`BookProof.ChapterF4.observable_matrix_identity` {dd kk : ℕ} (W : Matrix (Fin dd) (Fin kk) ℂ) (a : Fin dd) (r s : Fin kk) : Matrix.trace ((Matrix.single r s (1 : ℂ) : Matrix (Fin k

Proved

BookProof.ChapterF4.observable_matrix_identity

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterF4.

BookProof.ChapterF4.observable_matrix_identity {dd kk : ℕ} (W : Matrix (Fin dd) (Fin kk) ℂ) (a : Fin dd) (r s : Fin kk) : Matrix.trace ((Matrix.single r s (1 : ℂ) : Matrix (Fin kk) (Fin kk) ℂ)ᴴ * Wᴴ * (Matrix.single a a (1 : ℂ) : Matrix (Fin dd) (Fin dd) ℂ) * W) = (starRingEnd ℂ) (W a r) * W a s

Formalization note: Lean 4 identifier BookProof.ChapterF4.observable_matrix_identity.

Preamble

-- Generated from ChapterF4.lean — theorem BookProof.ChapterF4.observable_matrix_identity
import Mathlib
import Definitions.Def_ChapterF4
open BookProof.ChapterF4


open scoped BigOperators Matrix

variable {d k : ℕ}

Formal statement

theorem BookProof.ChapterF4.observable_matrix_identity {dd kk : ℕ} (W : Matrix (Fin dd) (Fin kk) ℂ)
    (a : Fin dd) (r s : Fin kk) :
    Matrix.trace ((Matrix.single r s (1 : ℂ) : Matrix (Fin kk) (Fin kk) ℂ)ᴴ * Wᴴ
        * (Matrix.single a a (1 : ℂ) : Matrix (Fin dd) (Fin dd) ℂ) * W)
      = (starRingEnd ℂ) (W a r) * W a s := by sorry

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