Use \sqcup and \sqcap

Project.toml

@@ -1,13 +1,15 @@
 name = "IntervalContractors"
 uuid = "15111844-de3b-5229-b4ba-526f2f385dc9"
-version = "0.4.7"
+version = "0.4.8"
 
 [deps]
 IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
+IntervalBoxes = "43d83c95-ebbb-40ec-8188-24586a1458ed"
 
 [compat]
-IntervalArithmetic = "0.16, 0.17, 0.18, 0.19, 0.20"
-julia = "1.3, 1.4, 1.5"
+IntervalArithmetic = "0.22"
+IntervalBoxes = "0.1"
+julia = "1"
 
 [extras]
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"

README.md

@@ -18,7 +18,7 @@
 
 The reverse function of a function `f` calculates the (interval hull of) its inverse function, `f⁻¹`.
 
-For example, `sin_rev(Y::Interval, X::Interval)` calculates the (interval hull of) those `x ∈ X` such that `sin(x) ∈ Y`. This can also be thought of as an inverse function, calculating `X_new := sin⁻¹(Y) ∩ X`.
+For example, `sin_rev(Y::Interval, X::Interval)` calculates the (interval hull of) those `x ∈ X` such that `sin(x) ∈ Y`. This can also be thought of as an inverse function, calculating `X_new := sin⁻¹(Y) ⊓ X`.
 The return value is `(Y, X_new)`.
 
 Functions such as `mul_rev(C::Interval, A::Interval, B::Interval)` take three arguments, and correspond to `C = A * B`; they return `(C, A_new, B_new)`, with `A_new` and `B_new` similarly defined to be the corresponding inverse images of the multiplication operator in each component.

examples/examples.jl

@@ -5,19 +5,19 @@ function sawtooth(X::IntervalBox)
 
     x, y = X
 
-    x = x ∩ (-1..1)
-    y = y ∩ (-2..2)
+    x = x ⊓ (-1..1)
+    y = y ⊓ (-2..2)
 
-    y = y ∩ (2x)
-    x = x ∩ (y/2)
+    y = y ⊓ (2x)
+    x = x ⊓ (y/2)
 
     return IntervalBox(x, y)
 end
 
 
 function constant_contractor(X, y_val)
     x, y = X
-    y = y ∩ Interval(y_val)
+    y = y ⊓ Interval(y_val)
     return IntervalBox(x, y)
 end
 
@@ -26,8 +26,8 @@ end
 function add_one(X)  # y = x + 1
     x, y = X
 
-    y = y ∩ (x + 1)
-    x = x ∩ (y - 1)
+    y = y ⊓ (x + 1)
+    x = x ⊓ (y - 1)
 
     return IntervalBox(x, y)
 end
@@ -38,10 +38,10 @@ function cube0(X::IntervalBox)  # contractor for y=x^3, x>=0
 
     x, y = X
 
-    x = x ∩ (0..∞)
+    x = x ⊓ (0..∞)
 
-    y = y ∩ (x ^ 3)
-    x = x ∩ Interval(y.lo ^ (1/3), y.hi^(1/3))   # not rigorous!
+    y = y ⊓ (x ^ 3)
+    x = x ⊓ Interval(y.lo ^ (1/3), y.hi^(1/3))   # not rigorous!
 
     return x × y
 end
@@ -53,7 +53,7 @@ odd(X::IntervalBox) = ( (x,y) = X; IntervalBox(-x, -y) )
 """
 
 cube_neg = symmetrise(cube0, odd)
-cube = cube0 ∪ cube_neg
+cube = cube0 ⊔ cube_neg
 
 ff(x) = x^2 - x^3
 
@@ -63,10 +63,10 @@ function ff(X::IntervalBox)
     a = x^2
     b = x^3
 
-    y = y ∩ (a - b)  # y = a - b
+    y = y ⊓ (a - b)  # y = a - b
 
-    a = a ∩ (y + b)
-    b = b ∩ (a - y)
+    a = a ⊓ (y + b)
+    b = b ⊓ (a - y)
 
     x, a = square(IntervalBox(x, a))
     x, b = cube(IntervalBox(x, b))

src/IntervalContractors.jl

@@ -17,12 +17,12 @@ export plus_rev, minus_rev, inv_rev,
 
 using IntervalArithmetic
 
-const half_pi = IntervalArithmetic.half_pi(Float64)  # interval
-const two_pi = IntervalArithmetic.two_pi(Float64)  # interval
+const half_pi = ExactReal(0.5) * interval(pi)
+const two_pi = ExactReal(2.0) * interval(pi)
 
 #
-# Base.:∪(f::Function, g::Function) = X -> ( f(X) ∪ g(X) )
-# Base.:∩(f::Function, g::Function) = X -> ( f(X) ∩ g(X) )  # or f(g(X)) for contractors
+# Base.:⊔(f::Function, g::Function) = X -> ( f(X) ⊔ g(X) )
+# Base.:⊓(f::Function, g::Function) = X -> ( f(X) ⊓ g(X) )  # or f(g(X)) for contractors
 
 include("arithmetic.jl")
 include("transformations.jl")
@@ -33,7 +33,6 @@ include("inverse_trig.jl")
 include("hyperbolic.jl")
 include("inverse_hyperbolic.jl")
 include("extrema.jl")
-include("decorated.jl")
 
 """
 Dictionary mapping functions to their reverse functions.

src/arithmetic.jl

@@ -13,9 +13,9 @@ The triplet `(a, b_new, c_new)` where
 - `c_new` is the interval hull of the set ``{y ∈ c : ∃ x ∈ b, x + y ∈ a}``
 """
 function plus_rev(a::Interval, b::Interval, c::Interval)  # a = b + c
-    # a = a ∩ (b + c)  # add this line for plus contractor (as opposed to reverse function)
-    b_new = b ∩ (a - c)
-    c_new = c ∩ (a - b)
+    # a = a ⊓ (b + c)  # add this line for plus contractor (as opposed to reverse function)
+    b_new = b ⊓ (a - c)
+    c_new = c ⊓ (a - b)
 
     return a, b_new, c_new
 end
@@ -37,16 +37,16 @@ The triplet `(a, b_new, c_new)` where
 """
 function minus_rev(a::Interval, b::Interval, c::Interval)  # a = b - c
 
-    b_new = b ∩ (a + c)
-    c_new = c ∩ (b - a)
+    b_new = b ⊓ (a + c)
+    c_new = c ⊓ (b - a)
 
     return a, b_new, c_new
 end
 
 minus_rev(a,b,c) = minus_rev(promote(a,b,c)...)
 
 function minus_rev(a::Interval, b::Interval)  # a = -b
-    b_new = b ∩ (-a)
+    b_new = b ⊓ (-a)
     return (a, b_new)
 end
 
@@ -56,25 +56,25 @@ Reverse multiplication
 """
 function mul_rev(a::Interval, b::Interval, c::Interval)  # a = b * c
 
-    # ((0.0 ∉ a) || (0.0 ∉ b)) && (c = c ∩ (a / b))
-    # ((0.0 ∉ a) || (0.0 ∉ c)) && (b = b ∩ (a / c))
+    # ((0.0 ∉ a) || (0.0 ∉ b)) && (c = c ⊓ (a / b))
+    # ((0.0 ∉ a) || (0.0 ∉ c)) && (b = b ⊓ (a / c))
 
-    # a = a ∩ (b * c)  # ?
+    # a = a ⊓ (b * c)  # ?
 
     if 0 ∈ b
-        temp = c .∩ extended_div(a, b)
+        temp = c .⊓ extended_div(a, b)
         c′ = union(temp[1], temp[2])
 
     else
-        c′ = c ∩ (a / b)
+        c′ = c ⊓ (a / b)
     end
 
     if 0 ∈ c
-        temp = b .∩ extended_div(a, c)
+        temp = b .⊓ extended_div(a, c)
         b′ = union(temp[1], temp[2])
 
     else
-        b′ = b ∩ (a / c)
+        b′ = b ⊓ (a / c)
     end
 
     return a, b′, c′
@@ -87,8 +87,8 @@ Reverse division
 """
 function div_rev(a::Interval, b::Interval, c::Interval)  # a = b / c
 
-    b = b ∩ (a * c)
-    c = c ∩ (b / a)
+    b = b ⊓ (a * c)
+    c = c ⊓ (b / a)
 
     return a, b, c
 end
@@ -109,7 +109,7 @@ Pair `(a, b_new)` where
 """
 function inv_rev(a::Interval, b::Interval)  # a = inv(b)
 
