Z. Luo. Computation and Reasoning: A Type Theory for Computer Science, volume 11 of International Series of Monographs on Computer Science. Oxford University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....topos that has no such embedding. 16 Although in this paper we use models of IZF set theory to achieve algebraic compactness, many other set theories and type theories appear rich enough to carry out the proofs in this paper. One such theory is the Extended Calculus of Constructions (ECC) [23], as used, for example, in [35] However, it appears that one does not need the full impredicativity of ECC. In fact, it seems likely that, with appropriate reformulations, the development of this paper could be carried out in the (predicative) context of Martin L of s Type Theory [26] making ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. OUP, 1994. 76

....the community has insuciently many trained individuals with the time, inclination and expertise required to check all the proof that the transformation developer community might be likely to produce. Fortunately, recent work on proof assistants such as HOL [3, 4] Isabelle [5, 6] PVS [7, 8] LEGO [9, 10], ALF [11, 12] and Coq [13, 14, 15] has opened the possibility for machine checked proofs, in which the human discharges the proof obligation, but the proof is checked by a machine. This paper shows an example how the mechanical proof assistant Coq is used as a tool to support program ....

Z. Luo, Computation and Reasoning: A Type Theory for Computer Science. No. 11 in International Series of Monographs on Computer Science, Oxford University Press, 1994.

....Section 7 concludes the whole paper. Additional technical detail can be found in Appendices. The technical development of this paper has been fully formalized and checked by Coq. The Coq scripts are available from http : www.dur.ac.uk xingyuan.zhang tphol 3 Conventions. Because type theory [4, 15, 11] was first proposed to formalize constructive mathematics, we are able to present the work in standard mathematical notation. For brevity, we use the name of a variable to suggest its type. For example, s, s # , s, in this paper always represent program stores. Type theory has a notion of ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.

....the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions (ECC) with inductive types [Luo94] LEGO is a refinement style proof checker, publicly available by ftp and WWW, with a User s Manual [LP92] and a large collection of examples. Section 1.3 contains Submitted to Journal of Automated Reasoning A version of this paper appears as technical report ECS LFCS 97 359, University of ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. International Series of Monographs on Computer Science. Oxford University Press, 1994.

....we could call Implicit Pure Type Systems. 2 The Implicit Calculus of Constructions 2.1 Syntax From a pure syntactical point of view, the Implicit Calculus of Constructions (ICC) or, shortly, the implicit calculus is a Curry style variant of the Calculus of Construction with universes a.k.a. ECC [9] in which we make a distinction between two forms of dependent products: the explicit product, denoted by x : T : U , and the implicit product, denoted by 8x : T : U . Formally, a term of the implicit calculus (see gure 1) is either: a variable x; a sort s; an explicit product x : T : ....

.... an application M N , where M and N are terms. The set of sorts of the implicit calculus is de ned by S = fProp; Setg [ fType i ; i 0g; where Prop and Set denote the impredicative sorts, and (Type i ) i 0 the usual predicative universe hierarchy of the Extended Calculus of Constructions [9]. Notice that here, we follow the convention of the Calculus of Inductive Constructions [18] by making a distinction between two impredicative sorts, since it is convenient to distinguish a sort for propositional types (Prop) from a sort for impredicative data types (Set) although both sorts are ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

....that are predicative and proof irrelevant, including (predicative) higher order predicate logic. Proof scripts Most of the results presented in this paper are being formalised in the proof assistant Coq. They will be included as supplementary material at a later stage. Notations Following [15], we use Prop for the universe of propositions and Type for the universe of types. Finally, we use the notation hl : L; r : Ri for a record type with two elds l of type L and r of type R and hl = a; r = bi for an inhabitant of that type. 2 Total setoids vs. partial setoids 2.1 Total setoids A ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.

....model in section 5, and we use it for proving the consistency of a restriction of the Implicit Calculus of Constructions, which contains ECC. 2 The implicit calculus 2. 1 Presentation Basically, the Implicit Calculus of Constructions (ICC) or, shortly, the implicit calculus, is a variant of ECC [9] in which we distinguish two kinds of products: the explicit product and the implicit product, denoted by x : T : U and 8x : T : U respectively. The explicit product is the usual dependent product of Pure Type Systems (PTS) whereas the implicit product is much more an intersection type binder, ....

....indexed over a cardinal c a such that b i a for all i 2 c, we have sup i2c b i a. Although the existence of inaccessible cardinals cannot be proven in the ZF set theory [8] 6 , such big cardinals have been used for a while, especially for building settheoretical models of type theories [9]. Lemma 3.5 Let A and B be two coherence spaces equipped with a stable functions, where a is either countable or inaccessible. If X is a small type basis of A and if (Y a ) a2X is a family of small type bases of B indexed over X , then the sets (a 2X ; Y a ) and 8(a 2X ; Y a ) are small type ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

.... Two Level Approach towards Lean Proof Checking Gilles Barthe # , Mark Ruys and Henk Barendregt May 15, 1996 Abstract We present a simple and e#ective methodology for equational reasoning in proof checkers. The method is based on a two level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with ....

