C. Paulin. Extraction de programmes dans le calcul des constructions. PhD thesis, Universite Paris 7, January 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that this flexibility is an inherent advantage of our approach compared to a priori separating propositions (inhabited by proofs that are always irrelevant) from specifications (inhabited by terms that are never irrelevant) This is the approach mostly taken in the literature (see, for example, [18] or, allowing even for some classical reasoning, 2] Our system is also interesting in its relation to intuitionistic modal logic when we ignore the objects. Our default judgment M : A can be interpreted as A is true . The judgment M : A can be read as A is valid . The judgment M A can be ....

C. Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, Universite Paris VII, Jan. 1989.

.... that this flexibility is an inherent advantage of our approach compared to a priori separating propositions (inhabited by proofs that are always irrelevant) from specifications (inhabited by terms that are never irrelevant) This is the approach mostly taken in the literature (see, for example, PM89] or, allowing even for some classical reasoning, BBS01] Our system is also interesting in its relation to intuitionistic modal logic when we ignore the objects. Our default judgment M : A can be interpreted as A is true . The judgment M : A can be read as A is valid and satisfies the ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, January 1989.

.... that this flexibility is an inherent advantage of our approach compared to a priori separating propositions (inhabited by proofs that are always irrelevant) from specifications (inhabited by terms that are never irrelevant) This is the approach mostly taken in the literature (see, for example, PM89] or, allowing even for some classical reasoning, BBS01] Our system is also interesting in its relation to intuitionistic modal logic when we ignore the objects. Our default judgment M : A can be interpreted as A is true . The judgment M : A can be read as A is valid and satisfies the ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, Universite Paris VII, January 1989.

....a realization of the specification. Realizability allows to eliminate non computational parts of proofs (to extract programs from proofs) and to certify extracted programs to be still correct with respect to the initial specification. Indeed, from proofs written in Coq, programs can be extracted [PM89b] into a typed functional language like ML. Some other systems like PX and NuPrl offer similar possibilities of extraction [HN88, Con86] Both of them are using untyped theories. More precisely, PX uses an untyped theory and, in NuPrl, the This research was partly supported by ESPRIT Basic ....

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....can be seen either as a method of proving programs or as a method of synthetically describing proofs. 1 Introduction This paper talks about automation of proofs about functional programs correctness with respect to a specification. The framework is the Calculus of Inductive Constructions [Coq85, PM89b] which is a typed calculus with polymorphism, higher order and dependent types. We focus on the Coq [DFH 93] system which is an implementation of this calculus. It is a system for formalizing and checking the mathematical reasoning [Bar91, ML84] It contains a specification language and a ....

....8x: P x) 9y: Q x y) A proof represents a functional program which validates the specification. However, in general, to have more efficient programs, all the parts of a term that compute the proof rather than the result itself are suppressed. This is the principle of an extraction function [PM89b, PM89a] that suppresses some parts of proofs and that we call the strong extraction. In the previous example, a large part of the proof is devoted to check that q and r verify the property a = b q r b r. Such a proof contains computational parts (to construct q and r in our example) and ....

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....due to a particular interpretation of Heyting s semantic for the Calculus of Constructions. 4.1 Reviewing Heyting s semantic The principle of the extraction mechanism relies on Heyting s semantic interpretation of predicate calculus. A more exhaustive description may be found in [PMW93, PM89a, PM89b] Following Heyting s interpretation of propositions, one can see through an object x of type T a program x that matches the specification T . From a logical point of view, x can be seen as a proof of the proposition T . A uniform notation for this ternary isomorphism is usually x:T, also called ....

C. Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, Universit'e Denis Diderot, Paris, Fr, 1989.

.... original approach of using Type Theory for program verification as proposed by Martin Lof and many others, e.g. see [NPS90, BCMS89] However, it has been proposed to reintroduce a difference between proofs and programs and between data types and propositions in Type Theory either for pragmatic [PM89] or for philosophical reasons [Luo94] We shall attempt to show by means of example that a pure approach is not only feasible but also preferable. Here we exploit the progress in the presentation of Type Theory which has been made, especially by using the pattern matching notation as proposed by ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, L'Universite Paris VII, 1989. This article was processed using the L A T E X macro package with LLNCS style

....irrelevant, that is parts which are only present to verify that the resulting list is really a permutation of the input list and that it is sorted. There are methods to extract pure programs without these computationally irrelevant parts from the programs derived from the specication ([PM89]) which are provably correct with respect to their specication. Without dependent function types, the specication of a sorting program would be the type List(N ) List(N ) which is not precise enough to only contain sorting programs. Since type theory is at the same time a specication language ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, Universite Paris VII, 1989.

.... P is a Harrop formula, then it can be demonstrated 1 that this optimization, usually called program extraction, yields a functional program f : A B which satisfies the original specification, in the sense that the following proposition is true: 8x : A:P (x) R(x; fx) 1 See for example [64]. 20 This shows how type theory and the proposition as types principle enable to do program derivation, where the program developed is proven correct by construction. It is also possible to go in the other way round, starting from a functional program f : A B already written, and trying to ....

