`BookProof.BrstReducedTransfer.infDist_exactStates_eq` (hzero : ∀ x : H, U 0 x = x) (hgroup : ∀ (s t : ℝ) (x : H), U s (U t x) = U (s + t) x) (hisom : ∀ (t : ℝ) (x : H), ‖U t x‖ =

Disproved

BookProof.BrstReducedTransfer.infDist_exactStates_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstReducedTransfer.

BookProof.BrstReducedTransfer.infDist_exactStates_eq (hzero : ∀ x : H, U 0 x = x) (hgroup : ∀ (s t : ℝ) (x : H), U s (U t x) = U (s + t) x) (hisom : ∀ (t : ℝ) (x : H), ‖U t x‖ = ‖x‖) (t : ℝ) (x : H) : Metric.infDist (U t x) (exactStates Om) = Metric.infDist x (exactStates Om)

Formalization note: Lean 4 identifier BookProof.BrstReducedTransfer.infDist_exactStates_eq.

Preamble

-- Generated from ChapterBrstReducedTransfer.lean — theorem BookProof.BrstReducedTransfer.infDist_exactStates_eq
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterBrstReducedTransfer
open BookProof.BrstReducedTransfer



open BookProof BookProof.ChapterStoneResolvent


variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H]

variable (Om : H →L[ℂ] H)

variable {Om}
variable (Om)
variable (U : ℝ → (H →L[ℂ] H)) (hcomm : ∀ (t : ℝ) (y : H), U t (Om y) = Om (U t y))

Formal statement

theorem BookProof.BrstReducedTransfer.infDist_exactStates_eq (hzero : ∀ x : H, U 0 x = x)
    (hgroup : ∀ (s t : ℝ) (x : H), U s (U t x) = U (s + t) x)
    (hisom : ∀ (t : ℝ) (x : H), ‖U t x‖ = ‖x‖) (t : ℝ) (x : H) :
    Metric.infDist (U t x) (exactStates Om) = Metric.infDist x (exactStates Om) := by sorry

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