Prove2Me

`BookProof.ChapterLogPartitionConvex.logPartition_convex_comb` (s : Fin m → ℝ) (i : Fin m) (b c t u : ℝ) (ht : 0 ≤ t) (hu : 0 ≤ u) (htu : t + u = 1) : logPartition (t * b + u * c)

Proved
BookProof.ChapterLogPartitionConvex.logPartition_convex_comb

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLogPartitionConvex.

BookProof.ChapterLogPartitionConvex.logPartition_convex_comb (s : Fin m → ℝ) (i : Fin m) (b c t u : ℝ) (ht : 0 ≤ t) (hu : 0 ≤ u) (htu : t + u = 1) : logPartition (t * b + u * c) s ≤ t * logPartition b s + u * logPartition c s

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

Preamble
-- Generated from ChapterLogPartitionConvex.lean — theorem BookProof.ChapterLogPartitionConvex.logPartition_convex_comb
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterLogPartitionConvex
import Definitions.Def_ChapterSoftmaxFluctuation
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_convex_comb (s : Fin m → ℝ) (i : Fin m) (b c t u : ℝ)
    (ht : 0 ≤ t) (hu : 0 ≤ u) (htu : t + u = 1) :
    logPartition (t * b + u * c) s ≤ t * logPartition b s + u * logPartition c s := by sorry

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