Thierry Coquand. An Algorithm for Testing Conversion in Type Theory. In G'erard Huet and G. Plotkin, editors, Logical frameworks, pages 255--277. Cambridge University Press, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....notion of equality is decidable. However, the conversions pose a special diculty because of the lack of con uence for reduction in the case of non well typed terms. Since LF typing is dependent on equality, the attempt to de ne an equality based on reduction leads to a Catch 22. Coquand [Coq91] tests convertibility in LF using untyped reduction and extensionality, which is applied when comparing a abstraction to a non abstraction. However, this method fails when a unit type is present as in LLF because it may be necessary to apply extensionality even when neither of the terms ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

....of fn fn x : #1 .e : #1 #1 # of app e1 e2 : #1 of z z : nat e : nat of s s e : nat e : nat # e1 : # #,x : nat of case case e of z of fix fix x : #. e : # Figure 1: Inference rules for Mini ML ## conversion as the notion of definitional equality [5, 3]. Dependent functions that map objects of type A1 to A2 are written as #x : A1 . A2 . Following standard terminology, types that are indexed by other LF objects are referred to as type families. In the LF logical framework we represent judgments as types and derivations as objects. For example, ....

T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....5. 2 LF The type theoretic foundation for this paper is the logical framework LF [8] In addition to the standard syntactic categories of objects, types, and kinds, we will also use substitutions in a critical way throughout this paper so we briefly introduce them here (see also, for example, [1]) imp : o o o. forall : i o) o. k : hil (imp A (imp B A) s : hil (imp (imp A (imp B C) imp (imp A B) imp A C) mp : hil (imp A B) hil A hil B. Fig. 1. The Propositional Hilbert Calculus Kinds K : type #x:A. K Atomic Types B : a B M Types A : B ....

....identify # equivalent terms. In order to state certain definitions and propositions more concisely, we write U to stand for either an object or a type and V for either a type or a kind and h for a family level or object level constant. We take ## conversion as the notion of definitional equality [8, 1], for which we write U # and V V # . Substitutions are capture avoiding and written as U [#] or V [#] with the special form U [M x] and V [M x] Often, we write # for contexts that are interpreted existentially, and # for universal ones. When we write # [#] it is a shorthand for applying # ....

T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....kinds are definable, and show that every constructor has a principal (most specific) kind. In Section 3 we present a sound algorithm for determining equivalence of well formed constructors. We were inspired by Coquand s approach to 3rt equivalence for a type theory with II types and one universe [3]. Coquand worked with an algorithm which directly decides equivalence, rather than using a confluent and strongly normalizing reduction relation. However, in contrast to Coquand s system we cannot compare terms by their shape alone; we must take ac count of both the context and the classifier. ....

....4 K and F, a:Kx b Lx 4 L Figure 8: A Simplified Algorithm be expressed, in comparison to the simpler defmitional constralnts allowed by singletons. The cost of identity types is that typechecking becomes substantially more difficult, if not undecidable. Our proof was inspired by that of Coquand [3], but as the equivalence considered there was not context sensitive in any way our algorithm and proof are substantially different. Because of the validity logical relations and the form of the symmetry and transitivity properties for logical equivalence, our attempts to use more traditional ....

Thierry Coquand. An Algorithm for Testing Conver- sion in Type Theory. In Grard Huet and G. Plotkin, editors, Logical frameworks, pages 255-277. Cambridge University Press, 1991.

....of equality is decidable. However, the # conversions pose a special di#culty because of the lack of confluence for ## reduction in the case of non well typed terms. Since LF typing is dependent on equality, the attempt to define an equality based on ## reduction leads to a Catch 22. Coquand [Coq91] tests ## convertibility in LF using untyped # reduction and extensionality, which is applied when comparing a # abstraction to a non abstraction. However, this method fails when a unit type is present as in LLF because it may be necessary to apply extensionality even when neither of the terms ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....# expansion of # . Our algorithm proceeds in the recursive call by declaring the shared parameter and unifying the body of the original abstraction with the body of the # expansion. This trick for dealing with # equivalence is inspired by Coquand s algorithm for testing conversion in Type Theory [12]. 304 (U 14) If both types are definable types, we simply unify the definable types. Q#d=#(#) # Q#d=# # Q##(#) d # Q##=d # (U 15) U 16) These rules are symmetric and similar to Rules (U 12) and (U 13) If one of the types is a definable type, but the other is a type name, then ....

Thierry Coquand. An algorithm for testing conversion in Type Theory. In G.HuetandG.D.Plotkin,editors,Logical Frameworks. Cambridge University Press, 1991.

....forms for well typed terms were then established by giving a set of algorithmic judgments that are sound and complete with respect to de nitional equality and can be instrumented to extract canonical forms from the terms they compare. The algorithm is similar to one introduced by Coquand [2], except that Coquand s algorithm performs expansion based on the shapes of terms, while HP s is directed by types and kinds. The type directed nature of the algorithm has the advantage of making it scalable to theories such as LLF that contain unit types. Apart from the typed, declarative ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

.... 2 LF The type theoretic foundation for this paper is the logical framework LF [9] In addition to the standard syntactic categories of objects, types, and kinds, we will also use substitutions in a critical way throughout this paper so we briefly introduce them here (see also, for example, [2]) Kinds K : type #x:A. K Atomic Types B : a B M Types A : B #x:A 1 . A 2 Objects M : x M 1 M 2 #x:A. M Signatures # : #, a : K #, c : A Contexts # : #, x : A Substitutions # : #, M x We write a for constant type families, x or u for object level ....

....identify # equivalent terms. In order to state certain definitions and propositions more concisely, we write U to stand for either an object or a type and V for either a type or a kind and h for a family level or object level constant. We take ## conversion as the notion of definitional equality [9, 2], for which we write U # and V V # . Substitutions are capture avoiding and written as U [#] or V [#] with the special form U [M x] and V [M x] Often, we write # for contexts that are interpreted existentially, and # for universal ones. When we write # [#] it is a shorthand for applying # ....

T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91]. Coquand s approach is based on analyzing the shapes of terms, building in the principle of extensionality instead of relying on # reduction or expansion. This algorithm improves on reduction based approaches by avoiding explicit computation of normal forms, and allowing for early termination ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991. 38

....kinds are definable, and show that every constructor has a principal (most specific) kind. In Section 3 we present a sound algorithm for determining equivalence of well formed constructors. Wewere inspired by Coquand s approachtofij equivalence for a type theory with Pi types and one universe [3]. Coquand worked with an algorithm which directly decides equivalence, rather than using a confluent and strongly normalizing reduction relation. However, in contrast to Coquand s system we cannot compare terms by their shape alone# wemust take account of both the context and the classifier. Where ....

.... A1 j A2 : K. 2. If ; p1 : K1 , p2 : K2,and; p1 p2 K then ; p1 j p2 : K. Theorem 5.4 (Decidability) 1. If ; A1 : K and ; A2 : K then ; A1 , A2 : K is decidable. 2. If ; K1 and ; K2 then ; K1 , K2 is decidable. 6 Related Work Our proof was inspired by that of Coquand [3], but because the equivalence considered there was not context sensitive in anyway our algorithm and proof are substantially different. Because of the validity logical relations and the form of the symmetry and transitivity properties for logical equivalence, our initial attempts to use more ....

Thierry Coquand. An algorithm for testing conversion in Type Theory.InG'erard Huet and G. Plotkin, editors, Logical frameworks, pages 255--277. Cambridge University Press, 1991.

