`BookProof.BrstLeakage.truncation_leakage_le_of_physical` {P H Om : E →L[ℂ] E} (hPi : IsIdempotentElem P) (hPs : IsSelfAdjoint P) (hH : IsSelfAdjoint H) (hcomm : Commute H Om) (t :

Proved

BookProof.BrstLeakage.truncation_leakage_le_of_physical

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstTruncationLeakage.

BookProof.BrstLeakage.truncation_leakage_le_of_physical {P H Om : E →L[ℂ] E} (hPi : IsIdempotentElem P) (hPs : IsSelfAdjoint P) (hH : IsSelfAdjoint H) (hcomm : Commute H Om) (t : ℝ) (ht : 0 ≤ t) {x : E} (hx : P x = x) (hOm : Om x = 0) : ‖Om (flow (truncGen P H) t x)‖ ≤ ‖Om‖ * (‖(1 - P) * H * P‖ * ‖x‖ * t)

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

Preamble

-- Generated from ChapterBrstTruncationLeakage.lean — theorem BookProof.BrstLeakage.truncation_leakage_le_of_physical
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.truncation_leakage_le_of_physical {P H Om : E →L[ℂ] E} (hPi : IsIdempotentElem P)
    (hPs : IsSelfAdjoint P) (hH : IsSelfAdjoint H) (hcomm : Commute H Om)
    (t : ℝ) (ht : 0 ≤ t) {x : E} (hx : P x = x) (hOm : Om x = 0) :
    ‖Om (flow (truncGen P H) t x)‖ ≤ ‖Om‖ * (‖(1 - P) * H * P‖ * ‖x‖ * t) := by sorry

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