| import Mathlib |
| import OAI.Geometry.Cannon.Modulus.BoundaryNormalizedAnnularCarriers |
| import OAI.Geometry.Cannon.Conformal.FinitePackingVertex |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Metric MeasureTheory |
| open scoped BigOperators OnePoint |
|
|
| namespace CannonRadialEnergy |
|
|
| lemma local_subdisk (c p : ℂ) {r R K : ℝ} (hr : 0 ≤ r) (hR : 0 ≤ R) (hK : 0 ≤ K) |
| (hnear : dist c p ≤ r+K*R) : |
| ∃ q : ℂ,ball q (min r R)⊆ball c r ∧ |
| ball q (min r R)⊆ball p ((K+2)*R) := by |
| let s:=min r R |
| have hs : 0 ≤ s:=le_min hr hR |
| have hsr : s ≤ r:=min_le_left _ _ |
| have hsR : s ≤ R:=min_le_right _ _ |
| have hn : ¬Disjoint (closedBall c (r-s)) (closedBall p (K*R+s)) := by |
| rw [disjoint_closedBall_closedBall_iff (sub_nonneg.mpr hsr) (by positivity)] |
| linarith |
| obtain ⟨q,hqc,hqp⟩:=Set.not_disjoint_iff_nonempty_inter.mp hn |
| refine ⟨q,?_,?_⟩ |
| · intro x hx |
| have ht:=dist_triangle x q c |
| have hx':=mem_ball.mp hx |
| have hq':=mem_closedBall.mp hqc |
| apply mem_ball.mpr |
| linarith |
| · intro x hx |
| have ht:=dist_triangle x q p |
| have hx':=mem_ball.mp hx |
| have hq':=mem_closedBall.mp hqp |
| apply mem_ball.mpr |
| nlinarith |
|
|
| theorem local_disjoint_disk_square_budget {ι : Type*} [Fintype ι] |
| (c : ι → ℂ) (r : ι → ℝ) (p : ℂ) {R K : ℝ} |
| (hr : ∀ i,0 ≤ r i) (hR : 0 ≤ R) (hK : 0 ≤ K) |
| (hdisj : Pairwise (fun i j=> Disjoint (ball (c i) (r i)) (ball (c j) (r j)))) |
| (hnear : ∀ i,dist (c i) p ≤ r i+K*R) : |
| (∑ i,(min (r i) R)^2) ≤ (K+2)^2*R^2 := by |
| choose q hqc hqp using fun i=> local_subdisk (c i) p (hr i) hR hK (hnear i) |
| have h:=disjoint_disk_square_budget q (fun i=> min (r i) R) p |
| (fun i=> le_min (hr i) hR) (by positivity) |
| (fun i j hij=> (hdisj hij).mono (hqc i) (hqc j)) hqp |
| convert h using 1 |
| ring |
|
|
| theorem active_local_disk_square_budget {ι : Type*} [Fintype ι] |
| (c : ι → ℂ) (r : ι → ℝ) (p : ℂ) (active : ι → Prop) [DecidablePred active] |
| {R K : ℝ} (hr : ∀ i,0 ≤ r i) (hR : 0 ≤ R) (hK : 0 ≤ K) |
| (hdisj : Pairwise (fun i j=> Disjoint (ball (c i) (r i)) (ball (c j) (r j)))) |
| (hnear : ∀ i,active i → dist (c i) p ≤ r i+K*R) : |
| (∑ i,if active i then (min (r i) R)^2 else 0) ≤ (K+2)^2*R^2 := by |
| have h:=local_disjoint_disk_square_budget (fun i : {i // active i}=> c i) |
| (fun i=> r i) p (fun i=> hr i) hR hK |
| (fun i j hij=> hdisj (Subtype.val_injective.ne hij)) (fun i=> hnear i i.property) |
| have he : (∑ i ∈ Finset.univ.filter active,(min (r i) R)^2)= |