....The two level approach was grew out from earlier work by P. Aczel and the first author on the formalization of (universal) algebra in type theory. The applications of universal algebra for equational reasoning were realized later by the first author and presented at the HISC meeting in Amsterdam in March 1994 (see [5, 4] After the completion of this work, H. Elbers and the first author have developed further the two level approach and provided an automatic procedure to solve equational problems in Lego [6] The work presented in this paper bears some similarities with the work of the NuPrl team on ....

[Article contains additional citation context not shown here]

Z. Luo. Computation and reasoning: a type theory for computer science, OUP, 1994.

.... we introduce a simple formal semantics of computation that includes time and space, and this is enough to demonstrate that mathematics and logics, although being enough to de ne what we are going to call mathematical computation and are still going to be used 3 as computation and reasoning[45], does not capture the broader notion of computation, in accord with our somewhat informal semantics. Concerning philosophical factors, we enter in a subjective, informal and psychological world, the real world, transcending the mathematical and logical language and traditional computer science ....

Z. Luo. Computation and Reasoning: a type theory for computer science, volume 11 of International Series of Monographs on Computer Science, edited by Gabbay, Hopcroft, Plotkin, Schwartz, Scott, Vuillemin and Galil. Oxford Science Publications, 1994.

....and it uses a very liberal notion of inductive definitions. PTS systems, in particular CC, provide higher order (dependent) types, but they are based on a fixed notion of computation, namely fi reduction. This unsatisfying situation has been addressed by addition of inductive definitions [55][40] and algebraic extensions in the style of abstract data type systems [6] Also, the idea of overcoming these limitations using some combination of membership equational logic with the calculus of constructions has been suggested as a long term goal in [39] To close the gap between these two ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. International Series of Monographs on Computer Science. Oxford University Press, 1994.

....results are to a large extent independent from the choice of a type system. 2 G. Barthe, V. Capretta and O. Pons Proof scripts Most of the results presented in the paper have been formalised in the proof assistant Coq. They will be made available on the web at a later stage. Notations Following [35], we use Prop for the universe of propositions, Type i for the i th universe of types. By abuse of notation, we write Type for Type 0 so we have Prop:Type and Type i : Type i 1 . Moreover, we use the notation hl : L; r : Ri for a record type with two elds l of type L and r of type R and hl = a; r ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994. 16

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994. 4

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

.... subtyping with a mixture of simple coercions, parameterised coercions, coercion rules for parameterised inductive types, and dependent coercions [LS99] A formal presentation Coercive subtyping is formally formulated as an extension of (type theories speci ed in) the logical framework LF [Luo94] whose rules are given in Appendix A. Types in LF are called kinds. The kind Type represents the conceptual universe of types and a kind of the form (x : K)K represents the dependent product with functional operations f as objects (e.g. abstraction [x : K]k ) which can be applied to ....

....[x : K]k ) which can be applied to objects of kind K to form application f(k) For every type (an object of kind Type) El(A) is the kind of objects of A. A kind is small if it does not contain Type. LF can be used to specify type theories, such as Martin Lf s type theory [NPS90] and UTT [Luo94] Notation We shall use the following notations: We often write (K)K for (x : K)K when x does not occur free in K , and A for El(A) and hence (A)B for (El(A) El(B) when no confusion may occur. Substitution: We sometimes use M [x] to indicate that variable x may occur free in M and ....

[Article contains additional citation context not shown here]

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

....the framework of coercive subtyping, which is also the basis for our development in latter sections. We shall be brief in this paper (for details and more explanations, see [Luo99] Coercive subtyping is formally formulated as an extension of (type theories speci ed in) the logical framework LF [Luo94] whose rules are given in Appendix A. In LF, Type represents the conceptual universe of types and (x : K)K 0 represents the dependent product with functional operations f as objects (e.g. abstraction [x : K]k) which can be applied to objects of kind K to form application f(k) LF can be used ....

.... universe of types and (x : K)K 0 represents the dependent product with functional operations f as objects (e.g. abstraction [x : K]k) which can be applied to objects of kind K to form application f(k) LF can be used to specify type theories, such as Martin L of s type theory [NPS90] and UTT [Luo94] For example, types, types of dependent functions, can be speci ed by introducing the constants for (1) formation: A;B) is a type for any type A and any family of types B, 2) introduction: A; B; f) is a function of type (A;B) if f is a functional operation of kind (x : A)B(x) and (3) ....

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science, volume 11 of International Series of Monographs on Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science, volume 11 of International Series of Monographs on Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. International Series of Monographs on Computer Science. Clarendon Press, Oxford, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. International Series of Monographs on Computer Science. Clarendon Press, Oxford, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science.In- ternational Series of onographs on Computer Science. Oxford Univ. Press, 1994.

No context found.

Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC