Lambda Calculus via C# (4) Tuple and Signed Numeral

Friday, November 10, 2023

[FP & LINQ via C# series]

[Lambda Calculus via C# series]

Besides modeling values like Boolean and numeral, anonymous function can also model data structures. In Church encoding, Church pair is an approach to use functions to represent a tuple of 2 items.

Church pair (2-tuple)

A tuple can be constructed with its first item x, its second item y, and a function f:

CreateTuple := λx.λy.λf.f x y

So a tuple is can be created by partially applying CreateTuple with 2 items x and y:

Tuple := CreateTuple x y
       ≡ (λx.λy.λf.f x y) x y
       ≡ λf.f x y

So a tuple is a higher-order function, which accept a function f, and apply it with its 2 items. So f accepts 2 arguments, it is in the form of λx.λy.E.

To get the tuple’s first item x, just apply the tuple function with a specific function f, where f simply accepts 2 items and returns the first item:

  Tuple (λx.λy.x)
≡ (λf.f x y) (λx.λy.x)
≡ (λx.λy.x) x y
≡ x

Similarly, to get the tuple’s second item y, just apply tuple function with a specific function f, where f simply accepts 2 items and returns the first item:

  Tuple (λx.λy.y)
≡ (λf.f x y) (λx.λy.y)
≡ (λx.λy.y) x y
≡ y

So the following Item1 function is defined to accept a tuple, apply the tuple function with function λx.λy.x, and return the tuple’s first item:

Item1 := λt.t (λx.λy.x)

Again, this is how it works:

  Item1 (CreateTuple x y)
≡ (λt.t (λx.λy.x)) (CreateTuple x y)
≡ (λt.t (λx.λy.x)) (λf.f x y)
≡ (λf.f x y) (λx.λy.x)
≡ (λx.λy.x) x y
≡ x

And Item2 function can ne defined in the same way to get the tuple’s second item:

Item2 := λt.t (λx.λy.y)

Notice functions λx.λy.x and λx.λy.y can be alpha converted to λt.λf.t and λt.λf.f, which are just Church Boolean True and False. So Item1 and Item2 can be defined as:

Item1 := λt.t True
Item2 := λt.t False

To implement tuple in C#, its function type needs to be identified. The tuple function accepts argument f, which is either function True of function False, so f is of function type Boolean. In the body of tuple function, f is applied, and f returns dynamic. So tuple virtually is of function type Boolean -> dynamic:

using static ChurchBoolean;

// Tuple is the alias of (dynamic -> dynamic -> dynamic) -> dynamic.
public delegate dynamic Tuple<out T1, out T2>(Boolean f);

public static partial class ChurchTuple<T1, T2>
{
    public static readonly Func<T1, Func<T2, Tuple<T1, T2>>> 
        Create = item1 => item2 => f => f(item1)(item2);

    // Item1 = tuple => tuple(True)
    public static readonly Func<Tuple<T1, T2>, T1> 
        Item1 = tuple => (T1)(object)tuple(True);

    // Item2 = tuple => tuple(False)
    public static readonly Func<Tuple<T1, T2>, T2> 
        Item2 = tuple => (T2)(object)tuple(False);
}

There are type conversions in Item1/Item2 functions. At compiled time, the tuple function returns dynamic, and at runtime it actually it calls True/False function to return either item1 or item2 . So the type conversions are always safe. Also notice here tuple function’s return value cannot be directly converted to T1 or T2, because of a bug of C# runtime binding layer. The workaround is to convert dynamic to object first, then convert to T1 or T2.

The following are the extension methods for convenience:

public static partial class TupleExtensions
{
    public static T1 Item1<T1, T2>(this Tuple<T1, T2> tuple) => ChurchTuple<T1, T2>.Item1(tuple);

    public static T2 Item2<T1, T2>(this Tuple<T1, T2> tuple) => ChurchTuple<T1, T2>.Item2(tuple);
}

For example, a point can be a tuple of 2 numerals:

internal static void Point(Numeral x, Numeral y)
{
    Tuple<Numeral, Numeral> point1 = ChurchTuple<Numeral, Numeral>.Create(x)(y);
    Numeral x1 = point1.Item1();
    Numeral y1 = point1.Item1();

    // Move up.
    Numeral y2 = y1.Increase();
    Tuple<Numeral, Numeral> point2 = ChurchTuple<Numeral, Numeral>.Create(x1)(y2);
}

Tuple operators

The Swap function accepts a tuple (x, y), swaps its first item and second item, and returns a new tuple (y, x):

Swap := λt.CreateTuple (Item2 t)(Item1 t)

Apparently, Swap is of function type Tuple<T1, T2> -> Tuple<T2, T1>:

// Swap = tuple => Create(tuple.Item2())(tuple.Item1())
public static readonly Func<Tuple<T1, T2>, Tuple<T2, T1>>
    Swap = tuple => ChurchTuple<T2, T1>.Create(tuple.Item2())(tuple.Item1());

The Shift function accepts a tuple (x, y) and a function f, and returns a new tuple (y, f y):

Shift := λf.λt.CreateTuple (Item2 t) (f (Item2 t))