| ∑ i : {i // active i},(min (r i) R)^2 := |
| Finset.sum_subtype _ (by simp) _ |
| rw [←he,Finset.sum_filter] at h |
| exact h |
|
|
| end CannonRadialEnergy |
|
|
| namespace CannonLocalPacking |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma compact_subset_closedBall_of_frontier_subset {K : Set ℂ} (hK : IsCompact K) |
| (c : ℂ) (R : ℝ) (hfront : frontier K ⊆ closedBall c R) : |
| K ⊆ closedBall c R := by |
| intro z hz |
| have hd : DiffContOnCl ℂ (fun w : ℂ=>w-c) K := |
| (differentiable_id.sub_const c).diffContOnCl |
| have h:=Complex.norm_le_of_forall_mem_frontier_norm_le hK.isBounded hd |
| (fun w hw=>by simpa only [mem_closedBall,dist_eq_norm] using hfront hw) |
| (subset_closure hz) |
| simpa only [mem_closedBall,dist_eq_norm] using h |
|
|
| lemma tile_ball_dichotomy {K : Set ℂ} (hK : IsClosed K) (p : ℂ) (R : ℝ) |
| (hmeet : (K∩closedBall p R).Nonempty) : |
| (frontier K∩closedBall p R).Nonempty ∨ closedBall p R ⊆ interior K := by |
| by_cases hf : (frontier K∩closedBall p R).Nonempty |
| · exact Or.inl hf |
| right |
| have hsub : closedBall p R ⊆ interior K∪Kᶜ := by |
| intro x hx |
| by_cases hxK : x∈K |
| · left |
| by_contra hn |
| exact hf ⟨x,⟨subset_closure hxK,hn⟩,hx⟩ |
| · exact Or.inr hxK |
| obtain hi|ho:=(convex_closedBall p R).isPreconnected.subset_or_subset |
| isOpen_interior hK.isOpen_compl |
| (disjoint_compl_right.mono interior_subset Subset.rfl) hsub |
| · exact hi |
| · obtain ⟨x,hxK,hxp⟩:=hmeet |
| exact (ho hxp hxK).elim |
|
|
| lemma tile_max_disk_control {ν : Type*} (c : ν → ℂ) (r : ν → ℝ) |
| (V : Set ν) (s : ν) {K : Set ℂ} (hK : IsCompact K) |
| (hmax : ∀ j∈V,r j ≤ r s) |
| (hpair : ∀ j∈V,dist (c j) (c s) ≤ r j+r s) |
| (hfront : ∀ x∈frontier K,∃ j∈V,x∈closedBall (c j) (r j)) : |
| K⊆closedBall (c s) (3*r s) := by |
| apply compact_subset_closedBall_of_frontier_subset hK |
| intro x hx |
| obtain ⟨j,hj,hxj⟩:=hfront x hx |
| have ht:=dist_triangle x (c j) (c s) |
| have hm:=hmax j hj |
| have hp:=hpair j hj |
| have hxj':=mem_closedBall.mp hxj |
| apply mem_closedBall.mpr |
| linarith |
|
|
| lemma frontier_disk_charge {ν : Type*} (c : ν → ℂ) (r : ν → ℝ) |
| (V : Set ν) (s : ν) (hs : s∈V) (p : ℂ) {R : ℝ} |
| (hpair : ∀ j∈V,dist (c j) (c s) ≤ r j+r s) |
| {K : Set ℂ} (hfront : ∀ x∈frontier K,∃ j∈V,x∈closedBall (c j) (r j)) |
| (hmeet : (frontier K∩closedBall p R).Nonempty) : |
| ∃ j∈V,dist (c j) p ≤ r j+3*R ∧ min (r s) R ≤ min (r j) R := by |
| obtain ⟨x,hxK,hxp⟩:=hmeet |
| obtain ⟨j,hj,hxj⟩:=hfront x hxK |
