`BookProof.ChapterAttentionCoarseGrain.sum_mergeWeights_log` (f : Fin m → Fin r) (p : Fin m → ℝ) : ∑ y, mergeWeights f p y * Real.log (mergeWeights f p y) = ∑ x, p x * Real.log (me

Proved

BookProof.ChapterAttentionCoarseGrain.sum_mergeWeights_log

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCoarseGrain.

BookProof.ChapterAttentionCoarseGrain.sum_mergeWeights_log (f : Fin m → Fin r) (p : Fin m → ℝ) : ∑ y, mergeWeights f p y * Real.log (mergeWeights f p y) = ∑ x, p x * Real.log (mergeWeights f p (f x))

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness

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

Formal statement

theorem BookProof.ChapterAttentionCoarseGrain.sum_mergeWeights_log (f : Fin m → Fin r) (p : Fin m → ℝ) :
    ∑ y, mergeWeights f p y * Real.log (mergeWeights f p y)
      = ∑ x, p x * Real.log (mergeWeights f p (f x)) := by sorry

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