`BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le` {f : Fin m → Fin r} {p : Fin m → ℝ} (hp : ∀ x, 0 ≤ p x) : shannonEntropy (mergeWeights f p) ≤ shannonEntropy

Proved

BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCoarseGrain.

BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le {f : Fin m → Fin r} {p : Fin m → ℝ} (hp : ∀ x, 0 ≤ p x) : shannonEntropy (mergeWeights f p) ≤ shannonEntropy p

Formalization note: Lean 4 identifier BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le.

Preamble

-- Generated from ChapterAttentionCoarseGrain.lean — theorem BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le
import Definitions.Def_ChapterObservableExpectation
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterAttentionCoarseGrain
import Definitions.Def_ChapterAttentionEntropy
open BookProof.ChapterAttentionCoarseGrain


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionEntropy

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

Formal statement

theorem BookProof.ChapterAttentionCoarseGrain.shannonEntropy_mergeWeights_le {f : Fin m → Fin r} {p : Fin m → ℝ}
    (hp : ∀ x, 0 ≤ p x) :
    shannonEntropy (mergeWeights f p) ≤ shannonEntropy p := by sorry

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