# [Lambda Calculus via C# series]

A third way to encode Church list, is to use fold function (also called aggregate in C#/.NET):

```CreateListNode3 = λv.λn.λf.λx.f v (n f x)
Null3 = λf.λx.x
IsNull3 = λl.l (λv.λx.False) True
Value3 = λl.λx.l (λv.λy.v) x
Next3 = λl.Item2 (l (λv.λt.Shift (CreateListNode3 v)) (CreateTuple Null3 Null3))
Index3 = λl.λi.i Next3 l```

# ListNode and wrapper

According to the definition, this is the ListNode in C#:

```// Curried from TResult ListNode<out T, TResult>(Func<T, TResult, TResult> f, TResult x)
public delegate Func<TAccumulate, TAccumulate> ListNode<out T, TAccumulate>(Func<TAccumulate, Func<T, TAccumulate>> f);
// ListNode is the alias of: Func<Func<T, Func<TResult, TResult>>, Func<TResult, TResult>>```

f is the fold/aggregate function. In .NET aggregate function is a Func<TAccumulate, T, TAccumulate>, here in lambda calculus it is curried to Func<TAccumulate, Func<T, TAccumulate>.

Just like the Church numeral wrapper _Numeral, a wrapper is needed to hide TAccumulate “inside”:

```public partial class _ListNode<T>
{

protected virtual _ListNode<T> Next { get; set; }

public _ListNode(T value, _ListNode<T> next)
{
this.value = value;
this.Next = next;
}

public virtual ListNode<T, TAccumulate> Node<TAccumulate>
() =>
f => x => f(this.Next.Node<TAccumulate>()(f)(x))(this.value);
}```

Again, class name is tagged with underscore since class is C# specific.

Null is a special case, so, exactly the same way how 0 is handled in Church numeral:

```public partial class _ListNode<T>
{
private _ListNode()
{
}

private class _NullListNode : _ListNode<T>
{
protected override _ListNode<T> Next { get { return this; } set { } }

public override ListNode<T, TAccumulate> Node<TAccumulate>
() =>
f => x => x;
}

public static _ListNode<T> Null { get; } = new _NullListNode();
}```

# IsNull

Null is _ListNode<T>.Null, so IsNull function is easy to implement:

```public static partial class _ListNodeExtensions
{
// IsNull = node => node(value => _ => ChurchBoolean.False)(ChurchBoolean.True)
public static Boolean IsNull<T>
(this _ListNode<T> node) =>
node.Node<Boolean>()(value => _ => ChurchBoolean.False)(ChurchBoolean.True);
}```

IsNull predicate is similar to the IsNull of Church list encoded with 1 tuple as each node.

# Create, value and Next

```public static partial class _ListNodeExtensions
{
// Create = value => next => f => x => f(value)(next(f)(x))
public static Func<_ListNode<T>, _ListNode<T>> Create<T>
(T value) => next => new _ListNode<T>(value, next);

// Value = node => anyValueToIgnore => node(value => _ => value)(anyValueToIgnore)
public static T Value<T>
(this _ListNode<T> node, T anyValueToIgnore = default(T)) =>
node.Node<T>()(_ => value => value)(anyValueToIgnore);

// Next = node => node(value => tuple => tuple.Shift(Create(value)))(ChurchTuple.Create(Null)(Null)).Item1()
public static _ListNode<T> Next<T>
(this _ListNode<T> node) =>
node.Node<Tuple<_ListNode<T>, _ListNode<T>>>()
(tuple => value => tuple.Shift(Create(value)))
(ChurchTuple.Create<_ListNode<T>, _ListNode<T>>(_ListNode<T>.Null)(_ListNode<T>.Null))
.Item1();
}```

Next is tricky but the same way as Church numeral’s Decrease2, which version with shifting the tuple.

# Index

The same as other list encodings:

```public static partial class _ListNodeExtensions
{
// Index = start => index => index(Next)(start)
public static _ListNode<T> Index<T>
(this _ListNode<T> start, _Numeral index) => index.Numeral<_ListNode<T>>()(Next)(start);
}```

# Unit tests

```[TestClass()]
public class _ListNodeTests
{
[TestMethod()]
public void CreateValueNextTest()
{
_ListNode<int> node1 = _ListNodeExtensions.Create(1)(_ListNodeExtensions.GetNull<int>());
_ListNode<int> node2 = _ListNodeExtensions.Create(2)(node1);
_ListNode<int> node3 = _ListNodeExtensions.Create(3)(node2);
Assert.AreEqual(1, node1.Value());
Assert.AreEqual(_ListNodeExtensions.GetNull<int>(), node1.Next());
Assert.AreEqual(2, node2.Value());
Assert.AreEqual(node1.Value(), node2.Next().Value());
Assert.AreEqual(3, node3.Value());
Assert.AreEqual(node2.Value(), node3.Next().Value());
Assert.IsTrue(_ListNodeExtensions.GetNull<object>().Next().IsNull()._Unchurch());
}

[TestMethod()]
public void NullIsNullTest()
{
_ListNode<int> node = _ListNodeExtensions.Create(1)(_ListNodeExtensions.GetNull<int>());
Assert.IsTrue(_ListNodeExtensions.GetNull<object>().IsNull()._Unchurch());
Assert.IsFalse(node.IsNull()._Unchurch());
}

[TestMethod()]
public void IndexTest()
{
_ListNode<int> node1 = _ListNodeExtensions.Create(1)(_ListNodeExtensions.GetNull<int>());
_ListNode<int> node2 = _ListNodeExtensions.Create(2)(node1);
_ListNode<int> node3 = _ListNodeExtensions.Create(3)(node2);
Assert.AreEqual(node3.Value(), node3.Index(0U._Church()).Value());
Assert.AreEqual(node2.Value(), node3.Index(1U._Church()).Value());
Assert.AreEqual(node1.Value(), node3.Index(2U._Church()).Value());
Assert.IsTrue(node3.Index(3U._Church()).IsNull()._Unchurch());
Assert.IsTrue(node3.Index(4U._Church()).IsNull()._Unchurch());
Assert.IsTrue(node3.Index(5U._Church()).IsNull()._Unchurch());
}
}```