   ## Functional core of Objective CAML

Like all functional languages, Objective CAML is an expression oriented language, where programming consists mainly of creating functions and applying them. The result of the evaluation of one of these expressions is a value in the language and the execution of a program is the evaluation of all the expressions which comprise it.

### Primitive values, functions, and types

Integers and floating-point numbers, characters, character strings, and booleans are predefined in Objective CAML.

#### Numbers

There are two kinds of numbers: integers1 of type int and floating-point numbers of type float. Objective CAML follows the IEEE 754 standard2 for representing double-precision floating-point numbers. The operations on integers and floating-point numbers are described in figure 2.1. Let us note that when the result of an integer operation is outside the interval on which values of type int are defined, this does not produce an error, but the result is an integer within the system's interval of integers. In other words, all integer operations are operations modulo the boundaries of the interval.

integer numbers floating-point numbers
 + addition - subtraction and unary negation * multiplication / integer division mod remainder of integer division
 +. addition -. subtraction and unary negation *. multiplication /. division ** exponentiation
 ```# `1`` `;;`- : int = 1`# `1`` ``+`` ``2`` `;;`- : int = 3`# `9`` ``/`` ``2`` `;;`- : int = 4`# `1``1`` `mod` ``3`` `;;`- : int = 2``(* limits of the representation *)``(* of integers *)`# `2``1``4``7``4``8``3``6``5``0`` `;;`- : int = 2` ``` ```# `2``.``0`` `;;`- : float = 2`# `1``.``1`` ``+.`` ``2``.``2`` `;;`- : float = 3.3`# `9``.``1`` ``/.`` ``2``.``2`` `;;`- : float = 4.13636363636`# `1``.`` ``/.`` ``0``.`` `;;`- : float = inf``(* limits of the representation *)``(* of floating-point numbers *)`# `2``2``2``2``2``2``2``2``2``2``2``2``.``1``1``1``1``1`` `;;`- : float = 222222222222` ```

Figure 2.1: Operations on numbers.

##### Differences between integers and floating-point numbers
Values having different types such as float and int can never be compared directly. But there are functions for conversion (float_of_int and int_of_float) between one and the other.
```# `2`` ``=`` ``2``.``0`` `;;`Characters 5-8:``This expression has type float but is here used with type int`# `3``.``0`` ``=`` `float_of_int` ``3`` `;;`- : bool = true`

```

In the same way, operations on floating-point numbers are distinct from those on integers.

```# `3`` ``+`` ``2`` `;;`- : int = 5`# `3``.``0`` ``+.`` ``2``.``0`` `;;`- : float = 5`# `3``.``0`` ``+`` ``2``.``0`` `;;`Characters 0-3:``This expression has type float but is here used with type int`# sin` ``3``.``1``4``1``5``9`` `;;`- : float = 2.65358979335e-06`

```

An ill-defined computation, such as a division by zero, will raise an exception (see page ??) which interrupts the computation. Floating-point numbers have a representation for infinite values (printed as Inf) and ill-defined computations (printed as NaN3). The main functions on floating-point numbers are described in figure 2.2.

functions on floats trigonometric functions
 ceil floor sqrt square root exp exponential log natural log log10 log base 10
 cos cosine sin sine tan tangent acos arccosine asin arcsine atan arctangent
 ```# ceil` ``3``.``4`` `;;`- : float = 4`# floor` ``3``.``4`` `;;`- : float = 3`# ceil` `(`-.``3``.``4`)` `;;`- : float = -3`# floor` `(`-.``3``.``4`)` `;;`- : float = -4` ``` ```# sin` ``1``.``5``7``0``7``8`` `;;`- : float = 0.999999999867`# sin` `(asin` ``0``.``7``0``7`)` `;;`- : float = 0.707`# acos` ``0``.``0`` `;;`- : float = 1.57079632679`# asin` ``3``.``1``4`` `;;`- : float = nan` ```

Figure 2.2: Functions on floats.

#### Characters and Strings

Characters, type char, correspond to integers between 0 and 255 inclusive, following the ASCII encoding for the first 128. The functions char_of_int and int_of_char support conversion between integers and characters. Character strings, type string, are sequences of characters of definite length (less than 224-6). The concatenation operator is  . The functions int_of_string, string_of_int, string_of_float and float_of_string carry out the various conversions between numbers and character strings.
```# `'B'`` `;;`- : char = 'B'`# int_of_char` ``'B'`` `;;`- : int = 66`# `"is a string"`` `;;`- : string = "is a string"`# (string_of_int` ``1``9``8``7`)` ``^`` ``" is the year Caml was created"`` `;;`- : string = "1987 is the year Caml was created"`

```

Even if a string contains the characters of a number, it won't be possible to use it in operations on numbers without carrying out an explicit conversion.
```# `"1999"`` ``+`` ``1`` `;;`Characters 1-7:``This expression has type string but is here used with type int`# (int_of_string` ``"1999"`)` ``+`` ``1`` `;;`- : int = 2000`

```
Numerous functions on character strings are gathered in the String module (see page ??).

#### Booleans

Booleans, of type bool, belong to a set consisting of two values: true and false. The primitive operators are described in figure 2.3. For historical reasons, the ``and'' and ``or'' operators each have two forms.

 not negation && sequential and || sequential or
 & synonym for && or synonym for ||

Figure 2.3: Operators on booleans.

```# true` `;;`- : bool = true`# not` `true` `;;`- : bool = false`# true` ``&&`` `false` `;;`- : bool = false`

```
The operators && and ||, or their synonyms, evaluate their left argument and then, depending on its value, evaluate their right argument. They can be rewritten in the form of conditional constructs (see page ??).

 = structural equality == physical equality <> negation of = != negation of ==
 < less than > greater than <= less than or equal to >= greater than or equal to

Figure 2.4: Equality and comparison operators.

The equality and comparison operators are described in figure 2.4. They are polymorphic, i.e., they can be used to compare two integers as well as two character strings. The only constraint is that their two operands must be of the same type (see page ??).
```# `1``<=``1``1``8`` ``&&`` `(`1``=``2`` ``||`` `not(`1``=``2`))` `;;`- : bool = true`# `1``.``0`` ``<=`` ``1``1``8``.``0`` ``&&`` `(`1``.``0`` ``=`` ``2``.``0`` ``||`` `not` `(`1``.``0`` ``=`` ``2``.``0`))` `;;`- : bool = true`# `"one"`` ``<`` ``"two"`` `;;`- : bool = true`# `0`` ``<`` ``'0'`` `;;`Characters 4-7:``This expression has type char but is here used with type int`

```

Structural equality tests the equality of two values by traversing their structure, whereas physical equality tests whether the two values occupy the same region in memory. These two equality operators return the same result for simple values: booleans, characters, integers and constant constructors (page ??).

### Warning

Floating-point numbers and character strings are considered structured values.

#### Unit

The unit type describes a set which possesses only a single element, denoted: ().
```# ()` `;;`- : unit = ()`

```
This value will often be used in imperative programs (3, page ??) for functions which carry out side effects. Functions whose result is the value () simulate the notion of procedure, which doesn't exist in Objective CAML, just as the type void does in the C language.

#### Cartesian product, tuple

Values of possibly different types can be gathered in pairs or more generally in tuples. The values making up a tuple are separated by commas. The type constructor * indicates a tuple. The type int * string is the type of pairs whose first element is an integer (of type int) and whose second is a character string (of type string).
```# (` ``1``2`` ``,`` ``"October"`` `)` `;;`- : int * string = 12, "October"`

```
When there is no ambiguity, it can be written more simply:
```# `1``2`` ``,`` ``"October"`` `;;`- : int * string = 12, "October"`

```
The functions fst and snd allow access to the first and second elements of a pair.
```# fst` `(` ``1``2`` ``,`` ``"October"`` `)` `;;`- : int = 12`# snd` `(` ``1``2`` ``,`` ``"October"`` `)` `;;`- : string = "October"`

```
These two functions accept pairs whose components are of any type whatsoever. They are polymorphic, in the same way as equality.
```# fst;;`- : 'a * 'b -> 'a = <fun>`# fst` `(` ``"October"``,`` ``1``2`` `)` `;;`- : string = "October"`

```
The type int * char * string is that of triplets whose first element is of type int, whose second is of type char, and whose third is of type string. Its values are written
```# (` ``6``5`` ``,`` ``'B'`` ``,`` ``"ascii"`` `)` `;;`- : int * char * string = 65, 'B', "ascii"`

```

### Warning

The functions fst and snd applied to a tuple, other than a pair, result in a type error.
```# snd` `(` ``6``5`` ``,`` ``'B'`` ``,`` ``"ascii"`` `)` `;;`Characters 7-25:``This expression has type int * char * string but is here used with type``  'a * 'b`

```
There is indeed a difference between the type of a pair and that of a triplet. The type int * int * int is different from the types (int * int) * int and int * (int * int). Functions to access a triplet (and other tuples) are not defined by the core library. One can use pattern matching to define them if need be (see page ??).

#### Lists

Values of the same type can be gathered into a list. A list can either be empty or consist of elements of the same type.
```# []` `;;`- : 'a list = []`# `[`` ``1`` `;` ``2`` `;` ``3`` ``]`` `;;`- : int list = [1; 2; 3]`# `[`` ``1`` `;` ``"two"`` `;` ``3`` ``]`` `;;`Characters 14-17:``This expression has type int list but is here used with type string list`

```

The function which adds an element at the head of a list is the infix operator :: . It is the analogue of Lisp's cons.
```# `1`` `::` ``2`` `::` ``3`` `::` `[]` `;;`- : int list = [1; 2; 3]`

