CReal.ivt_step: one bisection step that halves the bracket without ever deciding an exact sign

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

Recorded description

CReal.ivt_step is the single bisection step underlying the constructive approximate IVT (creal/ivt.rs). Given a function F, a bracket [P, Q] with P <= Q, and a fixed positive slack eps, such that F P <= eps and -eps <= F Q -- the WEAK sign invariant, not the classical exact F P <= 0 <= F Q -- the step produces a new bracket [P', Q'] nested inside [P, Q], of exactly half the width, satisfying the identical weak invariant. It never decides the sign of F at the midpoint: the case split comes from cotransitivity (CReal.lt_cotrans) applied to the FIXED, always-strict pair -eps < eps, which holds independently of F, so no undecidable comparison of F's value against zero is ever made. This is machinery, not an independent statement of IVT: it is the one step CReal.ivt_iter repeats n times and CReal.ivt_approx composes against a continuity modulus to reach a stated accuracy; on its own it makes no claim about roots existing.

Formal statement

theorem CReal.ivt_step : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : CReal.lt CReal.zero x3) -> ((x5 : CReal.le x1 x2) -> ((x6 : CReal.le (x0 x1) x3) -> ((x7 : CReal.le (CReal.neg x3) (x0 x2)) -> Exists.{1} CReal (fun (x8 : CReal) => Exists.{1} CReal (fun (x9 : CReal) => And (CReal.le x1 x8) (And (CReal.le x8 x9) (And (CReal.le x9 x2) (And (CReal.le (x0 x8) x3) (And (CReal.le (CReal.neg x3) (x0 x9)) (CReal.Equiv (CReal.add x9 (CReal.neg x8)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero)))))))))))))))))))

Dependencies

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

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

Evidence

kernel-CReal.ivt_step

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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

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_step. 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."
}