Lambda Calculus via C# (3) Numeral, Arithmetic and Predicate

[FP & LINQ via C# series]

[Lambda Calculus via C# series]

Anonymous functions can also model numerals and their arithmetic. In Church encoding, a natural number n is represented by a function that calls a given function for n times. This representation is called Church Numeral.

Church numerals

Church numerals are defined as:

0 := λfx.x                  ≡ λf.λx.x
1 := λfx.f x                ≡ λf.λx.f x
2 := λfx.f (f x)            ≡ λf.λx.f (f x)
3 := λfx.f (f (f x))        ≡ λf.λx.f (f (f x))
...
n := λfx.f (f ... (f x)...) ≡ λf.λx.f (f .u.. (f x)...)

So a Church numeral n is a higher order function, it accepts a function f and an argument x. When n is applied, it repeatedly applies f for n times by starting with x, and returns the result. If n is 0, f is not applied (in another word, f is applied 0 times), and x is directly returned.

0 f x ≡ x
1 f x ≡ f x
2 f x ≡ f (f x)
3 f x ≡ f (f (f x))
...
n f x ≡ f (f (... (f x)...))

According to the definition of function composition:

f (f x) ≡ (f ∘ f) x

This definition is equivalent to compose f for n time:

0 := λfx.x                  ≡ λf.λx.x                   ≡ λf.λx.f0 x
1 := λfx.f x                ≡ λf.λx.f x                 ≡ λf.λx.f1 x
2 := λfx.f (f x)            ≡ λf.λx.(f ∘ f) x           ≡ λf.λx.f2 x
3 := λfx.f (f (f x))        ≡ λf.λx.(f ∘ f ∘ f) x       ≡ λf.λx.f3 x
...
n := λfx.f (f ... (f x)...) ≡ λf.λx.(f ∘ f ∘ ... ∘ f) x ≡ λf.λx.fn x

The partial application with f is the composition of f, so Church numeral n can be simply read as – do something n times:

0 f ≡ f0
1 f ≡ f1
2 f ≡ f2
3 f ≡ f3
...
n f ≡ fn

In C#, x can be anything, so leave its type as dynamic. f can be viewed as a function accept a value x and returns something, and f can also accept its returned value again, so f is of type dynamic -> dynamic. And n’ return type is the same as f’s return type, so n returns dynamic too. As a result, n can be virtually viewed as curried function type (dynamic -> dynamic) –> dynamic -> dynamic, which in C# is represented by Func<Func<dynamic, dynamic>, Func<dynamic, dynamic>>. Similar to the C# implementation of Church Boolean, a alias Numeral can be defined:

// Curried from (dynamic -> dynamic, dynamic) -> dynamic.
// Numeral is the alias of (dynamic -> dynamic) -> dynamic -> dynamic.
public delegate Func<dynamic, dynamic> Numeral(Func<dynamic, dynamic> f);

Based on the definition:

public static partial class ChurchNumeral
{
    public static readonly Numeral
        Zero = f => x => x;

    public static readonly Numeral
        One = f => x => f(x);

    public static readonly Numeral
        Two = f => x => f(f(x));

    public static readonly Numeral
        Three = f => x => f(f(f(x)));

    // ...
}

Also since n f ≡ fn, n can be also implemented with composition of f:

public static readonly Numeral
    OneWithComposition = f => f;

// Two = f => f o f
public static readonly Numeral
    TwoWithComposition = f => f.o(f);

// Three = f => f o f o f
public static readonly Numeral
    ThreeWithComposition = f => f.o(f).o(f);

// ...

Here the o operator is the forward composition extension method defined previously. Actually, instead of defining each number individually, Church numeral can be defined recursively by increase or decrease.

Increase and decrease

By observing the definition and code, there are some patterns when the Church numeral increases from 0 to 3. In the definitions of Church numerals:

0 := λf.λx.x
1 := λf.λx.f (x)
2 := λf.λx.f (f x)
3 := λf.λx.f (f (f x))
...

The expressions in the parenthesis can be reduced from the following function applications expressions:

0 f x ≡ x
1 f x ≡ f x
2 f x ≡ f (f x)
...

With substitution, Church numerals’ definition become:

0 := λf.λx.x
1 := λf.λx.f (0 f x)
2 := λf.λx.f (1 f x)
3 := λf.λx.f (2 f x)
...

This shows how the the Church numerals increases. Generally, given a Church numeral n, the next numeral n + 1 is λf.λx.f (n f x). So:

Increase := λn.λf.λx.f (n f x)

In C#, this is:

public static Func<Numeral, Numeral> 
    Increase = n => f => x => f(n(f)(x));

In the other way, Church numeral n is to compose f for n times:

n f ≡ fn

So increasing n means to compose f for one more time:

Increase := λn.λf.f ∘ fn ≡ λn.λf.f ∘ (n f)

And in C#:

public static readonly Func<Numeral, Numeral> 
    IncreaseWithComposition = n => f => f.o(n(f));

To decrease a Church numeral n, when n is 0, the result is defined as 0, when n is positive, the result is n – 1. The Decrease function is more complex:

Decrease := λn.λf.λx.n (λg.λh.h (g f)) (λv.x) Id

When n is 0, regarding n f ≡ fn, applying Decrease with 0 can be reduced as:

  Decrease 0
≡ λf.λx.0 (λg.λh.h (g f)) (λv.x) Id
≡ λf.λx.(λg.λh.h (g f))0 (λv.x) Id
≡ λf.λx.(λv.x) Id
≡ λf.λx.x
≡ λf.λx.f0 x

The last expression is the definition of 0.

When n is positive, regarding function function composition is associative, the expression n (λg.λh.h (g f)) (λu.x) can be reduced first. When n is 1, 2, 3, ...:

  1 (λg.λh.h (g f)) (λv.x)
≡ (λg.λh.h (g f))1 (λv.x)
≡ (λg.λh.h (g f)) (λv.x)
≡ λh.h ((λv.x) f)
≡ λh.h x
≡ λh.h (f0 x) 

  2 (λg.λh.h (g f)) (λv.x)
≡ (λg.λh.h (g f))2 (λv.x)
≡ (λg.λh.h (g f)) ∘ (λg.λh.h (g f))1 (λv.x)
≡ (λg.λh.h (g f)) (λh.h (f0 x))
≡ λh.h (λh.h (f0 x) f)
≡ λh.h (f (f0 x))
≡ λh.h (f1 x)

  3 (λg.λh.h (g f)) (λv.x)
≡ (λg.λh.h (g f))3 (λv.x)
≡ (λg.λh.h (g f)) ∘ (λg.λh.h (g f))2 (λv.x)
≡ (λg.λh.h (g f)) (λh.h (f1 x))
≡ λh.h ((λh.h (f1 x)) f)
≡ λh.h (f (f1 x))
≡ λh.h (f2 x)

...

