| import Mathlib |
|
|
| 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 |
|
|
| namespace CannonFiniteFaces |
|
|
| theorem finite_components_of_dense_regions {X : Type*} [TopologicalSpace X] |
| [LocallyConnectedSpace X] {V : Set X} (hV : IsOpen V) |
| {ι : Type*} [Finite ι] (S : ι → Set X) |
| (hS : ∀ i, S i⊆V) (hc : ∀ i, IsPreconnected (S i)) |
| (hd : Dense (⋃ i, S i)) : Finite (ConnectedComponents V) := by |
| classical |
| let _ : LocallyConnectedSpace V := hV.locallyConnectedSpace |
| let f : V → ConnectedComponents V := ConnectedComponents.mk |
| have hf : Continuous f := ConnectedComponents.continuous_coe |
| let U : ι → Set V := fun i => Subtype.val ⁻¹' S i |
| have hUi (i : ι) : IsPreconnected (U i) := by |
| apply IsInducing.subtypeVal.isPreconnected_image.mp |
| have he : Subtype.val '' U i=S i := by |
| ext x |
| constructor |
| · rintro ⟨y,hy,rfl⟩ |
| exact hy |
| · intro hx |
| exact ⟨⟨x,hS i hx⟩,hx,rfl⟩ |
| rw [he] |
| exact hc i |
| have hfi (i : ι) : (f '' U i).Finite := |
| ((hUi i).image f hf.continuousOn).subsingleton.finite |
| have hfin : (f '' (⋃ i, U i)).Finite := by |
| rw [image_iUnion] |
| exact finite_iUnion hfi |
| have hden : Dense (⋃ i, U i) := by |
| have hh := hd.preimage hV.isOpenMap_subtype_val |
| simpa only [preimage_iUnion] using hh |
| have hdenf : Dense (f '' (⋃ i, U i)) := |
| ConnectedComponents.surjective_coe.denseRange.dense_image hf hden |
| have he : f '' (⋃ i, U i)=univ := hfin.isClosed.closure_eq.symm.trans hdenf.closure_eq |
| exact finite_univ_iff.mp (he ▸ hfin) |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| omit [CompleteSpace E] in |
| lemma nonzero_functional_surjective (f : E →L[ℝ] ℝ) (hf : f≠0) : |
| Function.Surjective f := by |
| obtain ⟨v,hv⟩ : ∃ v, f v≠0 := by |
| by_contra hn |
| push Not at hn |
| apply hf |
| ext v |
| exact hn v |
| intro y |
| refine ⟨(y/f v) • v,?_⟩ |
| simp only [map_smul,smul_eq_mul] |
| exact div_mul_cancel₀ y hv |
|
|
| theorem hyperplane_chambers {ι : Type*} [Finite ι] |
| (f : ι → E →L[ℝ] ℝ) (c : ι → ℝ) (hf : ∀ i, f i≠0) : |
| ∃ S : (ι → Bool) → Set E, |
| (∀ s, Convex ℝ (S s)) ∧ Dense (⋃ s, S s) ∧ |
| (∀ s, S s⊆{x | ∀ i, f i x≠c i}) := by |
| classical |
| let S : (ι → Bool) → Set E := fun s => ⋂ i, |
| if s i=true then {x | c i<f i x} else {x | f i x<c i} |
| have hconv (s : ι → Bool) : Convex ℝ (S s) := by |
| apply convex_iInter |
| intro i |
| split_ifs |
| · exact (convex_Ioi (c i)).linear_preimage (f i).toLinearMap |
| · exact (convex_Iio (c i)).linear_preimage (f i).toLinearMap |
| have heq : (⋃ s, S s)={x | ∀ i, f i x≠c i} := by |
| ext x |
| constructor |
| · intro hx i |
| obtain ⟨s,hs⟩ := mem_iUnion.mp hx |
| have hh := mem_iInter.mp hs i |
| split_ifs at hh with h |
| · exact ne_of_gt hh |
| · exact ne_of_lt hh |
| · intro hx |
| refine mem_iUnion.mpr ⟨fun i => decide (c i<f i x),?_⟩ |
| apply mem_iInter.mpr |
| intro i |
| dsimp |
| split_ifs with hi |
| · exact of_decide_eq_true hi |
| · have hh : ¬c i<f i x := by simpa using hi |
| exact lt_of_le_of_ne (le_of_not_gt hh) (hx i) |
| have hopen (i : ι) : IsOpen {x | f i x≠c i} := |
| isOpen_compl_singleton.preimage (f i).continuous |
| have hdense (i : ι) : Dense {x | f i x≠c i} := |
| (dense_compl_singleton (c i)).preimage |
| ((f i).isOpenMap (nonzero_functional_surjective (f i) (hf i))) |
| have hden : Dense {x | ∀ i, f i x≠c i} := by |
| have hh := dense_iInter_of_isOpen hopen hdense |
| simpa only [mem_iInter,ofPred_forall] using hh |
| refine ⟨S,hconv,heq ▸ hden,?_⟩ |
| intro s |
| rw [←heq] |
| exact subset_iUnion S s |
|
|
| theorem finite_components_hyperplane_support {K : Set E} (hK : IsClosed K) |
