The cube is a way of presenting the fine structure of Coquand's Calculus of Constructions relating that system to a variety of other typed lambda calculi. This idea is due to Henk Barendregt , and is presented in his Introduction to Generalised Type Systems which is my principle source in preparing this material. Pure Type Systems - distilling an essence of type assignment for the lambda calculus. These judgements assign a type to some term of the typed lambda calculus in the given context. The connections between the lambda-calculus and programming languages are diverse and pervasive.
Type systems are an important aspect of programming language design.
The list processing language LISP devised by John McCarthy in the late 1950's. LISP was the first of many functional programming languages which are closely related to the lambda calculus. It was and remains the dominant language for programming in the field of Artificial Intelligence. The lambda calculus was first devised by Alonzo Church , first to provide a foundation for mathematics and then to show the existence of unsolvable problems. It incorporated a polymorphic strict type system which owes its origin to work on the lambda calculus.