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

finite-set-exercises
Joshua Potter 2023-05-15 14:07:42 -06:00
parent c0f8895686
commit 68d46e1a7d
4 changed files with 19 additions and 163 deletions

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

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@ -1,5 +1,4 @@
import Mathlib.Data.Fin.Basic import Mathlib.Data.Fin.Basic
import Mathlib.Tactic.LibrarySearch
import Common.Geometry.Point import Common.Geometry.Point
import Common.Geometry.Segment import Common.Geometry.Segment
@ -13,176 +12,27 @@ A characterization of an orthogonal rectangle.
namespace Geometry.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 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 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. `Orthogonal` rectangle. Edges of the rectangle are included in the result set.
-/ -/
def toSet (r : Orthogonal) : Set Point := 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 end Orthogonal

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import Common.Finset import Common.Finset
import Common.Geometry.Point
import Common.Geometry.Rectangle.Orthogonal import Common.Geometry.Rectangle.Orthogonal
import Common.List.Basic import Common.List.Basic
import Common.Set.Partition import Common.Set.Partition
@ -34,7 +33,15 @@ def toSet (sf : StepFunction) : Set Point :=
i ∈ Finset.finRange sf.p.ivls.length, i ∈ Finset.finRange sf.p.ivls.length,
let I := sf.p.ivls[i] let I := sf.p.ivls[i]
Rectangle.Orthogonal.toSet 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 end StepFunction

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