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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.DiskWinding
namespace OAI
section
section
noncomputable section
open LinearMap
namespace CriticalPercolation.EllipseLinear
def det (a b c d : ℝ) : ℝ := a*d-b*c
def time (a b c d x y : ℝ) : ℝ := ((d-b)*x+(a-c)*y)/det a b c d
def height (a b c d x y : ℝ) : ℝ := ((d+b)*x-(c+a)*y)/det a b c d
lemma center_left (a b c d : ℝ) (h : det a b c d≠0) :
time a b c d a b=1 ∧ height a b c d a b=1 := by
constructor <;> dsimp only [time,height] <;> field_simp [h] <;> unfold det <;> ring
lemma center_right (a b c d : ℝ) (h : det a b c d≠0) :
time a b c d c d=1 ∧ height a b c d c d= -1 := by
constructor <;> dsimp only [time,height] <;> field_simp [h] <;> unfold det <;> ring
lemma det_lower_of_separation (θ ψ : ℝ)
(hsep : Real.pi/4≤|DiskWinding.increment ((2*θ : ℝ) : Real.Angle)
((2*ψ : ℝ) : Real.Angle)|) :
(1 : ℝ)/4≤|det (Real.cos θ) (Real.sin θ) (Real.cos ψ) (Real.sin ψ)| := by
let t : Real.Angle := ((2*ψ : ℝ) : Real.Angle)-((2*θ : ℝ) : Real.Angle)
have ht : t=((2*(ψ-θ) : ℝ) : Real.Angle) := by
dsimp [t]
rw [←Real.Angle.coe_sub]
congr 1
ring
have hcos := Real.cos_le_cos_of_nonneg_of_le_pi
(by positivity : 0≤Real.pi/4) (Real.Angle.abs_toReal_le_pi t) hsep
rw [Real.cos_abs,Real.Angle.cos_toReal,ht,Real.Angle.cos_coe,
Real.cos_two_mul,Real.cos_pi_div_four] at hcos
have hsqrt : Real.sqrt 2≤(7 : ℝ)/4 := by
nlinarith [Real.sq_sqrt (by norm_num : (0 : ℝ)≤2),Real.sqrt_nonneg 2]
have hsq := Real.sin_sq_add_cos_sq (ψ-θ)
have hsin : (1 : ℝ)/4≤|Real.sin (ψ-θ)| := by
nlinarith [sq_abs (Real.sin (ψ-θ)),abs_nonneg (Real.sin (ψ-θ))]
have heq : det (Real.cos θ) (Real.sin θ) (Real.cos ψ) (Real.sin ψ)=
Real.sin (ψ-θ) := by rw [Real.sin_sub]; unfold det; ring
rwa [heq]
lemma row_bound {a b c d : ℝ} (ha : |a|≤1) (hb : |b|≤1)
(hc : |c|≤1) (hd : |d|≤1) (hdet : (1 : ℝ)/4≤|det a b c d|)
(x y q : ℝ) (hq : 0≤q) (hx : |x|≤q) (hy : |y|≤q) :
|time a b c d x y|≤16*q ∧ |height a b c d x y|≤16*q := by
have hpos : 0 < |det a b c d| := by linarith
have hdb : |d-b|≤2 := (abs_sub d b).trans (by linarith)
have hac : |a-c|≤2 := (abs_sub a c).trans (by linarith)
have hdb' : |d+b|≤2 := (abs_add_le d b).trans (by linarith)
have hca : |c+a|≤2 := (abs_add_le c a).trans (by linarith)
have hprod : ∀ z w : ℝ, |z|≤2 → |w|≤q → |z*w|≤2*q := by
intro z w hz hw
rw [abs_mul]
exact mul_le_mul hz hw (abs_nonneg _) (by norm_num)
constructor
· unfold time
rw [abs_div]
apply (div_le_iff₀ hpos).mpr
calc
|(d-b)*x+(a-c)*y|≤|(d-b)*x|+|(a-c)*y| := abs_add_le _ _
_ ≤4*q := by linarith [hprod (d-b) x hdb hx,hprod (a-c) y hac hy]
_ ≤16*q*|det a b c d| := by nlinarith
· unfold height
rw [abs_div]
apply (div_le_iff₀ hpos).mpr
calc
|(d+b)*x-(c+a)*y|≤|(d+b)*x|+|(c+a)*y| := abs_sub _ _
_ ≤4*q := by linarith [hprod (d+b) x hdb' hx,hprod (c+a) y hca hy]
_ ≤16*q*|det a b c d| := by nlinarith
lemma time_sub (a b c d x y x' y' : ℝ) :
time a b c d x y-time a b c d x' y'=time a b c d (x-x') (y-y') := by
unfold time; ring
lemma height_sub (a b c d x y x' y' : ℝ) :
height a b c d x y-height a b c d x' y'=height a b c d (x-x') (y-y') := by
unfold height; ring
def equiv (a b c d : ℝ) (h : det a b c d≠0) : (ℝ×ℝ) ≃ₗ[ℝ] (ℝ×ℝ) where
toFun p := (time a b c d p.1 p.2,height a b c d p.1 p.2)
invFun p := ((a*(p.1+p.2)+c*(p.1-p.2))/2,(b*(p.1+p.2)+d*(p.1-p.2))/2)
left_inv p := by
apply Prod.ext <;> dsimp only [time,height] <;> field_simp [h] <;> unfold det <;> ring
right_inv p := by
apply Prod.ext <;> dsimp only [time,height] <;> field_simp [h] <;> unfold det <;> ring
map_add' p q := by apply Prod.ext <;> dsimp [time,height] <;> ring
map_smul' r p := by apply Prod.ext <;> dsimp [time,height] <;> ring
lemma source_budgets {eps : ℝ} (heps : 0<eps) (heps' : eps<1/10) :
16*(4*(eps/1000))<eps/8 ∧
16*((1+eps/1000)^2)<30 := by constructor <;> nlinarith
end CriticalPercolation.EllipseLinear
end
end
end
end OAI
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