We present a new and formal coinductive proof of confluence and normalisation of B "ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result. The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.
Infinitary lambda calculus was introduced in @cite_12 @cite_4 . Meaningless terms were defined in @cite_22 . The confluence and normalisation results of this paper were already obtained in @cite_12 @cite_22 , by a different proof method. See also @cite_39 @cite_36 @cite_18 for an overview of various results in infinitary lambda calculus and infinitary rewriting.
@cite_18 @cite_4 @cite_22 @cite_36 @cite_39 @cite_12
1970779642 1976336764 2403774400 1968168625
We present an introduction to infinitary lambda calculus, highlighting its main properties. Subsequently we give three applications of infinitary lambda calculus. The first addresses the non-definability of Surjective Pairing, which was shown by the first author not to be definable in lambda calculus. We show how this result follows easily as an application of Berry's Sequentiality Theorem, which itself can be proved in the setting of infinitary lambda calculus. The second pertains to the notion of relative recursiveness of number-theoretic functions. The third application concerns an explanation of counterexamples to confluence of lambda calculus extended with non-left-linear reduction rules: Adding non-left-linear reduction rules such as @dxx->x or the reduction rules for Surjective Pairing to the lambda calculus yields non-confluence, as proved by the second author. We discuss how an extension to the infinitary lambda calculus, where Bohm trees can be directly manipulated as infinite terms, yields a more simple and intuitive explanation of the correctness of these Church-Rosser counterexamples. We show the existence of an infinitary confluen t and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom. As corollaries we obtain a simple, syntax based construction of an extensional Bohm model of the finite lambda calculus; and a simple, syntax based proof that two lambda terms have the same semantics in this model if and only if theyha ve the same eta-Bohm tree if and only if they are observationally equivalent wrt to beta normal forms. The confluence proof reduces confluence of beta, bottom and eta via infinitary commutation and postponement arguments to confluence of beta and bottom and confluence of eta.We give counterexamples against confluence of similar extensions based on the identification of the terms without weak head normal form and the terms without top normal form (rootactive terms) respectively. We provide a coinductive definition of strongly convergent reductions between infinite lambda terms. This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept. As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types. The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step. In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Bohm model of the lambda calculus can be seen as an infinitary term model. In contrast to term rewriting, there are several different possible notions of infinite term, which give rise to different Bohm-like models, which embody different notions of lazy or eager computation.