## Syntax and Semantics of Dependent Types (1997)

Venue: | Semantics and Logics of Computation |

Citations: | 40 - 4 self |

### BibTeX

@INPROCEEDINGS{Hofmann97syntaxand,

author = {Martin Hofmann},

title = {Syntax and Semantics of Dependent Types},

booktitle = {Semantics and Logics of Computation},

year = {1997},

pages = {79--130},

publisher = {Cambridge University Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The El-operator is omitted. For example the \Pi-type is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...

### Citations

695 | A framework for defining logics - Harper, Honsell, et al. - 1993 |

471 |
The calculus of constructions
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Citation Context ...ical systems such as predicate logic and modal logic. The interested reader is referred to (Harper, Honsell, and Plotkin 1993). 2.2.3 The Calculus of Constructions The Calculus of Constructions (CC) (=-=Coquand and Huet 1988-=-) is a type theory with \Pi-types and a universe closed under impredicative quantification (U-Impr). The universe is traditionally denoted by Prop and the correspondingsEl-operator is either written P... |

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Category Theory for Computing Science
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Citation Context ...category of stages or worlds and of a presheaf as a set varying with these stages. The Yoneda embedding y : K ! K sending I to K(\Gamma; I) is a full and faithful embedding of the stages into K. See (=-=Barr and Wells 1990-=-) for more information on presheaves. 4.1 Presheaves as a CwF Our aim is to construct a CwF which has K as the category of contexts. This category has a terminal object given by ? I = f?g. We define t... |

265 |
Constructive mathematics and computer programming
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Citation Context ... one which even in the presence of identity types remains confined to intensional equality. Therefore, type theory together with identity types as defined here is called intensional type theory, see (=-=Martin-Lof 1982-=-). There exists another formulation of identity types in which one may conclude \Gamma ` M = N : oe from \Gamma ` P : Id oe (M; N ). This rule is called equality reflection and makes it possible to de... |

263 | Foundations of Constructive Mathematics - Beeson - 1985 |

261 | Programming in Martin-Löf’s Type Theory: An Introduction - Nordstrom, Petersson, et al. - 1990 |

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An intuitionistic theory of types: Predicative part
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Citation Context ... one wants to see type theory as a foundation of constructive mathematics which accordingly is to be justified by a philosophical argument rather than via an interpretation in some other system, see (=-=Martin-Lof 1975-=-;(1984)). For us the distinction between canonical and noncanonical elements is not important. However, we will use it to motivate further definitional equalities. A more pragmatic explanation for \Pi... |

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Citation Context ...flection. Martin-Lof invented his type theories with the aim of extending the propositions as types correspondence to predicate logic and to provide a universal language for constructive mathematics (=-=Martin-Lof 1984-=-; Martin-Lof 1975). A standard reference on Martin-Lof's type theories is (Nordstrom, Petersson, and Smith 1990). An implementation of extensional Martin-Lof type theory is the Nuprl system (Constable... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
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Citation Context ...basic notions, in particular categories, functors, natural transformations, isomorphisms, terminal objects, and the category of sets and functions. Reading the relevant parts of the first chapter of (=-=Lambek and Scott 1985-=-), for instance, should suffice to attain the required state of knowledge. If C is a category we write jCj or C for its collection of objects and C(A; B) for the collection of morphisms from A to B. W... |

134 | Higher-order modules and the phase distinction - Harper, Mitchell, et al. - 1990 |

109 |
Irlductively defined types
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Citation Context ...ructions: Luo's Extended Calculus of Constructions (ECC) implemented in the Lego system (Luo 1994; Luo and Pollack 1992) and the Calculus of Inductive Definitions (CID) implemented in the Coq system (=-=Coquand and Paulin-Mohring 1989-=-; Dowek et al. 1991). ECC extends the Calculus of Constructions by a sequence of universes U 0 , U 1 , : : : where each U i+1 contains a code for U i and Prop is contained in U 0 . Datatypes reside in... |

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- 1993
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Citation Context ...so form \SigmaX : U :SEM (X); the type of semigroups. More complex examples of this kind may be found in (Luo 1991). An application of this pattern to semantics of modules in functional languages is (=-=Harper and Mitchell 1993-=-). Under the propositions-as-types analogy we can view a universe also as a type of propositions. For instance, the type oe ! U can be viewed as an analogue to the power-set of oe. 2.1.7 Miscellaneous... |