| have hxp':=mem_closedBall.mp hxp |
| have hxj':=mem_closedBall.mp hxj |
| by_cases hjR : r j ≤ R |
| · refine ⟨s,hs,?_,le_rfl⟩ |
| have ht:=dist_triangle (c s) (c j) p |
| have ht':=dist_triangle (c j) x p |
| have hp:=hpair j hj |
| rw [dist_comm (c j) (c s)] at hp |
| rw [dist_comm (c j) x] at ht' |
| linarith |
| · refine ⟨j,hj,?_,?_⟩ |
| · have ht:=dist_triangle (c j) x p |
| rw [dist_comm (c j) x] at ht |
| have hR : 0 ≤ R:=(dist_nonneg.trans hxp') |
| linarith |
| · rw [min_eq_right (le_of_not_ge hjR)] |
| exact min_le_right _ _ |
|
|
| lemma clipped_tile_control {ν : Type*} (c : ν → ℂ) (r : ν → ℝ) |
| (V : Set ν) (s : ν) {K : Set ℂ} (hK : IsCompact K) |
| (hmax : ∀ j∈V,r j ≤ r s) |
| (hpair : ∀ j∈V,dist (c j) (c s) ≤ r j+r s) |
| (hfront : ∀ x∈frontier K,∃ j∈V,x∈closedBall (c j) (r j)) |
| (p : ℂ) {R : ℝ} (hR : 0 ≤ R) : |
| K∩closedBall p R⊆closedBall (if r s ≤ R then c s else p) (3*min (r s) R) := by |
| by_cases hr : r s ≤ R |
| · rw [ite_eq_left hr,min_eq_left hr] |
| exact inter_subset_left.trans (tile_max_disk_control c r V s hK hmax hpair hfront) |
| · rw [ite_eq_right hr,min_eq_right (le_of_not_ge hr)] |
| exact inter_subset_right.trans (closedBall_subset_closedBall (by linarith)) |
|
|
| end CannonLocalPacking |
|
|
| namespace CannonLocalPacking |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma sum_indicator_le_one {ι : Type*} [Fintype ι] (P : ι → Prop) [DecidablePred P] |
| {A : ℝ} (hA : 0 ≤ A) (hP : ∀ i j,P i → P j → i=j) : |
| (∑ i,if P i then A else 0) ≤ A := by |
| classical |
| by_cases h : ∃ i,P i |
| · obtain ⟨i,hi⟩:=h |
| have he : (∑ j,if P j then A else 0)=A := by |
| rw [Finset.sum_eq_single i] |
| · simp [hi] |
| · intro j _ hji |
| have hj : ¬P j:=fun hj=>hji (hP j i hj hi) |
| simp [hj] |
| · simp |
| exact he.le |
| · have hn : ∀ i,¬P i:=by simpa only [not_exists] using h |
| simpa [hn] using hA |
|
|
| lemma incidence_sum_le {κ ν : Type*} [Fintype κ] [Fintype ν] [DecidableEq ν] |
| (V : κ → Finset ν) (g : ν → ℝ) (hg : ∀ j,0 ≤ g j) (D : ℕ) |
| (hinc : ∀ j,(Finset.univ.filter (fun d=>j∈V d)).card ≤ D) : |
| (∑ d,∑ j∈V d,g j) ≤ (D:ℝ)*∑ j,g j := by |
| classical |
| have hs (d : κ) : (∑ j∈V d,g j)=∑ j,if j∈V d then g j else 0 := by simp |
| simp_rw [hs] |
| rw [Finset.sum_comm,Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro j _ |
| rw [←Finset.sum_filter] |
| simp only [Finset.sum_const,nsmul_eq_mul] |
| exact mul_le_mul_of_nonneg_right (by exact_mod_cast hinc j) (hg j) |
|
|
