Abstract
This chapter discusses the strong normalization in a typed lambda calculus with lambda structured types. The importance of lambda calculus for the development of recursive functions is also discussed. The calculus has also been brought in relation with the theory of ordinal numbers, predicate calculus, and other theories. The important issue in lambda calculus is the question of the normalization of terms. In lambda calculus, which allows all functions as arguments of functions, such a termination of the reduction is not guaranteed. If there is some reduction sequence which terminates, a term in lambda calculus is called normalizable. Normalization problems also arise in systems of typed lambda calculus investigated a lambda calculus with types and found all terms in this calculus to be strongly normalizable. A typed lambda calculus, in which the types themselves have lambda structure, is described. The typed lambda calculus Λ, has a large overlap with the mathematical language Automath. Normalization and strong normalization for the system is also discussed. An introduction of a method for deriving strong normalization from normalization together with the uniqueness of normal forms is also provided.
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