`BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff` : S.Undecidable ↔ (∃ a : H, a ≠ 0 ∧ S.op a = a) ∧ (∃ b : H, b ≠ 0 ∧ S.op b = 0)

Proved

BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterStatementOperator.

BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff : S.Undecidable ↔ (∃ a : H, a ≠ 0 ∧ S.op a = a) ∧ (∃ b : H, b ≠ 0 ∧ S.op b = 0)

Formalization note: Lean 4 identifier BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff.

Preamble

-- Generated from ChapterStatementOperator.lean — theorem BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff
import Mathlib
import Definitions.Def_ChapterStatementOperator
open BookProof.ChapterStatementOperator
open BookProof.ChapterStatementOperator



open ContinuousLinearMap

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (S : ModelStatement H) (ψ : H)

Formal statement

theorem BookProof.ChapterStatementOperator.ModelStatement.undecidable_iff :
    S.Undecidable ↔ (∃ a : H, a ≠ 0 ∧ S.op a = a) ∧ (∃ b : H, b ≠ 0 ∧ S.op b = 0) := by sorry

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