....stand for one of the constants Prop or Type, called sorts. In some versions of the calculus, the sort Prop is duplicated into two different sorts called Set and Prop. This duplication is used for marking those parts of the proof that must be erased during the process of program extraction, see [64] for details. Traditionally, the intensional equality A = B between types is not included as a form of judgment, but explained as the symmetric closure of a notion of computation, which is external to the calculus. This notion of computing corresponds to the minimal reflexive, transitive and ....

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....Nqthm, PVS, HOL, Nuprl and Coq. Some 2 of them implement highly expressive logics, whose counterpart is a relatively poor automation. The system Coq implements the Calculus of Inductive Constructions, an extension of J. Y. Girard s system F [12] developed by G. Huet, T. Coquand and C. Paulin [5, 23], which belongs to the family of type theories. Type theory identi es types with propositions and terms with proofs, through the widely known Curry Howard isomorphism. There is no real di erence between the usual rst order objects of the mathematical discourse such as integers, sets and so ....

....like building a function together with a proof of its correctness, and consequently type theory appears as naturally suited for the proof of purely functional programs. Moreover, there is a systematic way to extract the underlying program from such a proof, as has been demonstrated by C. Paulin [22, 23]. Conversely, C. Parent showed that there is a way to partly reconstruct a proof of (1) from a given program of the right type, which leads to the expected proof obligations [20, 21] Our objective is to cope with imperative programs in this context. We propose an interpretation of the Hoare ....

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. These de doctorat, Universite de Paris VII, Janvier 1989. 34

....of (R n) R m) where we evaluate (R n) and (R m) What we want to do here, is to use a very nice property of the Coq system: its implementation language is almost a sublanguage of the metalanguage. Coq is implemented in Objective Caml [15] a dialect of ML; and Coq has an extraction process [17] from the Coq object language to Objective Caml. So we can use this extraction process to translate deleteneutral and R from Coq to Objective Caml. This extraction process is automatic and its correctness is gauranteed. So, using the extraction process, we want to perform the computation of (R m) ....

C. Paulin-Mohring, Extraction de programmes dans le Calcul des Constructions, Thesis, Paris 7, 1989.

....the second order : 5 2.1.4 Using subtyping for pruning [3] 8 2.1.5 Some comments on Turin school : 10 2.2 C. Paulin s approach [9] : 10 2.2.1 The Calculus of Constructions : 10 2.2.2 Extraction in the Calculus of Constructions : 12 2.2.3 ....

....get a good program. We will use the word of extraction to characterize this work. Several extraction techniques have been studied. One can approximately divide those ones into two ways. The first point of view is to realize the searched optimization within the syntax of the terms. C. Paulin in [9] develops this approach for a higher order system: the Calculus of Construction. The basic idea is to duplicate the system. One part is used for the logical annotations without computational meaning and the second is used to express the parts of the program useful for the computation . This ....

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C. Paulin, Extraction de programmes dans le calcul des constructions, PhD thesis, Unniversit'e Paris 7, 1989.

....like Coq and NuPRL, are also programming languages. The program is automatically extracted from the proof of its specification, and a metatheorical result Submitted to the 24th Annual SIGPLAN SIGACT Symposium on Principles of Programming Languages, Paris, France, January 15 17, 1997. [13] insures that the extracted program obeys its specification. However there are two obstacles to the application of proof checkers to large scale formal developments. The first one is that the proof activity is tedious because one has to justify each reasoning step. This obstacle is (partially) ....

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit 'e Paris 7, January 1989.

....to compute the final result. For instance a proof of the Euclidean division gives for any integers a, b a couple q, r which verify a = bq r and r # b 1. From a computational point of view only q, r are valuable, their properties might be seen as dead code. The work of C. Paulin (see [Pau89, PM89] might be seen as a forerunner of type based systems for this kind of dead code analysis. In this work a system to extract F# programs from Calculus of Constructions (CC for short) proofs is developed. From a programming language point of view, there is no di#erence between CC and F# . ....

....of two di#erent universes from which types may be built. Following C. Paulin, those two universes may be seen as P rop and Spec. In the following of the paper we will rather use the notation # and # for P rop and Spec, since our use of those two universes is rather di#erent from the one of [Pau89] The originality of our system lies on two points: 1. We define an inclusion relation between the two universes, namely ##:#, from which we derive a subtyping relation on types. 2. We introduce a notion of universe variable, which is a refinement of [Pro97] properties variables. We develop a ....

C. Paulin. Extraction de programmes dans le calcul des constructions. PhD thesis, Universite Paris 7, January 1989.

....isomorphic type [A : Set]A. So, a proof of the specification 8x : A: P x) fy : B j (Q x y)g extracts to a function f : A B. The correctness of the extractum is justified by the realisability 1 provided they are typable in Caml, which is the case for most practical applications interpretation [16, 17], ensuring in this case that f satisfies 8x : A: I x) Q x (f x) Decision procedures are specified by a variant of logical disjunction (with arity of sort Set) given by: Inductive sumbool [A : Prop; B : Prop] Set : left : A (sumbool A B) j right : B (sumbool A B) The notation for ....

