`BookProof.ChapterH8.sirk_band_refinement_poly` (Vn : F →L[ℂ] E) (Vm : G →L[ℂ] E) (J : F →L[ℂ] G) (X : E →L[ℂ] E) (hJ : Vn = Vm.comp J) (hVm : (adjoint Vm).comp Vm = ContinuousLine

Proved

BookProof.ChapterH8.sirk_band_refinement_poly

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

timepiece

Prove the following Lean 4 theorem from ChapterH8.

BookProof.ChapterH8.sirk_band_refinement_poly (Vn : F →L[ℂ] E) (Vm : G →L[ℂ] E) (J : F →L[ℂ] G) (X : E →L[ℂ] E) (hJ : Vn = Vm.comp J) (hVm : (adjoint Vm).comp Vm = ContinuousLinearMap.id ℂ G) (hJJ : (adjoint J).comp J = ContinuousLinearMap.id ℂ F) (hinvn : ∀ x : F, ∃ y : F, X (Vn x) = Vn y) (hinvm : ∀ x : G, ∃ y : G, X (Vm x) = Vm y) (p : Polynomial ℂ) (v : E) (hv : Vn ((adjoint Vn) v) = v) : Vm ((Polynomial.aeval (compress Vm X) p) ((adjoint Vm) v)) = Vn ((Polynomial.aeval (compress Vn X) p) ((adjoint Vn) v))

Formalization note: Lean 4 identifier BookProof.ChapterH8.sirk_band_refinement_poly.

Preamble

-- Generated from ChapterH8.lean — theorem BookProof.ChapterH8.sirk_band_refinement_poly
import Definitions.Def_ChapterH5
import Definitions.Def_ChapterH6
import Mathlib
import Definitions.Def_ChapterH8
import Definitions.Def_ChapterH4
open BookProof.ChapterH4
open BookProof.ChapterH8


noncomputable section


open BookProof.ChapterH4 BookProof.ChapterH5 BookProof.ChapterH6
open ContinuousLinearMap

variable {K E : Type*} [Field K] [AddCommGroup E] [Module K E]
variable {E F G : Type*}
  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]

Formal statement

theorem BookProof.ChapterH8.sirk_band_refinement_poly (Vn : F →L[ℂ] E) (Vm : G →L[ℂ] E) (J : F →L[ℂ] G)
    (X : E →L[ℂ] E) (hJ : Vn = Vm.comp J)
    (hVm : (adjoint Vm).comp Vm = ContinuousLinearMap.id ℂ G)
    (hJJ : (adjoint J).comp J = ContinuousLinearMap.id ℂ F)
    (hinvn : ∀ x : F, ∃ y : F, X (Vn x) = Vn y)
    (hinvm : ∀ x : G, ∃ y : G, X (Vm x) = Vm y)
    (p : Polynomial ℂ) (v : E) (hv : Vn ((adjoint Vn) v) = v) :
    Vm ((Polynomial.aeval (compress Vm X) p) ((adjoint Vm) v))
      = Vn ((Polynomial.aeval (compress Vn X) p) ((adjoint Vn) v)) := by sorry

Source

https://github.com/leonardopedro/timepiece/blob/61595bc/BookProof/ChapterH8.lean

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