Redefine an `Orthogonal` rectangle as a subtype of a `Skew` one.

Common/Geometry/Area.lean

@@ -1,5 +1,5 @@
 import Common.Geometry.Basic
-import Common.Geometry.Rectangle.Orthogonal
+import Common.Geometry.Rectangle.Skew
 import Common.Geometry.StepFunction
 
 /-! # Common.Geometry.Area

Common/Geometry/Rectangle/Orthogonal.lean

@@ -1,5 +1,4 @@
 import Mathlib.Data.Fin.Basic
-import Mathlib.Tactic.LibrarySearch
 
 import Common.Geometry.Point
 import Common.Geometry.Segment
@@ -13,176 +12,27 @@ A characterization of an orthogonal rectangle.
 namespace Geometry.Rectangle
 
 /--
-An `Orthogonal` rectangle is characterized by two points on opposite corners. It
+An `Orthogonal` rectangle is a `Skew` rectangle with edges parallel to the
-is assumed the edges of the rectangle are parallel to the coordinate axes.
+coordinate axes.
-
-A `Point` can alternatively be viewed as an `Orthogonal` rectangle in which the
-two points coincide. A horizontal or vertical `Segment` can alternatively be
-viewed as an `Orthogonal` rectangle with width or height (but not both) `0`.
 -/
-structure Orthogonal where
+def Orthogonal := { r : Skew // r.bl.x = r.tl.x ∧ r.bl.y = r.br.y }
-  bl : Point  -- bottom left
-  tr : Point  -- top right
 
 namespace Orthogonal
 
-/--
-The width of the `Orthogonal` rectangle.
--/
-def width (r : Orthogonal) := r.tr.x - r.bl.x
-
-/--
-The height of the `Orthogonal` rectangle.
--/
-def height (r : Orthogonal) := r.tr.y - r.bl.y
-
-/--
-The top-left corner of the `Orthogonal` rectangle.
--/
-def tl (r : Orthogonal) : Point := ⟨r.bl.x, r.bl.y + r.height⟩
-
-/--
-The bottom-right corner of the `Orthogonal` rectangle.
--/
-def br (r : Orthogonal) : Point := ⟨r.bl.x + r.width, r.bl.y⟩
-
 /--
 The `Set` of `Point`s enclosed in the region determined by the edges of the
 `Orthogonal` rectangle. Edges of the rectangle are included in the result set.
 -/
 def toSet (r : Orthogonal) : Set Point :=
-  { p | r.bl.x ≤ p.x ∧ p.x ≤ r.br.x ∧ r.bl.y ≤ p.y ∧ p.y ≤ r.tl.y }
+  { p | r.val.bl.x ≤ p.x ∧ p.x ≤ r.val.br.x ∧
+        r.val.bl.y ≤ p.y ∧ p.y ≤ r.val.tl.y }
 
