Program Extraction from Normalization Proofs

Abstract

This paper describes formalizations of Tait's normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.

A Context-based Approach to Proving Termination of Evaluation

“…Erasing the logical parts, we obtain a lambda term that only contains computationally relevant parts of the original proof, and it is this term that we call the "extracted" program-in our case, an evaluator, i.e., a program computing weak head normal forms of lambda terms. This is essentially what the modified realizability interpretation does to a proof term to extract its computational content [3,9].…”

“…The specification of the normalization problem and the proof of Theorem 2.5 can be formalized in a number of ways and its computational content can be extracted in the form of a lambda term that can be interpreted as an evaluator for the object language [3][4][5][6]9]. In this work, our interest lies not in completely formalizing the problem-it can easily be done, e.g., along the lines of the work cited above-but in showing another way of proving normalization using a context-based approach.…”

“…The syntactic representations of contexts we used in the proof can be optimized away (i.e., simply erased) since they do not play any role in the evaluator. This optimization is not arbitrary-it is provably correct and it corresponds to Berger's optimization to eliminate unused object variables, based on distinguishing between computationally relevant and irrelevant variables [3]. Without this optimization, the extracted evaluator would thread an additional argument-the context-which is never used.…”

“…Without this optimization, the extracted evaluator would thread an additional argument-the context-which is never used. 4 The function eval is the computational content of the proof of Lemma 2.4. In the proof of Theorem 2.5, we apply Lemma 2.4 with the empty sequence of values and with the empty context to obtain the proof of N ( t, • ).…”

“…Berger and Schwichtenberg identified the computational content of their constructive proof of strong normalization that uses the reducibility method to be an instance of normalization by evaluation, and subsequently this observation has been applied to proofs of weak head normalization by Coquand and Dybjer for combinatory logic [10] and by Biernacka et al for the lambda calculus [9]. Some of the proofs have been formalized in proof assistants and normalizers have been extracted from them in the form of functional programs [4,6]. Not surprisingly, the computational content of our proofs are instances of normalization by evaluation; the extracted programs are evaluators in continuation-passing style, whose continuations arise by extraction from a context reducibility predicate.…”