| import Mathlib | |
| import OAI.Geometry.Cannon.PlaneTopology.RegionEdges | |
| import OAI.Geometry.Cannon.PlaneTopology.VertexLines | |
| import OAI.Geometry.Cannon.PlaneTopology.OtherComponentDeleteArc | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| open Set unitInterval Topology | |
| noncomputable section | |
| noncomputable section | |
| noncomputable section | |
| open Set Topology unitInterval | |
| noncomputable section | |
| open Set Metric Topology unitInterval | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| noncomputable section | |
| open Set unitInterval | |
| noncomputable section | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology unitInterval | |
| noncomputable section | |
| namespace CannonPlanarEuler | |
| open Set Topology unitInterval CannonEmbeddedArc CannonEmbeddedArc.Arc CannonFaceIncidence | |
| variable {ι : Type*} [DecidableEq ι] | |
| def support (F : ι → Arc ℂ) (S : Finset ι) : Set ℂ := ⋃ i∈S,range (F i) | |
| def vertices (F : ι → Arc ℂ) (S : Finset ι) : Finset ℂ := by | |
| classical | |
| exact S.biUnion fun i => {F i 0,F i 1} | |
| def faces (F : ι → Arc ℂ) (S : Finset ι) : ℕ := | |
| Nat.card (ConnectedComponents ((support F S)ᶜ : Set ℂ)) | |
| lemma support_insert (F : ι → Arc ℂ) (S : Finset ι) (i : ι) : | |
| support F (insert i S)=support F S∪range (F i) := by | |
| simp only [support,Finset.mem_insert,iUnion_iUnion_eq_or_left,union_comm] | |
| lemma vertices_insert (F : ι → Arc ℂ) (S : Finset ι) (i : ι) : | |
| vertices F (insert i S)=insert (F i 0) (insert (F i 1) (vertices F S)) := by | |
| classical | |
| ext x | |
| simp only [vertices,Finset.biUnion_insert,Finset.mem_union,Finset.mem_insert,Finset.mem_singleton] | |
| tauto | |
| omit [DecidableEq ι] in | |
| lemma support_closed (F : ι → Arc ℂ) (S : Finset ι) : IsClosed (support F S) := | |
| isClosed_biUnion_finset fun i _ => (F i).compact_range.isClosed | |
| omit [DecidableEq ι] in | |
| lemma mem_vertices (F : ι → Arc ℂ) (S : Finset ι) (x : ℂ) : | |
| x∈vertices F S ↔ ∃ i∈S,F i 0=x ∨ F i 1=x := by | |
| classical | |
| simp [vertices,eq_comm] | |
| omit [DecidableEq ι] in | |
| lemma vertices_subset (F : ι → Arc ℂ) (S : Finset ι) : (vertices F S:Set ℂ)⊆support F S := by | |
| intro x hx | |
| obtain ⟨i,hi,hx⟩ := (mem_vertices F S x).mp hx | |
| exact mem_iUnion.mpr ⟨i,mem_iUnion.mpr ⟨hi,hx.elim (fun h => ⟨0,h⟩) (fun h => ⟨1,h⟩)⟩⟩ | |
| omit [DecidableEq ι] in | |
| lemma support_subtype (F : ι → Arc ℂ) (S : Finset ι) : | |
| support F S=⋃i:S,range (F i.val) := by ext x; simp [support] | |
| omit [DecidableEq ι] in | |
| lemma support_tame (F : ι → Arc ℂ) (S : Finset ι) (Q : Finset (ℂ × ℂ)) | |
| (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) : | |
| ∃ P : Finset (ℂ × ℂ),support F S=⋃p∈P,segment ℝ p.1 p.2 := by | |
| rw [support_subtype] | |
| exact finite_graph_exact_segments (fun i:S => F i.val) Q (fun i => hQ i.val) | |
| omit [DecidableEq ι] in | |
| lemma faces_finite (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) | |
| (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) : | |
| Finite (ConnectedComponents ((support F S)ᶜ : Set ℂ)) := by | |
| let _ : Nonempty S := hS.to_subtype | |
| rw [support_subtype] | |
| exact CannonEmbeddedArc.Arc.faces_finite (fun i:S => F i.val) | |
| (fun _ _ h => hF (fun he => h (Subtype.ext he))) Q (fun i => hQ i.val) | |
| omit [DecidableEq ι] in | |
