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`BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive` (mu : Measure X) : ¬ ((atoms mu).Finite ∧ (atoms mu).Infinite) ∧ ¬ ((atoms mu = ∅) ∧ (atoms mu).Nonempty) ∧ ¬ (

Proved
BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAtomicDecomposition.

BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive (mu : Measure X) : ¬ ((atoms mu).Finite ∧ (atoms mu).Infinite) ∧ ¬ ((atoms mu = ∅) ∧ (atoms mu).Nonempty) ∧ ¬ ((atoms mu = ∅) ∧ (atoms mu).Infinite) ∧ ¬ (continuousPart mu = 0 ∧ continuousPart mu ≠ 0)

Formalization note: Lean 4 identifier BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive.

Preamble
-- Generated from ChapterAtomicDecomposition.lean — theorem BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive
import Mathlib
import Definitions.Def_ChapterAtomicDecomposition
open BookProof.ChapterAtomicDecomposition


open MeasureTheory


variable {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
Formal statement
theorem BookProof.ChapterAtomicDecomposition.five_types_pairwise_exclusive (mu : Measure X) :
    ¬ ((atoms mu).Finite ∧ (atoms mu).Infinite) ∧
      ¬ ((atoms mu = ∅) ∧ (atoms mu).Nonempty) ∧
      ¬ ((atoms mu = ∅) ∧ (atoms mu).Infinite) ∧
      ¬ (continuousPart mu = 0 ∧ continuousPart mu ≠ 0) := by sorry

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