Thierry Coquand. Metamathematical Investigations of a Calculus of Constructions. In Piergiuorgio Odifreddi, editor, Logic and Computer Science, pages 91-122. Academic Press, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(M0 x) M ) The suitable term should be obtained from the hypothesis H. However, Coq does not allow us to eliminate a Proposition (like H) to build a term in a Set (M in tm) Such eliminations of strong # types may lead to inconsistencies, and hence are ruled out by the type theory CIC [3]. The solution we adopt is to move the whole proof in the Set realm, and then to lift the result to Prop. Therefore, we introduce a Set typed version of the induction principle which, equivalently, can be seen as a recursor with 14 Honsell et al. dependent types: Axiom tm rec1 : ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In Logic and Computer Science, volume 31, pages 91--122. Academic Press, 1990.

....by a grant from the Australian Research Council. Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Preprint submitted to Elsevier Science 6 April 2004 versions published by Coquand himself, one appears in [1,4,5] and another appears in [2], and still another appears in [3] One of the most distinctive versions in the literature is due to Seldin, which di#ers from the others in some important ways. Seldin had first learned of the calculus of constructions in early 1986, when he was working for Odyssey Research Associates. Richard ....

....of N for all occurrences of x in M with bound variables being changed as necessary to avoid collision. The corresponding conversion relation will be written M = # N. We do not consider # reduction or conversion. The two constants Prop and Type are called sorts. They are called kinds in [2] and earlier papers by Seldin. Unspecified sorts will be denoted here by s, s # , etc. so we always have s, s # , # Prop : Type . Formulas (called statements in PTSs) are of the form M : A, where M and A are pseudoterms. All of the versions have the same axiom: PT) Prop : Type. In ....

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T. Coquand, Metamathematical investigations of a calculus of constructions, february 9, 1987.

....proved in the next section. 3.3 Normalization We prove the Subject Reduction, Church Rosser and Strong Normalization properties simultaneously, following an idea proposed by Goguen [Gog94] for the Calculus of Constructions with reduction. Our proof is inspired by Coquand and Gallier s proofs [Gal90, Coq87]. 3.3.1 Elementary Properties of We rst give a few elementary properties which will be useful in the following sections. Lemma 2 For any context and any type A, A A holds. Lemma 3 For any context , and any types A 1 and A 2 such that A 1 . c A 2 , then A 1 A 2 and A 2 A 1 ....

Thierry Coquand. A meta-mathematical investigation of a Calculus of Constructions. Private Communication, 1987.

....rise to several typed calculi, where terms are decorated with types in various ways. Examples of typed calculi are the simply typed calculus ( of Church [Chu41] the second order calculus of Girard and Reynolds (2) Gir86, Rey74] and the Calculus of Constructions (P ) of Coquand and Huet [Coq91, CH88] Barendregt gave in [Bar92] a compact and appealing presentation of a class of typed systems, arranging them in a cube. In this cube, every vertex represents a different typed system. One vertex is the origin and represents the simply typed calculus; the three dimensions of the cube ....

T. Coquand. Metamathematical Investigations of a Calculus of Constructions. In P. Odifreddi, editor, Logic and Computer Science, pages 91--122. Academic press, 1991.

....large case, even constructing a reasonable candidate CPS counterpart fails. But one can actually prove a crisp result: no type correct CPS translation for CC can exist extending that for CC. Recall that in CC, a classical operator is inconsistent with dependently eliminated sum types, see e.g. [14, 21]. Indeed, such a combination allows one to construct a retract pair from to the type 1 1 of booleans. In CC , the type X : X (X 0) is inhabited by lem = X: x: x inr( z: x inl(z) The two functions : Bool , E : Bool forming the retract pair are = x: case (x; u: 1; ....

....(x; u: 1; u: 0) E = X: case (lem X; u: inl(hi) u: inr(hi) The proofs p1 : X : X (E X) p2 : X : E X) X showing that and E give a retract pair indeed are p1 = X: x: case (lem X; u: hi; u: u x) p2 = X: case (lem X; u: x: u; u: x: r(x) Further, T. Coquand [14] shows that retract pairs from to a small type yield an inconsistency in CC. Hence CC is inconsistent. Because of the inconsistency, there is a closed object M in CC such that M : 0. Now, if a type correct CPS translation existed for CC and therefore also for CC , then, in CC , we ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In [37], pp. 91-122.

