In the pure Calculus of Constructions, it is possible to represent data structures and predicates using higher-order quantification. However, this representation is not satisfactory, from the point of view of both the efficiency of the underlying programs and the power of the logical system. For these reasons, the calculus was extended with a primitive notion of inductive definitions [8]. This paper describes the rules for inductive definitions in the system Coq. They are general enough to be seen as one formulation of adding inductive definitions to a typed lambda-calculus. We prove strong normalization for a subsystem of Coq corresponding to the pure Calculus of Constructions plus Inductive Definitions with only weak non-dependent eliminations.

this paper is similar to the one in [2]. In this paper they define a normalization function for simply typed

This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...

In various lectures given during 1992, Martin-Lof has presented the main ideas of a formulation of the theory of types in which substitution is not left unspecied and instead appears explicit in the expressions of the language. We here present this formulation in a complete manner. The various forms of judgements are given their semantical explanations and then a calculus is exhibited and explained in detail. Contents 1 Introduction 2 2 The forms of judgement 5 Types and families of types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 Intuitive considerations about expressions containing variables : : : : : : : : : : 7 Contexts, environments and the relative forms of judgement : : : : : : : : : : : : 8 Substitutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3 The calculus 13 General rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 Special rules for substitutions : : : : : : : : : : : : : : :...

. Interactive theorem proving gives a general approach for modelling and verification of both hardware and software systems but requires significant human efforts to deal with many tedious proofs. To be used in practical, we need some automatic tools such as model checkers to deal with those tedious proofs. In this paper, we formalise a verification system of both CCS and an imperative language in LEGO which can be used to verify both finite and infinite problems. Then a model checker, LegoMC, is implemented to generate the LEGO proof terms of finite models automatically. Therefore people can use LEGO to verify a general problem and throw some finite sub-problems to be verified by LegoMC. On the other hand, this integration extends the power of model checking to verify more complicated and infinite models as well. 1 Introduction Interactive theorem proving gives a general approach for modelling and verification of both hardware and software systems but requires significant human effor...

We describe a complete formalization of a strong normalization proof for the Curry style presentation of System F in LEGO. The underlying type theory is the Calculus of Constructions enriched by inductive types. The proof follows Girard et al [GLT89], i.e. we use the notion of candidates of reducibility, but we make essential use of general inductive types to simplify the presentation. We discuss extensions and variations of the proof: the extraction of a normalization function, the use of saturated sets instead of candidates, and the extension to a Church Style presentation. We conclude with some general observations about Computer Aided Formal Reasoning.

We present a formalization of the set Z of integers using Martin-Lof's type theory. In particular we focus on the task of proving that this set with the operations + and form an Integral Domain. The proofs are developed for an inductive definition of Z, but we also discuss what kind of proofs could be obtained for a formulation where the set is defined as a quotient. The differences between both approaches when one is interested in regarding the computational meaning of proofs are pointed out. In order to better reason about the proofs of the properties following from the postulates of an integral domain, an abstract formalization of this algebraic system is also proposed. With this, we aimed at not just being able to formally reflect the derivation of the properties independently of the concrete representation we were interested in, but also to translate these results to every algebraic structure satisfying those postulates. Keywords and phrases: integers, type theory, integral dom...

We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in Martin-Lof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almost-fullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almost-fullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almost-fullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...

Interactive theorem proving provides a general approach to modeling and verification of both finite-state and infinite-state systems but requires significant human efforts to deal with many tedious proofs. On the other hand, modelchecking is limited to some application domain with small finite-state space. A natural thought for this problem is to integrate these two approaches. To keep the consistency of the integration and ensure the correctness of verification, we suggest to use type theory based theorem provers (e.g. Lego) as the platform for the integration and build a model-checker to do parts of the verification automatically. We formalise a verification system of both CCS and an imperative language in the proof development system Lego which can be used to verify both finite-state and infinite-state problems. Then a model-checker, LegoMC, is implemented to generate Lego proof terms for finite-state problems automatically. Therefore people can use Lego to verify a general problem ...

This paper demonstrates a method of extracting programs from formal deductions represented in the Edinburgh Logical Framework, using the Elf programming language. Deductive systems are given for the extraction of simple types from formulas of first-order arithmetic and of -calculus terms from natural deduction proofs. These systems are easily encoded in Elf, yielding an implementation of extraction that corresponds to modified realizability. Because extraction is itself implemented as a set of formal deductive systems, some of its correctness properties can be partially represented and mechanically checked in the Elf language.