And generally:

  n (λg.λh.h (g f)) (λv.x)
≡ λh.h (fn - 1 x)

So when Decrease is applied with positive n:

  Decrease n
≡ λf.λx.n (λg.λh.h (g f)) (λv.x) Id
≡ λf.λx.(λh.h (fn - 1 x)) Id
≡ λf.λx.Id (fn - 1 x)
≡ λf.λx.fn - 1 x

The returned result is the definition of n – 1. In the following C# implementation, a lot of noise of type information is involved to implement complex lambda expression:

// Decrease = n => f => x => n(g => h => h(g(f)))(_ => x)(Id)
public static readonly Func<Numeral, Numeral> 
    Decrease = n => f => x => n(g => new Func<Func<dynamic, dynamic>, dynamic>(h => h(g(f))))(new Func<Func<dynamic, dynamic>, dynamic>(_ => x))(new Func<dynamic, dynamic>(Functions<dynamic>.Id));

Here are the actual types of the elements in above lambda expression at runtime:

  • g: (dynamic -> dynamic) -> dynamic
  • h: dynamic -> dynamic
  • g(f): dynamic
  • h(g(f)): dynamic
  • h => h(g(f)): (dynamic -> dynamic) -> dynamic
  • g => h => h(g(f)): ((dynamic -> dynamic) -> dynamic) -> (dynamic -> dynamic) -> dynamic
  • n(g => h => h(g(f))): ((dynamic -> dynamic) -> dynamic) -> (dynamic -> dynamic) -> dynamic
  • _ => x: (dynamic -> dynamic) -> dynamic
  • n(g => h => h(g(f)))(_ => x): (dynamic -> dynamic) -> dynamic
  • Id: dynamic -> dynamic
  • n(g => h => h(g(f)))(_ => x)(Id): dynamic

At compile time, function types must be provided for a few elements. When n is applied, C# compiler expects its first argument g => h => h(g(f)) to be of type dynamic => dynamic. So C# compiler infers g to dynamic, but cannot infer the type of h => h(g(f)), which can be expression tree or anonymous function, so the constructor call syntax is used here to specify it is a function of type (dynamic -> dynamic) -> dynamic. Similarly, C# compiler expects n’s second argument to be dynamic, and C# compiler cannot infer the type of _ => x, so the constructor syntax is used again for _ => x. Also, Functions<dynamic>.Id is of Unit<dynamic> type, while at runtime a dynamic -> dynamic function is expected. Unit<dynamic> is alias of function type dynamic –> dynamic, but the conversion does not happen automatically at runtime, so the constructor syntax is used once again to indicate the function type conversion.

Later after introducing Church pair, a cleaner version of Decrease will be implemented.

Arithmetic operators

To implement add operation, according to the definition, Church numeral a adding Church numeral b means to apply f for a times, then apply f again for b times:

Add := λa.λb.λf.λx.b f (a f x)

With the definition of function composition, Add can be also defined as:

Add := λa.λb.λf.fa ∘ fb ≡ λa.λb.λf.(a f) ∘ (b f)

So in C#:

public static readonly Func<Numeral, Func<Numeral, Numeral>>  
    Add = a => b => f => x => b(f)(a(f)(x));

public static readonly Func<Numeral, Func<Numeral, Numeral>> 
    AddWithComposition = a => b => f => a(f).o(b(f));

With Increase function, Add can also be defined as increase a for b times:

Add := λa.λb.b Increase a

In C#, there are some noise of type information again:

public static readonly Func<Numeral, Func<Numeral, Numeral>>
    AddWithIncrease = a => b => b(Increase)(a);

Unfortunately, the above code cannot be compiled, because b is a function of type (dynamic -> dynamic) -> dynamic x -> dynamic. So its first argument f must be a function of type dynamic -> dynamic. Here, Increase is of type Numeral -> Numeral, and b(Increase) cannot be compiled. The solution is to eta convert Increase to a wrapper function λn.Increase n:

Add := λa.λb.a (λn.Increase n) b

So that in C#:

// Add = a => b => b(Increase)(a)
// η conversion:
// Add = a => b => b(n => Increase(n))(a)
public static readonly Func<Numeral, Func<Numeral, Numeral>>
    AddWithIncrease = a => b => b(n => Increase(n))(a);

Since a dynamic -> dynamic function is expected and the wrapper function n => Increase(n), n inferred to be of type dynamic. Increase(n) still returns Numeral, so the wrapper function is of type dynamic -> Numeral. Regarding dynamic is just object, and Numeral derives from object, with support covariance in C#, the wrapper function is implicitly converted to dynamic -> dynamic, so calling b with the wrapper function can be compiled.

Similarly, Church numeral a subtracting b can be defined as decrease a for b times, a multiplying b can be defined as adding a for b times to 0, and raising a to the power b can be defined as multiplying a for n times with 1:

Subtract := λa.λb.b Decrease a
Multiply := λa.λb.b (Add a) 0
Power := λa.λb.b (Multiply a) 1

The C# implementation are in the same pattern:

// Subtract = a => b => b(Decrease)(a)
// η conversion:
// Subtract = a => b => b(n => Decrease(n))(a)
public static readonly Func<Numeral, Func<Numeral, Numeral>>
    Subtract = a => b => b(n => Decrease(n))(a);

// Multiply = a => b => b(Add(a))(a)
// η conversion:
// Multiply = a => b => b(n => Add(a)(n))(Zero)
public static readonly Func<Numeral, Func<Numeral, Numeral>>
    Multiply = a => b => b(n => Add(a)(n))(Zero);

// Pow = a => b => b(Multiply(a))(a)
// η conversion:
// Pow = a => b => b(n => Multiply(a)(n))(1)
public static readonly Func<Numeral, Func<Numeral, Numeral>>
    Pow = a => b => b(n => Multiply(a)(n))(One);

Similar to Church Boolean operators, the above arithmetic operators can also be wrapped as extension method for convenience:

public static partial class NumeralExtensions
{
    public static Numeral Increase(this Numeral n) => ChurchNumeral.Increase(n);

    public static Numeral Decrease(this Numeral n) => ChurchNumeral.Decrease(n);

    public static Numeral Add(this Numeral a, Numeral b) => ChurchNumeral.Add(a)(b);

    public static Numeral Subtract(this Numeral a, Numeral b) => ChurchNumeral.Subtract(a)(b);

    public static Numeral Multiply(this Numeral a, Numeral b) => ChurchNumeral.Multiply(a)(b);

    public static Numeral Pow(this Numeral mantissa, Numeral exponent) => ChurchNumeral.Pow(mantissa)(exponent);
}

