`BookProof.ChapterAttentionCoarseGrain.mergeWeights_scoreSoftmax_of_fiber_const` (beta : ℝ) (f : Fin m → Fin r) (t : Fin r → ℝ) (y : Fin r) : mergeWeights f (scoreSoftmax beta (fun

Proved

BookProof.ChapterAttentionCoarseGrain.mergeWeights_scoreSoftmax_of_fiber_const

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCoarseGrain.

BookProof.ChapterAttentionCoarseGrain.mergeWeights_scoreSoftmax_of_fiber_const (beta : ℝ) (f : Fin m → Fin r) (t : Fin r → ℝ) (y : Fin r) : mergeWeights f (scoreSoftmax beta (fun x => t (f x))) y = ((Finset.univ.filter (fun x => f x = y)).card : ℝ) * Real.exp (beta * t y) / ∑ z, ((Finset.univ.filter (fun x => f x = z)).card : ℝ) * Real.exp (beta * t z)

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

Preamble

-- Generated from ChapterAttentionCoarseGrain.lean — theorem BookProof.ChapterAttentionCoarseGrain.mergeWeights_scoreSoftmax_of_fiber_const
import Definitions.Def_ChapterObservableExpectation
import Mathlib
import Definitions.Def_ChapterAttentionCoarseGrain
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
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.mergeWeights_scoreSoftmax_of_fiber_const (beta : ℝ) (f : Fin m → Fin r)
    (t : Fin r → ℝ) (y : Fin r) :
    mergeWeights f (scoreSoftmax beta (fun x => t (f x))) y
      = ((Finset.univ.filter (fun x => f x = y)).card : ℝ) * Real.exp (beta * t y)
        / ∑ z, ((Finset.univ.filter (fun x => f x = z)).card : ℝ) * Real.exp (beta * t z) := by sorry

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