Merge pull request #316 from JuliaReach/mforets/300

#300 - Correct usage of IntervalArithmetic

docs/src/lib/representations.md

@@ -135,7 +135,6 @@ low(::Hyperrectangle{Float64})
 
 ```@docs
 Interval
-IA
 dim(::Interval)
 σ(::AbstractVector{Float64}, ::Interval{Float64, IntervalArithmetic.AbstractInterval{Float64}})
 center(::Interval)

src/Interval.jl

@@ -1,24 +1,18 @@
-using IntervalArithmetic
+import IntervalArithmetic
+import IntervalArithmetic: AbstractInterval
 
-"""
-    IA = IntervalArithmetic
-
-Name alias for the interval arithmetic library.
-"""
-const IA = IntervalArithmetic
-
-export Interval, IA,
+export Interval,
        dim, σ, center, +, -, *, ∈, ⊆,
        low, high, vertices_list
 
 """
-    Interval{N<:Real, IN <: IA.AbstractInterval{N}} <: AbstractPointSymmetricPolytope{N}
+    Interval{N<:Real, IN <: AbstractInterval{N}} <: AbstractPointSymmetricPolytope{N}
 
 Type representing an interval on the real line. Mathematically, it is of the
 form
 
 ```math
-[a, b] := \{ a \le x \le b \} \subseteq \mathbb{R}.
+[a, b] := \\{ a ≤ x ≤ b \\} ⊆ \\mathbb{R}.
 ```
 
 ### Fields
@@ -72,20 +66,27 @@ julia> center(x)
 This type is such that the usual pairwise arithmetic operators `+`, `-`, `*` trigger
 the corresponding interval arithmetic backend method, and return a new
 `Interval` object. For the symbolic Minkowksi sum, use `MinkowskiSum` or `⊕`.
+
+Interval of other numeric types can be created as well, eg. a rational interval:
+
+```jldoctest interval_constructor
+julia> Interval(0//1, 2//1)
+LazySets.Interval{Rational{Int64},IntervalArithmetic.Interval{Rational{Int64}}}([0//1, 2//1])
+```
 """
-struct Interval{N<:Real, IN <: IA.AbstractInterval{N}} <: AbstractPointSymmetricPolytope{N}
+struct Interval{N<:Real, IN <: AbstractInterval{N}} <: AbstractPointSymmetricPolytope{N}
    dat::IN
 end
 # type-less convenience constructor
-Interval(interval::IN) where {N, IN <: IA.AbstractInterval{N}} = Interval{N, IN}(interval)
+Interval(interval::IN) where {N, IN <: AbstractInterval{N}} = Interval{N, IN}(interval)
 
 # TODO: adapt show method
 
 # constructor that takes two numbers
-Interval(lo::N, hi::N) where {N} = Interval(IA.Interval(lo, hi))
+Interval(lo::N, hi::N) where {N} = Interval(IntervalArithmetic.Interval(lo, hi))
 
 # constructor from a vector
-Interval(x::AbstractVector{N}) where {N} = Interval(IA.Interval(x[1], x[2]))
+Interval(x::AbstractVector{N}) where {N} = Interval(IntervalArithmetic.Interval(x[1], x[2]))
 
 """
     dim(x::Interval)::Int
@@ -103,7 +104,7 @@ The integer 1.
 dim(x::Interval)::Int = 1
 
 """
-    σ(d::V, x::Interval{N, IN})::V where {N, IN <: IA.AbstractInterval{N}, V<:AbstractVector{N}}
+    σ(d::V, x::Interval{N, IN})::V where {N, IN <: AbstractInterval{N}, V<:AbstractVector{N}}
 
 Return the support vector of an ellipsoid in a given direction.
 
@@ -116,7 +117,7 @@ Return the support vector of an ellipsoid in a given direction.
 
 Support vector in the given direction.
 """
-function σ(d::V, x::Interval{N, IN})::V where {N, IN <: IA.AbstractInterval{N}, V<:AbstractVector{N}}
+function σ(d::V, x::Interval{N, IN})::V where {N, IN <: AbstractInterval{N}, V<:AbstractVector{N}}
     @assert length(d) == dim(x)
     return d[1] > 0 ? [x.dat.hi] : [x.dat.lo]
 end
@@ -134,7 +135,7 @@ Return the interval's center.
 
 The center, or midpoint, of ``x``.
 """
-center(x::Interval) = [IA.mid(x.dat)]
+center(x::Interval) = [IntervalArithmetic.mid(x.dat)]
 
 import Base:+, -, *, ∈, ⊆
 

src/convert.jl

@@ -84,10 +84,10 @@ function convert(::Type{Zonotope}, H::AbstractHyperrectangle{N}) where {N}
     return Zonotope{N}(center(H), diagm(radius_hyperrectangle(H)))
 end
 
-@require IntervalArithmetic begin
+import IntervalArithmetic.AbstractInterval
 
 """
-    convert(::Type{Hyperrectangle}, x::LazySets.Interval{N, IN}) where {N, IN <: IA.AbstractInterval{N}}
+    convert(::Type{Hyperrectangle}, x::LazySets.Interval{N, IN}) where {N, IN <: AbstractInterval{N}}
 
 Converts a unidimensional interval into a hyperrectangular set.
 
@@ -107,7 +107,6 @@ julia> convert(Hyperrectangle, Interval(0.0, 1.0))
 LazySets.Hyperrectangle{Float64}([0.5], [0.5])
 ```
 """
-function convert(::Type{Hyperrectangle}, x::LazySets.Interval{N, IN}) where {N, IN <: IA.AbstractInterval{N}}
+function convert(::Type{Hyperrectangle}, x::LazySets.Interval{N, IN}) where {N, IN <: AbstractInterval{N}}
     return Hyperrectangle(low=[low(x)], high=[high(x)])
 end
-end