Predicate and relational operators

Predicate is function returning Church Boolean. For example, the following function predicate whether a Church numeral n is 0:

IsZero := λn.n (λx.False) True

When n is 0, (λx.False) is not applied, and IsZero directly returns True. When n is positive, (λx.False) is applied for n times. (λx.False) always return False, so IsZero returns False. The following are the implementation and extension method:

public static partial class ChurchPredicate
{
    public static readonly Func<Numeral, Boolean> 
        IsZero = n => n(_ => False)(True);
}

public static partial class NumeralExtensions
{
    public static Boolean IsZero(this Numeral n) => ChurchPredicate.IsZero(n);
}

With IsZero, it is easy to define functions to compare 2 Church numerals a and b. According the to definition of Decrease and Subtract, when a – b is 0, a is either equal to b, or less than b. So IsLessThanOrEqualTo can be defined with IsZero and Subtract:

IsLessThanOrEqualTo := λa.λb.IsZero (Subtract a b)

IsGreaterThanOrEqualTo is similar:

IsGreaterThanOrEqualTo := λa.λb.IsZero (Subtract b a)

Then these 2 functions can define IsEqualTo:

IsEqualTo := λa.λb.And (IsLessThanOrEqualTo a b) (IsGreaterThanOrEqualTo a b)

The opposite of these functions are IsGreaterThan, IsLessThan, IsNotEqual. They can be defined with Not:

IsGreaterThan := λa.λb.Not (IsLessThanOrEqualTo a b)
IsLessThan := λa.λb.Not (IsGreaterThanOrEqualTo a b)
IsNotEqualTo := λa.λb.Not (IsEqualTo a b)

The following are the C# implementation of these 6 predicates:

public static partial class ChurchPredicate
{
    public static readonly Func<Numeral, Func<Numeral, Boolean>> 
        IsLessThanOrEqualTo = a => b => a.Subtract(b).IsZero();

    public static readonly Func<Numeral, Func<Numeral, Boolean>> 
        IsGreaterThanOrEqualTo = a => b => b.Subtract(a).IsZero();

    public static readonly Func<Numeral, Func<Numeral, Boolean>>
        IsEqualTo = a => b => IsLessThanOrEqualTo(a)(b).And(IsGreaterThanOrEqualTo(a)(b));

    public static readonly Func<Numeral, Func<Numeral, Boolean>>
        IsGreaterThan = a => b => IsLessThanOrEqualTo(a)(b).Not();

    public static readonly Func<Numeral, Func<Numeral, Boolean>> 
        IsLessThan = a => b => IsGreaterThanOrEqualTo(a)(b).Not();

    public static readonly Func<Numeral, Func<Numeral, Boolean>>
        IsNotEqualTo = a => b => IsEqualTo(a)(b).Not();
}

public static partial class NumeralExtensions
{
    public static Boolean IsLessThanOrEqualTo(this Numeral a, Numeral b) => ChurchPredicate.IsLessThanOrEqualTo(a)(b);

    public static Boolean IsGreaterThanOrEqualTo(this Numeral a, Numeral b) => ChurchPredicate.IsGreaterThanOrEqualTo(a)(b);

    public static Boolean IsEqualTo(this Numeral a, Numeral b) => ChurchPredicate.IsEqualTo(a)(b);

    public static Boolean IsGreaterThan(this Numeral a, Numeral b) => ChurchPredicate.IsGreaterThan(a)(b);

    public static Boolean IsLessThan(this Numeral a, Numeral b) => ChurchPredicate.IsLessThan(a)(b);

    public static Boolean IsNotEqualTo(this Numeral a, Numeral b) => ChurchPredicate.IsNotEqualTo(a)(b);
}

Attempt of recursion

The division of natural numbers can be defined with arithmetic and relation operators:

a / b := if a >= b then 1 + (a – b) / b else 0

This is a recursive definition. If defining division in this way lambda calculus, the function name is referred in its own body:

DivideBy := λa.λb.If (IsGreaterThanOrEqualTo a b) (λx.Add One (DivideBy (Subtract a b) b)) (λx.Zero)

As fore mentioned, in lambda calculus, functions are anonymously by default, and names are just for readability. Here the self reference does not work with anonymous function:

λa.λb.If (IsGreaterThanOrEqualTo a b) (λx.Add One (? (Subtract a b) b)) (λx.Zero)

So the above DivideBy function definition is illegal in lambda calculus. The recursion implementation with anonymous function will be discussed later in this chapter.

In C#, recursion is a basic feature, so the following self reference is supported:

using static ChurchBoolean;

public static partial class ChurchNumeral
{
    // Divide = dividend => divisor => 
    //    If(dividend >= divisor)
    //        (_ => 1 + DivideBy(dividend - divisor)(divisor))
    //        (_ => 0);
    public static readonly Func<Numeral, Func<Numeral, Numeral>>
        DivideBy = dividend => divisor =>
            If(dividend.IsGreaterThanOrEqualTo(divisor))
                (_ => One.Add(DivideBy(dividend.Subtract(divisor))(divisor)))
                (_ => Zero);
}

Here using static directive is used so that ChurchBoolean.If function can be called directly. DivideBy is compiled to a field definition and field initialization code in static constructor, and apparently referencing to a field in the constructor is allowed:

using static ChurchBoolean;
using static ChurchNumeral;

public static partial class CompiledChurchNumeral
{
    public static readonly Func<Numeral, Func<Numeral, Numeral>> DivideBySelfReference;

    static CompiledChurchNumeral()
    {
        DivideBySelfReference = dividend => divisor =>
            If(dividend.IsGreaterThanOrEqualTo(divisor))
                (_ => One.Add(DivideBySelfReference(dividend.Subtract(divisor))(divisor)))
                (_ => Zero);
    }
}

The self reference also works for named function:

public static partial class ChurchNumeral
{
    public static Func<Numeral, Numeral> DivideByMethod(Numeral dividend) => divisor =>
        If(dividend.IsGreaterThanOrEqualTo(divisor))
            (_ => One.Add(DivideByMethod(dividend.Subtract(divisor))(divisor)))
            (_ => Zero);
}

The only exception is, when this function is a local variable instead of field, then the inline self reference cannot be compiled:

internal static void Inline()
{
    Func<Numeral, Func<Numeral, Numeral>> divideBy = dividend => divisor =>
        If(dividend.IsGreaterThanOrEqualTo(divisor))
            (_ => One.Add(divideBy(dividend.Subtract(divisor))(divisor)))
            (_ => Zero);
}

The reason is, the value of the local variable is compiled before the local variable is compiled. when the anonymous function is compiled, the referenced divideBy function is not defined yet, and C# compiler gives CS0165 error: Use of unassigned local variable 'divideBy'. To resolve this problem, divideBy can be first initialized with default value null. When divideBy is initialized again with the anonymous function, it is already defined, so the lambda expression can be compiled:

internal static void Inline()

{
    Func<Numeral, Func<Numeral, Numeral>> divideBy = null;
    divideBy = dividend => divisor =>
        If(dividend.IsGreaterThanOrEqualTo(divisor))
            (_ => One.Add(divideBy(dividend.Subtract(divisor))(divisor)))
            (_ => Zero);
}

The above division operator DivideBy will be used temporarily. Later after introducing fixed point combinator, the division can be implemented with an anonymous function without self reference at all.

Conversion between Church numeral and System.UInt32

In .NET, natural number can be represented with unit (System.UInt32). It would be intuitive if Church numeral and uint can be converted to each other. Similar to the conversion between Church Boolean and bool, the following extension methods can be defined:

public static partial class ChurchEncoding
{
    public static Numeral Church(this uint n) => n == 0U ? ChurchNumeral.Zero : Church(n - 1U).Increase();

    public static uint Unchurch(this Numeral n) => (uint)n(x => (uint)x + 1U)(0U);
}

Converting uint to Church numeral is recursive. When n is 0, Zero is returned directly. When n is positive, n is decreased and converted recursively. The recursion terminates when n is decreased to 0, then Increase is called for n times with Zero, and Church numeral n is calculated. And converting Church numeral n to uint just need to add 1U for n times to 0U.

The following code demonstrate how the operators and conversions work:

[TestClass]
public partial class ChurchNumeralTests
{
    [TestMethod]
    public void IncreaseTest()
    {
        Numeral numeral = 0U.Church();
        Assert.AreEqual(0U + 1U, (numeral = numeral.Increase()).Unchurch());
        Assert.AreEqual(1U + 1U, (numeral = numeral.Increase()).Unchurch());
        Assert.AreEqual(2U + 1U, (numeral = numeral.Increase()).Unchurch());
        Assert.AreEqual(3U + 1U, (numeral = numeral.Increase()).Unchurch());
        numeral = 123U.Church();
        Assert.AreEqual(123U + 1U, numeral.Increase().Unchurch());
    }

    [TestMethod]
    public void AddTest()
    {
        Assert.AreEqual(0U + 0U, 0U.Church().Add(0U.Church()).Unchurch());
        Assert.AreEqual(0U + 1U, 0U.Church().Add(1U.Church()).Unchurch());
        Assert.AreEqual(10U + 0U, 10U.Church().Add(0U.Church()).Unchurch());
        Assert.AreEqual(0U + 10U, 0U.Church().Add(10U.Church()).Unchurch());
        Assert.AreEqual(1U + 1U, 1U.Church().Add(1U.Church()).Unchurch());
        Assert.AreEqual(10U + 1U, 10U.Church().Add(1U.Church()).Unchurch());
        Assert.AreEqual(1U + 10U, 1U.Church().Add(10U.Church()).Unchurch());
        Assert.AreEqual(3U + 5U, 3U.Church().Add(5U.Church()).Unchurch());
        Assert.AreEqual(123U + 345U, 123U.Church().Add(345U.Church()).Unchurch());
    }

    [TestMethod]
    public void DecreaseTest()
    {
        Numeral numeral = 3U.Church();
        Assert.AreEqual(3U - 1U, (numeral = numeral.Decrease()).Unchurch());
        Assert.AreEqual(2U - 1U, (numeral = numeral.Decrease()).Unchurch());
        Assert.AreEqual(1U - 1U, (numeral = numeral.Decrease()).Unchurch());
        Assert.AreEqual(0U, (numeral = numeral.Decrease()).Unchurch());
        numeral = 123U.Church();
        Assert.AreEqual(123U - 1U, numeral.Decrease().Unchurch());
    }

    [TestMethod]
    public void SubtractTest()
    {
        Assert.AreEqual(0U - 0U, 0U.Church().Subtract(0U.Church()).Unchurch());
        Assert.AreEqual(0U, 0U.Church().Subtract(1U.Church()).Unchurch());
        Assert.AreEqual(10U - 0U, 10U.Church().Subtract(0U.Church()).Unchurch());
        Assert.AreEqual(0U, 0U.Church().Subtract(10U.Church()).Unchurch());
        Assert.AreEqual(1U - 1U, 1U.Church().Subtract(1U.Church()).Unchurch());
        Assert.AreEqual(10U - 1U, 10U.Church().Subtract(1U.Church()).Unchurch());
        Assert.AreEqual(0U, 1U.Church().Subtract(10U.Church()).Unchurch());
        Assert.AreEqual(0U, 3U.Church().Subtract(5U.Church()).Unchurch());
        Assert.AreEqual(0U, 123U.Church().Subtract(345U.Church()).Unchurch());
    }

    [TestMethod]
    public void MultiplyTest()
    {
        Assert.AreEqual(0U*0U, 0U.Church().Multiply(0U.Church()).Unchurch());
        Assert.AreEqual(0U*1U, 0U.Church().Multiply(1U.Church()).Unchurch());
        Assert.AreEqual(10U*0U, 10U.Church().Multiply(0U.Church()).Unchurch());
        Assert.AreEqual(0U*10U, 0U.Church().Multiply(10U.Church()).Unchurch());
        Assert.AreEqual(1U*1U, 1U.Church().Multiply(1U.Church()).Unchurch());
        Assert.AreEqual(10U*1U, 10U.Church().Multiply(1U.Church()).Unchurch());
        Assert.AreEqual(1U*10U, 1U.Church().Multiply(10U.Church()).Unchurch());
        Assert.AreEqual(3U*5U, 3U.Church().Multiply(5U.Church()).Unchurch());
        Assert.AreEqual(12U*23U, 12U.Church().Multiply(23U.Church()).Unchurch());
    }

    [TestMethod]
    public void PowTest()
    {
        Assert.AreEqual(Math.Pow(0U, 1U), 0U.Church().Pow(1U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(10U, 0U), 10U.Church().Pow(0U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(0U, 10U), 0U.Church().Pow(10U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(1U, 1U), 1U.Church().Pow(1U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(10U, 1U), 10U.Church().Pow(1U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(1U, 10U), 1U.Church().Pow(10U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(3U, 5U), 3U.Church().Pow(5U.Church()).Unchurch());
        Assert.AreEqual(Math.Pow(5U, 3U), 5U.Church().Pow(3U.Church()).Unchurch());
    }

