`BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability` [Fintype Model] (seedProb : Seed → ℝ) (train : Seed → Model) (hnonneg : ∀ s, 0 ≤ seedProb s) (hsum : ∑ s, seedPro

Proved

BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDeepLearningSampling.

BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability [Fintype Model] (seedProb : Seed → ℝ) (train : Seed → Model) (hnonneg : ∀ s, 0 ≤ seedProb s) (hsum : ∑ s, seedProb s = 1) : (∀ m, 0 ≤ inducedPrior seedProb train m) ∧ ∑ m, inducedPrior seedProb train m = 1

Formalization note: Lean 4 identifier BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability.

Preamble

-- Generated from ChapterDeepLearningSampling.lean — theorem BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability
import Mathlib
import Definitions.Def_ChapterDeepLearningSampling
open BookProof.ChapterDeepLearningSampling


open scoped BigOperators


variable {Seed Model Data : Type*}
variable [Fintype Seed] [DecidableEq Model]

Formal statement

theorem BookProof.ChapterDeepLearningSampling.inducedPrior_isProbability [Fintype Model]
    (seedProb : Seed → ℝ) (train : Seed → Model)
    (hnonneg : ∀ s, 0 ≤ seedProb s) (hsum : ∑ s, seedProb s = 1) :
    (∀ m, 0 ≤ inducedPrior seedProb train m) ∧
      ∑ m, inducedPrior seedProb train m = 1 := by sorry

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