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 ...

A new typed, higher-order logic is described which appears particularly well fitted to reasoning about forms of computation whose operational behaviour can be specified using the Natural Semantics style of structural operational semantics [5]. The logic's underlying type system is Moggi's computational metalanguage [11], which enforces a distinction between computations and values via the categorical structure of a strong monad. This is extended to a (constructive) predicate logic with modal formulas about evaluation of computations to values, called evaluation modalities. The categorical structure corresponding to this kind of logic is explained and a couple of examples of categorical models given. As a first example of the naturalness and applicability of this new logic to program semantics, we investigate the translation of a (tiny) fragment of Standard ML into a theory over the logic, which is proved computationally adequate for ML's Natural Semantics [10]. Whilst it is tiny, the M...

This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of Martin-Lof's `iteration type' [11]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a category-theoretic version of the `logical relations' method. 1 Computation types It is well known that primitive recursion at higher types can be given a categorical characterisation in terms of Lawvere's concept of natural number object [6]. We show that a similar characterisation can be given for general recursion via fixpoint operators of higher types, in terms of a new concept---that of a fixpoint object in ...

The theorem prover Isabelle and several of its object-logics are described. Where