`BookProof.ChapterH3.sirk_krylov_mem_adjoin` (Xm : Module.End ℂ E) (Y X : ℕ → Module.End ℂ E) (hX : ∀ j, X j = Y j * Xm) (v : E) (j : ℕ) : ∃ r ∈ Algebra.adjoin ℂ ({Xm} ∪ Set.range

Proved

BookProof.ChapterH3.sirk_krylov_mem_adjoin

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterH3.

BookProof.ChapterH3.sirk_krylov_mem_adjoin (Xm : Module.End ℂ E) (Y X : ℕ → Module.End ℂ E) (hX : ∀ j, X j = Y j * Xm) (v : E) (j : ℕ) : ∃ r ∈ Algebra.adjoin ℂ ({Xm} ∪ Set.range Y), sirkKrylov X v j = r v

Formalization note: Lean 4 identifier BookProof.ChapterH3.sirk_krylov_mem_adjoin.

Preamble

-- Generated from ChapterH3.lean — theorem BookProof.ChapterH3.sirk_krylov_mem_adjoin
import Mathlib
import Definitions.Def_ChapterH3
open BookProof.ChapterH3


open scoped BigOperators
open intervalIntegral


noncomputable section

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]

Formal statement

theorem BookProof.ChapterH3.sirk_krylov_mem_adjoin
    (Xm : Module.End ℂ E) (Y X : ℕ → Module.End ℂ E)
    (hX : ∀ j, X j = Y j * Xm) (v : E) (j : ℕ) :
    ∃ r ∈ Algebra.adjoin ℂ ({Xm} ∪ Set.range Y),
      sirkKrylov X v j = r v := by sorry

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