`BookProof.BrstUnboundedLeakage.defect_orbit_le` {x : H} (hx : x ∈ V) (s : ℝ) : ‖T.op ⟨flow (truncGen T V hV) s x, flow_truncGen_mem_domain T V hV s hx⟩ - truncGen T V hV (flow (tr

Proved

BookProof.BrstUnboundedLeakage.defect_orbit_le

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstUnboundedLeakage.

BookProof.BrstUnboundedLeakage.defect_orbit_le {x : H} (hx : x ∈ V) (s : ℝ) : ‖T.op ⟨flow (truncGen T V hV) s x, flow_truncGen_mem_domain T V hV s hx⟩ - truncGen T V hV (flow (truncGen T V hV) s x)‖ ≤ ‖truncDefect T V hV‖ * ‖x‖

Formalization note: Lean 4 identifier BookProof.BrstUnboundedLeakage.defect_orbit_le.

Preamble

-- Generated from ChapterBrstUnboundedLeakage.lean — theorem BookProof.BrstUnboundedLeakage.defect_orbit_le
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterBrstUnboundedLeakage
import Definitions.Def_ChapterBrstTruncationLeakage
import Definitions.Def_ChapterStoneResolvent
import Theorems.Thm_BookProof_BrstUnboundedLeakage_flow_truncGen_mem_domain
open BookProof.BrstLeakage
open BookProof.BrstUnboundedLeakage


open NormedSpace Filter Topology
open scoped InnerProductSpace


open BookProof.BrstLeakage BookProof.ChapterStoneResolvent

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (T : UnboundedSelfAdjoint H)
variable (V : Submodule ℂ H) [FiniteDimensional ℂ V] (hV : V ≤ T.domain)

Formal statement

theorem BookProof.BrstUnboundedLeakage.defect_orbit_le {x : H} (hx : x ∈ V) (s : ℝ) :
    ‖T.op ⟨flow (truncGen T V hV) s x, flow_truncGen_mem_domain T V hV s hx⟩
        - truncGen T V hV (flow (truncGen T V hV) s x)‖ ≤ ‖truncDefect T V hV‖ * ‖x‖ := by sorry

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