COINDUCTIVE FOUNDATIONS OF INFINITARY REWRITING AND INFINITARY EQUATIONAL LOGIC

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define 2, the infinitary extension of a given equational theory =(R), and ->(infinity) , the standard notion of infinitary rewriting associated to a reduction relation ->(R), as follows: (sic) := vR. (=(R) boolean OR (R) over bar* ->(infinity) := mu R.vS. (->(R) boolean OR (R) over bar* ; (S) over bar Here mu and v are the least and greatest fixed-point operators, respectively, and (R) over bar :(-) { < f(s(1) , ... , s(n)), f(t(1) , ... , t(n))> broken vertical bar f epsilon Sigma, s(1) R t(1) , ... , s(n) R t(n) } boolean OR 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.