| import Mathlib |
| import OAI.Geometry.Cannon.PlaneTopology.Crossing |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
|
|
| theorem cyclic_folded_frames_positions (n:ℕ) (hn:0<n) (L α d:ℕ→ℝ) |
| (hL:∀ k≤n,0<L k) (hd:∀ k<n,0<d k) |
| (hα:∀ k<n,α k∈Ioo (0:ℝ) Real.pi) |
| (hs:∑ k∈Finset.range n,α k=2*Real.pi) (hp:L n=L 0) |
| (hcos:∀ k<n,2*L k*L (k+1)*Real.cos (α k)=(L k)^2+(L (k+1))^2-(d k)^2) |
| (A B C:Fin n→ℂ) |
| (hAB:∀ i,dist (A i) (B i)=L i.val) |
| (hAC:∀ i,dist (A i) (C i)=L (i.val+1)) |
| (hBC:∀ i,dist (B i) (C i)=d i.val) |
| (K:Fin n→(ℂ≃ᵢℂ)) |
| (hKA:∀ i,K i (A i)=A (fanSucc hn i)) |
| (hKC:∀ i,K i (C i)=B (fanSucc hn i)) |
| (hKB:∀ i,orientedArea (A (fanSucc hn i)) (B (fanSucc hn i)) (K i (B i))* |
| orientedArea (A (fanSucc hn i)) (B (fanSucc hn i)) (C (fanSucc hn i))<0) : |
| ∃ e:Fin n→(ℂ≃ᵢℂ), |
| (∀ i,e i (A i)=0 ∧ e i (B i)=fanRay L α i.val ∧ e i (C i)=fanRay L α (i.val+1)) ∧ |
| ∀ i,K i*(e i).symm=(e (fanSucc hn i)).symm := by |
| obtain ⟨hclose,hR,hD⟩ := closed_euclidean_fan n L α d hL hd hα hs hp hcos |
| obtain ⟨e,he⟩ := fan_triangle_frames n L α d hL hd hα hs hp hcos A B C hAB hAC hBC |
| have harea (i:Fin n) : orientedArea (A i) (B i) (C i)≠0 := by |
| have hh := isometry_orientedArea_product (e i) (A i) (B i) (C i) (C i) |
| rw [(he i).1,(he i).2.1,(he i).2.2] at hh |
| intro hz |
| rw [hz,mul_zero] at hh |
| exact mul_ne_zero (hD i.val i.isLt).2.ne' (hD i.val i.isLt).2.ne' hh |
| refine ⟨e,he,fun i=>?_⟩ |
| let j := fanSucc hn i |
| have hray : fanRay L α (i.val+1)=fanRay L α j.val := fanRay_succ hn L α hclose i |
| apply folded_frames_compatible (a:=A i) (b:=C i) (c:=B i) |
| (A:=A j) (B:=B j) (C:=C j) (P:=0) (Q:=fanRay L α j.val) |
| (e:=e i) (f:=e j) |
| · intro hh |
| have hh' := hAC i |
| rw [hh,dist_self] at hh' |
| exact (hL (i.val+1) (by omega)).ne' hh'.symm |
| · calc dist (A i) (C i) = dist ((e i) (A i)) ((e i) (C i)) := |
| ((e i).dist_eq _ _).symm |
| _ = dist ((e j) (A j)) ((e j) (B j)) := by |
| rw [(he i).1,(he i).2.2,(he j).1,(he j).2.1,hray] |
| _ = dist (A j) (B j) := (e j).dist_eq _ _ |
| · rw [orientedArea_swap] |
| exact neg_ne_zero.mpr (harea i) |
| · exact harea j |
| · exact hKA i |
| · exact hKC i |
| · exact hKB i |
| · exact (he i).1 |
| · exact (he i).2.2.trans hray |
| · exact (he j).1 |
| · exact (he j).2.1 |
| · rw [(he i).2.1,(he j).2.2,←hray,orientedArea_swap 0 (fanRay L α (i.val+1))] |
| exact mul_neg_of_neg_of_pos (neg_neg_of_pos (hD i.val i.isLt).2) |
| (by rw [hray] |
|
|
| theorem cyclic_fan_global_frame (n : ℕ) (hn : 0<n) (L α d : ℕ → ℝ) |
| (hL : ∀ k≤n,0<L k) (hd : ∀ k<n,0<d k) |
| (hα : ∀ k<n,α k∈Ioo (0:ℝ) Real.pi) |
| (hs : ∑ k∈Finset.range n,α k=2*Real.pi) (hp : L n=L 0) |
| (hcos : ∀ k<n,2*L k*L (k+1)*Real.cos (α k)=(L k)^2+(L (k+1))^2-(d k)^2) |
| (A : ℂ) (B : Fin n → ℂ) |
| (hAB : ∀ i,dist A (B i)=L i.val) |
| (hBC : ∀ i,dist (B i) (B (fanSucc hn i))=d i.val) |
| (hLnext : ∀ i,L (i.val+1)=L (fanSucc hn i).val) |
| (hop : ∀ i, orientedArea A (B (fanSucc hn i)) (B i)* |
| orientedArea A (B (fanSucc hn i)) (B (fanSucc hn (fanSucc hn i)))<0) : |
| ∃ e : ℂ≃ᵢℂ,e A=0 ∧ ∀ i,e (B i)=fanRay L α i.val := by |
| obtain ⟨e,he,hstep⟩ := cyclic_folded_frames_positions n hn L α d hL hd hα hs hp hcos |
| (fun _=>A) B (fun i=>B (fanSucc hn i)) hAB |
| (fun i=>(hAB (fanSucc hn i)).trans (hLnext i).symm) hBC |
| (fun _=>1) (fun _=>rfl) (fun _=>rfl) hop |
| have heq (i j : Fin n) : e i=e j := by |
| have H := CannonCycleOrder.eq_of_steps hn (fun i=>(e i).symm) |
| (fun i=>by simpa only [one_mul,fanSucc,CannonCycleOrder.next] using hstep i) i j |
| exact congrArg IsometryEquiv.symm H |
| let i : Fin n := ⟨0,hn⟩ |
| refine ⟨e i,(he i).1,?_⟩ |
| intro j |
| rw [heq i j] |
| exact (he j).2.1 |
| |
| end CannonPackingGeometry |
|
|
| end |
| end |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
|
|
| lemma sum_range_partition (a : ℕ → ℝ) (n : ℕ) {t : ℝ} |
| (ht : 0≤t) (hn : t<∑ i∈Finset.range n,a i) : |
| ∃ k,k<n ∧ (∑ i∈Finset.range k,a i)≤t ∧ t<∑ i∈Finset.range (k+1),a i := by |
| induction n with |
| | zero => simp only [Finset.range_zero,Finset.sum_empty] at hn; exact (not_lt_of_ge ht hn).elim |
| | succ n ih => |
| by_cases h : t<∑ i∈Finset.range n,a i |
| · obtain ⟨k,hk,hlo,hhi⟩ := ih h |
| exact ⟨k,by omega,hlo,hhi⟩ |
| · exact ⟨n,by omega,le_of_not_gt h,hn⟩ |
|
|
| lemma polar_nonnegative_arg (p : ℂ) : |
| ∃ t : ℝ,0≤t ∧ t<2*Real.pi ∧ p=(‖p‖:ℂ)*(Circle.exp t:ℂ) := by |
| have hpi := Real.pi_pos |
| by_cases h : Complex.arg p<0 |
| · refine ⟨Complex.arg p+2*Real.pi,by linarith [Complex.neg_pi_lt_arg p],by linarith,?_⟩ |
| rw [Circle.exp_add,Circle.exp_two_pi,mul_one] |
| simpa only [Circle.coe_exp] using (Complex.norm_mul_exp_arg_mul_I p).symm |
| · refine ⟨Complex.arg p,le_of_not_gt h,by linarith [Complex.arg_le_pi p],?_⟩ |
| simpa only [Circle.coe_exp] using (Complex.norm_mul_exp_arg_mul_I p).symm |
|
|
| theorem closed_fan_cones_cover (n : ℕ) (L α : ℕ → ℝ) |
| (hL : ∀ k≤n,0<L k) (hα : ∀ k<n,α k∈Ioo (0:ℝ) Real.pi) |
| (hs : ∑ k∈Finset.range n,α k=2*Real.pi) (p : ℂ) : |
| ∃ k,k<n ∧ 0≤orientedArea 0 (fanRay L α k) p ∧ |
| 0≤orientedArea 0 p (fanRay L α (k+1)) := by |
| obtain ⟨t,ht,hthi,hp⟩ := polar_nonnegative_arg p |
| obtain ⟨k,hk,hlo,hhi⟩ := sum_range_partition α n ht (by rwa [hs]) |
| let s := ∑ j∈Finset.range k,α j |
| have hnext : (∑ j∈Finset.range (k+1),α j)=s+α k := Finset.sum_range_succ α k |
| rw [hnext] at hhi |
| have hleft : t-s∈Icc (0:ℝ) Real.pi := ⟨sub_nonneg.mpr hlo,by dsimp [s] at * |
| have hright : s+α k-t∈Icc (0:ℝ) Real.pi := ⟨by linarith,by linarith [(hα k hk).2]⟩ |
| refine ⟨k,hk,?_,?_⟩ |
| · rw [hp] |
| change 0≤orientedArea 0 ((L k:ℂ)*(Circle.exp s:ℂ)) ((‖p‖:ℂ)*(Circle.exp t:ℂ)) |
| rw [show t=s+(t-s) by ring,rotated_polar_orientedArea] |
| exact mul_nonneg (mul_nonneg (hL k (by omega)).le (norm_nonneg p)) |
| (Real.sin_nonneg_of_nonneg_of_le_pi hleft.1 hleft.2) |
| · rw [hp] |
| change 0≤orientedArea 0 ((‖p‖:ℂ)*(Circle.exp t:ℂ)) |
| ((L (k+1):ℂ)*(Circle.exp (∑ j∈Finset.range (k+1),α j):ℂ)) |
| rw [hnext,show s+α k=t+(s+α k-t) by ring,rotated_polar_orientedArea] |
| exact mul_nonneg (mul_nonneg (norm_nonneg p) (hL (k+1) (by omega)).le) |
| (Real.sin_nonneg_of_nonneg_of_le_pi hright.1 hright.2) |
|
|
| end CannonPackingGeometry |
|
|
| end |
| end |
| section |
| noncomputable section |
| namespace CannonTangentWedges |
| open Set |
|
|
| def det (a b c : ℂ) : ℝ := (b.re-a.re)*(c.im-a.im)-(b.im-a.im)*(c.re-a.re) |
| |
| lemma barycentric {a b c p : ℂ} (hd : det a b c≠0) : |
| p=a+(det c a p/det a b c) • (b-a)+(det a b p/det a b c) • (c-a) := by |
| apply Complex.ext |
| all_goals simp only [Complex.add_re,Complex.smul_re,Complex.sub_re,Complex.add_im,Complex.smul_im,Complex.sub_im,smul_eq_mul] |
| all_goals field_simp |
| all_goals dsimp [det] |
|
|
| lemma combination_norm_sq (b c : ℂ) (u v : ℝ) : |
| ‖u • b+v • c‖^2=u^2*‖b‖^2+v^2*‖c‖^2+u*v*(‖b‖^2+‖c‖^2-‖b-c‖^2) := by |
| simp only [Complex.sq_norm,Complex.normSq_apply,Complex.add_re,Complex.add_im, |
| Complex.smul_re,Complex.smul_im,Complex.sub_re,Complex.sub_im,smul_eq_mul] |
| ring |
|
|
| theorem wedge_ball_opposite_positive {a b c p : ℂ} {r s t : ℝ} |
| (hr : 0<r) (hs : 0<s) (ht : 0<t) (hab : dist a b=r+s) |
| (hac : dist a c=r+t) (hbc : dist b c=s+t) (hD : 0<det a b c) |
| (hp : dist a p<r) (h1 : 0≤det a b p) (h2 : 0≤det c a p) : |
| 0<det b c p := by |
| let u := det c a p/det a b c |
| let v := det a b p/det a b c |
| have hu : 0≤u := div_nonneg h2 hD.le |
| have hv : 0≤v := div_nonneg h1 hD.le |
| have hp' : p-a=u • (b-a)+v • (c-a) := by |
| have H := barycentric (p:=p) hD.ne' |
| change p=a+u • (b-a)+v • (c-a) at H |
| rw [H] |
| have hn := combination_norm_sq (b-a) (c-a) u v |
| rw [←hp',←dist_eq_norm b a,←dist_eq_norm c a, |
| show b-a-(c-a)=b-c by abel,←dist_eq_norm b c, |
| dist_comm b a,dist_comm c a,hab,hac,hbc,←dist_eq_norm p a,dist_comm p a] at hn |
| have heq : dist a p^2=r^2*(u+v)^2+2*r*(u+v)*(u*s+v*t)+(u*s-v*t)^2 := by |
| rw [hn]; ring |
| by_contra h |
| have hsum : det a b p+det b c p+det c a p=det a b c := by dsimp [det]; ring |
| have huv : 1≤u+v := by |
| dsimp [u,v] |
| rw [←add_div] |
| apply (le_div_iff₀ hD).mpr |
| linarith [le_of_not_gt h] |
| have hsq : (1:ℝ)≤(u+v)^2 := by nlinarith |
| have hterm := mul_le_mul_of_nonneg_left hsq (sq_nonneg r) |
| have hpos : 0≤2*r*(u+v)*(u*s+v*t) := by positivity |
| have hrem := sq_nonneg (u*s-v*t) |
| have hdist := dist_nonneg (x:=a) (y:=p) |
| nlinarith |
| |
| end CannonTangentWedges |
| |
| end |
| end |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
| |
| lemma orientedArea_eq_det (a b c : ℂ) : orientedArea a b c=CannonTangentWedges.det a b c := by |
| simp only [orientedArea,CannonTangentWedges.det,Complex.mul_im,Complex.conj_re,Complex.conj_im, |
| Complex.sub_re,Complex.sub_im] |
| ring |
| lemma orientedArea_cyclic (a b c : ℂ) : orientedArea b c a=orientedArea a b c := by |
| simp only [orientedArea_eq_det,CannonTangentWedges.det]; ring |
| lemma orientedArea_swap_first (a b c : ℂ) : orientedArea b a c= -orientedArea a b c := by |
| simp only [orientedArea_eq_det,CannonTangentWedges.det]; ring |
| lemma winding_area_eq (a b c : ℂ) : CannonTriangleWinding.area a b c=orientedArea a b c := by |
| rw [orientedArea_eq_det]; rfl |
| |
| def TriInterior (a b c p : ℂ) : Prop := |
| 0<orientedArea a b p*orientedArea a b c ∧ |
| 0<orientedArea b c p*orientedArea b c a ∧ |
| 0<orientedArea c a p*orientedArea c a b |
| |
| lemma TriInterior.isometry (e : ℂ≃ᵢℂ) (a b c p : ℂ) : |
| TriInterior (e a) (e b) (e c) (e p) ↔ TriInterior a b c p := by |
| simp only [TriInterior,isometry_orientedArea_product] |
| |
| lemma TriInterior.of_positive {a b c p : ℂ} (hD : 0<orientedArea a b c) |
| (h1 : 0<orientedArea a b p) (h2 : 0<orientedArea b c p) (h3 : 0<orientedArea c a p) : |
| TriInterior a b c p := by |
| exact ⟨mul_pos h1 hD,mul_pos h2 (by rwa [orientedArea_cyclic]), |
| mul_pos h3 (by rwa [orientedArea_cyclic,orientedArea_cyclic])⟩ |
| |
| lemma TriInterior.inside {a b c p : ℂ} (hD : 0<CannonTriangleWinding.area a b c) |
| (h : TriInterior a b c p) : CannonTriangleWinding.Inside a b c p := by |
| simp only [winding_area_eq] at hD |
| change 0<CannonTriangleWinding.area a b p ∧ 0<CannonTriangleWinding.area b c p ∧ 0<CannonTriangleWinding.area c a p |
| simp only [winding_area_eq] |
| exact ⟨pos_of_mul_pos_left h.1 hD.le, |
| pos_of_mul_pos_left h.2.1 (by rw [orientedArea_cyclic]; exact hD.le), |
| pos_of_mul_pos_left h.2.2 (by rw [orientedArea_cyclic,orientedArea_cyclic]; exact hD.le)⟩ |
| |
| lemma TriInterior.edge_positive {a b c p x y z : ℂ} (h : TriInterior a b c p) |
| (hx : x∈({a,b,c}:Finset ℂ)) (hy : y∈({a,b,c}:Finset ℂ)) (hz : z∈({a,b,c}:Finset ℂ)) |
| (hxy : x≠y) (hxz : x≠z) (hyz : y≠z) : 0<orientedArea x y p*orientedArea x y z := by |
| classical |
| have hrev (a b c : ℂ) : orientedArea b a p*orientedArea b a c=orientedArea a b p*orientedArea a b c := by |
| rw [orientedArea_swap_first a b p,orientedArea_swap_first a b c,neg_mul_neg] |
| simp only [Finset.mem_insert,Finset.mem_singleton] at hx hy hz |
| rcases hx with rfl|rfl|rfl <;> rcases hy with rfl|rfl|rfl <;> rcases hz with rfl|rfl|rfl |
| all_goals first |
| | exact (hxy rfl).elim |
| | exact (hxz rfl).elim |
| | exact (hyz rfl).elim |
| | exact h.1 |
| | exact h.2.1 |
| | exact h.2.2 |
| | rw [hrev]; exact h.1 |
| | rw [hrev]; exact h.2.1 |
| | rw [hrev]; exact h.2.2 |
| |
| lemma TriInterior.of_vertices {a b c p x y z : ℂ} (h : TriInterior a b c p) |
| (hx : x∈({a,b,c}:Finset ℂ)) (hy : y∈({a,b,c}:Finset ℂ)) (hz : z∈({a,b,c}:Finset ℂ)) |
| (hxy : x≠y) (hxz : x≠z) (hyz : y≠z) : TriInterior x y z p := |
| ⟨h.edge_positive hx hy hz hxy hxz hyz, |
| h.edge_positive hy hz hx hyz hxy.symm hxz.symm, |
| h.edge_positive hz hx hy hxz.symm hyz.symm hxy⟩ |
| |
| lemma isometry_area_ne_zero (e : ℂ≃ᵢℂ) {a b c : ℂ} (h : orientedArea a b c≠0) : |
| orientedArea (e a) (e b) (e c)≠0 := by |
| intro hh |
| have H := isometry_orientedArea_product e a b c c |
| rw [hh,zero_mul] at H |
| exact mul_ne_zero h h H.symm |
| |
| end CannonPackingGeometry |
| |
| end |
| end |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
| |
| theorem finite_star_ball_cover {n : ℕ} (hn : 0<n) (r : ℝ) (s α : Fin n → ℝ) |
| (hr : 0<r) (hs : ∀ i,0<s i) (hα : ∀ i,α i∈Ioo (0:ℝ) Real.pi) |
| (hsum : ∑ i,α i=2*Real.pi) |
| (hcos : ∀ i,2*(r+s i)*(r+s (CannonCycleOrder.next hn i))*Real.cos (α i)= |
| (r+s i)^2+(r+s (CannonCycleOrder.next hn i))^2-(s i+s (CannonCycleOrder.next hn i))^2) |
| (A : ℂ) (B : Fin n → ℂ) |
| (hAB : ∀ i,dist A (B i)=r+s i) |
| (hBC : ∀ i,dist (B i) (B (CannonCycleOrder.next hn i))=s i+s (CannonCycleOrder.next hn i)) |
| (hop : ∀ i,orientedArea A (B (CannonCycleOrder.next hn i)) (B i)* |
| orientedArea A (B (CannonCycleOrder.next hn i)) (B (CannonCycleOrder.next hn (CannonCycleOrder.next hn i)))<0) |
| {p : ℂ} (hp : dist A p<r) (hgen : ∀ i,orientedArea A (B i) p≠0) : |
| ∃ i,TriInterior A (B i) (B (CannonCycleOrder.next hn i)) p := by |
| let nx:=CannonCycleOrder.next hn |
| let atN (k:ℕ):Fin n := ⟨k%n,Nat.mod_lt _ hn⟩ |
| have hat (i:Fin n):atN i.val=i := Fin.ext (Nat.mod_eq_of_lt i.isLt) |
| have hat' (i:Fin n):atN (i.val+1)=nx i := Fin.ext (CannonCycleOrder.next_mod hn i).symm |
| let L : ℕ → ℝ := fun k=>r+s (atN k) |
| let al : ℕ → ℝ := fun k=>α (atN k) |
| let d : ℕ → ℝ := fun k=>s (atN k)+s (nx (atN k)) |
| have hL (k : ℕ) (hk : k≤n) : 0<L k := add_pos hr (hs _) |
| have hd (k : ℕ) (hk : k<n) : 0<d k := add_pos (hs _) (hs _) |
| have hal (k : ℕ) (hk : k<n) : al k∈Ioo (0:ℝ) Real.pi := hα _ |
| have htotal : ∑ k∈Finset.range n,al k=2*Real.pi := by |
| rw [Finset.sum_range] |
| simpa only [al,hat] using hsum |
| have hperiod : L n=L 0 := by simp [L,atN] |
| have hcosN (k : ℕ) (hk : k<n) : |
| 2*L k*L (k+1)*Real.cos (al k)=(L k)^2+(L (k+1))^2-(d k)^2 := by |
| have h0:=hat ⟨k,hk⟩ |
| have h1:=hat' ⟨k,hk⟩ |
| simpa only [L,al,d,h0,h1] using hcos ⟨k,hk⟩ |
| obtain ⟨e,heA,heB⟩ := cyclic_fan_global_frame n hn L al d hL hd hal htotal hperiod hcosN A B |
| (fun i=>by simpa only [L,hat] using hAB i) |
| (fun i=>by simpa only [d,hat,nx,fanSucc,CannonCycleOrder.next] using hBC i) |
| (fun i=>by simp only [L,hat',hat] |
| obtain ⟨hclose,hR,hD⟩:=closed_euclidean_fan n L al d hL hd hal htotal hperiod hcosN |
| obtain ⟨k,hk,h1,h2⟩:=closed_fan_cones_cover n L al hL hal htotal (e p) |
| let i : Fin n:=⟨k,hk⟩ |
| have hnext : fanRay L al (k+1)=e (B (nx i)) := by |
| rw [heB] |
| exact fanRay_succ hn L al hclose i |
| have hcur : fanRay L al k=e (B i) := (heB i).symm |
| have ha : 0<orientedArea 0 (e (B i)) (e p) := by |
| rw [hcur] at h1 |
| exact lt_of_le_of_ne h1 (by simpa only [heA] using (isometry_area_ne_zero e (hgen i)).symm) |
| have hb : 0<orientedArea (e (B (nx i))) 0 (e p) := by |
| rw [hnext,orientedArea_swap,neg_nonneg] at h2 |
| have hh : orientedArea 0 (e (B (nx i))) (e p)≠0 := by |
| simpa only [heA] using isometry_area_ne_zero e (hgen (nx i)) |
| rw [orientedArea_cyclic,orientedArea_cyclic,orientedArea_swap] |
| exact neg_pos.mpr (lt_of_le_of_ne h2 hh) |
| have hD' : 0<orientedArea 0 (e (B i)) (e (B (nx i))) := by |
| rw [←hcur,←hnext]; exact (hD k hk).2 |
| have hc : 0<orientedArea (e (B i)) (e (B (nx i))) (e p) := by |
| rw [orientedArea_eq_det] |
| apply CannonTangentWedges.wedge_ball_opposite_positive (a:=0) hr (hs i) (hs (nx i)) |
| · rw [←heA,e.dist_eq]; exact hAB i |
| · rw [←heA,e.dist_eq]; exact hAB (nx i) |
| · rw [e.dist_eq]; exact hBC i |
| · rwa [←orientedArea_eq_det] |
| · rwa [←heA,e.dist_eq] |
| · rw [←orientedArea_eq_det]; exact ha.le |
| · rw [←orientedArea_eq_det]; exact hb.le |
| refine ⟨i,?_⟩ |
| apply (TriInterior.isometry e A (B i) (B (nx i)) p).mp |
| rw [heA] |
| exact TriInterior.of_positive hD' ha hc hb |
|
|
| end CannonPackingGeometry |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
