A coinductive framework for infinitary rewriting and equational reasoning

We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way. We define the relation ∞=, a notion of infinitary equational reasoning, and →∞, the standard notion of infinitary rewriting as follows: ∞=:= νR. (=R ≊ R)→∞ := μR. νS.(→R ≊ R)o S where μ and ν are the least and greatest fixed-point operators, respectively, and where R := {f(s₁, . . . , s_n), f(t₁, . . . , t_n) f∈∑ s₁ R t₁, . . . , s_nR t_n } ≊ Id . The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.