crossingUpper: a slack upper bound at the sampled crossing bucket

Identifier: F:creal-crossingupper
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a<=... (any a,b), eps>0: b <= a + eps*(crossingIndex(a,b,eps)+1)/2, roughly. This is a deliberately SLACK bracket, not the tight a+i0*eps <= c <= a+(i0+1)*eps: the tight bracket is not constructible because deciding which side of an exact crossing c falls on IS the undecidable comparison (creal/ivt.rs refutes the analogous exact-root construction with two kernel-computed counterexamples).

Formal statement

theorem CReal.crossingUpper : ((x0 : CReal) -> ((x1 : CReal) -> ((x2 : Rat) -> ((x3 : Rat.lt Rat.zero x2) -> CReal.le x1 (CReal.add x0 (CReal.mul (CReal.ofRat x2) (CReal.ofRat (Rat.add (Rat.natDivSucc (AxNat.succ (CReal.bucketIndex (CReal.mul (CReal.ofRat (Rat.inv x2)) (CReal.add x1 (CReal.neg x0))) AxNat.zero)) AxNat.zero) (Rat.natDivSucc (AxNat.succ (AxNat.succ AxNat.zero)) (AxNat.mul (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero)))))))))))

Dependencies

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

Evidence

kernel-CReal.crossingUpper

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.crossingUpper through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted theorem cannot read as a re-derived one), and grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command prints exactly one matching line for CReal.crossingUpper. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.crossingUpper

Kind: exhaustive-enumeration
Status: checked

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

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.crossingUpper, 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/crossing.rs, declare_crossing_upper)",
  "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()) -- the same Kernel::render_lean canonical form nat_theorem_inventory prints, just not filtered to Declaration::Theorem. That output was piped to a scratchpad file and the exact row for this declaration's own prelude label was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct dependency edges (where applicable) were cross-read from the same run's direct_theorems column and independently confirmed against theorem_dependency_inventory's own output on this tree. No new probe binary was written for this batch, since the in-tree tool already emits the canonical type; crates/ source was not touched to produce this batch."
}