Size Relationships in Abstract Cyclic Entailment Systems

A cyclic proof system generalises the standard notion of a proof as a finite tree of locally sound inferences by allowing proof objects to be potentially infinite. Regular infinite proofs can be finitely represented as graphs. To preclude spurious cyclic reasoning, cyclic proof systems come equipped with a well-founded notion of 'size' for the models that interpret their logical statements. A global soundness condition on proof objects, stated in terms of this notion of size, ensures that any non-well-founded paths in the proof object can be disregarded. We give an abstract definition of a subclass of such cyclic proof systems: cyclic entailment systems. In this setting, we consider the problem of comparing the size of a model when interpreted in relation to the antecedent of an entailment, with that when interpreted in relation to the consequent. Specifically, we give a further condition on proof objects which ensures that models of a given entailment are always 'smaller' when interpreted with respect to the consequent than when interpreted with respect to the antecedent. Knowledge of such relationships is useful in a program verification setting