`BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull` (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n))) (v : Fin m → ℝ) (q : EuclideanSpace ℂ (Fin

Proved

BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterObservableOperator.

BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n))) (v : Fin m → ℝ) (q : EuclideanSpace ℂ (Fin n)) (hq : ‖q‖ = 1) : BookProof.ChapterObservableExpectation.observableExpectation (bornProb (fun j => b j) q) v ∈ convexHull ℝ (Set.range v)

Formalization note: Lean 4 identifier BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull.

Preamble

-- Generated from ChapterObservableOperator.lean — theorem BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull
import Mathlib
import Definitions.Def_ChapterObservableOperator
import Definitions.Def_ChapterDoubleSlit
import Definitions.Def_ChapterObservableExpectation
open BookProof.ChapterDoubleSlit
open BookProof.ChapterObservableExpectation
open BookProof.ChapterObservableOperator


open scoped BigOperators

noncomputable section


variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterObservableOperator.observableOp_expectation_mem_convexHull
    (b : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin n))) (v : Fin m → ℝ)
    (q : EuclideanSpace ℂ (Fin n)) (hq : ‖q‖ = 1) :
    BookProof.ChapterObservableExpectation.observableExpectation
        (bornProb (fun j => b j) q) v ∈ convexHull ℝ (Set.range v) := by sorry

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