| theorem local_active_tile_square_budget {κ ν : Type*} [Fintype κ] [Fintype ν] |
| [DecidableEq ν] (c : ν → ℂ) (r : ν → ℝ) (hr : ∀ j,0 ≤ r j) |
| (V : κ → Finset ν) (s : κ → ν) (hs : ∀ d,s d∈V d) |
| (K : κ → Set ℂ) (hclosed : ∀ d,IsClosed (K d)) |
| (htile : Pairwise (fun d e=>Disjoint (interior (K d)) (interior (K e)))) |
| (hpair : ∀ d,∀ j∈V d,dist (c j) (c (s d)) ≤ r j+r (s d)) |
| (hfront : ∀ d,∀ x∈frontier (K d),∃ j∈V d,x∈closedBall (c j) (r j)) |
| (hdisj : Pairwise (fun i j=>Disjoint (ball (c i) (r i)) (ball (c j) (r j)))) |
| (p : ℂ) {R : ℝ} (hR : 0 ≤ R) (D : ℕ) |
| (hinc : ∀ j,(Finset.univ.filter (fun d=>j∈V d)).card ≤ D) : |
| (∑ d,if (K d∩closedBall p R).Nonempty then (min (r (s d)) R)^2 else 0) ≤ |
| (25*(D:ℝ)+1)*R^2 := by |
| classical |
| let near : ν → Prop:=fun j=>dist (c j) p ≤ r j+3*R |
| let g : ν → ℝ:=fun j=>if near j then (min (r j) R)^2 else 0 |
| let whole : κ → Prop:=fun d=>closedBall p R⊆interior (K d) |
| have hg : ∀ j,0 ≤ g j:=by intro j; dsimp [g]; split_ifs <;> positivity |
| have hnear : (∑ j,g j) ≤ 25*R^2 := by |
| have hh:=CannonRadialEnergy.active_local_disk_square_budget c r p near hr hR |
| (show (0:ℝ) ≤ 3 by norm_num) hdisj (fun _ hj=>hj) |
| norm_num [g] at hh ⊢ |
| exact hh |
| have hw : (∑ d,if whole d then R^2 else 0) ≤ R^2 := by |
| apply sum_indicator_le_one whole (sq_nonneg R) |
| intro d e hd he |
| by_contra hde |
| have hpd:=hd (mem_closedBall_self hR) |
| have hpe:=he (mem_closedBall_self hR) |
| exact Set.disjoint_left.mp (htile hde) hpd hpe |
| have hf (d : κ) : |
| (if (K d∩closedBall p R).Nonempty then (min (r (s d)) R)^2 else 0) ≤ |
| (∑ j∈V d,g j)+(if whole d then R^2 else 0) := by |
| have hsg : 0 ≤ ∑ j∈V d,g j:=Finset.sum_nonneg (fun j _=>hg j) |
| have hwi : 0 ≤ if whole d then R^2 else 0:=by split_ifs <;> positivity |
| by_cases ha : (K d∩closedBall p R).Nonempty |
| · rw [ite_eq_left ha] |
| obtain hb|hb:=tile_ball_dichotomy (hclosed d) p R ha |
| · obtain ⟨j,hj,hjn,hle⟩:=frontier_disk_charge c r (V d) (s d) (hs d) p |
| (hpair d) (hfront d) hb |
| have hle': (min (r (s d)) R)^2 ≤ (min (r j) R)^2:= |
| pow_le_pow_left₀ (le_min (hr _) hR) hle 2 |
| have hgj : g j=(min (r j) R)^2:=ite_eq_left hjn |
| have hsum:=Finset.single_le_sum (fun j _=>hg j) hj |
| rw [hgj] at hsum |
| exact (hle'.trans hsum).trans (le_add_of_nonneg_right hwi) |
| · have hb' : whole d:=hb |
| rw [ite_eq_left hb'] |
| have hle:=pow_le_pow_left₀ (le_min (hr (s d)) hR) (min_le_right (r (s d)) R) 2 |
| exact hle.trans (le_add_of_nonneg_left hsg) |
