`BookProof.BrstLeakage.leakage_le` {H B Om : E →L[ℂ] E} (hH : IsSelfAdjoint H) (hcomm : Commute H Om) (t : ℝ) (ht : 0 ≤ t) (x : E) (K : ℝ) (hK : ∀ s ∈ Set.Icc (0 : ℝ) t, ‖(H - B) (

Proved

BookProof.BrstLeakage.leakage_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstTruncationLeakage.

BookProof.BrstLeakage.leakage_le {H B Om : E →L[ℂ] E} (hH : IsSelfAdjoint H) (hcomm : Commute H Om) (t : ℝ) (ht : 0 ≤ t) (x : E) (K : ℝ) (hK : ∀ s ∈ Set.Icc (0 : ℝ) t, ‖(H - B) (flow B s x)‖ ≤ K) : ‖Om (flow B t x)‖ ≤ ‖Om x‖ + ‖Om‖ * (K * t)

Formalization note: Lean 4 identifier BookProof.BrstLeakage.leakage_le.

Preamble

-- Generated from ChapterBrstTruncationLeakage.lean — theorem BookProof.BrstLeakage.leakage_le
import Mathlib
import Definitions.Def_ChapterBrstTruncationLeakage
open BookProof.BrstLeakage


open NormedSpace


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

Formal statement

theorem BookProof.BrstLeakage.leakage_le {H B Om : E →L[ℂ] E} (hH : IsSelfAdjoint H) (hcomm : Commute H Om)
    (t : ℝ) (ht : 0 ≤ t) (x : E) (K : ℝ)
    (hK : ∀ s ∈ Set.Icc (0 : ℝ) t, ‖(H - B) (flow B s x)‖ ≤ K) :
    ‖Om (flow B t x)‖ ≤ ‖Om x‖ + ‖Om‖ * (K * t) := by sorry

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