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16
Equality In Lazy Computation Systems
, 1998
"... In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equ ..."
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Cited by 95 (6 self)
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In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equality reasoning are valid for it. This condition is a simple syntactic one, and is easy to verify for the various lazy computation systems we have considered so far. We also give conditions under which the equivalence coincides with observational congruence. These results have some important consequences for type theories like those of MartinLöf and Nuprl.
The Semantics of Reflected Proof
 IN PROC. OF FIFTH SYMP. ON LOGIC IN COMP. SCI
, 1990
"... We begin to lay the foundations for reasoning about proofs whose steps include both invocations of programs to build subproofs (tactics) and references to representations of proofs themselves (reflected proofs). The main result is the definition of a single type of proof which can mention itself, ..."
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Cited by 88 (11 self)
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We begin to lay the foundations for reasoning about proofs whose steps include both invocations of programs to build subproofs (tactics) and references to representations of proofs themselves (reflected proofs). The main result is the definition of a single type of proof which can mention itself, using a new technique which finds a fixed point of a mapping between metalanguage and object language. This single type contrasts with hierarchies of types used in other approaches to accomplish the same classification. We show that these proofs are valid, and that every proof can be reduced to a proof involving only primitive inference rules. We also show how to extend the results to proofs from which programs (such as tactics) can be derived, and to proofs that can refer to a library of definitions and previously proven theorems. We believe that the mechanism of reflection is fundamental in building proof development systems, and we illustrate its power with applications to automating reasoning and describing modes of computation.
Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 57 (15 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
The BoyerMoore Prover and Nuprl: An Experimental Comparison
 LOGICAL FRAMEWORKS
, 1991
"... We use an example to compare the BoyerMoore Theorem Prover and the Nuprl Proof Development System. The respective machine verifications of a version of Ramsey's theorem illustrate similarities and differences between the two systems. The proofs are compared using both quantitative and nonquantitat ..."
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Cited by 24 (8 self)
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We use an example to compare the BoyerMoore Theorem Prover and the Nuprl Proof Development System. The respective machine verifications of a version of Ramsey's theorem illustrate similarities and differences between the two systems. The proofs are compared using both quantitative and nonquantitative measures, and we examine difficulties in making such comparisons.
A TwoLevel Approach towards Lean ProofChecking
, 1996
"... We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle t ..."
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Cited by 16 (4 self)
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We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our twolevel approach is illustrated by some examples developed in Lego.
Reasoning About Functional Programs in Nuprl
 In Functional Programming, Concurrency, Simulation and Automated Reasoning
, 1993
"... . There are two ways of reasoning about functional programs in the constructive type theory of the Nuprl proof development system. Nuprl can be used in a conventional programverification mode, in which functional programs are written in a familiar style and then proven to be correct. It can als ..."
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Cited by 12 (0 self)
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. There are two ways of reasoning about functional programs in the constructive type theory of the Nuprl proof development system. Nuprl can be used in a conventional programverification mode, in which functional programs are written in a familiar style and then proven to be correct. It can also be used in an extraction mode, where programs are not written explicitly, but instead are extracted from mathematical proofs. Nuprl is the only constructive type theory to support both of these approaches. These approaches are illustrated by applying Nuprl to Boyer and Moore's "majority" algorithm. 1 Introduction A type system for a functional programming language can be syntactic or semantic. In a syntactically typed language, such as SML 1 [25], typing is a property of the syntax of expressions. Only certain combinations of language constructs are designated "welltyped", and only welltyped expressions are given a meaning. Each welltyped expression has a type which can be derive...
Partial computations in constructive type theory
 JOURNAL OF LOGIC AND COMPUTATION
, 1991
"... Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects ..."
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Cited by 7 (5 self)
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Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A fixed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of fixed points. There is a Scott fixed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.
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...
Distributed Theorem Proving for Distributed Hybrid Systems ⋆
"... Abstract. Distributed hybrid systems present extraordinarily challenging problems for verification. On top of the notorious difficulties associated with distributed systems, they also exhibit continuous dynamics described by quantified differential equations. All serious proofs rely on decision proc ..."
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Cited by 6 (5 self)
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Abstract. Distributed hybrid systems present extraordinarily challenging problems for verification. On top of the notorious difficulties associated with distributed systems, they also exhibit continuous dynamics described by quantified differential equations. All serious proofs rely on decision procedures for real arithmetic, which can be extremely expensive. Quantified Differential Dynamic Logic (QdL) has been identified as a promising approach for getting a handle in this domain. QdL has been proved to be complete relative to quantified differential equations. But important questions remain as to how best to translate this theoretical result into practice: how do we succinctly specify a proof search strategy, and how do we control the computational cost? We address the problem of automated theorem proving for distributed hybrid systems. We identify a simple mode of use of QdL that cuts down on the enormous number of choices that it otherwise allows during proof search. We have designed a powerful strategy and tactics language for directing proof search. With these techniques, we have implemented a new automated theorem prover called KeYmaeraD. To overcome the high computational complexity of distributed hybrid systems verification, KeYmaeraD uses a distributed proving backend. We have experimentally observed that calls to the real arithmetic decision procedure can effectively be made in parallel. In this paper, we demonstrate these findings through an extended case study where we prove absence of collisions in a distributed car control system with a varying number of arbitrarily many cars. 1
Program development through proof transformation
 CONTEMPORARY MATHEMATICS
, 1990
"... We present a methodology for deriving verified programs that combines theorem proving and proof transformation steps. It extends the paradigm employed in systems like NuPrl where a program is developed and verified through the proof of the specification in a constructive type theory. We illustrate ..."
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Cited by 4 (1 self)
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We present a methodology for deriving verified programs that combines theorem proving and proof transformation steps. It extends the paradigm employed in systems like NuPrl where a program is developed and verified through the proof of the specification in a constructive type theory. We illustrate our methodology through an extended example  a derivation of Warshall's algorithm for graph reachability. We also outline how our framework supports the definition, implementation, and use of abstract data types.