A Cauchy witness transports across a pointwise-Equiv sequence

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

Recorded description

forall G F, (forall n, Equiv (G n) (F n)) -> Cauchy G -> Cauchy F -- the general lemma this lane built to scale a Cauchy witness across a pointwise Equiv, e.g. CReal.mul_sumRange's index-shifted mul c (sumRange f n) ~ sumRange (scaled f) n bridge. Named explicitly in this batch's brief as part of the exponential domination bridge.

Formal statement

theorem CReal.cauchyOfPointwiseEquiv : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : ((x1 : AxNat) -> CReal)) -> ((x2 : ((x2 : AxNat) -> CReal.Equiv (x0 x2) (x1 x2))) -> ((x3 : CReal.Cauchy x0) -> CReal.Cauchy x1))))

Dependencies

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7 direct dependencies 2 direct dependents

Evidence

kernel-CReal.cauchyOfPointwiseEquiv

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.cauchyOfPointwiseEquiv through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command currently prints exactly one matching line for CReal.cauchyOfPointwiseEquiv. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.cauchyOfPointwiseEquiv

Kind: exhaustive-enumeration
Status: checked

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

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. --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/exponential.rs, declare_cauchy_of_pointwise_equiv)",
  "source": "theorem name and canonical type read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint(), theorem_dependencies()) -- no in-tree example prints this theorem's own type directly. Direct dependency edges cross-checked against theorem_dependency_inventory's own output on the same tree. The probe was built and run in the session scratchpad and deleted after use; crates/ source was not touched to produce this batch."
}