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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
namespace OAI
section
section
noncomputable section
open scoped Topology
namespace CriticalPercolation.EuclideanEllipse
def diagonal (a b : ℝ) (z : ℂ) : ℂ := ⟨a*z.re,b*z.im⟩
lemma diagonal_add (a b : ℝ) (z w : ℂ) :
diagonal a b (z+w)=diagonal a b z+diagonal a b w := by
apply Complex.ext <;> simp [diagonal,mul_add]
lemma diagonal_sub (a b : ℝ) (z w : ℂ) :
diagonal a b (z-w)=diagonal a b z-diagonal a b w := by
apply Complex.ext <;> simp [diagonal,mul_sub]
lemma diagonal_comp (a b c d : ℝ) (z : ℂ) :
diagonal a b (diagonal c d z)=diagonal (a*c) (b*d) z := by
apply Complex.ext <;> simp [diagonal,mul_assoc]
lemma diagonal_one (z : ℂ) : diagonal 1 1 z=z := by apply Complex.ext <;> simp [diagonal]
lemma diagonal_neg (a b : ℝ) (z : ℂ) : diagonal a b (-z)= -diagonal a b z := by
apply Complex.ext <;> simp [diagonal]
lemma diagonal_norm_le (a b M : ℝ) (hM : 0≤M) (ha : |a|≤M) (hb : |b|≤M) (z : ℂ) :
‖diagonal a b z‖≤M*‖z‖ := by
have ha2 : a^2≤M^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr ha
have hb2 : b^2≤M^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr hb
have hsq : ‖diagonal a b z‖^2≤(M*‖z‖)^2 := by
rw [Complex.sq_norm,Complex.normSq_apply,mul_pow,Complex.sq_norm,Complex.normSq_apply]
dsimp [diagonal]
nlinarith [mul_le_mul_of_nonneg_right ha2 (sq_nonneg z.re),
mul_le_mul_of_nonneg_right hb2 (sq_nonneg z.im)]
exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hM (norm_nonneg _))).mp hsq
def normalize (R W : ℝ) (c : Circle) (z : ℂ) : ℂ :=
(c : ℂ)*diagonal (1/R) (1/W) ((c : ℂ)⁻¹*z)
def expand (R W : ℝ) (c : Circle) (z : ℂ) : ℂ :=
(c : ℂ)*diagonal R W ((c : ℂ)⁻¹*z)
lemma normalize_sub (R W : ℝ) (c : Circle) (z w : ℂ) :
normalize R W c (z-w)=normalize R W c z-normalize R W c w := by
simp only [normalize,mul_sub,diagonal_sub]
lemma expand_normalize (R W : ℝ) (hR : R≠0) (hW : W≠0) (c : Circle) (z : ℂ) :
expand R W c (normalize R W c z)=z := by
simp only [normalize,expand,←mul_assoc,inv_mul_cancel₀ c.coe_ne_zero,one_mul,
diagonal_comp,mul_one_div_cancel hR,mul_one_div_cancel hW,diagonal_one]
rw [mul_assoc,mul_inv_cancel_left₀ c.coe_ne_zero]
lemma normalize_expand (R W : ℝ) (hR : R≠0) (hW : W≠0) (c : Circle) (z : ℂ) :
normalize R W c (expand R W c z)=z := by
simp only [normalize,expand,←mul_assoc,inv_mul_cancel₀ c.coe_ne_zero,one_mul,
diagonal_comp,one_div_mul_cancel hR,one_div_mul_cancel hW,diagonal_one]
rw [mul_assoc,mul_inv_cancel_left₀ c.coe_ne_zero]
lemma normalize_norm_le (R W : ℝ) (hW : 0<W) (hWR : W≤R) (c : Circle) (z : ℂ) :
‖normalize R W c z‖≤‖z‖/W := by
have hR : 0<R := hW.trans_le hWR
have h1 : |1/R|≤1/W := by rw [abs_of_pos (one_div_pos.mpr hR)]; exact one_div_le_one_div_of_le hW hWR
have h2 : |1/W|≤1/W := (abs_of_pos (one_div_pos.mpr hW)).le
have h := diagonal_norm_le (1/R) (1/W) (1/W) (by positivity) h1 h2 ((c : ℂ)⁻¹*z)
simpa [normalize,norm_mul,Circle.norm_coe,div_eq_mul_inv,mul_comm]
using h
lemma expand_norm_le (R W : ℝ) (hW : 0≤W) (hWR : W≤R) (c : Circle) (z : ℂ) :
‖expand R W c z‖≤R*‖z‖ := by
have hR := hW.trans hWR
have h := diagonal_norm_le R W R hR (by simp [abs_of_nonneg hR])
(by simpa [abs_of_nonneg hW] using hWR) ((c : ℂ)⁻¹*z)
simpa only [expand,norm_mul,Circle.norm_coe,norm_inv,inv_one,one_mul] using h
lemma normalize_center (W : ℝ) (c : Circle) (z : ℂ) :
normalize W W c z=(1/W : ℝ)•z := by
have he : ∀ w : ℂ, diagonal (1/W) (1/W) w=(1/W : ℝ)•w := by intro w; apply Complex.ext <;> simp [diagonal]
rw [normalize,he,mul_smul_comm,←mul_assoc,mul_inv_cancel₀ c.coe_ne_zero,one_mul]
lemma normalize_neg (R W : ℝ) (c : Circle) (z : ℂ) :
normalize R W (-c) z=normalize R W c z := by
simp only [normalize,Circle.coe_neg,inv_neg,neg_mul,diagonal_neg,mul_neg,neg_neg]
lemma normalize_components (R W : ℝ) (c : Circle) (z : ℂ) :
(((c : ℂ)⁻¹*normalize R W c z).re=(1/R)*((c : ℂ)⁻¹*z).re) ∧
(((c : ℂ)⁻¹*normalize R W c z).im=(1/W)*((c : ℂ)⁻¹*z).im) := by
simp [normalize,diagonal]
lemma normalize_origin (R W : ℝ) (c : Circle) : normalize R W c 0=0 := by
simp [normalize,diagonal,Complex.ext_iff]
lemma normalize_real_linear (R W : ℝ) (c : Circle) (z w : ℂ) (a b : ℝ) :
normalize R W c (a•z+b•w)=a•normalize R W c z+b•normalize R W c w := by
have hd : ∀ q : ℂ, ∀ t : ℝ, diagonal (1/R) (1/W) (t•q)=t•diagonal (1/R) (1/W) q := by
intro q t
apply Complex.ext <;> simp [diagonal,Complex.mul_re,Complex.mul_im] <;> ring
simp only [normalize,mul_add,mul_smul_comm,diagonal_add,hd]
end CriticalPercolation.EuclideanEllipse
end
end
end
end OAI
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