An affine cover by twisted sections

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.SectionOpens MaximalSeshadri.Projective

variable {X : Scheme.{0}}

theorem LineBundle.twist_affine_cover [IsIntegral X] [CompactSpace X]
    (P J : LineBundle X) {κ : Type} (t : κ → (O X ⟶ P.sheaf))
    (ht : (⨆ i,isoOpen (t i)) = ⊤)
    (hlocal : ∀ i, ∃ (α : Type)
      (w : α → (O (isoOpen (t i)).toScheme ⟶ J.sheaf.restrict (isoOpen (t i)).ι)),
        (⨆ a,isoOpen (w a)) = ⊤ ∧ ∀ a, IsAffineOpen (isoOpen (w a))) :
    ∃ n : ℕ, 0 < n ∧ ∃ (σ : Type) (_ : Fintype σ)
      (s : σ → (O X ⟶ ((P.pow n).tensor J).sheaf)),
        (⨆ a,isoOpen (s a)) = ⊤ ∧
        (∀ a, IsAffineOpen (isoOpen (s a))) ∧
        (∀ a, (isoOpen (s a) : Set X).Nonempty) := by
  sorry

end
end MaximalSeshadri.Geometry
end
end OAI