`BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias` {w : Fin m → ℝ} (hw : ∀ j, 0 < w j) {beta : ℝ} (hb : beta ≠ 0) (s : Fin m → ℝ) (j : Fin m) : priorSoftmax w beta

Proved

BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionPrior.

BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias {w : Fin m → ℝ} (hw : ∀ j, 0 < w j) {beta : ℝ} (hb : beta ≠ 0) (s : Fin m → ℝ) (j : Fin m) : priorSoftmax w beta s j = scoreSoftmax beta (fun l => s l + Real.log (w l) / beta) j

Formalization note: Lean 4 identifier BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias.

Preamble

-- Generated from ChapterAttentionPrior.lean — theorem BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionPrior
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionPrior


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionPrior.priorSoftmax_eq_scoreSoftmax_bias {w : Fin m → ℝ} (hw : ∀ j, 0 < w j)
    {beta : ℝ} (hb : beta ≠ 0) (s : Fin m → ℝ) (j : Fin m) :
    priorSoftmax w beta s j
      = scoreSoftmax beta (fun l => s l + Real.log (w l) / beta) j := by sorry

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