import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Sheaves.ChartModuleMap

namespace OAI

section
noncomputable section
                                        
section

namespace MaximalSeshadri.Geometry.BaseSections
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite

variable {K : Type} [CommRing K] {X : Scheme.{0}}

lemma congr_apply (k : K →+* Γ(X,⊤)) (M : X.Modules) {U V : X.Opens}
    (h : U = V) (m : Sections k M U) :
    congr k M h m = M.presheaf.map (eqToHom h.symm).op m := by
  subst V
  change m = M.presheaf.map (𝟙 (op U)) m
  exact congrArg (fun f : M.presheaf.obj (op U) ⟶ M.presheaf.obj (op U) => f m)
    (M.presheaf.map_id (op U)).symm

def nestedRestriction (M : X.Modules) {U W : X.Opens} (h : W ≤ U) :
    (M.restrict U.ι).restrict (X.homOfLE h) ≅ M.restrict W.ι :=
  ((Scheme.Modules.restrictFunctorComp (X.homOfLE h) U.ι).app M).symm ≪≫
    (Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M

lemma nestedBase (k : K →+* Γ(X,⊤)) {U W : X.Opens} (h : W ≤ U) :
    (X.homOfLE h).appTop.hom.comp (U.ι.appTop.hom.comp k) = W.ι.appTop.hom.comp k := by
  rw [← RingHom.comp_assoc, ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop,
    Scheme.homOfLE_ι]

lemma chartTop_nested (k : K →+* Γ(X,⊤)) (M : X.Modules)
    {U W : X.Opens} (h : W ≤ U)
    (m : Sections (U.ι.appTop.hom.comp k) (M.restrict U.ι) ⊤) :
    chartTop k M W (chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k)
      (X.homOfLE h) (nestedBase k h) (M.restrict U.ι) (nestedRestriction M h) m) =
        res k M h (chartTop k M U m) := by
  simp only [chartTop, LinearEquiv.trans_apply, restrictEquiv, congr_apply,
    chartMap, chartModuleMap, LinearMap.coe_comp,
    chartModuleEquiv, moduleIsoSections, openImmersionTop, restrictedOpenSections,
    nestedRestriction, Iso.trans_hom, Iso.symm_hom, Iso.app_hom]
  change (M.presheaf.map _ ≫ M.presheaf.map _ ≫ M.presheaf.map _ ≫
    M.presheaf.map _) m = (M.presheaf.map _ ≫ M.presheaf.map _) m
  simp only [← M.presheaf.map_comp]
  apply congrArg (fun morphism :
    M.presheaf.obj (op (U.ι ''ᵁ (⊤ : U.toScheme.Opens))) ⟶ M.presheaf.obj (op W) =>
      morphism m)
  exact (M.presheaf.map_comp _ _).symm.trans
    ((congrArg M.presheaf.map (Subsingleton.elim _ _)).trans
      (M.presheaf.map_comp _ _))
lemma chartMap_range_coherent (k : K →+* Γ(X,⊤)) (M : X.Modules)
    {U W : X.Opens} (h : W ≤ U) :
    (chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k) (X.homOfLE h)
      (nestedBase k h) (M.restrict U.ι) (nestedRestriction M h)).range.map
        (chartTop k M W).toLinearMap = (res k M h).range := by
  ext x
  constructor
  · rintro ⟨y,⟨z,rfl⟩,rfl⟩
    exact ⟨chartTop k M U z, (chartTop_nested k M h z).symm⟩
  · rintro ⟨z,rfl⟩
    refine ⟨_, ⟨(chartTop k M U).symm z,rfl⟩, ?_⟩
    exact (chartTop_nested k M h _).trans
      (congrArg (res k M h) ((chartTop k M U).apply_symm_apply z))

end
end MaximalSeshadri.Geometry.BaseSections
end


end
end

end OAI