`BookProof.ChapterAttentionMixture.multiHead_isProb` {w : Fin H → ℝ} (hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1) (beta : Fin H → ℝ) (s : Fin H → Fin m → ℝ) (j₀ : Fin m) : (∀...

Proved

BookProof.ChapterAttentionMixture.multiHead_isProb

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMixture.

BookProof.ChapterAttentionMixture.multiHead_isProb {w : Fin H → ℝ} (hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1) (beta : Fin H → ℝ) (s : Fin H → Fin m → ℝ) (j₀ : Fin m) : (∀ j, 0 ≤ multiHead w beta s j) ∧ ∑ j, multiHead w beta s j = 1

Formalization note: Lean 4 identifier BookProof.ChapterAttentionMixture.multiHead_isProb.

Preamble

-- Generated from ChapterAttentionMixture.lean — theorem BookProof.ChapterAttentionMixture.multiHead_isProb
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionMixture
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionMixture


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m H : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionMixture.multiHead_isProb {w : Fin H → ℝ} (hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1)
    (beta : Fin H → ℝ) (s : Fin H → Fin m → ℝ) (j₀ : Fin m) :
    (∀ j, 0 ≤ multiHead w beta s j) ∧ ∑ j, multiHead w beta s j = 1 := by sorry

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