```

Concatenation of two lists is also an infix operator @.
```# `[`` ``1`` ``]`` `` ``@`` `` ``[`` ``2`` `;` ``3`` ``]`` `;;`- : int list = [1; 2; 3]`# `[`` ``1`` `;` ``2`` ``]`` `` ``@`` `` ``[`` ``3`` ``]`` `;;`- : int list = [1; 2; 3]`

```

The other list manipulation functions are defined in the List library. The functions hd and tl from this library give respectively the head and the tail of a list when these values exist. These functions are denoted by List.hd and List.tl to indicate to the system that they can be found in the module List4.
```# List.hd` ``[`` ``1`` `;` ``2`` `;` ``3`` ``]`` `;;`- : int = 1`# List.hd` `[]` `;;`Uncaught exception: Failure("hd")`

```
In this last example, it is indeed problematic to request retrieval of the first element of an empty list. It is for this reason that the system raises an exception (see page ??).

### Conditional control structure

One of the indispensable control structures in any programming language is the structure called conditional (or branch) which guides the computation as a function of a condition.

### Syntax

if expr1 then expr2 else expr3
The expression expr1 is of type bool. The expressions expr2 and expr3 must be of the same type, whatever it may be.

```# if` ``3``=``4`` `then` ``0`` `else` ``4`` `;;`- : int = 4`# if` ``3``=``4`` `then` ``"0"`` `else` ``"4"`` `;;`- : string = "4"`# if` ``3``=``4`` `then` ``0`` `else` `` ``"4"`;;`Characters 20-23:``This expression has type string but is here used with type int`

```

A conditional construct is itself an expression and its evaluation returns a value.

```# (if` ``3``=``5`` `then` ``8`` `else` ``1``0`)` ``+`` ``5`` `;;`- : int = 15`

```

### Note

The else branch can be omitted, but in this case it is implicitly replaced by else` `(). Consequently, the type of the expression expr2 must be unit (see page ??).

### Value declarations

A declaration binds a name to a value. There are two types: global declarations and local declarations. In the first case, the declared names are known to all the expressions following the declaration; in the second, the declared names are only known to one expression. It is equally possible to simultaneously declare several name-value bindings.

### Syntax

let name = expr ;;
A global declaration defines the binding between the name name and the value of the expression expr which will be known to all subsequent expressions.
```# let` `yr` ``=`` ``"1999"`` `;;`val yr : string = "1999"`# let` `x` ``=`` `int_of_string(yr)` `;;`val x : int = 1999`# x` `;;`- : int = 1999`# x` ``+`` ``1`` `;;`- : int = 2000`# let` `new_yr` ``=`` `string_of_int` `(x` ``+`` ``1`)` `;;`val new_yr : string = "2000"`

```

### Syntax

 let name1 = expr1 and name2 = expr2 : and namen = exprn ;;

A simultaneous declaration declares different symbols at the same level. They won't be known until the end of all the declarations.
```# let` `x` ``=`` ``1`` `and` `y` ``=`` ``2`` `;;`val x : int = 1``val y : int = 2`# x` ``+`` `y` `;;`- : int = 3`# let` `z` ``=`` ``3`` `and` `t` ``=`` `z` ``+`` ``2`` `;;` ``Characters 18-19:``Unbound value z`

```

It is possible to gather several global declarations in the same phrase; then printing of their types and their values does not take place until the end of the phrase, marked by double ``;;''. These declarations are evaluated sequentially, in contrast with a simultaneous declaration.
```# let` `x` ``=`` ``2`` `` `let` `y` ``=`` `x` ``+`` ``3`` `` `;;`val x : int = 2``val y : int = 5`

```
A global declaration can be masked by a new declaration of the same name (see page ??).

### Syntax

let name = expr1 in expr2;;
The name name is only known during the evaluation of expr2. The local declaration binds it to the value of expr1.
```# let` `xl` ``=`` ``3`` `in` `xl` ``*`` `xl` `;;`- : int = 9`

```
The local declaration binding xl to the value 3 is only in effect during the evaluation of xl` ``*`` `xl.
```# xl` `;;`Characters 1-3:``Unbound value xl`

```
A local declaration masks all previous declarations of the same name, but the previous value is reinstated upon leaving the scope of the local declaration:
```# let` `x` ``=`` ``2`` `;;`val x : int = 2`# let` `x` ``=`` ``3`` `in` `x` ``*`` `x` `;;`- : int = 9`# x` ``*`` `x` `;;`- : int = 4`

```
A local declaration is an expression and can thus be used to construct other expressions:
```# (let` `x` ``=`` ``3`` `in` `x` ``*`` `x)` ``+`` ``1`` `;;`- : int = 10`

```

Local declarations can also be simultaneous.

### Syntax

 let name1 = expr1 and name2 = expr2 : and namen = exprn in expr ;;

```# let` `a` ``=`` ``3``.``0`` `and` `b` ``=`` ``4``.``0`` `in` `sqrt` `(a`*.`a` ``+.`` `b`*.`b)` `;;`- : float = 5`# b` `;;`Characters 0-1:``Unbound value b`

```

### Function expressions, functions

A function expression consists of a parameter and a body. The formal parameter is a variable name and the body an expression. The parameter is said to be abstract. For this reason, a function expression is also called an abstraction.