    [TestMethod]
    public void DivideByRecursionTest()
    {
        Assert.AreEqual(1U / 1U, 1U.Church().DivideBy(1U.Church()).Unchurch());
        Assert.AreEqual(1U / 2U, 1U.Church().DivideBy(2U.Church()).Unchurch());
        Assert.AreEqual(2U / 2U, 2U.Church().DivideBy(2U.Church()).Unchurch());
        Assert.AreEqual(2U / 1U, 2U.Church().DivideBy(1U.Church()).Unchurch());
        Assert.AreEqual(10U / 3U, 10U.Church().DivideBy(3U.Church()).Unchurch());
        Assert.AreEqual(3U / 10U, 3U.Church().DivideBy(10U.Church()).Unchurch());
    }
}

87 Comments

  • برند <a href="https://hitema.co">هیتما</a> برای اولین بار 25 سال قبل در کشور ایتالیا به ثبت رسید و با تولید سیستم های سردکننده و تهویه مطبوع شروع به کار کرد. از ویژگی های محصولات هیتما، طراحی منحصر به فرد و راندمان بالای این محصول و خدمات پس از فروش گسترده آن میباشد. شما میتوانید برای تهیه محصولات یا کسب اطلاعات بیشتر به وبسایت هیتما مراجعه فرمایید.

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

  • Thanks for sharing.I found a lot of interesting information here. A really good post, very thankful and hopeful that you will write many more

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

  • awesome post. I’m a normal visitor of your web site and appreciate you taking the time to maintain the nice site. I’ll be a frequent visitor for a long time.

  • This post is really astounding one! I was delighted to read this, very much useful. Many thanks

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

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

  • This post is truly inspiring. I like your post and everything you share with us is current and very informative

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

  • کلینیک لیزر شفا به عنوان بهترین مرکز درمان بیماری های گوارشی و نشیمن گاهی با مدیریت دکتر داود تاج بخش در رابطه با بیماریهای ناحیه مقعد خدماتی از جمله درمان بیماری هایی مثل هموروئید، فیستول، شقاق ارائه می دهد. کلینیک لیزر شفا فقط و فقط بر روی بیماری های مقعدی تمرکز داشته و تمامی متخصصین در این رابطه دور هم گرد آورده است.

  • https://ma-study.blogspot.com/

  • Those developers who have some issues in creating the "Lambda Calculus" program, they can get easily help and get out complete code for the program.

    We are a registered company in Eastvale, We offer Painting Services Eastvale to residential and commercial.

  • In the USA, we gathered in a large company and, naturally, the question arose of how to get from the airport and not get lost along the way. We used a car rental company <a href="https://rental24h.com/usa/group/7-seater">Rental24h.com</a>. I will say that their service is excellent, reasonable prices and good cars. Next time I'll just go to them.

  • Your writing is perfect and complete. Keonhacai 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?


  • Goldendoodle Puppies and Cockapoo Puppies in Virginia by Carriage House - breeder of Goldendoodle puppies specializing in English Teddy Bear Goldendoodles and Cockapoos for sale. We hand-deliver our puppies to Virginia, Maryland, West Virginia, Pennsylvania, Ohio, New Jersey, North Carolina, South Carolina, Delaware and Washington, D.C. Committed to raising healthy happy puppies

  • If you are going to travel by car I recommend you to use a car rental service. Because it’s very comfortable and profitable. You can rent any car and start your trip with a good mood. Last time I used this option.

  • I'm writing on this topic these days, <a href="http://google.com.pr/url?sa=t&url=https%3A%2F%2Foncasino.biz">safetoto</a>, but I have stopped writing because there is no reference material. Then I accidentally found your article. I can refer to a variety of materials, so I think the work I was preparing will work! Thank you for your efforts.

  • Predicate, Numeral, and Arithmetic The Numeral and Arithmetic is a long-time favourite among math enthusiasts and a fun approach to keep your youngster interested in the lesson. The game is meant to be played in groups, but it may also be used as a centre activity for math groups, a fun game at home, or a quick game for your next game night.

  • I want to talk to you and tell you that I love the article you wrote. want to see you your article is good i want to talk to you.

  • When I read an article on this topic, "bitcoincasino" 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?

  • Great post! Thanks for sharing

  • Great post! Thanks for sharing

  • Install the Disney Plus application on your Smart tv. For this, visit the application store of your tv and then search for the Disney Plus application on it. Install the application with the help of the install button provided.

  • Thank you for this great post

  • You have written content that is very interesting.

  • I sincerely enjoy the design and layout of your site. It is so comfortable for the eyes that it makes it much more enjoyable for me to come and visit here more often. Did you hire a designer to create your topic? It's great! 메이저토토사이트 https://szzzac.com/

  • I enjoyed your blog post full of happiness. I hope your life is filled with luck777
    http://www.golden-evolution.com/

  • This is a great post. It helped me a lot. I hope you have a happy life goldenevolution10
    http://www.golden-evolution.com/

  • This is exceptionally educational and supportive post in the blog posting world thank you All Your posts consistently give us some subjective things I hope you have a happy life goldenevolution10
    http://www.golden-evolution.com/

  • I enjoyed your blog post full of happiness. I hope your life is filled with luck792
    <a href="https://www.golden-evolution.com/"><title=카지노콤프 콤프주는사이트" title="카지노콤프 콤프주는사이트" rel="nofollow ugc">카지노콤프 콤프주는사이트</a>

  • 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>

  • آزمایشات قبل از کاشت ابرو
    کاشت ابرو یک عمل زیبایی میباشد که با عوارض جانبی محدودی همراه است اما افرادی هستند که پزشکان آنان را از انجام این عمل منع میکنند. مهم‌ترین دلیل پزشکان آزمایشات قبل از کاشت ابرو میباشد. این آزمایش اطلاعات بسیار زیادی را به دکتر می‌دهد همچنین به نرخ موفقیت جراحی هم می‌افزاید. در ادامه به اکثر سوالاتی مانند دلیل مهم بودن آزمایشات قبل کاشت ابرو چیست؟ اقداماتی که باید قبل از کاشت ابرو انجام داد و غیره پاسخ داده ایم

    برای آزمایش کاشت ابرو باید ناشتا بود؟
    در برخی آزمایشات مانند خون، قند، و دیگر آزمایشاتی از این قبیل نیاز است تا 8 الی 12 ساعت قبل از آزمایش چیزی مصرف نشود (غذا، آب) اما در آزمایشاتی که معمولا سر و کار آنان با پوست و زیبایی می‌باشد نیازی به ناشتا بودن نیست. البته در این مورد باید با پزشک متخصص خود مشورت کنید تا بنا بر آزمایشی که برایتان تجویز می‌گردد مشخص شود که باید ناشتا باشید یا خیر. بهتر است قبل از هرگونه اقدامی به دکتر خود درباره داروهایی که مصرف می‌کنید بگویید. این اطلاعات کمک بسیاری در تعیین آزمایش شما می‌کند.

