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e320e82 | 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 | import OAI.Combinatorics.HalvingLines.Model
namespace OAI
namespace PlanarHalving
theorem critical_image_null (f : Point → Point) :
MeasureTheory.volume (f '' {p | DifferentiableAt ℝ f p ∧
(fderiv ℝ f p).det = 0}) = 0 := by
apply MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
MeasureTheory.volume
· intro p hp
exact hp.1.hasFDerivAt.hasFDerivWithinAt
· intro p hp
exact hp.2
theorem orient_swap (p q r : Point) : orient q p r = -orient p q r := by
unfold orient
ring
@[simp] theorem orient_left (p q : Point) : orient p q p = 0 := by
simp [orient]
@[simp] theorem orient_right (p q : Point) : orient p q q = 0 := by
unfold orient
ring
@[simp] theorem switchCount_zero {n : ℕ} (P : Fin n → Point) :
switchCount P 0 = 0 := by
simp [switchCount]
@[simp] theorem switchCount_top {n : ℕ} (P : Fin n → Point) :
switchCount P n = 0 := by
simp [switchCount]
theorem orient_eq_proj_diff (p q r : Point) (hx : q.1 ≠ p.1) :
orient p q r = (q.1 - p.1) *
((r.2 - slope p q * r.1) - (p.2 - slope p q * p.1)) := by
unfold orient slope
field_simp
ring
lemma orient_zero_iff {n : ℕ} {P : Fin n → Point} (hP : GeneralPosition P)
{i j : Fin n} (hij : i ≠ j) (l : Fin n) :
orient (P i) (P j) (P l) = 0 ↔ l = i ∨ l = j := by
constructor
· intro hl
by_contra! hh
exact hP.2 i j l hij hh.2.symm hh.1.symm hl
· rintro (rfl | rfl) <;> simp
lemma sideCount_add_reverse {n : ℕ} {P : Fin n → Point} (hP : GeneralPosition P)
{i j : Fin n} (hij : i ≠ j) : sideCount P i j + sideCount P j i = n - 2 := by
classical
let A := Finset.univ.filter (fun l => 0 < orient (P i) (P j) (P l))
let B := Finset.univ.filter (fun l => 0 < orient (P j) (P i) (P l))
have hd : Disjoint A B := by
apply Finset.disjoint_left.mpr
intro l hlA hlB
have ha : 0 < orient (P i) (P j) (P l) := (Finset.mem_filter.mp hlA).2
have hb : 0 < orient (P j) (P i) (P l) := (Finset.mem_filter.mp hlB).2
rw [orient_swap] at hb
linarith only [ha, hb]
have hu : A ∪ B = (Finset.univ.erase i).erase j := by
ext l
simp only [A, B, Finset.mem_union, Finset.mem_filter, Finset.mem_univ,
true_and, Finset.mem_erase]
rw [orient_swap (P i) (P j) (P l)]
have hz := orient_zero_iff hP hij l
constructor
· intro hl
have hn : orient (P i) (P j) (P l) ≠ 0 := by
rcases hl with hl | hl <;> linarith only [hl]
have hn' : ¬ (l = i ∨ l = j) := fun he => hn (hz.mpr he)
exact ⟨fun h => hn' (Or.inr h), fun h => hn' (Or.inl h), trivial⟩
· rintro ⟨hlj, hli, _⟩
have hn : orient (P i) (P j) (P l) ≠ 0 := by
intro hh
rcases hz.mp hh with hh | hh
· exact hli hh
· exact hlj hh
rcases lt_or_gt_of_ne hn with hl | hl
· right; linarith only [hl]
· exact Or.inl hl
change A.card + B.card = n - 2
rw [← Finset.card_union_of_disjoint hd, hu,
Finset.card_erase_of_mem (by simp [hij.symm]),
Finset.card_erase_of_mem (Finset.mem_univ _), Finset.card_univ, Fintype.card_fin]
omega
lemma belowCount_eq_sideCount_of_x_lt {n : ℕ} (P : Fin n → Point) (i j : Fin n)
(hx : (P i).1 < (P j).1) : belowCount P i j = sideCount P j i := by
classical
unfold belowCount sideCount
dsimp only
apply congrArg Finset.card
apply Finset.filter_congr
intro l _
rw [orient_swap, orient_eq_proj_diff _ _ _ hx.ne']
have hx' : 0 < (P j).1 - (P i).1 := sub_pos.mpr hx
rw [neg_pos]
simp only [mul_neg_iff, hx', hx'.not_gt, true_and, false_and,
or_false, sub_neg]
lemma belowCount_eq_sideCount_of_x_gt {n : ℕ} (P : Fin n → Point) (i j : Fin n)
(hx : (P j).1 < (P i).1) : belowCount P i j = sideCount P i j := by
classical
unfold belowCount sideCount
dsimp only
apply congrArg Finset.card
apply Finset.filter_congr
intro l _
rw [orient_eq_proj_diff _ _ _ hx.ne]
have hx' : (P j).1 - (P i).1 < 0 := sub_neg.mpr hx
simp only [mul_pos_iff, hx', hx'.not_gt, true_and, false_and,
false_or, sub_neg]
theorem halvingCount_eq_middle_switchCount {m : ℕ} (hm : 0 < m)
(P : Fin (2 * m) → Point) (hP : Generic P) :
halvingCount P = switchCount P m := by
classical
unfold halvingCount switchCount
congr 1
apply Finset.filter_congr
intro ij _
have hm' : m < 2 * m := by omega
simp only [hm, hm', true_and]
by_cases hij : ij.1 < ij.2
· simp only [hij, true_and]
have hs := sideCount_add_reverse hP.1 hij.ne
have hdiv : (2 * m - 2) / 2 = m - 1 := by omega
rw [hdiv]
rcases lt_or_gt_of_ne (hP.2.1 ij.1 ij.2 hij.ne) with hx | hx
· rw [belowCount_eq_sideCount_of_x_lt P _ _ hx]
omega
· rw [belowCount_eq_sideCount_of_x_gt P _ _ hx]
omega
· simp only [hij, false_and]
end PlanarHalving
end OAI
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