### Syntax

function p -> expr
Thus the function which squares its argument is written:
```# function` `x` `->` `x`*`x` `;;`- : int -> int = <fun>`

```
The Objective CAML system deduces its type. The function type int -> int indicates a function expecting a parameter of type int and returning a value of type int.

Application of a function to an argument is written as the function followed by the argument.
```# (function` `x` `->` `x` ``*`` `x)` ``5`` `;;`- : int = 25`

```
The evaluation of an application amounts to evaluating the body of the function, here x` ``*`` `x, where the formal parameter, x, is replaced by the value of the argument (or the actual parameter), here `5`.

In the construction of a function expression, expr is any expression whatsoever. In particular, expr may itself be a function expression.

```# function` `x` `->` `(function` `y` `->` ``3``*`x` ``+`` `y)` `;;`- : int -> int -> int = <fun>`

```

The parentheses are not required. One can write more simply:
```# function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y` `;;`- : int -> int -> int = <fun>`

```
The type of this expression can be read in the usual way as the type of a function which expects two integers and returns an integer value. But in the context of a functional language such as Objective CAML we are dealing more precisely with the type of a function which expects an integer and returns a function value of type int -> int:
```# (function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` ``5`` `;;`- : int -> int = <fun>`

```

One can, of course, use the function expression in the usual way by applying it to two arguments. One writes:
```# (function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` ``4`` ``5`` `;;`- : int = 17`

```
When one writes f` `a` `b, there is an implicit parenthesization on the left which makes this expression equivalent to: (f` `a)` `b.

Let's examine the application
(function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` ``4`` ``5`
in detail. To compute the value of this expression, it is necessary to compute the value of
(function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` ``4`
which is a function expression equivalent to
function` `y` `->` ``3``*``4`` ``+`` `y
obtained by replacing x by `4` in `3``*`x` ``+`` `y. Applying this value (which is a function) to `5` we get the final value `3``*``4``+``5` = `1``7`:
```# (function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` ``4`` ``5`` `;;`- : int = 17`

```

#### Arity of a function

The number of arguments of a function is called its arity. Usage inherited from mathematics demands that the arguments of a function be given in parentheses after the name of the function. One writes: f(4,5). We've just seen that in Objective CAML, one more usually writes: f 4 5. One can, of course, write a function expression in Objective CAML which can be applied to (4,5):
```# function` `(x`,`y)` `->` ``3``*`x` ``+`` `y` `;;`- : int * int -> int = <fun>`

```
But, as its type indicates, this last expression expects not two, but only one argument: a pair of integers. Trying to pass two arguments to a function which expects a pair or trying to pass a pair to a function which expects two arguments results in a type error:
```# (function` `(x`,`y)` `->` ``3``*`x` ``+`` `y)` ``4`` ``5`` `;;`Characters 29-30:``This expression has type int but is here used with type int * int`# (function` `x` `->` `function` `y` `->` ``3``*`x` ``+`` `y)` `(`4``,`` ``5`)` `;;`Characters 39-43:``This expression has type int * int but is here used with type int`

```

#### Alternative syntax

There is a more compact way of writing function expressions with several parameters. It is a legacy of former versions of the Caml language. Its form is as follows:

### Syntax

fun p1 ...pn -> expr
It allows one to omit repetitions of the function keyword and the arrows. It is equivalent to the following translation:
function p1 -> -> function pn -> expr
```# fun` `x` `y` `->` ``3``*`x` ``+`` `y` `;;`- : int -> int -> int = <fun>`# (fun` `x` `y` `->` ``3``*`x` ``+`` `y)` ``4`` ``5`` `;;`- : int = 17`

```
This form is still encountered often, in particular in the libraries provided with the Objective CAML distribution.

#### Closure

Objective CAML treats a function expression like any other expression and is able to compute its value. The value returned by the computation is a function expression and is called a closure. Every Objective CAML expression is evaluated in an environment consisting of name-value bindings coming from the declarations preceding the expression being computed. A closure can be described as a triplet consisting of the name of the formal parameter, the body of the function, and the environment of the expression. This environment needs to be preserved because the body of a function expression may use, in addition to the formal parameters, every other variable declared previously. These variables are said to be ``free'' in the function expression. Their values will be needed when the function expression is applied.
```# let` `m` ``=`` ``3`` `;;`val m : int = 3`# function` `x` `->` `x` ``+`` `m` `;;`- : int -> int = <fun>`# (function` `x` `->` `x` ``+`` `m)` ``5`` `;;`- : int = 8`

```

When application of a closure to an argument returns a new closure, the latter possesses within its environment all the bindings necessary for a future application. The subsection on the scope of variables (see page ??) details this notion. We will return to the memory representation of a closure in chapter 4 (page ??) as well as chapter 12 (page ??).