....is normalising, then it is decidable whether Gamma e : A is derivable. See e.g. 6] for a recent survey of type checking algorithms for dependent type theory. Finally, the calculus is consistent. This can be established by a standard model construction, e.g. the proof irrelevance model of [17]. Proposition 12. There is no M such that A : Prop M : A. We conjecture, but do not prove, that every legal term is fifi oe strongly normalising. 4 Conclusion We have detailed a number of situations in which (a computational interpretation of) type isomorphisms allow proof reuse and ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In P. Odifreddi, editor, Logic and Computer Science, pages 91122. Academic Press, 1990.

....Research (N.W.O. y Partly supported by HCM project No. ERBCHRXCT920046 Typed Lambda Calculus z Partly supported by grants NSF CCR 9113196, KBN 2 1192 91 01 and by a grant from the Commission of The European Communities ERB CIPA CT92 2266(294) 11, 15] and the calculus of constructions [5, 6]. Barendregt [1] gave a compact and appealing presentation of a class of typed systems, arranging them in a cube. In this cube, every vertex represents a di erent typed system. One vertex is the origin and represents the simply typed calculus of Church; the edges represent the introduction of ....

Coquand, T., Metamathematical Investigations of a Calculus of Constructions, Logic and Computer Science, Odifreddi ed., Academic Press, 91-122, 1990.

.... for types is given by the context Gamma ACT ACTmake : Pi U; V : Type: tr U V ) U V; ACT check : 8U; V : Type:8R : tr U V:8x : U: R Delta rel x (ACTmake U V R x) It is well known that Gamma ACT is consistent, but not inhabited in the Calculus of Inductive Constructions, see e.g. [10]. 3.2 The axiom of choice for total setoids The axiom of choice for total setoids states that every total relation between total setoids induces a map of total setoids. Recall that a relation from A to B, where A and B are total setoids, consists of a type theoretical relation R : el t A) el ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In P. Odifreddi, editor, Logic and Computer Science, pages 91122. Academic Press, 1990.

....large case, even constructing a reasonable candidate CPS counterpart fails. But one can actually prove a crisp result: no type correct CPS translation for CC can exist extending that for CC. Recall that in CC, a classical operator is inconsistent with dependently eliminated sum types, see e.g. [14, 21]. Indeed, such a combination allows one to construct a retract pair from to the type 1 1 of booleans. In CC , the type X : X (X 0) is inhabited by lem = X: x: x inr( z: x inl(z) The two functions : Bool , E : Bool forming the retract pair are = x: case 2 (x; u: 1; ....

....(x; u: 1; u: 0) E = X: case (lem X; u: inl(hi) u: inr(hi) The proofs p1 : X : X (E X) p2 : X : E X) X showing that and E give a retract pair indeed are p1 = X: x: case (lem X; u: hi; u: u x) p2 = X: case (lem X; u: x: u; u: x: r(x) Further, T. Coquand [14] shows that retract pairs from to a small type yield an inconsistency in CC. Hence CC is inconsistent. Because of the inconsistency, there is a closed object M in CC such that M : 0. Now, if a type correct CPS translation existed for CC and therefore also for CC , then, in CC , we ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In [37], pp. 91-122.

....of type checking algorithms for dependent type theory. Proposition 11. If every legal term is normalising, then it is decidable whether Gamma e : A is derivable. Finally, the calculus is consistent. This can be established by a standard model construction, e.g. the proof irrelevance model of [19]. Proposition 12. There is no M such that A : Prop M : A. We conjecture, but do not prove, that every legal term is fifi oe strongly normalising. The most direct way to prove the conjecture seems to adapt the model Type Isomorphisms and Proof Reuse in Dependent Type Theory 13 ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In P. Odifreddi, editor, Logic and Computer Science, pages 91122. Academic Press, 1990.

.... type tr U V = h rel : U V Prop; total : 8x : U:9y : V: rel x yi The axiom of choice for types is given by the context ACT ACTmake : U;V : Type: tr U V ) U V; ACT check : 8U; V : Type:8R : tr U V:8x : U: R rel x (ACTmake U V R x) The following result is well known, see e.g. [21] and also [47] where an easy model of the Calculus of Inductive Constructions is given in which the ACT is valid. Proposition 1. ACT is consistent, but not inhabited in the Calculus of Inductive Constructions. We conclude this subsection with two observations on the validity of the axiom of ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In P. Odifreddi, editor, Logic and Computer Science, pages 91-122. Academic Press, 1990.

....(M0 x) M ) The suitable term should be obtained from the hypothesis H. However, Coq does not allow us to eliminate a Proposition (like H) to build a term in a Set (M in tm) Such eliminations of strong # types may lead to inconsistencies, and hence are ruled out by the type theory CIC [5]. The solution we adopt is to move the whole proof in the Set realm, and then to lift the result to Prop. Therefore, we introduce a Set typed version of the induction principle which, equivalently, can be seen as a recursor with dependent types: 11 Axiom tm rec1 : P: var tm) Set) P ....

