Prove2Me

`BookProof.ChapterGaugeComprehensiveFixing.exists_physical_ne_agreeing_of_not_comprehensive` (hS : ¬ IsComprehensiveGaugeFixing G S) : ∃ f f' : X → ℝ, IsPhysicalObservable G f ∧ Is

Proved
BookProof.ChapterGaugeComprehensiveFixing.exists_physical_ne_agreeing_of_not_comprehensive

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeComprehensiveFixing.

BookProof.ChapterGaugeComprehensiveFixing.exists_physical_ne_agreeing_of_not_comprehensive (hS : ¬ IsComprehensiveGaugeFixing G S) : ∃ f f' : X → ℝ, IsPhysicalObservable G f ∧ IsPhysicalObservable G f' ∧ (∀ s ∈ S, f s = f' s) ∧ f ≠ f'

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

Preamble
-- Generated from ChapterGaugeComprehensiveFixing.lean — theorem BookProof.ChapterGaugeComprehensiveFixing.exists_physical_ne_agreeing_of_not_comprehensive
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] {S : Set X}
Formal statement
theorem BookProof.ChapterGaugeComprehensiveFixing.exists_physical_ne_agreeing_of_not_comprehensive
    (hS : ¬ IsComprehensiveGaugeFixing G S) :
    ∃ f f' : X → ℝ, IsPhysicalObservable G f ∧ IsPhysicalObservable G f' ∧
      (∀ s ∈ S, f s = f' s) ∧ f ≠ f' := by sorry

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