CReal.ivt_iter: n-fold bisection, bracket width shrinking geometrically as (1/2)^n

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

Recorded description

CReal.ivt_iter iterates CReal.ivt_step n times by structural induction on a Nat, starting from an original bracket [P0, Q0] satisfying the weak sign invariant at slack eps. For every n it produces a bracket [P, Q] nested inside [P0, Q0], still satisfying the same weak invariant, with width EXACTLY (Q0 - P0) * (1/2)^n -- plain geometric bisection, not merely a worst-case bound. This is still pure machinery: it is the geometric-shrinking half of the approximate IVT construction, and it does not by itself connect the shrinking bracket to a stated accuracy on abs (F x) -- that combination step, which needs the continuity modulus and an Archimedean bound on the initial width, is CReal.ivt_approx, built on top of this fact rather than inside it.

Formal statement

theorem CReal.ivt_iter : ((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)) -> ((x8 : AxNat) -> Exists.{1} CReal (fun (x9 : CReal) => Exists.{1} CReal (fun (x10 : CReal) => And (CReal.le x1 x9) (And (CReal.le x9 x10) (And (CReal.le x10 x2) (And (CReal.le (x0 x9) x3) (And (CReal.le (CReal.neg x3) (x0 x10)) (CReal.Equiv (CReal.add x10 (CReal.neg x9)) (CReal.mul (CReal.add x2 (CReal.neg x1)) (CReal.pow (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero))) x8))))))))))))))))))

Dependencies

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

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

Evidence

kernel-CReal.ivt_iter

Kind: kernel-term
Status: checked

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

Kind: exhaustive-enumeration
Status: checked

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

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_iter. 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."
}