R. Harper and R. Pollack. "Type checking with universes." Theoretical Computer Science 89, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....but relatively small categories. Thus the construction of CAT above is consistent with the analysis by Coquand[16] of paradoxes related to the category of categories. 50 It is to be remarked that this example justifies the mechanism called universe polymorphism defined by Harper and Pollack[32]. That is, with universe polymorphism, we could directly define CAT as a Category, without having to make an explicit copy of the notion, the copying being done implicitly for each occurrence of the name Category. Coq does not implement universe polymorphism at present, because this mechanism is ....

R. Harper and R. Pollack. "Type checking with universes." Theoretical Computer Science 89, 1991.

....but relatively small categories. Thus the construction of CAT above is consistent with the analysis by Coquand[16] of paradoxes related to the category of categories. It is to be remarked that this example justifies the mechanism called universe polymorphism defined by Harper and Pollack[32]. That is, with universe polymorphism, we could directly define CAT as a Category, without having to make an explicit copy of the notion, the copying being done implicitly for each occurrence of the name Category. Coq does not implement universe polymorphism at present, because this mechanism is ....

R. Harper and R. Pollack. "Type checking with universes." Theoretical Computer Science 89, 1991.

....to the type theory. 1.2. Related Work This section surveys the implementations of universes in other proof assistants, and then compares these to Plastic in more general terms. ECC and its Russell style universes are implemented in Lego [35] together with a form of universe polymorphism [17]; the facility of typical ambiguity allows one to omit explicit universe level information and the implementation ensures that an consistent set of levels is used. It allows the flexibility of Type : Type, but without the inconsistency. The universes are integrated with Lego s inductive types; ....

....[Setoid:Type] Constructors [mk: ca:El Type0) eqr:El (EqRel (T0 ca) Setoid] 4.6. Discussion Readers may have realised that Setoid above is only usable with types having names in Type 0 . This is less attractive than ECC Lego with the universe polymorphism of Harper and Pollack [17] (henceforth H P) In Lego, one can use the symbol Type and the implementation assigns appropriate universe levels internally each time the definition is used. What can we do for setoids in LF First note that a small number of universe levels is sufficient in practice, and that the full power of ....

Harper, R. and R. Pollack: 1991, `Type Checking with Universes'. Theoretical Computer Science 89(1), 107--136.

....but relatively small categories. Thus the construction of CAT above is consistent with the analysis by Coquand[4] of paradoxes related to the category of categories. It is to be remarked that this example justifies the mechanism called universe polymorphism defined by Harper and Pollack[6]. That is, with universe polymorphism, we could directly define CAT as a Category, without having to make an explicit copy of the notion, the copying being done implicitly for each occurrence of the name Category. Coq does not implement universe polymorphism at present, because this mechanism is ....

R. Harper and R. Pollack. "Type checking with universes." Theoretical Computer Science 89, 1991.

.... extends the official syntax of ECC in (at least) four significant ways: Typical ambiguity The syntax is extended with an anonymous universe symbolType, freeing the user from having to specify universe levels, which are inferred by the system, subject to the constraints of predicativity [36, 86]; for example, Chapter 2. Type theoretic preliminaries: ECC and LEGO 26 [t:Type] x:t]x defines a polymorphic identity function at all levels; Definitions Both local and global definitions are available to the user. They are denoted by [x = M ] uniformly, which acts as a binder for local ....

R.Harper and R.A.Pollack, Type checking with universes, in: Theoretical Computer Science, Vol. 89, North-Holland, Amsterdam, 1991.

....and a pseudoterm M , it is decidable whether there exists a term A with Gamma M : A. If such a term A exists, it can be computed effectively. Some hints towards a proof can be found in [Coquand and Huet 1988] and more details in [Coquand 1985] and especially in [Martin Lof 1971] See also [Harper and Pollack 1991] for an exposition on the decidability of typing for an extended version of CC, which also describes an algorithm for computing a type. 4 The formulas as types embedding from higher order predicate logic into CC The Curry Howard embedding from higher order predicate logic into CC makes an ....

....one can t allow an impredicative Sigma type like Sigmaff:Prop: Prop. This means it is not possible, as can be done in the first order case, to represent the higher order existential quantification by a Sigma type. See [Coquand 1986] for a discussion on inconsistent extensions of CC, and [Harper and Pollack 1991] for a description of CC , CC extended with universes and universe inclusion. As a final extension of CC we want to point at the possibility of adding inductive types to the system. This can be useful because, although the system is very powerful from an extensional point of view (all ....

R. Harper and R. Pollack, Type checking with universes, TCS 89, pp 107-136.

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