....E) T. Hypothetical judgments that add new hypotheses to # are encoded using higher order techniques, as the functional argument to tp lam shows. Following standard practice [Pfe91] we omit all implicit # abstractions from types and we take ## conversion as the notion of definitional equality [Coq91] Finally, we give an operational semantics for this language which is a standard and call by value. eval : exp type. ev lam : eval (lam E) lam E) ev app : eval (E 1 # V 2 ) V eval E 2 V 2 eval E 1 (lam E 1 # ) eval (app E 1 E 2 ) V. ev fix : eval (E (fix E) V eval (fix ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....be unique and may vary during reduction see [18] for more details. The solution may be to follow the same approach as indicated for the empty type and consider rewrite relations acting, not on terms, but on term judgements. However some partial results have been obtained by other researchers [12,26,35] (some using j contractions) and we hope that we shall be able to put a case forward for j expansions. Another way to increase the expressivity of our example calculi is to add recursive types such as natural numbers and lists. The following expansionary j rewrite rule was derived in Chapter 2 by ....

T. Coquand. An algorithm for testing conversion in type theory. In G. P. Huet and G.D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....kinds are definable, and show that every constructor has a principal (most specific) kind. In Section 3 we present a sound algorithm for determining equivalence of well formed constructors. We were inspired by Coquand s approach to fij equivalence for a type theory with Pi types and one universe [3]. Coquand worked with an algorithm which directly decides equivalence, rather than using a confluent and strongly normalizing reduction relation. However, in contrast to Coquand s system we cannot compare terms by their shape alone; we must take account of both the context and the classifier. ....

....: K2 , and Gamma p1 p2 K then Gamma p1 j p2 : K. Theorem 5.4 (Decidability) 1. If Gamma A1 : K and Gamma A2 : K then Gamma A1 , A2 : K is decidable. 2. If Gamma K1 and Gamma K2 then Gamma K1 , K2 is decidable. 6 Related Work Our proof was inspired by that of Coquand [3], but because the equivalence considered there was not context sensitive in any way our algorithm and proof are substantially different. Because of the validity logical relations and the form of the symmetry and transitivity properties for logical equivalence, our initial attempts to use more ....

Thierry Coquand. An algorithm for testing conversion in Type Theory. In G'erard Huet and G. Plotkin, editors, Logical frameworks, pages 255--277. Cambridge University Press, 1991.

....is di erent from other presentations, which usually require those types to be syntactically identical, not just equivalent; they then use a separate rule of conversion to reclassify N with type A, if indeed A . The test used for equivalence is the term directed, context independent one of [7], whose description is omitted here. It should be possible to extend the results to a context dependent test like that of [11] I. Classi cations: v : A A = lookup(v; type) type : kind ; x : A M : B x : A: B : x : A: M : x : A: B 2 ftype; kindg M : x : A: B N : ....

T. Coquand. An algorithm for testing conversion in Type Theory, pages 255-79. In Huet and Plotkin [12], 1991.

....is the b conversion of terms at all three levels. The definitional equality relation, j, between terms at each respective level is defined to be the symmetric and transitive closure of the parallel nested reduction relation. There is little difficulty (other than that for the lP calculus [6, 20]) in strengthening the definitional equality relation by the h rule. 2.1 Context Joining The method of joining two contexts is a ternary relation [X;G;D] defined as follows: JOIN) hi; hi;hi] JOIN ) X;x A;G;x A;D;x A] X;G;D] JOIN L) X;x:A;G;x:A;D] X;G;D] JOIN R) ....

T Coquand. An algorithm for testing conversion in type theory. In G Huet and G Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

....in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91]. Coquand s approach is based on analyzing the shapes of terms, building in the principle of extensionality instead of relying on j reduction or expansion. This algorithm improves on reduction based approaches by avoiding explicit computation of normal forms, and allowing for early termination ....

Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991. 12

....canonical forms for well typed terms were then established by giving a set of algorithmic judgments that are sound and complete with respect to de nitional equality and can be instrumented to extract canonical forms from the terms they compare. The algorithm is similar to one introduced by Coquand [2], except that Coquand s algorithm performs expansion based on the shapes of terms, while HP s is directed by types and kinds. The type directed nature of the algorithm has the advantage of making it scalable to theories such as LLF that contain unit types. Apart from the typed, declarative ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

....variables become bound. In our formalization, syntactic terms using named variables are themselves concrete: the names of bound variables actually occur (parametrically) in meta formulas containing them, just as the names of free variables do. This is done using a formulation suggested by Coquand [Coq91] based on syntactically distinguishing free from bound variables . Other work on formalization of binding and substitution using names includes [Coq96b, GM96, Owe95, Sat83, Sto88] but these do not work out any large examples using their binding notions. It would be interesting to compare our ....

Thierry Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....We write M : A for the LF typing judgment where is the standard LF context, M is an object and A its type. Every object in LF has a canonical ( normal long) form for which we write M A. LF s formulation is standard and we take conversion as the notion of de nitional equality [HHP93,Coq91] An induction principle for natural numbers in the LF setting is slightly more complicated then the one above, because dependencies among the declarations in must be respected. 1 ; 2 (z) P(z) 1 ; n 0 : nat; x 2 P(n 0 ) 2 (s n 0 ) P(s n 0 ) ind nat n 0 ;x : 1 ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255{ 279. Cambridge University Press, 1991.

....2 q imp : o o o imp is used as an in x operator throughout this paper. In LF, judgments are represented as types, and derivations as objects. Following standard practice [Pfe91] we omit all implicit abstractions from types and we take conversion as the notion of de nitional equality [Coq91] Example 1 (Representation of Figure 1 in LF) nd : o type impi : nd G 1 nd G 2 ) nd (G 1 imp G 2 ) impe : nd (G 1 imp G 2 ) nd G 1 nd G 2 A is a hypothetical judgment since the premiss to I in Figure 1 discharges the hypothesis u. A good choice for the representation of is ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255{ 279. Cambridge University Press, 1991.

....The signature is used below to declare the constants related to our encoding. Following standard practice [14] we assume substitutions to be capture avoiding and we omit all leading abstractions pre xes from types. convertibility is taken as the underlying notion of de nitional equality [2]. A K and A 1 A 2 are used as abbreviations for x : A: K and x : A 1 : A 2 if x does not occur free in K and A 2 , respectively. Sometimes we write A 2 A 1 for A 1 A 2 . As typing judgments for LF we write M : A if object M has type A in context , and M : c A if M is well typed and ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

....number(or literally, a value of type int(a) for some natural number a) and returns an integer. After elaboration, f is transformed into the following explicitly typed expression e f in Church typing style. x f : a : nat:int(a) int: i a : nat: e x : int(a) if( a] 0] x; 0) 1; f [a 1] a][1](x; 1) Note that we have assigned = and the following types, respectively. a : int: b : int:int(a) int(b) int(eq(a; b) and a : int: b : int:int(a) int(b) int(a b) We use eq(a; b) for the function that returns 1 and 0 depending on whether a equals b. In order to type check e f , ....

....All LF encodings presented in this paper are adequate, which means that canonical LF objects stand in one to one relation with derivations they represent. Following standard practice [11] we omit all implicit abstractions from types and we take conversion as the notion of de nitional equality [1]. For brevity reasons we discuss only one case: trans app : transform (of app D 1 D 2 ) r app P 1 P 2 ) r Q 0 2 ) of lam ( k: u: of app R 1 (of lam ( x 1 : u x1 : of app R 2 (of lam ( x 2 : u x2 : of app (of app u x1 u x2 ) u) transform D 2 P 2 (r Q 2 ) R 2 transform D 1 P ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gordon Plotkin and Gerard Huet, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

....of the type represented by s) Constructively, that function has really two arguments: the realisor, and the proof that it satisfies I , and the value of the function will depend on this proof. However the predicates arising from any pair of proofs should be extensionally equal. Coquand s model in [18] can be viewed in this way. 1 Here and below, the scope of quantifiers should be taken as extending as far as possible to the right. 28 These suggestions for a universal category determine metamathematical models, in which the judgments are interpreted as propositions about a mathematical ....

