The cross-width crossing bound, with both domain hypotheses discharged by construction

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

Recorded description

CReal.crossingClose restated with samplePt replaced by clampedPt := min samplePt b: given F uniformly continuous on [a,b], 0 < delta, a <= c <= b, and the two crossing-slack bounds already within the continuity modulus at accuracy e, |F(c) - F(clampedPt)| <= 1/e. BOTH domain-membership hypotheses CReal.crossingClose needs (a <= samplePt, samplePt <= b) are GONE from this statement -- discharged BY CONSTRUCTION rather than assumed, no comparison decided anywhere: clampedPt <= b is min_le_right, unconditional; a <= clampedPt is le_min applied to CReal.crossingSampleGeA (this batch, a <= samplePt) and a <= b (le_trans on the two hypotheses this theorem already carries). samplePt <= b is NOT itself provable in general (per creal/integral.rs's 2026-08-27 module doc entries), but the theorem never needed samplePt un-clamped: clamping into range costs nothing (min is fully constructive) and the closeness bound survives the substitution via the same le_min move applied to c - bound_embed (giving the upper half) and via plain transitivity for the lower half (clampedPt <= samplePt is min_le_left, transferring crossingSampleLower's existing bound with no case split).

Formal statement

theorem CReal.crossingCloseClamped : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : Rat) -> ((x5 : AxNat) -> ((x6 : CReal.UniformlyContinuousOn x0 x1 x2) -> ((x7 : Rat.lt Rat.zero x4) -> ((x8 : CReal.le x1 x3) -> ((x9 : CReal.le x3 x2) -> ((x10 : CReal.le (CReal.add (CReal.ofRat x4) (CReal.mul (CReal.ofRat x4) (CReal.ofRat (Rat.natDivSucc (AxNat.succ (AxNat.succ AxNat.zero)) (AxNat.mul (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero)))))) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x6 x5)))) -> ((x11 : CReal.le (CReal.neg (CReal.mul (CReal.ofRat x4) (CReal.ofRat (Rat.neg (Rat.natDivSucc (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.mul (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero))))))) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (CReal.UniformlyContinuousOn.modulus x0 x1 x2 x6 x5)))) -> CReal.le (CReal.abs (CReal.add (x0 x3) (CReal.neg (x0 (CReal.min (CReal.add x1 (CReal.mul (CReal.ofNat (CReal.bucketIndex (CReal.mul (CReal.ofRat (Rat.inv x4)) (CReal.add x3 (CReal.neg x1))) AxNat.zero)) (CReal.ofRat x4))) x2))))) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x5))))))))))))))

Dependencies

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

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

Evidence

kernel-CReal.crossingCloseClamped

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.crossingCloseClamped 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. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.crossingCloseClamped

Kind: exhaustive-enumeration
Status: checked

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

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.crossingCloseClamped, 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_close_clamped)",
  "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 theorem dependencies were cross-read from the same run's direct_theorems column (field 6) and matched against the ledger's own registered kernel_theorem/formal.statement names to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}