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.
Confluence and normalisation results in infinitary rewriting and infinitary lambda calculus have been generalised to the framework of infinitary combinatory reduction systems @cite_1 @cite_13 @cite_27 .
@cite_13 @cite_27 @cite_1
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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 study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs. These facts properly generalise a number of results on normalising strategies in first-order infinitary rewriting and provide the first examples of normalising strategies for infinitary lambda-calculus. 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 define infinitary Combinatory Reduction Systems (iCRSs), thus providing the first notion of infinitary higher-order rewriting. The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary @l-calculus are special cases. Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary @l-calculus to iCRSs. In particular, for fully-extended, left-linear iCRSs we prove the well-known compression property, and for orthogonal iCRSs we prove that (1) if a set of redexes U has a complete development, then all complete developments of U end in the same term and that (2) any tiling diagram involving strongly convergent reductions S and T can be completed iff at least one of S T and T S is strongly convergent. We also prove an ancillary result of independent interest: a set of redexes in an orthogonal iCRS has a complete development iff the set has the so-called finite jumps property.