| import Mathlib |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace CannonTileTopology |
| open Set Metric Topology |
| open scoped OnePoint |
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| lemma tangent_disk_meet {a b : ℂ} {r s : ℝ} (hr : 0<r) (hs : 0<s) |
| (hd : dist a b=r+s) : (closedBall a r∩closedBall b s).Nonempty := by |
| let q:=AffineMap.lineMap a b (r/(r+s)) |
| have hsum : 0<r+s:=add_pos hr hs |
| have ht : 0≤r/(r+s):=div_nonneg hr.le hsum.le |
| have ht' : r/(r+s)≤1:=(div_le_one hsum).mpr (by linarith) |
| refine ⟨q,?_,?_⟩ |
| · change dist q a≤r |
| dsimp [q] |
| rw [dist_lineMap_left,Real.norm_eq_abs,abs_of_nonneg ht,hd,div_mul_cancel₀ _ hsum.ne'] |
| · change dist q b ≤ s |
| dsimp [q] |
| rw [dist_lineMap_right,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr ht'),hd] |
| have he : (1-r/(r+s))*(r+s)=s:=by field_simp; ring |
| exact he.le |
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| lemma transformed_disks_touch (N : OnePoint ℂ ≃ₜ OnePoint ℂ) |
| {a b a' b' : ℂ} {r s r' s' : ℝ} (hr : 0<r) (hs : 0<s) |
| (hd : dist a b=r+s) |
| (ha : N '' ((↑) '' closedBall a r : Set (OnePoint ℂ))=(↑) '' closedBall a' r') |
| (hb : N '' ((↑) '' closedBall b s : Set (OnePoint ℂ))=(↑) '' closedBall b' s') : |
| dist a' b'≤r'+s' := by |
| obtain ⟨q,hqa,hqb⟩:=tangent_disk_meet hr hs hd |
| have ha' : N (q:OnePoint ℂ)∈((↑) '' closedBall a' r' : Set (OnePoint ℂ)):= |
| ha ▸ ⟨(q:OnePoint ℂ),⟨q,hqa,rfl⟩,rfl⟩ |
| have hb' : N (q:OnePoint ℂ)∈((↑) '' closedBall b' s' : Set (OnePoint ℂ)):= |
| hb ▸ ⟨(q:OnePoint ℂ),⟨q,hqb,rfl⟩,rfl⟩ |
| obtain ⟨u,hu,heu⟩:=ha' |
| obtain ⟨v,hv,hev⟩:=hb' |
| have huv : u=v:=OnePoint.coe_injective (heu.trans hev.symm) |
| subst v |
| have ht:=dist_triangle a' u b' |
| rw [dist_comm a' u] at ht |
| exact ht.trans (add_le_add hu hv) |
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| lemma packing_balls_disjoint {ν : Type*} (z : ν → ℂ) (r : ν → ℝ) |
| (hsep : ∀ i j,i≠j → r i+r j≤dist (z i) (z j)) : |
| Pairwise fun i j=>Disjoint (ball (z i) (r i)) (ball (z j) (r j)) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| intro q hqi hqj |
| have h1:=mem_ball.mp hqi |
| have h2:=mem_ball.mp hqj |
| have ht:=dist_triangle (z i) q (z j) |
| rw [dist_comm (z i) q] at ht |
| have hs:=hsep i j hij |
| linarith |
| end CannonTileTopology |
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| end |
| end |
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| end OAI |
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