Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on F follows the rewriting system RLKsp and hence its interpretation in the algebra of terms on F follows the rewriting system R LKsp . This mix elimination system ELKsp is based on a mix elimination system proposed in [Pab90] and is in the tradition of those studied in [Gen38] Gir87] [GLT89], Tah92] Gal93] and [CRS96] We show in a later section that the set of transformations given in this section is exhaustive, which means that each mix inference with non mix inference premises occurring in a proof matches at least one left hand side of a mix elimination rule. We point out also ....

J.Y. Girard, Y. Lafont and P. Taylor, Proofs and types, Vol. 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge university press, (1989).

....to ML T . The most basic of these was the addition of function types, resulting in the computational calculus ML T . Adding a fixpoint type fix , coproduct types ff fi and a natural number type nat to the computational calculus ML T , we arrive at a system FIX= which extends Godel s system T [Gir89]. FIX= admits sound translations of Plotkin s PCF [Plo77] and we shall return to the topic of PCF translations in Chapter 6. We now formally define FIX= Signatures for FIX= Definition 2.5.1 A FIX= signature, denoted by Sg; is specified by: ffl A collection of types. The types are built up in ....

J.-Y. Girard. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989. Translated and with appendices by P. Taylor and Y. Lafont.

....and perhaps more surprising, result of this paper is then that all normal characterisations of (strong head) normalisation are consequences of the strong normalisation of cut elimination. Many strong normalisation results in the context of types use the technique of Computability Predicates [24, 18], which provides a means for proving termination of typeable terms using a predicate defined by induction on the structure of types. This technique has been widely used to study normalisation properties (or similar results) as for example in [20, 12, 15, 22, 19, 1, 2, 17, 7, 4, 16, 5] this list ....

....are straightforward by Definition 15. 4. STRONG NORMALISATION OF DERIVATION REDUCTION In this subsection, we will prove a strong normalisation result for derivation reduction. In order to prove that each derivation in is strongly normalisable with respect to D , a notion of computable [24, 18] derivations will be introduced. We will show that all computable derivations are strongly normalisable with respect to derivation reduction, and then that all derivations in are computable. Definition 19 (Computability Predicate) Comp (D) is defined recursively on types by: Comp (D : B M ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....intuitionistic logic with intuitionistic linear logic. The natural deduction presentation of the new logic then gave, by the Curry Howard correspondence, a mixed linear and non linear lambda calculus. At rst sight, one might be tempted to regard LNL logic as a logical atrocity without interest [GLT89]. I hope, however, that I have shown that this is not the case. LNL logic has a very natural class of categorical models and a well behaved proof theory in both its sequent calculus and natural deduction formulations. Given this, and the links with other research which were mentioned in the ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [8, 23, 61], combined with process algebraic reasoning techniques [11, 51, 53, 57, 66] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to significant ....

....via their normal forms. 4) Embedding, using Milner s encoding [49] of the simply typed l calculus with sums and products (l ; into our typed p calculus. The embedding is fully abstract w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [6, 18, 19, 23], automatically leading to SN in the source calculus. 5) Establishment of a basic interaction based liveness property in linear processes via their strong normalisability, bridging the traditional notion of SN and one of the basic properties in concurrent, interactive computation. Related Work. ....

Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989.

....Lemma 2 (Subject Reduction) If the typing judgment F M: F is derivable and M N then F N: F is derivable. Proposition 3 (Strong Normalization) If M is typable, then it is strongly normalizing, i.e. every reduction sequence starting from M is finite. Proof. By the reducibility method [18]. Let SN be the set of strongly normalizing terms, and define an interpretation of types as sets of terms as follows: for every base type A, IF D GI[ M N[whenever M Ass. M then for every N C IIFIl, mm x : N c IIFII M SSlwhenever M . 0 then MxO IIFII Observe that: CR1) IIFII ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, 1989.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 15, 42], combined with process algebraic reasoning [9, 35, 37, 41, 47] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....reasoning. 3. Embedding, using Milner s encoding [33] of the simply typed l calculus with sums and products (l ; into our typed p calculus. The embedding is fully abstract w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [15] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 12] Abramsky extends the CurryHoward correspondence to linear logic [14] using proof expressions and proves ....

Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989. 12

....a derivation of #. We shall refer to the eliminated formula # as the cut formula, the derivation # # as the ticket , and the step of replacing # # by a derivation # # # as a conversion. It is a rule over derivations, not a rule within a derivation. For discussion, see Girard [GLT89] Troelstra and Schwichtenberg [TS00] D #D # means that converts to # (in one step) while # means that there is a finite sequence of conversions 0 #D 1 # . #D n 1 = # . We let #D # mean that or # . A sequence of conversions is called a reduction ....

.... an introduction rule and the major premise of an elimination rule ( IE formulae ) Basing our conversion procedure on the elimination of IE formulae would make # non trivial, and make # both well founded and confluent (or Church Rosser) for proofs and discussion, see Tennant [Ten78] Girard [GLT89] Troelstra and Schwichtenberg [TS00] Note that premise minimality is just a special case of derivational minimality: the case where none of the formulae which occur at the leaf nodes of the derivation tree can be completely eliminated. Proof Trivial, from the definitions of argument, ....

J-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989.

....of strategies which gives us a notion of composition. Consider the example of addition again, with the type We would like to combine this with 3 : IN : 3 : IN x : IN; y : IN x y : IN y : IN 3 y : IN This is just substitution of 3 for x, or in logical terms, the Cut rule [16]. In order to represent this composition, we let the two strategies interact with one another. When add plays a move in the first copy of IN (corresponding to asking for the value of x) we feed it as an O move to the strategy for 3; conversely, when this strategy responds with 3 in IN , we feed ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1989. Cambridge Tracts in Theoretical Computer Science 7.

....the rewrite rules of Combinatory Logic are not recursive, so, in particular, satisfy the scheme. 3.2 The strong normalisation theorem We shall prove that, when the rewrite rules satisfy the general schema, every typeable term is strongly normalisable. This will be done using Tait Girard s method [22] and the techniques devised in [26] in order to cope with some of the difficulties that the presence of algebraic rewriting makes arise. From now on all the rewrite rules will be assumed to satisfy the general schema. In the following, a sequence P QP of elements will be denoted by P . ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....of jQj Theta jRj. Definition 6 Let Q and R be qualitative domains. A rigid embedding of Q into R is an injection f : jQj jRj such that, for all u jQj, one has u 2 Q iff f(u) 2 R. Thomas Ehrhard It is the canonical notion of substructure for qualitative domains. For more details, see [GLT]. Now we recall some definitions and results of [BE] Definition 7 A qualitative domain with coherence (qDC) is a pair (Q; C (Q) where Q is a qualitative domain and C (Q) is a subset of P fin (Q) satisfying the following properties: ffl if x 2 Q then fxg 2 C (Q) ffl if A 2 C (Q) and if B 2 ....

J.-Y. Girard, Y. Lafont, P. Taylor. Proofs and Types. vol. 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1991.

....true, this property must be formalized explicitly in LF. For a complete development of this (and related) theorems and their proofs consult [17] 7 Example F can be used to de ne new types, values, and the corresponding elimination principles for Booleans, natural numbers, pairs, and sum types [6]. By construction all formally encoded objects, types and kinds are well typed and well kinded, respectively. We demonstrate how to use our encoding of F by de ning Booleans values and the corresponding elimination principle as depicted in Figure 6. bool = 8 0 ( a : tp o: u : at a: j ) j ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....station can service cars with di erent owners using the same method. When using this function, rst supply it with the appropriate context. Then, apply the resulting function to the argument. This is the same way we use polymorphic functions in an explicitly polymorphic calculus (such as System F [9] or Abadi and Cardelli s second order object calculus [1] ss ll( car ) eff = gg)ff car = x gg The most important application of context polymorphic methods is in class constructors. In the object calculus [1] a constructor is a method of the object representing a class. ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....of proof nets, describing parsing processes. We want to present here our vision of the generation process based on the proposal of [dG et al..96] for semantic proof nets. The main features of this paper consist in the use of proof nets for lambek calculus, of the Curry Howard isomorphism [Howard80, Girard et al..88] of semantic proof nets with semantic expressions a la Montague [Montague74, Dowty et al..81] and in an algorithm for proof search with a target proof net. In this work, we do not consider the choice of lexical items from a given semantic expression the syntactic realization of which we want to ....

J.-Y. Girard, Y. Lafont et P. Taylor. -- Proofs and Types. -- Cambridge University Press, 1988, Cambridge Tracts in Theoretical Computer Science 7.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....reasoning. 3. Embedding, using Milner s encoding [34] of the simply typed l calculus with sums and products (l ; into our typed p calculus. The embedding is fully abstract w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [16] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 13] Abramsky extends the CurryHoward correspondence to linear logic [15] using proof expressions and proves ....

Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....reasoning. 3. Embedding, using Milner s encoding [34] of the simply typed l calculus with sums and products (l ; into our typed p calculus. The embedding is fully abstract w.r.t. the observational congruence of l ; justifying all commutative conversions and h equations [16] and automatically leads to SN in the source calculus. Related Work Strong normalisation in typed l calculi has been studied extensively in the past; detailed surveys can be found in [7, 13] Abramsky extends the CurryHoward correspondence to linear logic [15] using proof expressions and proves ....

Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989.

....in two di#erent ways how trees can be represented as lambda terms and how operations like f mir on these objects become lambda definable. The first method is from [22] The resulting data objects and functions can be represented by lambda terms typeable in the second order lambda calculus #2, see [51] or [8] 190 HENK BARENDREGT Definition 3.1. i) Let b, l, p be variables (used as mnemonics for bud, leaf and plus) Define # = # b,l,p : tree # term, where term is the collection of untyped lambda terms, as follows. #( b; #(leaf n) l #n#; #(t 1 t 2 ) p #(t 1 )#(t 2 ) Here ....

J-Y. Girard, Y. G. A. Lafont, and P. Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, 1989.

....of proof nets, describing parsing processes. We want to present here our vision of the generation process based on the proposal of [dG et al..96] for semantic proof nets. The main features of this paper consist in the use of proof nets for lambek calculus, of the Curry Howard isomorphism [Howard80, Girard et al..88] of semantic proof nets with semantic expressions alaMontague [Montague74, Dowty et al..81] and in an algorithm for proof search with a target proof net. In this work, we do not consider the choice of lexical items from a given semantic expression the syntactic realization of which we want to ....

J.-Y. Girard, Y. Lafont et P. Taylor. -- Proofs and Types. -- Cambridge University Press, 1988, Cambridge Tracts in Theoretical Computer Science 7.

....normalisation process: normalising a term in the language of sequent calculus takes a term involving cuts and yields a term containing no cuts. Equivalently, a proof involving cuts becomes a proof with no cuts. Full explanations of cut elimination can be found in Girard, Lafont and Taylor s book [GLT89] and Gallier s tutorial [Gal93] To a logician, the important property of sequent calculus is that in all the rules in the logic except cut, the formulas appearing above the line are proper subformulas of those below it. While natural deduction is more intuitive as a proof methodology, sequent ....

....in the polymorphic lambda calculus, which is known to be strongly normalising, we believe there is a proof of termination for this algorithm that enables the sum of product types to be positively recursive. Another proof of termination for cut elimination provided by Girard, Lafont and Taylor [GLT89] 3.5 Interlude: Recursion In this Section we describe two methods for adding recursion to a typed lambda calculus based language. The goal is to find a suitable framework for extending cut elimination to recursive programs. 3.5.1 Cyclic Terms The first method is strikingly simple: we just ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press (1989).

....context variable, allows expressions which are applicable to any context. Context abstraction is used to write methods which are polymorphic in or indi erent to the owners of its arguments. This is analogous to the use of polymorphic functions in an explicitly polymorphic calculus such as System F [19] or Abadi and Cardelli s second order object calculus [1] and bounded universal quanti cation [9, 10] There are two kinds of context abstraction, namely, a, p)a, and of course an application expression, b(p) We have found no application for the form ( p)a, except in its ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types.Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J-Y. Girard, P. Taylor and Y. Lafont, Proofs and Types, Cambridge Tracts in Theoretical Computer Science (Cambridge University Press, 1989).

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types.Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

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J.-Y. Girard et al., Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7 (Cambridge Univ. Press 1989)

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J.-Y. Girard. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989. Translated and with appendices by P. Taylor and Y. Lafont.

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Jean-Yves Girard, Paul Taylor, and Yves Lafont, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, 1989.

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1989. (p. 5)

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J.-Y. Girard. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989. Translated and with appendices by P. Taylor and Y. Lafont.

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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J.-Y. Girard, Y. Lafont, P. Taylor. Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, 1989.

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Girard J. Y., Lafont Y., and Taylor P. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press (1989).

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Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, Cambridge, UK, 1989.

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Girard, Lafont, Taylor [1989] Proofs and Types, Cambridge University Press (Cambridge Tracts in Theoretical Computer Science 7), Cambridge.

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3.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1989.

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Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989.

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J-Y. Girard, Y. Lafont, P. Taylor. Proofs and Types. vol. 7 of Cambridge Tracts in Theoretical Computer Science, 1991.

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

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Girard, J.-Y., Lafont Y. and Taylor, P., Proofs and Types, vol. 7 of Cambridge Tracts in Theoretical Computer Science, CUP, 1989.

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J. Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

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J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

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