# Equational inference, canonical proofs, and proof orderings

@article{Bachmair1994EquationalIC, title={Equational inference, canonical proofs, and proof orderings}, author={Leo Bachmair and Nachum Dershowitz}, journal={J. ACM}, year={1994}, volume={41}, pages={236-276} }

We describe the application of proof orderings—a technique for reasoning about inference systems-to various rewrite-based theorem-proving methods, including refinements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion (a refutationally complete extension of standard completion); and a proof by consistency procedure for proving inductive theorems.

## 97 Citations

Proof Transformation for Non-Compatible Rewriting

- Computer ScienceAISMC
- 1996

The completeness of this inference system is shown by proof transformation techniques using a very powerful new proof ordering that can handle semi-compatible reduction relations and explains all known complete redundancy criteria in a uniform framework.

Abstract canonical inference

- Mathematics, Computer ScienceTOCL
- 2007

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation,…

A New and Formalized Proof of Abstract Completion

- Computer ScienceITP
- 2014

A new proof of the correctness of abstract completion that is based on peak decreasingness, a special case of decreasing diagrams, which replaces Newman’s Lemma and allows us to avoid proof orders in the correctness proof of completion.

Paramodulation, Superposition, and Simplification

- MathematicsKurt Gödel Colloquium
- 1997

Equational reasoning techniques are a key component in many automated theorem provers and interactive proof and verification systems and have been a notable recent success in equational theorem proving.

Abstract Completion, Formalized

- Computer Science, MathematicsLog. Methods Comput. Sci.
- 2019

This paper presents new correctness proofs of abstract completion, both for finite and infinite runs, and extends the results to ordered completion, an important extension of completion that aims to produce ground-complete presentations of the initial equations.

Completion Is an Instance of Abstract Canonical System Inference

- Computer ScienceEssays Dedicated to Joseph A. Goguen
- 2006

Abstract canonical systems and inference (ACSI) were introduced to formalize the intuitive notions of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like…

A Taste of Rewrite Systems

- Computer ScienceFunctional Programming, Concurrency, Simulation and Automated Reasoning
- 1993

This survey of the theory and applications of rewriting with equations discusses the existence and uniqueness of normal forms, the KnuthBendix completion procedure and its variations, as well as…

Efficient Encodings of First-Order Horn Formulas in Equational Logic

- Computer ScienceIJCAR
- 2018

We present several translations from first-order Horn formulas to equational logic. The goal of these translations is to allow equational theorem provers to efficiently reason about non-equational…

## References

SHOWING 1-10 OF 149 REFERENCES

Proof methods for equational theories

- Computer Science, Mathematics
- 1987

An extension of standard completion, completion without failure, that often succeeds in constructing a canonical system when standard completion fails is described and a refutationally complete theorem prove for purely equational theories is proved.

How to Prove Algebraic Inductive Hypotheses Without Induction

- Computer Science, MathematicsCADE
- 1980

This paper proves the correctness of algebraic methods for deciding the equivalence of expressions by applying rewrite rules, and for proving inductive equational hypotheses without using induction;…

On proving inductive properties of abstract data types

- Computer SciencePOPL '80
- 1980

The equational axioms of an algebraic specification of a data type often can be formed into a convergent set of rewrite rules, which leads to a new method of proof of inductive properties--not requiring the explicit invocation of an inductive rule of inference.

Proofs by induction in equational theories with constructors

- Mathematics, Computer Science21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
- 1980

Termination of Rewriting'

- Computer Science

This survey describes methods for proving that systems of rewrite rules are terminating programs, including polynomial interpretations and path orderings, and illustrates the use in termination proofs of various kinds of orderings on terms.

Existence, Uniqueness, and Construction of Rewrite Systems

- Computer ScienceSIAM J. Comput.
- 1988

The construction of term-rewriting systems, specifically by the Knuth–Bendix completion procedure, is considered, and several notions of equivalence between rewriting systems in the ordinary and modulo case are defined.

A Completion Procedure for Conditional Equations

- MathematicsCTRS
- 1987

Techniques for simplification of conditional equations and rules, so that the procedure terminates on more specifications, are presented.