The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed -calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed.

Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a meta-logic (or `logical framework') in which the object-logics are formalized. Isabelle is now based on higher-order logic --- a precise and well-understood foundation. Examples illustrate use of this meta-logic to formalize logics and proofs. Axioms for first-order logic are shown sound and complete. Backwards proof is formalized by meta-reasoning about object-level entailment. Higher-order logic has several practical advantages over other meta-logics. Many proof techniques are known, such as Huet's higher-order unification procedure. Key words: higher-order logic, higher-order unification, Isabelle, LCF, logical frameworks, meta-reasoning, natural deduction Contents 1 History and overview 2 2 The meta-logic M 4 2.1 Syntax of the meta-logic ......................... 4 2.2 ...

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Several related typed languages for modular programming and data abstraction have been proposed recently, including Pebble, SOL, and ML modules. We review and compare the basic type-theoretic ideas behind these languages and evaluate how they

. We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type...

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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.

Proof-Carrying Code (PCC) is a general framework for verifying the safety properties of machine-language programs. PCC proofs are usually written in a logic extended with language-specific typing rules. In Foundational Proof-Carrying Code (FPCC), on the other hand, proofs are constructed and verified using strictly the foundations of mathematical logic, with no type-specific axioms. FPCC is more flexible and secure because it is not tied to any particular type system and it has a smaller trusted base. Foundational proofs, however, are much harder to construct. Previous efforts on FPCC all required building sophisticated semantic models for types. In this paper, we present a syntactic approach to FPCC that avoids the difficulties of previous work. Under our new scheme, the foundational proof for a typed machine program simply consists of the typing derivation plus the formalized syntactic soundness proof for the underlying type system. We give a translation from a typed assembly language into FPCC and demonstrate the advantages of our new system via an implementation in the Coq proof assistant. 1.

A certified binary is a value together with a proof that the value satisfies a given specification. Existing compilers that generate certified code have focused on simple memory and control-flow safety rather than more advanced properties. In this paper, we present a general framework for explicitly representing complex propositions and proofs in typed intermediate and assembly languages. The new framework allows us to reason about certified programs that involve effects while still maintaining decidable typechecking. We show how to integrate an entire proof system (the calculus of inductive constructions) into a compiler intermediate language and how the intermediate language can undergo complex transformations (CPS and closure conversion) while preserving proofs represented in the type system. Our work provides a foundation for the process of automatically generating certified binaries in a type-theoretic framework. 1