T. Coquand. Metamathematical investigations of a calculus of constructions. In P. Odifreddi, editor, Logic and Computer Science, volume 31, pages 91--122. Academic Press, 1990.

....best way to promote flexibility in notation because it can lead to confusion. In any case, it is not a decisive reason to prefer a typed language to an untyped one, so we will not discuss it further. 3. 6 Constructive Type Theories A number of type theories, such as the Calculus of Constructions [Coquand 1990], have been designed as constructive alternatives to classical set theory. Constructive reasoning whether typed or not is concerned with what we can know, as opposed to what might be true out there [Dummett 1977] This shift of emphasis rejects basic laws of classical logic, even the ....

Coquand, T. 1990. Metamathematical investigations of a calculus of constructions. In P. Odifreddi (Ed.), Logic and Computer Science, pp. 91--122. Academic Press.

.... order dependent type theory The system of second order dependent type theory, P 2, is an extension of the polymorphic calculus with dependent types and it was rst introduced in [Longo and Moggi 1988] It can be seen as a subsystem of the Calculus of Constructions ( Coquand and Huet 1988] [Coquand 1990]) where the operations of forming type constructors are restricted to second order ones. So, one can quantify over type constructors of kind , but one can not form type constructors of kind ( It can also be seen as an extension of the rst order system P , where quanti cation over ....

Th. Coquand, Metamathematical investigations of a calculus of constructions. In Logic and Computer Science, ed. P.G. Odifreddi, APIC series, vol. 31, Academic Press, pp 91-122.

....within an index free setting. A larger class of algebraic types is de ned. Logical aspects need more examination. But we already give a syntactic way for dealing with partial and total objects, leading to the notion of generic proof. 1 Introduction The Calculus of Constructions (CC for short) 8] 9] is a typed high order functional calculus which provides a nice formalism for constructive proofs in natural deduction style. It can also be seen as a high level functional programming language. Since F is embeddable in it, we already know that any fonction over integers is de nable on ....

....constructive proofs in natural deduction style. It can also be seen as a high level functional programming language. Since F is embeddable in it, we already know that any fonction over integers is de nable on Nat (C : P rop)C (C C) C if it is provably total in higher order arithmetic [8] However Church s representation of natural numbers is not satisfactory. Computing the predecessor of n requires n steps of reduction. The principle of induction is neither provable nor realizable [14] The representation of recursive inductive types [15] seems to be very inecient from the ....

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Coquand T. (1990) Metamathematical investigations of a Calculus of Constructions. In Logic and Computer science. P. Odifreddi editor. Academic Press Limited. London.

....c) has property P = M has property P , ii) for any type M the set formed by all terms of type M having proprety P is saturated. Then any provably well formed term has property P . The proof requires many steps and de nitions fully given elsewhere ( 2] We give a sketch of it below, following [7] 3.3 Full premodel P rop; P rop; Type; Type are special constants in . Let C be a xed set of constants. Then I = C [ V is de ned as the set of identi ers of . A premodel is a well formed environment such that V ( C. Given a premodel 0 a list is a relative environment if ....

....x) can be seen as a generic proof . 6.2.4 Remark All the previous results can be given in a topological setting. Therefore terms in ANF are to be understood as compact terms for the corresponding topology. Since we restrict ourselves to syntactic tools this interpretation is studied in [2] 7 Conclusion Introduction of xpoints really enriches the Calculus of Constructions. As far as p.r. functions are concerned we get al..l the power of type free calculus with the advantages of type checking. Computations, like that of the predecessor of type Nat Nat, are made more ecient. Using ....

Coquand T. (1987) Metamathematical investigations of a Calculus of Constructions. Part I: Syntax. Unpublished ?

....can build arities on Type and there is no problem in the expression of the elimination scheme. Other possible strong eliminations. We cannot allow strong eliminations for non small inductive types without getting an inconsistency. This can be shown with an adaptation of the argument developed in [6] to this system. Assume the rule Nodep Set;Type is allowed for non small inductive definitions. Then we can construct B : Set with one constructor : Prop B. With the rule Nodep Set;Type we can build a projection E : B Prop and we will have (E ( A) and A convertible for each A of type ....

