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