`BookProof.ChapterE.collapse_density` (t : ℝ) : (!![Real.cos t ^ 2, 0; 0, Real.sin t ^ 2] : Matrix (Fin 2) (Fin 2) ℝ) = (1 / 2 : ℝ) • (1 : Matrix (Fin 2) (Fin 2) ℝ) + (1 / 2 * Real

Proved

BookProof.ChapterE.collapse_density

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterE.

BookProof.ChapterE.collapse_density (t : ℝ) : (!![Real.cos t ^ 2, 0; 0, Real.sin t ^ 2] : Matrix (Fin 2) (Fin 2) ℝ) = (1 / 2 : ℝ) • (1 : Matrix (Fin 2) (Fin 2) ℝ) + (1 / 2 * Real.cos (2 * t)) • !![1, 0; 0, -1]

Formalization note: Lean 4 identifier BookProof.ChapterE.collapse_density.

Preamble

-- Generated from ChapterE.lean — theorem BookProof.ChapterE.collapse_density
import Mathlib
import Definitions.Def_ChapterE
open BookProof.ChapterE


open scoped Matrix BigOperators
open Filter
open scoped Topology

Formal statement

theorem BookProof.ChapterE.collapse_density (t : ℝ) :
    (!![Real.cos t ^ 2, 0; 0, Real.sin t ^ 2] : Matrix (Fin 2) (Fin 2) ℝ)
      = (1 / 2 : ℝ) • (1 : Matrix (Fin 2) (Fin 2) ℝ)
        + (1 / 2 * Real.cos (2 * t)) • !![1, 0; 0, -1] := by sorry

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