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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | import OAI.Geometry.CylinderCovering.Caps.CapsDefinitions
namespace OAI
namespace CylinderCovering.HalfHeightCaps.CompensationEstimates
lemma square_le_one {x : ℝ} (hx : |x| ≤ 1) : x^2 ≤ 1 := by
nlinarith only [sq_abs x, abs_nonneg x, hx]
lemma alpha_bounds {r : ℝ} (hr : |r| ≤ 1) : 0 ≤ alpha r ∧ alpha r ≤ 1/2 := by
have h := square_le_one hr
unfold alpha
constructor
· positivity
· linarith
lemma profile_bounds {r : ℝ} (hr : |r| ≤ 1) : 0 ≤ F r ∧ F r ≤ 1/8 := by
have h := square_le_one hr
have hprod := mul_nonneg (sq_nonneg r) (sub_nonneg.mpr h)
unfold F
constructor
· positivity
· nlinarith only [h, sq_nonneg r, hprod]
lemma abs_product_le_one {x r : ℝ} (hx : |x| ≤ 1) (hr : |r| ≤ 1) :
|x*r| ≤ 1 := by
rw [abs_mul]
nlinarith only [hx, hr, abs_nonneg x, abs_nonneg r,
mul_nonneg (sub_nonneg.mpr hx) (abs_nonneg r)]
lemma weight_bounds {δ x r : ℝ} (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2)
(hx : |x| ≤ 1) (hr : |r| ≤ 1) :
1-δ ≤ 1-δ*x*r ∧ 1-δ ≤ 1+δ*x*r ∧
1/2 ≤ 1-δ*x*r ∧ 1/2 ≤ 1+δ*x*r := by
have hp := abs_product_le_one hx hr
have hlow := mul_le_mul_of_nonneg_left (abs_le.mp hp).1 hδ
have hhigh := mul_le_mul_of_nonneg_left (abs_le.mp hp).2 hδ
refine ⟨?_, ?_, ?_, ?_⟩ <;> nlinarith only [hlow,hhigh,hδsmall]
lemma weighted_square_le_half {x r : ℝ} (hx : |x| ≤ 1) (hr : |r| ≤ 1) :
x^2 * alpha r ≤ 1/2 := by
have hs := square_le_one hx
obtain ⟨ha,ha'⟩ := alpha_bounds hr
have hm := mul_le_mul_of_nonneg_right hs ha
nlinarith only [hm,ha']
lemma identity {δ X Y Z r s u v : ℝ}
(hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r)
(hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) :
(u-1/2)*(1-δ*X*r)+(v-1/2)*(1+δ*Y*s) =
δ^2*(X^2*alpha r+Y^2*alpha s) := by
unfold alpha beta at *
linear_combination δ*X*hY-δ*Y*hX-hZ-hZv
theorem radial_excess_bounds {δ X Y Z r s u v : ℝ}
(hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2)
(hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1)
(hu : 1/2 ≤ u) (hv : 1/2 ≤ v)
(hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r)
(hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) :
0 ≤ u-1/2 ∧ u-1/2 ≤ 2*δ^2 ∧ 0 ≤ v-1/2 ∧ v-1/2 ≤ 2*δ^2 := by
have hcomp := identity hY hZ hX hZv
have hwU := (weight_bounds hδ hδsmall hXb hr).2.2.1
have hwV := (weight_bounds hδ hδsmall hYb hs).2.2.2
have hA : X^2*alpha r+Y^2*alpha s ≤ 1 := by
linarith only [weighted_square_le_half hXb hr, weighted_square_le_half hYb hs]
have hsum := mul_le_mul_of_nonneg_left hA (sq_nonneg δ)
have hu0 : 0 ≤ u-1/2 := by linarith
have hv0 : 0 ≤ v-1/2 := by linarith
have hU := mul_le_mul_of_nonneg_left hwU hu0
have hV := mul_le_mul_of_nonneg_left hwV hv0
exact ⟨hu0, by nlinarith only [hcomp,hsum,hU,hV,hv0],
