Type theory has been used in organizing and clarifying programming language features. As such features become more complex we will need more advanced and powerful type systems like Martin-Loef's intuitionistic theory of types.
This paper investigates the use of such powerful type systems from a programming language perspective. To satisfy language design needs, these type systems must be extended so that their ordinary semantic theories are no longer applicable. A semantics is developed that justifies the extensions of Martin-Loef's type system with recursion and the Type:Type property.
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