On lists and other abstract data types in the calculus of constructions

The representation of the inductively defined abstract data type for lists was left incomplete in [2, §9]. Here that representation is completed, and it is proved that all extra axioms needed are consistent. Among the innovations of this paper is a definition of cdr, whose definition was left for future work in [2, §9]. The results are then extended to other abstract data types, those of [1]. The method used to define cdr for lists is extended to obtain the definition of an inverse for each argument of each constructor of an abstract data type. These inverses are used to prove the injective property for the constructors. Also, Dedekind’s method of defining the natural numbers is used to define a predicate associated with each abstract data type, and the use of this predicate makes it unnecessary to postulate the induction principle. The only axioms left to be proved are those asserting the disjointness of the co-domains of different constructors, and it is shown that those axioms can be proved consistent.