Type checking and normalisation

This thesis is about Martin-Löf's intuitionistic theory of types (type theory). Type theory is at the same time a formal system for mathematical proof and a dependently typed programming language. Dependent types are types which depend on data and therefore to type check dependently typed programming we need to perform computation(normalisation) in types. Implementations of type theory (usually some kind of automatic theorem prover or interpreter) have at their heart a type checker. Implementations of type checkers for type theory have at their heart a normaliser. In this thesis I consider type checking as it might form the basis of an implementation of type theory in the functional language Haskell and then normalisation in the more rigorous setting of the dependently typed languages Epigram and Agda. I investigate a method of proving normalisation called Big-Step Normalisation (BSN). I apply BSN to a number of calculi of increasing sophistication and provide machine checked proofs of meta theoretic properties

Formalization of Universal Algebra in Agda

In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Aspects of the theory of containers within automated theorem proving

This thesis explores applications of the theory of containers within automated theorem proving. Container theory provides a foundational analysis of data types as containers, specified by a type $S$ of shapes and a function P assigning to each shape its set of positions for data.More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. Container theory is interesting, in this context, for the following reasons. A mechanism for representing and reasoning with ellipsis (the dots in x_1, x_2, ... , x_n) in lists, existing in the literature, has proved to be very useful for formalisations involving abstractions. Success with this mechanism came by means of a meta-level representation through which many functions that normally require recursive definitions can be given explicit ones. As a result, not only can induction and generalisation be eliminated from proofs but, by means of an associated portrayal system, the resulting proofs are also intuitive and much closer to informal mathematical proofs. This ellipsis mechanism, however, is not based on any formal theory, making it rather exiguous in comparison with rival techniques. There also remains questions about its scope and applications. Our aim is to improve this ellipsis mechanism. In this connection, we hypothesize that the theory of containers provides a formal underpinning for such representations. In order to test our hypothesis, we identify limitations of the ellipsis mechanism and show how they can be addressed within the theory of containers. We subsequently develop a new reasoning system based on containers, which does not suffer from these limitations. This judicious container-based system endorses representations of polymorphic rewrite rules using arithmetic, which naturally lends itself to applications of arithmetic decision procedures. We exploit this facet to develop a new technique for deciding properties of lists. Our technique is developed within a quasi-container setting: shape maps are given as piecewise-linear functions, while a new representation is derived for re-indexing functions that obviates the need for dependent types, which are fundamental in a judicious container approach. We show that this new setting enables us to represent and reason about a large class of properties