Term rewriting has been shown to be a good environment for both programming and proving. For analysing and debugging rule-based programs, we propose in this work a formalism based on the rewriting calculus with explicit substitutions ( -calculus). This formalism also allows us to build the proof terms of rewriting derivations. Therefore, term rewriting proofs can be exported to other systems by translating them into the corresponding syntaxes. That is, using a proof checker, one can certify these proofs and vice versa, this method allows us to get term rewriting in proof assistants using an external system. Our method not only works with syntactic rewriting but also with rewriting modulo a set of axioms (e.g. associativity-commutativity).

We introduce a polymorphic #-calculus that features inductive types and that enforces termination of recursive definitions through typing. Then, we define a sound and complete type inference algorithm that computes a set of constraints to be satisfied for terms to be typable.

In proof systems like Coq [15], proof-checking involves comparing types modulo #-conversion, which is potentially a time-consuming task. Significant speed-ups are achieved by compiling proof terms, see [8].

We present a formalization of a constructive proof of weak normalization for the simply-typed λ-calculus in the theorem prover Isabelle/HOL, and show how a program can be extracted from it. Unlike many other proofs of weak normalization based on Tait’s strong computability predicates, which require a logic supporting strong eliminations and can give rise to dependent types in the extracted program, our formalization requires only relatively simple proof principles. Thus, the program obtained from this proof is typable in simply-typed higher-order logic as implemented in Isabelle/HOL, and a proof of its correctness can automatically be derived within the system.

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Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of formalization (encoding proofs in the computer) in general. Keywords: formalization of mathematics, Mizar, social choice theory, Arrow’s theorem, Gibbard-Satterthwaite theorem, proof errors.