| import Mathlib |
| import OAI.Geometry.Cannon.Modulus.CarrierHomeomorph |
| import OAI.Geometry.Cannon.Boundary.ThreeMarksTwoMetrics |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonRelease |
| open Set Metric Topology CannonGraph CannonCarrierCharging CannonUniformization |
|
|
| variable {G : Type*} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
|
| theorem boundary_normalized_annular_carriers (φ : Boundary D ≃ₜ SphereTwo) : |
| let _ := visualMetricSpace D hδ |
| ∃ L c : ℝ,1 ≤ L ∧ 0 < c ∧ c ≤ 1 ∧ ∀ {Y κ : Type*},∀ [PseudoMetricSpace Y], |
| ∀ A : κ → Set (Boundary D),∀ K : κ → Set Y, |
| ∀ f : C(Boundary D,Y), |
| (∀ S : Set Y,f ⁻¹' S ⊆ |
| ⋃(d : {d : κ // (S∩K d).Nonempty}),A d.val) → |
| ∀ v : Fin 3 → Boundary D,∀ w : Fin 3 → Y,∀ s t : ℝ, |
| (∀ i j,i ≠ j → s ≤ dist (v i) (v j)) → |
| (∀ i j,i ≠ j → t ≤ dist (w i) (w j)) → |
| (∀ i,f (v i)=w i) → |
| ∀ (o a₀ b₀ : Boundary D) (r R : ℝ),0 < r → 0 < R → R ≤ s/2 → |
| dist o a₀ ≤ r → R ≤ dist o b₀ → |
| ∀ a b : ℝ,0 ≤ a → b ≤ t/2 → |
| b ≤ dist (f a₀) (f o) → dist (f b₀) (f o) ≤ a → |
| ∀ q∈Icc a b, |
| ∃ x∈⋃(d : {d : κ // (sphere (f o) q∩K d).Nonempty}),A d.val, |
| ∃ y∈⋃(d : {d : κ // (sphere (f o) q∩K d).Nonempty}),A d.val, |
| ∀ γ : Path x y,range γ∈annulusFamily o (L*r) (c*R) := by |
| let _ := visualMetricSpace D hδ |
| dsimp only |
| obtain ⟨L,c,hL,hc,hc1,hcross⟩:=boundary_annular_metric_carriers D hδ φ |
| refine ⟨L,c,hL,hc,hc1,?_⟩ |
| intro Y κ _ A K f hpre v w s t hv hw hmark o a₀ b₀ r R hr hR hRs ha₀ hb₀ |
| a b ha hb hfa hfb |
| obtain ⟨z,hz,hfz⟩:=CannonFiniteNormalization.normalized_outer_endpoint |
| f v w hv hw hmark hRs hb o |
| exact hcross A K f hpre o a₀ b₀ z r R hr hR ha₀ hb₀ hz |
| (f o) a b (by simpa using ha) hfa hfb hfz |
| end CannonRelease |
|
|
| end |
| end |
| end |
| end |
| section |
| noncomputable section |
| namespace CannonTileTopology |
| open Set Topology Metric |
| open scoped OnePoint |
|
|
| lemma regular_tiles_inter_frontier {X : Type*} [TopologicalSpace X] |
| {A B : Set X} (hA : closure (interior A)=A) (hB : closure (interior B)=B) |
| (hd : Disjoint (interior A) (interior B)) {x : X} (hxA : x∈A) (hxB : x∈B) : |
| x∈frontier A ∩ frontier B := by |
| have hda : Disjoint (interior A) B := hB ▸ hd.closure_right isOpen_interior |
| have hdb : Disjoint A (interior B) := hA ▸ hd.closure_left isOpen_interior |
| exact ⟨⟨subset_closure hxA,fun h=>Set.disjoint_left.mp hda h hxB⟩, |
| ⟨subset_closure hxB,fun h=>Set.disjoint_left.mp hdb hxA h⟩⟩ |
|
|
| lemma disk_center_tile_incidence {X κ ν : Type*} [TopologicalSpace X] |
| (K : κ → Set X) (V : κ → Set ν) (B : ν → Set X) (p : ν → X) |
| (hreg : ∀ d,closure (interior (K d))=K d) |
| (hd : Pairwise (fun d e=>Disjoint (interior (K d)) (interior (K e)))) |
| (hfront : ∀ d,frontier (K d)⊆⋃ j∈V d,B j) |
| (hsep : ∀ i j,i≠j → p i∉B j) |
| (hvertex : ∀ j,∃ d,j∈V d ∧ p j∈K d) : |
| ∀ j d,p j∈K d → j∈V d := by |
| intro j d hj |
| by_contra hn |
| obtain ⟨e,hje,hpe⟩:=hvertex j |
| have hde : d≠e:=by intro h; subst e; exact hn hje |
| have hf:=(regular_tiles_inter_frontier (hreg d) (hreg e) (hd hde) hj hpe).1 |
| obtain ⟨i,hi,hpi⟩:=mem_iUnion₂.mp (hfront d hf) |
| exact hsep j i (fun h=>hn (h ▸ hi)) hpi |
|
|
| lemma compact_trace {K : Set (OnePoint ℂ)} (hc : IsClosed K) (hpole : ∞∉K) : |
| IsCompact (((↑) : ℂ → OnePoint ℂ) ⁻¹' K) := by |
| have he : (↑) '' (((↑) : ℂ → OnePoint ℂ) ⁻¹' K)=K := by |
| apply image_preimage_eq_of_subset |
| intro x hx |