    آزمایشات قبل از کاشت ابرو بسیار مهم می‌باشد زیرا ممکن است با عوارض جبران ناپذیری همراه باشد، در جواب سوال دلیل مهم بودن آزمایشات قبل از کاشت ابرو چیست باید بگوییم این ازمایش مشخص میکند که افراد کاندید مناسبی برای کاشت ابرو هستند یا خیر. افرادی که نمیتوانند کاشت ابرو را انجام دهند به چهار الی پنج دسته تقسیم می‌شوند، این پنج دسته عبارنتد از:

    افراد دارای بیماری‌های خاص
    مشکلات پوستی، سرطان، دیابت، و بیماری‌هایی از این قبیل می‌تواند مشکلات بیشتری را برای فرد فراهم کند.
    افراد دارای بیماری‌های پوستی و عفونی
    بیماری‌های پوستی یکی از دلایلی است که کاشت ابرو را با شکست مواجه میکند. افرادی‌که دارای بیماری‌هایی مانند اگزما، پسوریازیس، التهاب پوست، و غیره می‌باشند کاندید مناسبی نیستند.
    افرادی با بیماری‌های هورمونی
    بیماری‌های هورمونی مانند تیروئید کم کار و پرکار، ریزش مو، ضایعات پوستی، عرق شبانه، سردرد تیره شدن پوست، و دیگر بیماریی‌های هورمونی.
    همچنین افرادی با بیماری‌های مانند قبلی مانند فشار خون بالا، و افرادی که دارای عوارض جانبی مانند بی‌خوابی، استرس شدید، افسردگی و مصرف مواد مخدر می‌باشند کاندید مناسبی جهت کاشت ابرو نمی‌باشند.

    همانطور که در موارد بالا ذکر شد، آزمایشات قبل از کاشت ابرو بسیار مهم می‌باشد. تا اینجای مقاله شاید به جواب سوال برای کاشت ابرو چه آزمایشی لازم است رسیده ایم اما باز هم این بحث را تخصصی تر میکنیم. هدف از چیستی آزمایشات قبل از کاشت ابرو تعیین وضعیت سلامتی فرد می‌باشد. آزمایشات مهمی که میتوان به آنها اشاره کرد عبارتند از:

    آزمایش قند خون
    آزمایش ایدز یا HIV
    آزمایش HCV یا هپاتیت C
    آزمایش انعقادی
    آزمایش . بررسی فشار خون
    تست قند خون
    هرکدام از این آزمایش ها جدا از تعیین سلامت فرد با هدف خاصی گرفته می‌شود. همانطور که میدانید آزمایش قند خون جهت اطلاع از دیابت فرد می‌باشد. اگر فرد دارای قند خون بالایی باشد یا به عبارتی دیگر دارای هپاتیت باشد قبل از کاشت مو و ابرو باید تحت درمان قرار گیرد. در اکثر مواقع بیماران هپاتیتی نمی‌توانند کاشت ابرو انجام دهند مگر اینکه تحت درمان پزشک قرار گیرند و داروهای مصرفی را طبق برنامه جلو ببرند.

    یکی دیگر از دغدغه های بیماران دیابتی برداشتن گرافت از بانک موی آنها می‌باشد. با برداشتن گرافت جای زخم شکل میگیرد و در بیماران دیابتی جای زخم دیرتر از زمان معمول خوب میشود. این کار قطعا به تبحر و تخصص بالای پزشک نیاز دارد. حال با تمام این تفاسیر شاید درک بهتری از سوال برای کاشت ابرو چه آزمایشی لازم است پیدا کرده ایم

    آزمایش ایدز قبل از کاشت ابرو
    ایدز یکی از شایع ترین بیماری های جنسی میباشد. این بیماری سلول های فرد و ایمنی بدن فرد را مورد هدف قرار داده و با حمله به آنها باعث ضعیف شدن بیمار می‌گردد همچنین بدن فرد را در مقابل مبارزه با بیماری ها و عفونت ها بسیار ضعیف میکند. در کلینیک های حرفه‌ای معمولا یکی از آزمایش هایی که تجویز میشود آزمایش ایدز یا HIV میباشد. در صورت مثبت بودن بیماری از کاشت ابرو جلوگیری میشود

    آزمایش هپاتیت قبل از کاشت ابرو
    یکی دیگر از آزمایشات قبل از کاشت ابرو آزمایش هپاتیت میباشد. هپاتیت یک ویروس بسیار خطرناک است که مستقیما به کبد حمله میکند. هپاتیت های نوع B و C قبل از کاشت مو و ابرو گرفته میشود و به جرات میتوان گفت یکی از آزمایش‌های مهم قبل از کاشت ابرو میباشد. این نوع بیماری ها معمولا از طریق وسایل جراحی به فرد منتقل می‌شوند، اگر فرد به این نوع بیماری‌ها مبتلا باشد به هیچ عنوان کاندید مناسبی نمی‌باشد



    آزمایش خون قبل از کاشت ابرو
    آزمایش خون میتواند یکی از مهم ترین تست هایی باشد که فرد باید قبل از کاشت ابرو بدهد. این تست زمان دقیق لخته شدن خون را تعیین می‌کند و به دکتر متخصص کمک میکند تا در حین عمل بتواند از خطرات احتمالی خونریزی جلوگیری کند. بسیار مهم می‌باشد که قبل از انجام کاشت مو و ابرو این تست را گذرانده باشید و از میزان انعقاد خون با خبر باشید.

    آزمایش ادرار قبل از کاشت ابرو
    یکی دیگر از آزمایش های مهم قبل از کاشت ابرو آزمایش ادرار میباشد. جراح و دکتر جهت تشخیص بیماری های کلیوی، مشکلات کبدی، عفونت های ادراری درخواست میشود. اطلاعات فوق میتواند کمک بزرگی به جراح جهت تجویز داروهای بعد از عمل میکند به طور کلی هر‌چقدر اطلاعات دارویی شما بیشتر باشد بعد جراحی دوران نقاهت راحت تری را سپری می‌کنید

    اقداماتی که باید قبل از کاشت ابرو انجام داد
    قبل از کاشت ابرو نیاز است تا اقداماتی را انجام داد تا بعد از عمل مشکلی پیش نیاید بهتر است استعمال دخانیات را قطع کنید همچنین از مصرف ویتامین های E خودداری شود مصرف این نوع ویتامین باعث رقیق شدن خون میشود. توصیه می‌شود از داروهایی مانند آسپرین که باعث رقیق شدن خون میشود خودداری شود همچنین مشروبات الکلی نیز مشکل ساز می‌باشند. با رعایت نکات بالا احتمال زیاد مشکلی در هنگام عمل نخواهید داشت.

