| import Mathlib |
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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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| namespace CannonFaceIncidence |
| variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] |
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| def region (V : Set X) (π : V → D) (d : D) : Set X := |
| {x | ∃ h : x∈V, π ⟨x,h⟩=d} |
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| def incident (V : Set X) (π : V → D) (x : X) : Set D := |
| {d | x∈closure (region V π d)} |
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| omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] in |
| lemma region_subset (V : Set X) (π : V → D) (d : D) : region V π d⊆V := by |
| rintro x ⟨hx,_⟩ |
| exact hx |
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| lemma region_isOpen {V : Set X} (hV : IsOpen V) {π : V → D} (hπ : Continuous π) (d : D) : |
| IsOpen (region V π d) := by |
| have he : region V π d=Subtype.val '' (π ⁻¹' {d}) := by |
| ext x |
| simp only [region,mem_ofPred_eq,mem_image,mem_preimage,mem_singleton_iff] |
| constructor |
| · rintro ⟨hx,he⟩ |
| exact ⟨⟨x,hx⟩,he,rfl⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| exact ⟨x.property,hx⟩ |
| rw [he] |
| exact hV.isOpenMap_subtype_val _ (hπ.isOpen_preimage _ (isOpen_discrete _)) |
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| lemma label_on_connected {V S : Set X} {π : V → D} (hπ : Continuous π) |
| (hS : IsConnected S) (hSV : S⊆V) : ∃ d : D, S⊆region V π d := by |
| let _ : PreconnectedSpace S := isPreconnected_iff_preconnectedSpace.mp hS.isPreconnected |
| obtain ⟨x,hx⟩ := hS.nonempty |
| let f : S → D := fun y => π ⟨y.val,hSV y.property⟩ |
| have hf : Continuous f := hπ.comp (by fun_prop) |
| refine ⟨f ⟨x,hx⟩,fun y hy => ⟨hSV hy,?_⟩⟩ |
| exact (IsLocallyConstant.iff_continuous f).mpr hf |>.apply_eq_of_preconnectedSpace ⟨y,hy⟩ ⟨x,hx⟩ |
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| theorem sector_incidence {V W B : Set X} (hW : IsOpen W) (hBW : B⊆W) |
| {π : V → D} (hπ : Continuous π) {ι : Type*} [Finite ι] |
| (S : ι → Set X) (hS : ∀ i, IsConnected (S i)) |
| (hcover : W∩V=⋃ i, S i) (hclose : ∀ i, B⊆closure (S i)) : |
| ∃ labels : ι → D, (∀ i, S i⊆region V π (labels i)) ∧ |
| ∀ x∈B, incident V π x=range labels := by |
| classical |
| have hSV (i : ι) : S i⊆V := by |
| intro x hx |
| exact (hcover.symm ▸ mem_iUnion_of_mem i hx).2 |
| choose labels hlabels using fun i => label_on_connected hπ (hS i) (hSV i) |
| refine ⟨labels,hlabels,?_⟩ |
| intro x hx |
| ext d |
| constructor |
| · intro hd |
| by_contra hn |
| have hnone : W∩region V π d=∅ := by |
| apply eq_empty_iff_forall_notMem.mpr |
| rintro y ⟨hyW,hyR⟩ |
| have hyWV : y∈W∩V := ⟨hyW,region_subset V π d hyR⟩ |
| rw [hcover] at hyWV |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hyWV |
| obtain ⟨hy,he⟩ := hlabels i hi |
| obtain ⟨hy',he'⟩ := hyR |
| apply hn |
| exact ⟨i,he.symm.trans he'⟩ |
| have hm : (W∩region V π d).Nonempty := |
| (mem_closure_iff_nhds.mp hd) W (hW.mem_nhds (hBW hx)) |
| rw [hnone] at hm |
| exact not_nonempty_empty hm |
| · rintro ⟨i,rfl⟩ |
| exact closure_mono (hlabels i) (hclose i hx) |
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| omit [TopologicalSpace D] [DiscreteTopology D] in |
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| theorem incident_constant_on_preconnected {V A : Set X} {π : V → D} |
| (hA : IsPreconnected A) |
| (hloc : ∀ x∈A, ∃ W : Set X, IsOpen W ∧ x∈W ∧ |
| ∀ y∈A∩W, incident V π y=incident V π x) : |
| ∀ x∈A, ∀ y∈A, incident V π x=incident V π y := by |
| let _ : PreconnectedSpace A := isPreconnected_iff_preconnectedSpace.mp hA |
| let f : A → Set D := fun x => incident V π x.val |
| have hf : IsLocallyConstant f := by |
| apply (IsLocallyConstant.iff_eventually_eq f).mpr |
| intro x |
| obtain ⟨W,hW,hxW,heq⟩ := hloc x.val x.property |
| filter_upwards [(continuous_subtype_val.isOpen_preimage W hW).mem_nhds hxW] with y hy |
| exact heq y.val ⟨y.property,hy⟩ |
| intro x hx y hy |
| exact hf.apply_eq_of_preconnectedSpace ⟨x,hx⟩ ⟨y,hy⟩ |
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| end CannonFaceIncidence |
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| namespace CannonFaceIncidence |
| open Set Metric Topology |
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| def openBox (r : ℝ) : Set (ℝ × ℝ) := Ioo (-r) r ×ˢ Ioo (-r) r |
| def halfBox (r : ℝ) (b : Bool) : Set (ℝ × ℝ) := |
| Ioo (-r) r ×ˢ (if b then Ioo 0 r else Ioo (-r) 0) |
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| lemma openBox_eq_ball (r : ℝ) : openBox r=ball (0,0) r := by |
| rw [←ball_prod_same,Real.ball_eq_Ioo] |
| simp [openBox] |
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| lemma halfBox_connected {r : ℝ} (hr : 0<r) (b : Bool) : IsConnected (halfBox r b) := by |
| cases b |
| · exact (isConnected_Ioo (by linarith : -r<r)).prod (isConnected_Ioo (neg_neg_of_pos hr)) |
| · exact (isConnected_Ioo (by linarith : -r<r)).prod (isConnected_Ioo hr) |
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| lemma halfBox_closure {r : ℝ} (hr : 0<r) {p : ℝ × ℝ} |
| (hp : p∈openBox r) (hp₂ : p.2=0) (b : Bool) : p∈closure (halfBox r b) := by |
| have hrr : -r<r := by linarith |
| cases b <;> simp only [halfBox,Bool.false_eq_true,↓reduceIte,closure_prod_eq] |
| · rw [closure_Ioo hrr.ne,closure_Ioo (neg_neg_of_pos hr).ne] |
| exact ⟨⟨hp.1.1.le,hp.1.2.le⟩,by rw [hp₂]; exact ⟨(neg_neg_of_pos hr).le,le_rfl⟩⟩ |
| · rw [closure_Ioo hrr.ne,closure_Ioo hr.ne] |
| exact ⟨⟨hp.1.1.le,hp.1.2.le⟩,by rw [hp₂]; exact ⟨le_rfl,hr.le⟩⟩ |
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| variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] |
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| theorem flat_chart_incidence {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) : |
| ∃ U : Set X, IsOpen U ∧ H (0,0)∈U ∧ U⊆W ∧ |
| ∃ labels : Bool → D, ∀ x∈U∩Vᶜ, incident V π x=range labels := 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,_,hlabels⟩ := sector_incidence hU inter_subset_left hπ S hSc hcover hclose |
| exact ⟨U,hU,h₀U,hUW,labels,hlabels⟩ |
| end CannonFaceIncidence |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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