Here assume the argument tuple (x, y) is of type Tuple<T1, T2>, regarding f is applied with y, assume f returns TResult type, then f is of function type T2 -> TResult, so that the returned new tuple (y, f y) is of type Tuple<T2, TResult>. As a result, Shift is of type Tuple<T1, T2> -> (T2 -> TResult) -> Tuple<T2, TResult>:

public static partial class ChurchTuple<T1, T2, TResult>
{
    // Shift = f => tuple => Create(tuple.Item2())(f(tuple.Item1()))
    public static readonly Func<Func<T2, TResult>, Func<Tuple<T1, T2>, Tuple<T2, TResult>>>
        Shift = f => tuple => ChurchTuple<T2, TResult>.Create(tuple.Item2())(f(tuple.Item2()));
}

And their extension methods:

public static Tuple<T2, T1> Swap<T1, T2>(this Tuple<T1, T2> tuple) => ChurchTuple<T1, T2>.Swap(tuple);

public static Tuple<T2, TResult> Shift<T1, T2, TResult>(this Tuple<T1, T2> tuple, Func<T2, TResult> f) => 
    ChurchTuple<T1, T2, TResult>.Shift(f)(tuple);

Here Shift function can be used to define the Subtract function for Church numerals. Remember a Church numeral n can be viewed as to apply Increase n times from 0:

  n Increase 0
≡ n

Applying Shift with Increase and a tuple of Church numerals, it returns a new tuple of Church numerals, so this application can repeat forever:

  Shift Increase (0, 0)
≡ (0, Increase 0)
≡ (0, 1)

  Shift Increase (0, 1)
≡ (1, Increase 1)
≡ (1, 2)

  Shift Increase (1, 2)
≡ (2, Increase 2)
≡ (2, 3)

...

In another word, partially applying Shift with Increase is a function that can be repeatedly applied with a tuple of Church numerals:

  (Shift Increase) (0, 0)                                       ≡ (Shift Increase)¹ (0, 0) ≡ 1 (Shift Increase) (0, 0) 
≡ (0, 1)

  (Shift Increase) (0, 1)
≡ (Shift Increase) ((Shift Increase) (0, 0))
≡ (Shift Increase) ∘ (Shift Increase) (0, 0)                    ≡ (Shift Increase)² (0, 0) ≡ 2 (Shift Increase) (0, 0) 
≡ (1, 2)

  (Shift Increase) (1, 2)
≡ (Shift Increase) ((Shift Increase) ∘ (Shift Increase) (0, 0))
≡ (Shift Increase) ∘ (Shift Increase) ∘ (Shift Increase) (0, 0) ≡ (Shift Increase)³ (0, 0) ≡ 3 (Shift Increase) (0, 0) 
≡ (2, 3)

...

So generally:

  n (Shift Increase) (0, 0)
≡ (n - 1, n)

As a result, to decrease n to n – 1, just apply n with function (Shift Increase) and tuple (0, 0), get the result tuple (n – 1, n), and return its first item:

  Item1 (n (Shift Increase) (0, 0))
≡ Item1 (n - 1, n)
≡ n - 1

So Decrease can be defined as:

Decrease := λn.Item1 (n (Shift Increase) (CreateTuple 0 0))

And C#:

// Decrease = n => n(tuple => tuple.Shift(Increase))(0, 0).Item1();
public static readonly Func<Numeral, Numeral> Decrease = n =>
    ((Tuple<Numeral, Numeral>)n
        (tuple => ((Tuple<Numeral, Numeral>)tuple).Shift(Increase))
        (ChurchTuple<Numeral, Numeral>.Create(Zero)(Zero)))
    .Item1();

N-tuple

An easy way is to model n-tuple as a 2-tuple of the first value and a (n-1)-tuple of the rest values. A 3 tuple of values 1, 2, 3 can be represented by nested 2-tuples as (a, (b, c)), a 4-tuple of values 1, 2, 3, 4 can be represented by nested 2-tuples (1, (2, (3, 4))), etc., and a n tuple of values 1, 2, 3, …, n can be represented by nested 2 tuples (1, (2, (3, (…(n-1, n)…)))). For example, the following is the definition of 3 tuple:

Create3Tuple := λx.λy.λz.CreateTuple x (CreateTuple y z)

3TupleItem1 := λt.Item1 t
3TupleItem2 := λt.Item1 (Item2 t)
3TupleItem3 := λt.Item2 (Item2 t)

And in C#:

public delegate dynamic Tuple<out T1, out T2, out T3>(Boolean f);

public static partial class ChurchTuple<T1, T2, T3>
{
    // Create = item1 => item2 => item3 => Create(item1)(Create(item2)(item3))
    public static readonly Func<T1, Func<T2, Func<T3, Tuple<T1, T2, T3>>>>
        Create = item1 => item2 => item3 => new Tuple<T1, T2, T3>(ChurchTuple<T1, Tuple<T2, T3>>.Create(item1)(ChurchTuple<T2, T3>.Create(item2)(item3)));

    // Item1 = tuple.Item1()
    public static readonly Func<Tuple<T1, T2, T3>, T1>
        Item1 = tuple => new Tuple<T1, Tuple<T2, T3>>(tuple).Item1();

    // Item2 = tuple.Item2().Item1()
    public static readonly Func<Tuple<T1, T2, T3>, T2>
        Item2 = tuple => new Tuple<T1, Tuple<T2, T3>>(tuple).Item2().Item1();

    // Item3 = tuple.Item2().Item2()
    public static readonly Func<Tuple<T1, T2, T3>, T3>
        Item3 = tuple => new Tuple<T1, Tuple<T2, T3>>(tuple).Item2().Item2();
}

public static partial class TupleExtensions
{
    public static T1 Item1<T1, T2, T3>(this Tuple<T1, T2, T3> tuple) => ChurchTuple<T1, T2, T3>.Item1(tuple);

    public static T2 Item2<T1, T2, T3>(this Tuple<T1, T2, T3> tuple) => ChurchTuple<T1, T2, T3>.Item2(tuple);

    public static T3 Item3<T1, T2, T3>(this Tuple<T1, T2, T3> tuple) => ChurchTuple<T1, T2, T3>.Item3(tuple);
}

Signed numeral

With tuple, a signed numeral (integer) can be modeled by a pair of Church numerals (natural numbers), where the first item represents the positive value, and the second item represents the negative value:

SignedNumeral := Tuple

For example (1, 0) and (2, 1) models 1, (0, 2) and (1, 3) models –2, (0, 0) and (1, 1) models 0, etc.:

 1 := (1, 0) ≡ (2, 1) ≡ (3, 2) ≡ (4, 3) ≡ ...
 0 := (0, 0) ≡ (1, 1) ≡ (2, 2) ≡ (3, 3) ≡ ...
-2 := (0, 2) ≡ (1, 3) ≡ (2, 4) ≡ (3, 5) ≡ ...

In C#, the function type SignedNumeral is the same as Tuple, except SignedNumeral is not open generic type:

// SignedNumeral is the alias of Tuple<Numeral, Numeral>.
public delegate dynamic SignedNumeral(Boolean f);

Church numeral represents natural number. So Converting a Church numeral n to signed number is easy, just make it a tuple (n, 0):

Sign := λn.CreateTuple n 0

To negate a signed numeral, just swap its positive value and negative value:

Negate := Swap

And it is straightforward to get the positive value and the negative value from a signed number:

Positive := Item1
Negative := Item2

Signed numeral like (4, 3), (3, 3), (3, 5) can be formatted to have at least one 0: (1, 0), (0, 0), (0, 2). For a signed number s represented by (p, n), If p >= n, then it is (p - n, 0), otherwise it is (0, n – p):

Format := λs.If (s_p >=  s_n) (λx.(s_p - s_n, 0)) (λx.(0, s_n - s_p))

Here S_p is the positive value of s, and s_n the the negative value of s.

The following are these function’s C# implementation, and the extension methods:

using static ChurchBoolean;
using static ChurchNumeral;

public static partial class ChurchSignedNumeral
{
    // Sign = n => (n, 0)
    public static readonly Func<Numeral, SignedNumeral>
        Sign = n => new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create(n)(Zero));

    // Negate = signed => signed.Swap()
    public static readonly Func<SignedNumeral, SignedNumeral>
        Negate = signed => new SignedNumeral(new Tuple<Numeral, Numeral>(signed).Swap());

    // Positive = signed => signed.Item1()
    public static readonly Func<SignedNumeral, Numeral>
        Positive = signed => new Tuple<Numeral, Numeral>(signed).Item1();

    // Negative = signed => signed.Item2()
    public static readonly Func<SignedNumeral, Numeral>
        Negative = signed => new Tuple<Numeral, Numeral>(signed).Item2();

    // Format = signed =>
    //    If(positive >= negative)
    //        (_ => (positive - negative, 0))
    //        (_ => (0, negative - positive))
    public static readonly Func<SignedNumeral, SignedNumeral>
        Format = signed =>
            If(signed.Positive().IsGreaterThanOrEqualTo(signed.Negative()))
                (_ => signed.Positive().Subtract(signed.Negative()).Sign())
                (_ => signed.Negative().Subtract(signed.Positive()).Sign().Negate());
}

public static partial class SignedNumeralExtensions
{
    public static SignedNumeral Sign(this Numeral n) => ChurchSignedNumeral.Sign(n);

    public static SignedNumeral Negate(this SignedNumeral signed) => ChurchSignedNumeral.Negate(signed);

    public static Numeral Positive(this SignedNumeral signed) => ChurchSignedNumeral.Positive(signed);

    public static Numeral Negative(this SignedNumeral signed) => ChurchSignedNumeral.Negative(signed);

    public static SignedNumeral Format(this SignedNumeral signed) => ChurchSignedNumeral.Format(signed);
}

Arithmetic operators

Naturally, for signed numbers a, b:

  a + b
≡ (a_p, a_n) + (b_p, b_n)
≡ (a_p - a_n) + (b_p - b_n)
≡ (a_p + b_p, a_n + b_n)

  a - b
≡ (a_p, a_n) - (b_p, b_n)
≡ (a_p - a_n) - (b_p - b_n)
≡ (a_p + b_n, a_n + b_p)

  a * b
≡ (a_p, a_n) * (b_p, b_n)
≡ (a_p - a_n) * (b_p - b_n)
≡ (a_p * b_p + a_n * b_n, a_p * b_n + a_n * b_p)

  a / b
≡ (a_p, a_n) / (b_p, b_n)
≡ (a_p - a_n) / (b_p - b_n)
≡ (a_p / b_p + a_n / b_n, a_p / b_n + a_n / b_p)

So in lambda calculus:

AddSigned := λa.λb.Format (CreateTuple (a_p + b_p) (a_n + b_n))
SubtractSigned := λa.λb.Format (CreateTuple (a_p + b_n) (a_n + b_p))
MultiplySigned := λa.λb.Format (CreateTuple (a_p * b_p + a_n * b_n) (a_p * b_n + a_n * b_p))

Division is more tricky because a and b’s positive and negative values can be 0. In this case, just return 0 when dividing by 0:

DivideByIgnoreZero := λa.λb.If (IsZero b) (λx.0) (λx.DivideBy a b)

Here the DivideBy function for Church numeral is used. As fore mentioned, this DivideBy function is not well defined. It is temporarily used here and will be revisited later. So division can be defined as:

DivideBySigned := λa.λb.Format (CreateTuple ((DivideByIgnoreZero a_p b_p) + (DivideByIgnoreZero a_n b_n)) ((DivideByIgnoreZero a_p b_n) + (DivideByIgnoreZero a_n b_p)))

The following are the C# implementations and the extension methods:

public static partial class ChurchSignedNumeral
{
    // Add = a => b => (a.Positive() + b.Positive(), a.Negative() + b.Negative()).Format()
    public static readonly Func<SignedNumeral, Func<SignedNumeral, SignedNumeral>>
        Add = a => b => new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create
                (a.Positive().Add(b.Positive()))
                (a.Negative().Add(b.Negative())))
            .Format();

    // Subtract = a => b => (a.Positive() + b.Negative(), a.Negative() + b.Positive()).Format()
    public static readonly Func<SignedNumeral, Func<SignedNumeral, SignedNumeral>>
        Subtract = a => b => new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create
                (a.Positive().Add(b.Negative()))
                (a.Negative().Add(b.Positive())))
            .Format();

    // Multiply = a => b => (a.Positive() * b.Positive() + a.Negative() * b.Negative(), a.Positive() * b.Negative() + a.Negative() * b.Positive()).Format()
    public static readonly Func<SignedNumeral, Func<SignedNumeral, SignedNumeral>>
        Multiply = a => b => new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create
                (a.Positive().Multiply(b.Positive()).Add(a.Negative().Multiply(b.Negative())))
                (a.Positive().Multiply(b.Negative()).Add(a.Negative().Multiply(b.Positive()))))
            .Format();

    // / = dividend => divisor => If(divisor.IsZero())(_ => 0)(_ => dividend.DivideBy(divisor))
    private static readonly Func<Numeral, Func<Numeral, Numeral>> 
        DivideByIgnoreZero = dividend => divisor =>
            ChurchBoolean<Numeral>.If(divisor.IsZero())
                (_ => Zero)
                (_ => dividend.DivideBy(divisor));

    // DivideBy = dividend => divisor => (dividend.Positive() / divisor.Positive() + dividend.Negative() / divisor.Negative(), dividend.Positive() / divisor.Negative() + dividend.Negative() / divisor.Positive()).Format();
    public static readonly Func<SignedNumeral, Func<SignedNumeral, SignedNumeral>>
        DivideBy = dividend => divisor => new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create
                (DivideByIgnoreZero(dividend.Positive())(divisor.Positive()).Add(DivideByIgnoreZero(dividend.Negative())(divisor.Negative())))
                (DivideByIgnoreZero(dividend.Positive())(divisor.Negative()).Add(DivideByIgnoreZero(dividend.Negative())(divisor.Positive()))))
            .Format();
}

public static partial class SignedNumeralExtensions
{
    public static SignedNumeral Add(this SignedNumeral a, SignedNumeral b) => ChurchSignedNumeral.Add(a)(b);

    public static SignedNumeral Subtract(this SignedNumeral a, SignedNumeral b) => ChurchSignedNumeral.Subtract(a)(b);

    public static SignedNumeral Multiply(this SignedNumeral a, SignedNumeral b) => ChurchSignedNumeral.Multiply(a)(b);

    public static SignedNumeral DivideBy(this SignedNumeral dividend, SignedNumeral divisor) => ChurchSignedNumeral.DivideBy(dividend)(divisor);
}

And the following code demonstrate how these operators work:

[TestClass]
public class ChurchSignedNumeralTests
{
    [TestMethod]
    public void SignNegatePositiveNegativeTest()
    {
        SignedNumeral signed = 0U.Church().Sign();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());
        signed = signed.Negate();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());

        signed = 1U.Church().Sign();
        Assert.IsTrue(1U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());
        signed = signed.Negate();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(1U == signed.Negative().Unchurch());

        signed = 2U.Church().Sign();
        Assert.IsTrue(2U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());
        signed = signed.Negate();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(2U == signed.Negative().Unchurch());

        signed = 123U.Church().Sign();
        Assert.IsTrue(123U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());
        signed = signed.Negate();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(123U == signed.Negative().Unchurch());

        signed = new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create(12U.Church())(23U.Church()));
        Assert.IsTrue(12U == signed.Positive().Unchurch());
        Assert.IsTrue(23U == signed.Negative().Unchurch());
        signed = signed.Negate();
        Assert.IsTrue(23U == signed.Positive().Unchurch());
        Assert.IsTrue(12U == signed.Negative().Unchurch());
    }

    [TestMethod]
    public void FormatWithZeroTest()
    {
        SignedNumeral signed = new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create(12U.Church())(23U.Church()));
        signed = signed.Format();
        Assert.IsTrue(0U == signed.Positive().Unchurch());
        Assert.IsTrue(11U == signed.Negative().Unchurch());

        signed = new SignedNumeral(ChurchTuple<Numeral, Numeral>.Create(23U.Church())(12U.Church()));
        signed = signed.Format();
        Assert.IsTrue(11U == signed.Positive().Unchurch());
        Assert.IsTrue(0U == signed.Negative().Unchurch());
    }

