`BookProof.ChapterGaugeUnconstrainedSpectrum.isPhysicalFunction_iff_factors` (ρ : G →* Equiv.Perm X) (d : X → ℂ) : IsPhysicalFunction ρ d ↔ ∃ D : observableSpectrum ρ → ℂ, ∀ x : X,

Proved

BookProof.ChapterGaugeUnconstrainedSpectrum.isPhysicalFunction_iff_factors

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeUnconstrainedSpectrum.

BookProof.ChapterGaugeUnconstrainedSpectrum.isPhysicalFunction_iff_factors (ρ : G →* Equiv.Perm X) (d : X → ℂ) : IsPhysicalFunction ρ d ↔ ∃ D : observableSpectrum ρ → ℂ, ∀ x : X, d x = D (Quotient.mk (orbitSetoid ρ) x)

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

Preamble

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



variable {X : Type*}

variable {G : Type*} [Group G]

Formal statement

theorem BookProof.ChapterGaugeUnconstrainedSpectrum.isPhysicalFunction_iff_factors (ρ : G →* Equiv.Perm X) (d : X → ℂ) :
    IsPhysicalFunction ρ d ↔
      ∃ D : observableSpectrum ρ → ℂ, ∀ x : X, d x = D (Quotient.mk (orbitSetoid ρ) x) := by sorry

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