`BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half` (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ) (hk : ∀ l, ‖k l‖ = r) (j : Fin m) : bornW

Proved

BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxBorn.

BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ) (hk : ∀ l, ‖k l‖ = r) (j : Fin m) : bornWeight q k j = Real.exp (inner ℝ q (k j) / (1 / 2)) / ∑ l, Real.exp (inner ℝ q (k l) / (1 / 2))

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half.

Preamble

-- Generated from ChapterSoftmaxBorn.lean — theorem BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half
import Definitions.Def_ChapterCoherentOverlap
import Mathlib
import Definitions.Def_ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxBorn


open scoped BigOperators

noncomputable section


open BookProof.ChapterCoherentOverlap

variable {n m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxBorn.coherentBorn_temperature_half (q : EuclideanSpace ℝ (Fin n))
    (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ) (hk : ∀ l, ‖k l‖ = r) (j : Fin m) :
    bornWeight q k j =
      Real.exp (inner ℝ q (k j) / (1 / 2)) / ∑ l, Real.exp (inner ℝ q (k l) / (1 / 2)) := by sorry

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