    نتیجه گیری
    جدی گرفتن این آزمایشات کمک بسیار شایانی به شما عزیزان میکند. دلیل مهم بودن آزمایشات قبل از کاشت ابرو متوجه شدن از وضعیت سلامتی و جسمانی خود می‌باشد همچنین به دکتر و جراح کمک می‌کنید تا بعد از عمل داروهایی را متناسب با وضعیت سلامتی و جسمانی خود تجویز کند. جراحی و کاشت ابرو هم مانند دیگر عمل ها نیاز به یک سری آزمایش ها جهت جلوگیری از بروز مشکلات بعد از عمل دارد امیدواریم توانسته باشیم جواب سوالات شما را تا حد ممکن داده باشیم

    سوالات متداول
    قبل از کاشت ابرو مصرف چه داروهایی باید قطع شود ؟
    به طور کلی داروهایی مانند آسپرین که خون را رقیق میکنند باید قطع شود زیرا در هنگام عمل ممکن است مشکلاتی را ایجاد کند.
    دلیل مهم بودن آزمایشات قبل از کاشت ابرو چیست ؟
    آزمایش ها تا حد بسیار زیادی به شما و به دکتر در انتخاب نوع جراحی کمک میکند . این نوع آزمایش ها به شما کمک میکند تا دچار مشکلات بعد از عمل نشوید.



  • I think your article will give those people a good reminding. And they will express thanks to you later.

  • <a href="https://www.golden-evolution.com"><title="에볼루션바카라" rel="nofollow ugc">에볼루션바카라</a>
    This is a great post. It helped me a lot 에볼루션바카라 I hope you have a happy life.

  • Hello, thank you for seeing the good posts,에볼루션카지노
    and I look forward to seeing more good posts in the future
    https://www.golden-evolution.com

  • I hope your life is filled with luck 에볼루션슬롯
    https://www.golden-evolution.com
    https://www.golden-evolution.com/evobacarat
    https://www.golden-evolution.com/onlinecasino
    https://www.golden-evolution.com/casinosite
    https://www.golden-evolution.com/casino

  • I enjoyed your blog post full of happiness
    슬롯사이트
    https://www.golden-evolution.com

  • This is a great post. It helped me a lot. I hope you have a happy life
    에볼루션
    https://www.golden-evolution.com

  • I have been looking for articles on these topics for a long time. <a href="https://images.google.no/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">bitcoincasino</a> 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

  • I've been searching for hours on this topic and finally found your post. <a href="https://images.google.nl/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">safetoto</a>, I have read your post and I am very impressed. We prefer your opinion and will visit this site frequently to refer to your opinion. When would you like to visit my site?

  • I'm writing on this topic these days, <a href="https://images.google.ne/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">totosite</a>, but I have stopped writing because there is no reference material. Then I accidentally found your article. I can refer to a variety of materials, so I think the work I was preparing will work! Thank you for your efforts.

  • I've been looking for photos and articles on this topic over the past few days due to a school assignment, <a href="https://images.google.mw/url?sa=t&url=https%3A%2F%2Fwww.mtclean.blog/">baccarat online</a> and I'm really happy to find a post with the material I was looking for! I bookmark and will come often! Thanks :D

  • برای ارائه <a href="https://mag.gozine2.ir/%D8%AE%D8%AF%D9%85%D8%A7%D8%AA-%D8%AF%D9%87%D9%85-%DB%8C%D8%A7%D8%B2%D8%AF%D9%87%D9%85-1403-1402/">خدمات سال تحصیلی 1403-1402</a> به سایت گزینه دو مراجعه کنید. باتشکر

  • I think your article will give those people a good reminding. And they will express thanks to you later.

  • I think your article will give those people a good reminding. And they will express thanks to you later.

  • I think your article will give those people a good reminding.

  • This is a great post. It helped me a lot. I hope you have a happy life goldenevolution10

  • Board Model Papers 2024 provide all states of 6th to 10th text books 2024 Candidates who are Searching for 6th to 10th and 11th to 12th text books and syllabus, sample questions, exam pattern, and Co-Curricular Subject textbooks can refer to this entire article. https://boardmodelpaper.com/ and question papers for following the website and Arts, Science, Commerce Stream Subject Wise Solved Question Bank for Hindi & English Medium Students with Exam Pattern & Blueprint and subject Wise with 11th & 12th Question Bank 2024 for General & Vocational Course. Here, we have gathered all subjects of Board textbooks for all Class along with the direct download links.

  • Pavzi.com is a startup by passionate webmasters and bloggers who have a passion for providing engaging content that is accurate, interesting, and worthy to read. https://pavzi.com/ We are more like a web community where you can find different information, resources, and topics on day-to-day incidents or news. We provide you with the finest web content on every topic possible with the help of the editorial and content team.

  • store shelving with best prices. if you want to equip your supermarket with rack and stands, you can visit us on Daycompany.ir.

  • <a href="https://chandjalnelagaa.com/">Chand Jalne Laga</a>" offered a riveting combination of emotions, drawing viewers deeper into the intricate relationships and unfolding drama. The romantic subplot took center stage, showcasing the chemistry between the lead characters and adding a touch of tenderness to the storyline. Against the backdrop of a celestial event, the show's unique premise added a mystical charm, making it a one-of-a-kind experience for the audience.

  • In C# Lambda Calculus, numeral representation uses functions like succ for successor and zero for 0. Arithmetic operations can be defined using these functions, such as addition and multiplication. Predicates can be implemented with conditional expressions like ifThenElse to check for zero or non-zero values. For example, representing the numeral 2 as succ(succ(zero)), addition as repeated successor applications, and a predicate for checking if a numeral is zero could be implemented using ifThenElse(isZero(n), trueBranch, falseBranch). Download TikTok Mp3 is not relevant to Lambda Calculus or C# programming.

  • For example, representing the numeral 2 as succ(succ(zero)), addition as repeated successor applications, and a predicate for checking if a numeral is zero could be implemented using ifThenElse(isZero(n), trueBranch, falseBranch). Download TikTok Mp3 is not relevant to Lambda Calculus or C# programming.