The function expressions used until now are anonymous. It is rather useful to be able to name them.

#### Function value declarations

Function values are declared in the same way as other language values, by the let construct.
```# let` `succ` ``=`` `function` `x` `->` `x` ``+`` ``1`` `;;`val succ : int -> int = <fun>`# succ` ``4``2``0`` `;;`- : int = 421`# let` `g` ``=`` `function` `x` `->` `function` `y` `->` ``2``*`x` ``+`` ``3``*`y` `;;`val g : int -> int -> int = <fun>`# g` ``1`` ``2`;;`- : int = 8`

```

To simplify writing, the following notation is allowed:

### Syntax

let name p1 ... pn = expr
which is equivalent to the following form:
let name = function p1 -> -> function pn -> expr

The following declarations of succ and g are equivalent to their previous declaration.
```# let` `succ` `x` ``=`` `x` ``+`` ``1`` `;;`val succ : int -> int = <fun>`# let` `g` `x` `y` ``=`` ``2``*`x` ``+`` ``3``*`y` `;;`val g : int -> int -> int = <fun>`

```

The completely functional character of Objective CAML is brought out by the following example, in which the function h1 is obtained by the application of g to a single integer. In this case one speaks of partial application:
```# let` `h1` ``=`` `g` ``1`` `;;`val h1 : int -> int = <fun>`# h1` ``2`` `;;`- : int = 8`

```

One can also, starting from g, define a function h2 by fixing the value of the second parameter, y, of g:
```# let` `h2` ``=`` `function` `x` `->` `g` `x` ``2`` `;;`val h2 : int -> int = <fun>`# h2` ``1`` `;;`- : int = 8`

```

#### Declaration of infix functions

Certain functions taking two arguments can be applied in infix form. This is the case with addition of integers. One writes `3`` ``+`` ``5` for the application of + to `3` and `5`. To use the symbol + as a regular function value, this must be syntactically indicated by surrounding the infix symbol with parentheses. The syntax is as follows:

### Syntax

( op )

The following example defines the function succ using (` ``+`` `).
```# (` ``+`` `)` `;;`- : int -> int -> int = <fun>`# let` `succ` ``=`` `(` ``+`` `)` ``1`` `;;`val succ : int -> int = <fun>`# succ` ``3`` `;;`- : int = 4`

```

It is also possible to define new operators. We define an operator ++, addition on pairs of integers
```# let` `(` ``++`` `)` `c1` `c2` ``=`` `(fst` `c1)`+`(fst` `c2)`,`` `(snd` `c1)`+`(snd` `c2)` `;;`val ++ : int * int -> int * int -> int * int = <fun>`# let` `c` ``=`` `(`2``,``3`)` `;;`val c : int * int = 2, 3`# c` ``++`` `c` `;;`- : int * int = 4, 6`

```

There is an important limitation on the possible operators. They must contain only symbols (such as *, +, `@`, etc. ) and not letters or digits. Certain functions predefined as infixes are exceptions to the rule. They are listed as follows: or mod land lor lxor lsl lsr asr.

#### Higher order functions

A function value (a closure) can be returned as a result. It can equally well be passed as an argument to a function. Functions taking function values as arguments or returning them as results are called higher order.
```# let` `h` ``=`` `function` `f` `->` `function` `y` `->` `(f` `y)` ``+`` `y` `;;`val h : (int -> int) -> int -> int = <fun>`

```

### Note

Application is implicitly parenthesized to the left, but function types are implicitly parenthesized to the right. Thus the type of the function h can be written
(int -> int) -> int -> int  or  (int -> int) -> (int -> int)

Higher order functions offer elegant possibilities for dealing with lists. For example the function List.map can apply a function to all the elements of a list and return the results in a list.
```# List.map` `;;`- : ('a -> 'b) -> 'a list -> 'b list = <fun>`# let` `square` `x` ``=`` `string_of_int` `(x`*`x)` `;;`val square : int -> string = <fun>`# List.map` `square` ``[``1`;` ``2`;` ``3`;` ``4``]`` `;;`- : string list = ["1"; "4"; "9"; "16"]`

```

As another example, the function List.for_all can find out whether all the elements of a list satisfy a given criterion.
```# List.for_all` `;;`- : ('a -> bool) -> 'a list -> bool = <fun>`# List.for_all` `(function` `n` `->` `n`<>``0`)` ``[-``3`;` ``-``2`;` ``-``1`;` ``1`;` ``2`;` ``3``]`` `;;`- : bool = true`# List.for_all` `(function` `n` `->` `n`<>``0`)` ``[-``3`;` ``-``2`;` ``0`;` ``1`;` ``2`;` ``3``]`` `;;`- : bool = false`

