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34
A Survey on the Theorema Project
 In International Symposium on Symbolic and Algebraic Computation
, 1997
"... The Theorema project aims at extending current computer algebra systems by facilities for supporting mathematical proving. The present earlyprototype version of the Theorema software system is implemented in Mathematica 3.0. The system consists of a general higherorder predicate logic prover and ..."
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Cited by 47 (10 self)
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The Theorema project aims at extending current computer algebra systems by facilities for supporting mathematical proving. The present earlyprototype version of the Theorema software system is implemented in Mathematica 3.0. The system consists of a general higherorder predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing humanreadable proofs in natural language presented in nested cells that facilitate studying the computergenerated proofs at various levels of detail. The special provers are intimately connected with the functors that build up the various mathematical domains. 1 The Objectives of the Theorema Project The Theorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a "knowledge base" K of formulae (and a logical / computat...
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 46 (18 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
A Mechanised Proof System for Relation Algebra using Display Logic
 In Proc. JELIA98, LNAI
, 1997
"... . We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory. Our implementation is the first mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra. The inference rules of Display Logic are coded ..."
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Cited by 16 (10 self)
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. We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory. Our implementation is the first mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra. The inference rules of Display Logic are coded directly as Isabelle theorems, thereby guaranteeing the correctness of all derivations. We describe various tactics and derived rules developed for simplifying proof search, including an automatic cutelimination procedure, and example theorems proved using Isabelle. We show how some relation algebraic theorems proved using our system can be put in the form of structural rules of Display Logic, facilitating later reuse. We then show how the implementation can be used to prove results comparing alternative formalizations of relation algebra from a prooftheoretic perspective. Keywords: logical frameworks, higherorder logic, relation algebra, display logic 1 Introduction Relation algebras a...
An Overview of the Tecton Proof System
, 1992
"... The Tecton Proof System is an experimental tool for constructing proofs of first order logic formulas and of program specifications expressed using formulas in Hoare's axiomatic proof formalism. It is designed to make interactive proof construction easier than with previous proof tools, by mainta ..."
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Cited by 13 (5 self)
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The Tecton Proof System is an experimental tool for constructing proofs of first order logic formulas and of program specifications expressed using formulas in Hoare's axiomatic proof formalism. It is designed to make interactive proof construction easier than with previous proof tools, by maintaining multiple proof attempts internally in a structured form called a proof forest; displaying them in an easy to comprehend form, using a combination of tabular formats, graphical representations, and hypertext links; and automating substantial parts of proofs through rewriting, induction, case analysis, and generalization inference mechanisms, along with a linear arithmetic decision procedure. Further development of the system is planned as part of an overall framework aimed at supporting the kind of abstractions and specializations necessary for building libraries of generic software and hardware components. Partially supported by National Science Foundation Grants CCR8906678...
Logic Frameworks for Logic Programs
, 1994
"... . We show how logical frameworks can provide a basis for logic program synthesis. With them, we may use firstorder logic as a foundation to formalize and derive rules that constitute program development calculi. Derived rules may be in turn applied to synthesize logic programs using higherorder re ..."
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Cited by 12 (7 self)
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. We show how logical frameworks can provide a basis for logic program synthesis. With them, we may use firstorder logic as a foundation to formalize and derive rules that constitute program development calculi. Derived rules may be in turn applied to synthesize logic programs using higherorder resolution during proof that programs meet their specifications. We illustrate this using Paulson's Isabelle system to derive and use a simple synthesis calculus based on equivalence preserving transformations. 1 Introduction Background In 1969 Dana Scott developed his Logic for Computable Functions and with it a model of functional program computation. Motivated by this model, Robin Milner developed the theorem prover LCF whose logic PP used Scott's theory to reason about program correctness. The LCF project [13] established a paradigm of formalizing a programming logic on a machine and using it to formalize different theories of functional programs (e.g., strict and lazy evaluation) and the...
A Case Study of Coinduction in Isabelle
, 1995
"... The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, the w ..."
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Cited by 7 (0 self)
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The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, the work presented here deals with the formalisation of this result in the generic theorem prover Isabelle. Contents 1 Introduction 1 2 Coinduction in Relation Semantics 2 2.1 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.2 The Language : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.3 Dynamic Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.4 Static Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.5 Consistency : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3 Isabelle 7 3.1 Documentation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3.2 Notation : : : : : : : : : : : : : : : : : : : : : ...