| · rw [ite_eq_right ha] |
| exact add_nonneg hsg hwi |
| calc |
| _ ≤ ∑ d,((∑ j∈V d,g j)+(if whole d then R^2 else 0)) := |
| Finset.sum_le_sum (fun d _=>hf d) |
| _ = (∑ d,∑ j∈V d,g j)+(∑ d,if whole d then R^2 else 0) := Finset.sum_add_distrib |
| _ ≤ (D:ℝ)*(∑ j,g j)+R^2 := add_le_add (incidence_sum_le V g hg D hinc) hw |
| _ ≤ (D:ℝ)*(25*R^2)+R^2 := by gcongr |
| _ = _ := by ring |
|
|
| end CannonLocalPacking |
|
|
| namespace CannonTileTopology |
| open CannonLocalPacking |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem selected_tile_budget {κ ν : Type*} [Fintype κ] [Fintype ν] [DecidableEq ν] |
| (V : κ → Finset ν) (hV : ∀ d,(V d).Nonempty) (pole : ν) |
| (K : κ → Set ℂ) (hc : ∀ d,pole∉V d → IsCompact (K d)) |
| (hd : Pairwise fun d e=>Disjoint (interior (K d)) (interior (K e))) |
| (c : ν → ℂ) (r : ν → ℝ) (hr : ∀ j,0≤r j) |
| (hdisj : Pairwise fun i j=>Disjoint (ball (c i) (r i)) (ball (c j) (r j))) |
| (hfront : ∀ d,pole∉V d → ∀ x∈frontier (K d),∃ j∈V d,x∈closedBall (c j) (r j)) |
| (hpair : ∀ d,pole∉V d → ∀ i∈V d,∀ j∈V d,dist (c i) (c j)≤r i+r j) |
| (D : ℕ) (hinc : ∀ j,(Finset.univ.filter (fun d=>j∈V d)).card≤D) : |
| ∃ s : κ → ν,(∀ d,s d∈V d) ∧ |
| (∀ d,pole∉V d → K d⊆closedBall (c (s d)) (3*r (s d))) ∧ |
| (∀ p : ℂ,∀ R : ℝ,0≤R → |
| (∑ d,if pole∉V d ∧ (K d∩closedBall p R).Nonempty then (min (r (s d)) R)^2 else 0) |
| ≤(25*(D:ℝ)+1)*R^2) ∧ |
| (Finset.univ.filter (fun d=>¬pole∉V d)).card≤D := by |
| classical |
| have hm (d : κ) : ∃ j∈V d,∀ i∈V d,r i≤r j := Finset.exists_max_image (V d) r (hV d) |
| choose s hs hm using hm |
| refine ⟨s,hs,?_,?_,?_⟩ |
| · intro d hdp |
| exact tile_max_disk_control c r (V d) (s d) (hc d hdp) (hm d) |
| (fun j hj=>hpair d hdp j hj (s d) (hs d)) (hfront d hdp) |
| · intro p R hR |
| let good : κ → Prop:=fun d=>pole∉V d |
| have hi (j : ν) : |
| (Finset.univ.filter (fun d : {d // good d}=>j∈V d.val)).card≤D := by |
| have he : (Finset.univ.filter (fun d : {d // good d}=>j∈V d.val)).card= |
| (Finset.univ.filter (fun d : κ=>good d ∧ j∈V d)).card := by |
| apply Finset.card_bij (fun d _=>d.val) |
| · intro d hd |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hd ⊢ |
| exact ⟨d.property,hd⟩ |
| · intro d hd e he hde |
| exact Subtype.ext hde |
| · intro d hd |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hd |
| exact ⟨⟨d,hd.1⟩,by simpa using hd.2,rfl⟩ |
| rw [he] |
| apply (Finset.card_le_card (show Finset.univ.filter (fun d=>good d ∧ j∈V d)⊆ |
| Finset.univ.filter (fun d=>j∈V d) from ?_)).trans (hinc j) |
