Orthogonality and Weak Convergence in Infinitary Rewriting. Stefan K ahrs. What you probably know. in orthogonal iTRSs weak/strong reduction fails to be confluent in the presence of collapsing rules (for metric ) is a counterexample
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for even k
for odd k
The latter are really minor points, except the infiniteness issue: with infinitely many rules we can make iTRSs behave non-continuously.
The second rule is not actually necessary, it is just used here for presentational purposes.
Think of as a list-cons.
Imagine we have an infinite list, containing the values A, S(A), S(S(A)), etc. (in that order)
Not only that – this relation is even confluent for our very first counterexample, with the collapsing rules!