| lemma edge_interior_avoids (F : ι → Arc ℂ) (S : Finset ι) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) : | |
| F i '' Ioo (0:I) 1⊆(support F S)ᶜ := by | |
| rintro x ⟨t,ht,rfl⟩ hx | |
| simp only [support,mem_iUnion] at hx | |
| obtain ⟨j,hj,s,hs⟩ := hx | |
| have hij : i≠j := fun h => hi (h ▸ hj) | |
| rcases (hF hij).1 t s hs.symm |>.1 with he|he | |
| · exact ht.1.ne' he | |
| · exact ht.2.ne he | |
| omit [DecidableEq ι] in | |
| lemma edge_end_mem_support (F : ι → Arc ℂ) (S : Finset ι) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) (t : I) : | |
| F i t∈support F S ↔ F i t∈vertices F S := by | |
| constructor | |
| · intro h | |
| simp only [support,mem_iUnion] at h | |
| obtain ⟨j,hj,s,hs⟩ := h | |
| refine (mem_vertices F S (F i t)).mpr ⟨j,hj,?_⟩ | |
| rcases (hF (show i≠j from fun h => hi (h ▸ hj))).1 t s hs.symm |>.2 with he|he | |
| · exact Or.inl (he ▸ hs) | |
| · exact Or.inr (he ▸ hs) | |
| · exact fun h => vertices_subset F S h | |
| omit [DecidableEq ι] in | |
| lemma edge_face (F : ι → Arc ℂ) (S : Finset ι) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) {i : ι} (hi : i∉S) : | |
| ∃ d : ConnectedComponents ((support F S)ᶜ : Set ℂ), | |
| F i '' Ioo (0:I) 1⊆region _ ConnectedComponents.mk d ∧ | |
| ∀ t:I, F i t∉support F S → F i t∈region _ ConnectedComponents.mk d := by | |
| let _ : LocallyConnectedSpace ((support F S)ᶜ : Set ℂ) := | |
| (support_closed F S).isOpen_compl.locallyConnectedSpace | |
| let _ : DiscreteTopology (ConnectedComponents ((support F S)ᶜ : Set ℂ)) := inferInstance | |
| have hc : IsConnected (F i '' Ioo (0:I) 1) := | |
| (isConnected_Ioo (show (0:I)<1 from zero_lt_one)).image _ (F i).continuous.continuousOn | |
| obtain ⟨d,hd⟩ := label_on_connected ConnectedComponents.continuous_coe hc (edge_interior_avoids F S hF hi) | |
| have hcl : range (F i)⊆closure (region _ ConnectedComponents.mk d) := | |
| range_in_closed_of_interior (F i) isClosed_closure (fun t ht₀ ht₁ => subset_closure (hd ⟨t,⟨ht₀,ht₁⟩,rfl⟩)) | |
| refine ⟨d,hd,fun t ht => ?_⟩ | |
| exact closure_region_inside (support_closed F S).isOpen_compl ConnectedComponents.continuous_coe | |
| (hcl (mem_range_self t)) ht | |
| lemma single_arc_faces (a : Arc ℂ) (Q : Finset (ℂ × ℂ)) | |
| (hQ : range a⊆⋃q∈Q,segment ℝ q.1 q.2) : | |
| Nat.card (ConnectedComponents ((range a)ᶜ : Set ℂ))=1 := by | |
| have hconn : IsPreconnected ({a 1}ᶜ : Set ℂ) := by | |
| convert CannonPlanarNeighborhood.open_preconnected_remove_point isOpen_univ isPreconnected_univ (a 1) using 1 | |
| ext x | |
| simp only [mem_compl_iff,mem_singleton_iff,mem_sdiff,mem_univ,true_and] | |
| have ha₀ : a 0∈({a 1}ᶜ : Set ℂ) := fun h => zero_ne_one (a.injective h) | |
| have hUa : a '' Ioo (0:I) 1⊆({a 1}ᶜ : Set ℂ) := by | |
| rintro x ⟨t,ht,rfl⟩ h | |
| exact ht.2.ne (a.injective h) | |
| have hc := tame_slit_connected a Q hQ isClosed_singleton.isOpen_compl hconn ha₀ hUa (by simp) | |
| have heq : ({a 1}ᶜ : Set ℂ)\range a=(range a)ᶜ := by | |
| ext z | |
| constructor | |
| · exact fun h => h.2 | |
| · exact fun h => ⟨fun he => h (he ▸ mem_range_self 1),h⟩ | |
| rw [heq] at hc | |
| exact connected_components_card_one hc | |
| lemma attached_edge_euler (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty) | |
| (hconn : IsPreconnected (support F S)) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) | |
| (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) | |
| {i : ι} (hi : i∉S) (hattach : (support F S∩range (F i)).Nonempty) : | |
| faces F (insert i S)+(vertices F (insert i S)).card= | |
| faces F S+(vertices F S).card+1 := by | |
| classical | |
| let _ := faces_finite F S hS hF Q hQ | |
| obtain ⟨d,hd,hdend⟩ := edge_face F S hF hi | |
| have hdist : F i 0≠F i 1 := fun h => zero_ne_one ((F i).injective h) | |
| have hatt : F i 0∈support F S ∨ F i 1∈support F S := by | |
| obtain ⟨x,hx,t,rfl⟩ := hattach | |
| by_cases ht₀ : t=0 | |
| · exact Or.inl (ht₀ ▸ hx) | |
| by_cases ht₁ : t=1 | |
| · exact Or.inr (ht₁ ▸ hx) | |
| exact (edge_interior_avoids F S hF hi ⟨t,⟨lt_of_le_of_ne (nonneg t) (Ne.symm ht₀), | |
| lt_of_le_of_ne (le_one t) ht₁⟩,rfl⟩ hx).elim | |
| by_cases h₀ : F i 0∈support F S | |
| · have hv₀ := (edge_end_mem_support F S hF hi 0).mp h₀ | |
| by_cases h₁ : F i 1∈support F S | |
| · have hv₁ := (edge_end_mem_support F S hF hi 1).mp h₁ | |
| obtain ⟨P,hP⟩ := support_tame F S Q hQ | |
| have hf : faces F (insert i S)=faces F S+1 := by | |
| unfold faces | |
| rw [support_insert] | |
| have aux : ∀ d : ConnectedComponents ((support F S)ᶜ : Set ℂ), | |
| F i '' Ioo (0:I) 1⊆region _ ConnectedComponents.mk d → | |
| F i 0∈support F S → F i 1∈support F S → | |
| Nat.card (ConnectedComponents ((support F S∪range (F i))ᶜ : Set ℂ))= | |
| Nat.card (ConnectedComponents ((support F S)ᶜ : Set ℂ))+1 := by | |
| rw [hP] | |
| let _ : Finite (ConnectedComponents ((⋃p∈P,segment ℝ p.1 p.2)ᶜ : Set ℂ)) := by | |
| rw [←hP] | |
| exact faces_finite F S hS hF Q hQ | |
| intro e he he₀ he₁ | |
| exact crosscut_face_count P Q (hP ▸ hconn) (F i) (hQ i) e he₀ he₁ he | |
| exact aux d hd h₀ h₁ | |
| rw [hf,vertices_insert,Finset.insert_eq_of_mem hv₁,Finset.insert_eq_of_mem hv₀] | |
| omega | |
| · have hv₁ : F i 1∉vertices F S := fun h => h₁ (vertices_subset F S h) | |
| have hf : faces F (insert i S)=faces F S := by | |
| unfold faces | |
| rw [support_insert,←reverse_range (F i)] | |
| apply leaf_face_count (support F S) (support_closed F S) (F i).reverse Q | |
| (by simpa only [reverse_range] using hQ i) d | |
| · simpa only [reverse_zero] using hdend 1 h₁ | |
| · simpa only [reverse_one] using h₀ | |
| · rintro x ⟨t,ht,rfl⟩ | |
| apply hd | |
| refine ⟨unitInterval.symm t,⟨?_,?_⟩,rfl⟩ | |
| · change (0:ℝ)<1-(t:ℝ) | |
| have hh : (t:ℝ)<1 := ht.2 | |
| linarith | |
| · change 1-(t:ℝ)<(1:ℝ) | |
| have hh : (0:ℝ)<t := ht.1 | |
| linarith | |
| rw [hf,vertices_insert,Finset.insert_eq_of_mem (Finset.mem_insert_of_mem hv₀),Finset.card_insert_of_notMem hv₁] | |
| omega | |
| · have h₁ := hatt.resolve_left h₀ | |
| have hv₀ : F i 0∉vertices F S := fun h => h₀ (vertices_subset F S h) | |
| have hv₁ := (edge_end_mem_support F S hF hi 1).mp h₁ | |
| have hf : faces F (insert i S)=faces F S := by | |
| unfold faces | |
| rw [support_insert] | |
| exact leaf_face_count _ (support_closed F S) (F i) Q (hQ i) d (hdend 0 h₀) h₁ hd | |
| rw [hf,vertices_insert,Finset.insert_eq_of_mem hv₁,Finset.card_insert_of_notMem hv₀] | |
| omega | |
| omit [DecidableEq ι] in | |
| lemma mem_support (F : ι → Arc ℂ) (S : Finset ι) (x : ℂ) : | |
| x∈support F S ↔ ∃ i∈S, x∈range (F i) := by simp only [support,mem_iUnion,exists_prop] | |
| omit [DecidableEq ι] in | |
| lemma support_mono (F : ι → Arc ℂ) {S T : Finset ι} (h : S⊆T) : support F S⊆support F T := by | |
| intro x hx | |
| obtain ⟨i,hi,hx⟩ := (mem_support F S x).mp hx | |
| exact (mem_support F T x).mpr ⟨i,h hi,hx⟩ | |
| omit [DecidableEq ι] in | |
| lemma support_nonempty (F : ι → Arc ℂ) (S : Finset ι) (hS : S.Nonempty) : (support F S).Nonempty := by | |
| obtain ⟨i,hi⟩ := hS | |
| exact ⟨F i 0,(mem_support ..).mpr ⟨i,hi,mem_range_self 0⟩⟩ | |
| lemma exists_attached_edge [Fintype ι] (F : ι → Arc ℂ) (S : Finset ι) | |
| (hconn : IsPreconnected (support F Finset.univ)) (hS : S.Nonempty) (hSne : S≠Finset.univ) : | |
| ∃ i∉S, (support F S∩range (F i)).Nonempty := by | |
| classical | |
| have hT : (Finset.univ\S).Nonempty := Finset.sdiff_nonempty.mpr (by simpa only [Finset.univ_subset_iff] using hSne) | |
| have hcover : support F Finset.univ⊆support F S∪support F (Finset.univ\S) := by | |
| intro x hx | |
| obtain ⟨i,_,hi⟩ := (mem_support F _ x).mp hx | |
| by_cases h : i∈S | |
| · exact Or.inl ((mem_support F S x).mpr ⟨i,h,hi⟩) | |
| · exact Or.inr ((mem_support F _ x).mpr ⟨i,by simp [h],hi⟩) | |
| obtain ⟨u,hu⟩ := support_nonempty F S hS | |
| obtain ⟨v,hv⟩ := support_nonempty F (Finset.univ\S) hT | |
| obtain ⟨x,_,hxS,hxT⟩ := isPreconnected_closed_iff.mp hconn | |
| (support F S) (support F (Finset.univ\S)) (support_closed F S) (support_closed F _) | |
| hcover ⟨u,support_mono F (Finset.subset_univ S) hu,hu⟩ | |
| ⟨v,support_mono F (Finset.subset_univ _) hv,hv⟩ | |
| obtain ⟨i,hi,hix⟩ := (mem_support F _ x).mp hxT | |
| exact ⟨i,(Finset.mem_sdiff.mp hi).2,x,hxS,hix⟩ | |
| theorem planar_euler [Fintype ι] [Nonempty ι] (F : ι → Arc ℂ) | |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) | |
| (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) | |
| (hconn : IsPreconnected (support F Finset.univ)) : | |
| faces F Finset.univ+(vertices F Finset.univ).card=Fintype.card ι+2 := by | |
| classical | |
| let good : Finset ι → Prop := fun S => S.Nonempty ∧ IsPreconnected (support F S) ∧ | |
| faces F S+(vertices F S).card=S.card+2 | |
| let A : Finset (Finset ι) := Finset.univ.powerset.filter good | |
| have hA (S : Finset ι) : S∈A ↔ good S := by simp [A] | |
| obtain ⟨i⟩ := ‹Nonempty ι› | |
| have hsupp : support F {i}=range (F i) := by | |
| simp only [support,Finset.mem_singleton,iUnion_iUnion_eq_left] | |
| have hv : (vertices F {i}).card=2 := by | |
| have hn : F i 0≠F i 1 := fun h => zero_ne_one ((F i).injective h) | |
| simp [vertices,hn] | |
| have hf : faces F {i}=1 := by | |
| unfold faces | |
| rw [hsupp] | |
| exact single_arc_faces (F i) Q (hQ i) | |
| have hg : good {i} := ⟨Finset.singleton_nonempty _,hsupp ▸ isPreconnected_range (F i).continuous,by simp [hf,hv]⟩ | |
| obtain ⟨S,hSA,hmax⟩ := A.exists_max_image Finset.card ⟨{i},(hA {i}).mpr hg⟩ | |
| obtain ⟨hSne,hSC,hSE⟩ := (hA S).mp hSA | |
| have hSuniv : S=Finset.univ := by | |
| by_contra hn | |
| obtain ⟨j,hj,hjS⟩ := exists_attached_edge F S hconn hSne hn | |
| have hnew : good (insert j S) := by | |
| refine ⟨Finset.insert_nonempty ..,?_,?_⟩ | |
| · rw [support_insert] | |
| exact hSC.union' hjS (isPreconnected_range (F j).continuous) | |
| · rw [attached_edge_euler F S hSne hSC hF Q hQ hj hjS,hSE,Finset.card_insert_of_notMem hj] | |
| have hm := hmax (insert j S) ((hA _).mpr hnew) | |
| rw [Finset.card_insert_of_notMem hj] at hm | |
| omega | |
| subst S | |
| simpa only [Finset.card_univ] using hSE | |
| end CannonPlanarEuler | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |