`BookProof.ChapterGaugeComprehensiveFixing.physical_extension_iff_complete` : (∀ h : X → ℝ, ∃ f : X → ℝ, IsPhysicalObservable G f ∧ ∀ s ∈ S, f s = h s) ↔ IsCompleteGaugeFixing' G S

Proved

BookProof.ChapterGaugeComprehensiveFixing.physical_extension_iff_complete

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeComprehensiveFixing.

BookProof.ChapterGaugeComprehensiveFixing.physical_extension_iff_complete : (∀ h : X → ℝ, ∃ f : X → ℝ, IsPhysicalObservable G f ∧ ∀ s ∈ S, f s = h s) ↔ IsCompleteGaugeFixing' G S

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

Preamble

-- Generated from ChapterGaugeComprehensiveFixing.lean — theorem BookProof.ChapterGaugeComprehensiveFixing.physical_extension_iff_complete
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.physical_extension_iff_complete :
    (∀ h : X → ℝ, ∃ f : X → ℝ, IsPhysicalObservable G f ∧ ∀ s ∈ S, f s = h s) ↔
      IsCompleteGaugeFixing' G S := by sorry

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