import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Region

namespace OAI

section
section
section
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open Set unitInterval Topology

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open Set Topology unitInterval

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open Set Metric Topology unitInterval

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open Set Metric Topology

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open Set unitInterval

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noncomputable section
open Set Metric Topology

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open Set Metric Topology

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open Set Metric Topology

namespace CannonFaceIncidence
open Set Topology

variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]
variable {Y : Type*} [TopologicalSpace Y]
def chartLabel (e : Y → X) {V : Set X} (π : V → D) : (e ⁻¹' V) → D :=
  fun x => π ⟨e x.val,x.property⟩

omit [DiscreteTopology D] in
lemma chartLabel_continuous (e : Y → X) (he : Continuous e) {V : Set X}
    {π : V → D} (hπ : Continuous π) : Continuous (chartLabel e π) :=
  hπ.comp ((he.comp continuous_subtype_val).subtype_mk _)

omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] [TopologicalSpace Y] in
lemma region_chartLabel (e : Y → X) {V : Set X} (π : V → D) (d : D) :
    region _ (chartLabel e π) d=e ⁻¹' region V π d := by
  ext x
  rfl

omit [TopologicalSpace D] [DiscreteTopology D] in
lemma incident_chartLabel (e : Y → X) (he : IsOpenMap e) (hc : Continuous e)
    {V : Set X} (π : V → D) (x : Y) :
    incident _ (chartLabel e π) x=incident V π (e x) := by
  ext d
  change x∈closure (region _ (chartLabel e π) d) ↔ e x∈closure (region V π d)
  rw [region_chartLabel]
  rw [← he.preimage_closure_eq_closure_preimage hc (region V π d)]
  rfl

omit [TopologicalSpace Y] in
lemma chartLabel_surjective (e : Y → X) (he : DenseRange e) {V : Set X}
    (hV : IsOpen V) {π : V → D} (hπ : Continuous π) (hsur : Function.Surjective π) :
    Function.Surjective (chartLabel e π) := by
  intro d
  obtain ⟨x,hx⟩ := hsur d
  obtain ⟨y,hy,hyd⟩ := he.exists_mem_open (region_isOpen hV hπ d) ⟨x.val,x.property,hx⟩
  exact ⟨⟨y,hy⟩,hyd⟩
end CannonFaceIncidence

namespace CannonFaceIncidence
open Set Topology

variable {X Y D : Type*} [TopologicalSpace X] [TopologicalSpace Y]
  [TopologicalSpace D] [DiscreteTopology D]

def labelPull (e : Y → X) {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V)
    (π : V → D) : W → D := chartLabel e π ∘ (Homeomorph.setCongr hW)

omit [DiscreteTopology D] in
lemma labelPull_continuous (e : Y → X) (he : Continuous e)
    {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) {π : V → D} (hπ : Continuous π) :
    Continuous (labelPull e hW π) :=
  (chartLabel_continuous e he hπ).comp (Homeomorph.setCongr hW).continuous

omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] in
lemma region_labelPull (e : Y → X) {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V)
    (π : V → D) (d : D) : region W (labelPull e hW π) d=e ⁻¹' region V π d := by
  subst W
  exact region_chartLabel e π d

omit [TopologicalSpace D] [DiscreteTopology D] in
lemma incident_labelPull (e : Y → X) (he : IsOpenMap e) (hc : Continuous e)
    {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) (π : V → D) (x : Y) :
    incident W (labelPull e hW π) x=incident V π (e x) := by
  ext d
  change x∈closure (region W (labelPull e hW π) d) ↔ e x∈closure (region V π d)
  rw [region_labelPull, ← he.preimage_closure_eq_closure_preimage hc (region V π d)]
  rfl

lemma labelPull_surjective (e : Y → X) (he : DenseRange e)
    {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) (hV : IsOpen V)
    {π : V → D} (hπ : Continuous π) (hsur : Function.Surjective π) :
    Function.Surjective (labelPull e hW π) :=
  (chartLabel_surjective e he hV hπ hsur).comp (Homeomorph.setCongr hW).surjective
end CannonFaceIncidence

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI