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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import Mathlib
namespace OAI
section
noncomputable section
namespace CannonTileTopology
open Set Metric Topology
open scoped OnePoint
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
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)
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
end
end
end OAI
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