`BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing` {S : Set ℝ} (hcomp : IsComprehensiveGaugeFixing (Multiplicative ℤ) S) (hcompl : IsCompleteGaugeFixi

Proved

BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeComprehensiveFixing.

BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing {S : Set ℝ} (hcomp : IsComprehensiveGaugeFixing (Multiplicative ℤ) S) (hcompl : IsCompleteGaugeFixing' (Multiplicative ℤ) S) : ¬ IsClopen S

Formalization note: Lean 4 identifier BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing.

Preamble

-- Generated from ChapterGaugeComprehensiveFixing.lean — theorem BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing
import Mathlib
import Definitions.Def_ChapterGaugeComprehensiveFixing
import Definitions.Def_ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeComprehensiveFixing



open BookProof.ChapterGaugeIncompleteFixing

variable {X : Type*} (G : Type*) [Group G] [MulAction G X]
variable {X : Type*} {G : Type*} [Group G] [MulAction G X] {S : Set X}
variable {X : Type*} [TopologicalSpace X] {G : Type*} [Group G] [MulAction G X]
variable (X : Type*) (G : Type*) [Group G] [MulAction G X]

Formal statement

theorem BookProof.ChapterGaugeComprehensiveFixing.shift_no_clopen_complete_gaugeFixing {S : Set ℝ}
    (hcomp : IsComprehensiveGaugeFixing (Multiplicative ℤ) S)
    (hcompl : IsCompleteGaugeFixing' (Multiplicative ℤ) S) :
    ¬ IsClopen S := by sorry

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