`BookProof.ChapterStatementOperator.approximate_proof_error` (S : ModelStatement H) (A : H →L[ℂ] H) (ε : ℝ) (hA : ‖A - S.op‖ ≤ ε) (ψ : H) : |RCLike.re (inner ℂ ψ (A ψ)) - S.truthVa

Proved

BookProof.ChapterStatementOperator.approximate_proof_error

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterStatementOperator.

BookProof.ChapterStatementOperator.approximate_proof_error (S : ModelStatement H) (A : H →L[ℂ] H) (ε : ℝ) (hA : ‖A - S.op‖ ≤ ε) (ψ : H) : |RCLike.re (inner ℂ ψ (A ψ)) - S.truthValue ψ| ≤ ε * ‖ψ‖ ^ 2

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

Preamble

-- Generated from ChapterStatementOperator.lean — theorem BookProof.ChapterStatementOperator.approximate_proof_error
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.approximate_proof_error (S : ModelStatement H) (A : H →L[ℂ] H) (ε : ℝ)
    (hA : ‖A - S.op‖ ≤ ε) (ψ : H) :
    |RCLike.re (inner ℂ ψ (A ψ)) - S.truthValue ψ| ≤ ε * ‖ψ‖ ^ 2 := by sorry

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