-    b_new = b ∩ inv(a)
+    b_new = b ⊓ inv(a)
 
     return a, b_new
 end
@@ -132,27 +132,27 @@ The triplet `(a, b_new, n)` where
 function power_rev(a::Interval{T}, b::Interval{T}, n::Integer) where T  # a = b^n,  log(a) = n.log(b),  b = a^(1/n)
 
     if iszero(n)
-        1 ∈ a && return (a, entireinterval(T) ∩ b, n)
+        1 ∈ a && return (a, entireinterval(T) ⊓ b, n)
         return (a, emptyinterval(T), n)
     end
 
     if n == 2  # a = b^2
         root = √a
-        b1 = b ∩ root
-        b2 = b ∩ (-root)
+        b1 = b ⊓ root
+        b2 = b ⊓ (-root)
 
     elseif iseven(n)
         root = a^(1//n)
 
-        b1 = b ∩ root
-        b2 = b ∩ (-root)
+        b1 = b ⊓ root
+        b2 = b ⊓ (-root)
 
     elseif isodd(n)
-        pos_root = (a ∩ (0..∞)) ^ (1//n)
-        neg_root = -( ( (-a) ∩ (0..∞) ) ^ (1//n) )
+        pos_root = (a ⊓ (0..∞)) ^ (1//n)
+        neg_root = -( ( (-a) ⊓ (0..∞) ) ^ (1//n) )
 
-        b1 = b ∩ pos_root
-        b2 = b ∩ neg_root
+        b1 = b ⊓ pos_root
+        b2 = b ⊓ neg_root
 
     end
 
@@ -170,8 +170,8 @@ function power_rev(a::Interval, b::Interval, c::Interval)  # a = b^c
         return (temp[1], temp[2], interval(temp[3]))
     end
 
-    b_new = b ∩ ( a^(inv(c) ))
-    c_new = c ∩ (log(a) / log(b))
+    b_new = b ⊓ ( a^(inv(c) ))
+    c_new = c ⊓ (log(a) / log(b))
 
     return a, b_new, c_new
 end
@@ -193,7 +193,7 @@ The pair `(a, b_new)` where
 """
 function sqrt_rev(a::Interval, b::Interval)  # a = sqrt(b)
 
-    b_new = b ∩ (a^2)
+    b_new = b ⊓ (a^2)
 
     return a, b_new
 end
@@ -220,8 +220,8 @@ function sqr_rev(c, x)   # c = x^2;  refine x
 
     root = sqrt(c)
 
-    x1 = x ∩ root
-    x2 = x ∩ (-root)
+    x1 = x ⊓ root
+    x2 = x ⊓ (-root)
 
     return (c, hull(x1, x2))
 end
@@ -241,10 +241,10 @@ The pair `(c, x_new)` where
 """
 function abs_rev(y, x)   # y = abs(x); refine x
 
-    y_new = y ∩ (0..∞)
+    y_new = y ⊓ (0..∞)
 
-    x1 = y_new ∩ x
-    x2 = -(y_new ∩ (-x))
+    x1 = y_new ⊓ x
+    x2 = -(y_new ⊓ (-x))
 
     return (y, hull(x1, x2))
 end
@@ -254,9 +254,9 @@ Reverse sign
 """
 function sign_rev(a::Interval, b::Interval)  # a = sqrt(b)
 
-    (a == 1.0) && b = b ∩ (0..∞)
-    (a == 0.0) && b = b ∩ (0.0..0.0)
-    (a == -1.0) && b = b ∩ (-∞..0.0)
+    (a == 1.0) && b = b ⊓ (0..∞)
+    (a == 0.0) && b = b ⊓ (0.0..0.0)
+    (a == -1.0) && b = b ⊓ (-∞..0.0)
 
     return a, b
 end
@@ -288,7 +288,7 @@ IEEE 1788-2015 standard for interval arithmetic.
 - `x_new` the interval hull of the set ``{t ∈ x : ∃ y ∈ b, tʸ ∈ c}
 """
 function pow_rev1(b, c, x)   # c = x^b
-    return x ∩ c^(1/b)  # replace by 1//b
+    return x ⊓ c^(1/b)  # replace by 1//b
 end
 
 """
@@ -302,7 +302,7 @@ byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 st
 - `x_new` the interval hull of the set ``{t ∈ x : ∃ y ∈ b, tʸ ∈ c}
 """
 function pow_rev2(a, c, x)   # c = a^x
-    return x ∩ (log(c) / log(a))
+    return x ⊓ (log(c) / log(a))
 end
 
 """