`BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull` (p : Fin m → ℝ) (hp : ∀ j, 0 ≤ p j) (hp1 : ∑ j, p j = 1) (v : Fin m → E) : observableExpectation p v ∈ con

Proved

BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterObservableExpectation.

BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull (p : Fin m → ℝ) (hp : ∀ j, 0 ≤ p j) (hp1 : ∑ j, p j = 1) (v : Fin m → E) : observableExpectation p v ∈ convexHull ℝ (Set.range v)

Formalization note: Lean 4 identifier BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull.

Preamble

-- Generated from ChapterObservableExpectation.lean — theorem BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull
import Definitions.Def_ChapterSoftmaxBorn
import Mathlib
import Definitions.Def_ChapterObservableExpectation
open BookProof.ChapterObservableExpectation


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxBorn

variable {m n : ℕ}
variable {E : Type*} [AddCommGroup E] [Module ℝ E]

Formal statement

theorem BookProof.ChapterObservableExpectation.prob_weighted_sum_mem_convexHull (p : Fin m → ℝ) (hp : ∀ j, 0 ≤ p j)
    (hp1 : ∑ j, p j = 1) (v : Fin m → E) :
    observableExpectation p v ∈ convexHull ℝ (Set.range v) := by sorry

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