`BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax` (nbar : ℝ≥0) (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m) : dtBornMulti nbar q

Proved

BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDisplacedThermalMulti.

BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax (nbar : ℝ≥0) (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m) : dtBornMulti nbar q k j = BookProof.ChapterSoftmaxSharpness.scoreSoftmax (inverseTemperature nbar) (fun l => -‖q - k l‖ ^ 2) j

Formalization note: Lean 4 identifier BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax.

Preamble

-- Generated from ChapterDisplacedThermalMulti.lean — theorem BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax
import Mathlib
import Definitions.Def_ChapterDisplacedThermalMulti
import Definitions.Def_ChapterDisplacedThermalOverlap
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterDisplacedThermalOverlap
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterDisplacedThermalMulti


noncomputable section

open MeasureTheory ProbabilityTheory Real
open scoped NNReal


open BookProof.ChapterDisplacedThermalOverlap

variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterDisplacedThermalMulti.dtBornMulti_eq_softmax (nbar : ℝ≥0) (q : EuclideanSpace ℝ (Fin n))
    (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m) :
    dtBornMulti nbar q k j
      = BookProof.ChapterSoftmaxSharpness.scoreSoftmax (inverseTemperature nbar)
          (fun l => -‖q - k l‖ ^ 2) j := by sorry

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