`BookProof.ChapterSoftmaxDivergence.klDiv_nonneg` {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) : 0 ≤ klDiv p q

Proved

BookProof.ChapterSoftmaxDivergence.klDiv_nonneg

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxDivergence.

BookProof.ChapterSoftmaxDivergence.klDiv_nonneg {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) : 0 ≤ klDiv p q

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxDivergence.klDiv_nonneg.

Preamble

-- Generated from ChapterSoftmaxDivergence.lean — theorem BookProof.ChapterSoftmaxDivergence.klDiv_nonneg
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterSoftmaxDivergence
open BookProof.ChapterSoftmaxDivergence


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxDivergence.klDiv_nonneg {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1)
    (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) : 0 ≤ klDiv p q := by sorry

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