On Extensibility of Proof Checkers
 in Dybjer, Nordstrom and Smith (eds), Types for Proofs and Programs: International Workshop TYPES'94, Bastad
, 1995
"... This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. Howeve ..."
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Cited by 6 (2 self)
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This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. However, we are quite rigid about this: only a derivation in our given formal system will do; nothing else counts as evidence! Thus it is not a collection of judgements (provability), or a consequence relation [Avr91] (derivability) we are interested in, but the derivations themselves; the formal system used to present a logic is important. This viewpoint seems forced on us by our intention to actually do formal mathematics. There is still a question, however, revolving around whether we insist on objects that are immediately recognisable as proofs (direct proofs), or will accept some metanotations that only compute to proofs (indirect proofs). For example, we informally refer to previously proved results, lemmas and theorems, without actually inserting the texts of their proofs in our argument. Such an argument could be made into a direct proof by replacing all references to previous results by their direct proofs, so it might be accepted as a kind of indirect proof. In fact, even for very simple formal systems, such an indirect proof may compute to a very much bigger direct proof, and if we will only accept a fully expanded direct proof (in a mechanical proof checker for example), we will not be able to do much mathematics. It is well known that this notion of referring to previous results can be internalized in a logic as a cut rule, or Modus Ponens. In a logic containing a cut rule, proofs containing cuts are considered direct proofs, and can be directly accepted by a proof ch...
A Case Study of Coinduction in Isabelle HOL
, 1993
"... The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, th ..."
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Cited by 5 (1 self)
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The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, the work presented here deals with the formalisation of this result in the higherorder logic of the generic theorem prover Isabelle. 1 Introduction In the paper Coinduction in Relational Semantics [1], R.Milner and M.Tofte prove the dynamic and static semantics for a small functional programming language consistent. The dynamic semantics associates a value to an expression of the language, while the static semantics associates a type. A value has a type. Consistency requires that the value of an expression has the type of the expression. Values can be infinite or nonwellfounded because the language contains recursive functions. Nonwellfounded values are handled using coinductive def...
Efficient Substitution in Hoare Logic Expressions
, 2000
"... Substitution plays an important role in Hoare Logic, as it is used in interpreting assignments. When writing a computerbased realization of Hoare Logic, it is therefore important to choose a good implementation for it. In this paper we compare di#erent definitions and implementations of substitutio ..."
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Cited by 4 (2 self)
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Substitution plays an important role in Hoare Logic, as it is used in interpreting assignments. When writing a computerbased realization of Hoare Logic, it is therefore important to choose a good implementation for it. In this paper we compare di#erent definitions and implementations of substitution in a logical framework, in an e#ort to maximize e#ciency. We start by defining substitution as a logical formula. In a conventional approach, this is done by specifying the syntactic changes substitution performs on expressions. We choose instead a semantic definition that describes the behavioral relation between the original expression and its substituted counterpart. Next, we use this semantic definition as an abstract specification, and compare two of its concrete implementations. The first we consider is the usual one, that operates recursively over the structure of the term. This requires a number of inference steps proportional to the size of the expression, which is unacceptable ...
Towards Machinechecked Compiler Correctness for Higherorder Pure Functional Languages
 CSL '94, European Association for Computer Science Logic, Springer LNCS
, 1994
"... . In this paper we show that the critical part of a correctness proof for implementations of higherorder functional languages is amenable to machineassisted proof. An extended version of the lambdacalculus is considered, and the congruence between its direct and continuation semantics is proved. ..."
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Cited by 4 (1 self)
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. In this paper we show that the critical part of a correctness proof for implementations of higherorder functional languages is amenable to machineassisted proof. An extended version of the lambdacalculus is considered, and the congruence between its direct and continuation semantics is proved. The proof has been constructed with the help of a generic theorem prover  Isabelle. The major part of the problem lies in establishing the existence of predicates which describe the congruence. This has been solved using Milne's inclusive predicate strategy [5]. The most important intermediate results and the main theorem as derived by Isabelle are quoted in the paper. Keywords: Compiler Correctness, Theorem Prover, Congruence Proof, Denotational Semantics, Lambda Calculus 1 Introduction Much of the work done previously in compiler correctness concerns restricted subsets of imperative languages. Some studies involve machinechecked correctnesse.g. Cohn [1], [2]. A lot of research h...