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Citation Context ...y the two rules \Gamma ` oe type \Gamma ` L(oe) type \Gamma ` M : oe \Gamma ` l(M) : L(oe) In LF it would have to be rendered by two constants L : (Set)Set and l : (oe: Set; oe)L(oe). The Alf system (=-=Magnusson and Nordstrom 1994-=-) is based on the Logical Framework. It allows for the definition of types in Set simply by giving their 1 In loc. cit. and in the Alf system the type annotations in functional abstractions are omitte... |

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- Pitts
- 1993
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Citation Context ...ral device of pattern-matching on the form of the constructors. Finally, we can consider arbitrary theories of dependent types defined by type symbols, constants, and equations. This is described in (=-=Pitts 1997). 2.2 Exa-=-mples of type theories In this section we briefly describe some "named" type theories and how they fit into the formal framework described here. 2.2.1 Martin-Lof's type theory This is a coll... |

80 |
Categories for Types
- Crole
- 1993
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Citation Context ...fine syntax. As long as these type formers are supported by T (\Sigma and N are) the conservativity theorem continues to hold by the same proof. Using a dependent version of the glueing construction (=-=Crole 1993-=-) it is possible to obtain the stronger property that the term M 0 in Thm. 4.1 is T LF -equal to M .sHofmann We also remark that we have not used any particular properties of the term model in the con... |

74 | Implementing Mathematics with the Nuprl Development System - Constable - 1986 |

60 |
Metamathematical investigations of a calculus of constructions. Rapport de recherche de l’INRIA
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Citation Context ...e:P ) \Gamma ` Choice(H) : oe It is known that such choice function becomes inconsistent if in addition one imposes the equation ` Choice(9-I(M; N)) = M : oe for all M : oe and N : Prf (P [M ]), see (=-=Coquand 1990-=-). 4 Extended example: presheaf models In this section we encounter a family of interpretations of type theory which generalises the set-theoretic model in that types are interpreted as variable sets ... |

56 | Locally cartesian closed categories and type theory - Seely - 1984 |

49 |
Extensional concepts in intensional type theory
- Hofmann
- 1995
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Citation Context ... congruence property in the case thatsdepends on x; for this and other derived properties and combinators for propositional equality we refer to (Nordstrom, Petersson, and Smith 1990; Streicher 1993; =-=Hofmann 1995-=-a). We have now collected enough material to carry out the promised example of a proof by induction. We wish to construct an element of the type m: N ` Id N (m + 0; m) typesHofmann where + is the addi... |

43 |
Extracting F# 's programs from proofs in the Calculus of Constructions
- Paulin-Mohring
- 1989
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Citation Context ...o's Extended Calculus of Constructions (ECC) implemented in the Lego system (Luo 1994; Luo and Pollack 1992) and the Calculus of Inductive Definitions (CID) implemented in the Coq system (Coquand and =-=Paulin-Mohring 1989-=-; Dowek et al. 1991). ECC extends the Calculus of Constructions by a sequence of universes U 0 , U 1 , : : : where each U i+1 contains a code for U i and Prop is contained in U 0 . Datatypes reside in... |

42 |
The theory of constructions: categorical semantics and topostheoretic models
- Hyland, Pitts
- 1987
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Citation Context ... are essentially equivalent as far as interpretation of type theory in them is concerned. Locallycartesian closed categories (Seely 1984) and categories with display maps (Taylor 1986; Lamarche 1987; =-=Hyland and Pitts 1989-=-) are less general than CwFs because semantic types are identified with their associated projections. Usually, in these models the conditions corresponding to (Ty-Comp) and (Ty-Id) only hold up to iso... |

42 |
Categorical Type Theory
- Jacobs
- 1991
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Citation Context ...ponding to (Ty-Comp) and (Ty-Id) only hold up to isomorphism, which makes the definition of the interpretation function more complicated. See (Hofmann 1995b; Curien 1993). Models based on fibrations (=-=Jacobs 1991-=-; Jacobs 1993; Ehrhard 1988) and indexed categories (Curien 1989; Obtu/lowicz 1989) are more general since they allow for morphisms between semantic types. These morphisms make it possible to describe... |

