A Coherence Theorem for Martin-ll Of's Type Theory

In type theory a proposition is represented by a type, the type of its proofs. As a consequence the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality, which we also call identity, is inductively deened as the \smallest reeexive relation". It is sometimes convenient to know that the type representing a proposition is collapsed in the sense that all its inhabitants are identical. Although uniqueness of identity proofs for an arbitrary type is not derivable inside type theory, there is a large class of types for which it may be proved. Our main result is a proof that any type with decidable identity has unique identity proofs. This result is convenient for proving that the class of types with decidable identities is closed under indexed sum. Our proof of the main result is completely formalized within a kernel fragment of Martin-LL of's type theory and mechanized using ALF. Proofs of auxiliary lemmas are explained in terms of the category theoretical properties of identity. These suggest two coherence theorems as the result of rephrasing the main result in a context of conventional equality, where the inductive equality has been replaced by, in the rst, an initial category structure and, in the second, a smallest reeexive relation. 1 Introduction 1.1 Type theory. Martin-LL of's type theory 13, 14] may be seen as a framework for expressing constructive mathematical reasoning. The interactive proof assistant ALF 2, 1, 12] can be used to implement a formalization of type theory and to develop formal type theory proofs. The fundamental ingredients of type theory are those of set, element of a set, family of sets and dependently typed 1 function. 1 The type of the result of a function application may depend on the values given as arguments to the function, and in an m-place function f(x 1 ; : : : ; xm) or family of sets A(x 1 ; : : : ; xm), the range of the variable x i may depend on the values assigned to the preceding variables x 1 , .