```

#### Scope of variables

In order for it to be possible to evaluate an expression, all the variables appearing therein must be defined. This is the case in particular for the expression e in the declaration let` `p` ``=`` `e. But since p is not yet known within this expression, this variable can only be present if it refers to another value issued by a previous declaration.
```# let` `p` ``=`` `p` ``^`` ``"-suffix"`` `;;`Characters 9-10:``Unbound value p`# let` `p` ``=`` ``"prefix"`` `;;`val p : string = "prefix"`# let` `p` ``=`` `p` ``^`` ``"-suffix"`` `;;`val p : string = "prefix-suffix"`

```

In Objective CAML, variables are statically bound. The environment used to execute the application of a closure is the one in effect at the moment of its declaration (static scope) and not the one in effect at the moment of application (dynamic scope).

```# let` `p` ``=`` ``1``0`` `;;`val p : int = 10`# let` `k` `x` ``=`` `(x`,`` `p`,`` `x`+`p)` `;;`val k : int -> int * int * int = <fun>`# k` `p` `;;`- : int * int * int = 10, 10, 20`# let` `p` ``=`` ``1``0``0``0`` `;;`val p : int = 1000`# k` `p` `;;`- : int * int * int = 1000, 10, 1010`

```
The function k contains a free variable: p. Since the latter is defined in the global environment, the definition of k is legal. The binding between the name p and the value `1``0` in the environment of the closure k is static, i.e., does not depend on the most recent definition of p.

#### Recursive declarations

A variable declaration is called recursive if it uses its own identifier in its definition. This facility is used mainly for functions, notably to simulate a definition by recurrence. We have just seen that the let declaration does not support this. To declare a recursive function we will use a dedicated syntactic construct.

### Syntax

let rec name = expr ;;
We can equally well use the syntactic facility for defining function values while indicating the function parameters:

### Syntax

let rec name p1 ...pn = expr ;;

By way of example, here is the function sigma which computes the sum of the (non-negative) integers between `0` and the value of its argument, inclusive.
```# let` `rec` `sigma` `x` ``=`` `if` `x` ``=`` ``0`` `then` ``0`` `else` `x` ``+`` `sigma` `(x`-``1`)` `;;`val sigma : int -> int = <fun>`# sigma` ``1``0`` `;;`- : int = 55`

```
It may be noted that this function does not terminate if its argument is strictly negative.

A recursive value is in general a function. The compiler rejects some recursive declarations whose values are not functions:
```# let` `rec` `x` ``=`` `x` ``+`` ``1`` `;;`Characters 13-18:``This kind of expression is not allowed as right-hand side of `let rec'`

```
We will see however that in certain cases such declarations are allowed (see page ??).

The let rec declaration may be combined with the and construction for simultaneous declarations. In this case, all the functions defined at the same level are known within the bodies of each of the others. This permits, among other things, the declaration of mutually recursive functions.
```# let` `rec` `even` `n` ``=`` `(n`<>``1`)` ``&&`` `((n`=``0`)` `or` `(odd` `(n`-``1`)))` `and` `` `` `` `` `odd` `` `n` ``=`` `(n`<>``0`)` ``&&`` `((n`=``1`)` `or` `(even` `(n`-``1`)))` `` `` `` `;;`val even : int -> bool = <fun>``val odd : int -> bool = <fun>`# even` ``4`` `;;`- : bool = true`# odd` ``5`` `;;`- : bool = true`

```

In the same way, local declarations can be recursive. This new definition of sigma tests the validity of its argument before carrying out the computation of the sum defined by a local function sigma_rec.
```# let` `sigma` `x` ``=`` `` `` `let` `rec` `sigma_rec` `x` ``=`` `` `if` `x` ``=`` ``0`` `then` ``0`` `else` `x` ``+`` `sigma_rec` `(x`-``1`)` `in` `` `` `` `if` `(x`<``0`)` `then` ``"error: negative argument"`` `` `` `` `else` ``"sigma = "`` ``^`` `(string_of_int` `(sigma_rec` `x))` `;;`val sigma : int -> string = <fun>`

```

### Note

The need to give a return value of the same type, whether the argument is negative or not, has forced us to give the result in the form of a character string. Indeed, what value should be returned by sigma when its argument is negative? We will see the proper way to manage this problem, using exceptions (see page ??).

### Polymorphism and type constraints

Some functions execute the same code for arguments having different types. For example, creation of a pair from two values doesn't require different functions for each type known to the system5. In the same way, the function to access the first field of a pair doesn't have to be differentiated according to the type of the value of this first field.
```# let` `make_pair` `a` `b` ``=`` `(a`,`b)` `;;`val make_pair : 'a -> 'b -> 'a * 'b = <fun>`# let` `p` ``=`` `make_pair` ``"paper"`` ``4``5``1`` `;;`val p : string * int = "paper", 451`# let` `a` ``=`` `make_pair` ``'B'`` ``6``5`` `;;`val a : char * int = 'B', 65`# fst` `p` `;;`- : string = "paper"`# fst` `a` `;;`- : char = 'B'`

```

Functions are called polymorphic if their return value or one of their parameters is of a type which need not be specified. The type synthesizer contained in the Objective CAML compiler finds the most general type for each expression. In this case, Objective CAML uses variables, here 'a and 'b, to designate these general types. These variables are instantiated to the type of the argument during application of the function.