| exact OnePoint.ne_infty_iff_exists.mp (fun h=>hpole (h ▸ hx)) |
| have hc' : IsClosed (((↑) : ℂ → OnePoint ℂ) '' (((↑) : ℂ → OnePoint ℂ) ⁻¹' K)) := by rw [he]; exact hc |
| exact (OnePoint.isClosed_image_coe.mp hc').2 |
|
|
| lemma image_regular {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| (f : X ≃ₜ Y) {K : Set X} (hk : closure (interior K)=K) : |
| closure (interior (f '' K))=f '' K := by |
| rw [←f.image_interior,←f.image_closure,hk] |
|
|
| lemma trace_frontier (K : Set (OnePoint ℂ)) : |
| frontier (((↑) : ℂ → OnePoint ℂ) ⁻¹' K)= |
| ((↑) : ℂ → OnePoint ℂ) ⁻¹' frontier K := |
| (OnePoint.isOpenMap_coe.preimage_frontier_eq_frontier_preimage |
| OnePoint.continuous_coe K).symm |
|
|
| lemma trace_regular {K : Set (OnePoint ℂ)} (hk : closure (interior K)=K) : |
| closure (interior (((↑) : ℂ → OnePoint ℂ) ⁻¹' K))= |
| ((↑) : ℂ → OnePoint ℂ) ⁻¹' K := by |
| rw [←OnePoint.isOpenMap_coe.preimage_interior_eq_interior_preimage |
| OnePoint.continuous_coe, |
| ←OnePoint.isOpenMap_coe.preimage_closure_eq_closure_preimage |
| OnePoint.continuous_coe,hk] |
| end CannonTileTopology |
|
|
| namespace CannonTileTopology |
| open Set Metric Topology |
| open scoped OnePoint |
|
|
| lemma coe_image_interior (K : Set ℂ) : |
| interior (((↑) : ℂ → OnePoint ℂ) '' K)=((↑) : ℂ → OnePoint ℂ) '' interior K := by |
| apply Subset.antisymm |
| · intro x hx |
| obtain ⟨z,hz,rfl⟩:=interior_subset hx |
| have h:=OnePoint.isOpenMap_coe.preimage_interior_eq_interior_preimage OnePoint.continuous_coe |
| (((↑) : ℂ → OnePoint ℂ) '' K) |
| rw [Set.preimage_image_eq _ OnePoint.coe_injective] at h |
| exact ⟨z,h ▸ hx,rfl⟩ |
| · exact OnePoint.isOpenMap_coe.image_interior_subset _ |
|
|
| lemma coe_closure_interior {K : Set ℂ} (hc : IsCompact K) |
| (hr : closure (interior K)=K) : |
| closure (((↑) : ℂ → OnePoint ℂ) '' interior K)=((↑) : ℂ → OnePoint ℂ) '' K := by |
| apply Subset.antisymm |
| · exact closure_minimal (image_mono interior_subset) (hc.image OnePoint.continuous_coe).isClosed |
| · simpa only [hr] using (image_closure_subset_closure_image (OnePoint.continuous_coe (X := ℂ)) (s := interior K)) |
|
|
| lemma coe_regular {K : Set ℂ} (hc : IsCompact K) (hr : closure (interior K)=K) : |
| closure (interior (((↑) : ℂ → OnePoint ℂ) '' K))=((↑) : ℂ → OnePoint ℂ) '' K := by |
| rw [coe_image_interior,coe_closure_interior hc hr] |
|
|
| lemma coe_frontier {K : Set ℂ} (hc : IsCompact K) : |
| frontier (((↑) : ℂ → OnePoint ℂ) '' K)=((↑) : ℂ → OnePoint ℂ) '' frontier K := by |
| rw [frontier,frontier,(hc.image OnePoint.continuous_coe).isClosed.closure_eq, |
| hc.isClosed.closure_eq,coe_image_interior,Set.image_sdiff OnePoint.coe_injective] |
|
|
| lemma coe_interior_frontier {K : Set ℂ} (hc : IsCompact K) |
| (hr : closure (interior K)=K) : |
| frontier (((↑) : ℂ → OnePoint ℂ) '' interior K)=((↑) : ℂ → OnePoint ℂ) '' frontier K := by |
| rw [frontier,coe_closure_interior hc hr, |
| (OnePoint.isOpenMap_coe _ isOpen_interior).interior_eq, |
| frontier,hc.isClosed.closure_eq,Set.image_sdiff OnePoint.coe_injective] |
|
|
| lemma segment_two_closedBalls {a b : ℂ} {r s : ℝ} |
| (h : dist a b=r+s) : segment ℝ a b⊆closedBall a r∪closedBall b s := by |
| intro z hz |
| have he:=dist_add_dist_of_mem_segment hz |
| rw [h,dist_comm a z] at he |
| by_cases hza : dist z a≤r |
| · exact Or.inl hza |
| · exact Or.inr (show dist z b ≤ s by linarith) |
| end CannonTileTopology |
|
|
| end |
| end |
|
|
| end OAI |
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|
