`BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase` (q k : EuclideanSpace ℂ (Fin n)) : coherentOverlapC q k = (Real.exp (-‖q‖ ^ 2 / 2 - ‖k‖ ^ 2 / 2 + (i

Proved

BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCoherentOverlapComplex.

BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase (q k : EuclideanSpace ℂ (Fin n)) : coherentOverlapC q k = (Real.exp (-‖q‖ ^ 2 / 2 - ‖k‖ ^ 2 / 2 + (inner ℂ q k : ℂ).re) : ℂ) * Complex.exp (((inner ℂ q k : ℂ).im : ℂ) * Complex.I)

Formalization note: Lean 4 identifier BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase.

Preamble

-- Generated from ChapterCoherentOverlapComplex.lean — theorem BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase
import Mathlib
import Definitions.Def_ChapterCoherentOverlapComplex
open BookProof.ChapterCoherentOverlapComplex


open scoped BigOperators

noncomputable section


variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterCoherentOverlapComplex.coherentOverlapC_eq_modulus_mul_phase (q k : EuclideanSpace ℂ (Fin n)) :
    coherentOverlapC q k =
      (Real.exp (-‖q‖ ^ 2 / 2 - ‖k‖ ^ 2 / 2 + (inner ℂ q k : ℂ).re) : ℂ) *
        Complex.exp (((inner ℂ q k : ℂ).im : ℂ) * Complex.I) := by sorry

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