| intro d hd |
| simpa only [Finset.mem_filter,Finset.mem_univ,true_and] using |
| (Finset.mem_filter.mp hd).2.2 |
| have hh:=local_active_tile_square_budget c r hr (fun d : {d // good d}=>V d.val) |
| (fun d=>s d.val) (fun d=>hs d.val) (fun d=>K d.val) (fun d=>(hc d.val d.property).isClosed) |
| (fun d e hde=>hd (Subtype.val_injective.ne hde)) |
| (fun d j hj=>hpair d.val d.property j hj (s d.val) (hs d.val)) |
| (fun d=>hfront d.val d.property) hdisj p hR D hi |
| have he : (∑ d∈Finset.univ.filter good, |
| if (K d∩closedBall p R).Nonempty then (min (r (s d)) R)^2 else 0)= |
| ∑ d : {d // good d},if (K d.val∩closedBall p R).Nonempty then (min (r (s d.val)) R)^2 else 0 := |
| Finset.sum_subtype _ (by simp) _ |
| rw [←he,Finset.sum_filter] at hh |
| simpa only [good,ite_and] using hh |
| · simpa only [not_not] using hinc pole |
| end CannonTileTopology |
|
|
| namespace CannonTileTopology |
| open Topology CannonMobius |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem pole_chart_tile_budget {κ ν : Type*} [Fintype κ] [Fintype ν] [DecidableEq ν] |
| (V : κ → Finset ν) (hV : ∀ d,(V d).Nonempty) (pole : ν) |
| (K : κ → Set (OnePoint ℂ)) (hclosed : ∀ d,IsClosed (K d)) |
| (hd : Pairwise fun d e=>Disjoint (interior (K d)) (interior (K e))) |
| (z : ν → ℂ) (ρ : ν → ℝ) (hρ : ∀ j,0<ρ j) |
| (hsep : ∀ i j,i≠j → ρ i+ρ j≤dist (z i) (z j)) |
| (hfront : ∀ d,frontier (K d)⊆⋃ j∈V d,((↑) '' closedBall (z j) (ρ j) : Set (OnePoint ℂ))) |
| (hpair : ∀ d,∀ i∈V d,∀ j∈V d,i≠j → dist (z i) (z j)=ρ i+ρ j) |
| (hpinc : ∀ d,(z pole:OnePoint ℂ)∈K d → pole∈V d) |
| (N : OnePoint ℂ ≃ₜ OnePoint ℂ) (hN : N∈biholomorphic) (hpole : N (z pole)=∞) |
| (D : ℕ) (hinc : ∀ j,(Finset.univ.filter (fun d=>j∈V d)).card≤D) : |
| let T : κ → Set ℂ:=fun d=>((↑) : ℂ → OnePoint ℂ) ⁻¹' (N '' K d) |
| ∃ c : κ → ℂ,∃ r : κ → ℝ,∃ s : κ → ν, |
| (∀ d,s d∈V d) ∧ (∀ d,0≤r d) ∧ |
| (∀ d,pole∉V d → T d⊆closedBall (c d) (3*r d)) ∧ |
| (∀ p : ℂ,∀ R : ℝ,0≤R → |
| (∑ d,if pole∉V d ∧ (T d∩closedBall p R).Nonempty then (min (r d) R)^2 else 0) |
| ≤(25*(D:ℝ)+1)*R^2) ∧ |
| (Finset.univ.filter (fun d=>¬pole∉V d)).card≤D := by |
| classical |
| dsimp only |
| let T : κ → Set ℂ:=fun d=>((↑) : ℂ → OnePoint ℂ) ⁻¹' (N '' K d) |
| obtain ⟨c₀,r₀,hr₀,hb₀,hc₀,hd₀⟩:=finite_packing_at_vertex z ρ hρ hsep N hN pole hpole |
| let c : ν → ℂ:=fun j=>if h:j=pole then 0 else c₀ ⟨j,h⟩ |
| let r : ν → ℝ:=fun j=>if h:j=pole then 0 else r₀ ⟨j,h⟩ |
| have hr (j : ν) : 0≤r j := by |
| dsimp [r] |
| split_ifs with h |
| · exact le_rfl |