hv0, by nlinarith only [hcomp,hsum,hU,hV,hu0]⟩
lemma radial_coordinate_error {δ x r u y : ℝ}
(hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2)
(hx : |x| ≤ 1) (hr : |r| ≤ 1)
(hu0 : 0 ≤ u-1/2) (hu : u-1/2 ≤ 2*δ^2)
(hy : y = u*r+δ*x*alpha r) : |y-r/2| ≤ 3*δ/2 := by
obtain ⟨ha,ha'⟩ := alpha_bounds hr
have heq : y-r/2 = (u-1/2)*r+δ*x*alpha r := by rw [hy]; ring
have hfirst : |(u-1/2)*r| ≤ 2*δ^2 := by
rw [abs_mul,abs_of_nonneg hu0]
calc
(u-1/2)*|r| ≤ (u-1/2)*1 := mul_le_mul_of_nonneg_left hr hu0
_ ≤ 2*δ^2 := by simpa using hu
have hsecond : |δ*x*alpha r| ≤ δ/2 := by
rw [abs_mul,abs_mul,abs_of_nonneg hδ,abs_of_nonneg ha]
calc
δ * |x| * alpha r ≤ δ*1*(1/2) := by gcongr
_ = δ/2 := by ring
rw [heq]
have htriangle := abs_add_le ((u-1/2)*r) (δ*x*alpha r)
have hquad := mul_le_mul_of_nonneg_left hδsmall hδ
nlinarith only [htriangle,hfirst,hsecond,hquad]
theorem angular_errors {δ X Y Z r s u v : ℝ}
(hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2)
(hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1)
(hu : 1/2 ≤ u) (hv : 1/2 ≤ v)
(hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r)
(hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) :
|Y-r/2| ≤ 3*δ/2 ∧ |X-s/2| ≤ 3*δ/2 := by
obtain ⟨hu0,hu',hv0,hv'⟩ := radial_excess_bounds hδ hδsmall hXb hYb hr hs hu hv hY hZ hX hZv
refine ⟨radial_coordinate_error hδ hδsmall hXb hr hu0 hu' hY, ?_⟩
apply radial_coordinate_error hδ hδsmall (x := -Y) (by simpa using hYb) hs hv0 hv'
nlinarith only [hX]
lemma squared_error {δ x s : ℝ} (hδ : 0 ≤ δ) (hx : |x| ≤ 1) (hs : |s| ≤ 1)
(he : |x-s/2| ≤ 3*δ/2) : x^2 ≤ s^2/4+9*δ/4 := by
have hsum : |x+s/2| ≤ 3/2 := by
calc
|x+s/2| ≤ |x|+|s/2| := abs_add_le _ _
_ ≤ 1+1/2 := by rw [abs_div]; norm_num; linarith
_ = 3/2 := by norm_num
have hprod : |(x-s/2)*(x+s/2)| ≤ (3*δ/2)*(3/2) := by
rw [abs_mul]
exact mul_le_mul he hsum (abs_nonneg _) (by positivity)
have hid : (x-s/2)*(x+s/2)=x^2-s^2/4 := by ring
rw [hid] at hprod
nlinarith only [(abs_le.mp hprod).2]
lemma monotone_profile_comparison (r s : ℝ) :
(s^2*alpha r+r^2*alpha s)/4 ≤ F r+F s := by
unfold alpha F
nlinarith only [sq_nonneg (r^2-s^2)]
theorem weighted_comparison {δ X Y r s : ℝ} (hδ : 0 ≤ δ)
(hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1)
(hYr : |Y-r/2| ≤ 3*δ/2) (hXs : |X-s/2| ≤ 3*δ/2) :
X^2*alpha r+Y^2*alpha s ≤ (s^2*alpha r+r^2*alpha s)/4+9*δ/4 ∧
X^2*alpha r+Y^2*alpha s ≤ F r+F s+9*δ/4 := by
obtain ⟨har,har'⟩ := alpha_bounds hr
obtain ⟨has,has'⟩ := alpha_bounds hs
have hx := mul_le_mul_of_nonneg_right (squared_error hδ hXb hs hXs) har
have hy := mul_le_mul_of_nonneg_right (squared_error hδ hYb hr hYr) has
have hsum := mul_le_mul_of_nonneg_left (add_le_add har' has') hδ
have hfirst : X^2*alpha r+Y^2*alpha s ≤
