In the context of a general approach of studying properties of rewriting under strategies, we focus in this paper on the property of completeness of function definitions. We propose a procedure proving sufficient completeness of rewriting in the specific case of the innermost strategy, under the only assumption that the rewriting system innermost terminates on ground terms. It relies on an inductive proof that every ground term rewrites into a constructor term. The innermost rewriting relation on ground terms is simulated through an abstraction mechanism and narrowing. The inductive hypothesis allows assuming that terms smaller than the starting terms for an induction ordering rewrite into a constructor term. The existence of the induction ordering is checked during the proof process, by ensuring satisfiability of ordering constraints. An extension of this procedure is also proposed, to establish in the same time sufficient completeness and the needed innermost termination property.

We propose a transformation based method for proving termination of ELAN strategies. We first give a sufficient criterion for ELAN strategies to terminate, only lying on rewrite rules involved in the strategy. We then give a simplification process of strategies, itself described by rewriting, to empower the previous criterion. This simplification, beyond easing termination proof of strategies, can both facilitate elaboration of specifications and ease proofs of other program properties.

From an inductive method for proving weak innermost termination of rule-based programs, we automatically infer, for each successful proof, a finite strategy for data evaluation. The proof principle uses an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we analyse proof trees built from the rewrite system, schematizing the innermost derivations of any ground term. These proof trees are issued from patterns representing sets of ground terms. They are generated using two mechanisms, namely abstraction, introducing variables that represent ground terms in normal form, and narrowing, schematizing rewriting on ground terms. From the proof trees, we can extract for any given ground term, a rewriting strategy to compute one of its normal form, even if the ground term admits infinite rewriting derivations.

From an inductive method for proving weak innermost termination of rule-based programs, we automatically infer, for each successful proof, a finite strategy for data evaluation. The proof principle uses an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we analyse proof trees built from the rewrite system, schematizing the innermost derivations of any ground term. These proof trees are issued from patterns representing sets of ground terms. They are generated using two mechanisms, namely abstraction, introducing variables that represent ground terms in normal form, and narrowing, schematizing rewriting on ground terms. From the proof trees, we can extract for any given ground term, a rewriting strategy to compute one of its normal form, even if the ground term admits infinite rewriting derivations.

We propose an original approach to prove weak termination of innermost rewriting on ground term algebras, based on induction, abstraction and narrowing. The abstraction mechanism schematizes the application of the induction hypothesis on terms, and narrowing schematizes reductions. The induction relation, an F-stable ordering having the subterm property, is not given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints. An extension of the method is given, where the information given by abstraction are memorized and used. Our method is more specific than methods proving strong termination of rewriting under strategies, which also give a weak termination proof for the standard rewriting. Indeed, ours can in addition handle TRSs that are not strongly innermost terminating.

We propose a semi-automatic inductive process to prove termination of outermost rewriting on ground term algebras. The method is based on an abstraction mechanism, schematizing normalisation of subterms, and on narrowing, schematizing reductions on ground terms. The induction ordering is not needed a priori, but is a solution of constraints set along the proof. Our method applies in particular to systems that are non-terminating for the standard strategy nor even for the lazy strategy.

We describe Cariboo, the implementation of an inductive process recently proposed to prove termination of rewriting under strategies on ground term algebras. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. It applies in particular to non-terminating systems which are terminating with innermost or local strategies. The narrowing process, well known to easily diverge, is controlled by using appropriate abstraction constraints. The proof mechanism lies on abstraction and ordering constraints satisfiability problems. Thanks to the power of induction, ordering constraints are often simple and automatically solved by our system. Otherwise, they can be treated by the user or any external automatic solver. On many examples, Cariboo even enables to succeed without considering any constraint at all; the process is then completely automatic. Cariboo offers a graphical view of the proof process. It is implemented in ELAN, a rule based programming environment, and so can be used for proving termination of ELAN programs.

We propose an original approach to prove termination of innermost rewriting on ground term algebras, based on induction, abstraction and narrowing. Our method applies in particular to non-terminating systems which are innermost terminating, and to systems that do not innermost terminate on the free term algebra but do on the ground term one. The induction relation, an F-stable ordering having the subterm property, is not given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. An extension of the method is given, where the noetherian induction is strengthened by a structural induction. A variant is also proposed, to characterize terminating subset of the ground term algebra, for non-innermost terminating system. Finally, the method is adapted in a natural way to outermost termination.

In this paper, we propose a method for specifically proving termination of rewriting with particular strategies: local strategies on operators. An inductive proof procedure is proposed, based on an explicit induction on the termination property. Given a term, the proof principle relies on alternatively applying the induction hypothesis on its subterms, by abstracting the subterms with induction variables, and narrowing the obtained terms in one step, according to the strategy. The induction relation, an F-stable ordering having the subterm property, is not given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints.

We propose a new approach to prove termination of innermost rewriting, based on induction, abstraction and narrowing. The induction ordering is not explicitly given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints. A rule-based description of the technique is given, as well as a few examples to illustrate the method.