`BookProof.ChapterAttentionMixture.le_shannonEntropy_mixture` {w : Fin H → ℝ} {p : Fin H → Fin m → ℝ} (hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1) (hp0 : ∀ h j, 0 ≤ p h j) : ∑...
ProvedBookProof.ChapterAttentionMixture.le_shannonEntropy_mixturetheoremstimepiece
Prove the following Lean 4 theorem from ChapterAttentionMixture.
BookProof.ChapterAttentionMixture.le_shannonEntropy_mixture {w : Fin H → ℝ} {p : Fin H → Fin m → ℝ} (hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1) (hp0 : ∀ h j, 0 ≤ p h j) : ∑ h, w h * shannonEntropy (p h) ≤ shannonEntropy (mixture w p)
Formalization note: Lean 4 identifier BookProof.ChapterAttentionMixture.le_shannonEntropy_mixture.
Preamble
-- Generated from ChapterAttentionMixture.lean — theorem BookProof.ChapterAttentionMixture.le_shannonEntropy_mixture
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionMixture
import Definitions.Def_ChapterAttentionEntropy
open BookProof.ChapterAttentionMixture
open scoped BigOperators
noncomputable section
open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionEntropy
variable {m H : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]Formal statement
theorem BookProof.ChapterAttentionMixture.le_shannonEntropy_mixture {w : Fin H → ℝ} {p : Fin H → Fin m → ℝ}
(hw0 : ∀ h, 0 ≤ w h) (hw : ∑ h, w h = 1) (hp0 : ∀ h j, 0 ≤ p h j) :
∑ h, w h * shannonEntropy (p h) ≤ shannonEntropy (mixture w p) := by sorry