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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 | import OAI.Geometry.CylinderCovering.Rational.InversionDefinitions
import OAI.Geometry.CylinderCovering.Convexity.ThresholdInversion
namespace OAI
open Set
open scoped NNReal
namespace SlopeRational
lemma inverseRectangle_eq_box (d : ℝ) :
inverseRectangle d =
{p : Plane | ∀ i : Fin 2, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} := by
ext p
simp [inverseRectangle, Fin.forall_fin_two, abs_le, and_assoc]
lemma inverseRectangle_isClosed (d : ℝ) : IsClosed (inverseRectangle d) := by
unfold inverseRectangle
have h0 : Continuous (fun p : Plane => |p 0|) := by fun_prop
have h1 : Continuous (fun p : Plane => p 1) := by fun_prop
exact (isClosed_le h0 continuous_const).inter
((isClosed_le continuous_const h1).inter (isClosed_le h1 continuous_const))
lemma inverseRectangle_isCompact {d : ℝ} (hd : 0 < d) :
IsCompact (inverseRectangle d) := by
apply Metric.isCompact_iff_isClosed_bounded.mpr
refine ⟨inverseRectangle_isClosed d, isBounded_iff_forall_norm_le.mpr ?_⟩
refine ⟨2 + d, ?_⟩
intro p hp
rcases hp with ⟨hp0, hp1lo, hp1hi⟩
have hp1 : |p 1| ≤ 1 + d := by
apply abs_le.mpr
constructor <;> linarith
have hnorm := EuclideanSpace.norm_sq_eq p
simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hnorm
have hp0sq : (p 0) ^ 2 ≤ 1 := by
have hs := (sq_le_sq₀ (abs_nonneg (p 0)) (by norm_num : (0 : ℝ) ≤ 1)).mpr hp0
simpa only [sq_abs, one_pow] using hs
have hp1sq : (p 1) ^ 2 ≤ (1 + d) ^ 2 := by
have hs := (sq_le_sq₀ (abs_nonneg (p 1)) (by positivity : 0 ≤ 1 + d)).mpr hp1
simpa only [sq_abs] using hs
nlinarith [norm_nonneg p]
lemma inverseRectangle_convex (d : ℝ) : Convex ℝ (inverseRectangle d) := by
rw [inverseRectangle_eq_box]
have heq : {p : Plane | ∀ i : Fin 2, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} =
⋂ i : Fin 2, {p : Plane | p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} := by
ext p
simp
rw [heq]
exact convex_iInter fun i =>
(convex_Icc _ _).linear_preimage (PiLp.projₗ 2 (fun _ : Fin 2 => ℝ) i)
lemma inverseRectangle_denominator_pos {d : ℝ} (hd : 0 < d)
{p : Plane} (hp : p ∈ inverseRectangle d) : 0 < radius (p 1) + d := by
have ht := hp.2.2
unfold radius
linarith
lemma field_contDiffOn_inverseRectangle {d : ℝ} (hd : 0 < d) :
ContDiffOn ℝ 1 (field d) (inverseRectangle d) := by
intro p hp
apply ContDiffAt.contDiffWithinAt
have hden : 0 < 1 / 2 - p 1 + d := inverseRectangle_denominator_pos hd hp
unfold field vec2
apply PiLp.contDiff_toLp.contDiffAt.comp p
apply contDiffAt_pi.mpr
intro i
fin_cases i
· change ContDiffAt ℝ 1 (fun z : Plane => a d (z 0) (z 1)) p
unfold a radius
fun_prop (disch := positivity)
· change ContDiffAt ℝ 1 (fun z : Plane => b d (z 0) (z 1)) p
unfold b radius
fun_prop (disch := positivity)
lemma rectangleInversion_of_contDiff {d : ℝ} (hd : 0 < d)
(hregular : ContDiffOn ℝ 1 (field d) (inverseRectangle d)) : RectangleInversionAt d := by
