| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Region |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| noncomputable section |
| open Set unitInterval |
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| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology unitInterval |
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| namespace CannonFaceIncidence |
| open Set Topology Metric |
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| variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] |
| theorem flat_chart_boundary_patch {V W : Set X} {π : V → D} (hπ : Continuous π) |
| (H : (ℝ × ℝ) ≃ₜ X) (hW : IsOpen W) (h₀ : H (0,0)∈W) |
| (hflat : ∀ z∈W, z∈V ↔ (H.symm z).2≠0) |
| {d : D} (hd : d∈incident V π (H (0,0))) : |
| ∃ U B : Set X, IsOpen U ∧ H (0,0)∈U ∧ U⊆W ∧ |
| IsConnected B ∧ B⊆W∩region V π d ∧ U∩Vᶜ⊆closure B := by |
| obtain ⟨r,hr,hball⟩ := Metric.mem_nhds_iff.mp |
| ((hW.preimage H.continuous).mem_nhds h₀) |
| let C := openBox r |
| let U := H '' C |
| have hCU : C⊆H ⁻¹' W := by rw [show C=openBox r from rfl,openBox_eq_ball]; exact hball |
| have hUW : U⊆W := by rintro _ ⟨p,hp,rfl⟩; exact hCU hp |
| have hU : IsOpen U := H.isOpenMap C (by rw [show C=openBox r from rfl,openBox_eq_ball]; exact isOpen_ball) |
| have h₀U : H (0,0)∈U := ⟨(0,0),by rw [show C=openBox r from rfl,openBox_eq_ball]; exact mem_ball_self hr,rfl⟩ |
| let S : Bool → Set X := fun b => H '' halfBox r b |
| have hSc (b : Bool) : IsConnected (S b) := |
| (halfBox_connected hr b).image H H.continuous.continuousOn |
| have hcover : U∩V=⋃ b, S b := by |
| ext x |
| constructor |
| · rintro ⟨⟨p,hp,rfl⟩,hpV⟩ |
| have hp₂ : p.2≠0 := by simpa using (hflat (H p) (hCU hp)).mp hpV |
| rcases lt_or_gt_of_ne hp₂ with hl|hr' |
| · exact mem_iUnion.mpr ⟨false,⟨p,⟨hp.1,hp.2.1,hl⟩,rfl⟩⟩ |
| · exact mem_iUnion.mpr ⟨true,⟨p,⟨hp.1,hr',hp.2.2⟩,rfl⟩⟩ |
| · intro hx |
| obtain ⟨b,p,hp,rfl⟩ := mem_iUnion.mp hx |
| have hpC : p∈C := by |
| cases b |
| · exact ⟨hp.1,hp.2.1,hp.2.2.trans hr⟩ |
| · exact ⟨hp.1,(neg_neg_of_pos hr).trans hp.2.1,hp.2.2⟩ |
| refine ⟨⟨p,hpC,rfl⟩,(hflat (H p) (hCU hpC)).mpr ?_⟩ |
| simp only [H.symm_apply_apply] |
| cases b |
| · exact hp.2.2.ne |
| · exact hp.2.1.ne' |
| have hclose (b : Bool) : U∩Vᶜ⊆closure (S b) := by |
| rintro x ⟨⟨p,hp,hpx⟩,hxV⟩ |
| subst x |
| have hp₂ : p.2=0 := by |
| by_contra h |
| apply hxV |
| apply (hflat (H p) (hCU hp)).mpr |
| simpa using h |
| rw [show S b=H '' halfBox r b from rfl,←H.image_closure] |
| exact ⟨p,halfBox_closure hr hp hp₂ b,rfl⟩ |
| obtain ⟨labels,hsub,hlabels⟩ := sector_incidence hU inter_subset_left hπ S hSc hcover hclose |
| have hnv : H (0,0)∉V := by |
| intro h |
| have hh := (hflat _ h₀).mp h |
| simp at hh |
| have hh := hlabels (H (0,0)) ⟨h₀U,hnv⟩ |
| obtain ⟨b,hb⟩ := hh ▸ hd |
| refine ⟨U,S b,hU,h₀U,hUW,hSc b,?_,hclose b⟩ |
| intro z hz |
| refine ⟨hUW ?_,hb ▸ hsub b hz⟩ |
| exact (hcover.symm ▸ mem_iUnion_of_mem b hz).1 |
| end CannonFaceIncidence |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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