## Program Development in Constructive Type Theory (1992)

Venue: | Theoretical Computer Science |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Galmiche92programdevelopment,

author = {Didier Galmiche},

title = {Program Development in Constructive Type Theory},

journal = {Theoretical Computer Science},

year = {1992},

volume = {94},

pages = {94--2}

}

### OpenURL

### Abstract

We present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications. We are interested in two important facts that are the mechanization of the proof construction and the possibility to express in the theory significiant concepts for programming (like inductively deøned types and general recursion). We give here a survey on some results and problems appearing in logical frameworks devoted to the programming with proofs approach.

### Citations

485 |
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(Show Context)
Citation Context ...tive frameworks. Dioeerents theories have been studied in this way like MartinL #f's constructive type theory [3] [9], [18], Feferman's theory T o and its applications [13], Calculus of Constructions =-=[4] or -=-variants of intuitionnistic ZF set theory [14]. Their applications and corresponding systems have been developed towards veri��- cation of proofs and, more generally, interactive development of co... |

159 | Implementing mathematics with the Nuprl proof development system
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(Show Context)
Citation Context ...tion There has been lately a great deal of interest in development of programs in constructive frameworks. Dioeerents theories have been studied in this way like MartinL #f's constructive type theory =-=[3]-=- [9], [18], Feferman's theory T o and its applications [13], Calculus of Constructions [4] or variants of intuitionnistic ZF set theory [14]. Their applications and corresponding systems have been dev... |

93 |
1980], A survey of the project AUTOMATH
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(Show Context)
Citation Context ...2 A logical framework We use, here, a presentation of intuitionnistic type theory in a lambda calculus framework [7] like in [8], which has an important connexion with ELF [12] and Automath languages =-=[2]-=-. This formulation shows a presentation of logical systems in typed theory. 2.2.1 Type theory framework We give here a classical presentation of the theory considering the following notationssx 1 : : ... |

56 |
Recursive programming with proofs
- Parigot
- 1992
(Show Context)
Citation Context ...tive type theory [18] and want to show how to use it to derive programs (through some necessary extensions to obtain interesting programs) . We can mention other theories to program with proofs [14], =-=[15]-=- confronted to similar problems (perhaps under another form but) with the same essential constraint to have extensions or reformulations that preserve properties like correctness or termination of the... |

43 |
A synthesis of several sorting algorithms
- Darlington
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(Show Context)
Citation Context ...proof derivations A good example of synthesis is the derivation of the well known sorting algorithms from a common speci��cation. We have some results of such exercise using transformation techniq=-=ues [5]. He-=-re we want to take up this example in a programming with proofs context with a view to de��ning new formulations or extensions to derive eOEcient programs. With the two following examples, we can ... |

14 |
The system F of variable types, fteen years later
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Citation Context ...n. This is unlike another theories like Calculus of Construction [4] where inductively generated types can be de��ned using second order quanti��cation or in other systems issued of Girard's s=-=ystem F [11]-=-. The commun recursive data structure introduced as a primitive is that of natural numbers. The other recursive data types are represented as powerfull well-ordering. Regarding program development, co... |

13 |
A framework for de��ning logics
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- 1942
(Show Context)
Citation Context ... set theory [14]. Their applications and corresponding systems have been developed towards veri��- cation of proofs and, more generally, interactive development of correct programs in speci��c=-= logics [12]-=-, [20]. The use (and the adaptation) of proof concepts for programming and more precisely for program synthesis [13] leads to the programming with proofs notion which will be illustrated in this paper... |

13 |
Intuitionistic Type Theory
- Martin-Lf
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(Show Context)
Citation Context ...e has been lately a great deal of interest in development of programs in constructive frameworks. Dioeerents theories have been studied in this way like MartinL #f's constructive type theory [3] [9], =-=[18]-=-, Feferman's theory T o and its applications [13], Calculus of Constructions [4] or variants of intuitionnistic ZF set theory [14]. Their applications and corresponding systems have been developed tow... |