    [TestMethod]
    public void AddTest()
    {
        SignedNumeral a = 0U.Church().Sign();
        SignedNumeral b = 0U.Church().Sign();
        SignedNumeral result = a.Add(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 1U.Church().Sign();
        b = 1U.Church().Sign().Negate();
        result = a.Add(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 3U.Church().Sign();
        b = 5U.Church().Sign().Negate();
        result = a.Add(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(2U == result.Negative().Unchurch());
    }

    [TestMethod]
    public void SubtractTest()
    {
        SignedNumeral a = 0U.Church().Sign();
        SignedNumeral b = 0U.Church().Sign();
        SignedNumeral result = a.Subtract(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 1U.Church().Sign();
        b = 1U.Church().Sign().Negate();
        result = a.Subtract(b);
        Assert.IsTrue(2U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 3U.Church().Sign();
        b = 5U.Church().Sign().Negate();
        result = a.Subtract(b);
        Assert.IsTrue(8U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());
    }

    [TestMethod]
    public void MultiplyTest()
    {
        SignedNumeral a = 0U.Church().Sign();
        SignedNumeral b = 0U.Church().Sign();
        SignedNumeral result = a.Multiply(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 1U.Church().Sign();
        b = 1U.Church().Sign().Negate();
        result = a.Multiply(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(1U == result.Negative().Unchurch());

        a = 3U.Church().Sign();
        b = 5U.Church().Sign().Negate();
        result = a.Multiply(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(15U == result.Negative().Unchurch());
    }

    [TestMethod]
    public void DivideByTest()
    {
        SignedNumeral a = 0U.Church().Sign();
        SignedNumeral b = 0U.Church().Sign();
        SignedNumeral result = a.DivideBy(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(0U == result.Negative().Unchurch());

        a = 1U.Church().Sign();
        b = 1U.Church().Sign().Negate();
        result = a.DivideBy(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(1U == result.Negative().Unchurch());

        a = 11U.Church().Sign();
        b = 5U.Church().Sign().Negate();
        result = a.DivideBy(b);
        Assert.IsTrue(0U == result.Positive().Unchurch());
        Assert.IsTrue(2U == result.Negative().Unchurch());
    }
}

81 Comments

جراحی لاغری

اسلیومعده - Monday, August 3, 2020 10:45:12 AM

Many gamblers will agree that playing and winning real money is a far better experience at any online casino. This is why you should choose the right casino for you that suits your needs and has a quality reputation for keeping your info and money safe. Our list has the best online casinos around and they offer the top online gaming experience for gamblers of all types <a href="https://www.totosite365.info" target="_blank" title="토토">토토</a>

토토 - Saturday, December 19, 2020 9:17:12 AM

All online casinos have wagering requirements in place; this means any winnings made using your bonus money will only become available after wagering a certain amount. <a href="https://www.slotmachine777.site" target="_blank" title="릴게임">릴게임</a>

릴게임 - Saturday, December 19, 2020 9:18:39 AM

thats awesome!

حمل خودرو ، امداد خودرو ، خودرو بر ، یدک کش در شیراز - Tuesday, February 16, 2021 9:12:47 PM

very good

شماره امداد خودرو شیراز - Tuesday, February 16, 2021 9:14:16 PM

great

حمل خودرو از شیراز به بوشهر - Tuesday, February 16, 2021 9:15:50 PM

This article is really fantastic and thanks for sharing the valuable post.

Love problem solutions - Tuesday, April 27, 2021 1:18:51 AM

Great article with excellent idea! I have bookmarked your site since this site contains important data in it.

Vashikaran Specialist Astrologer Baba ji - Tuesday, April 27, 2021 4:43:04 AM

Wow, incredible blog layout! How long have you been blogging for you made blogging look easy. The overall look of your site is excellent, let alone the content!

Muslim Vashikaran Specialist - Tuesday, April 27, 2021 4:57:27 AM

Great Article it its really informative and innovative keep us posted with new updates. its was really valuable.

Vashikaran Specialist Astrologers - Tuesday, April 27, 2021 5:50:59 AM

Thanks for writing such a good article, I stumbled onto your blog and read a few post. I like your style of writing...

Love Problem Solutions - Tuesday, April 27, 2021 6:10:16 AM

Very interesting topic will bookmark your site to check if you Post more about in the future.

Husband Wife Problem Solution In India - Tuesday, April 27, 2021 6:58:51 AM

فیلتر پرس :خرید و فروش دستگاه فیلترپرس ، پارچه فیلتر پرس ، صفحه فیلتر پرس ، قیمت و هزینه مناسب (مستقیم از تولید کننده)،بهترین قطعات با کیفیت و ارائه خدمات تضمین شده در زمینه لوازم جانبی و مواد و تجهیزات مصرفی دستگاه فیلتر پرس ، با فیلتر پرس صفا به آسودگی به تجارت خود بپردازید

فیلتر پرس - Monday, May 3, 2021 2:19:48 AM

if you are looking pet shop dogs for your animal, you can visit our site

petshop dog - Tuesday, May 18, 2021 1:45:08 AM

https://ma-study.blogspot.com/

medicalphd - Monday, December 13, 2021 11:25:48 AM

Many developers are getting help from program solutions websites like Github etc. But post. I want to prefer your "Lambda Calculus" post.

Local Movers Services in Atlanta GA is an essential procedure for removing your furniture.