T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279, Cambridge, 1991. Cambridge University Press.

....syntax of LF, with algorithms for manipulating it, plus a standard algorithms for type inference and convertibility. It follows [Hue89] in many details; one difference is that the Constructive Engine does not implement j conversion, which we require in LF (we use the technique due to Coquand [Coq91] Another difference is the implementation language: we use the non strict functional language Haskell [Has] One might think that use of lazy languages implies slower programs, but with current technology (e.g. the GHC compiler [P a] we have no reason to regret our choice. The laziness ....

Th. Coquand. An algorithm for testing conversion in Type Theory. In G. Huet and G. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....7 before we conclude and assess results in Section 8. 2 Hypothetical Judgments In this paper, we restrict ourselves to the dependently typed meta language also known under the name LF. Its formulation is standard and we take conversion as the notion of de nitional equality [HHP93,Coq91] As running examples throughout this paper we use two proof calculi for the implicational fragment of intuitionistic logic: natural deduction and sequent calculus, and their representation in the logical framework. Although we present only one connective, the example scales to full rst order ....

Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255{ 279. Cambridge University Press, 1991.

....However, once record inclusion is formally stipulated it is also required that rules of subtyping have to be given for the rest of the type formers. In [Bet98] we have investigated an alternative formulation of the extended theory. In that formulation we make use of parameters, in the sense of [Coq91, Pol94b], to stand for generic objects of the various types. The introduction of the notion of parameter allows us to give a solution to the problems posed by the manipulation of free names in the presence of dependent types. We also present in [Bet98] the procedures for the mechanical verification of ....

....x:A with the declaration x:Px, but we are restrained from doing this by the criterion for context formation above. Relatively recent works on the construction of proof checkers for type theories with dependent types have addressed (in a direct manner or not) the problems presented above. In [Coq91] Coquand investigates the question of checking the formal correctness of judgements of type and object equality in a formulation of Martin Lof s set theory with generalized cartesian product and one universe. The notion of context in this theory is that of a list of assumptions of the form p:ff, ....

Th. Coquand. An algorithm for testing conversion in type theory. In Logical Frameworks, Huet G., Plotkin G. (eds.), pages 71--92. Cambridge University Press, 1991. 18

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Coquand, T. 1991. An algorithm for testing conversion in type theory. In Logical Frameworks, G. Huet and G. Plotkin, Eds. Cambridge University Press, 255--279.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255-279. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G. Huet and G.D. Plotk in, editors, Logical Frameworks. Cambridge Univ. Press, 1991.

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Thierry Coquand. An Algorithm for Testing Conversion in Type Theory. In G'erard Huet and G. Plotkin, editors, Logical frameworks, pages 255--277. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991. To appear.

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Thierry Coquand. An Algorithm for Testing Conversion in Type Theory. In G'erard Huet and G. Plotkin, editors, Logical frameworks, pages 255--277. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-- 279. Cambridge University Press, 1991.

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Coquand, Thierry. (1991). An algorithm for testing conversion in type theory. Logical Frameworks, 255 -- 279.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin (Eds.): Logical Frameworks, pp. 255--280, Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, ed., Logical Frameworks, p 255279. Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet, G. Plotkin, editors, Logical Frameworks, p. 255279. Cambridge Univ. Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255279. Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255279. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-- 279. Cambridge University Press, 1991.

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Coquand, Th. (1991). An algorithm for testing conversion in type theory. Pages 71--92 of: L Martin-Lofogical Frameworks, Huet G., Plotkin G. (eds.). Cambridge University Press.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge University Press, 1991.

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T. Coquand. An algorithm for testing conversion in type theory. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 255--279, Cambridge, 1991. Cambridge University Press.

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Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--277. Cambridge University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255--279. Cambridge 14 University Press, 1991.

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Thierry Coquand. An algorithm for testing conversion in type theory. In Gerard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 255-- 279. Cambridge University Press, 1991.

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