File size: 7,602 Bytes
785c727 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | import OAI.Geometry.CylinderCovering.Convexity.NormalFrameDefinitions
namespace OAI
noncomputable section
open Finset
open scoped BigOperators InnerProductSpace
namespace CylinderNormalFrame
theorem positive_mass_eq_negative_mass {ι : Type*} [Fintype ι]
(x : ι → ℝ) (hx : ∑ i, x i = 0) :
∑ i, max (x i) 0 = ∑ i, max (-x i) 0 := by
have hpoint (i : ι) : max (x i) 0 - max (-x i) 0 = x i := by
by_cases hi : 0 ≤ x i
· rw [max_eq_left hi, max_eq_right (by linarith)]
ring
· rw [max_eq_right (le_of_not_ge hi), max_eq_left (by linarith)]
ring
have hsum := congrArg (fun f : ι → ℝ => ∑ i, f i) (funext hpoint)
simp only [sum_sub_distrib, hx] at hsum
linarith
theorem sum_abs_eq_twice_positive_mass {ι : Type*} [Fintype ι]
(x : ι → ℝ) (hx : ∑ i, x i = 0) :
∑ i, |x i| = 2 * ∑ i, max (x i) 0 := by
have hpoint (i : ι) : |x i| = max (x i) 0 + max (-x i) 0 := by
by_cases hi : 0 ≤ x i
· rw [abs_of_nonneg hi, max_eq_left hi, max_eq_right (by linarith)]
ring
· rw [abs_of_nonpos (le_of_not_ge hi), max_eq_right (le_of_not_ge hi),
max_eq_left (by linarith)]
ring
simp_rw [hpoint, sum_add_distrib]
rw [positive_mass_eq_negative_mass x hx]
ring
theorem sum_sq_le_half_sq_sum_abs {ι : Type*} [Fintype ι]
(x : ι → ℝ) (hx : ∑ i, x i = 0) :
∑ i, x i ^ 2 ≤ (∑ i, |x i|) ^ 2 / 2 := by
have hpoint (i : ι) : x i ^ 2 = (max (x i) 0) ^ 2 + (max (-x i) 0) ^ 2 := by
by_cases hi : 0 ≤ x i
· rw [max_eq_left hi, max_eq_right (by linarith)]
ring
· rw [max_eq_right (le_of_not_ge hi), max_eq_left (by linarith)]
ring
have hp : ∑ i, (max (x i) 0) ^ 2 ≤ (∑ i, max (x i) 0) ^ 2 :=
sum_sq_le_sq_sum_of_nonneg (fun i _ => le_max_right _ _)
have hn : ∑ i, (max (-x i) 0) ^ 2 ≤ (∑ i, max (-x i) 0) ^ 2 :=
sum_sq_le_sq_sum_of_nonneg (fun i _ => le_max_right _ _)
rw [← positive_mass_eq_negative_mass x hx] at hn
rw [sum_abs_eq_twice_positive_mass x hx]
simp_rw [hpoint, sum_add_distrib]
nlinarith
theorem sum_normal : ∑ i : Fin 4, normal i = 0 := by
ext j
fin_cases j <;> simp [normal, Fin.sum_univ_succ]
theorem inner_normal_zero (u : Space) :
inner ℝ u (normal 0) = (Real.sqrt 2 * u 1 - u 2) / Real.sqrt 3 := by
simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv]
ring
theorem inner_normal_one (u : Space) :
inner ℝ u (normal 1) = (-Real.sqrt 2 * u 1 - u 2) / Real.sqrt 3 := by
simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv]
ring
theorem inner_normal_two (u : Space) :
inner ℝ u (normal 2) = (Real.sqrt 2 * u 0 + u 2) / Real.sqrt 3 := by
simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv]
ring
theorem inner_normal_three (u : Space) :
inner ℝ u (normal 3) = (-Real.sqrt 2 * u 0 + u 2) / Real.sqrt 3 := by
simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv]
ring
theorem sum_sq_inner_normal (u : Space) :
∑ i : Fin 4, (inner ℝ u (normal i)) ^ 2 = (4 / 3 : ℝ) * ‖u‖ ^ 2 := by
rw [EuclideanSpace.real_norm_sq_eq]
simp only [Fin.sum_univ_four, inner_normal_zero, inner_normal_one,
inner_normal_two, inner_normal_three]
simp only [Fin.sum_univ_three]
have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num)
have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num)
simp only [div_pow, h3]
nlinarith [sq_nonneg (u 0), sq_nonneg (u 1), sq_nonneg (u 2)]
theorem normal_unit (i : Fin 4) : ‖normal i‖ = 1 := by
have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num)
have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num)
have hs : ‖normal i‖ ^ 2 = 1 := by
rw [EuclideanSpace.real_norm_sq_eq]
fin_cases i <;> norm_num [normal, Fin.sum_univ_succ, mul_pow, h3, h2]
nlinarith [norm_nonneg (normal i)]
theorem frameCost_lower_bound (u : Space) (hu : ‖u‖ = 1) :
Real.sqrt 2 ≤ frameCost u := by
have hsum : ∑ i : Fin 4, inner ℝ u (normal i) = 0 := by
rw [← inner_sum, sum_normal, inner_zero_right]
have hsq := sum_sq_le_half_sq_sum_abs (fun i : Fin 4 => inner ℝ u (normal i)) hsum
rw [sum_sq_inner_normal, hu] at hsq
have hnonneg : 0 ≤ ∑ i : Fin 4, |inner ℝ u (normal i)| :=
sum_nonneg (fun i _ => abs_nonneg _)
have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num)
apply Real.sqrt_le_iff.mpr
constructor
· unfold frameCost
positivity
· unfold frameCost
nlinarith [sq_nonneg (∑ i : Fin 4, |inner ℝ u (normal i)|)]
theorem edgeDirection_unit : ‖edgeDirection‖ = 1 := by
simp [edgeDirection, EuclideanSpace.norm_eq, Fin.sum_univ_succ]
theorem frameCost_edgeDirection : frameCost edgeDirection = Real.sqrt 2 := by
have h3 : Real.sqrt 3 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num))
have h2 : 0 ≤ Real.sqrt 2 / Real.sqrt 3 := by positivity
simp only [frameCost, Fin.sum_univ_four, inner_normal_zero, inner_normal_one,
inner_normal_two, inner_normal_three]
simp [edgeDirection, abs_of_nonneg h2, neg_div]
field_simp
norm_num
theorem barycentricCost_lower_bound (u : BarycentricSpace)
(hu : ‖u‖ = 1) (hsum : ∑ i : Fin 4, u i = 0) :
1 / Real.sqrt 2 ≤ barycentricCost u := by
have hsq := sum_sq_le_half_sq_sum_abs (fun i : Fin 4 => u i) hsum
rw [← EuclideanSpace.real_norm_sq_eq, hu] at hsq
have hnonneg : 0 ≤ ∑ i : Fin 4, |u i| := sum_nonneg (fun i _ => abs_nonneg _)
have hbound : Real.sqrt 2 ≤ ∑ i : Fin 4, |u i| := by
apply Real.sqrt_le_iff.mpr
exact ⟨hnonneg, by nlinarith⟩
have h2pos : 0 < Real.sqrt 2 := Real.sqrt_pos.2 (by norm_num)
have h2sq : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num)
have hid : 1 / Real.sqrt 2 = Real.sqrt 2 / 2 := by
apply (div_eq_iff h2pos.ne').2
nlinarith
rw [hid]
unfold barycentricCost
linarith
theorem sum_normal_mul (j k : Fin 3) :
∑ i : Fin 4, normal i j * normal i k = if j = k then (4 / 3 : ℝ) else 0 := by
have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num)
have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num)
have h3ne : Real.sqrt 3 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num))
fin_cases j <;> fin_cases k <;>
simp [normal, Fin.sum_univ_succ] <;> field_simp <;> nlinarith
theorem isLeast_frameCost :
IsLeast {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = frameCost u} (Real.sqrt 2) := by
constructor
· exact ⟨edgeDirection, edgeDirection_unit, frameCost_edgeDirection.symm⟩
· rintro a ⟨u, hu, rfl⟩
exact frameCost_lower_bound u hu
theorem barycentricEdge_sum : ∑ i : Fin 4, barycentricEdge i = 0 := by
simp [barycentricEdge, Fin.sum_univ_succ]
theorem barycentricEdge_unit : ‖barycentricEdge‖ = 1 := by
have hs : ‖barycentricEdge‖ ^ 2 = 1 := by
rw [EuclideanSpace.real_norm_sq_eq]
norm_num [barycentricEdge, Fin.sum_univ_succ, div_pow, Real.sq_sqrt]
nlinarith [norm_nonneg barycentricEdge]
theorem barycentricCost_edge : barycentricCost barycentricEdge = 1 / Real.sqrt 2 := by
simp [barycentricCost, barycentricEdge, Fin.sum_univ_succ,
abs_of_nonneg (Real.sqrt_nonneg (2 : ℝ))]
ring
theorem isLeast_barycentricCost :
IsLeast {a : ℝ | ∃ u : BarycentricSpace,
‖u‖ = 1 ∧ ∑ i : Fin 4, u i = 0 ∧ a = barycentricCost u} (1 / Real.sqrt 2) := by
constructor
· exact ⟨barycentricEdge, barycentricEdge_unit, barycentricEdge_sum,
barycentricCost_edge.symm⟩
· rintro a ⟨u, hu, hsum, rfl⟩
exact barycentricCost_lower_bound u hu hsum
end CylinderNormalFrame
end
end OAI
|