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e de Paris VII, Jan. 1989.

.... typed terms, Sol95, Hei95] have used annotated type system to perform many analyses like control flow, binding time, strictness, in [Pro95] dead code analysis is treated using marks (not only limited to type) The idea of using types for the analysis of typed terms was already present in [Pau89, Ber93, Boe94, Tak91] A similar technique is used in [DG97] the refinement types are used for strictness analysis. This latter version enlightens well a key point of annotated based system. It is the coexistence of two attributes linked to a term: its type, considered in the usual way, and its ....

C. Paulin. Extraction de programmes dans le calcul des constructions. PhD thesis, Universit'e Paris 7, January 1989.

....j C. The existence of values of open objects is of interest, for instance, in partial evaluation [7] and pattern matching [2, 4] When optimizing programs extracted from proofs, an important role is played by sets corresponding to Harrop formulas [6] since they are without computational content [12, 16]. We will define what it means for a set to be without computational content and then show that a set is without computational content if and only if it slashes itself; the sets satisfying this condition strictly contain the Harrop sets introduced in [13] In the formulation of Martin Lof s type ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, L'Universite Paris VII, 1989.

....interesting) To obtain more realistic programs, the real constructive part of the proof has to be distinguished from the correctness proof part. The first part is the computational part of the proof, the second the logical. Obviously, only the computational part needs to be kept as a program. In [PM89b] this operation is called program extraction . This approach validates a program by proving a specification and synthesizing a program from this proof. A logical language called the Calculus of Inductive Constructions supports proof development and the programming language is the F Ind ....

....language such as Verifying programs in the Calculus of Inductive Constructions 5 the system F Ind . This requires a method to retrieve and reuse such annotations in a program. Inversion of the extraction It is then clear that we should look for a process that inverts program extraction [PM89b] The extraction omits logical information from proofs to obtain programs and we want to retrieve logical information from programs and specifications to build proofs. Then, as we said before, we look for a language richer than F Ind . In this paper, we modify the extraction procedure of ....

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....of the minimality. The latter component is useful for building the proof smoothly, but from a computational point of view it is redundant. Once we have completed the proof we would like to strip it off before we use the program. We can here only indicate the ideas behind such program extraction [39, 3]. If we are only interested in the first component of min(ns) we would like to erase the second component everywhere. In this particular case it easy to see that this is possible, since the second component h of the recursive call is only used to build the second component of the result in the ....

Christine Paulin-Mohring. Extraction de Programmes dans le Calcul des Constructions. PhD thesis, L'Universite Paris VII, 1989.

.... Programmation . the program is represented by a logical formula [Con86, NPS90] Developing a program then corresponds to prove this formula [How80] In the state of the art, different methods exist to extract the computational part representing the program from a mathematical proof [Ber94, PM89b, PM89a] The correctness of the resulting functional program is then certified by construction. We focus on a particular framework that is the Calculus of Inductive Constructions [Coq85, Coq89] and a particular implementation that is the Coq system [CCF 94] Programs can be extracted from ....

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....of sub programs. The system Coq is an implementation of a particular type theory called the Calculus of Constructions enhanced with inductive definitions [PM93, DFH 93] As we explained earlier, proofs of specifications can be developed and programs can be extracted from these proofs [PM89b] The extraction in Coq is based on a realizability notion [PM89a] Realizability is an interpretation of the computational content of intuitionistic proofs as programs satisfying a given specification. Such a program is called a realization of the specification. Realizability allows to eliminate ....

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris VII, 1989.

....Part 3 gives the rules used in the system Coq and examples of inductive definitions. In part 4 the strong normalization of a subsystem coq of Coq is established. In part 5 we shall discuss our choices and make comparisons with other systems. 1. 2 Impredicative inductive definitions In [4, 21, 22] a systematic way to generate a representation of an inductive definition from a description of the generative rules for it was described. But this representation is not really adequate. We list some problems : ffl We can represent a type of boolean with two elements true and false but the fact ....

....of natural numbers with what is known as Church s numerals. ffl Some structures represented with an impredicative coding (like the products of two types) can contain more elements (closed normal terms) than the one built from the constructors of the type. It is explained for instance in [21]. Some points concerning the representation of inductive definitions in an impredicative type theory are reflected in our rules for primitive inductive definitions. For instance an inductive definition makes sense in any context (and not just at toplevel) the names of the type and of its ....

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris 7, January 1989.

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C. Paulin. Extraction de programmes dans le calcul des constructions. PhD thesis, Universite Paris 7, January 1989.

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris 7, January 1989.

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C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Universit'e Paris 7, January 1989.

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