`BookProof.ChapterLogPartitionConvex.logPartition_isGreatest` (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : IsGreatest {x : ℝ | ∃ p : Fin m → ℝ, (∀ j, 0 ≤ p j) ∧ (∑ j, p j) = 1...

Proved

BookProof.ChapterLogPartitionConvex.logPartition_isGreatest

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLogPartitionConvex.

BookProof.ChapterLogPartitionConvex.logPartition_isGreatest (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : IsGreatest {x : ℝ | ∃ p : Fin m → ℝ, (∀ j, 0 ≤ p j) ∧ (∑ j, p j) = 1 ∧ x = beta * (∑ j, p j * s j) + shannonEntropy p} (logPartition beta s)

Formalization note: Lean 4 identifier BookProof.ChapterLogPartitionConvex.logPartition_isGreatest.

Preamble

-- Generated from ChapterLogPartitionConvex.lean — theorem BookProof.ChapterLogPartitionConvex.logPartition_isGreatest
import Definitions.Def_ChapterAttentionEntropy
import Mathlib
import Definitions.Def_ChapterLogPartitionConvex
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxFluctuation
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterLogPartitionConvex


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterLogPartitionConvex.logPartition_isGreatest (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    IsGreatest {x : ℝ | ∃ p : Fin m → ℝ, (∀ j, 0 ≤ p j) ∧ (∑ j, p j) = 1 ∧
        x = beta * (∑ j, p j * s j) + shannonEntropy p} (logPartition beta s) := by sorry

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