Further Operations on Cartesian Lists. We refer to Example. Define the following operations on Cartesian lists: (1) length of a list: The number of (zero, one or more) elements that it contains. (2) index set (inds) of a list: The set of indices, from 1 to and including the length of the list. If the list is empty then the index set is the empty set. (3) element set (elems) of a list: The set of distinct elements of the list. If the list is empty then the element set is the empty set.

Example 4.2 A Simple Language of Cartesian Lists: Consider and ) to be delimiters of list expressions, that is.(a,b,c) designates the list of elements a, b and c, and in that order: a being the first list element, b being the second and c being the third element. Now consider using just pairs of Cartesians to designate lists: (token,token) () ((a) ,token)-?a) ((a),((b),token)) - (a,b) ((a),((b),((c),token))) (a,b,c) That is, (token,token) designates the empty list, and (a), ) designates the list whose first element is a and whose tail is the Cartesian list l. Does this work? Well, only if the pairs obey, for example, this restricted syntax: ::= (token,token) | ( , ) A is any set (i.e., type) of, for example, atomic (non-Cartesian) values With this language of Cartesian lists, how do we express concatenation, (a,b,c)^ ?d,e)= ?a,b,c,d,e) ? ^: (token ,token) ^((a),1) ((a),1) , of two lists: Well, let us try: : ((a),e)(token,token) E ((a),t) Example 4.2 A Simple Language of Cartesian Lists: Consider and ) to be delimiters of list expressions, that is.(a,b,c) designates the list of elements a, b and c, and in that order: a being the first list element, b being the second and c being the third element. Now consider using just pairs of Cartesians to designate lists: (token,token) () ((a) ,token)-?a) ((a),((b),token)) - (a,b) ((a),((b),((c),token))) (a,b,c) That is, (token,token) designates the empty list, and (a), ) designates the list whose first element is a and whose tail is the Cartesian list l. Does this work? Well, only if the pairs obey, for example, this restricted syntax: ::= (token,token) | ( , ) A is any set (i.e., type) of, for example, atomic (non-Cartesian) values With this language of Cartesian lists, how do we express concatenation, (a,b,c)^ ?d,e)= ?a,b,c,d,e) ? ^: (token ,token) ^((a),1) ((a),1) , of two lists: Well, let us try: : ((a),e)(token,token) E ((a),t)