bookshelf/Bookshelf/Real/Geometry/Rectangle.lean

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import Bookshelf.Real.Geometry.Basic
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/-! # Bookshelf.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
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/--
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']
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/--
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) : :=
dist r.bottom_left r.bottom_right
/--
Computes the height of a `Rectangle`.
-/
noncomputable def height (r : Rectangle) : :=
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
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/--
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
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/--
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