CReal.ivt_bisect_approx: ivt_approx's accuracy bound restated at a named, computable point

Identifier: F:creal-ivt-bisect-approx
Proof route: kernel-lean
External status: Not recorded
Axiom footprint: Empty

Recorded description

CReal.ivt_bisect_approx states the identical accuracy bound CReal.ivt_approx proves -- abs (F x) <= 1/(e+1) for x in [a, b], at every accuracy e -- but for the CONCRETE point CReal.ivt_bisect_hi computes at the depth ivt_approx's own schedule chooses, instead of an existentially quantified x. The mathematical content is exactly ivt_approx's: no new accuracy, no exact root, and the same restriction to a fixed target of zero and a fixed orientation. What is different, and the only reason this fact exists separately, is that the witness is now NAMED data rather than an opaque existential -- which is what lets `fun e => ivt_bisect_hi F a b ... e` become an ordinary Nat -> CReal sequence, the necessary ingredient for CReal.ivt_bisect_cauchy and ultimately CReal.ivt_exact_root.

Formal statement

theorem CReal.ivt_bisect_approx : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x4 : CReal.le x1 x2) -> ((x5 : CReal.le (x0 x1) CReal.zero) -> ((x6 : CReal.le CReal.zero (x0 x2)) -> ((x7 : AxNat) -> And (CReal.le x1 (CReal.ivt_bisect_hi x0 x1 x2 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)) (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x3 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)))) (CReal.bound (CReal.add x2 (CReal.neg x1)))))) (And (CReal.le (CReal.ivt_bisect_hi x0 x1 x2 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)) (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x3 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)))) (CReal.bound (CReal.add x2 (CReal.neg x1))))) x2) (CReal.le (CReal.abs (x0 (CReal.ivt_bisect_hi x0 x1 x2 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)) (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x2 (CReal.neg x1)))) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x3 (AxNat.succ (AxNat.mul (AxNat.succ (AxNat.succ AxNat.zero)) x7)))) (CReal.bound (CReal.add x2 (CReal.neg x1))))))) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x7))))))))))))

Dependencies

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

38 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.ivt_bisect_approx

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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

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.ivt_bisect_approx. 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": "curated",
  "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."
}