_A_l_g_e_b_r_a_i_c_ _t_y_p_e_ _d_e_f_i_n_i_t_i_o_n_s

The simplest method of introducing a new data type into a Miranda script
is  by  means of an algebraic type definition.  This enables the user to
introduce a new concrete  data  type  with  specified  constructors.   A
simple example would be
	tree ::= Nilt | Node num tree tree

The `::=' sign is used  to  introduce  an  algebraic  data  type.   This
definition introduces three new identifiers
    `tree' a typename
    `Nilt' a nullary constructor (i.e. an atom), of type tree
    `Node' a constructor, of type num->tree->tree->tree

Now we can define trees using constructors Nilt & Node, for example
	t = Node 3 Nilt Nilt

It is not necessary to have names for  selector  functions  because  the
constructors  can  be  used in pattern matching.  For example a function
for counting the number of nodes in a tree could be written
	size Nilt = 0
	size (Node a x y) = 1 + size x + size y

Note that the names of constructors _m_u_s_t_ _b_e_g_i_n_ _w_i_t_h_ _a_n_ _u_p_p_e_r_ _c_a_s_e_ _l_e_t_t_e_r
(and  conversely,  any identifier beginning with an upper case letter is
assumed to be a constructor).

An algebraic type can have any number (>=1)  of  constructors  and  each
constructor  can  have any number (>=0) fields, of specified types.  The
number of fields taken by  a  constructor  is  called  its  `arity'.   A
constructor  of  arity zero is said to be atomic.  Algebraic types are a
very general idea and include a number of special cases  that  in  other
languages require separate constructions.

One  interesting case that all of the constructors can be atomic, giving
us what is called in PASCAL a `scalar enumeration type'.  Example
	day ::= Mon|Tue|Wed|Thu|Fri|Sat|Sun

The union of two types can also be represented as an algebraic data type
- for example here is a union of num and bool.
	boolnum ::= Left bool | Right num

Notice  that  this  is  a `labelled union type' (the other kind of union
type, in which the parts of the union are not distinguished  by  tagging
information, is not permitted in Miranda).

An  algebraic typename can take parameters, thus introducing a family of
types.   This  is  done  be  using  generic  type  variables  as  formal
parameters  of the `::=' definition.  To modify our definition of `tree'
to allow trees with different types of labels at the nodes  (instead  of
all `num' as above) we would write
	tree * ::= Nilt | Node * (tree *) (tree *)

Now  we  have  many  different  tree  types  -  `tree num', `tree bool',
`tree([char]->[char])', and so on.  The constructors `Node'  and  `Nilt'
are  both  polymorphic, with types `tree *' and `*->tree *->tree *->tree
*' respectively.

Notice that in Miranda objects of a recursive user defined type are  not
restricted  to  being  finite.   For example we can define the following
infinite tree of type `tree num'
	bigtree = Node 1 bigtree bigtree

_C_o_n_t_r_o_l_l_i_n_g_ _t_h_e_ _s_t_r_i_c_t_n_e_s_s_ _o_f_ _c_o_n_s_t_r_u_c_t_o_r_s

Definition - a function f is strict iff
	f _| = _|
where _| is the value attributed to expressions which fail  to  terminate
or  terminate with an error. To support non-strict functions the calling
mechanism must not evaluate the arguments before  passing  them  to  the
function - this is what is meant by "lazy evaluation".

In Miranda constructors are, by default, non-strict in all their fields.
Example

	pair ::= PAIR num num
	fst (PAIR a b) = a
	snd (PAIR a b) = b

First note that there is a predefined identifier "undef"  which  denotes
undefined  -  evaluating  "undef"  in  a  Miranda session gives an error
message. Consider the following Miranda expressions:

	fst (PAIR 1 undef)
	snd (PAIR undef 1)

Both evaluate to `1', that is `PAIR' is non-strict in both arguments.

The primary reason for making constructors non-strict in Miranda is that
it is necessary to support equational reasoning on Miranda scripts.  (In
the example given, elementary equational reasoning from  the  definition
of  `fst'  implies  that  `fst(PAIR  1 anything)' should always have the
value `1'.) It is also as a consequence of constructors being non-strict
that Miranda scripts are able to define infinite data structures.

It is, however, possible to specify  that  a  given  constructor  of  an
algebraic data type is strict in one or more fields by putting `!' after
the field in the `::=' definition of  the  type.   For  example  we  can
change the above script to make PAIR strict in both fields, thus

	pair ::= PAIR num! num!
	fst (PAIR a b) = a
	snd (PAIR a b) = b

Now `fst (PAIR 1 undef)' and `snd  (PAIR  undef  1)'  both  evaluate  to
undefined.   It  is a consequence of the `!' annotations that `PAIR a b'
is undefined when either a or b is undefined.  It is  also  possible  to
make PAIR strict in just one of its fields by having only one `!' in the
type definition.

In the case  of  a  recursively  defined  algebraic  type,  if  all  the
constructors  having recursive fields are made strict in those fields it
ceases to be possible to construct infinite objects of that type.  It is
also  possible  to  deny  the possibility of certain infinite structures
while permitting others.  For example if we modify the definition of the
tree type first given above as follows
	tree ::= Nilt | Node num tree! tree

then it is still possible to construct trees which are infinite in their
right branch but not "left-infinite" ones.

The main reason for  allowing  `!'  annotations  on  Miranda  data  type
definitions is that one of the intended uses of Miranda is as a SEMANTIC
METALANGUAGE, in which to express the denotational  semantics  of  other
programming languages.