Partial inductive definitions as type-systems for λ-terms (original) (raw)

On the Correspondence Between Proofs and lambda-Terms

Technical Reports (CIS), 1993

The correspondence between natural deduction proofs and ��-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) ��-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed ��-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors���,��,+,���,��� and���(falsity)(with or without ��-like rules).

Inductive Definitions and Type Theory: An Introduction

2007

Martin-Lof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a full-scale system for the formalization of constructive mathematics, but has also proved to be a powerful framework for programming. The theory integrates an expressive specification language (its type system) and a functional programming language (where all programs terminate). There now exist several proof-assistants based on type theory, and many non-trivial examples from programming, computer science, logic, and mathematics have been implemented using these. In this series of lectures we shall describe type theory as a theory of inductive definitions. We emphasize its open nature: much like in a standard functional language such as ML or Haskell the user can add new types whenever there is a need for them. We discuss the syntax and semantics of the theory. Moreover, we present some examples ...

Soundness and completeness of simply typed λ-calculus

The λ-calculus was first published in , with the aim of providing a foundation for logic which would be an alternative to Russell's type theory or Zermelo's set theory. The theory is based on the concept of functions rather than sets and it is about functions as rules 1 rather than as graphs 2 . Almost immediately after publication a contradiction was found in it. Church revised the theory a year later, the inconsistency of which was discovered by two students of Church, Stephen . Discouraged in his foundational project, Church gave a consistent subtheory, known as λI -calculus in ].

A semantic characterization of the well-typed formulae of λ-calculus

Theoretical Computer Science, 1993

A model-theoretic operation is characterised that preserves the property of being a model of typed λ-calculus. (i.e., the result of applying it to a model of typed λ-calculus is another model of typed λ-calculus.) An expression is well-typed iff the class of its models is closed under this operation.

Strong normalization and equi-(co) inductive types

2007

A type system for the lambda-calculus enriched with recursive and corecursive functions over equi-inductive and-coinductive types is presented in which all well-typed programs are strongly normalizing. The choice of equi-inductive types, instead of the more common isoinductive types, influences both reduction rules and the strong normalization proof. By embedding iso-into equi-types, the latter ones are recognized as more fundamental. A model based on orthogonality is constructed where a semantical type corresponds to a set of observations, and soundness of the type system is proven.

A Categorical Semantics for Inductive-Inductive Definitions

Lecture Notes in Computer Science, 2011

Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A Ñ Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.

Mechanical procedure for proof construction via closed terms in typed λ calculus

Journal of Automated Reasoning, 1988

In this paper is presented an algorithm for constructing natural deduction proofs in the propositional intuitionistic and classical logics according to the analogy relating intuitionistic propositional formulas and natural deduction proofs, respectively, to types and terms of simple type theory. Proofs are constructed as closed terms in the simple typed λ calculus. The soundness and completeness of this method are proved.

Inductive, coinductive, and pointed types

ACM SIGPLAN Notices, 1996

An extension of the simply-typed lambda calculus is presented which contains both well-structured inductive and coinductive types, and which also identifies a class of types for which general recursion is possible. The motivations for this work are certain natural constructions in category theory, in particular the notion of an algebraically bounded functor, due to Freyd. We propose that this is a particularly elegant core language in which to work with recursive objects, since the potential for general recursion is contained in a single operator which interacts well with the facilities for bounded iteration and coiteration.

Strong Normalization of the Typed λ ws -Calculus

Lecture Notes in Computer Science, 2003

The λws-calculus is a λ-calculus with explicit substitutions introduced in [4]. It satisfies the desired properties of such a calculus: step by step simulation of β, confluence on terms with meta-variables and preservation of the strong normalization. It was conjectured in [4] that simply typed terms of λws are strongly normalizable. This was proved in [7] by Di Cosmo & al. by using a translation of λws into the proof nets of linear logic. We give here a direct and elementary proof of this result. The strong normalization is also proved for terms typable with second order types (the extension of Girard's system F). This is a new result.

A Predicative Strong Normalisation Proof for a λCalculus with Interleaving Inductive Types

Lecture Notes in Computer Science, 2000

We present a new strong normalisation proof for a λ-calculus with interleaving strictly positive inductive types λ µ which avoids the use of impredicative reasoning, i.e., the theorem of Knaster-Tarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based-a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel.