`BookProof.ChapterAttentionSink.shannonEntropy_sink` (beta s0 : ℝ) (s : Fin m → ℝ) (i : Fin m) : shannonEntropy (scoreSoftmax beta (Fin.cons s0 s)) = (-sinkWeight beta s0 s * Real.

Proved

BookProof.ChapterAttentionSink.shannonEntropy_sink

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSink.

BookProof.ChapterAttentionSink.shannonEntropy_sink (beta s0 : ℝ) (s : Fin m → ℝ) (i : Fin m) : shannonEntropy (scoreSoftmax beta (Fin.cons s0 s)) = (-sinkWeight beta s0 s * Real.log (sinkWeight beta s0 s) - (1 - sinkWeight beta s0 s) * Real.log (1 - sinkWeight beta s0 s)) + (1 - sinkWeight beta s0 s) * shannonEntropy (scoreSoftmax beta s)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionSink.shannonEntropy_sink.

Preamble

-- Generated from ChapterAttentionSink.lean — theorem BookProof.ChapterAttentionSink.shannonEntropy_sink
import Definitions.Def_ChapterObservableExpectation
import Mathlib
import Definitions.Def_ChapterAttentionSink
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionEntropy
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionSink


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionEntropy

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

Formal statement

theorem BookProof.ChapterAttentionSink.shannonEntropy_sink (beta s0 : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    shannonEntropy (scoreSoftmax beta (Fin.cons s0 s))
      = (-sinkWeight beta s0 s * Real.log (sinkWeight beta s0 s)
          - (1 - sinkWeight beta s0 s) * Real.log (1 - sinkWeight beta s0 s))
        + (1 - sinkWeight beta s0 s) * shannonEntropy (scoreSoftmax beta s) := by sorry

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