Modularity of Convergence in Infinitary Rewriting

Modularity of Convergence and Strong Convergence in Infinitary Rewriting

“…This slightly extends the result from [10] as the proof here also covers generalised iTRSs with extra variables. Another consequence of Propositions 9.9 and 4.6 is that the question whether such a sequence f is converging under metric d m reduces to the question whether its limit t under d ∞ is a term in Ter m (Σ), and by Proposition 4.8 this is equivalent to whether all its infinite paths converge.…”

“…Proof. Proposition 4 in [10] claimed this for MARS, but the proof works unchanged for TARS, with no constraints on the topology. Proposition 3.8.…”

“…The argument of the proof is essentially the same as in [10], but reworded to avoid reference to distances. In [10] it was said that it was "not clear" if the proposition could be generalised to non-Hausdorff spaces.…”

“…Before [10] there were few positive results about convergence in the literature, instead research has focussed on strong convergence which in addition requires that redex positions in a reduction sequence cannot stay short: if we replace each contracted redex with a random term we would still have a converging sequence.…”

“…Simonsen [19] proved that strong convergence is a modular property for left-linear iTRSs (for metric d ∞ ); the left-linearity condition is indeed unnecessary [10]. Proving such results for ordinary convergence is more complicated.…”