obtain ⟨L, hL⟩ := hregular.exists_lipschitzOnWith (by norm_num)
(inverseRectangle_convex d) (inverseRectangle_isCompact hd)
obtain ⟨B₀, hB₀⟩ := (inverseRectangle_isCompact hd).exists_bound_of_continuousOn
hregular.continuousOn
let B := max B₀ 0
have hB : 0 ≤ B := le_max_right _ _
have hbound : ∀ p ∈ inverseRectangle d, ‖field d p‖ ≤ B := by
intro p hp
exact (hB₀ p hp).trans (le_max_left _ _)
let margin := min (1 / 2 : ℝ) (d / 2)
have hm : 0 < margin := lt_min (by norm_num) (by positivity)
obtain ⟨δ, hδ, hsmall⟩ := CylinderCovering.ThresholdInversion.exists_small_tilt
(L : ℝ) B margin L.coe_nonneg hB hm
refine ⟨δ, hδ, ?_⟩
intro s hs
obtain ⟨hsL, hsB⟩ := hsmall s hs
constructor
· intro y t hy ht
have hbox := CylinderCovering.ThresholdInversion.existsUnique_inverse_on_box_smul
(![-1, -1] : Fin 2 → ℝ) (![1, 1 / 2 + d / 2] : Fin 2 → ℝ) hm.le
(W := field d) (L := (L : ℝ)) (B := B) (t := s) L.coe_nonneg hB
have hlip : ∀ p : Plane, (∀ i, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) →
∀ q : Plane, (∀ i, q i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) →
‖field d p - field d q‖ ≤ (L : ℝ) * ‖p - q‖ := by
intro p hp q hq
have hp' : p ∈ inverseRectangle d := by rw [inverseRectangle_eq_box]; exact hp
have hq' : q ∈ inverseRectangle d := by rw [inverseRectangle_eq_box]; exact hq
simpa only [dist_eq_norm] using hL.dist_le_mul p hp' q hq'
have hb : ∀ p : Plane, (∀ i, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) →
‖field d p‖ ≤ B := by
intro p hp
exact hbound p (by rwa [inverseRectangle_eq_box])
have htarget : ∀ i : Fin 2,
![-1, -1] i + margin ≤ vec2 y t i ∧
vec2 y t i ≤ ![1, 1 / 2 + d / 2] i - margin := by
have hm1 : margin ≤ 1 / 2 := min_le_left _ _
have hmd : margin ≤ d / 2 := min_le_right _ _
have hy' := abs_le.mp hy
have ht' := abs_le.mp ht
intro i
fin_cases i
· change -1 + margin ≤ y ∧ y ≤ 1 - margin
constructor <;> linarith
· change -1 + margin ≤ t ∧ t ≤ 1 / 2 + d / 2 - margin
constructor <;> linarith
have hex := hbox hlip hb (by linarith) hsB (vec2 y t) htarget
simpa only [inverseRectangle_eq_box, tiltedMap, Set.mem_ofPred_eq] using hex
· intro p hp q hq hhorizontal hvertical
apply CylinderCovering.ThresholdInversion.inverse_coordinate_strict (i := (1 : Fin 2))
(c := |s| * (L : ℝ)) (mul_nonneg (abs_nonneg _) L.coe_nonneg) hsL
(g := fun z => s • field d z) (y := tiltedMap d s p) (z := tiltedMap d s q)
rfl rfl
· rw [← smul_sub, norm_smul, Real.norm_eq_abs]
have hLip := hL.dist_le_mul q hq p hp
simp only [dist_eq_norm] at hLip
calc
|s| * ‖field d q - field d p‖ ≤ |s| * ((L : ℝ) * ‖q - p‖) :=
mul_le_mul_of_nonneg_left hLip (abs_nonneg _)
_ = |s| * (L : ℝ) * ‖q - p‖ := by ring
· intro j hj
fin_cases j
· exact hhorizontal.symm
· exact (hj rfl).elim
· exact hvertical
theorem rectangleInversion_main : RectangleInversionStatement := by
intro d hd
exact rectangleInversion_of_contDiff hd (field_contDiffOn_inverseRectangle hd)
end SlopeRational
end OAI
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