39 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1995
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Citation Context ... congruence property in the case thatsdepends on x; for this and other derived properties and combinators for propositional equality we refer to (Nordstrom, Petersson, and Smith 1990; Streicher 1993; =-=Hofmann 1995-=-a). We have now collected enough material to carry out the promised example of a proof by induction. We wish to construct an element of the type m: N ` Id N (m + 0; m) typesHofmann where + is the addi... |

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Citation Context ...rmal proposition then we define the set [OE] by [OE] j f?g if OE is true and [OE] j ; if OE is false. 3.1 Categories with families We choose the semantic framework of categories with families (CwFs) (=-=Dybjer 1996-=-) a variant of Cartmell's categories with attributes which have the advantage of being equationally defined, rather than using conditional equations. Furthermore, CwFs are closer to the syntax than ca... |

28 |
Semantics of Type Theory
- Streicher
- 1991
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Citation Context ...ablish the subject reduction property for an untyped rewrite system derived from definitional equality. Also, in many models rule U-\Pi-Ty is not valid under the canonical interpretation of \Pi, see (=-=Streicher 1991-=-) and the example following Def. 3.20. Closure under natural numbers is described by ` \Gamma ctxt \Gamma ` N : U U-N and further rules introducing term formers 0,sSuc, andsR N witnessing that El( N) ... |

28 |
Investigations into intensional type theory
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- 1993
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Citation Context ...derive a similar congruence property in the case thatsdepends on x; for this and other derived properties and combinators for propositional equality we refer to (Nordstrom, Petersson, and Smith 1990; =-=Streicher 1993-=-; Hofmann 1995a). We have now collected enough material to carry out the promised example of a proof by induction. We wish to construct an element of the type m: N ` Id N (m + 0; m) typesHofmann where... |

25 |
The groupoid model refutes uniqueness of identity proofs
- Hofmann, Streicher
- 1994
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Citation Context ...tic methods We give some directions for further reading on the subject of semantical methods in the study of theories of dependent types. Independence results are the subject of (Streicher 1992) and (=-=Hofmann and Streicher 1994-=-). Semantical methods in order to derive syntactic properties of type theories like strong normalisation and thus decidability of type checking have been used in (Hyland and Ong 1993; Altenkirch 1994;... |

24 | An operational semantics for - Borba, Goguen - 1994 |

23 | A category-theoretic account of program modules
- Moggi
- 1989
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Citation Context ...ons-R) = M so we are done.sThe pullback property of substitution may be taken as primitive thereby making substitution on terms superfluous. Definition 3.10 A category with attributes (Cartmell 1978; =-=Moggi 1991-=-; Pitts 1997) consists of ffl A category C with terminal object ?. ffl A functor Ty : C op ! Set , i.e., a set Ty (\Gamma) for each \Gamma 2 C and a function \Gammaff g : Ty (\Gamma) ! Ty(B) for each ... |

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Categorical abstract machines for higher-order typed lambdacalculi
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Citation Context ...gnment D to a contravariant functor from C to the category of categories. Such a functor is called an indexed category and forms the heart of Curien and Ehrhard's notion of D-categories (Curien 1989; =-=Ritter 1992-=-). 3.3 Semantic type formers In order to interpret a type theory in a CwF we must specify how the various type and term formers are to be interpreted. This results in certain requirements on a CwF whi... |

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Substitution up to isomorphism
- Curien
- 1993
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Citation Context ...lly, in these models the conditions corresponding to (Ty-Comp) and (Ty-Id) only hold up to isomorphism, which makes the definition of the interpretation function more complicated. See (Hofmann 1995b; =-=Curien 1993-=-). Models based on fibrations (Jacobs 1991; Jacobs 1993; Ehrhard 1988) and indexed categories (Curien 1989; Obtu/lowicz 1989) are more general since they allow for morphisms between semantic types. Th... |