| {ι : Type*} [Finite ι] (f : ι → E →L[ℝ] ℝ) (c : ι → ℝ) |
| (hf : ∀ i, f i≠0) (hs : K⊆⋃ i, {x | f i x=c i}) : |
| Finite (ConnectedComponents (Kᶜ : Set E)) := by |
| obtain ⟨S,hconv,hden,havoid⟩ := hyperplane_chambers f c hf |
| apply finite_components_of_dense_regions hK.isOpen_compl S _ |
| (fun s => (hconv s).isPreconnected) hden |
| intro s x hx hxK |
| obtain ⟨i,hi⟩ := mem_iUnion.mp (hs hxK) |
| exact havoid s hx i hi |
|
|
| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
|
|
| theorem segment_supporting_line (a b : Plane) : |
| ∃ f : Plane →L[ℝ] ℝ, f≠0 ∧ segment ℝ a b⊆{x | f x=f a} := by |
| classical |
| let x₀ : Plane →L[ℝ] ℝ := EuclideanSpace.proj 0 |
| let x₁ : Plane →L[ℝ] ℝ := EuclideanSpace.proj 1 |
| by_cases hd : b 0=a 0 |
| · refine ⟨x₀,?_,?_⟩ |
| · intro h |
| have hh := congrArg (fun f : Plane →L[ℝ] ℝ => f (EuclideanSpace.single 0 1)) h |
| norm_num [x₀] at hh |
| · have hcv : Convex ℝ {x | x₀ x=x₀ a} := |
| (convex_singleton (x₀ a)).linear_preimage x₀.toLinearMap |
| exact hcv.segment_subset rfl hd |
| · let f := x₁-((b 1-a 1)/(b 0-a 0)) • x₀ |
| refine ⟨f,?_,?_⟩ |
| · intro h |
| have hh := congrArg (fun f : Plane →L[ℝ] ℝ => f (EuclideanSpace.single 1 1)) h |
| norm_num [f,x₀,x₁,PiLp.single_apply] at hh |
| · have hab : f b=f a := by |
| change b 1-((b 1-a 1)/(b 0-a 0))*b 0=a 1-((b 1-a 1)/(b 0-a 0))*a 0 |
| have hn : b 0-a 0≠0 := sub_ne_zero.mpr hd |
| field_simp [hn] |
| ring |
| have hcv : Convex ℝ {x | f x=f a} := |
| (convex_singleton (f a)).linear_preimage f.toLinearMap |
| exact hcv.segment_subset rfl hab |
|
|
| theorem finite_components_segment_support {K : Set Plane} (hK : IsClosed K) |
| (P : Finset (Plane×Plane)) (hs : K⊆⋃ p∈P, segment ℝ p.1 p.2) : |
| Finite (ConnectedComponents (Kᶜ : Set Plane)) := by |
| classical |
| choose f hf hseg using fun p : P => segment_supporting_line p.val.1 p.val.2 |
| apply finite_components_hyperplane_support hK f (fun p => f p p.val.1) hf |
| intro x hx |
| obtain ⟨p,hp,hxp⟩ := mem_iUnion₂.mp (hs hx) |
| exact mem_iUnion.mpr ⟨⟨p,hp⟩,hseg ⟨p,hp⟩ hxp⟩ |
|
|
| theorem finite_components_chart_support {X : Type*} [TopologicalSpace X] |
| [T2Space X] [PreconnectedSpace X] [LocallyConnectedSpace X] |
| (e : Plane → X) (he : IsOpenEmbedding e) (p : X) (heR : range e={p}ᶜ) |
| {K : Set X} (hK : IsClosed K) (P : Finset (Plane×Plane)) |
| (hs : K⊆e '' (⋃ q∈P, segment ℝ q.1 q.2)) : |
| Finite (ConnectedComponents (Kᶜ : Set X)) := by |
| classical |
| choose f hf hseg using fun q : P => segment_supporting_line q.val.1 q.val.2 |
| obtain ⟨S,hconv,hden,havoid⟩ := hyperplane_chambers f (fun q => f q q.val.1) hf |
| have hsub (s : P → Bool) : e '' S s⊆Kᶜ := by |
| rintro _ ⟨x,hx,rfl⟩ hxK |
| obtain ⟨y,hy,hexy⟩ := hs hxK |
| have hey : y=x := he.injective hexy |
| subst y |
| obtain ⟨q,hq,hxq⟩ := mem_iUnion₂.mp hy |
| exact havoid s hx ⟨q,hq⟩ (hseg ⟨q,hq⟩ hxq) |
| have hde : DenseRange e := by |
| change Dense (range e) |
| rw [heR,dense_compl_singleton_iff_not_open] |
| intro hopen |
| have hall : ({p} : Set X)=univ := |
| (show IsClopen ({p} : Set X) from ⟨isClosed_singleton,hopen⟩).eq_univ |
| (singleton_nonempty p) |
| have hep : e 0=p := mem_singleton_iff.mp (hall.symm ▸ mem_univ (e 0)) |
| have hn : e 0∈({p}ᶜ : Set X) := heR ▸ mem_range_self 0 |
| exact hn (mem_singleton_iff.mpr hep) |
| apply finite_components_of_dense_regions hK.isOpen_compl (fun s => e '' S s) |
| hsub (fun s => (hconv s).isPreconnected.image e he.continuous.continuousOn) |
| rw [←image_iUnion] |
| exact hde.dense_image he.continuous hden |
|
|
| end CannonFiniteFaces |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