Local Movers Services in Atlanta GA - Thursday, May 19, 2022 10:06:48 AM

This article is very helpful and interesting too. Keep doing this in future. I will support you. <a href="https://powerballcity.com" title="파워볼사이트">파워볼사이트</a>

파워볼사이트 - Monday, June 20, 2022 10:10:46 PM

Hi! this is often nice article you shared with great information. Thanks for giving such an exquisite informative information. <a href="https://holdem777.com" title="홀덤사이트">홀덤사이트</a>

홀덤사이트 - Monday, July 4, 2022 11:01:16 PM

The 60-day game time is currently the only game time provided by the blizzard company for the players of the game, Word of Warcraft. In the past, game times such as 30 days and 180 days were also available, but due to the new policies of this company and the policy it has considered, the only game time that is currently available for dear gamers is Game Time 60. It is fasting. In the following, we have collected interesting explanations about Game Time for you, which are worth reading.

Game time application for two months

Currently, 2-month game time is used in all areas of World of Warcraft. But if you want to experience a series of interesting and new experiences, you should buy this game time. These experiences include:
Using new extensions
Play on new maps
Lollup in a new style
Change in the shape of the game
Produced from the Jet Game website. Please support the Jet Game website.

https://jetgames.ir - Thursday, July 7, 2022 4:46:48 AM

I accidentally searched and visited your site. I still saw several posts during my visit, but the text was neat and readable. I will quote this post and post it on my blog. Would you like to visit my blog later? <a href="https://twintoto.com" title="토토사이트">토토사이트</a>

토토사이트 - Sunday, July 17, 2022 10:25:57 PM

It's difficult to find well-informed people about this topic, but you seem like you know what you're talking about! Thanks, <a href="https://badug-i.com" title="바둑이사이트">바둑이사이트</a>

바둑이사이트 - Friday, July 22, 2022 10:10:19 PM

Of course, your article is good enough, Keonhacai but I thought it would be much better to see professional photos and videos together. There are articles and photos on these topics on my homepage, so please visit and share your opinions.

Keonhacai - Sunday, August 7, 2022 1:04:49 AM

I have been looking for articles on these topics for a long time.Keo nha cai I don't know how grateful you are for posting on this topic. Thank you for the numerous articles on this site, I will subscribe to those links in my bookmarks and visit them often. Have a nice day

Keo nha cai - Tuesday, August 16, 2022 6:59:11 PM

خرید بازی دراگون فلایت جت گیم سری بازی ورلد آف وارکرافت یکی از قدیمی ترین گیم هایی است که هم از نظر محبوبیت و هم از نظر شکل بازی نزدیک به دو دهه است که با ارائه انواع بسته های الحاقی برای دوستداران این گیم سرپا است و به کار خود ادامه می دهد .
ورلد آف وارکرافت توسط شرکت بلیزارد ارائه شده و بدلیل سبک بازی و گرافیک بالا در سرتاسر جهان طرفداران زیادی را به خود جذب کرده است.
این بازی محبوب دارای انواع بسته های الحاقی میباشد که جدید ترین آن که به تازگی رونمائی شده و در حال حاضر صرفا امکان تهیه پیش فروش آن فراهم میباشد دراگون فلایت است
این بازی که از نظر سبک بازی با سایر نسخه ها متفاوت بوده و جذابیت خاص خود را دارد که در ادامه به آن می پردازیم . همچنین برای تهیه نسخه های این گیم جذاب می توانید به سایت جت گیم مراجعه نمائید. در ادامه بیشتر در مورد بازی و سیستم مورد نیاز بازی می پردازیم

https://jetgames.ir - Tuesday, September 6, 2022 8:26:38 AM

What is this article? There's a lot going on in this article. I can't stop reading it. I really want to know the person who wrote this article.

ดาวน์โหลด wm - Wednesday, March 15, 2023 10:18:00 PM

Wow another very good knowledge Thanks for the great information. And it's quite useful.
<a href="https://www.ebet-789.com/apply/">สมัคร ebet</a>

สมัคร ebet - Monday, April 24, 2023 1:30:36 AM

Thank you for your very good site, as well as the variety of content that you provide to users.
<a href="https://asia-panel.com/roof-sandwichpanel/">ساندویچ پانل سقفی</a>
<a href="https://asia-panel.com/polyurthane-sandwichpanel/">ساندویچ پانل پلی اورتان</a>
<a href="https://asia-panel.com/runner/">رانر ساندویچ پانل</a>

آسیا پانل - Thursday, May 4, 2023 12:27:41 AM

check this out if you are interested in the net worth of celebrities
https://lavishcelebrities.com/

julia - Tuesday, June 6, 2023 9:35:43 PM

First, you must confirm whether your device supports the Disney Plus application and service. You can confirm it with the list of devices provided below.

Disney Plus Begin - Wednesday, July 5, 2023 3:36:30 AM

thank you for lambda calculus. Lambda can sometimes confusing me.

okey oyna - Wednesday, July 26, 2023 3:58:48 AM

Wow, amazing blog layout! How long have you been blogging? You make blogs easily. The overall look and content of your website is fantastic! 토토사이트 https://szzzac.com/

토토사이트 - Thursday, September 7, 2023 10:40:28 PM

Tak mungkin kamu menemukan situs terbaik selain di <a href="https://bursa188.pro/"rel="dofollow">BURSA188</a> <a href="https://bursa188.store/"rel="dofollow">BURSA188</a>

BURSA188 - Saturday, September 23, 2023 2:23:19 AM

Smadav is capable of restoring deleted files, whereas other antivirus programs only detect infections. Smadav can also be used as an additional antivirus alongside Windows Defender. Because Windows Defender can sometimes be overwhelmed by a large number of pathogen attacks.

