyClose. This is a whole-prelude bound, not a per-declaration measurement; the per-declaration figure is the footprint column of kernel_declaration_projection, measured 0 for this row.

Provenance

{
  "date": "2026-08-27",
  "curation": "generated-unreviewed",
  "generated_by": "scripts/gen-kernel-facts.py",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/)",
  "source": "Derived mechanically from the unfiltered emit of `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`, which prints one TSV row per declaration whose fields are (prelude, kind, display name, axiom-footprint size, direct type declarations, direct declarations, direct theorems, Kernel::render_lean(declaration.ty())). formal.statement is that last field verbatim; depends_on is the direct-theorem column intersected with this ledger's registered facts; axiom_footprint is the footprint-size column, cross-checked by the whole-prelude nat_axiom_inventory run recorded in the second evidence row. No field was hand-transcribed and no prose was authored."
}