. We present a denition of untyped -terms using a heterogeneous datatype, i.e. an inductively dened operator. This operator can be extended to a Kleisli triple, which is a concise way to verify the substitution laws for -calculus. We also observe that repetitions in the denition of the monad as well as in the proofs can be avoided by using well-founded recursion and induction instead of structural induction. We extend the construction to the simply typed -calculus using dependent types, and show that this is an instance of a generalization of Kleisli triples. The proofs for the untyped case have been checked using the LEGO system. Keywords. Type Theory, inductive types, -calculus, category theory. 1 Introduction The metatheory of substitution for -calculi is interesting maybe because it seems intuitively obvious but becomes quite intricate if we take a closer look. [Hue92] states seven formal properties of substitution which are then used to prove a general substitution theor...

Machine-checked proofs of properties of programming languages have become a critical need, both for increased confidence in large and complex designs and as a foundation for technologies such as proof-carrying code. However, constructing these proofs remains a black art, involving many choices in the formulation of definitions and theorems that make a huge cumulative difference in the difficulty of carrying out large formal developments. The representation and manipulation of terms with variable binding is a key issue. We propose a novel style for formalizing metatheory, combining locally nameless representation of terms and cofinite quantification of free variable names in inductive definitions of relations on terms (typing, reduction,...). The key technical insight is that our use of cofinite quantification obviates the need for reasoning about equivariance (the fact that free names can be renamed in derivations); in particular, the structural induction principles of relations

Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

We investigate semantical normalization proofs for typed combinatory logic and weak -calculus. One builds a model and a function `quote' which inverts the interpretation function. A normalization function is then obtained by composing quote with the interpretation function. Our models are just like the intended model, except that the function space includes a syntactic component as well as a semantic one. We call this a `glued' model because of its similarity with the glueing construction in category theory. Other basic type constructors are interpreted as in the intended model. In this way we can also treat inductively defined types such as natural numbers and Brouwer ordinals. We also discuss how to formalize -terms, and show how one model construction can be used to yield normalization proofs for two different typed -calculi -- one with explicit and one with implicit substitution. The proofs are formalized using Martin-Lof's type theory as a meta language and mechanized using the A...

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...

This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents Martin-Lof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.

Abstract. We present the complete development, in Gallina, of the residual theory of β-reduction in pure λ-calculus. The main result is the Prism Theorem, and its corollary Lévy’s Cube Lemma, a strong form of the parallel-moves lemma, itself a key step towards the confluence theorem and its usual corollaries (Church-Rosser, uniqueness of normal forms). Gallina is the specification language of the Coq Proof Assistant[7, 11]. It is a specific concrete syntax for its abstract framework, the Calculus of Inductive Constructions[15]. It may be thought of as a smooth mixture of higher-order predicate calculus with recursive definitions, inductively defined data-types, and inductive predicate definitions reminiscent of logic programming. The development presented here was fully checked in the current distribution version Coq V5.8. We just state the lemmas in the order in which they are proved, omitting the proof justifications. The full transcript is available as a standard library in the distribution of Coq. 1

We present parametric higher-order abstract syntax (PHOAS), a new approach to formalizing the syntax of programming languages in computer proof assistants based on type theory. Like higherorder abstract syntax (HOAS), PHOAS uses the meta language’s binding constructs to represent the object language’s binding constructs. Unlike HOAS, PHOAS types are definable in generalpurpose type theories that support traditional functional programming, like Coq’s Calculus of Inductive Constructions. We walk through how Coq can be used to develop certified, executable program transformations over several statically-typed functional programming languages formalized with PHOAS; that is, each transformation has a machine-checked proof of type preservation and semantic preservation. Our examples include CPS translation and closure conversion for simply-typed lambda calculus, CPS translation for System F, and translation from a language with ML-style pattern matching to a simpler language with no variable-arity binding constructs. By avoiding the syntactic hassle associated with first-order representation techniques, we achieve a very high degree of proof automation. Categories and Subject Descriptors F.3.1 [Logics and meanings

Abstract. Higher-order logical frameworks provide a powerful technology to reason about object languages with binders. This will be demonstrated for the case of the λµ-calculus with two different binders which can most elegantly be represented using a third-order constant. Since cases of third- and higher-order encodings are very rare in comparison with those of second order, a second-order representation is given as well and equivalence to the third-order representation is proven formally. 1

Contents 1 Introduction 2 2 Type Theory as a Programming Language 3 2.1 Hello World in Type Theory . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Hiding and argument synthesis . . . . . . . . . . . . . . . . . . . . . 4 2.3 Using dependent types in programming . . . . . . . . . . . . . . . . 4 2.4 Higher-order sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Logic for free 8 3.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 Inductively defined relations . . . . . . . . . . . . . . . . . . . . . . . 13 4 ALF's Type Theory 14 4.1 Judgements of Type Theory . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Conventions