  • This is a great post. It helped me a lot. I hope you have a happy life goldenevolution10

  • For example, representing the numeral 2 as succ(succ(zero)), addition as repeated successor applications, and a predicate for checking if a numeral is zero could be implemented using ifThenElse(isZero(n), trueBranch, falseBranch). Download TikTok Mp3 is not relevant to Lambda Calculus or C# programming.

  • Perhaps the most prominent advantage is their convenience. Disposable vape Products bundles come pre-packaged and pre-filled with e-liquid, eliminating the need for any additional accessories or maintenance. This convenience makes them ideal for both beginners and experienced vapers who want a hassle-free experience.

  • To ensure transparency and foster trust in AI technologies, it's crucial for researchers and developers to prioritize explainability and interpretability in model design and implementation. This involves making efforts to elucidate how AI models arrive at their predictions or decisions, enabling stakeholders to understand and scrutinize their behavior effectively.

    By enhancing transparency in AI models, stakeholders can better assess their reliability, mitigate potential risks, and address societal concerns related to privacy, fairness, and ethical considerations.

    For more information, you can visit https://subwaymenusprice.com/subway-menu-prices/






  • <p><a href="https://www.bamintahvie.com/category/storage-tank/tanks/tabarestan-horizontal-tank">منبع آب</a></p>

    <p><a href="https://www.bamintahvie.com/category/storage-tank/tanks/tabarestan-horizontal-tank">مخزن آب</a></p>

    <p><a href="https://www.bamintahvie.com/category/storage-tank/tanks/tabarestan-horizontal-tank">تانکر آب</a></p>

  • تولیدی کیف ابزار زارا اولین تولید کننده کیف ابزار بصورت عمده در ایران می باشد که بیش از 10 سال سابقه در زمینه صادرات به کشورهای خاورمیانه و سابقه تولید بیش از 12 سال در زمینه کیف ابزار را دارد.

  • بهترین و مجهزترین کلینیک دامپزشکی در آستانه اشرفیه

    کلینیک دامپزشکی دکتر وحید شمس نصرتی، با افتخار به عنوان تراز ارائه خدمات دامپزشکی در منطقه شرق گیلان مطرح می‌شود.

  • کاور لباس وسیله پر کاربرد با محفظه ای معمولا شفاف و کناره های نرم و منعطف برای حمل و نگهداری لباسهاست که در فروشگاه نظم دهنده قابل خرید است

  • ظرف درب دار فریزری نظم دهنده , ظروف فریزری

  • Excellent website you have here, so much cool information!..

  • I have read a few of the articles on your website now, and I really like your style. Thanks a million and please keep up the effective work.

  • Thanks for publishing such a unique and great article. We are very happy reading your article sir

  • Great job for publishing such a beneficial web site. Your web log isn’t only useful but it is additionally really creative too.

  • I found your this post while searching for information about blog-related research .. It’s a good post .. keep posting and updating information.

  • Each of your solutions are acceptable and also strongly practical. My buddies will also feel fortuitous to read this post.

  • While the Lambda Calculus in C# involves representing numerals using functions such as 'succ' for successor and 'zero' for 0, arithmetic operations like addition and multiplication can be defined accordingly. Predicates are implemented using conditional expressions like 'ifThenElse' to ascertain zero or non-zero values. For instance, numeral 2 is represented as 'succ(succ(zero))', addition involves repeated successor applications, and a predicate for checking if a numeral is zero can be achieved with 'ifThenElse(isZero(n), trueBranch, falseBranch)'. Although gaining TikTok likes is unrelated to Lambda Calculus or C# programming, feel free to reach out to us for further information

  • Al Ahly news is a topic of great anticipation and interest among football enthusiasts. The latest updates on player transfers, match results, and team strategies keep fans on the edge of their seats. With a rich history and a track record of success, Al Ahly's performance on the field often dominates sports headlines. Whether it's a thrilling victory or a challenging defeat, supporters eagerly await every news bulletin to stay connected with their beloved team.

  • Das ausflug hurghada tal der könige eine faszinierende Reise durch die Geschichte des alten Ägypten. Besucher haben die Möglichkeit, die imposanten Gräber und Monumente zu erkunden, die einst für die Pharaonen errichtet wurden.

  • I think this is a real great article post

  • Hello, I enjoy reading through your post. I wanted to writea little comment to support you.

  • Say, you got a nice blog.Really looking forward to read more. Awesome.

  • I just stumbled upon your blog and wanted to say that I really enjoy browsing your blog posts.

  • i really liked this blog, it helped me alot tnx for sharing

  • Your article has enriched my understanding of [topic] in ways I never thought possible.

  • I think your article will give those people a good reminding. And they will express thanks to you later.

  • Install the Disney Plus application on your Smart tv. For this, visit the application store of your tv and then search for the Disney Plus application on it. Install the application with the help of the install button provided.


  • Perhaps streaming services feel like a form of cultural désherbant, homogenizing content and limiting exploration. The algorithmic recommendations often prioritize popular trends over diverse choices, leaving some feeling disconnected from unique or niche content. Additionally, the constant barrage of options can be overwhelming, diminishing the joy of discovering new entertainment.

  • Dixin's Blog features a variety of fashion topics, including tips on styling stacked jeans in light blue hues. Explore their insightful posts for creative ways to incorporate this trendy denim into your wardrobe, whether you're aiming for a casual daytime look or a chic evening ensemble.

  • Downing Street opening its doors symbolizes transparency in governance. Meanwhile, in Manhattan,bathroom remodeling Manhattan offers top-notch services, transforming bathrooms into luxurious spaces.


  • In C#, you can implement Lambda Calculus concepts like numerals, arithmetic, and predicates using functional programming techniques. For numeral representation, you can define Church numerals as functions. Arithmetic operations can be performed by composing and applying these functions. Predicates, such as equality or less than, can be defined similarly using functions. Integrated within a property management system, these Lambda Calculus concepts can streamline data manipulation and validation processes, enhancing efficiency and flexibility.

  • Lambda Calculus via C# (3) Numeral, Arithmetic and Predicate is a nice post that is providing us the right results.There are a lot of people searching for the right results. and when I used the <a href="https://pressurecleaningdarwin.com.au/solar-panel-cleaning/">roof and solar panel cleaning</a> I saw the best services with the best results.

  • Lambda Calculus via C# (3) Numeral, Arithmetic and Predicate is a nice post that is providing us the right results.There are a lot of people searching for the right results. and when I used the roof and solar panel cleaning I saw the best services with the best results.

  • Thanks for sharing this detail about it here and we can find more and more ideas and codes that we are in need of. Also, people can learn more here to get the right solutions.

Add a Comment

As it will appear on the website

Not displayed

Your website