6 |
A constructive set theory for program development
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- 1988
(Show Context)
Citation Context ...udied in this way like MartinL #f's constructive type theory [3] [9], [18], Feferman's theory T o and its applications [13], Calculus of Constructions [4] or variants of intuitionnistic ZF set theory =-=[14]. Their -=-applications and corresponding systems have been developed towards veri��- cation of proofs and, more generally, interactive development of correct programs in speci��c logics [12], [20]. The ... |

5 |
Constructive Mathematics and Computer Programming
- Martin-Lf
- 1982
(Show Context)
Citation Context ...s, natural numbers or binary trees, are important in programming and the corresponding notion in Martin-L#f's type theory is that of an inductively de��ned type. In a ��rst version of the type=-= theory [17], there -=-is no general scheme to have an inductive de��nition. This is unlike another theories like Calculus of Construction [4] where inductively generated types can be de��ned using second order quan... |

4 |
PX: a system extracting programs from proofs
- Hayashi
- 1986
(Show Context)
Citation Context ...elopment of programs in constructive frameworks. Dioeerents theories have been studied in this way like MartinL #f's constructive type theory [3] [9], [18], Feferman's theory T o and its applications =-=[13], Ca-=-lculus of Constructions [4] or variants of intuitionnistic ZF set theory [14]. Their applications and corresponding systems have been developed towards veri��- cation of proofs and, more generally... |

4 |
Well-founded Recursion in Type theory
- Saaman, Malcolm
- 1987
(Show Context)
Citation Context ... the general recursion operator rec. 5.4 Program development examples In this section we derive programs using the general recursion and more precisely the Acc type. These examples are extracted from =-=[16] and are-=- signi��ciant concerning the program construction.. 5.4.1 Example 1 We discuss here about sorting algorithms of lists of naturals. The general speci��cation is 8x : L(N )fy : L(N )=sorted(y)sp... |

3 |
The power of constructive framework for program synthesis
- Galmiche
- 1988
(Show Context)
Citation Context ... There has been lately a great deal of interest in development of programs in constructive frameworks. Dioeerents theories have been studied in this way like MartinL #f's constructive type theory [3] =-=[9]-=-, [18], Feferman's theory T o and its applications [13], Calculus of Constructions [4] or variants of intuitionnistic ZF set theory [14]. Their applications and corresponding systems have been develop... |

2 |
On the meaning and construction of the rules in Martin-L#f's theory of types
- Backhouse
- 1987
(Show Context)
Citation Context ... add new rules for inductively de��ned types is solved by a general scheme meant to introduce new type operators by giving formation and introduction rules and deriving elimination and equality ru=-=les [1]. Anothe-=-r approach based on a ��xed point operator is also possible. We discuss the problem of representation as wellorderings and show that all inductively de��ned type can be represented as well ord... |

1 |
Program Veri��cation in a Logical Theory of Constructions
- Dybjer
- 1985
(Show Context)
Citation Context ...utational ones. For instance, treating the proposition x OE x as a type can be complicated, and even the ��nal term can be computed without computations of elements of x OE x [21]. The alternative=-= of [6] is to r-=-eason about programs in an untyped logical theory of propositions where propositions are primitive and with types as predicate. Here the ��xed-point Y exists and can de��ne recursive functions... |

1 |
Inductively de��ned sets in Martin-L#f's type theory
- Dybjer
- 1987
(Show Context)
Citation Context ... (denoted by MLTT), underlying the identi��cation between propositions types and speci��cations and the relationships between proofs and programs. Secondly we present a more general logical fr=-=amework [7]-=- and the formulation of MLTT in it illustrated by representation of datas like naturals or lists. 2.1 General ideas about type theory Martin-L#f's type theory can be seen as a programming logic for fu... |

1 | Formalisation de la th#orie intuitionniste des types : Contribution # l'#tude de la d#monstration en programmation. Th#se de troisi#me cycle, Universit# Paris VI.1985 - Galmiche |

1 |
A programming with proofs system based on type theory
- Galmiche
- 1989
(Show Context)
Citation Context ...be able to characterize the proofs with algorithmic information but also to know how to extract the programs. Some works start from existing logical framework (and theorem provers) to derive programs =-=[10], [2-=-0] and others from more pragmatic programming approach de��ned on adapted logical systems [26]. Research in program development using such theories has lighted a number of interesting problems. Th... |