From 0864bf1ebe37c9a0d1c2a880ae9fc7612144f708 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 3 Jun 2024 23:03:50 +0200
Subject: [PATCH 01/17] sort/add docs for Interval

---
 docs/src/lib/sets/Interval.md | 31 ++++++++++++++++++++++++++++---
 1 file changed, 28 insertions(+), 3 deletions(-)

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 7686577da4..6fe7bdb8e0 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -31,8 +31,6 @@ high(::Interval)
 isflat(::Interval)
 linear_map(::AbstractMatrix, ::Interval)
 low(::Interval)
-min(::Interval)
-max(::Interval)
 ngens(::Interval)
 radius_hyperrectangle(::Interval)
 radius_hyperrectangle(::Interval{N}, ::Int) where {N}
@@ -52,6 +50,8 @@ isdisjoint(::Interval, ::Interval, ::Bool=false)
 minkowski_difference(::Interval, ::Interval)
 minkowski_sum(::Interval, ::Interval)
 plot_recipe(::Interval{N}, ::Any=zero(N)) where {N}
+min(::Interval)
+max(::Interval)
 -(::Interval, ::Interval)
 *(::Interval, ::Interval)
 ```
@@ -61,6 +61,7 @@ CurrentModule = LazySets.API
 ```
 
 Undocumented implementations:
+* [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
 * [`radius`](@ref radius(::LazySet, ::Real))
 
 ```@meta
@@ -68,25 +69,49 @@ CurrentModule = LazySets
 ```
 
 Inherited from [`LazySet`](@ref):
+* [`complement`](@ref complement(::LazySet))
+* [`concretize`](@ref concretize(::LazySet))
+* [`constraints`](@ref constraints(::LazySet))
+* [`eltype`](@ref eltype(::Type{<:LazySet}))
+* [`eltype`](@ref eltype(::LazySet))
+* [`isoperation`](@ref isoperation(::LazySet))
 * [`singleton_list`](@ref singleton_list(::LazySet))
+* [`vertices`](@ref vertices(::LazySet))
+* [`affine_map`](@ref affine_map(::AbstractMatrix, ::LazySet, ::AbstractVector))
+* [`exponential_map`](@ref exponential_map(::AbstractMatrix, ::LazySet))
+* [`is_interior_point`](@ref is_interior_point(::AbstractVector, ::LazySet))
+* [`sample`](@ref sample(::LazySet, ::Int))
+* [`==`](@ref ==(::LazySet, ::LazySet))
+* [`isequivalent`](@ref isequivalent(::LazySet, ::LazySet))
+* [`⊂`](@ref ⊂(::LazySet, ::LazySet))
+
+Inherited from [`AbstractPolyhedron`](@ref):
+* [`is_polyhedral`](@ref is_polyhedral(::AbstractPolyhedron))
 
 Inherited from [`AbstractPolytope`](@ref):
 * [`isbounded`](@ref isbounded(::AbstractPolytope))
-* [`isuniversal`](@ref isuniversal(::AbstractPolytope{N}, ::Bool=false) where {N})
+* [`isboundedtype`](@ref isboundedtype(::Type{<:AbstractPolytope}))
 
 Inherited from [`AbstractCentrallySymmetricPolytope`](@ref):
 * [`isempty`](@ref isempty(::AbstractCentrallySymmetricPolytope))
+* [`isuniversal`](@ref isuniversal(::AbstractCentrallySymmetricPolytope{N}, ::Bool=false) where {N})
 
 Inherited from [`AbstractZonotope`](@ref):
 * [`order`](@ref order(::AbstractZonotope))
 * [`togrep`](@ref togrep(::AbstractZonotope))
 
 Inherited from [`AbstractHyperrectangle`](@ref):
+* [`extrema`](@ref extrema(::AbstractHyperrectangle))
+* [`extrema`](@ref extrema(::AbstractHyperrectangle, ::Int))
 * [`generators`](@ref generators(::AbstractHyperrectangle))
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
 * [`high`](@ref high(::AbstractHyperrectangle, ::Int))
+* [`isconvextype`](@ref isconvextype(::Type{<:AbstractHyperrectangle}))
 * [`low`](@ref low(::AbstractHyperrectangle, ::Int))
 * [`norm`](@ref norm(::AbstractHyperrectangle, ::Real))
+* [`volume`](@ref volume(::AbstractHyperrectangle))
+* [`cartesian_product`](@ref cartesian_product(::AbstractHyperrectangle, ::AbstractHyperrectangle))
+* [`distance`](@ref distance(::AbstractHyperrectangle, ::AbstractHyperrectangle))
 
 Some additional functionality is available for `IntervalArithmetic.Interval`s:
 

From 0a2efc1a76952cad0528528db3eaf44a33193cf3 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sat, 1 Jun 2024 12:22:29 +0200
Subject: [PATCH 02/17] add permute for Interval

---
 docs/src/lib/sets/Interval.md       | 1 +
 src/Sets/Interval/IntervalModule.jl | 5 +++--
 src/Sets/Interval/permute.jl        | 4 ++++
 test/Sets/Interval.jl               | 6 ++++++
 4 files changed, 14 insertions(+), 2 deletions(-)
 create mode 100644 src/Sets/Interval/permute.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 6fe7bdb8e0..f85b459f9e 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -62,6 +62,7 @@ CurrentModule = LazySets.API
 
 Undocumented implementations:
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
+* [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
 
 ```@meta
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index b32ca3be44..826af2325d 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -13,8 +13,8 @@ using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
 @reexport import ..API: an_element, center, constraints_list, diameter, dim, high, ∈,
-                        isoperationtype, linear_map, low, rand, rectify, reflect, scale, ρ, σ,
-                        translate, vertices_list,
+                        isoperationtype, linear_map, low, permute, rand, rectify, reflect,
+                        scale, ρ, σ, translate, vertices_list,
                         difference, intersection, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -39,6 +39,7 @@ include("linear_map.jl")
 include("low.jl")
 include("isoperationtype.jl")
 include("ngens.jl")
+include("permute.jl")
 include("radius.jl")
 include("radius_hyperrectangle.jl")
 include("rand.jl")
diff --git a/src/Sets/Interval/permute.jl b/src/Sets/Interval/permute.jl
new file mode 100644
index 0000000000..b6b989562f
--- /dev/null
+++ b/src/Sets/Interval/permute.jl
@@ -0,0 +1,4 @@
+function permute(X::Interval, p::AbstractVector{Int})
+    @assert length(p) == 1 && p[1] == 1 "invalid permutation vector $p"
+    return X
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 3db9b7a9a4..d079a37ab5 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -288,4 +288,10 @@ for N in [Float64, Float32, Rational{Int}]
     end
     res, w = ⊆(I13, I13, true)
     @test ⊆(I13, I13) && res && w == N[]
+
+    # permute
+    @test permute(x, [1]) == x
+    @test_throws AssertionError permute(x, Int[])
+    @test_throws AssertionError permute(x, Int[2])
+    @test_throws AssertionError permute(x, Int[1, 1])
 end

From e5e36127716c847f5f5c0a204aa8cff2849b1d0c Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sat, 1 Jun 2024 13:00:05 +0200
Subject: [PATCH 03/17] add project for Interval

---
 docs/src/lib/sets/Interval.md       | 1 +
 src/Sets/Interval/IntervalModule.jl | 5 +++--
 src/Sets/Interval/project.jl        | 4 ++++
 test/Sets/Interval.jl               | 6 ++++++
 4 files changed, 14 insertions(+), 2 deletions(-)
 create mode 100644 src/Sets/Interval/project.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index f85b459f9e..7c3510e668 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -63,6 +63,7 @@ CurrentModule = LazySets.API
 Undocumented implementations:
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
+* [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
 
 ```@meta
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 826af2325d..dcb5d53209 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -13,8 +13,8 @@ using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
 @reexport import ..API: an_element, center, constraints_list, diameter, dim, high, ∈,
-                        isoperationtype, linear_map, low, permute, rand, rectify, reflect,
-                        scale, ρ, σ, translate, vertices_list,
+                        isoperationtype, linear_map, low, permute, project, rand, rectify,
+                        reflect, scale, ρ, σ, translate, vertices_list,
                         difference, intersection, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -40,6 +40,7 @@ include("low.jl")
 include("isoperationtype.jl")
 include("ngens.jl")
 include("permute.jl")
+include("project.jl")
 include("radius.jl")
 include("radius_hyperrectangle.jl")
 include("rand.jl")
diff --git a/src/Sets/Interval/project.jl b/src/Sets/Interval/project.jl
new file mode 100644
index 0000000000..5073c518f7
--- /dev/null
+++ b/src/Sets/Interval/project.jl
@@ -0,0 +1,4 @@
+function project(X::Interval, block::AbstractVector{Int}; kwargs...)
+    @assert length(block) == 1 && block[1] == 1 "invalid permutation vector $block"
+    return X
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index d079a37ab5..749bf3715a 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -294,4 +294,10 @@ for N in [Float64, Float32, Rational{Int}]
     @test_throws AssertionError permute(x, Int[])
     @test_throws AssertionError permute(x, Int[2])
     @test_throws AssertionError permute(x, Int[1, 1])
+
+    # project
+    @test project(x, [1]) == x
+    @test_throws AssertionError project(x, Int[])
+    @test_throws AssertionError project(x, Int[2])
+    @test_throws AssertionError project(x, Int[1, 1])
 end

From 60fb487dcc19a79dcf5c0e1233c638ef3dae7813 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sat, 1 Jun 2024 19:58:22 +0200
Subject: [PATCH 04/17] add isapprox for Interval

---
 docs/src/lib/sets/Interval.md       | 1 +
 src/Sets/Interval/IntervalModule.jl | 3 ++-
 src/Sets/Interval/isapprox.jl       | 3 +++
 test/Sets/Interval.jl               | 4 ++++
 4 files changed, 10 insertions(+), 1 deletion(-)
 create mode 100644 src/Sets/Interval/isapprox.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 7c3510e668..82b393628c 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -65,6 +65,7 @@ Undocumented implementations:
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
+* [`≈`](@ref ≈(::LazySet, ::LazySet))
 
 ```@meta
 CurrentModule = LazySets
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index dcb5d53209..e9c886dc48 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -15,7 +15,7 @@ import IntervalArithmetic as IA
 @reexport import ..API: an_element, center, constraints_list, diameter, dim, high, ∈,
                         isoperationtype, linear_map, low, permute, project, rand, rectify,
                         reflect, scale, ρ, σ, translate, vertices_list,
-                        difference, intersection, isdisjoint, ⊆, minkowski_difference,
+                        difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
                              radius_hyperrectangle, split
@@ -55,6 +55,7 @@ include("vertices_list.jl")
 
 include("difference.jl")
 include("intersection.jl")
+include("isapprox.jl")
 include("isdisjoint.jl")
 include("issubset.jl")
 include("minkowski_difference.jl")
diff --git a/src/Sets/Interval/isapprox.jl b/src/Sets/Interval/isapprox.jl
new file mode 100644
index 0000000000..199eaaf835
--- /dev/null
+++ b/src/Sets/Interval/isapprox.jl
@@ -0,0 +1,3 @@
+function ≈(I1::Interval, I2::Interval)
+    return _isapprox(min(I1), min(I2)) && _isapprox(max(I1), max(I2))
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 749bf3715a..532bec4737 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -300,4 +300,8 @@ for N in [Float64, Float32, Rational{Int}]
     @test_throws AssertionError project(x, Int[])
     @test_throws AssertionError project(x, Int[2])
     @test_throws AssertionError project(x, Int[1, 1])
+
+    # isapprox
+    @test x ≈ x ≈ translate(x, [1e-8])
+    @test !(x ≈ translate(x, [1e-4]))
 end

From 73134cbcf08316ca002f216cbbb413646b51b909 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sat, 1 Jun 2024 20:15:24 +0200
Subject: [PATCH 05/17] add linear_combination for convex sets

---
 docs/src/lib/sets/Interval.md                | 12 ++++++++++++
 src/API/Binary/linear_combination.jl         |  2 ++
 src/ConcreteOperations/linear_combination.jl |  4 ++++
 test/Sets/Interval.jl                        |  5 +++++
 4 files changed, 23 insertions(+)

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 82b393628c..0ae7bd8e02 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -91,6 +91,18 @@ Inherited from [`LazySet`](@ref):
 Inherited from [`AbstractPolyhedron`](@ref):
 * [`is_polyhedral`](@ref is_polyhedral(::AbstractPolyhedron))
 
+Inherited from [`ConvexSet`](@ref):
+
+```@meta
+CurrentModule = LazySets.API
+```
+
+* [`linear_combination`](@ref linear_combination(::LazySet, ::LazySet))
+
+```@meta
+CurrentModule = LazySets
+```
+
 Inherited from [`AbstractPolytope`](@ref):
 * [`isbounded`](@ref isbounded(::AbstractPolytope))
 * [`isboundedtype`](@ref isboundedtype(::Type{<:AbstractPolytope}))
diff --git a/src/API/Binary/linear_combination.jl b/src/API/Binary/linear_combination.jl
index 7781f7a7ac..5aa683d7b1 100644
--- a/src/API/Binary/linear_combination.jl
+++ b/src/API/Binary/linear_combination.jl
@@ -19,5 +19,7 @@ The linear combination of two sets ``X`` and ``Y`` is defined as
 ```math
     \\left\\{\\frac{1}{2}(1+λ)x + \\frac{1}{2}(1-λ)y \\mid x ∈ X, y ∈ Y, λ ∈ [-1, 1]\\right\\}.
 ```
+
+If ``X`` and ``Y`` are convex, their linear combination is identical with their convex hull.
 """
 function linear_combination(::LazySet, ::LazySet) end
diff --git a/src/ConcreteOperations/linear_combination.jl b/src/ConcreteOperations/linear_combination.jl
index 762cb581cf..e22edc91e1 100644
--- a/src/ConcreteOperations/linear_combination.jl
+++ b/src/ConcreteOperations/linear_combination.jl
@@ -39,3 +39,7 @@ function linear_combination(P1::SimpleSparsePolynomialZonotope,
 
     return SimpleSparsePolynomialZonotope(c, G, E)
 end
+
+function linear_combination(X::ConvexSet, Y::ConvexSet)
+    return convex_hull(X, Y)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 532bec4737..7b9b9054e4 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -304,4 +304,9 @@ for N in [Float64, Float32, Rational{Int}]
     # isapprox
     @test x ≈ x ≈ translate(x, [1e-8])
     @test !(x ≈ translate(x, [1e-4]))
+
+    # convex_hull & linear_combination
+    I1 = Interval(N(0), N(1))
+    I2 = Interval(N(2), N(3))
+    @test convex_hull(I1, I2) == linear_combination(I1, I2) == Interval(N(0), N(3))
 end

From d82fb2a7e85e0f9894177c4a5630893e113527be Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sat, 1 Jun 2024 20:18:54 +0200
Subject: [PATCH 06/17] better convex_hull for Interval

---
 docs/src/lib/sets/Interval.md       | 1 +
 src/Sets/Interval/IntervalModule.jl | 7 ++++---
 src/Sets/Interval/convex_hull.jl    | 3 +++
 3 files changed, 8 insertions(+), 3 deletions(-)
 create mode 100644 src/Sets/Interval/convex_hull.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 0ae7bd8e02..61a1250387 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -65,6 +65,7 @@ Undocumented implementations:
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
+* [`convex_hull`](@ref convex_hull(::LazySet, ::LazySet))
 * [`≈`](@ref ≈(::LazySet, ::LazySet))
 
 ```@meta
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index e9c886dc48..4878f812b4 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -12,9 +12,9 @@ using ReachabilityBase.Comparison
 using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
-@reexport import ..API: an_element, center, constraints_list, diameter, dim, high, ∈,
-                        isoperationtype, linear_map, low, permute, project, rand, rectify,
-                        reflect, scale, ρ, σ, translate, vertices_list,
+@reexport import ..API: an_element, center, constraints_list, convex_hull, diameter, dim,
+                        high, ∈, isoperationtype, linear_map, low, permute, project,
+                        rand, rectify, reflect, scale, ρ, σ, translate, vertices_list,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -30,6 +30,7 @@ include("center.jl")
 include("chebyshev_center_radius.jl")
 include("constraints_list.jl")
 include("convert.jl")
+include("convex_hull.jl")
 include("diameter.jl")
 include("dim.jl")
 include("high.jl")
diff --git a/src/Sets/Interval/convex_hull.jl b/src/Sets/Interval/convex_hull.jl
new file mode 100644
index 0000000000..46b39e5076
--- /dev/null
+++ b/src/Sets/Interval/convex_hull.jl
@@ -0,0 +1,3 @@
+function convex_hull(I1::Interval, I2::Interval)
+    return Interval(min(min(I1), min(I2)), max(max(I1), max(I2)))
+end

From 3017cda67e0f5137b38e8a6bbae87e0f7218f4ce Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Thu, 6 Jun 2024 05:47:09 +0200
Subject: [PATCH 07/17] better complement for Interval

---
 docs/src/lib/sets/Interval.md       | 2 +-
 src/Sets/Interval/IntervalModule.jl | 8 +++++---
 src/Sets/Interval/complement.jl     | 6 ++++++
 test/Sets/Interval.jl               | 6 ++++++
 4 files changed, 18 insertions(+), 4 deletions(-)
 create mode 100644 src/Sets/Interval/complement.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 61a1250387..50741f3e92 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -61,6 +61,7 @@ CurrentModule = LazySets.API
 ```
 
 Undocumented implementations:
+* [`complement`](@ref complement(::LazySet))
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
@@ -73,7 +74,6 @@ CurrentModule = LazySets
 ```
 
 Inherited from [`LazySet`](@ref):
-* [`complement`](@ref complement(::LazySet))
 * [`concretize`](@ref concretize(::LazySet))
 * [`constraints`](@ref constraints(::LazySet))
 * [`eltype`](@ref eltype(::Type{<:LazySet}))
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 4878f812b4..c2ad63a720 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -12,9 +12,10 @@ using ReachabilityBase.Comparison
 using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
-@reexport import ..API: an_element, center, constraints_list, convex_hull, diameter, dim,
-                        high, ∈, isoperationtype, linear_map, low, permute, project,
-                        rand, rectify, reflect, scale, ρ, σ, translate, vertices_list,
+@reexport import ..API: an_element, center, complement, constraints_list, convex_hull,
+                        diameter, dim, high, ∈, isoperationtype, linear_map, low,
+                        permute, project, rand, rectify, reflect, scale, ρ, σ, translate,
+                        vertices_list,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -28,6 +29,7 @@ include("Interval.jl")
 include("an_element.jl")
 include("center.jl")
 include("chebyshev_center_radius.jl")
+include("complement.jl")
 include("constraints_list.jl")
 include("convert.jl")
 include("convex_hull.jl")
diff --git a/src/Sets/Interval/complement.jl b/src/Sets/Interval/complement.jl
new file mode 100644
index 0000000000..9d4f919b1f
--- /dev/null
+++ b/src/Sets/Interval/complement.jl
@@ -0,0 +1,6 @@
+function complement(X::Interval)
+    N = eltype(X)
+    L = HalfSpace(SingleEntryVector(1, 1, one(N)), min(X))
+    H = HalfSpace(SingleEntryVector(1, 1, -one(N)), -max(X))
+    return UnionSet(L, H)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 7b9b9054e4..241a6bc0cb 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -309,4 +309,10 @@ for N in [Float64, Float32, Rational{Int}]
     I1 = Interval(N(0), N(1))
     I2 = Interval(N(2), N(3))
     @test convex_hull(I1, I2) == linear_combination(I1, I2) == Interval(N(0), N(3))
+
+    # complement
+    C = complement(I2)
+    L = HalfSpace(N[1], N(2))
+    H = HalfSpace(N[-1], N(-3))
+    @test length(C) == 2 && ispermutation([C[1], C[2]], [L, H])
 end

From 04b0d5a5439b1ebd35de235cedcb0e767170a049 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sun, 9 Jun 2024 17:48:32 +0200
Subject: [PATCH 08/17] better extrema for Interval

---
 docs/src/lib/sets/Interval.md       |  4 ++--
 src/Sets/Interval/IntervalModule.jl |  7 ++++---
 src/Sets/Interval/extrema.jl        | 12 ++++++++++++
 test/Sets/Interval.jl               |  7 +++++++
 4 files changed, 25 insertions(+), 5 deletions(-)
 create mode 100644 src/Sets/Interval/extrema.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 50741f3e92..47f9ba7d56 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -62,6 +62,8 @@ CurrentModule = LazySets.API
 
 Undocumented implementations:
 * [`complement`](@ref complement(::LazySet))
+* [`extrema`](@ref extrema(::LazySet))
+* [`extrema`](@ref extrema(::LazySet, ::Int))
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
@@ -117,8 +119,6 @@ Inherited from [`AbstractZonotope`](@ref):
 * [`togrep`](@ref togrep(::AbstractZonotope))
 
 Inherited from [`AbstractHyperrectangle`](@ref):
-* [`extrema`](@ref extrema(::AbstractHyperrectangle))
-* [`extrema`](@ref extrema(::AbstractHyperrectangle, ::Int))
 * [`generators`](@ref generators(::AbstractHyperrectangle))
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
 * [`high`](@ref high(::AbstractHyperrectangle, ::Int))
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index c2ad63a720..ce60a5fa81 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -13,9 +13,9 @@ using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
 @reexport import ..API: an_element, center, complement, constraints_list, convex_hull,
-                        diameter, dim, high, ∈, isoperationtype, linear_map, low,
-                        permute, project, rand, rectify, reflect, scale, ρ, σ, translate,
-                        vertices_list,
+                        diameter, dim, extrema, high, ∈, isoperationtype, linear_map,
+                        low, permute, project, rand, rectify, reflect, scale, ρ, σ,
+                        translate, vertices_list,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -35,6 +35,7 @@ include("convert.jl")
 include("convex_hull.jl")
 include("diameter.jl")
 include("dim.jl")
+include("extrema.jl")
 include("high.jl")
 include("in.jl")
 include("isflat.jl")
diff --git a/src/Sets/Interval/extrema.jl b/src/Sets/Interval/extrema.jl
new file mode 100644
index 0000000000..676819291d
--- /dev/null
+++ b/src/Sets/Interval/extrema.jl
@@ -0,0 +1,12 @@
+# the implementations here are equivalent to the default implementation for
+# `LazySet`, but more efficient than the more specific implementations for
+# subtypes
+
+function extrema(X::Interval, i::Int)
+    @assert i == 1 "an interval has dimension 1, but the index is $i"
+    return (min(X), max(X))
+end
+
+function extrema(X::Interval)
+    return ([min(X)], [max(X)])
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 241a6bc0cb..f8f0b2d4c3 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -315,4 +315,11 @@ for N in [Float64, Float32, Rational{Int}]
     L = HalfSpace(N[1], N(2))
     H = HalfSpace(N[-1], N(-3))
     @test length(C) == 2 && ispermutation([C[1], C[2]], [L, H])
+
+    # low, high, extrema
+    @test extrema(I1) == (low(I1), high(I1)) == (N[0], N[1])
+    @test extrema(I1, 1) == (low(I1, 1), high(I1, 1)) == (N(0), N(1))
+    @test_throws AssertionError low(I1, 2)
+    @test_throws AssertionError high(I1, 2)
+    @test_throws AssertionError extrema(I1, 2)
 end

From 97d138ae8ec774a86f8c271fb8e9b2456b14a41c Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sun, 9 Jun 2024 17:59:30 +0200
Subject: [PATCH 09/17] better low/high for Interval

---
 docs/src/lib/sets/Interval.md |  8 ++++----
 src/Sets/Interval/high.jl     | 20 ++++++--------------
 src/Sets/Interval/low.jl      | 20 ++++++--------------
 3 files changed, 16 insertions(+), 32 deletions(-)

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 47f9ba7d56..ba62deb64a 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -26,11 +26,9 @@ chebyshev_center_radius(::Interval)
 constraints_list(::Interval)
 diameter(::Interval, ::Real=Inf)
 dim(::Interval)
-high(::Interval)
 ∈(::AbstractVector, ::Interval)
 isflat(::Interval)
 linear_map(::AbstractMatrix, ::Interval)
-low(::Interval)
 ngens(::Interval)
 radius_hyperrectangle(::Interval)
 radius_hyperrectangle(::Interval{N}, ::Int) where {N}
@@ -64,7 +62,11 @@ Undocumented implementations:
 * [`complement`](@ref complement(::LazySet))
 * [`extrema`](@ref extrema(::LazySet))
 * [`extrema`](@ref extrema(::LazySet, ::Int))
+* [`high`](@ref high(::LazySet, ::Int))
+* [`high`](@ref high(::LazySet))
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
+* [`low`](@ref low(::LazySet, ::Int))
+* [`low`](@ref low(::LazySet))
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
@@ -121,9 +123,7 @@ Inherited from [`AbstractZonotope`](@ref):
 Inherited from [`AbstractHyperrectangle`](@ref):
 * [`generators`](@ref generators(::AbstractHyperrectangle))
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
-* [`high`](@ref high(::AbstractHyperrectangle, ::Int))
 * [`isconvextype`](@ref isconvextype(::Type{<:AbstractHyperrectangle}))
-* [`low`](@ref low(::AbstractHyperrectangle, ::Int))
 * [`norm`](@ref norm(::AbstractHyperrectangle, ::Real))
 * [`volume`](@ref volume(::AbstractHyperrectangle))
 * [`cartesian_product`](@ref cartesian_product(::AbstractHyperrectangle, ::AbstractHyperrectangle))
diff --git a/src/Sets/Interval/high.jl b/src/Sets/Interval/high.jl
index 7b3dd7e382..ed40748d31 100644
--- a/src/Sets/Interval/high.jl
+++ b/src/Sets/Interval/high.jl
@@ -1,16 +1,8 @@
-"""
-    high(x::Interval)
-
-Return the higher coordinate of an interval set.
-
-### Input
-
-- `x` -- interval
-
-### Output
+function high(X::Interval, i::Int)
+    @assert i == 1 "an interval has dimension 1, but the index is $i"
+    return max(X)
+end
 
-A vector with the higher coordinate of the interval.
-"""
-function high(x::Interval)
-    return [x.dat.hi]
+function high(X::Interval)
+    return [max(X)]
 end
diff --git a/src/Sets/Interval/low.jl b/src/Sets/Interval/low.jl
index 6f18a067d3..7d2012a92b 100644
--- a/src/Sets/Interval/low.jl
+++ b/src/Sets/Interval/low.jl
@@ -1,16 +1,8 @@
-"""
-    low(x::Interval)
-
-Return the lower coordinate of an interval set.
-
-### Input
-
-- `x` -- interval
-
-### Output
+function low(X::Interval, i::Int)
+    @assert i == 1 "an interval has dimension 1, but the index is $i"
+    return min(X)
+end
 
-A vector with the lower coordinate of the interval.
-"""
-function low(x::Interval)
-    return [x.dat.lo]
+function low(X::Interval)
+    return [min(X)]
 end

From 7e5e5dbdbf6c7e6d8c3b5b89d81075b2e04d8df6 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sun, 9 Jun 2024 18:06:42 +0200
Subject: [PATCH 10/17] better norm for Interval

---
 docs/src/lib/sets/Interval.md       | 2 +-
 src/Sets/Interval/IntervalModule.jl | 3 ++-
 src/Sets/Interval/norm.jl           | 3 +++
 test/Sets/Interval.jl               | 4 ++++
 4 files changed, 10 insertions(+), 2 deletions(-)
 create mode 100644 src/Sets/Interval/norm.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index ba62deb64a..7cc54cf3ec 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -67,6 +67,7 @@ Undocumented implementations:
 * [`isoperationtype`](@ref isoperationtype(::Type{<:LazySet}))
 * [`low`](@ref low(::LazySet, ::Int))
 * [`low`](@ref low(::LazySet))
+* [`norm`](@ref norm(::LazySet, ::Real))
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
@@ -124,7 +125,6 @@ Inherited from [`AbstractHyperrectangle`](@ref):
 * [`generators`](@ref generators(::AbstractHyperrectangle))
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
 * [`isconvextype`](@ref isconvextype(::Type{<:AbstractHyperrectangle}))
-* [`norm`](@ref norm(::AbstractHyperrectangle, ::Real))
 * [`volume`](@ref volume(::AbstractHyperrectangle))
 * [`cartesian_product`](@ref cartesian_product(::AbstractHyperrectangle, ::AbstractHyperrectangle))
 * [`distance`](@ref distance(::AbstractHyperrectangle, ::AbstractHyperrectangle))
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index ce60a5fa81..b8b86ab8e2 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -14,7 +14,7 @@ import IntervalArithmetic as IA
 
 @reexport import ..API: an_element, center, complement, constraints_list, convex_hull,
                         diameter, dim, extrema, high, ∈, isoperationtype, linear_map,
-                        low, permute, project, rand, rectify, reflect, scale, ρ, σ,
+                        low, norm, permute, project, rand, rectify, reflect, scale, ρ, σ,
                         translate, vertices_list,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
@@ -43,6 +43,7 @@ include("linear_map.jl")
 include("low.jl")
 include("isoperationtype.jl")
 include("ngens.jl")
+include("norm.jl")
 include("permute.jl")
 include("project.jl")
 include("radius.jl")
diff --git a/src/Sets/Interval/norm.jl b/src/Sets/Interval/norm.jl
new file mode 100644
index 0000000000..dc213f6edc
--- /dev/null
+++ b/src/Sets/Interval/norm.jl
@@ -0,0 +1,3 @@
+function norm(X::Interval, p::Real=Inf)
+    return max(abs(min(X)), abs(max(X)))
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index f8f0b2d4c3..ccb3f09125 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -322,4 +322,8 @@ for N in [Float64, Float32, Rational{Int}]
     @test_throws AssertionError low(I1, 2)
     @test_throws AssertionError high(I1, 2)
     @test_throws AssertionError extrema(I1, 2)
+
+    # norm
+    @test norm(I1) == N(1)
+    @test norm(Interval(N(-2), N(1))) == N(2)
 end

From a7a57f1138201fed2e7af266ee0f737f97132893 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Sun, 9 Jun 2024 18:09:59 +0200
Subject: [PATCH 11/17] better volume for Interval

---
 docs/src/lib/sets/Interval.md       | 2 +-
 src/Sets/Interval/IntervalModule.jl | 3 ++-
 src/Sets/Interval/volume.jl         | 3 +++
 test/Sets/Interval.jl               | 4 ++++
 4 files changed, 10 insertions(+), 2 deletions(-)
 create mode 100644 src/Sets/Interval/volume.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 7cc54cf3ec..dd6afda6f8 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -71,6 +71,7 @@ Undocumented implementations:
 * [`permute`](@ref permute(::LazySet, ::AbstractVector{Int}))
 * [`project`](@ref project(::LazySet, ::AbstractVector{Int}))
 * [`radius`](@ref radius(::LazySet, ::Real))
+* [`volume`](@ref volume(::LazySet))
 * [`convex_hull`](@ref convex_hull(::LazySet, ::LazySet))
 * [`≈`](@ref ≈(::LazySet, ::LazySet))
 
@@ -125,7 +126,6 @@ Inherited from [`AbstractHyperrectangle`](@ref):
 * [`generators`](@ref generators(::AbstractHyperrectangle))
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
 * [`isconvextype`](@ref isconvextype(::Type{<:AbstractHyperrectangle}))
-* [`volume`](@ref volume(::AbstractHyperrectangle))
 * [`cartesian_product`](@ref cartesian_product(::AbstractHyperrectangle, ::AbstractHyperrectangle))
 * [`distance`](@ref distance(::AbstractHyperrectangle, ::AbstractHyperrectangle))
 
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index b8b86ab8e2..6376e9dc7d 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -15,7 +15,7 @@ import IntervalArithmetic as IA
 @reexport import ..API: an_element, center, complement, constraints_list, convex_hull,
                         diameter, dim, extrema, high, ∈, isoperationtype, linear_map,
                         low, norm, permute, project, rand, rectify, reflect, scale, ρ, σ,
-                        translate, vertices_list,
+                        translate, vertices_list, volume,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -57,6 +57,7 @@ include("support_vector.jl")
 include("support_function.jl")
 include("translate.jl")
 include("vertices_list.jl")
+include("volume.jl")
 
 include("difference.jl")
 include("intersection.jl")
diff --git a/src/Sets/Interval/volume.jl b/src/Sets/Interval/volume.jl
new file mode 100644
index 0000000000..393a7475db
--- /dev/null
+++ b/src/Sets/Interval/volume.jl
@@ -0,0 +1,3 @@
+function volume(X::Interval)
+    return max(X) - min(X)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index ccb3f09125..d0bd302b63 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -326,4 +326,8 @@ for N in [Float64, Float32, Rational{Int}]
     # norm
     @test norm(I1) == N(1)
     @test norm(Interval(N(-2), N(1))) == N(2)
+
+    # volume
+    @test volume(I1) == 1
+    @test volume(Interval(N(-2), N(1))) == N(3)
 end

From f390ec1af6e906ff880b600c53b6b33adfb144ba Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 10 Jun 2024 18:48:52 +0200
Subject: [PATCH 12/17] better affine_map for Interval

---
 docs/src/lib/sets/Interval.md       |  2 +-
 src/Sets/Interval/IntervalModule.jl |  9 +++++----
 src/Sets/Interval/affine_map.jl     | 10 ++++++++++
 test/Sets/Interval.jl               |  5 +++++
 4 files changed, 21 insertions(+), 5 deletions(-)
 create mode 100644 src/Sets/Interval/affine_map.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index dd6afda6f8..0c0bfc89de 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -59,6 +59,7 @@ CurrentModule = LazySets.API
 ```
 
 Undocumented implementations:
+* [`affine_map`](@ref affine_map(::AbstractMatrix, ::LazySet, ::AbstractVector))
 * [`complement`](@ref complement(::LazySet))
 * [`extrema`](@ref extrema(::LazySet))
 * [`extrema`](@ref extrema(::LazySet, ::Int))
@@ -87,7 +88,6 @@ Inherited from [`LazySet`](@ref):
 * [`isoperation`](@ref isoperation(::LazySet))
 * [`singleton_list`](@ref singleton_list(::LazySet))
 * [`vertices`](@ref vertices(::LazySet))
-* [`affine_map`](@ref affine_map(::AbstractMatrix, ::LazySet, ::AbstractVector))
 * [`exponential_map`](@ref exponential_map(::AbstractMatrix, ::LazySet))
 * [`is_interior_point`](@ref is_interior_point(::AbstractVector, ::LazySet))
 * [`sample`](@ref sample(::LazySet, ::Int))
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 6376e9dc7d..c5b7b3797c 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -12,10 +12,10 @@ using ReachabilityBase.Comparison
 using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
-@reexport import ..API: an_element, center, complement, constraints_list, convex_hull,
-                        diameter, dim, extrema, high, ∈, isoperationtype, linear_map,
-                        low, norm, permute, project, rand, rectify, reflect, scale, ρ, σ,
-                        translate, vertices_list, volume,
+@reexport import ..API: affine_map, an_element, center, complement, constraints_list,
+                        convex_hull, diameter, dim, extrema, high, ∈, isoperationtype,
+                        linear_map, low, norm, permute, project, rand, rectify, reflect,
+                        scale, ρ, σ, translate, vertices_list, volume,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -26,6 +26,7 @@ export Interval
 
 include("Interval.jl")
 
+include("affine_map.jl")
 include("an_element.jl")
 include("center.jl")
 include("chebyshev_center_radius.jl")
diff --git a/src/Sets/Interval/affine_map.jl b/src/Sets/Interval/affine_map.jl
new file mode 100644
index 0000000000..df7cfa63c2
--- /dev/null
+++ b/src/Sets/Interval/affine_map.jl
@@ -0,0 +1,10 @@
+function affine_map(M, X::Interval, v::AbstractVector; kwargs...)
+    @assert size(M, 2) == 1 "cannot apply an affine map of dimension $(size(M, 2)) " *
+                            "to an interval"
+    @assert size(M, 1) == length(v) "cannot apply an affine map of matrix dimension " *
+                                    "$(size(M, 1)) and translation dimension $(length(v))"
+    @inbounds if length(v) == 1
+        return Interval(M[1, 1] * X.dat + v[1])
+    end
+    return translate(linear_map(M, X; kwargs...), v)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index d0bd302b63..cbff1d5eb9 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -330,4 +330,9 @@ for N in [Float64, Float32, Rational{Int}]
     # volume
     @test volume(I1) == 1
     @test volume(Interval(N(-2), N(1))) == N(3)
+
+    # affine_map
+    M = hcat(N[2])
+    v = N[-3]
+    @test affine_map(M, I1, v) == Interval(N(-3), N(-1))
 end

From 90f2440cc6a35ee2973116b823ecf405d3c1c580 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 10 Jun 2024 19:05:48 +0200
Subject: [PATCH 13/17] better exponential_map for Interval

---
 docs/src/lib/sets/Interval.md        | 2 +-
 src/Interfaces/LazySet.jl            | 3 +++
 src/Sets/Interval/IntervalModule.jl  | 7 ++++---
 src/Sets/Interval/exponential_map.jl | 6 ++++++
 test/Sets/Interval.jl                | 3 +++
 5 files changed, 17 insertions(+), 4 deletions(-)
 create mode 100644 src/Sets/Interval/exponential_map.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 0c0bfc89de..1c95ee8f64 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -61,6 +61,7 @@ CurrentModule = LazySets.API
 Undocumented implementations:
 * [`affine_map`](@ref affine_map(::AbstractMatrix, ::LazySet, ::AbstractVector))
 * [`complement`](@ref complement(::LazySet))
+* [`exponential_map`](@ref exponential_map(::AbstractMatrix, ::LazySet))
 * [`extrema`](@ref extrema(::LazySet))
 * [`extrema`](@ref extrema(::LazySet, ::Int))
 * [`high`](@ref high(::LazySet, ::Int))
@@ -88,7 +89,6 @@ Inherited from [`LazySet`](@ref):
 * [`isoperation`](@ref isoperation(::LazySet))
 * [`singleton_list`](@ref singleton_list(::LazySet))
 * [`vertices`](@ref vertices(::LazySet))
-* [`exponential_map`](@ref exponential_map(::AbstractMatrix, ::LazySet))
 * [`is_interior_point`](@ref is_interior_point(::AbstractVector, ::LazySet))
 * [`sample`](@ref sample(::LazySet, ::Int))
 * [`==`](@ref ==(::LazySet, ::LazySet))
diff --git a/src/Interfaces/LazySet.jl b/src/Interfaces/LazySet.jl
index 640be7fb1f..c6df0a83cd 100644
--- a/src/Interfaces/LazySet.jl
+++ b/src/Interfaces/LazySet.jl
@@ -487,6 +487,9 @@ end
 The default implementation applies the functions `exp` and `linear_map`.
 """
 function exponential_map(M::AbstractMatrix, X::LazySet)
+    n = dim(X)
+    @assert size(M) == (n, n) "cannot apply an exponential map of dimension " *
+                              "$(size(M)) to an $n-dimensional set"
     return linear_map(exp(M), X)
 end
 
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index c5b7b3797c..e2f54f8b45 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -13,9 +13,9 @@ using ReachabilityBase.Distribution: reseed!
 import IntervalArithmetic as IA
 
 @reexport import ..API: affine_map, an_element, center, complement, constraints_list,
-                        convex_hull, diameter, dim, extrema, high, ∈, isoperationtype,
-                        linear_map, low, norm, permute, project, rand, rectify, reflect,
-                        scale, ρ, σ, translate, vertices_list, volume,
+                        convex_hull, diameter, dim, exponential_map, extrema, high, ∈,
+                        isoperationtype, linear_map, low, norm, permute, project, rand,
+                        rectify, reflect, scale, ρ, σ, translate, vertices_list, volume,
                         difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
                         minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
@@ -36,6 +36,7 @@ include("convert.jl")
 include("convex_hull.jl")
 include("diameter.jl")
 include("dim.jl")
+include("exponential_map.jl")
 include("extrema.jl")
 include("high.jl")
 include("in.jl")
diff --git a/src/Sets/Interval/exponential_map.jl b/src/Sets/Interval/exponential_map.jl
new file mode 100644
index 0000000000..368cc68ad5
--- /dev/null
+++ b/src/Sets/Interval/exponential_map.jl
@@ -0,0 +1,6 @@
+function exponential_map(M::AbstractMatrix, X::Interval)
+    @assert size(M) == (1, 1) "cannot apply an exponential map of dimension " *
+                              "$(size(M)) to an interval"
+    @inbounds e = exp(M[1, 1])
+    return Interval(e * X.dat)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index cbff1d5eb9..1231405e88 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -335,4 +335,7 @@ for N in [Float64, Float32, Rational{Int}]
     M = hcat(N[2])
     v = N[-3]
     @test affine_map(M, I1, v) == Interval(N(-3), N(-1))
+
+    # exponential_map
+    @test exponential_map(M, I1) == Interval(N(0), N(exp(N(2))))
 end

From 6866198cc2d1038de09db542a607e4eb575f3312 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 10 Jun 2024 20:40:42 +0200
Subject: [PATCH 14/17] fix alphabetic order

---
 src/Sets/Interval/IntervalModule.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index e2f54f8b45..1208832619 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -41,9 +41,9 @@ include("extrema.jl")
 include("high.jl")
 include("in.jl")
 include("isflat.jl")
+include("isoperationtype.jl")
 include("linear_map.jl")
 include("low.jl")
-include("isoperationtype.jl")
 include("ngens.jl")
 include("norm.jl")
 include("permute.jl")

From ce5801fe86dae07e9daa91a84dae978ca058a464 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 10 Jun 2024 21:00:23 +0200
Subject: [PATCH 15/17] better isequivalent for Interval

---
 docs/src/lib/sets/Interval.md       | 2 +-
 src/Sets/Interval/IntervalModule.jl | 5 +++--
 src/Sets/Interval/isequivalent.jl   | 3 +++
 test/Sets/Interval.jl               | 3 +++
 4 files changed, 10 insertions(+), 3 deletions(-)
 create mode 100644 src/Sets/Interval/isequivalent.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 1c95ee8f64..995d56a059 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -76,6 +76,7 @@ Undocumented implementations:
 * [`volume`](@ref volume(::LazySet))
 * [`convex_hull`](@ref convex_hull(::LazySet, ::LazySet))
 * [`≈`](@ref ≈(::LazySet, ::LazySet))
+* [`isequivalent`](@ref isequivalent(::LazySet, ::LazySet))
 
 ```@meta
 CurrentModule = LazySets
@@ -92,7 +93,6 @@ Inherited from [`LazySet`](@ref):
 * [`is_interior_point`](@ref is_interior_point(::AbstractVector, ::LazySet))
 * [`sample`](@ref sample(::LazySet, ::Int))
 * [`==`](@ref ==(::LazySet, ::LazySet))
-* [`isequivalent`](@ref isequivalent(::LazySet, ::LazySet))
 * [`⊂`](@ref ⊂(::LazySet, ::LazySet))
 
 Inherited from [`AbstractPolyhedron`](@ref):
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 1208832619..79531f9e80 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -16,8 +16,8 @@ import IntervalArithmetic as IA
                         convex_hull, diameter, dim, exponential_map, extrema, high, ∈,
                         isoperationtype, linear_map, low, norm, permute, project, rand,
                         rectify, reflect, scale, ρ, σ, translate, vertices_list, volume,
-                        difference, intersection, ≈, isdisjoint, ⊆, minkowski_difference,
-                        minkowski_sum
+                        difference, intersection, ≈, isdisjoint, isequivalent, ⊆,
+                        minkowski_difference, minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
                              radius_hyperrectangle, split
 import Base: convert, -, *, min, max
@@ -65,6 +65,7 @@ include("difference.jl")
 include("intersection.jl")
 include("isapprox.jl")
 include("isdisjoint.jl")
+include("isequivalent.jl")
 include("issubset.jl")
 include("minkowski_difference.jl")
 include("minkowski_sum.jl")
diff --git a/src/Sets/Interval/isequivalent.jl b/src/Sets/Interval/isequivalent.jl
new file mode 100644
index 0000000000..c0e2c7ad27
--- /dev/null
+++ b/src/Sets/Interval/isequivalent.jl
@@ -0,0 +1,3 @@
+function isequivalent(I1::Interval, I2::Interval)
+    return I1 ≈ I2
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 1231405e88..450e573ff5 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -338,4 +338,7 @@ for N in [Float64, Float32, Rational{Int}]
 
     # exponential_map
     @test exponential_map(M, I1) == Interval(N(0), N(exp(N(2))))
+
+    # isequivalent
+    @test isequivalent(I1, I1) && !isequivalent(I1, I2)
 end

From 9e720b4e88f30fbd12146b3ad5e31635f7960c17 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Mon, 10 Jun 2024 22:47:16 +0200
Subject: [PATCH 16/17] better isstrictsubset for Interval

---
 docs/src/lib/sets/Interval.md       |  2 +-
 src/Sets/Interval/IntervalModule.jl |  3 ++-
 src/Sets/Interval/isstrictsubset.jl | 11 +++++++++++
 test/Sets/Interval.jl               | 11 +++++++++++
 4 files changed, 25 insertions(+), 2 deletions(-)
 create mode 100644 src/Sets/Interval/isstrictsubset.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 995d56a059..9fd329c8d3 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -77,6 +77,7 @@ Undocumented implementations:
 * [`convex_hull`](@ref convex_hull(::LazySet, ::LazySet))
 * [`≈`](@ref ≈(::LazySet, ::LazySet))
 * [`isequivalent`](@ref isequivalent(::LazySet, ::LazySet))
+* [`⊂`](@ref ⊂(::LazySet, ::LazySet))
 
 ```@meta
 CurrentModule = LazySets
@@ -93,7 +94,6 @@ Inherited from [`LazySet`](@ref):
 * [`is_interior_point`](@ref is_interior_point(::AbstractVector, ::LazySet))
 * [`sample`](@ref sample(::LazySet, ::Int))
 * [`==`](@ref ==(::LazySet, ::LazySet))
-* [`⊂`](@ref ⊂(::LazySet, ::LazySet))
 
 Inherited from [`AbstractPolyhedron`](@ref):
 * [`is_polyhedral`](@ref is_polyhedral(::AbstractPolyhedron))
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 79531f9e80..0fe1bcc6de 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -16,7 +16,7 @@ import IntervalArithmetic as IA
                         convex_hull, diameter, dim, exponential_map, extrema, high, ∈,
                         isoperationtype, linear_map, low, norm, permute, project, rand,
                         rectify, reflect, scale, ρ, σ, translate, vertices_list, volume,
-                        difference, intersection, ≈, isdisjoint, isequivalent, ⊆,
+                        difference, intersection, ≈, isdisjoint, isequivalent, ⊂, ⊆,
                         minkowski_difference, minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
                              radius_hyperrectangle, split
@@ -66,6 +66,7 @@ include("intersection.jl")
 include("isapprox.jl")
 include("isdisjoint.jl")
 include("isequivalent.jl")
+include("isstrictsubset.jl")
 include("issubset.jl")
 include("minkowski_difference.jl")
 include("minkowski_sum.jl")
diff --git a/src/Sets/Interval/isstrictsubset.jl b/src/Sets/Interval/isstrictsubset.jl
new file mode 100644
index 0000000000..ea4bcbe02a
--- /dev/null
+++ b/src/Sets/Interval/isstrictsubset.jl
@@ -0,0 +1,11 @@
+function ⊂(X::Interval, Y::Interval, witness::Bool=false)
+    if min(X) < min(Y) || max(X) > max(Y)
+        return _witness_result_empty(witness, false, X, Y)
+    end
+    if min(X) > min(Y)
+        return witness ? (true, low(Y)) : true
+    elseif max(Y) > max(X)
+        return witness ? (true, high(Y)) : true
+    end
+    return _witness_result_empty(witness, false, X, Y)
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index 450e573ff5..fe1183a7ba 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -282,12 +282,23 @@ for N in [Float64, Float32, Rational{Int}]
     I13 = Interval(N(1), N(3))
     I02 = Interval(N(0), N(2))
     I24 = Interval(N(2), N(4))
+    I03 = Interval(N(0), N(3))
+    I14 = Interval(N(1), N(4))
     for I in (I02, I24)
         res, w = ⊆(I13, I, true)
         @test !(⊆(I13, I)) && !res && w ∈ I13 && w ∉ I
     end
     res, w = ⊆(I13, I13, true)
     @test ⊆(I13, I13) && res && w == N[]
+    # isstrictsubset
+    for I in (I02, I24, I13)
+        res, w = ⊂(I13, I, true)
+        @test !(⊂(I13, I)) && !res && w == N[]
+    end
+    for I in (I03, I14)
+        res, w = ⊂(I13, I, true)
+        @test ⊂(I13, I) && res && w ∈ I && w ∉ I13
+    end
 
     # permute
     @test permute(x, [1]) == x

From 36c3dba1b41967fc5dfe507c107271c1f5fa8158 Mon Sep 17 00:00:00 2001
From: schillic <git@christianschilling.net>
Date: Tue, 11 Jun 2024 18:55:31 +0200
Subject: [PATCH 17/17] better distance for Interval

---
 docs/src/lib/sets/Interval.md       | 2 +-
 src/Sets/Interval/IntervalModule.jl | 5 +++--
 src/Sets/Interval/distance.jl       | 7 +++++++
 test/Sets/Interval.jl               | 5 +++++
 4 files changed, 16 insertions(+), 3 deletions(-)
 create mode 100644 src/Sets/Interval/distance.jl

diff --git a/docs/src/lib/sets/Interval.md b/docs/src/lib/sets/Interval.md
index 9fd329c8d3..51d4009e02 100644
--- a/docs/src/lib/sets/Interval.md
+++ b/docs/src/lib/sets/Interval.md
@@ -75,6 +75,7 @@ Undocumented implementations:
 * [`radius`](@ref radius(::LazySet, ::Real))
 * [`volume`](@ref volume(::LazySet))
 * [`convex_hull`](@ref convex_hull(::LazySet, ::LazySet))
+* [`distance`](@ref distance(::LazySet, ::LazySet))
 * [`≈`](@ref ≈(::LazySet, ::LazySet))
 * [`isequivalent`](@ref isequivalent(::LazySet, ::LazySet))
 * [`⊂`](@ref ⊂(::LazySet, ::LazySet))
@@ -127,7 +128,6 @@ Inherited from [`AbstractHyperrectangle`](@ref):
 * [`genmat`](@ref genmat(::AbstractHyperrectangle))
 * [`isconvextype`](@ref isconvextype(::Type{<:AbstractHyperrectangle}))
 * [`cartesian_product`](@ref cartesian_product(::AbstractHyperrectangle, ::AbstractHyperrectangle))
-* [`distance`](@ref distance(::AbstractHyperrectangle, ::AbstractHyperrectangle))
 
 Some additional functionality is available for `IntervalArithmetic.Interval`s:
 
diff --git a/src/Sets/Interval/IntervalModule.jl b/src/Sets/Interval/IntervalModule.jl
index 0fe1bcc6de..1ebc741a40 100644
--- a/src/Sets/Interval/IntervalModule.jl
+++ b/src/Sets/Interval/IntervalModule.jl
@@ -16,8 +16,8 @@ import IntervalArithmetic as IA
                         convex_hull, diameter, dim, exponential_map, extrema, high, ∈,
                         isoperationtype, linear_map, low, norm, permute, project, rand,
                         rectify, reflect, scale, ρ, σ, translate, vertices_list, volume,
-                        difference, intersection, ≈, isdisjoint, isequivalent, ⊂, ⊆,
-                        minkowski_difference, minkowski_sum
+                        difference, distance, intersection, ≈, isdisjoint, isequivalent,
+                        ⊂, ⊆, minkowski_difference, minkowski_sum
 @reexport import ..LazySets: chebyshev_center_radius, isflat, ngens, plot_recipe,
                              radius_hyperrectangle, split
 import Base: convert, -, *, min, max
@@ -62,6 +62,7 @@ include("vertices_list.jl")
 include("volume.jl")
 
 include("difference.jl")
+include("distance.jl")
 include("intersection.jl")
 include("isapprox.jl")
 include("isdisjoint.jl")
diff --git a/src/Sets/Interval/distance.jl b/src/Sets/Interval/distance.jl
new file mode 100644
index 0000000000..bcd3392116
--- /dev/null
+++ b/src/Sets/Interval/distance.jl
@@ -0,0 +1,7 @@
+function distance(X::Interval, Y::Interval; p::Real=2)
+    d = max(min(X) - max(Y), min(Y) - max(X))
+    if d < 0
+        return zero(d)
+    end
+    return d
+end
diff --git a/test/Sets/Interval.jl b/test/Sets/Interval.jl
index fe1183a7ba..224acd85b6 100644
--- a/test/Sets/Interval.jl
+++ b/test/Sets/Interval.jl
@@ -352,4 +352,9 @@ for N in [Float64, Float32, Rational{Int}]
 
     # isequivalent
     @test isequivalent(I1, I1) && !isequivalent(I1, I2)
+
+    # distance
+    @test distance(I1, I2) == distance(I2, I1) == N(1)
+    I3 = Interval(N(1//2), N(2))
+    @test distance(I1, I3) == distance(I2, I3) == distance(I1, I1) == N(0)
 end