| import Mathlib | |
| namespace OAI | |
| namespace PlanarHalving | |
| abbrev Point := ℝ × ℝ | |
| def orient (p q r : Point) : ℝ := | |
| (q.1 - p.1) * (r.2 - p.2) - (q.2 - p.2) * (r.1 - p.1) | |
| def GeneralPosition {n : ℕ} (P : Fin n → Point) : Prop := | |
| Function.Injective P ∧ ∀ i j l, i ≠ j → j ≠ l → i ≠ l → | |
| orient (P i) (P j) (P l) ≠ 0 | |
| noncomputable def sideCount {n : ℕ} (P : Fin n → Point) (i j : Fin n) : ℕ := | |
| (Finset.univ.filter fun l => 0 < orient (P i) (P j) (P l)).card | |
| noncomputable def halvingCount {n : ℕ} (P : Fin n → Point) : ℕ := | |
| (Finset.univ.filter fun ij : Fin n × Fin n => | |
| ij.1 < ij.2 ∧ sideCount P ij.1 ij.2 = (n - 2) / 2 ∧ | |
| sideCount P ij.2 ij.1 = (n - 2) / 2).card | |
| noncomputable def slope (p q : Point) : ℝ := (q.2 - p.2) / (q.1 - p.1) | |
| def Generic {n : ℕ} (P : Fin n → Point) : Prop := | |
| GeneralPosition P ∧ | |
| (∀ i j, i ≠ j → (P i).1 ≠ (P j).1) ∧ | |
| (∀ i j a b, i < j → a < b → slope (P i) (P j) = slope (P a) (P b) → | |
| i = a ∧ j = b) ∧ | |
| (∀ i j a b c d, i < j → a < b → c < d → | |
| (i, j) ≠ (a, b) → (c, d) ≠ (i, j) → (c, d) ≠ (a, b) → | |
| ∀ z : Point, z ∈ openSegment ℝ (P i) (P j) → | |
| z ∈ openSegment ℝ (P a) (P b) → z ∉ segment ℝ (P c) (P d)) | |
| noncomputable def belowCount {n : ℕ} (P : Fin n → Point) (i j : Fin n) : ℕ := | |
| let s := slope (P i) (P j) | |
| (Finset.univ.filter fun l => (P l).2 - s * (P l).1 < | |
| (P i).2 - s * (P i).1).card | |
| noncomputable def switchCount {n : ℕ} (P : Fin n → Point) (k : ℕ) : ℕ := | |
| (Finset.univ.filter fun ij : Fin n × Fin n => | |
| ij.1 < ij.2 ∧ 0 < k ∧ k < n ∧ belowCount P ij.1 ij.2 = k - 1).card | |
| end PlanarHalving | |
| end OAI | |