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`BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum` : IsUnconstrainedGaugeFixing (fun m : Multiplicative ℤ => permOp (shiftPerm m)) ∧ constrain

Proved
BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeUnconstrainedSpectrum.

BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum : IsUnconstrainedGaugeFixing (fun m : Multiplicative ℤ => permOp (shiftPerm m)) ∧ constrainedSpectrum (fun m : Multiplicative ℤ => permOp (shiftPerm m)) = (Set.univ : Set ℤ) ∧ (Set.univ : Set ℤ).Infinite ∧ Subsingleton (observableSpectrum shiftPerm)

Formalization note: Lean 4 identifier BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum.

Preamble
-- Generated from ChapterGaugeUnconstrainedSpectrum.lean — theorem BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum
import Mathlib
import Definitions.Def_ChapterGaugeUnconstrainedSpectrum
open BookProof.ChapterGaugeUnconstrainedSpectrum



variable {X : Type*}

variable {G : Type*} [Group G]
Formal statement
theorem BookProof.ChapterGaugeUnconstrainedSpectrum.shift_full_spectrum_vs_observable_spectrum :
    IsUnconstrainedGaugeFixing (fun m : Multiplicative ℤ => permOp (shiftPerm m)) ∧
      constrainedSpectrum (fun m : Multiplicative ℤ => permOp (shiftPerm m)) =
        (Set.univ : Set ℤ) ∧
      (Set.univ : Set ℤ).Infinite ∧
      Subsingleton (observableSpectrum shiftPerm) := by sorry

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