`BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast` : (¬ ∃ ρ : Matrix (Fin 2) (Fin 2) ℝ, IsPureState ρ ∧ E ρ P0 = 1/2 ∧ E ρ Q = 1/2) ∧ (∃ ρ : Matrix (Fin 2) (Fin 2)

Proved

BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGleasonPureMixed.

BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast : (¬ ∃ ρ : Matrix (Fin 2) (Fin 2) ℝ, IsPureState ρ ∧ E ρ P0 = 1/2 ∧ E ρ Q = 1/2) ∧ (∃ ρ : Matrix (Fin 2) (Fin 2) ℝ, IsMixedState ρ ∧ E ρ P0 = 1/2 ∧ E ρ Q = 1/2)

Formalization note: Lean 4 identifier BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast.

Preamble

-- Generated from ChapterGleasonPureMixed.lean — theorem BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast
import Mathlib
import Definitions.Def_ChapterGleasonPureMixed
import Definitions.Def_ChapterB4
import Definitions.Def_ChapterBRSTNilpotent
open BookProof.ChapterB4
open BookProof.BRSTNilpotent
open BookProof.ChapterGleasonPureMixed


open scoped BigOperators
open Matrix

Formal statement

theorem BookProof.ChapterGleasonPureMixed.pure_vs_mixed_gleason_contrast :
    (¬ ∃ ρ : Matrix (Fin 2) (Fin 2) ℝ, IsPureState ρ ∧ E ρ P0 = 1/2 ∧ E ρ Q = 1/2) ∧
    (∃ ρ : Matrix (Fin 2) (Fin 2) ℝ, IsMixedState ρ ∧ E ρ P0 = 1/2 ∧ E ρ Q = 1/2) := by sorry

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