import Common.Real.Geometry.Basic /-! # Common.Real.Geometry.Rectangle A characterization of a rectangle. This follows the definition as outlined in [^1]. Note that a `Point` and a `LineSegment` are both considered rectangles, with one or both dimensions equal to `0` respectively. [^1]: Apostol, Tom M. Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991. -/ namespace Real /-- A `Rectangle` is characterized by three distinct points and the angle formed between line segments originating from the "bottom left" point. -/ structure Rectangle where top_left : ℝ² bottom_left : ℝ² bottom_right : ℝ² forms_right_angle : ∠ top_left bottom_left bottom_right = π / 2 namespace Rectangle /-- The top-right corner of the rectangle, oriented with respect to the other vertices. -/ def top_right (r : Rectangle) : ℝ² := ( r.top_left.fst + r.bottom_right.fst - r.bottom_left.fst , r.top_left.snd + r.bottom_right.snd - r.bottom_left.snd ) /-- A `Rectangle` is the locus of points bounded by its edges. -/ def set_def (r : Rectangle) : Set ℝ² := sorry /-- A `Rectangle`'s top side is equal in length to its bottom side. -/ theorem dist_top_eq_dist_bottom (r : Rectangle) : dist r.top_left r.top_right = dist r.bottom_left r.bottom_right := by unfold top_right dist repeat rw [add_comm, sub_right_comm, add_sub_cancel'] /-- A `Rectangle`'s left side is equal in length to its right side. -/ theorem dist_left_eq_dist_right (r : Rectangle) : dist r.top_left r.bottom_left = dist r.top_right r.bottom_right := by unfold top_right dist repeat rw [ sub_sub_eq_add_sub, add_comm, sub_add_eq_sub_sub, sub_right_comm, add_sub_cancel' ] /-- Computes the width of a `Rectangle`. -/ noncomputable def width (r : Rectangle) : ℝ := dist r.bottom_left r.bottom_right /-- Computes the height of a `Rectangle`. -/ noncomputable def height (r : Rectangle) : ℝ := dist r.bottom_left r.top_left end Rectangle /-- A `Point` is a `Rectangle` in which all points coincide. -/ abbrev Point := Subtype (fun r : Rectangle => r.top_left = r.bottom_left ∧ r.bottom_left = r.bottom_right) namespace Point /-- A `Point` is the set consisting of just itself. -/ def set_def (p : Point) : Set ℝ² := p.val.set_def /-- The width of a `Point` is `0`. -/ theorem width_eq_zero (p : Point) : p.val.width = 0 := by unfold Rectangle.width rw [p.property.right] unfold dist simp /-- The height of a `Point` is `0`. -/ theorem height_eq_zero (p : Point) : p.val.height = 0 := by unfold Rectangle.height rw [p.property.left] unfold dist simp end Point /-- A `LineSegment` is a `Rectangle` in which two of the three points coincide. -/ abbrev LineSegment := Subtype (fun r : Rectangle => (r.top_left = r.bottom_left ∧ r.bottom_left ≠ r.bottom_right) ∨ (r.top_left ≠ r.bottom_left ∧ r.bottom_left = r.bottom_right)) namespace LineSegment /-- A `LineSegment` `s` is the set of points corresponding to the shortest line segment joining the two distinct points of `s`. -/ def set_def (s : LineSegment) : Set ℝ² := s.val.set_def /-- Either the width or height of a `LineSegment` is zero. -/ theorem width_or_height_eq_zero (s : LineSegment) : s.val.width = 0 ∨ s.val.height = 0 := by apply Or.elim s.property · intro h refine Or.inr ?_ unfold Rectangle.height rw [h.left] unfold dist simp · intro h refine Or.inl ?_ unfold Rectangle.width rw [h.right] unfold dist simp end LineSegment end Real