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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 | import OAI.Geometry.CylinderCovering.Cartesian.CapsDefinitions
namespace OAI
namespace CylinderCovering.CartesianCaps
theorem discriminant_failures_location {τ η x y s : ℝ}
(hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hη : 0 ≤ η)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(hfirst : 1 / 4 + τ * x * y + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4) < s ^ 2)
(hsecond : 1 / 4 - τ * x * y + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4) <
(1 - s) ^ 2) :
|s - 1 / 2| ≤ 3 * τ ∧ |s - 1 / 2 - τ * x * y| ≤ 10 * τ ^ 2 := by
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 hDy : 0 ≤ D y := by unfold D; positivity
have hDx : 0 ≤ D x := by unfold D; positivity
have hB : -(1 / 4 : ℝ) ≤ D y + η / 2 - x ^ 2 / 4 := by linarith
have hC : -(1 / 4 : ℝ) ≤ D x + η / 2 - y ^ 2 / 4 := by linarith
have hBmul := mul_le_mul_of_nonneg_left hB (sq_nonneg τ)
have hCmul := mul_le_mul_of_nonneg_left hC (sq_nonneg τ)
have hxy : |τ * x * y| ≤ τ := by
rw [abs_mul, abs_mul, abs_of_nonneg hτ0]
calc
τ * |x| * |y| ≤ τ * 1 * 1 := by gcongr
_ = τ := by ring
have hτ2 : τ ^ 2 ≤ τ := by nlinarith
have hcoarse : |s - 1 / 2| ≤ 3 * τ := by
rw [abs_le]
constructor
· by_cases hs : s ≤ 1 / 2
· have hprod := mul_nonneg hs0 (sub_nonneg.mpr hs)
nlinarith only [hfirst, hBmul, (abs_le.mp hxy).1, hτ2, hprod, hτ0]
· linarith
· by_cases hs : 1 / 2 ≤ s
· have hprod := mul_nonneg (sub_nonneg.mpr hs) (sub_nonneg.mpr hs1)
nlinarith only [hsecond, hCmul, (abs_le.mp hxy).2, hτ2, hprod, hτ0]
· linarith
have hu2 : (s - 1 / 2) ^ 2 ≤ 9 * τ ^ 2 := by
have hh := (sq_le_sq₀ (abs_nonneg (s - 1 / 2)) (by positivity : 0 ≤ 3 * τ)).2 hcoarse
nlinarith only [hh, sq_abs (s - 1 / 2)]
refine ⟨hcoarse, ?_⟩
rw [abs_le]
constructor <;> nlinarith only [hfirst, hsecond, hBmul, hCmul, hu2, sq_nonneg τ]
theorem discriminant_failures_impossible {τ η x y s : ℝ}
(hτ : 0 < τ) (hτ1 : τ ≤ 1) (hsmall : τ ≤ η / 240)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1)
(hfirst : 1 / 4 + τ * x * y + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4) < s ^ 2)
(hsecond : 1 / 4 - τ * x * y + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4) <
(1 - s) ^ 2) : False := by
have hη : 0 ≤ η := by linarith
obtain ⟨hcoarse, hfine⟩ := discriminant_failures_location hτ.le hτ1 hη hx hy hs0 hs1 hfirst hsecond
have hxy : |τ * x * y| ≤ τ := by
rw [abs_mul, abs_mul, abs_of_nonneg hτ.le]
calc
τ * |x| * |y| ≤ τ * 1 * 1 := by gcongr
_ = τ := by ring
have hsum : |s - 1 / 2 + τ * x * y| ≤ 4 * τ :=
(abs_add_le _ _).trans (by linarith)
have herror : |(s - 1 / 2) ^ 2 - (τ * x * y) ^ 2| ≤ 40 * τ ^ 3 := by
calc
_ = |s - 1 / 2 - τ * x * y| * |s - 1 / 2 + τ * x * y| := by
rw [← abs_mul]
congr 1
ring
_ ≤ (10 * τ ^ 2) * (4 * τ) := mul_le_mul hfine hsum (abs_nonneg _) (by positivity)
_ = 40 * τ ^ 3 := by ring
have hquartic := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2))
have hmargin := mul_le_mul_of_nonneg_left hsmall (sq_nonneg τ)
have hτ3 : 0 < τ ^ 3 := pow_pos hτ 3
unfold D at hfirst hsecond
nlinarith only [hfirst, hsecond, (abs_le.mp herror).2, hquartic, hmargin, hτ3]
theorem discriminant_alternative {τ η x y s : ℝ}
(hτ : 0 < τ) (hτ1 : τ ≤ 1) (hsmall : τ ≤ η / 240)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hs0 : 0 ≤ s) (hs1 : s ≤ 1) :
Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4) ≤
1 / 2 + τ ^ 2 * (D y + η / 2) ∨
Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4) ≤
1 / 2 + τ ^ 2 * (D x + η / 2) := by
by_contra! hfail
have hη : 0 ≤ η := by linarith
have hDx : 0 ≤ D x := by unfold D; positivity
have hDy : 0 ≤ D y := by unfold D; positivity
have hBy : 0 ≤ 1 / 2 + τ ^ 2 * (D y + η / 2) := by positivity
have hBx : 0 ≤ 1 / 2 + τ ^ 2 * (D x + η / 2) := by positivity
have hQy : 0 ≤ s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4 :=
(Real.sqrt_pos.mp (hBy.trans_lt hfail.1)).le
have hQx : 0 ≤ (1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4 :=
(Real.sqrt_pos.mp (hBx.trans_lt hfail.2)).le
have hy2 := (sq_lt_sq₀ hBy (Real.sqrt_nonneg _)).2 hfail.1
have hx2 := (sq_lt_sq₀ hBx (Real.sqrt_nonneg _)).2 hfail.2
rw [Real.sq_sqrt hQy] at hy2
rw [Real.sq_sqrt hQx] at hx2
apply discriminant_failures_impossible hτ hτ1 hsmall hx hy hs0 hs1
· nlinarith only [hy2, sq_nonneg (τ ^ 2 * (D y + η / 2))]
· nlinarith only [hx2, sq_nonneg (τ ^ 2 * (D x + η / 2))]
theorem eventually_discriminant_alternative {η : ℝ} (hη : 0 < η) :
∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 < τ → τ ≤ τ₀ →
∀ x y s : ℝ, |x| ≤ 1 → |y| ≤ 1 → 0 ≤ s → s ≤ 1 →
Real.sqrt (s ^ 2 - τ * x * y + τ ^ 2 * x ^ 2 / 4) ≤
1 / 2 + τ ^ 2 * (D y + η / 2) ∨
Real.sqrt ((1 - s) ^ 2 + τ * x * y + τ ^ 2 * y ^ 2 / 4) ≤
1 / 2 + τ ^ 2 * (D x + η / 2) := by
refine ⟨min 1 (η / 240), lt_min (by norm_num) (by positivity), ?_⟩
intro τ hτ hsmall x y s hx hy hs0 hs1
exact discriminant_alternative hτ (hsmall.trans (min_le_left _ _))
(hsmall.trans (min_le_right _ _)) hx hy hs0 hs1
end CylinderCovering.CartesianCaps
end OAI
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