(s^2*alpha r+r^2*alpha s)/4+9*δ/4 := by nlinarith only [hx,hy,hsum]
exact ⟨hfirst, hfirst.trans (by linarith only [monotone_profile_comparison r s])⟩
theorem cubic_padding_lower {δ r s : ℝ} (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2)
(hr : |r| ≤ 1) (hs : |s| ≤ 1) :
F r+F s+15*δ/4 ≤ (1-δ)*(F r+F s+8*δ) := by
have hrF := (profile_bounds hr).2
have hsF := (profile_bounds hs).2
have hcoef : 0 ≤ 17/4-8*δ-(F r+F s) := by linarith
have hprod := mul_nonneg hδ hcoef
nlinarith only [hprod]
theorem cubic_failure_lower {δ X Y r s u v : ℝ}
(hδ : 0 < δ) (hδsmall : δ ≤ 1/2)
(hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1)
(hu : cubicCap δ r < u) (hv : cubicCap δ s < v) :
δ^2*(F r+F s+15*δ/4) <
(u-1/2)*(1-δ*X*r)+(v-1/2)*(1+δ*Y*s) := by
have hδone : 0 < 1-δ := by linarith
have hFr := (profile_bounds hr).1
have hFs := (profile_bounds hs).1
have hu' : δ^2*F r+4*δ^3 < u-1/2 := by unfold cubicCap at hu; linarith
have hv' : δ^2*F s+4*δ^3 < v-1/2 := by unfold cubicCap at hv; linarith
have hcapr : 0 ≤ δ^2*F r+4*δ^3 := by positivity
have hu0 : 0 ≤ u-1/2 := hcapr.trans hu'.le
have hcaps : 0 ≤ δ^2*F s+4*δ^3 := by positivity
have hv0 : 0 ≤ v-1/2 := hcaps.trans hv'.le
have hUstrict := mul_lt_mul_of_pos_right hu' hδone
have hVstrict := mul_lt_mul_of_pos_right hv' hδone
have hUweak := mul_le_mul_of_nonneg_left (weight_bounds hδ.le hδsmall hXb hr).1 hu0
have hVweak := mul_le_mul_of_nonneg_left (weight_bounds hδ.le hδsmall hYb hs).2.1 hv0
have hpad := mul_le_mul_of_nonneg_left (cubic_padding_lower hδ.le hδsmall hr hs)
(sq_nonneg δ)
nlinarith only [hUstrict,hVstrict,hUweak,hVweak,hpad]
theorem cubic_failures_incompatible {δ X Y Z r s u v : ℝ}
(hδ : 0 < δ) (hδsmall : δ ≤ 1/2)
(hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1)
(hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r)
(hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) :
¬ (cubicCap δ r < u ∧ cubicCap δ s < v) := by
rintro ⟨hu,hv⟩
have hFr := (profile_bounds hr).1
have hFs := (profile_bounds hs).1
have huc : 1/2 ≤ cubicCap δ r := by
have hnonneg : 0 ≤ δ^2*F r+4*δ^3 := by positivity
unfold cubicCap
linarith
have hvc : 1/2 ≤ cubicCap δ s := by
have hnonneg : 0 ≤ δ^2*F s+4*δ^3 := by positivity
unfold cubicCap
linarith
obtain ⟨hYr,hXs⟩ := angular_errors hδ.le hδsmall hXb hYb hr hs
(huc.trans hu.le) (hvc.trans hv.le) hY hZ hX hZv
have hupper := (weighted_comparison hδ.le hXb hYb hr hs hYr hXs).2
have hlower := cubic_failure_lower hδ hδsmall hXb hYb hr hs hu hv
rw [identity hY hZ hX hZv] at hlower
have hupper' := mul_le_mul_of_nonneg_left hupper (sq_nonneg δ)
have hcube : 0 < δ^3 := pow_pos hδ 3
nlinarith only [hlower,hupper',hcube]
end CylinderCovering.HalfHeightCaps.CompensationEstimates
end OAI
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