CReal.ivt_approx: the constructive approximate Intermediate Value Theorem -- roots to any requested accuracy

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

Recorded description

CReal.ivt_approx is the general constructive form of the Intermediate Value Theorem, ADR-0603 row 1. Let F be an arbitrary uniformly continuous real function on [a, b], with F a <= 0 <= F b. Then for EVERY accuracy target n, there exists x in [a, b] with abs (F x) <= 1/(n+1) -- an approximate root to any requested accuracy, all from one proof, since F, a, b are arbitrary and n is universally quantified. This is not a special case of a stronger unstated theorem: CReal.ivt_exact_root_decides_sign shows that an EXACT root (Equiv (F x) zero, no epsilon) cannot be reached from these hypotheses alone, because doing so for an arbitrary F would decide the sign of an arbitrary real. What this fact does NOT state: the target value is fixed at zero (Mathlib's Icc (f a) (f b) subset f '' Icc a b covers every intermediate value, not just a root -- reachable here via CReal.uniformly_continuous_sub/const but not registered as its own fact); the orientation is fixed at F a <= 0 <= F b (the reverse is reachable via CReal.uniformly_continuous_neg, also unregistered); and the hypothesis is UNIFORM continuity with an explicit modulus, not Mathlib's pointwise ContinuousOn -- this kernel has no pointwise-continuity predicate at all, so that gap cannot even be stated here.

Formal statement

theorem CReal.ivt_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) -> Exists.{1} CReal (fun (x8 : CReal) => And (CReal.le x1 x8) (And (CReal.le x8 x2) (CReal.le (CReal.abs (x0 x8)) (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 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.ivt_approx

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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