test/unit_Interval.jl

@@ -1,13 +1,13 @@
-using IntervalArithmetic
+import IntervalArithmetic
 
 # default constructor
-x = LazySets.Interval{Float64, IntervalArithmetic.Interval{Float64}}(IntervalArithmetic.Interval(0.0, 1.0))
+x = Interval{Float64, IntervalArithmetic.Interval{Float64}}(IntervalArithmetic.Interval(0.0, 1.0))
 
 # type-less constructor
-x = LazySets.Interval(0.0, 1.0)
+x = Interval(0.0, 1.0)
 
 # constructor with two numbers
-x = LazySets.Interval(0.0, 1.0)
+x = Interval(0.0, 1.0)
 
 @test dim(x) == 1
 @test center(x) == [0.5]
@@ -18,12 +18,12 @@ v = vertices_list(x)
 @test an_element(x) ∈ x
 # test containment
 @test (x ⊆ x) && !(x ⊆ 0.2 * x) && (x ⊆ 2. * x)
-@test issubset(x, LazySets.Interval(0.0, 2.0))
-@test !issubset(x, LazySets.Interval(-1.0, 0.5))
+@test issubset(x, Interval(0.0, 2.0))
+@test !issubset(x, Interval(-1.0, 0.5))
 
 
 # + operator (= concrete Minkowski sum of intervals)
-y = LazySets.Interval(-2.0, 0.5)
+y = Interval(-2.0, 0.5)
 m = x + y
 @test dim(m) == 1
 @test σ([1.0], m) == [1.5]

test/unit_decompose.jl

@@ -1,4 +1,3 @@
-# using IntervalArithmetic
 import LazySets.Approximations.decompose
 
 for N in [Float64, Float32] # TODO Rational{Int}
@@ -40,8 +39,9 @@ for N in [Float64, Float32] # TODO Rational{Int}
     @test d.array[1] isa Hyperrectangle && test_directions(d.array[1])
     d = decompose(b, set_type=HPolygon, ɛ=to_N(N, 1e-2))
     @test d.array[1] isa HPolygon && test_directions(d.array[1])
-    d = decompose(b, set_type=LazySets.Interval, blocks=ones(Int, 6))
-    @test d.array[1] isa LazySets.Interval &&
+
+    d = decompose(b, set_type=Interval, blocks=ones(Int, 6))
+    @test d.array[1] isa Interval &&
         σ(N[1], d.array[1])[1] == one(N) && σ(N[-1], d.array[1])[1] == -one(N)
 
     # ===================
@@ -69,8 +69,8 @@ for N in [Float64, Float32] # TODO Rational{Int}
         @test ai isa Hyperrectangle
     end
     # 1D intervals
-    d = decompose(b, set_type=LazySets.Interval)
+    d = decompose(b, set_type=Interval)
     for ai in array(d)
-        @test ai isa LazySets.Interval
+        @test ai isa Interval
     end
 end

test/unit_overapproximate.jl

@@ -1,4 +1,3 @@
-using IntervalArithmetic
 import LazySets.Approximations.overapproximate
 
 for N in [Float64, Float32] # TODO Rational{Int}