17 | The independence of Peano’s fourth axiom from Martin-Lof’s type theory without universes
- Smith
- 1988
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Citation Context ...using R N a function f : N ! U such that \Pi ` f0 =s1 : U and \Pi ` f(Suc(0)) =s0 : U . Later on we will show by a semantic argument that the above type is not inhabited in the absence of a universe (=-=Smith 1988-=-). E2.6 (Troelstra and van Dalen 1988) Show that in type theory without the empty type 0 such an empty type can be defined as Id N (0; Suc(0)). The elimination operator R 0 oe must then be defined by ... |

16 |
Computation and Reasoning
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- 1994
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Citation Context ... R \Sigma to those cases where the type ae does not depend on \Sigmax: oe: . One can then still define the first projection, but no longer the second one. This is called weak \Sigma-elimination, see (=-=Luo 1994-=-). Important special cases of \Pix: oe: and \Sigmax: oe: arise whensdoes not actually depend on oe. In this case, i.e. when \Gamma ` oe type and \Gamma `stype we write oe !sdef = \Pix: oe: and oe \The... |

16 | On functors expressible in the polymorphic typed lambda calculus
- Reynolds, Plotkin
- 1990
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Citation Context ...= [x = tt]. If p : \Gamma:oe ! Prop then we define 8 oe (p)(fl) def = ["p(fl; x) = tt for all x 2 oe fl "]. Notice that in this model all elements of a type Prf (A) are equal. By "Reyno=-=lds' paradox" (Reynolds and Plotkin 1988-=-) this has to be so in any set-theoretic interpretation of the Calculus of Constructions. In x3.4 below we describe a model where this is not the case. Notice that Prf f8 oe (p)g is not equal to \Pi(o... |

15 |
Program Verification in Synthetic Domain Theory
- Reus
- 1995
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Citation Context ...rpretations of type theory in order to establish the consistency of general recursion and fixpoint combinators with dependent types (Palmgren and StoltenbergHansen 1990). In a similar direction goes (=-=Reus 1995-=-) where an interpretation of type theory using synthetic domain theory has been employed to establish the consistency of a very powerful dependent type theory incorporating higher-order logic, general... |

14 |
Recursive domains, indexed category theory and polymorphism
- Taylor
- 1986
(Show Context)
Citation Context ...otions of model all of which are essentially equivalent as far as interpretation of type theory in them is concerned. Locallycartesian closed categories (Seely 1984) and categories with display maps (=-=Taylor 1986-=-; Lamarche 1987; Hyland and Pitts 1989) are less general than CwFs because semantic types are identified with their associated projections. Usually, in these models the conditions corresponding to (Ty... |

12 |
V.Stoltenberg-Hansen, Domain Interpretations of MartinLöfs Partial Type Theory
- Palmgren
- 1990
(Show Context)
Citation Context ...ice of pre-syntax to establish coherence of the semantics with respect to the conversion rules, but this would result in a more complicated proof. For a particular model this has been carried out in (=-=Palmgren and Stoltenberg-Hansen 1990-=-). Exercises E3.38 Extend the interpretation function to \Sigma-types E3.39 Formulate a completeness theorem for the semantics and use the term model to prove it. Deduce from the term model that if \G... |

11 | Proving strong normalization of CC by modifying realizability semantics. In Types for proofs and programs
- Altenkirch
- 1993
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Citation Context ...d Streicher 1994). Semantical methods in order to derive syntactic properties of type theories like strong normalisation and thus decidability of type checking have been used in (Hyland and Ong 1993; =-=Altenkirch 1994-=-; Goguen 1995). In (Moggi 1991) and (Harper, Mitchell, and Moggi 1990) categories with attributes are used to give an account of higher-order modules in functional programming. There is an intriguing ... |

10 |
Categorical Models of Polymorphism
- Asperti, Martini
- 1992
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Citation Context ...ory which generalises the set-theoretic model in that types are interpreted as variable sets (presheaves) or families of such. There are various applications of such interpretations, see for example (=-=Asperti and Martini 1992-=-) and (Altenkirch, Hofmann, and Streicher 1996) where they are used to define models of the polymorphic lambda calculus in which type quantification is interpreted as cartesian product. We use a presh... |

