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());
    }
}

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  • 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.



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  • 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.

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  • When working with Lambda Calculus in C#, numerals, arithmetic, and predicates play crucial roles in understanding and implementing functional programming concepts. Here's a brief overview:

    Numerals: In Lambda Calculus, numerals are represented using Church encoding. For example, the number 2 can be represented as a function that takes two arguments and applies a function twice. In C#, you would encode this using delegates or functions.

    Arithmetic: Arithmetic operations in Lambda Calculus are performed using these encoded numerals. For instance, addition can be represented by a function that applies a numeral's function to another numeral's function. Implementing these operations in C# involves creating functions that simulate these Lambda Calculus operations.

    Predicates: Predicates are functions that return true or false based on some condition. In Lambda Calculus, predicates can be used for conditional logic and decision-making. In C#, this would be represented by functions returning boolean values based on certain conditions.

    Incorporating these concepts in C# can deepen your understanding of functional programming and Lambda Calculus. For those interested in exploring this further, keeping up with "Top Follow" resources in functional programming and Lambda Calculus can provide valuable insights and updates.

  • When working with Lambda Calculus in C#, numerals, arithmetic, and predicates play crucial roles in understanding and implementing functional programming concepts. Here's a brief overview:

    Numerals: In Lambda Calculus, numerals are represented using Church encoding. For example, the number 2 can be represented as a function that takes two arguments and applies a function twice. In C#, you would encode this using delegates or functions.

    Arithmetic: Arithmetic operations in Lambda Calculus are performed using these encoded numerals. For instance, addition can be represented by a function that applies a numeral's function to another numeral's function. Implementing these operations in C# involves creating functions that simulate these Lambda Calculus operations.

    Predicates: Predicates are functions that return true or false based on some condition. In Lambda Calculus, predicates can be used for conditional logic and decision-making. In C#, this would be represented by functions returning boolean values based on certain conditions.

    Incorporating these concepts in C# can deepen your understanding of functional programming and Lambda Calculus. For those interested in exploring this further, keeping up with "Top Follow" resources in functional programming and Lambda Calculus can provide valuable insights and updates.

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  • "**Lambda Calculus via C# (3) Numeral, Arithmetic and Predicate**" delves into the application of lambda calculus concepts using C#. This tutorial explores the representation and manipulation of numerals, arithmetic operations, and predicates in lambda calculus, providing a detailed and practical approach for programmers. It's an excellent resource for those looking to deepen their understanding of functional programming and mathematical logic through real-world C# examples.

  • In Lambda Calculus via C#, you can implement numerals, arithmetic operations, and predicates as part of your functional programming explorations. Here's a brief overview:

    Numerals: In Lambda Calculus, numerals are typically represented using Church encoding. For example, the numeral 2 can be encoded as a lambda function that takes a function and an initial value and applies the function twice. In C#, this can be represented using delegates or lambda expressions.

    Arithmetic: Arithmetic operations can be implemented by defining functions that manipulate Church numerals. For instance, addition can be defined by composing two numerals and applying a function to the result.

    Predicate: Predicates in Lambda Calculus are functions that return true or false based on some condition. In C#, these can be represented using functions that return boolean values based on the input.

    Here is a simple C# example of Church numerals and addition:

    csharp
    Copy code
    // Define Church numerals
    Func<Func<int, int>, Func<int, int>> zero = f => x => x;
    Func<Func<int, int>, Func<int, int>> one = f => x => f(x);
    Func<Func<int, int>, Func<int, int>> two = f => x => f(f(x));

    // Addition function
    Func<Func<int, int>, Func<int, int>> add(Func<Func<int, int>, Func<int, int>> m, Func<Func<int, int>, Func<int, int>> n)
    => f => x => m(f)(n(f)(x));

    // Example usage
    var result = add(two, one)(x => x + 1)(0); // Should give 3
    Console.WriteLine(result); // Output: 3
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