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

Proved

BookProof.BrstLeakage.truncation_leakage_le_abs

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_abs {P H Om : E →L[ℂ] E} (hPi : IsIdempotentElem P) (hPs : IsSelfAdjoint P) (hH : IsSelfAdjoint H) (hcomm : Commute H Om) (t : ℝ) {x : E} (hx : P x = x) : ‖Om (flow (truncGen P H) t x)‖ ≤ ‖Om x‖ + ‖Om‖ * (‖(1 - P) * H * P‖ * ‖x‖ * |t|)

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

Preamble

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

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