`BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff` (beta : ℝ) (s : Fin m → ℝ) (y : Fin m) : crossEntropyLoss beta s y = 0 ↔ scoreSoftmax beta s y = 1

Proved

BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCrossEntropyGradient.

BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff (beta : ℝ) (s : Fin m → ℝ) (y : Fin m) : crossEntropyLoss beta s y = 0 ↔ scoreSoftmax beta s y = 1

Formalization note: Lean 4 identifier BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff.

Preamble

-- Generated from ChapterCrossEntropyGradient.lean — theorem BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterCrossEntropyGradient
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterCrossEntropyGradient


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterCrossEntropyGradient.crossEntropyLoss_eq_zero_iff (beta : ℝ) (s : Fin m → ℝ) (y : Fin m) :
    crossEntropyLoss beta s y = 0 ↔ scoreSoftmax beta s y = 1 := by sorry

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