`BookProof.ChapterStatementOperator.halfUncertain_not_idempotent` (h : ∃ ψ : H, ψ ≠ 0) : ¬ ((halfUncertain (H := H)).op ∘L (halfUncertain (H := H)).op = (halfUncertain (H := H)).op

Proved

BookProof.ChapterStatementOperator.halfUncertain_not_idempotent

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterStatementOperator.

BookProof.ChapterStatementOperator.halfUncertain_not_idempotent (h : ∃ ψ : H, ψ ≠ 0) : ¬ ((halfUncertain (H := H)).op ∘L (halfUncertain (H := H)).op = (halfUncertain (H := H)).op)

Formalization note: Lean 4 identifier BookProof.ChapterStatementOperator.halfUncertain_not_idempotent.

Preamble

-- Generated from ChapterStatementOperator.lean — theorem BookProof.ChapterStatementOperator.halfUncertain_not_idempotent
import Mathlib
import Definitions.Def_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.halfUncertain_not_idempotent (h : ∃ ψ : H, ψ ≠ 0) :
    ¬ ((halfUncertain (H := H)).op ∘L (halfUncertain (H := H)).op
        = (halfUncertain (H := H)).op) := by sorry

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