Lambda cube

Abstract: A higher-order calculus ECC (extended calculus of constructions) is presented which can be seen as an extension of the calculus of constructions by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice proof-theoretic properties. An omega -set (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically. >

Abstract: An elementary, purely algebraic definition of model for the untyped lambda calculus is given. This definition is shown to be equivalent to the natural semantic definition based on environments. These definitions of model are consistent with, and yield a completeness theorem for, the standard axioms for lambda convertibility. A simple construction of models for lambda calculus is reviewed. The algebraic formulation clarifies the relation between combinators and lambda terms.

Abstract: A typed lambda calculus with categorical type constructors is introduced. It has a uniform category theoretic mechanism to declare new types. Its type structure includes categorical objects like products and coproducts as well as recursive types like natural numbers and lists. It also allows duals of recursive types, i.e. lazy types, like infinite lists. It has generalized iterators for recursive types and duals of iterators for lazy types. We will give reduction rules for this simply typed lambda calculus and show that they are strongly normalizing even though it has infinite things like infinite lists.