Prove2Me

`BookProof.ChapterSoftmaxDivergence.klDiv_eq_crossEntropy_sub_shannonEntropy` {p q : Fin m → ℝ} (hq0 : ∀ j, 0 < q j) : klDiv p q = crossEntropy p q - shannonEntropy p

Proved
BookProof.ChapterSoftmaxDivergence.klDiv_eq_crossEntropy_sub_shannonEntropy

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxDivergence.

BookProof.ChapterSoftmaxDivergence.klDiv_eq_crossEntropy_sub_shannonEntropy {p q : Fin m → ℝ} (hq0 : ∀ j, 0 < q j) : klDiv p q = crossEntropy p q - shannonEntropy p

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

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxMaxEntropy

variable {m : ℕ}
Formal statement
theorem BookProof.ChapterSoftmaxDivergence.klDiv_eq_crossEntropy_sub_shannonEntropy {p q : Fin m → ℝ}
    (hq0 : ∀ j, 0 < q j) :
    klDiv p q = crossEntropy p q - shannonEntropy p := by sorry

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