With Objective CAML's polymorphic functions, we get the advantages of being able to write generic code usable for values of every type, while still preserving the execution safety of static typing. Indeed, although make_pair is polymorphic, the value created by (make_pair` ``'B'`` ``6``5`) has a well-specified type which is different from that of (make_pair` ``"paper"`` ``4``5``1`). Moreover, type verification is carried out on compilation, so the generality of the code does not hamper the efficiency of the program.

#### Examples of polymorphic functions and values

The following examples of polymorphic functions have functional parameters whose type is parameterized.

The app function applies a function to an argument.
```# let` `app` ``=`` `function` `f` `->` `function` `x` `->` `f` `x` `;;`val app : ('a -> 'b) -> 'a -> 'b = <fun>`

```
So it can be applied to the function odd defined previously:
```# app` `odd` ``2`;;`- : bool = false`

```

The identity function (id ) takes a parameter and returns it as is.
```# let` `id` `x` ``=`` `x` `;;`val id : 'a -> 'a = <fun>`# app` `id` `` ``1`` `;;`- : int = 1`

```

The compose function takes two functions and another value and composes the application of these two functions to this value.
```# let` `compose` `f` `g` `` `x` ``=`` `f` `(g` `x)` `;;`val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = <fun>`# let` `add1` `x` ``=`` `x`+``1`` `and` `mul5` `x` ``=`` `x`*``5`` `in` `compose` `` `mul5` `add1` `` ``9`` `;;`- : int = 50`

```
It can be seen that the result of g must be of the same type as the argument of f.

Values other than functions can be polymorphic as well. For example, this is the case for the empty list:
```# let` `l` ``=`` `[]` `;;`val l : 'a list = []`

```

The following example demonstrates that type synthesis indeed arises from resolution of the constraints coming from function application and not from the value obtained upon execution.
```# let` `t` ``=`` `List.tl` ``[``2``]`` `;;`val t : int list = []`

```
The type of List.tl is 'a list -> 'a list, so this function applied to a list of integers returns a list of integers. The fact that upon execution it is the empty list which is obtained doesn't change its type at all.

Objective CAML generates parameterized types for every function which doesn't use the form of its arguments. This polymorphism is called parametric polymorphism6.

#### Type constraint

As the Caml type synthesizer generates the most general type, it may be useful or necessary to specify the type of an expression.

The syntactic form of a type constraint is as follows:

### Syntax

( expr : t )
When it runs into such a constraint, the type synthesizer will take it into account while constructing the type of the expression. Using type constraints lets one:
• make the type of the parameters of a function visible;
• forbid the use of a function outside its intended context;
• specify the type of an expression, which will be particularly useful for mutable values (see page ??).
The following examples demonstrate the use of such type constraints
```# let` `add` `(x`:`int)` `(y`:`int)` ``=`` `x` ``+`` `y` `;;`val add : int -> int -> int = <fun>`# let` `make_pair_int` `(x`:`int)` `(y`:`int)` ``=`` `x`,`y;;`val make_pair_int : int -> int -> int * int = <fun>`# let` `compose_fn_int` `(f` ``:`` `int` `->` `int)` `(g` ``:`` `int` `->` `int)` `(x`:`int)` ``=`` `` `` `` `compose` `f` `g` `` `x;;`val compose_fn_int : (int -> int) -> (int -> int) -> int -> int = <fun>`# let` `nil` ``=`` `([]` ``:`` `string` `list);;`val nil : string list = []`# `'H'`::nil;;`Characters 5-8:``This expression has type string list but is here used with type char list`

```

Restricting polymorphism this way lets us control the type of an expression better by constraining the polymorphism of the type deduced by the system. Any defined type whatsoever may be used, including ones containing type variables, as the following example shows:
```# let` `llnil` ``=`` `([]` ``:`` `'a` `list` `list)` `;;`val llnil : 'a list list = []`# `[``1`;`2`;`3``]::`` `llnil` `;;`- : int list list = [[1; 2; 3]]`

```
The symbol llnil is a list of lists of any type whatsoever.

Here we are dealing with constraints, and not replacing Objective CAML's type synthesis with explicit typing. In particular, one cannot generalize types beyond what inference permits.
```# let` `add_general` `(x`:`'a)` `(y`:`'b)` ``=`` `add` `x` `y` `;;`val add_general : int -> int -> int = <fun>`

```

Type constraints will be used in module interfaces (14) as well as in class declarations (15).

### Examples

In this section we will give several somewhat elaborate examples of functions. Most of these functions are predefined in Objective CAML. We will redefine them for the sake of ``pedagogy''.