Smadav Latest Version - Sunday, October 1, 2023 1:05:00 AM

I know the more articles you write the more what you write It's definitely going to be even more wonderful. i will wait to read it

Pretty Baccarat ติดต่อเรา - Wednesday, November 1, 2023 12:02:46 AM

When I read an article on this topic, <a href="https://images.google.pn/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">slotsite</a> the first thought was profound and difficult, and I wondered if others could understand.. My site has a discussion board for articles and photos similar to this topic. Could you please visit me when you have time to discuss this topic?

slotsite - Wednesday, November 22, 2023 12:51:46 AM

Your writing is perfect and complete. <a href="https://images.google.pl/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">baccaratcommunity</a> However, I think it will be more wonderful if your post includes additional topics that I am thinking of. I have a lot of posts on my site similar to your topic. Would you like to visit once?

baccaratcommunity - Wednesday, November 22, 2023 12:52:13 AM

As I am looking at your writing, <a href="https://images.google.nu/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">casinocommunity</a> I regret being unable to do outdoor activities due to Corona 19, and I miss my old daily life. If you also miss the daily life of those days, would you please visit my site once? My site is a site where I post about photos and daily life when I was free.

casinocommunity - Wednesday, November 22, 2023 12:53:02 AM

Looking at this article, I miss the time when I didn't wear a mask. <a href="https://images.google.nr/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">majorsite</a> Hopefully this corona will end soon. My blog is a blog that mainly posts pictures of daily life before Corona and landscapes at that time. If you want to remember that time again, please visit us.

majorsite - Wednesday, November 22, 2023 12:53:39 AM

Great ! that’s what I was seeking for what a data! present here at this blog, thanks admin of this website.

ซื้อหวยผ่านคลิปโต - Thursday, November 30, 2023 9:38:04 PM

JNANABHUMI AP provides a CBSE syllabus for all classes for the academic year 2024 has been designed as per the guidelines of the CBSE Board. The syllabus offers a conceptual background and lays the groundwork for the Class 10 Board exams. https://jnanabhumiap.in/ By visiting the page, students will find the downloadable pdf of the reduced CBSE 10th Syllabus along with the deleted portion of the syllabus for each subject. So, students are advised to prepare for the exam, as per the syllabus mentioned here.

jnanabhumiap.in - Friday, January 12, 2024 12:28:00 AM

While electric smoke vapes Products are often considered a less harmful alternative to traditional smoking, they are not without risks. The long-term health effects of vaping are still being studied, and some concerns include nicotine addiction, lung issues, and potential exposure to harmful chemicals.

Fume Infinity Flavors - Thursday, February 15, 2024 10:28:08 AM

Indeed, exploring Lambda Calculus through C# offers insight into the versatile nature of anonymous functions. By employing Church encoding, one can represent data structures such as tuples using functions. This approach mirrors the concept of constructing fences in Minecraft, where individual blocks are used to create a larger structure. Similarly, in Lambda Calculus, functions serve as the building blocks for modeling complex data types, much like placing blocks in Minecraft to form a fence.

Drewq Finr - Saturday, March 9, 2024 8:51:35 AM

You have great knowledge and expertise in writing such blogs. Keep up the great work!

안전한 파워볼사이트 - Sunday, March 10, 2024 1:41:57 AM

I love the way you write and share your niche! Very interesting and different! Keep it coming!

안전 토토사이트 - Sunday, March 10, 2024 1:43:30 AM

This is the masterpiece I have ever read. Your presentation and writing style both are admirable. I’m also anxiously waiting for your upcoming updates.

온라인카지노사이트 - Sunday, March 10, 2024 1:45:38 AM

Hi there, I check your blog like every week. Your story-telling style is awesome, keep it up!

토토사이트 추천 - Sunday, March 10, 2024 1:47:14 AM

Its a great pleasure reading your post.Its full of information I am looking for and I love to post a comment that “The content of your post is awesome” Great work.

실시간 바카라사이트 - Sunday, March 10, 2024 1:49:07 AM

Innovative techniques with great selection of sentences summarizing the title in a best means. Happy to read it.

안전 토토사이트 - Sunday, March 10, 2024 1:51:04 AM

If you are having trouble calculating points, this is a useful tool for you.

final grade calculator - Monday, April 8, 2024 8:43:10 PM

Nice read, I just passed this onto a colleague who was doing a little research on that.

카지노사이트 - Monday, May 13, 2024 1:53:18 AM

That is a really good tip especially to those new to the blogosphere.

카지노사이트 - Monday, May 13, 2024 1:55:20 AM

I am glad that you shared this useful information withus. Please stay us up to date like this.

바카라사이트 - Monday, May 13, 2024 1:56:50 AM

I do believe all of the ideas you have offered on yourpost.

바카라사이트 - Monday, May 13, 2024 1:59:04 AM

Outstanding post once again. I am looking forward to more updates.

casinosite.one - Friday, June 7, 2024 1:08:26 AM

I want you to know that no matter what kind of article you write, you'll be able to write it. I will wait to read it every day and forever.

สมัคร bet game - Sunday, June 16, 2024 8:34:58 PM

The post has meaningful information, which helps to get better knowledge.

123plus สล็อต - Sunday, June 23, 2024 9:31:18 PM

This very nice post. I just stumbled upon your weblog and wished to say that I have truly enjoyed surfing around your blog posts.

