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The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational t ..."
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
Induction is Not Derivable in Second Order Dependent Type Theory
"... . This paper proves the nonderivability of induction in second order dependent type theory (P 2). This is done by providing a model construction for P 2, based on a saturated sets like interpretation of types as sets of terms of a weakly extensional combinatory algebra. We give countermodels i ..."
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. This paper proves the nonderivability of induction in second order dependent type theory (P 2). This is done by providing a model construction for P 2, based on a saturated sets like interpretation of types as sets of terms of a weakly extensional combinatory algebra. We give countermodels in which the induction principle over natural numbers is not valid. The proof does not depend on the specic encoding for natural numbers that has been chosen (like e.g. polymorphic Church numerals), so in fact we prove that there can not be an encoding of natural numbers in P2 such that the induction principle is satised. The method extends immediately to other data types, like booleans, lists, trees, etc. In the process of the proof we establish some general properties of the models, which we think are of independent interest. Moreover, we show that the Axiom of Choice is not derivable in P 2. 1 Introduction In second order dependent type theory, P 2, we can encode all kind...
Towards Normalization by Evaluation for the βηCalculus of Constructions
"... Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as part ..."
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Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as partial equivalence relations between values and type constructors as operators on PERs. This models also yields consistency of the betaetaCalculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq. 1