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### Completion for rewriting modulo a congruence [chapter]

Leo Bachmair, Nachum Dershowitz
1987 Lecture Notes in Computer Science
We present completion methods for rewriting modulo a congruence, generalizing previous methods by Peterson and Stickel (1981) and Jouannaud and Kirchner (1986) .  ...  We formalize our methods as equational inference systems and describe techniques for reasoning about such systems.  ...  Various methods have been proposed for rewriting modulo a congruence.  ...

### Completion for rewriting modulo a congruence

Leo Bachmair, Nachum Dershowitz
1989 Theoretical Computer Science
Completion modulo a congruence is a method for constructing a presentation of an equational theory as a rewrite system that defines unique normal forms with respect to the congruence.  ...  We formulate this completion method as an equational inference system and present techniques for proving the correctness of procedures based on the inference system.  ...  Completion for rewriting modulo a congruence was first shown to be practically feasible by Peterson and Stickel  , for associative-commutative congruences.  ...

### Automated Theorem Proving in First-Order Logic Modulo: On the Difference between Type Theory and Set Theory [chapter]

Gilles Dowek
2000 Lecture Notes in Computer Science
Resolution modulo is a first-order theorem proving method that can be applied both to first-order presentations of simple type theory (also called higher-order logic) and to set theory.  ...  Indeed, the reduction process is deterministic because the rewrite system is confluent. Cut elimination and completeness Resolution modulo is not complete for all congruences.  ...  Completeness Another difference is that, as type theory verifies the cut elimination property, resolution modulo this congruence is complete, while it is incomplete modulo the congruence of set theory.  ...

### Order-Sorted Rewriting and Congruence Closure [chapter]

José Meseguer
2016 Lecture Notes in Computer Science
The satisfiability problems for the theories of: (i) order-sorted uninterpreted function symbols, and (ii) of such symbols modulo a subset Δ of associative-commutative ones are reduced to the unsorted  ...  New results on order-sorted rewriting are needed to achieve this reduction.  ...  I thank Maria Paola Bonacina for suggested improvements.  ...

### Deduction modulo theory [article]

Gilles Dowek
2015 arXiv   pre-print
This paper is a survey on Deduction modulo theory  ...  A congruence ≡ is said to be valid in a model when A ≡ B implies A φ = B φ for all valuations φ, and a soundness and completeness theorem can be proved using standard methods.  ...  is performed modulo this congruence.  ...

### Semantic A-translations and Super-Consistency Entail Classical Cut Elimination [chapter]

Lisa Allali, Olivier Hermant
2013 Lecture Notes in Computer Science
Deduction Modulo is a formalism that aims at separating computation from reasoning in proofs by making inferences modulo some congruence.  ...  We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question.  ...  A rewrite system R (a congruence ≡) in deduction modulo is super-consistent if it has aB-valued model for all full, ordered and complete pseudo-Heyting algebraB.  ...

### Semantic A-translation and Super-consistency entail Classical Cut Elimination [article]

Lisa Allali
2014 arXiv   pre-print
We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question.  ...  For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R.  ...  A rewrite system R (a congruence ≡) in deduction modulo is super-consistent if it has aB-valued model for all full, ordered and complete pseudo-Heyting algebraB.  ...

### Schematization of infinite sets of rewrite rules generated by divergent completion processes

Hélène Kirchner
1989 Theoretical Computer Science
Infinite sets of rewrite rules may be generated for example by completion of termrewriting systems or by a narrowing process for solving equations in equational theories.  ...  This is a severe limitation to the practical use of these processes. We propose in this paper a notion of schematization for an infinite set of rewrite rules.  ...  Remy and to the referees for their valuable comments. I am grateful to M. Schmidt-Schauss for pointing out a problem with the associativity and idempotency axioms in a previous version of this paper.  ...

### Proof Search and Proof Check for Equational and Inductive Theorems [chapter]

Eric Deplagne, Claude Kirchner, Hélène Kirchner, Quang Huy Nguyen
2003 Lecture Notes in Computer Science
Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.  ...  In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs.  ...  Many thanks to the Protheo team for stimulating discussions on many subjects developed in this paper, and in particular to Jürgen Stuber.  ...

### Narrowing Based Inductive Proof Search [chapter]

Claude Kirchner, Hélène Kirchner, Fabrice Nahon
2013 Lecture Notes in Computer Science
We present in this paper a narrowing-based proof search method for inductive theorems.  ...  It has the specificity to be grounded on deduction modulo and to yield a direct translation from a successful proof search derivation to a proof in the sequent calculus.  ...  The reader can refer to  and to  for a detailed exposition. In deduction modulo, terms but also propositions can be identified modulo a congruence.  ...

### Deduction versus Computation: The Case of Induction [chapter]

Eric Deplagne, Claude Kirchner
2002 Lecture Notes in Computer Science
We extend slightly the original version of the deduction modulo framework and we provide modularity properties for it.  ...  We show how this applies to a uniform understanding of the so called induction by rewriting method and how this relates directly to the general use of an induction principle.  ...  It is thus not surprising to have our framework based on deduction modulo. This presentation of first-order logic relies on the sequent calculus modulo a congruence defined on terms and propositions.  ...

### Experimenting with Deduction Modulo [chapter]

Guillaume Burel
2011 Lecture Notes in Computer Science
For the integration of rewriting, we also compare several implementation techniques, based for instance on discrimination trees or on compilation.  ...  Deduction modulo is a generic framework to describe proofs in a theory better than using raw axioms. This is done by presenting the theory through rules rewriting terms and propositions.  ...  It consists in presenting a theory as a congruence over propositions, and in applying the inference rules of deductive systems modulo this congruence.  ...

### Page 6630 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews
process of rewriting modulo R, the word problem for ({a, b}; Ro) can be solved in time O(n?)  ...  Jantzen [“Thue congruences and complete string-rewriting systems”, Habilitationsschrift, Univ. Hamburg, Hamburg, 1986; per bibl.; see also Inform. Process.  ...

### Cut Admissibility by Saturation [chapter]

Guillaume Burel
2014 Lecture Notes in Computer Science
Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules.  ...  This work relies on a view of proposition rewriting rules as oriented clauses, like term rewriting rules can be seen as oriented equations.  ...  In Deduction Modulo 1 , a theory is represented by a congruence over formulae, and proofs are searched for modulo this congruence.  ...

### The problems of cyclic equality and conjugacy for finite complete rewriting systems

Paliath Narendran, Friedrich Otto
1986 Theoretical Computer Science
Thus, for a finite rewriting system, while the property of being complete is sufficient to guarantee the decidability of the word problem, it is not sufficient for the decidability of the conjugacy problem  ...  This does not hold for the problem of cyclic equality as is shown here by presenting a finite, length-reducing, and complete rewriting system with an undecidable problem of cyclic equality.  ...  CYCLIC Cyclic equality is undecidable for finite length-reducing complete rewriting systems Here we are going to present a finite length-reducing complete rewriting system T1 with the property that the  ...
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