 /--
-An `Orthogonal` rectangle's top side is equal in length to its bottom side.
+Show the `toSet` definition of an `Orthogonal` rectangle is in agreement with
+its `Skew` counterpart.
 -/
-theorem dist_top_eq_dist_bottom (r : Orthogonal)
+theorem orthogonal_set_eq_skew_set (r : Orthogonal)
-  : Point.dist r.tl r.tr = Point.dist r.bl r.br := by
+  : r.toSet = r.val.toSet := sorry
-  unfold tl br Point.dist width height
-  norm_num
-
-/--
-An `Orthogonal` rectangle's left side is equal in length to its right side.
--/
-theorem dist_left_eq_dist_right (r : Orthogonal)
-  : Point.dist r.tl r.bl = Point.dist r.tr r.br := by
-  unfold tl br Point.dist width height
-  norm_num
-
-/--
-Convert an `Orthogonal` rectangle into a `Skew` one.
--/
-def toSkew (r : Orthogonal) : Skew := ⟨r.tl, r.bl, r.br, sorry⟩
-
-/--
-The set of `Orthogonal` rectangles are embedded in the set of `Skew` rectangles.
--/
-def skewEmbedding : Orthogonal ↪ Skew :=
-  have : Function.Injective toSkew := by
-    unfold Function.Injective
-    intro r₁ r₂ h
-    unfold toSkew at h
-    have ⟨⟨blx₁, bly₁⟩, ⟨trx₁, try₁⟩⟩ := r₁
-    have ⟨⟨blx₂, bry₂⟩, ⟨trx₂, try₂⟩⟩ := r₂
-    simp
-    simp at h
-    unfold tl br width height at h
-    simp at h
-    exact ⟨⟨h.left.left, h.right.left.right⟩, ⟨h.right.right.left, h.left.right⟩⟩
-  ⟨toSkew, this⟩
-
-/-! ## Point -/
-
-/--
-A `Point` is an `Orthogonal` rectangle in which all points coincide.
--/
-abbrev AsPoint := Subtype (fun r : Orthogonal => r.bl = r.tr)
-
-namespace AsPoint
-
-/--
-The function mapping an `Orthogonal` rectangle with all points coinciding to a
-`Point`.
--/
-def toPoint (p : AsPoint) : Point := p.val.tl
-
-/--
-The function mapping a `Point` to an `Orthogonal` rectangle with all points
-coinciding.
--/
-def fromPoint (p : Point) : AsPoint := ⟨Orthogonal.mk p p, by simp⟩
-
-/--
-An isomorphism between an `Orthogonal` rectangle with all points coinciding and
-a `Point`.
--/
-def isoPoint : AsPoint ≃ Point :=
-  {
-    toFun := toPoint,
-    invFun := fromPoint,
-    left_inv := by
-      unfold Function.LeftInverse fromPoint toPoint
-      intro ⟨r, hr⟩
-      congr
-      repeat {
-        simp only
-        unfold tl height
-        rw [hr]
-        simp
-      }
-    right_inv := by
-      unfold Function.RightInverse Function.LeftInverse fromPoint toPoint
-      intro ⟨r, hr⟩
-      unfold tl height
-      simp
-  }
-
-/--
-The width of an `AsPoint` is `0`.
--/
-theorem width_eq_zero (p : AsPoint) : p.val.width = 0 := by
-  unfold Orthogonal.width
-  rw [p.property]
-  simp
-
-/--
-The height of an `AsPoint` is `0`.
--/
-theorem height_eq_zero (p : AsPoint) : p.val.height = 0 := by
-  unfold Orthogonal.height
-  rw [p.property]
-  simp
-
-end AsPoint
-
-/-! ## Segment -/
-
-/--
-A `Segment` is an `Orthogonal` rectangle either width or height equal to `0`.
--/
-abbrev AsSegment := Subtype (fun r : Orthogonal =>
-  (r.bl.x = r.tr.x ∧ r.bl.y ≠ r.tr.y) ∨ (r.bl.x ≠ r.tr.x ∧ r.bl.y = r.tr.y))
-
-namespace AsSegment
-
-/--
-Either the width or height of an `AsSegment` is zero.
--/
-theorem width_or_height_eq_zero (s : AsSegment)
-  : s.val.width = 0 ∨ s.val.height = 0 := by
-  apply Or.elim s.property
-  · intro h
-    refine Or.inl ?_
-    unfold width
-    rw [h.left]
-    simp
-  · intro h
-    refine Or.inr ?_
-    unfold height
-    rw [h.right]
-    simp
-
-end AsSegment
 
 end Orthogonal
 

Common/Geometry/StepFunction.lean

@@ -1,5 +1,4 @@
 import Common.Finset
-import Common.Geometry.Point
 import Common.Geometry.Rectangle.Orthogonal
 import Common.List.Basic
 import Common.Set.Partition
@@ -34,7 +33,15 @@ def toSet (sf : StepFunction) : Set Point :=
   ⋃ i ∈ Finset.finRange sf.p.ivls.length,
     let I := sf.p.ivls[i]
     Rectangle.Orthogonal.toSet
-      ⟨{ x := I.left, y := 0 }, { x := I.right, y := sf.toFun i }⟩
+      ⟨
+        {
+          tl := ⟨I.left, sf.toFun i⟩,
+          bl := ⟨I.left, 0⟩,
+          br := ⟨I.right, 0⟩,
+          has_right_angle := sorry
+        },
+        by simp
+      ⟩
 
 end StepFunction
 

Common/Real/Floor.lean

@@ -1,7 +1,6 @@
 import Mathlib.Algebra.BigOperators.Basic
 import Mathlib.Data.Real.Basic
 import Mathlib.Data.Finset.Basic
-import Mathlib.Tactic.LibrarySearch
 
 /-! # Common.Real.Floor