| import Mathlib |
| import OAI.Geometry.Cannon.CirclePacking.OpenEuclideanFan |
| import OAI.Geometry.Cannon.CirclePacking.PetalAngle |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
|
|
| def SectorRingData (n : ℕ) (t α : ℕ → ℝ) : Prop := |
| (∀ i,t i∈Icc (0:ℝ) 1) ∧ (∀ i,α i∈Icc (0:ℝ) Real.pi) ∧ |
| t 0=1/2 ∧ t n=1/2 ∧ α 0=0 ∧ α n=Real.pi/3 ∧ |
| (∀ i,i<n → α (i+1)=α i+petalAngle (t i) (t (i+1))) ∧ |
| ∀ i,i≤n → ∀ j,j≤n → i≠j → Real.cos (α i-α j)≤1-2*t i*t j |
|
|
| lemma SectorRingData.pos {n : ℕ} {t α : ℕ → ℝ} (h : SectorRingData n t α) |
| {i : ℕ} (hi : i≤n) : 0<t i := by |
| classical |
| rcases h with ⟨ht,_hα,ht0,htn,_hzero,_hend,hstep,hdisj⟩ |
| by_contra! hnpos |
| have hti : t i=0 := le_antisymm hnpos (ht i).1 |
| have hfirst : 0<t 0 := by rw [ht0]; norm_num |
| have hlast : 0<t n := by rw [htn]; norm_num |
| have hi0 : 0 < i := by |
| by_contra! hh |
| have heq : i=0 := by omega |
| subst i |
| linarith |
| have hin : i<n := by |
| by_contra! hh |
| have heq : i=n := by omega |
| subst i |
| linarith |
| let S := (Finset.range i).filter (fun j=>0<t j) |
| have hS : S.Nonempty := ⟨0,Finset.mem_filter.mpr ⟨Finset.mem_range.mpr hi0,hfirst⟩⟩ |
| let m := S.max' hS |
| have hm : m∈S := Finset.max'_mem S hS |
| have hmi : m < i := Finset.mem_range.mp (Finset.mem_filter.mp hm).1 |
| have hmp : 0<t m := (Finset.mem_filter.mp hm).2 |
| let T := (Finset.Icc (i+1) n).filter (fun j=>0<t j) |
| have hT : T.Nonempty := ⟨n,Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr ⟨by omega,le_rfl⟩,hlast⟩⟩ |
| let M := T.min' hT |
| have hM : M∈T := Finset.min'_mem T hT |
| have hiM : i < M := by have := (Finset.mem_Icc.mp (Finset.mem_filter.mp hM).1).1; omega |
| have hMn : M≤n := (Finset.mem_Icc.mp (Finset.mem_filter.mp hM).1).2 |
| have hMp : 0<t M := (Finset.mem_filter.mp hM).2 |
| have hnull : ∀ k,m < k → k < M → t k=0 := by |
| intro k hmk hkM |
| by_cases hki : k=i |
| · simpa only [hki] using hti |
| by_contra hk |
| have hkp : 0<t k := lt_of_le_of_ne (ht k).1 (Ne.symm hk) |
| by_cases hki' : k < i |
| · have H := Finset.le_max' S k (Finset.mem_filter.mpr ⟨Finset.mem_range.mpr hki',hkp⟩) |
| change k ≤ m at H |
| omega |
| · have H := Finset.min'_le T k (Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr ⟨by omega,by omega⟩,hkp⟩) |
| change M ≤ k at H |
| omega |
| have ham : α M=α m := interval_constant α (by omega) (fun k hmk hkM=>by |
| by_cases hkm : k=m |
| · have H:=hstep k (by omega) |
| have hz : t (k+1)=0 := hnull (k+1) (by omega) (by omega) |
| simpa only [hz,petalAngle_zero_right,add_zero] using H |
| · simpa only [hnull k (by omega) hkM,petalAngle_zero_left,add_zero] using hstep k (by omega)) |
| have H := hdisj m (by omega) M hMn (by omega) |
| rw [ham,sub_self,Real.cos_zero] at H |
| nlinarith [mul_pos hmp hMp] |
|
|
| lemma sectorRingData_closed (n : ℕ) : |
| IsClosed {p : (ℕ → ℝ) × (ℕ → ℝ) | SectorRingData n p.1 p.2} := by |
| unfold SectorRingData |
| simp only [ofPred_and] |
| refine IsClosed.inter ?_ (IsClosed.inter ?_ (IsClosed.inter ?_ (IsClosed.inter ?_ |
| (IsClosed.inter ?_ (IsClosed.inter ?_ (IsClosed.inter ?_ ?_)))))) |
| · simpa only [ofPred_forall,Set.preimage] using isClosed_iInter (fun i : ℕ => |
| isClosed_Icc.preimage (show Continuous (fun p : (ℕ → ℝ) × (ℕ → ℝ) => p.1 i) by fun_prop)) |
| · simpa only [ofPred_forall,Set.preimage] using isClosed_iInter (fun i : ℕ => |
| isClosed_Icc.preimage (show Continuous (fun p : (ℕ → ℝ) × (ℕ → ℝ) => p.2 i) by fun_prop)) |
| · exact isClosed_eq (by fun_prop) continuous_const |
| · exact isClosed_eq (by fun_prop) continuous_const |
| · exact isClosed_eq (by fun_prop) continuous_const |
| · exact isClosed_eq (by fun_prop) continuous_const |
| · simp only [ofPred_forall] |
| exact isClosed_iInter (fun i => isClosed_iInter (fun _hi => |
| isClosed_eq (by fun_prop) (by unfold petalAngle; fun_prop))) |
| · simp only [ofPred_forall] |
| exact isClosed_iInter (fun i => isClosed_iInter (fun _hi => |
| isClosed_iInter (fun j => isClosed_iInter (fun _hj => |
| isClosed_iInter (fun _hij => isClosed_le (by fun_prop) (by fun_prop)))))) |
| |
| lemma sectorRingData_compact (n : ℕ) : |
| IsCompact {p : (ℕ → ℝ) × (ℕ → ℝ) | SectorRingData n p.1 p.2} := by |
| have ht : IsCompact (pi univ (fun _ : ℕ => Icc (0:ℝ) 1)) := isCompact_univ_pi (fun _ => isCompact_Icc) |
| have hα : IsCompact (pi univ (fun _ : ℕ => Icc (0:ℝ) Real.pi)) := isCompact_univ_pi (fun _ => isCompact_Icc) |
| apply (ht.prod hα).of_isClosed_subset (sectorRingData_closed n) |
| intro p hp |
| exact ⟨by simpa only [mem_pi,mem_univ,true_implies] using hp.1, |
| by simpa only [mem_pi,mem_univ,true_implies] using hp.2.1⟩ |
|
|
| theorem exists_bounded_sector_constant (N : ℕ) : ∃ c : ℝ,0<c ∧ c≤1 ∧ |
| ∀ n≤N,∀ t α : ℕ → ℝ,SectorRingData n t α → ∀ i≤n,c≤t i := by |
| classical |
| have hex (n : Fin (N+1)) (i : Fin (n.val+1)) : ∃ c : ℝ,0<c ∧ |
| ∀ t α : ℕ → ℝ,SectorRingData n.val t α → c≤t i.val := by |
| let K := {p : (ℕ → ℝ) × (ℕ → ℝ) | SectorRingData n.val p.1 p.2} |
| by_cases hK : K.Nonempty |
| · obtain ⟨p,hp,hmin⟩ := (sectorRingData_compact n.val).exists_isMinOn hK |
| (show ContinuousOn (fun p : (ℕ → ℝ) × (ℕ → ℝ) => p.1 i.val) K from (by fun_prop : Continuous (fun p : (ℕ → ℝ) × (ℕ → ℝ) => p.1 i.val)).continuousOn) |
| exact ⟨p.1 i.val,SectorRingData.pos hp (by omega),fun t α h=>hmin (show (t,α)∈K from h)⟩ |
| · exact ⟨1,zero_lt_one,fun t α h=>(hK ⟨(t,α),h⟩).elim⟩ |
| choose c hc using hex |
| let S := Finset.univ.image (fun p : (n : Fin (N+1)) × Fin (n.val+1)=>c p.1 p.2) |
| let T := insert 1 S |
| have hT : T.Nonempty := ⟨1,Finset.mem_insert_self _ _⟩ |
| refine ⟨T.min' hT,?_,Finset.min'_le T 1 (Finset.mem_insert_self _ _),?_⟩ |
| · have hmem:=Finset.min'_mem T hT |
| rcases Finset.mem_insert.mp hmem with h|h |
| · rw [h]; exact zero_lt_one |
| · obtain ⟨p,-,hp⟩:=Finset.mem_image.mp h |
| rw [←hp]; exact (hc p.1 p.2).1 |
| · intro n hn t α h i hi |
| let nn : Fin (N+1) := ⟨n,by omega⟩ |
| let ii : Fin (nn.val+1) := ⟨i,by change i<n+1; omega⟩ |
| exact (Finset.min'_le T (c nn ii) |
| (Finset.mem_insert_of_mem (Finset.mem_image.mpr ⟨⟨nn,ii⟩,Finset.mem_univ _,rfl⟩))).trans |
| ((hc nn ii).2 t α h) |
| end CannonPackingGeometry |
| |
| end |
| end |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
| |
| theorem actual_open_fan_sectorData {n : ℕ} (hn : 0<n) (r : ℝ) (s α : ℕ → ℝ) |
| (hr : 0<r) (hs : ∀ i,0<s i) (hα : ∀ i,i<n → α i∈Ioo (0:ℝ) Real.pi) |
| (hsum : ∑ i∈Finset.range n,α i=Real.pi/3) (hs0 : s 0=r) (hsn : s n=r) |
| (hcos : ∀ i,i<n →2*(r+s i)*(r+s (i+1))*Real.cos (α i)= |
| (r+s i)^2+(r+s (i+1))^2-(s i+s (i+1))^2) |
| (A : ℂ) (B : ℕ → ℂ) |
| (hAB : ∀ i,i≤n →dist A (B i)=r+s i) |
| (hBC : ∀ i,i<n →dist (B i) (B (i+1))=s i+s (i+1)) |
| (hop : ∀ i,i+1<n → orientedArea A (B (i+1)) (B i)* |
| orientedArea A (B (i+1)) (B (i+2))<0) |
| (hdis : ∀ i,i≤n →∀ j,j≤n →i≠j →s i+s j≤dist (B i) (B j)) : |
| ∃ t β : ℕ → ℝ,SectorRingData n t β ∧ ∀ i≤n,t i=s i/(r+s i) := by |
| let L : ℕ → ℝ := fun k=>r+s k |
| let d : ℕ → ℝ := fun k=>s k+s (k+1) |
| obtain ⟨e,_heA,heB⟩ := open_fan_global_frame n hn L α d |
| (fun k _hk=>add_pos hr (hs _)) (fun k _hk=>add_pos (hs _) (hs _)) hα hcos A B hAB hBC hop |
| let t : ℕ → ℝ := fun k=>s (min k n)/(r+s (min k n)) |
| let β : ℕ → ℝ := fun k=>∑ j∈Finset.range (min k n),α j |
| have hb (k : ℕ) : β k∈Icc (0:ℝ) Real.pi := by |
| have hsmall : min k n≤n := min_le_right _ _ |
| have hle : β k≤Real.pi/3 := by |
| rw [←hsum] |
| exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono hsmall) |
| (fun j hj _=>(hα j (Finset.mem_range.mp hj)).1.le) |
| refine ⟨Finset.sum_nonneg (fun j hj=>(hα j (lt_of_lt_of_le (Finset.mem_range.mp hj) hsmall)).1.le),?_⟩ |
| linarith [Real.pi_pos] |
| refine ⟨t,β,⟨fun k=>⟨(radius_fraction_mem hr (hs _)).1.le,(radius_fraction_mem hr (hs _)).2.le⟩, |
| hb,?_,?_,by simp [β],by simpa [β] using hsum,?_,?_⟩,fun i hi=>by simp only [t,min_eq_left hi]⟩ |
| · simp only [t,zero_min,hs0] |
| field_simp |
| ring |
| · simp only [t,min_self,hsn] |
| field_simp |
| ring |
| · intro k hk |
| dsimp [β] |
| rw [min_eq_left (show k+1≤n by omega),min_eq_left hk.le,Finset.sum_range_succ] |
| congr 1 |
| simpa only [t,min_eq_left hk.le,min_eq_left (show k+1≤n by omega)] using |
| tangent_petal_angle hr (hs k) (hs (k+1)) (hα k hk) (hcos k hk) |
| · intro i hi j hj hij |
| have hsq := fanRay_dist_sq_pair L α i j |
| rw [←heB i hi,←heB j hj,e.dist_eq] at hsq |
| have H := tangent_nonoverlap_cosine hr (hs i) (hs j) (hdis i hi j hj hij) hsq |
| simpa only [β,t,min_eq_left hi,min_eq_left hj] using H |
| end CannonPackingGeometry |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
