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Moving proofsasprograms into practice
 In: Proceedings of the 12 th IEEE International Conference on Automated Software Engineering, IEEE Computer Society
, 1997
"... Proofs in the Nuprl system, an implementation of a constructive type theory, yield “correctbyconstruction ” programs. In this paper a new methodology is presented for extracting efficient and readable programs from inductive proofs. The resulting extracted programs are in a form suitable for use i ..."
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Proofs in the Nuprl system, an implementation of a constructive type theory, yield “correctbyconstruction ” programs. In this paper a new methodology is presented for extracting efficient and readable programs from inductive proofs. The resulting extracted programs are in a form suitable for use in hierarchical verifications in that they are amenable to clean partial evaluation via extensions to the Nuprl rewrite system. The method is based on two elements: specifications written with careful use of the Nuprl settype to restrict the extracts to strictly computational content; and on proofs that use induction tactics that generate extracts using familiar fixedpoint combinators of the untyped lambda calculus. In this paper the methodology is described and its application is illustrated by example. 1.
The structure of nuprl’s type theory
, 1997
"... on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) ..."
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Cited by 9 (3 self)
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on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)
Search algorithms in type theory
, 2000
"... In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by par ..."
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Cited by 8 (2 self)
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In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by parameterizing the speci cation of the problem. A constructive proof is presented which has as its computational content a correct search procedure for these problems. We show how a classical extension to an otherwise constructive system can be used to describe a typical use of the nonlocal control operator call/cc. Using the classical typing of nonlocal control we extend our purely constructive proof to incorporate a sophisticated backtracking technique known as ‘con ictdirected backjumping’ (CBJ). A variant of this proof is formalized in Nuprl yielding a correctbyconstruction implementation of CBJ. The extracted program has been translated into Scheme and serves as the basis for an implementation of a new solution to the Hamiltonian circuit problem. This paper demonstrates a nontrivial application of the proofsasprograms paradigm by applying the technique to the derivation of a sophisticated search algorithm; also, it shows the generality of the resulting implementation by demonstrating its application in a new problem
Verifying programs in the Calculus of Inductive Constructions
, 1997
"... . This paper deals with a particular approach to the verification of functional programs. A specification of a program can be represented by a logical formula [Con86, NPS90]. In a constructive framework, developing a program then corresponds to proving this formula. Given a specification and a progr ..."
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. This paper deals with a particular approach to the verification of functional programs. A specification of a program can be represented by a logical formula [Con86, NPS90]. In a constructive framework, developing a program then corresponds to proving this formula. Given a specification and a program, we focus on reconstructing a proof of the specification whose algorithmic contents corresponds to the given program. The best we can hope is to generate proof obligations on atomic parts of the program corresponding to logical properties to be verified. First, this paper studies a weak extraction of a program from a proof that keeps track of intermediate specifications. From such a program, we prove the determinism of retrieving proof obligations. Then, heuristic methods are proposed for retrieving the proof from a natural program containing only partial annotations. Finally, the implementation of this method as a tactic of the Coq proof assistant is presented. 1. Introduction A large p...
Decidability Extracted: SYNTHESIZING “CORRECTBYCONSTRUCTION” DECISION PROCEDURES FROM CONSTRUCTIVE PROOFS
, 1998
"... The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of ..."
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The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of Nuprl’s set type and techniques for extracting efficient programs from induction principles. The constructive formal theories required to express the decidability theorems are of independent interest. They formally circumscribe the mathematical knowledge needed to understand the derived algorithms. The formal theories express concepts that are taught at the senior college level. The decidability proofs themselves, depending on this material, are of interest and are presented in some detail. The proof of decidability of classical propositional logic is relative to a semantics based on Kleene’s strong threevalued logic. The constructive proof of intuitionistic decidability presented here is the first machine formalization of this proof. Theexposition reveals aspects of the Nuprl tactic collection relevant to the creation of readable proofs; clear extracts and efficient code are illustrated in the discussion of the proofs.
Extracting Recursion Operators in Nuprl's Type Theory
 Eleventh International Workshop on Logic based Program Synthesis, LOPSTR02, volume 2372 of LNCS
, 2001
"... In this paper we describe the extraction of efficient recursion schemes from proofs of wellfounded induction principles. This is part of a larger methodology ..."
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In this paper we describe the extraction of efficient recursion schemes from proofs of wellfounded induction principles. This is part of a larger methodology
Formalization of Divisibility Theory in Nuprl
"... The formalization of divisibility theory over cancellation monoids in Nuprl is described. The main theorems presented concern the existence and uniqueness of factorisations. Issues addressed include how to make formalized mathematics readable and the use of automated inference. The constructive nat ..."
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The formalization of divisibility theory over cancellation monoids in Nuprl is described. The main theorems presented concern the existence and uniqueness of factorisations. Issues addressed include how to make formalized mathematics readable and the use of automated inference. The constructive nature of mathematics in Nuprl is also discussed.
Constructive Membership and Indexes in Trees
, 2009
"... Trees carrying information stored in their nodes are a fundamental abstract data type. Approaching trees in a formal constructive environment allows us to realize properties of trees, inherent in their structure. Specifically we will look at the evidence provided by the predicates which operate on t ..."
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Trees carrying information stored in their nodes are a fundamental abstract data type. Approaching trees in a formal constructive environment allows us to realize properties of trees, inherent in their structure. Specifically we will look at the evidence provided by the predicates which operate on these trees. This evidence, expressed in terms of logical and programming languages, is realizable only in a constructive context. In the constructive setting, membership predicates over recursive types are inhabited by terms indexing the elements that satisfy the criteria for membership. In this paper, we motivate and explore this idea in the concrete setting of lists and trees. We first provide a background in constructive type theory and show relavent properties of trees. We present and define the concept of inhabitants of a generic shape type that corresponds naturally and exactly to the inhabitants of a membership predicate. In this context, (λx.T rue) ∈ S is the set of all indexes into S, but we show that not all subsets of indexes are expressible by strictly local predicates. Accordingly, we extend our membership predicates to predicates that compute and hold the state “from above” as well as allow “looking below”. The modified predicates of this form are complete in the sense that they can express every subset of indexes in S. These ideas are motivated by experience programming in Nuprl’s constructive type theory and the theorems for lists and trees have been formalized and mechanically checked. 1