The app function I defined has order two according to the definition of order in the WP article you cited. I have often been careless about the conventions regarding use of the terms rank, order, and degree: I apologise for any confusion that might result.Įither way, the apply function does not have higher rank, it's perfectly rank 1. But that is meta notation, and A and B are meta-level variables standing for an arbitrary simple type.īoth Church-style and Curry-style formalisms use schemes. It's true that when people talk about STLC types they often write down type "schemes" like A->B->B. So ML-style polymorphism, and system F, don't have much to do with the point I was making. Whereas terms with ML-style let statements cannot be translated into the STLC without blowing up the size of the term. Hindley-Milner-style STLC are exactly equivalent in terms of the lambda terms they define. Note that this is a more sophisticated form of polymorphism. I should have made this clear.Ī lambda calculus with H-M-style implicit let polymorphism is typically just called ML in the PL literature I invented the term, because I thought Hindley-Milner polymorphism would be more immediately comprehensible than the explanation I gave above about Church-style and Curry-style typing. There is no such thing as a "Hindley-Milner STLC" either. The original reference for the simply-typed lambda calculus is Curry & Feys (1958), which used Curry-style typing - this was natural since they wanted to model Hilbert-style proof theory, which uses a precisely analogous form of formula scheme. With the benefit of the Hindley-Milner algorithm, we see that there is a principal type, and this is the type obtained from algorithm W. Whereas Curry-style typing does not provide type information to lambda terms, being in a sense underspecified, which means that typable terms may admit many possible type assignments. Church-style typing decorates lambda terms with enough type annotations so that each typable term has exactly one non-schematic type. I don't think this assertion is defensible. The Curry–Howard correspondence implies that types can be constructed that express arbitrarily complex mathematical properties.The very meaning of "simply typed" is no polymorphism Simultaneous programming language and logic The system corresponds to the calculus of constructions whose derivative, the calculus of inductive constructions is the underlying system of the Coq proof assistant. For instance, a function that takes a positive integer n to all four forms of abstraction from the lambda cube: functions from terms to terms, types to types, terms to types and types to terms. The return type of a dependent function may depend on the value (not just type) of one of its arguments. Two common examples of dependent types are dependent functions and dependent pairs. In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". It is an overlapping feature of type theory and type systems. In computer science and logic, a dependent type is a type whose definition depends on a value.
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