# Functional Modelling System

### Open Source Your Knowledge, Become a Contributor ## Tuples and Constructors

FMS can handle more data than just numbers, strings and Booleans. In this section we introduce more advanced types.

### Tuples

Constructing tuples is straightforward: just write a tuple of expressions like this: `(x,y,z)`. Tuples of any arity (except 1) are supported. This means that the empty tuple `()` exists. As wel as the 8-ary tuple `(0,3,4,9,7,6,1,5)` exists.

When you get a tuple as input you should also be able to deconstruct it. You can do this in a number of ways.

• The simplest one is to use the tuple notation at the lefthand side of a definition. `(a,b) := x`. This means that you define `a` to be the first component of the binary tuple `x` and `b` to be the second one.
• Tuple deconstruction is also possible as locally scoped argument of a function definiton, in which case you assume a function to take tuples as its input. For instance, `f (a,b) := a*b` is a unary function which takes a binary tuple and multiplies its components. This also works with anonymous functions `\(a,b) -> a * b`.
• You can use a case-construct to deconstruct the tuple. `case x of (a,b) -> a+b;`. This expressions deconstructs the tuple `x` and locally introduces the new variables `a` and `b` to evaluate the expression `a+b`, which is the evaluation of this tuple.
Tuples

Warning: Tuple deconstructions are always definitions, which never form constraints on their own. As such, you can not deconstruct a tuple with `(a,a) := x` to enforce that the components of `x` are equal. Use `(a,b) := x` and `a = b` instead.

### Constructors

It is easy to get lost in the data when a lot of tuples are nested. For this reason we can use constructors. They are declared like this: `s/1 :: constructor` for a unary constructor named `s`. Constructors without arguments are also supported `nil/0 :: constructor`.

All the techniques for deconstructing tuples generalize to constructors, except that you need to use square brackets `[` `]` instead of round brackets `(` `)`.

• You can pattern match the left side of the definition like this: `s [x] := y.`
• Using patterns in the function definition is also allowed.
• Case-constructs also work for constructed types, it is even possible to match multiple patterns in one case like this `case x of s[y] -> y ; nil [] -> z;.` Patterns are handled top to bottom, and the first pattern that matches corresponds to the value of the case expression. So the expression `case x of s[s[nil []]] -> 5 ; s[x] -> 2 ; x -> 0;.` will map the Peano representation of 2 to 5, all other successors to 2, and everything which is not a successor of something to 0.

Note again that square brackets are used when deconstructing constructors. This serves to differentiate between `f x := y.` which declares a new function `f` which is the constant function to `y` and `f [x] := y.` which declares the new symbol `x` as removing the outer constructor `f` from `y`.

Using `s` and `nil` we can now do some Peano arithmetic.

Constructors

### More cases

The case-construct can be used in even more circumstances. In the matching part of a case expression you can also write a fixed number or string. An underscore is handled as a don't care-pattern and matches everything.

You can split definitions in to multiple lines as syntactical sugar for a case-construct. This allows us to for example define the factorial function.

More Patterns
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