Termination Proofs for Higher-order Rewrite Systems

  title={Termination Proofs for Higher-order Rewrite Systems},
  author={Jaco van de Pol},
This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [12]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique “Termination by interpretation in a well-founded monotone algebra”, described in [8, 19]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in… 

Polymorphic higher-order recursive path orderings

A family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols is defined, which can be generated from two given well-founded orderings, on the function symbols and on the type constructors.

Polynomial Interpretations for Higher-Order Rewriting

This work adapt and extend the termination method of weakly monotonic algebras to the alternative formalism of algebraic functional systems, where the simply-typed lambda-calculus is combined with algebraic reduction.

Size-based termination: Semantics and generalizations

A structured approach to proving the correctness of size-based termination and a modification of the classical size-types approach allows us to perform a fine control-flow analysis in a higher-order language.

Higher-Order Proof by Consistency

An integration of the first-order method of proof by consistency (PBC), also known as term rewriting induction, into theorem proving in higher-order specifications is investigated, which yields a proof procedure which has several advantages over conventional induction.

Strict Functionals for Termination Proofs

A semantical method to prove termination of higher order rewrite systems (HRS) is presented, which makes it possible to transfer intuitions about why an HRS should be terminating into a proof.

Computability Closure: Ten Years Later

  • F. Blanqui
  • Computer Science
    Rewriting, Computation and Proof
  • 2007
It is shown how the computability closure can easily be turned into a reduction ordering which contains Jean-Pierre Jouannaud and Albert Rubio's higher-order recursive path ordering and, in the firstorder case, is equal to the usual first-order recursion path ordering.

Termination of Curryfied Rewrite Systems

This paper studies termination of curryfied term rewriting systems (CTRSs), where functional values are introduced by “partial application” The limitations of syntactic simplification orderings for

Modular Termination Checking Theorems for Second-Order Computation

New theorems of modular termination checking for second-order computation are presented, useful for proving termination of higher-order programs and foundational calculi and offering a decomposition technique for difficult termination problems.

Proving termination of normalization functions for conditional expressions

The recursion relation approach seems flexible enough to handle subtle termination proofs where previously domain theory seemed essential, and an obviously total variant of the normalize function is presented as the ‘computational meaning’ of those two proofs.

Semantic Labelling for Proving Termination of Combinatory Reduction Systems

A novel transformation method for proving termination of higher-order rewrite rules in Klop's format called Combinatory Reduction System (CRS), an extension of a method known in the theory of term rewriting.



A computation model for executable higher-order algebraic specification languages

  • J. JouannaudM. Okada
  • Computer Science, Mathematics
    [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
  • 1991
A general modularity result, which allows as particular cases primitive recursive functionals of higher types, transfinite recursion of highertypes, and inheritance for all types, is proved.

The Clausal Theory of Types

This book introduces just such a theory, based on a lambda-calculus formulation of a clausal logic with equality, known as the Clausal Theory of Types, which is a concise form of logic programming that incorporates functional programming.

Comparing Combinatory Reduction Systems and Higher-order Rewrite Systems

It is concluded that as far as rewrite theory is concerned, Combinatory Reduction Systems and Higher-Order Rewrite Systems are equivalent, the only difference being that Combinatories Reduction Systems employ a more ‘lazy’ evaluation strategy.

Higher-order critical pairs

  • T. Nipkow
  • Computer Science
    [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
  • 1991
The notion of critical pair is generalized to higher-order rewrite systems, and the analog of the critical pair lemma is proved.

Orthogonal Higher-Order Rewrite Systems are Confluent

The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more

Equations and rewrite rules: a survey

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Termination of Term Rewriting by Interpretation

A transformation on term rewriting systems eliminating distributive rules is introduced and a new termination proof of SUBST from [10] is given.

Isabelle: The Next 700 Theorem Provers

A thorough history of Isabelle is given, beginning with its origins in the LCF system, and an account of how logics are represented is presented, illustrated using classical logic.

The lambda calculus - its syntax and semantics

  • H. Barendregt
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1985