Congruence Types

We introduce a type theoretical framework in which canon ical term rewriting systems can be represented faithfully both from the logical and the computational points of view The framework is based on congruence types a new syntax which combines inductive algebraic and quotient types Congruence types improve on existing work to combine type theories with algebraic rewriting by making explicit the fact that the term rewriting systems under consideration are initial models of an equational theory As a result the interaction gustavo thesisween the type theory and the algebraic types rewriting systems is much more power ful than in previous work Congruence types can be used i to introduce initial models of canonical term rewriting systems ii to obtain a suit able computational behavior of a de nable operation iii to provide an elegant solution to the problem of equational reasoning in type theory