`BookProof.ChapterReconstruct.offDiag_eq` (U : Fin n → Fin n → ℂ) (a : Fin n) (Ψ : Fin n → ℂ) : offDiag U a Ψ = (∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k) * (∑ b : Fin n, (starRi

Proved

BookProof.ChapterReconstruct.offDiag_eq

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterReconstruct.

BookProof.ChapterReconstruct.offDiag_eq (U : Fin n → Fin n → ℂ) (a : Fin n) (Ψ : Fin n → ℂ) : offDiag U a Ψ = (∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k) * (∑ b : Fin n, (starRingEnd ℂ) (Ψ b) * U b a) - ∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k * (starRingEnd ℂ) (Ψ k) * U k a

Formalization note: Lean 4 identifier BookProof.ChapterReconstruct.offDiag_eq.

Preamble

-- Generated from ChapterReconstruct.lean — theorem BookProof.ChapterReconstruct.offDiag_eq
import Mathlib
import Definitions.Def_ChapterReconstruct
open BookProof.ChapterReconstruct


open scoped BigOperators
open Finset


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterReconstruct.offDiag_eq (U : Fin n → Fin n → ℂ) (a : Fin n) (Ψ : Fin n → ℂ) :
    offDiag U a Ψ =
      (∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k) *
        (∑ b : Fin n, (starRingEnd ℂ) (Ψ b) * U b a) -
      ∑ k : Fin n,
        (starRingEnd ℂ) (U k a) * Ψ k * (starRingEnd ℂ) (Ψ k) * U k a := by sorry

View graph