We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science --- LICS'96 (E. Clarke editor), pp. 264--275, New Brunswick, NJ, July 27--30 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination. 1 Introduction A logical framework is a formal system desig...

We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of functional languages. Our main technical result is a conservative embedding of Nielson & Nielson's two-level functional language in our language Mini-ML, which in

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We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing -calculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers and at last replacing typing predicates by membership to some sets. The theory obtained this way has both a type theoretical flavor and a set theoretical one. Like set theory, it is a first order theory, and it uses only one notion of collection. Like type theory, it gives an explicit notation for objects, a primitive notion of function and propositions are objects.

Reasoning with equality in first order logic can be accomplished axiomatically. That is, we can simply add reflexivity, symmetry, transitivity, and congruence rules for each predicate and function symbol and use the standard theorem proving technology developed in the previous chapters. This approach, however, does not take strong advantage of inherent properties of equality and leads to a very large and inefficent search space. While there has been a deep investigation of equality reasoning in classical logic, much less is known for intuitionistic logic. Some recent references are [Vor96, DV99]. In this chapter we develop some of the techniques of equational reasoning, starting again from first principles in the definition of logic. We therefore recapitulate some of the material in earlier chapters, now adding equality as a new primitive predicate symbol. 7.1 Natural Deduction We characterize equality by its introduction rule, which simply states that s. = s for any term s. =I ⊢ s. = s We have already seen this introduction rule in unification logic in Section 4.3. In the context of unification logic, however, we did not consider hypothetical judgments, so we did not need or specify elimination rules for equality. If we know s. = t we can replace any number of occurrences of s in a true proposition and obtain another true proposition. ⊢ s. = t ⊢ [s/x]A.=E1 ⊢ [t/x]A

After the definition of logic via natural deduction, we have developed a succession of techniques for theorem proving based on sequent calculi. We considered asequentΓ= ⇒ C as a goal, to be solved by backwards-directed search which was modeled by the bottom-up construction of a derivation. The critical choices were disjunctive non-determinism (resolved by guessing and backtracking) and existential non-determinism (resolved by introducing existential variables and unification). The limiting factor in more refined theorem provers based on this method is generally the number of disjunctive choices which have to be made. It is complicated by the fact that existential variables are global in a partial derivation, which means that choices in one conjunctive branch have effects in other branches. This effects redundancy elimination, since subgoals are not independent of each other. The diametrically opposite approach would be to work forward from the initial sequents until the goal sequent is reached. If we guarantee a fair strategy in the selection of axioms and inference rules, every goal sequent can be derived this way. Without further improvements, this is clearly infeasible, since there are too many derivations for us to hope that we can find one for the goal sequent in this manner. The inverse method is based on the property that in a cut-free derivation of a goal sequent, we only need to consider subformulas of the goal and their substitution instances. For example, when we have derived both A and B in the forward direction, we only derive their conjunction A ∧ B if A ∧ B is a subformula of the goal sequent. The nature of forward search under these restrictions is quite different from the backward search. Since we always add new consequences to the sequents already derived, knowledge grows monotonicallyand no disjunctive non-determinism arises. Similarly for existential non-determinism, if we keep sequents in their maximally general form. On the other hand, there is a potentially very large amount of conjunctive non-determinism, since we have to apply all applicable rules to all sequents in a fair manner in order to guarantee completeness. The critical factor in forward search is to limit conjunctive non-determinism. We

The inference rules so far only model intuitionistic logic, and some classically true propositions such as A ∨¬A (for an arbitrary A) are not derivable, as we will see in Section??. There are three commonly used ways one can construct a system of classical natural deduction by adding one additional rule of inference. ⊥C is called Proof by Contradiction or Rule of Indirect Proof, ¬¬C is the Double Negation Rule, and XM is referred to as Excluded Middle.

Since hypotheses and their restrictions are critical for linear logic, we give here a formulation of natural deduction for intuitionistic logic with localized hypotheses, but not parameters. For this we need a notation for hypotheses which we call a context. Contexts \Gamma ::= \Delta j \Gamma; u:A Here, "\Delta" represents the empty context, and \Gamma; u:A adds hypothesis ` A labelled u to \Gamma. We assume that each label u occurs at most once in a context in order to avoid ambiguities. The main judgment can then be written as \Gamma ` A, where \Delta; u 1 :A 1 ; : : : ; un :An ` A stands for u 1 ` A 1 : : :<F43.12

2 3 Objectives 2 4 Background and State of the Art 3 5 Project Organization 6 5.1 Technical Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.1 Logical Representation of the Specification Paradigms . . . . 6 5.1.2 Integration with the Proof Assistant Systems . . . . . . . . . 8 5.1.3 Publications and Cooperation . . . . . . . . . . . . . . . . . . 8 5.2 Task Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Work Planning 13 7 Deliverables 14 8 Project Management 14 9 Publication of Results 14 10 Ethical, Social and Environmental Impact 14 11 Team Member Experience 15 12 Available and Requested Resources 19 Research and Development medium size project proposal submitted to the PRAXIS XXI Program, in the area of Information Technology and Telecommunications, in August 1995 1 1 Title LOGCOMP-- Logic and Computation Integration of Proof Assistants with Symbolic Systems for Specification and Prototyping. 2 Abstract The project aims to integra...

108 Equality Symmetrically, we can also replaces of occurrences of t by s. ` s : = t ` [t=x]A : = E 2 ` [s=x]A It might seem that this second rule is redundant, and in some sense it is. In particular, it is a derivable rule of the calculus with only : = E 1 : ` s : = t : = I ` s : = s : = E 1 ` t : = s ` [t=x]A : = E 1 ` [s=x]A However, this deduction is not normal (as defined below), and without the second elimination rule the normalization theorem would not hold and cut elimination in the sequent calculus would f