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`BookProof.ChapterCayleyTransform.add_cayley_shift` (x : T.domain) : T.shift (-1) x + cayley T (T.shift (-1) x) = (2 : ℂ) • T.op x

Proved
BookProof.ChapterCayleyTransform.add_cayley_shift

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCayleyTransform.

BookProof.ChapterCayleyTransform.add_cayley_shift (x : T.domain) : T.shift (-1) x + cayley T (T.shift (-1) x) = (2 : ℂ) • T.op x

Formalization note: Lean 4 identifier BookProof.ChapterCayleyTransform.add_cayley_shift.

Preamble
-- Generated from ChapterCayleyTransform.lean — theorem BookProof.ChapterCayleyTransform.add_cayley_shift
import Definitions.Def_ChapterUnitaryTransport
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterCayleyTransform
import Definitions.Def_ChapterStoneResolvent
open BookProof.ChapterCayleyTransform


open scoped InnerProductSpace
open Filter Topology


open BookProof.ChapterUnitaryTransport BookProof.ChapterStoneResolvent

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : UnboundedSelfAdjoint H)
Formal statement
theorem BookProof.ChapterCayleyTransform.add_cayley_shift (x : T.domain) :
    T.shift (-1) x + cayley T (T.shift (-1) x) = (2 : ℂ) • T.op x := by sorry

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