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External Rewriting for Skeptical Proof Assistants
, 2002
"... This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a ..."
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Cited by 21 (3 self)
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This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a speci c and ecient environment and to check the computations later in a proof assistant.
Equational Reasoning via Partial Reflection
"... We modify the reection method to enable it to deal with partial functions like division. The idea behind reflection is to program a tactic for a theorem prover not in the implementation language but in the object language of the theorem prover itself. The main ingredients of the reflection metho ..."
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Cited by 11 (7 self)
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We modify the reection method to enable it to deal with partial functions like division. The idea behind reflection is to program a tactic for a theorem prover not in the implementation language but in the object language of the theorem prover itself. The main ingredients of the reflection method are a syntactic encoding of a class of problems, an interpretation function (mapping the encoding to the problem) and a decision function, written on the encodings. Together with a correctness proof of the decision function, this gives a fast method for solving problems. The contribution of this work lies in the extension of the reflection method to deal with equations in algebraic structures where some functions may be partial. The primary example here is the theory of fields. For the reflection method, this yields the problem that the interpretation function is not total. In this paper we show how this can be overcome by defining the interpretation as a relation. We give the precise details, both in mathematical terms and in Coq syntax. It has been used to program our own tactic `Rational', for verifying equations between field elements.
Universal Algebra in Type Theory
 Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS
, 1999
"... We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ... ..."
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Cited by 8 (6 self)
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We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ...
Changing Data Structures in Type Theory: a study of natural numbers
 Types for Proofs and Programs, Intl. Workshop (TYPES 2000), LNCS 2277
, 2000
"... In typetheory based proof systems that provide inductive structures, computation tools are automatically associated to inductive de nitions. Choosing a particular representation for a given concept has a strong inuence on proof structure. We propose a method to make the change from one represe ..."
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Cited by 3 (0 self)
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In typetheory based proof systems that provide inductive structures, computation tools are automatically associated to inductive de nitions. Choosing a particular representation for a given concept has a strong inuence on proof structure. We propose a method to make the change from one representation to another easier, by systematically translating proofs from one context to another. We show how this method works by using it on natural numbers, for which a unary representation (based on Peano axioms) and a binary representation are available. This method leads to an automatic translation tool that we have implemented in Coq and successfully applied to several arithmetical theorems.
Brokers and Webservices for automatic deduction: a case study
 In Therese Hardin and Renaud Rioboo, editors, Calculemus 2003
"... Abstract. We present a planning broker and several WebServices for automatic deduction. Each WebService implements one of the tactics usually available in interactive proofassistants. When the broker is submitted a \proof status " (an incomplete proof tree and a focus on an open goal) it di ..."
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Abstract. We present a planning broker and several WebServices for automatic deduction. Each WebService implements one of the tactics usually available in interactive proofassistants. When the broker is submitted a \proof status " (an incomplete proof tree and a focus on an open goal) it dispatches the proof to the WebServices, collects the successful results, and send them back to the client as \hints " as soon as they are available. In our experience this architecture turns out to be helpful both for experienced users (who can take benet of distributing heavy computations) and beginners (who can learn from it). 1
Proof by Computation in the Coq system
 in Theoretical Computer Science
, 2000
"... In informal mathematics, statements involving computations are seldom proved. Instead, it is assumed that readers of the proof can carry out the computations on their own. However, when using an automated proof development system based on type theory, the user is forced to nd proofs for all claimed ..."
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In informal mathematics, statements involving computations are seldom proved. Instead, it is assumed that readers of the proof can carry out the computations on their own. However, when using an automated proof development system based on type theory, the user is forced to nd proofs for all claimed propositions, including computational statements. This paper presents a method to automatically prove statements from primitive recursive arithmetic. The method replaces logical formulas by boolean expressions. A correctness proof is constructed, which states that the original formula is derivable, if and only if the boolean expression equals true. Because the boolean expression reduces to true, the conversion rule yields a trivial proof of the equality. By combining this proof with the correctness proof, we get a proof for the original statement. 1 Introduction This paper presents a method to automatically prove statements from rst order primitive recursive arithmetic, in the context o...
Certifying Term Rewriting Proofs in ELAN
, 2001
"... Term rewriting has been shown to be a good environment for both programming and proving. For analysing and debugging rulebased programs, we propose in this work a formalism based on the rewriting calculus with explicit substitutions ( calculus). This formalism also allows us to build the proof ter ..."
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Cited by 3 (0 self)
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Term rewriting has been shown to be a good environment for both programming and proving. For analysing and debugging rulebased programs, we propose in this work a formalism based on the rewriting calculus with explicit substitutions ( calculus). This formalism also allows us to build the proof terms of rewriting derivations. Therefore, term rewriting proofs can be exported to other systems by translating them into the corresponding syntaxes. That is, using a proof checker, one can certify these proofs and vice versa, this method allows us to get term rewriting in proof assistants using an external system. Our method not only works with syntactic rewriting but also with rewriting modulo a set of axioms (e.g. associativitycommutativity).
Recursive Families of Inductive Types
, 2000
"... Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong p ..."
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Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong positivity, that characterizes these families. Then we investigate its solutions. First, we construct a model using wellorderings. Second, we use an extension...
Equational Reasoning in Type Theory
, 2000
"... We dene the notions of equational theory and equational logic in Type Theory using the development of Universal Algebra presented in a previous paper. The main result is the formal proof of Birkho's validity and completeness theorem, that gives a theoretical basis to the two level approach ..."
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We dene the notions of equational theory and equational logic in Type Theory using the development of Universal Algebra presented in a previous paper. The main result is the formal proof of Birkho's validity and completeness theorem, that gives a theoretical basis to the two level approach to interactive theorem proving. The whole development has been implemented using the proof assistant Coq. Keywords: Type Theory, Universal Algebra, Equational Logic, Interactive Theorem Proving. 1