.... the system with a proof irrelevance semantics (all proofs of a proposition are identified) However this rule forbids the use of other natural principle for instance the axiom for extensionality (A; B : Prop) A B) B A) A = B or the excluded middle (A : Prop)A :A as shown by Berardi or Coquand [6]. We do not allow the rule Nodep Set;Type S although it could have very interesting applications. But this rule destroys the interpretation of proofs in Coq as non dependent programs of F extended with inductive definitions. We could for instance build a type T (n) such that T ( n) is the n ary ....

Th. Coquand. Metamathematical investigations of a Calculus of Constructions. In P. Oddifredi, editor, Logic and Computer Science. Academic Press, 1990. Rapport de recherche INRIA 1088, also in [11].

.... order dependent type theory The system of second order dependent type theory, P 2, is an extension of the polymorphic calculus with dependent types and it was rst introduced in [Longo and Moggi 1988] It can be seen as a subsystem of the Calculus of Constructions ( Coquand and Huet 1988] [Coquand 1990]) where the operations of forming type constructors are restricted to second order ones. So, one can quantify over type constructors of kind , but one can not 2 form type constructors of kind ( It can also be seen as an extension of the rst order system P , where quanti cation over ....

Th. Coquand, Metamathematical investigations of a calculus of constructions. In Logic and Computer Science, ed. P.G. Odifreddi, APIC series, vol. 31, Academic Press, pp 91-122.

....Paradigm Type theories were originally conceived by logicians at the beginning of this century, during the discussion about the foundations of Mathematics, with the aim of providing a well suited mathematical 18 language. Although several type theories have been developed since those years [60, 59, 16, 50], and there exists important differences between them, we can recognize a common doctrine which is usually repeated in most of them. A central idea in type theories is that, in order to explain the universe of objects under consideration (of programs, of proofs, of specifications) this universe is ....

....of the proposition stated above. This point of view corresponds to a program verification approach, and has been studied in [61] 2.3 The Calculus of Constructions The particular type theory on which we will set our work is the Calculus of Constructions (CC) developed by Th. Coquand and G. Huet [16]. The expressive power of this theory is considerable: it can be seen as the expression of the propositions as types principle for intuitionistic higher order logic. The calculus consists of a collection of rules for deriving the form of judgment F a: A, where P is a list 37 1 : A1, 37 , ....

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T. Coquand. Metamathematical Investigations of a Calculus of Constructions. In P. Odifreddi, editor, Logic and Computer' Science, volume 31 of The APIC series, pages 91 122. Academic Press, 1990.

....a fundamental role in the propositions as types paradigm of Logic, since it provides the consistency proof of the type theory as a logic. Usually, it is also the most difficult meta theoretical property to prove. In this section we show how to extend to our system the proof developed by Coquand in [15] for the pure Calculus of Constructions. The proof of strong normalization is carried out for the calculus C C (c.f. Definition 3.3.3) and with respect to the notion of computation A B introduced in Definition 3.2.4. It is clear that the larger relation A B, where fixpoints may be freely ....

....or thn R or R, respectively. In order to avoid dealing at the semantic level with valid contexts, context extensions and weakening properties, a notion of infinite context is introduced. This notion is of common use in proofs based on the reducibility method, and it is usually called a premodel [15] or an environment 12 [50] Informally, an environment is an infinite list of type assignments for variables, such that any finite segment of it is a valid context, and it contains an infinity of variables of each type. The following definition introduce this notion formally. Definition 3.6.2 ....

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T. Coquand. Metamathematical investigations of a calculus of constructions. Unpublished. Do not confuse with the article of the same name published in [16]., 1987. 181

No context found.

Thierry Coquand. Metamathematical Investigations of a Calculus of Constructions. In Piergiuorgio Odifreddi, editor, Logic and Computer Science, pages 91-122. Academic Press, 1990.

No context found.

Thierry Coquand. A meta-mathematical investigation of a Calculus of Constructions. Private Communication, 1987.

No context found.

-149. Coquand, T., Metamathematical investigations of a calculus of constructions, in \Logic and Computer Science ", P. Odifreddi ed., Academic Press, London, 1990, pp. 91-122. Coquand, T., An Algorithm for Type-Checking Dependent Types, Science of Computer Programming 26 (1{

No context found.

T. Coquand. "Metamathematical Investigations of a Calculus of Constructions." Rapport de recherche INRIA 1088, Sept. 89. In "Logic and Computer Science," ed. P. Odifreddi, Academic Press, 1990, 91--122.

No context found.

Th. Coquand. Metamathematical Investigations of a Calculus of Constructions. In P. Oddifredi, editor, Logic and Computer Science. Academic Press, 1990. INRIA Research Report 1088, also in [41].

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