Prove2Me

`BookProof.NonnegSemigroup.norm_expNeg_sub_self_le` (A : F →L[ℂ] F) (hA : 0 ≤ A) {t : ℝ} (ht : 0 ≤ t) (x : F) : ‖expNeg A t x - x‖ ≤ t * ‖A x‖

Proved
BookProof.NonnegSemigroup.norm_expNeg_sub_self_le

by leonardopedro · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNonnegSemigroup.

BookProof.NonnegSemigroup.norm_expNeg_sub_self_le (A : F →L[ℂ] F) (hA : 0 ≤ A) {t : ℝ} (ht : 0 ≤ t) (x : F) : ‖expNeg A t x - x‖ ≤ t * ‖A x‖

Formalization note: Lean 4 identifier BookProof.NonnegSemigroup.norm_expNeg_sub_self_le.

Preamble
-- Generated from ChapterNonnegSemigroup.lean — theorem BookProof.NonnegSemigroup.norm_expNeg_sub_self_le
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterPositiveSquareRootUnique
import Definitions.Def_ChapterNonnegResolvent
import Definitions.Def_ChapterNonnegUnitaryGroup
import Mathlib
import Definitions.Def_ChapterNonnegSemigroup



open BookProof.ClosureUniqueness
open Filter Topology NormedSpace
open scoped InnerProductSpace

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
Formal statement
theorem BookProof.NonnegSemigroup.norm_expNeg_sub_self_le (A : F →L[ℂ] F) (hA : 0 ≤ A) {t : ℝ} (ht : 0 ≤ t) (x : F) :
    ‖expNeg A t x - x‖ ≤ t * ‖A x‖ := by sorry

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