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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import Mathlib
namespace OAI
noncomputable section
namespace Problem367
def binaryFusion (u v : ℝ) : ℝ := (u + v) / (1 + u * v)
def binaryFusionLoss (u v : ℝ) : ℝ :=
u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) / (1 - u ^ 2 * v ^ 2)
theorem binaryFusion_abs_mul_lt_one {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : |u * v| < 1 := by
rw [abs_mul]
calc
|u| * |v| ≤ |u| * 1 := mul_le_mul_of_nonneg_left hv.le (abs_nonneg u)
_ < 1 := by simpa using hu
theorem binaryFusion_denominator_pos {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : 0 < 1 + u * v := by
have h := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).1
linarith
theorem binaryFusionLoss_denominator_pos {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : 0 < 1 - u ^ 2 * v ^ 2 := by
have h := (sq_lt_one_iff_abs_lt_one (u * v)).mpr
(binaryFusion_abs_mul_lt_one hu hv)
nlinarith
theorem one_sub_binaryFusion_sq {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) :
1 - binaryFusion u v ^ 2 =
(1 - u ^ 2) * (1 - v ^ 2) / (1 + u * v) ^ 2 := by
have hd := ne_of_gt (binaryFusion_denominator_pos hu hv)
unfold binaryFusion
field_simp
ring
theorem abs_binaryFusion_lt_one {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : |binaryFusion u v| < 1 := by
apply (sq_lt_one_iff_abs_lt_one _).mp
have hu' : 0 < 1 - u ^ 2 := by
have := (sq_lt_one_iff_abs_lt_one u).mpr hu
linarith
have hv' : 0 < 1 - v ^ 2 := by
have := (sq_lt_one_iff_abs_lt_one v).mpr hv
linarith
have hd := binaryFusion_denominator_pos hu hv
have hpos : 0 < 1 - binaryFusion u v ^ 2 := by
rw [one_sub_binaryFusion_sq hu hv]
positivity
linarith
theorem binaryFusionLoss_nonneg {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : 0 ≤ binaryFusionLoss u v := by
have hu' := (sq_lt_one_iff_abs_lt_one u).mpr hu
have hv' := (sq_lt_one_iff_abs_lt_one v).mpr hv
have hf : 0 ≤ 2 - u ^ 2 - v ^ 2 := by linarith
have hd := binaryFusionLoss_denominator_pos hu hv
unfold binaryFusionLoss
positivity
theorem binaryFusion_symmetric_square {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) :
((u + v) ^ 2 / (1 + u * v) + (u - v) ^ 2 / (1 - u * v)) / 2 =
u ^ 2 + v ^ 2 - binaryFusionLoss u v := by
have hp := ne_of_gt (binaryFusion_denominator_pos hu hv)
have hm : 1 - u * v ≠ 0 := by
have := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).2
linarith
have hq := ne_of_gt (binaryFusionLoss_denominator_pos hu hv)
unfold binaryFusionLoss
field_simp
ring
theorem binaryFusion_symmetric_square_le {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) :
((u + v) ^ 2 / (1 + u * v) + (u - v) ^ 2 / (1 - u * v)) / 2 ≤
u ^ 2 + v ^ 2 := by
rw [binaryFusion_symmetric_square hu hv]
exact sub_le_self _ (binaryFusionLoss_nonneg hu hv)
theorem binaryFusionLoss_lower_bound {u v theta : ℝ}
(hu : |u| ≤ theta) (hv : |v| ≤ theta) (htheta : theta < 1) :
2 * (1 - theta ^ 2) * u ^ 2 * v ^ 2 ≤ binaryFusionLoss u v := by
have ht0 : 0 ≤ theta := le_trans (abs_nonneg u) hu
have hu' : u ^ 2 ≤ theta ^ 2 := by
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg u) ht0).mpr hu
have hv' : v ^ 2 ≤ theta ^ 2 := by
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg v) ht0).mpr hv
have ht2 : theta ^ 2 < 1 := (sq_lt_one_iff₀ ht0).mpr htheta
have hp : 0 ≤ u ^ 2 * v ^ 2 := mul_nonneg (sq_nonneg u) (sq_nonneg v)
have hf : 0 ≤ 2 - u ^ 2 - v ^ 2 := by linarith
have hd := binaryFusionLoss_denominator_pos (lt_of_le_of_lt hu htheta)
(lt_of_le_of_lt hv htheta)
have hfactor : 2 * (1 - theta ^ 2) ≤ 2 - u ^ 2 - v ^ 2 := by linarith
have hnum : 0 ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) := mul_nonneg hp hf
unfold binaryFusionLoss
calc
2 * (1 - theta ^ 2) * u ^ 2 * v ^ 2 =
(u ^ 2 * v ^ 2) * (2 * (1 - theta ^ 2)) := by ring
_ ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) :=
mul_le_mul_of_nonneg_left hfactor hp
_ ≤ u ^ 2 * v ^ 2 * (2 - u ^ 2 - v ^ 2) / (1 - u ^ 2 * v ^ 2) := by
apply (le_div_iff₀ hd).mpr
exact mul_le_of_le_one_right hnum (by linarith)
theorem binaryFusion_integrand_le_two {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : (u + v) ^ 2 / (1 + u * v) ≤ 2 := by
apply (div_le_iff₀ (binaryFusion_denominator_pos hu hv)).mpr
have hu' := (sq_lt_one_iff_abs_lt_one u).mpr hu
have hv' := (sq_lt_one_iff_abs_lt_one v).mpr hv
nlinarith
theorem binaryFusionLoss_le_sum_sq {u v : ℝ}
(hu : |u| < 1) (hv : |v| < 1) : binaryFusionLoss u v ≤ u ^ 2 + v ^ 2 := by
have hp : 0 ≤ (u + v) ^ 2 / (1 + u * v) :=
div_nonneg (sq_nonneg _) (binaryFusion_denominator_pos hu hv).le
have hmden : 0 < 1 - u * v := by
have := (abs_lt.mp (binaryFusion_abs_mul_lt_one hu hv)).2
linarith
have hm : 0 ≤ (u - v) ^ 2 / (1 - u * v) :=
div_nonneg (sq_nonneg _) hmden.le
have hid := binaryFusion_symmetric_square hu hv
linarith
end Problem367
end
end OAI
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