10 |
A uniform approach to type theory
- Huet
- 1990
(Show Context)
Citation Context ... assume the existence of a capture-free substitution function on these equivalence classes. One can use a de Bruijn style presentation of the syntax to avoid this identification. A good reference is (=-=Huet 1990-=-). The de Bruijn presentation gives rise to canonical representatives of the equivalence classes and yields an algorithm implementing capture-free substitution.sHofmann 2.1 Type formers Type and term ... |

8 | Conservativity of equality reflection over intensional type theory - Hofmann - 1995 |

6 |
Une s'emantique cat'egorique des types d'ependants. Application au calcul des constructions. Th`ese de doctorat, Universit'e de Paris VII
- Ehrhard
- 1988
(Show Context)
Citation Context ...Ty-Id) only hold up to isomorphism, which makes the definition of the interpretation function more complicated. See (Hofmann 1995b; Curien 1993). Models based on fibrations (Jacobs 1991; Jacobs 1993; =-=Ehrhard 1988-=-) and indexed categories (Curien 1989; Obtu/lowicz 1989) are more general since they allow for morphisms between semantic types. These morphisms make it possible to describe certain type formers more ... |

4 |
Comprehension categories and the semantics of type theory
- Jacobs
- 1993
(Show Context)
Citation Context ...y-Comp) and (Ty-Id) only hold up to isomorphism, which makes the definition of the interpretation function more complicated. See (Hofmann 1995b; Curien 1993). Models based on fibrations (Jacobs 1991; =-=Jacobs 1993-=-; Ehrhard 1988) and indexed categories (Curien 1989; Obtu/lowicz 1989) are more general since they allow for morphisms between semantic types. These morphisms make it possible to describe certain type... |

3 |
Deliverables: An approach to program semantics in constructions
- Burstall, McKinna
- 1990
(Show Context)
Citation Context ...ules in functional programming. There is an intriguing connection with Paulin's work on program extraction in type theory (Paulin-Mohring 1989) and the "deliverables" approach to program dev=-=elopment (Burstall and McKinna 1993-=-). In each of these works a type is modelled as a type or a set together with a predicate or a dependent type and terms are modelled as terms which preserve these predicates. A similar interpretation ... |

3 |
A simple model for the theory of constructions
- Lamarche
- 1989
(Show Context)
Citation Context ...el all of which are essentially equivalent as far as interpretation of type theory in them is concerned. Locallycartesian closed categories (Seely 1984) and categories with display maps (Taylor 1986; =-=Lamarche 1987-=-; Hyland and Pitts 1989) are less general than CwFs because semantic types are identified with their associated projections. Usually, in these models the conditions corresponding to (Ty-Comp) and (Ty-... |

3 |
Dependence and independence results for (impredicative) calculi of dependent types
- Streicher
- 1992
(Show Context)
Citation Context ...applications of semantic methods We give some directions for further reading on the subject of semantical methods in the study of theories of dependent types. Independence results are the subject of (=-=Streicher 1992-=-) and (Hofmann and Streicher 1994). Semantical methods in order to derive syntactic properties of type theories like strong normalisation and thus decidability of type checking have been used in (Hyla... |

2 | Categorical and algebraic aspects of martin-lof type theory - Obtulowicz - 1989 |

1 |
A non-type-theoretic account of Martin-Lof's types
- Allen
- 1987
(Show Context)
Citation Context ...e might want to consider App [x:N]N (0; 0) as a term albeit not a well-typed one. These preterms, -types, and-contexts have been used to give semantics to type theory in terms of untyped computation (=-=Allen 1987-=-; Martin-Lof 1984); we will use them as an auxiliary device in the definition of the interpretation of type theory in semantic structures and also in the definition of context morphisms below. The pre... |

1 | Semantics of Dependent Types 51 - Syntax - 1996 |

1 |
Generalized Algebraic Theories and Contextual
- Cartmell
- 1978
(Show Context)
Citation Context ...ion of q and (Cons-R) = M so we are done.sThe pullback property of substitution may be taken as primitive thereby making substitution on terms superfluous. Definition 3.10 A category with attributes (=-=Cartmell 1978-=-; Moggi 1991; Pitts 1997) consists of ffl A category C with terminal object ?. ffl A functor Ty : C op ! Set , i.e., a set Ty (\Gamma) for each \Gamma 2 C and a function \Gammaff g : Ty (\Gamma) ! Ty(... |