| 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 | |