import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Geometry.CurveFinite

namespace OAI

section
noncomputable section
                                        
section

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Set

lemma curve_nonclosed_generic {X : Type*} [TopologicalSpace X] [T0Space X]
    [IrreducibleSpace X] (hd : topologicalKrullDim X ≤ 1) (x : X)
    (hx : ¬ IsClosed ({x} : Set X)) : IsGenericPoint x Set.univ := by
  by_contra h
  have hp : closure ({x} : Set X) ≠ Set.univ := h
  have hs := proper_irreducible_closed_subsingleton hd isClosed_closure
    isIrreducible_singleton.closure hp
  have he : closure ({x} : Set X) = {x} := by
    apply Set.Subset.antisymm
    · intro y hy
      exact hs hy (subset_closure (Set.mem_singleton x))
    · exact subset_closure
  exact hx (he ▸ isClosed_closure)

theorem proper_curve_nonconstant_isFinite {X Y : Scheme}
    [IsIntegral X] [IsNoetherian X] (f : X ⟶ Y) [IsProper f]
    (hd : topologicalKrullDim X ≤ 1) (hn : ∃ a b : X, f a ≠ f b) : IsFinite f := by
  let : LocallyQuasiFinite f := .of_finite_preimage_singleton f (by
    intro y
    by_cases hy : IsClosed ({y} : Set Y)
    · apply proper_closed_finite_of_dimension_one hd (hy.preimage f.continuous)
      intro he
      obtain ⟨a,b,hab⟩ := hn
      have ha : a ∈ f ⁻¹' ({y} : Set Y) := he ▸ Set.mem_univ a
      have hb : b ∈ f ⁻¹' ({y} : Set Y) := he ▸ Set.mem_univ b
      exact hab (ha.trans hb.symm)
    · apply Set.Subsingleton.finite
      intro a ha b hb
      apply (curve_nonclosed_generic hd a ?_).eq (curve_nonclosed_generic hd b ?_)
      · intro hc
        have hi := f.isClosedMap _ hc
        rw [Set.image_singleton, ha] at hi
        exact hy hi
      · intro hc
        have hi := f.isClosedMap _ hc
        rw [Set.image_singleton, hb] at hi
        exact hy hi)
  exact .of_isProper_of_locallyQuasiFinite f

end
end MaximalSeshadri.Geometry
end


end
end

end OAI