Section: New Results
In  , Frédéric Blanqui showed how to extend the notion of reducibility introduced by Girard for proving the termination of β-reduction in the polymorphic λ-calculus, to prove the termination of various kinds of rewrite relations on λ-terms, including rewriting modulo some equational theory and rewriting with matching modulo βη, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems.
In  , Frédéric Blanqui, together with Jean-Pierre Jouannaud and Albert Rubio, introduced the computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols. A first version, core CPO, is essentially obtained from the higher-order recursive path ordering (HORPO) by eliminating type checks from some recursive calls and by incorporating the treatment of bound variables as in the so-called computability closure. The well-foundedness proof shows that core CPO captures the essence of computability arguments à la Tait and Girard, therefore explaining its name. We further show that no more type check can be eliminated from its recursive calls without loosing well-foundedness, but one for which we found no counterexample yet. Two extensions of core CPO are then introduced which allow one to consider: the first, higher-order inductive types; the second, a precedence in which some function symbols are smaller than application and abstraction.
Another extension of CPO, to dependently typed terms, has been developed by Jean-Pierre Jouannaud and Jianqi Li in  .
Jean-Pierre Jouannaud and Albert Rubio showed in  how to modify recursive path orders for higher-order terms which, like CPO, include -reductions, into orders that are compatible with -conversion. The result is a powerful order for proving termination of higher-order rewrite rules based on higher-order pattern matching.
Gaëtan Gilbert and Olivier Hermant have introduced a constructive way to perform proof normalization through completeness proofs  .