A raw rational Cauchy bound at TWO DIFFERENT sample indices lifts to a CReal closeness bound

Identifier: F:creal-close-within-of-within-indexed
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

If the m-th rational sample of x and the n-th rational sample of y are within k of each other (the raw, Rat-level Within predicate, at potentially DIFFERENT indices m and n for the two sides), then x and y are close as CReal values: |x - y| <= ofRat(k + 1/(m+1) + 1/(n+1)). This generalizes the already-registered F:creal-close-within-of-within (which shares one index n for both x and y) to two INDEPENDENT sample indices, absorbing one self-approximation gap for whichever index each side actually used. This is the bridge CReal.riemannSum_sharedAccuracyClose_at's own kind of argument needs, where the two sides of a comparison are not sampled at the same accuracy-driven index.

Formal statement

theorem CReal.close_within_of_within_indexed : ((x0 : CReal) -> ((x1 : CReal) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : Rat) -> ((x5 : CReal.Within (Rat.sub (CReal.seq x0 x2) (CReal.seq x1 x3)) x4) -> CReal.le (CReal.abs (CReal.add x0 (CReal.neg x1))) (CReal.ofRat (Rat.add x4 (Rat.add (Rat.natDivSucc (AxNat.succ AxNat.zero) x2) (Rat.natDivSucc (AxNat.succ AxNat.zero) x3))))))))))

Dependencies

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

20 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.close_within_of_within_indexed

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.close_within_of_within_indexed through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c asserts the exact tab-anchored line. Verified with /usr/bin/grep directly (not the ugrep-backed interactive `grep` function) against a freshly built --release binary on this tree, returning count 1. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.close_within_of_within_indexed

Kind: exhaustive-enumeration
Status: checked

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

Checker command

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

Evidence notes

Re-measured on this tree: creal: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: creal trusted surface = 0'. That bounds every declaration in the creal environment, including CReal.close_within_of_within_indexed, since a declaration cannot depend on a trusted declaration the environment does not contain. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/integral.rs, declare_close_within_of_within_indexed)",
  "source": "canonical type read via kernel_declaration_projection's own UNFILTERED emit mode (cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection, no --require-declaration flag), which prints, per constructed prelude, one TSV row per declaration whose last field is kernel.render_lean(declaration.ty()). That output was piped to a scratchpad file and the exact row for this declaration's creal prelude row was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct theorem dependencies were cross-read from the same run's direct_theorems column (second-to-last field) and matched against the ledger's own registered facts (by parsing each candidate fact's formal.statement for its declared theorem/def name) to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}