# [Haskell-beginners] The Holy Trinity of Functional Programming

Costello, Roger L. costello at mitre.org
Wed Aug 22 11:27:08 CEST 2012

```Hi Folks,

In Richard Bird's book (Introduction to Functional Programming using Haskell) he has a fascinating discussion on three key aspects of functional programming. Below is my summary of his discussion.  /Roger

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The Holy Trinity of Functional Programming
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These three ideas constitute the holy trinity of functional programming:

1.  User-defined recursive data types.
2.  Recursively defined functions over recursive data types.
3.  Proof by induction: show that some property P(n) holds for each element of a recursive data type.

Here is an example of a user-defined recursive data type. It is a declaration for the natural numbers 0, 1, 2, ...:

data Nat = Zero | Succ Nat

The elements of this data type include:

Zero,  Succ Zero,  Succ (Succ Zero),  Succ (Succ (Succ Zero)),  ...

To understand this, when creating a Nat value we have a choice of either Zero or Succ Nat. Suppose we choose Succ Nat. Well, now we must choose a value for the Nat in Succ Nat. Again, we have a choice of either Zero or Succ Nat. Suppose this time we choose Zero, to obtain Succ Zero.

The ordering of the elements in the Nat data type can be specified by defining Nat to be a member of the Ord class:

instance Ord Nat where
Zero < Zero 		=  False
Zero < Succ n	=  True
Succ m < Zero	=  False
Succ m < Succ n 	=  (m < n)

Here is how the Nat version of the expression 2 < 3 is evaluated:

Succ (Succ Zero) < Succ (Succ (Succ Zero))	-- Nat version of 2 < 3

= Succ Zero < Succ (Succ Zero)			-- by the 4th equation defining order

= Zero < Succ Zero				-- by the 4th equation defining order

= True						-- by the 2nd equation defining order

Here is a recursively defined function over the data type; it adds two Nat elements:

(+) :: Nat -> Nat -> Nat
m + Zero = m
m + Succ n = Succ(m + n)

Here is how the Nat version of 0 + 1 is evaluated:

Zero + Succ Zero				-- Nat version of 0 + 1

= Succ (Zero + Zero)				-- by the 2nd equation defining +

= Succ Zero					-- by the 1st equation defining +

Here is another recursively defined function over the data type; it subtracts two Nat elements:

(-) :: Nat -> Nat -> Nat
m - Zero = m
Succ m - Succ n = m - n

If the Nat version of 0 - 1 is executed, then the result is undefined:

Zero - Succ Zero

The "undefined value" is denoted by this symbol: _|_ (also known as "bottom")

Important: _|_ is an element of *every* data type.

So we must expand the list of elements in Nat:

_|_,  Zero,  Succ Zero,  Succ (Succ Zero),  Succ (Succ (Succ Zero)),  ...

There are still more elements of Nat. Suppose we define a function that returns a Nat. Let's call the function undefined:

undefined :: Nat
undefined = undefined

It is an infinitely recursive function: when invoked it never stops until the program is interrupted.

This function undefined is denoted by the symbol _|_

Recall how we defined the ordering of values in Nat:

instance Ord Nat where
Zero < Zero 		=  False
Zero < Succ n	=  True
Succ m < Zero	=  False
Succ m < Succ n 	=  (m < n)

```