2023-05-15 01:32:18 +00:00
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic.LibrarySearch
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import Common.Geometry.Point
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import Common.Geometry.Segment
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import Common.Geometry.Rectangle.Skew
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/-! # Common.Geometry.Rectangle.Orthogonal
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A characterization of an orthogonal rectangle.
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-/
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namespace Geometry.Rectangle
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/--
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An `Orthogonal` rectangle is characterized by two points on opposite corners. It
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is assumed the edges of the rectangle are parallel to the coordinate axes.
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A `Point` can alternatively be viewed as an `Orthogonal` rectangle in which the
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two points coincide. A horizontal or vertical `Segment` can alternatively be
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viewed as an `Orthogonal` rectangle with width or height (but not both) `0`.
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-/
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structure Orthogonal where
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bl : Point -- bottom left
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tr : Point -- top right
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namespace Orthogonal
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/--
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The width of the `Orthogonal` rectangle.
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-/
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def width (r : Orthogonal) := r.tr.x - r.bl.x
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/--
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The height of the `Orthogonal` rectangle.
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-/
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def height (r : Orthogonal) := r.tr.y - r.bl.y
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/--
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The top-left corner of the `Orthogonal` rectangle.
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-/
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def tl (r : Orthogonal) : Point := ⟨r.bl.x, r.bl.y + r.height⟩
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/--
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The bottom-right corner of the `Orthogonal` rectangle.
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-/
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def br (r : Orthogonal) : Point := ⟨r.bl.x + r.width, r.bl.y⟩
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2023-05-15 14:00:01 +00:00
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/--
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The `Set` of `Point`s enclosed in the region determined by the edges of the
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`Orthogonal` rectangle. Edges of the rectangle are included in the result set.
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-/
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def toSet (r : Orthogonal) : Set Point :=
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{ p | r.bl.x ≤ p.x ∧ p.x ≤ r.br.x ∧ r.bl.y ≤ p.y ∧ p.y ≤ r.tl.y }
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2023-05-15 01:32:18 +00:00
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/--
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An `Orthogonal` rectangle's top side is equal in length to its bottom side.
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-/
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theorem dist_top_eq_dist_bottom (r : Orthogonal)
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: Point.dist r.tl r.tr = Point.dist r.bl r.br := by
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unfold tl br Point.dist width height
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norm_num
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/--
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An `Orthogonal` rectangle's left side is equal in length to its right side.
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-/
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theorem dist_left_eq_dist_right (r : Orthogonal)
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: Point.dist r.tl r.bl = Point.dist r.tr r.br := by
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unfold tl br Point.dist width height
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norm_num
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/--
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Convert an `Orthogonal` rectangle into a `Skew` one.
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-/
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def toSkew (r : Orthogonal) : Skew := ⟨r.tl, r.bl, r.br, sorry⟩
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/--
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The set of `Orthogonal` rectangles are embedded in the set of `Skew` rectangles.
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-/
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def skewEmbedding : Orthogonal ↪ Skew :=
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have : Function.Injective toSkew := by
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unfold Function.Injective
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intro r₁ r₂ h
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unfold toSkew at h
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have ⟨⟨blx₁, bly₁⟩, ⟨trx₁, try₁⟩⟩ := r₁
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have ⟨⟨blx₂, bry₂⟩, ⟨trx₂, try₂⟩⟩ := r₂
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simp
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simp at h
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unfold tl br width height at h
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simp at h
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exact ⟨⟨h.left.left, h.right.left.right⟩, ⟨h.right.right.left, h.left.right⟩⟩
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⟨toSkew, this⟩
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/-! ## Point -/
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/--
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A `Point` is an `Orthogonal` rectangle in which all points coincide.
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-/
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abbrev AsPoint := Subtype (fun r : Orthogonal => r.bl = r.tr)
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namespace AsPoint
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/--
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The function mapping an `Orthogonal` rectangle with all points coinciding to a
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`Point`.
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-/
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def toPoint (p : AsPoint) : Point := p.val.tl
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/--
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The function mapping a `Point` to an `Orthogonal` rectangle with all points
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coinciding.
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-/
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def fromPoint (p : Point) : AsPoint := ⟨Orthogonal.mk p p, by simp⟩
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/--
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An isomorphism between an `Orthogonal` rectangle with all points coinciding and
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a `Point`.
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-/
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def isoPoint : AsPoint ≃ Point :=
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{
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toFun := toPoint,
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invFun := fromPoint,
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left_inv := by
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unfold Function.LeftInverse fromPoint toPoint
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intro ⟨r, hr⟩
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congr
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repeat {
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simp only
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unfold tl height
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rw [hr]
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simp
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}
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right_inv := by
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unfold Function.RightInverse Function.LeftInverse fromPoint toPoint
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intro ⟨r, hr⟩
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unfold tl height
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simp
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}
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/--
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The width of an `AsPoint` is `0`.
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-/
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theorem width_eq_zero (p : AsPoint) : p.val.width = 0 := by
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unfold Orthogonal.width
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rw [p.property]
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simp
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/--
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The height of an `AsPoint` is `0`.
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-/
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theorem height_eq_zero (p : AsPoint) : p.val.height = 0 := by
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unfold Orthogonal.height
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rw [p.property]
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simp
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end AsPoint
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/-! ## Segment -/
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/--
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A `Segment` is an `Orthogonal` rectangle either width or height equal to `0`.
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-/
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abbrev AsSegment := Subtype (fun r : Orthogonal =>
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(r.bl.x = r.tr.x ∧ r.bl.y ≠ r.tr.y) ∨ (r.bl.x ≠ r.tr.x ∧ r.bl.y = r.tr.y))
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namespace AsSegment
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/--
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Either the width or height of an `AsSegment` is zero.
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-/
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theorem width_or_height_eq_zero (s : AsSegment)
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: s.val.width = 0 ∨ s.val.height = 0 := by
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apply Or.elim s.property
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· intro h
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refine Or.inl ?_
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unfold width
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rw [h.left]
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simp
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· intro h
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refine Or.inr ?_
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unfold height
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rw [h.right]
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simp
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end AsSegment
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end Orthogonal
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end Geometry.Rectangle
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