| · exact (hr₀ ⟨j,h⟩).le |
| have hc (j : ν) (hj : j≠pole) : |
| N '' ((↑) '' closedBall (z j) (ρ j) : Set (OnePoint ℂ))=(↑) '' closedBall (c j) (r j) := by |
| simpa only [c,r,dite_eq_right hj] using hc₀ ⟨j,hj⟩ |
| have hdisj : Pairwise fun i j=>Disjoint (ball (c i) (r i)) (ball (c j) (r j)) := by |
| intro i j hij |
| by_cases hi : i=pole |
| · simp [r,hi] |
| by_cases hj : j=pole |
| · simp [r,hj] |
| simpa only [c,r,dite_eq_right hi,dite_eq_right hj] using |
| hd₀ (show (⟨i,hi⟩ : {j // j≠pole})≠⟨j,hj⟩ from fun h=>hij (congrArg Subtype.val h)) |
| have hTc (d : κ) (hdp : pole∉V d) : IsCompact (T d) := by |
| apply compact_trace (N.isClosedMap _ (hclosed d)) |
| rintro ⟨x,hx,he⟩ |
| have hex : x=(z pole:OnePoint ℂ):=N.injective (he.trans hpole.symm) |
| subst x |
| exact hdp (hpinc d hx) |
| have hTd : Pairwise fun d e=>Disjoint (interior (T d)) (interior (T e)) := by |
| intro d e hde |
| have hh := (hd hde).image N.injective.injOn (subset_univ _) (subset_univ _) |
| rw [N.image_interior,N.image_interior] at hh |
| have hh':=hh.preimage ((↑) : ℂ → OnePoint ℂ) |
| rw [OnePoint.isOpenMap_coe.preimage_interior_eq_interior_preimage OnePoint.continuous_coe, |
| OnePoint.isOpenMap_coe.preimage_interior_eq_interior_preimage OnePoint.continuous_coe] at hh' |
| exact hh' |
| have hTf (d : κ) (hdp : pole∉V d) : |
| ∀ x∈frontier (T d),∃ j∈V d,x∈closedBall (c j) (r j) := by |
| intro x hx |
| rw [show frontier (T d)=((↑) : ℂ → OnePoint ℂ) ⁻¹' frontier (N '' K d) from trace_frontier _, |
| ←N.image_frontier] at hx |
| obtain ⟨y,hy,hey⟩:=hx |
| obtain ⟨j,hj,u,hu,heu⟩:=mem_iUnion₂.mp (hfront d hy) |
| have hjp : j≠pole:=fun h=>hdp (h ▸ hj) |
| have hh : (x:OnePoint ℂ)∈N '' ((↑) '' closedBall (z j) (ρ j) : Set (OnePoint ℂ)):= |
| ⟨y,⟨u,hu,heu⟩,hey⟩ |
| rw [hc j hjp] at hh |
| obtain ⟨v,hv,hev⟩:=hh |
| have hvx : v=x:=OnePoint.coe_injective hev |
| exact ⟨j,hj,hvx ▸ hv⟩ |
| have hTp (d : κ) (hdp : pole∉V d) : |
| ∀ i∈V d,∀ j∈V d,dist (c i) (c j)≤r i+r j := by |
| intro i hi j hj |
| by_cases hij : i=j |
| · subst j |
| simp only [dist_self] |
| exact add_nonneg (hr i) (hr i) |
| exact transformed_disks_touch N (hρ i) (hρ j) (hpair d i hi j hj hij) |
| (hc i (fun h=>hdp (h ▸ hi))) (hc j (fun h=>hdp (h ▸ hj))) |
| obtain ⟨s,hs,hcell,hbud,hbad⟩:=selected_tile_budget V hV pole T hTc hTd c r hr hdisj hTf hTp D hinc |
| exact ⟨(fun d=>c (s d)),(fun d=>r (s d)),s,hs,(fun d=>hr (s d)),hcell,hbud,hbad⟩ |
| end CannonTileTopology |
|
|
| end |
| end |
|
|
| end OAI |
|
|