Here, the test for the terminal case of recursive functions is implemented by a conditional. Hence a programming style closer to Lisp. We will see how to give a more ML character to these definitions when we present another way of defining functions by case (see page ??).

#### Length of a list

Let's start with the function null which tests whether a list is empty.
```# let` `null` `l` ``=`` `` `(l` ``=`` `[])` `;;`val null : 'a list -> bool = <fun>`

```
Next, we define the function size to compute the length of a list (i.e., the number of its elements).
```# let` `rec` `size` `l` ``=`` `` `` `if` `null` `l` `then` ``0`` `` `` `else` ``1`` ``+`` `(size` `(List.tl` `l))` `;;`val size : 'a list -> int = <fun>`# size` `[]` `;;`- : int = 0`# size` ``[``1`;`2`;`1``8`;`2``2``]`` `;;`- : int = 4`

```
The function size tests whether the list argument is empty. If so it returns 0, if not it returns 1 plus the value resulting from computing the length of the tail of the list.

#### Iteration of composition

The expression iterate` `n` `f computes the value f iterated n times.
```# let` `rec` `iterate` `n` `f` ``=`` `` `` `if` `n` ``=`` ``0`` `then` `(function` `x` `->` `x)` `` `` `else` `compose` `f` `(iterate` `(n`-``1`)` `f)` `;;`val iterate : int -> ('a -> 'a) -> 'a -> 'a = <fun>`

```
The iterate function tests whether n is 0, if yes it returns the identity function, if not it composes f with the iteration of f n-1 times.

Using iterate, one can define exponentiation as iteration of multiplication.
```` `# let` `rec` `power` `i` `n` ``=`` `` `` `let` `i_times` ``=`` `(` ``*`` `)` `i` `in` `` `` `` `iterate` `n` `i_times` ``1`` `;;`val power : int -> int -> int = <fun>`# power` ``2`` ``8`` `;;`- : int = 256`

```
The power function iterates n times the function expression i_times, then applies this result to 1, which does indeed compute the nth power of an integer.

#### Multiplication table

We want to write a function multab which computes the multiplication table of an integer passed as an argument.

First we define the function apply_fun_list such that, if f_list is a list of functions, apply_fun_list` `x` `f_list returns the list of results of applying each element of f_list to x.
```# let` `rec` `apply_fun_list` `x` `f_list` ``=`` `` `` `if` `null` `f_list` `then` `[]` `` `` `else` `((List.hd` `f_list)` `x)::(apply_fun_list` `x` `(List.tl` `f_list))` `;;`val apply_fun_list : 'a -> ('a -> 'b) list -> 'b list = <fun>`# apply_fun_list` ``1`` ``[`(` ``+`` `)` ``1`;(` ``+`` `)` ``2`;(` ``+`` `)` ``3``]`` `;;`- : int list = [2; 3; 4]`

```

The function mk_mult_fun_list returns the list of functions multiplying their argument by i, for i varying from `0` to n.
```# let` `mk_mult_fun_list` `n` ``=`` `` `` `` `let` `rec` `mmfl_aux` `p` ``=`` `` `` `` `` `` `` `if` `p` ``=`` `n` `then` ``[`` `(` ``*`` `)` `n` ``]`` `` `` `` `` `` `else` `((` ``*`` `)` `p)` `::` `(mmfl_aux` `(p`+``1`))` `` `` `` `in` `` `(mmfl_aux` ``1`)` `;;`val mk_mult_fun_list : int -> (int -> int) list = <fun>`

```

We obtain the multiplication table of 7 by:
```# let` `multab` `n` ``=`` `apply_fun_list` `n` `(mk_mult_fun_list` ``1``0`)` `;;`val multab : int -> int list = <fun>`# multab` ``7`` `;;`- : int list = [7; 14; 21; 28; 35; 42; 49; 56; 63; 70]`

```

#### Iteration over lists

The function call fold_left` `f` `a` ``[`e1;` `e2;` ``...`` `;` `en`]` returns f` ``...`` `(f` `(f` `a` `e1)` `e2)` ``...`` `en. So there are n applications.
```# let` `rec` `fold_left` `f` `a` `l` ``=`` `` `` `if` `null` `l` `then` `a` `` `` `else` `fold_left` `f` `(` `f` `a` `(List.hd` `l))` `(List.tl` `l)` `;;`val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b list -> 'a = <fun>`

```

The function fold_left permits the compact definition of a function to compute the sum of the elements of a list of integers:
```# let` `sum_list` ``=`` `fold_left` `(`+`)` ``0`` `;;`val sum_list : int list -> int = <fun>`# sum_list` ``[``2`;`4`;`7``]`` `;;`- : int = 13`

```

Or else, the concatenation of the elements of a list of strings:
```# let` `concat_list` ``=`` `fold_left` `(`^`)` ``""`` `;;`val concat_list : string list -> string = <fun>`# concat_list` ``[``"Hello "`;` ``"world"`` `;` ``"!"``]`` `;;`- : string = "Hello world!"`

```   