`BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le` {n : ℕ} {beta : ℝ} (hbeta : 0 < beta) (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (i j : Fin m) :

Proved

BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxOrder.

BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le {n : ℕ} {beta : ℝ} (hbeta : 0 < beta) (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (i j : Fin m) : softmax beta q k i ≤ softmax beta q k j ↔ (inner ℝ q (k i) : ℝ) ≤ inner ℝ q (k j)

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le.

Preamble

-- Generated from ChapterSoftmaxOrder.lean — theorem BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le
import Mathlib
import Definitions.Def_ChapterSoftmaxOrder
import Definitions.Def_ChapterSoftmaxBorn
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxOrder


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxBorn BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxOrder.softmax_le_iff_inner_le {n : ℕ} {beta : ℝ} (hbeta : 0 < beta)
    (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (i j : Fin m) :
    softmax beta q k i ≤ softmax beta q k j ↔
      (inner ℝ q (k i) : ℝ) ≤ inner ℝ q (k j) := by sorry

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