123vip ไลน์ - Thursday, June 27, 2024 1:14:22 AM

آموزش حسابداری در بهترین آموزشگاه خانه حسابداری در کرج ( شعه خانه حسابداری )

آموزشگاه خانه حسابداری در کرج ( - Tuesday, July 23, 2024 12:40:10 AM

Thank you for contributing to the body of knowledge in [field or industry].

ยี่ กี ล็ อ ต โต้ - Tuesday, September 3, 2024 10:11:25 PM

Great job! This post was very helpful and informative.

Area Code - Monday, September 9, 2024 11:37:14 PM

As with many MOD APKs, future updates may introduce new features or enhancements to improve the user experience. It’s important to stay updated with the latest versions to take advantage of these improvements.

Vmake ai - Friday, September 13, 2024 11:39:29 PM

eCourts Portal of Western Australia · Register · Log in · Manage Subscriptions · Accessibility · Help. Acknowledge Conditions Of Use. Small Text; Medium Text

NYSCEF - Thursday, September 19, 2024 12:28:05 AM

This is the article I've been looking for for a long time. I love every topic and every story you write about.

ลิงค์สมัครหวยฮ่องกง - Friday, October 18, 2024 4:37:42 PM

Toca Life is perfect for kids who love to explore and imagine. Each time my daughter plays, she builds entirely different stories, interacting with new characters and discovering hidden surprises within the game. It's a fantastic way to develop creativity.

Toca Download - Monday, October 21, 2024 3:55:17 AM

Fire Kirin stands out because of its variety of game modes and the chance to win big. The special events and bonuses are a great way to keep players engaged, and the regular updates add new features to enjoy.

Online Fire Kirin - Monday, October 21, 2024 3:55:52 AM

Great blog i think your blog so useful for us thank a lot i hope you will share your article to us again.

tarea อุบล - Tuesday, October 29, 2024 3:41:22 AM

Your passion for [subject matter] shines through in your writing, and it is truly inspiring.

หวย ขูด เกาหลี - Thursday, November 7, 2024 11:20:02 PM

Users can create unlimited playlists and enjoy the unlimited skip count and play the song in any order as per their preference. Unlike other music streaming platforms, it does not pose any restrictions on users and does not ask you to purchase the premium plan.’

vimusic - Friday, November 8, 2024 12:06:43 PM

You have really creative ideas. It's all interesting I'm not tired of reading at all.

สมัคร sa gaming - Thursday, November 14, 2024 6:20:45 PM

i must say and thanks for the information.I appreciate your post and look forward to more.

123bet สล็อต - Thursday, November 14, 2024 8:37:23 PM

Minecraft Education Edition is a free version of Minecraft designed for students and educators, focusing on learning and collaboration.

Minecraft Download - Monday, November 18, 2024 11:39:31 PM

Whether you’re tackling puzzles solo or sharing the fun with friends, Dream Pet Link offers an enjoyable and engaging experience for everyone.

dream pet link kostenlos spielen - Wednesday, November 27, 2024 8:41:33 PM

Gran blog, Dixin. Excelente combinación de experiencia técnica y creatividad. Los ejemplos en GitHub son un recurso valioso. ¿Planeas incluir más contenido sobre desarrollo avanzado?

M.henderson - Thursday, November 28, 2024 5:21:46 AM

Gran blog, Dixin. Excelente combinación de experiencia técnica y creatividad. Los ejemplos en GitHub son un recurso valioso. ¿Planeas incluir más contenido sobre desarrollo avanzado?

M.henderson - Thursday, November 28, 2024 5:22:11 AM

I’m thankful for the wide variety of resources available here.

ติดต่อหวย huay bee - Thursday, November 28, 2024 9:14:41 PM

"Dreams often hold clues to our thoughts and emotions, offering a glimpse into our inner world. If you're interested in learning more about their meanings and interpretations, check out <a href="https://dreamsgenius.com">this insightful website</a>. It provides valuable information to help you decode what your dreams might be trying to tell you."

willsmith - Tuesday, December 3, 2024 9:37:53 PM

"Sometimes, understanding the differences between similar concepts or terms can make a huge difference in learning and decision-making. I came across a really helpful site that explains these distinctions clearly. Check it out here: <a href="https://askanydifference.com">this comparison resource</a>. It's an excellent tool for anyone looking to enhance their knowledge."

Raba - Tuesday, December 10, 2024 7:29:18 AM

The DGME Paystub Portal, a key component of Dollar General's employee management system, stands out as an exemplary model in the retail sector.

dgme - Tuesday, December 10, 2024 11:15:14 PM

In this Game, Players dunk themselves into the dynamic world of real excitement and challenges, where games of each category offer their own set of unique themes and hurdles.

fire kirin apk - 2 days ago

You are very talented in your writing. I went through all of your articles, they were very interesting.

ลิงค์รับทรัพย์ sexy gaming - 19 hours ago

यदि आप श्री खाटू श्याम जी के भक्त हैं और उनके चालीसा का पाठ करना चाहते हैं, तो एक बहुत ही उपयोगी वेबसाइट है जो आपको सभी जानकारी प्रदान करती है। आप इसे यहाँ देख सकते हैं: <a href="https://khatushyamchalisa.com">हमारी वेबसाइट</a> और श्री खाटू श्याम जी के आशीर्वाद से अपने जीवन को सुखी बनाएं। जय श्री श्याम!

Laila - 12 hours ago

Dixin's Blog