This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associative-commutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a speci c and ecient environment and to check the computations later in a proof assistant.

We modify the reection method to enable it to deal with partial functions like division. The idea behind reflection is to program a tactic for a theorem prover not in the implementation language but in the object language of the theorem prover itself. The main ingredients of the reflection method are a syntactic encoding of a class of problems, an interpretation function (mapping the encoding to the problem) and a decision function, written on the encodings. Together with a correctness proof of the decision function, this gives a fast method for solving problems. The contribution of this work lies in the extension of the reflection method to deal with equations in algebraic structures where some functions may be partial. The primary example here is the theory of fields. For the reflection method, this yields the problem that the interpretation function is not total. In this paper we show how this can be overcome by defining the interpretation as a relation. We give the precise details, both in mathematical terms and in Coq syntax. It has been used to program our own tactic `Rational', for verifying equations between field elements.

In informal mathematics, statements involving computations are seldom proved. Instead, it is assumed that readers of the proof can carry out the computations on their own. However, when using an automated proof development system based on type theory, the user is forced to nd proofs for all claimed propositions, including computational statements. This paper presents a method to automatically prove statements from primitive recursive arithmetic. The method replaces logical formulas by boolean expressions. A correctness proof is constructed, which states that the original formula is derivable, if and only if the boolean expression equals true. Because the boolean expression reduces to true, the conversion rule yields a trivial proof of the equality. By combining this proof with the correctness proof, we get a proof for the original statement. 1 Introduction This paper presents a method to automatically prove statements from rst order primitive recursive arithmetic, in the context o...

This paper presents several technical results concerning the operational semantics of the reflective functional programming language reFLect, currently being developed at Intel [7]. The design of reFLect...