On a Generalised Logicality Theorem

Structures for Abstract Rewriting

“…Let for given terms t 1 , t 2 and t 3 (which is then the only instance of Down ¡ ). We easily show that all the appropriate instances of symmetry, context and substitution rules semi-commute with this instance (see [3]). Moreover, we have:…”

“…defined by a combination of good proofs and proofs to simplify) can be identified with a good proof 2 2 In the equational rewriting setting, this inclusion is called the provided that proofs to simplify are identified with good proofs 3 , and conversely. We will name this result Church-Rosser's result 4 .…”

“…As a result of the rewriting abstraction defined in the paper, all the results as well as the decision procedure and completion method established here are de facto generalizations of standard ones we find in different rewriting theories. 3 In the equational rewriting setting, this inclusion is called the confluence property and is expressed as follows: Actually, all rewriting theories that we know satisfy the axioms given in this paper, and thus enter our framework (see the numerous examples developed in the paper).…”

“…Classically, completeness is obtained by defining basic proof tree transformations (see the example of such a transformation just below). In [1][2][3]12], we have studied how to define the three sets RS, De and Rmv from a generalization of these transformations that we have called semi-commutation. Let us recall briefly this work in this section.…”

“…These conditions are based on basic properties of elementary combinations of inference rules which assure that the induced "global" proof tree transformation processes do terminate. We refer the reader to terms of this generalized version of the logicality theorem as well as its proof in [3,4]. Here, we suppose that convertibility coincides with deductions, that is:…”