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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 | import Mathlib
namespace OAI
namespace AngularCartesianValidation
theorem cartesian_failure_displacement
{x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1)
(hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1)
(hη : 0 ≤ η)
(hfirst : 1 / 4 + τ * x * y +
τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) < s ^ 2)
(hsecond : 1 / 4 - τ * x * y +
τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) < (1 - s) ^ 2) :
|s - 1 / 2| ≤ 3 * τ ∧ |s - 1 / 2 - τ * x * y| ≤ 10 * τ ^ 2 := by
let u := s - 1 / 2
have hu0 : -(1 / 2 : ℝ) ≤ u := by dsimp [u]; linarith
have hu1 : u ≤ (1 / 2 : ℝ) := by dsimp [u]; linarith
have hx2 : x ^ 2 ≤ 1 := by
nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2]
have hy2 : y ^ 2 ≤ 1 := by
nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2]
have hxy : |x * y| ≤ 1 := by
rw [abs_mul]
nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)]
have htxyl : -τ ≤ τ * x * y := by
nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp hxy).1 hτ.le]
have htxyr : τ * x * y ≤ τ := by
nlinarith only [mul_le_mul_of_nonneg_left (abs_le.mp hxy).2 hτ.le]
have hτsq : τ ^ 2 ≤ τ := by
nlinarith only [mul_nonneg hτ.le (sub_nonneg.mpr hτ1)]
have hA : -(1 / 4 : ℝ) ≤ (y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4 := by
nlinarith only [sq_nonneg y, sq_nonneg (y ^ 2), hx2, hη]
have hB : -(1 / 4 : ℝ) ≤ (x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4 := by
nlinarith only [sq_nonneg x, sq_nonneg (x ^ 2), hy2, hη]
have hfirst' : τ * x * y - τ ^ 2 / 4 < u + u ^ 2 := by
dsimp [u]
nlinarith only [hfirst, mul_le_mul_of_nonneg_left hA (sq_nonneg τ)]
have hsecond' : -τ * x * y - τ ^ 2 / 4 < -u + u ^ 2 := by
dsimp [u]
nlinarith only [hsecond, mul_le_mul_of_nonneg_left hB (sq_nonneg τ)]
have hul : -3 * τ ≤ u := by
by_cases hu : 0 ≤ u
· linarith
· have hmul := mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hu)
(show 0 ≤ u + 1 / 2 by linarith)
nlinarith only [hmul, hfirst', htxyl, hτsq, hτ.le]
have hur : u ≤ 3 * τ := by
by_cases hu : u ≤ 0
· linarith
· have hmul := mul_nonpos_of_nonneg_of_nonpos (le_of_not_ge hu)
(show u - 1 / 2 ≤ 0 by linarith)
nlinarith only [hmul, hsecond', htxyr, hτsq, hτ.le]
have hu2 : u ^ 2 ≤ 9 * τ ^ 2 := by
nlinarith only [mul_nonneg (sub_nonneg.mpr hur)
(show 0 ≤ u + 3 * τ by linarith only [hul])]
constructor
· exact abs_le.mpr ⟨by simpa only [neg_mul] using hul, hur⟩
· change |u - τ * x * y| ≤ 10 * τ ^ 2
apply abs_le.mpr
constructor
· nlinarith only [hfirst', hu2, sq_nonneg τ]
· nlinarith only [hsecond', hu2, sq_nonneg τ]
theorem cartesian_squared_failures_impossible
{x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1)
(hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1)
(hη : 0 < η) (hsmall : τ ≤ η / 80)
(hfirst : 1 / 4 + τ * x * y +
τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) < s ^ 2)
(hsecond : 1 / 4 - τ * x * y +
τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) < (1 - s) ^ 2) :
False := by
obtain ⟨hu, herr⟩ := cartesian_failure_displacement hx hy hs0 hs1 hτ hτ1 hη.le
hfirst hsecond
let u := s - 1 / 2
change |u| ≤ 3 * τ at hu
change |u - τ * x * y| ≤ 10 * τ ^ 2 at herr
have hxy : |x * y| ≤ 1 := by
rw [abs_mul]
nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)]
have htx : |τ * x * y| ≤ τ := by
calc
|τ * x * y| = τ * |x * y| := by rw [mul_assoc, abs_mul, abs_of_pos hτ]
_ ≤ τ * 1 := mul_le_mul_of_nonneg_left hxy hτ.le
_ = τ := mul_one τ
have hsum : |u + τ * x * y| ≤ 4 * τ := by
exact (abs_add_le _ _).trans (by linarith)
have hsq : |u ^ 2 - (τ * x * y) ^ 2| ≤ 40 * τ ^ 3 := by
calc
|u ^ 2 - (τ * x * y) ^ 2| = |u - τ * x * y| * |u + τ * x * y| := by
rw [← abs_mul]
congr 1
ring
_ ≤ (10 * τ ^ 2) * (4 * τ) :=
mul_le_mul herr hsum (abs_nonneg _) (by positivity)
_ = 40 * τ ^ 3 := by ring
have hsumfail : τ ^ 2 * (x ^ 4 + y ^ 4 + η) < 2 * u ^ 2 := by
dsimp [u]
nlinarith only [hfirst, hsecond]
have hquartic : 2 * x ^ 2 * y ^ 2 ≤ x ^ 4 + y ^ 4 := by
nlinarith only [sq_nonneg (x ^ 2 - y ^ 2)]
have hgap : η * τ ^ 2 < 2 * (u ^ 2 - (τ * x * y) ^ 2) := by
nlinarith only [hsumfail, mul_le_mul_of_nonneg_left hquartic (sq_nonneg τ)]
have hwindow : 80 * τ ^ 3 ≤ η * τ ^ 2 := by
nlinarith only [mul_le_mul_of_nonneg_right hsmall (sq_nonneg τ)]
nlinarith only [hgap, hwindow, (abs_le.mp hsq).2]
theorem cartesian_squared_lower_of_sqrt {Q τ c : ℝ} (hc : 0 ≤ c)
(hroot : 1 / 2 + τ ^ 2 * c < Real.sqrt Q) :
1 / 4 + τ ^ 2 * c < Q := by
have hbase : 0 < 1 / 2 + τ ^ 2 * c := by positivity
have hQ : 0 ≤ Q := (Real.sqrt_pos.mp (hbase.trans hroot)).le
have hsq := mul_self_lt_mul_self hbase.le hroot
rw [Real.mul_self_sqrt hQ] at hsq
nlinarith only [hsq, sq_nonneg (τ ^ 2 * c)]
theorem cartesian_sqrt_failures_impossible
{x y s τ η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1)
(hs0 : 0 ≤ s) (hs1 : s ≤ 1) (hτ : 0 < τ) (hτ1 : τ ≤ 1)
(hη : 0 < η) (hsmall : τ ≤ η / 80)
(hfirst : 1 / 2 + τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2) <
Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4))
(hsecond : 1 / 2 + τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2) <
Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4)) :
False := by
have hfirst' := cartesian_squared_lower_of_sqrt (by positivity) hfirst
have hsecond' := cartesian_squared_lower_of_sqrt (by positivity) hsecond
apply cartesian_squared_failures_impossible hx hy hs0 hs1 hτ hτ1 hη hsmall
· nlinarith only [hfirst']
· nlinarith only [hsecond']
theorem cartesian_edge_one_obstruction (ε x y : ℝ) :
(ε / 2) ^ 2 * (((2 * x) ^ 2 - (2 * y) ^ 2) ^ 2 + 1 / 1000) =
ε ^ 2 * (4 * (x ^ 2 - y ^ 2) ^ 2 + 1 / 4000) := by
ring
end AngularCartesianValidation
end OAI
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