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.
In the @math -calculus, after addition of appropriate @math -rules, every finite term has its B "o hm tree @cite_4 as the normal form. In @math and @math , the normal forms are, respectively, Berarducci trees and Levy-Longo trees @cite_12 @cite_4 @cite_33 @cite_6 @cite_11 . With the addition of infinite @math - or @math -reductions it is possible to also capture, repsectively, @math -B "ohm or @math -B "o hm trees as normal forms @cite_0 @cite_28 .
@cite_4 @cite_33 @cite_28 @cite_6 @cite_0 @cite_12 @cite_11
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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. Despite @l-calculus is now three quarters of a century old, no formula counting @l-terms has been proposed yet, and the combinatorics of @l-calculus is considered a hard problem. The difficulty lies in the fact that the recursive expression of the numbers of terms of size n with at most m free variables contains the number of terms of size n-1 with at most m+1 variables. This leads to complex recurrences that cannot be handled by classical analytic methods. Here based on de Bruijn indices (another presentation of @l-calculus) we propose several results on counting untyped lambda terms, i.e., on telling how many terms belong to such or such class, according to the size of the terms and or to the number of free variables. We extend the results to normal forms. This paper describes two new bisimulation equivalences for the pure untyped call-by-value spl lambda -calculus, called enf bisimilarity and enf bisimilarity up to spl eta . They are based on eager reduction of terms to eager normal form (enf), analogously to co-inductive bisimulation characterizations of Levy-Longo tree equivalence and Bohm tree equivalence (up to spl eta ). We argue that enf bisimilarity is the call-by-value analogue of Levy-Longo tree equivalence. Enf bisimilarity (up to spl eta ) is the congruence on source terms induced by the call-by-value CPS transform and Bohm tree equivalence (up to spl eta ) on target terms. Enf bisimilarity and enf bisimilarity up to spl eta enjoy powerful bisimulation proof principles which, among other things, can be used to establish a retraction theorem for the call-by-value CPS transform. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive.