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f205175 | 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.MovingStrips
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma exists_int_coefficient_gt (a r : ℝ) (ha : a≠0) : ∃ m : ℤ, r<a*m := by
rcases lt_or_gt_of_ne ha with ha|ha
· obtain ⟨m,hm⟩ := exists_int_lt (r/a)
exact ⟨m,by have := (lt_div_iff_of_neg ha).mp hm; nlinarith only [this]⟩
· obtain ⟨m,hm⟩ := exists_int_gt (r/a)
exact ⟨m,by have := (div_lt_iff₀ ha).mp hm; nlinarith only [this]⟩
lemma integer_coordinates_direction_unbounded (pi : V → ℂ)
(honto : ∀ m n : ℤ, ∃ x, (pi x).re=m ∧ (pi x).im=n)
(a b : ℝ) (hab : a^2+b^2=1) :
∀ r : ℕ, ∃ x, (r : ℝ)<normalCoordinate (fun x => (pi x).re) (fun x => (pi x).im) a b x := by
intro r
by_cases ha : a=0
· have hb : -b≠0 := by intro hh; have : b=0 := neg_eq_zero.mp hh; simp [ha,this] at hab
obtain ⟨m,hm⟩ := exists_int_coefficient_gt (-b) r hb
obtain ⟨x,hx,hy⟩ := honto m 0
exact ⟨x,by simpa [normalCoordinate,hx,hy] using hm⟩
· obtain ⟨n,hn⟩ := exists_int_coefficient_gt a r ha
obtain ⟨x,hx,hy⟩ := honto 0 n
exact ⟨x,by simpa [normalCoordinate,hx,hy] using hn⟩
lemma rotated_coordinate_inverse (u v : V → ℝ) (a b : ℝ) (hab : a^2+b^2=1) (x : V) :
u x=a*parallelCoordinate u v a b x-b*normalCoordinate u v a b x ∧
v x=b*parallelCoordinate u v a b x+a*normalCoordinate u v a b x := by
constructor <;> dsimp [parallelCoordinate,normalCoordinate] <;>
nlinarith [congrArg (fun t => t*u x) hab,congrArg (fun t => t*v x) hab]
lemma rotated_box_bound (u v : V → ℝ) (a b U W : ℝ) (hab : a^2+b^2=1) (x : V)
(hx : |parallelCoordinate u v a b x|≤U) (hy : |normalCoordinate u v a b x|≤W) :
|u x|≤|a| *U+|b| *W ∧ |v x|≤|b| *U+|a| *W := by
obtain ⟨he,hf⟩ := rotated_coordinate_inverse u v a b hab x
constructor
· rw [he]
apply (abs_sub _ _).trans
rw [abs_mul,abs_mul]
exact add_le_add (mul_le_mul_of_nonneg_left hx (abs_nonneg a))
(mul_le_mul_of_nonneg_left hy (abs_nonneg b))
· rw [hf]
apply (abs_add_le _ _).trans
rw [abs_mul,abs_mul]
exact add_le_add (mul_le_mul_of_nonneg_left hx (abs_nonneg b))
(mul_le_mul_of_nonneg_left hy (abs_nonneg a))
lemma rotated_no_bounded_ray (u v : V → ℝ) (a b : ℝ) (hab : a^2+b^2=1) (ω : Set E)
(h : ¬∃ r : G.OpenRay ω, ∃ U W : ℝ, ∀ n, |u (r.vertex n)|≤U ∧ |v (r.vertex n)|≤W) :
¬∃ r : G.OpenRay ω, ∃ U W : ℝ, ∀ n,
|parallelCoordinate u v a b (r.vertex n)|≤U ∧ |normalCoordinate u v a b (r.vertex n)|≤W := by
rintro ⟨r,U,W,hr⟩
exact h ⟨r,|a| *U+|b| *W,|b| *U+|a| *W,fun n => rotated_box_bound u v a b U W hab _ (hr n).1 (hr n).2⟩
lemma coordinate_linear_difference (u v : V → ℝ) (a b : ℝ) (x y : V) :
parallelCoordinate u v a b x-parallelCoordinate u v a b y=a*(u x-u y)+b*(v x-v y) ∧
normalCoordinate u v a b x-normalCoordinate u v a b y= -b*(u x-u y)+a*(v x-v y) := by
constructor <;> dsimp [parallelCoordinate,normalCoordinate] <;> ring
lemma rotated_coordinate_increments (u v : V → ℝ) (a b D : ℝ) (hab : a^2+b^2=1) (hD : 0≤D)
(hu : ∀ ⦃x y⦄, G.fullGraph.Adj x y → |u x-u y|≤D)
(hv : ∀ ⦃x y⦄, G.fullGraph.Adj x y → |v x-v y|≤D) :
(∀ ⦃x y⦄, G.fullGraph.Adj x y → |parallelCoordinate u v a b x-parallelCoordinate u v a b y|≤2*D) ∧
(∀ ⦃x y⦄, G.fullGraph.Adj x y → |normalCoordinate u v a b x-normalCoordinate u v a b y|≤2*D) := by
obtain ⟨ha,ha',hb,hb'⟩ := unit_pair_bounds hab
have haa : |a|≤1 := abs_le.mpr ⟨ha,ha'⟩
have hbb : |b|≤1 := abs_le.mpr ⟨hb,hb'⟩
have bound : ∀ A B : ℝ, |A|≤1 → |B|≤1 → ∀ x y, G.fullGraph.Adj x y →
|A*(u x-u y)+B*(v x-v y)|≤2*D := by
intro A B hA hB x y hxy
apply (abs_add_le _ _).trans
rw [abs_mul,abs_mul]
have hu' := mul_le_mul_of_nonneg_left (hu hxy) (abs_nonneg A)
have hv' := mul_le_mul_of_nonneg_left (hv hxy) (abs_nonneg B)
nlinarith [mul_le_mul_of_nonneg_right hA hD,mul_le_mul_of_nonneg_right hB hD]
constructor
· intro x y hxy
rw [(coordinate_linear_difference u v a b x y).1]
exact bound a b haa hbb x y hxy
· intro x y hxy
rw [(coordinate_linear_difference u v a b x y).2]
exact bound (-b) a (by simpa only [abs_neg] using hbb) haa x y hxy
lemma rotated_aut_differences (u v : V → ℝ) (a b : ℝ) (phi : G.Aut)
(hu : ∀ x y, u (phi.vertex x)-u (phi.vertex y)=u x-u y)
(hv : ∀ x y, v (phi.vertex x)-v (phi.vertex y)=v x-v y) :
(∀ x y, parallelCoordinate u v a b (phi.vertex x)-parallelCoordinate u v a b (phi.vertex y)=
parallelCoordinate u v a b x-parallelCoordinate u v a b y) ∧
(∀ x y, normalCoordinate u v a b (phi.vertex x)-normalCoordinate u v a b (phi.vertex y)=
normalCoordinate u v a b x-normalCoordinate u v a b y) := by
constructor
· intro x y
rw [(coordinate_linear_difference u v a b _ _).1,(coordinate_linear_difference u v a b x y).1,hu,hv]
· intro x y
rw [(coordinate_linear_difference u v a b _ _).2,(coordinate_linear_difference u v a b x y).2,hu,hv]
end CriticalPercolation.BondGraph
end
end
end
end OAI
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