## Variants of the Basic Calculus of Constructions

Citations: | 1 - 0 self |

### BibTeX

@MISC{Bunder_variantsof,

author = {M. W. Bunder and Jonathan P. Seldin},

title = {Variants of the Basic Calculus of Constructions},

year = {}

}

### OpenURL

### Abstract

In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.

### Citations

595 | Untersuchungen über das logische Schließen - Gentzen - 1935 |

552 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...s and the other ones. After Coquand introduced the calculus of constructions, Berardi [17] and Terlouw [18] introduced the concept of a PTS as a generalization of the GTSs of the Barendregt cube (see =-=[19]-=-). A comparison of the definition of a PTS to the definitions of the original versions of the calculus introduced by Coquand shows that the latter is a special case of the former. More recently, Bunde... |

496 |
The Calculus of Constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...rted 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... |

112 | A set of postulates for the foundation of logic - Church - 1933 |

87 | ECC: An Extended Calculus of Constructions
- Luo
- 1990
(Show Context)
Citation Context ...ults of this paper will help researchers make this choice. There are di#erent kinds of extensions of the calculus of constructions that we do not consider here: the extended calculus of constructions =-=[23,24]-=-, the calculus of constructions with inductive types [25,26], the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction [34,35], and the calculus of constru... |

68 |
Une théorie des constructions inductives
- Werner
- 1994
(Show Context)
Citation Context ...here are di#erent kinds of extensions of the calculus of constructions that we do not consider here: the extended calculus of constructions [23,24], the calculus of constructions with inductive types =-=[25,26]-=-, the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction [34,35], and the calculus of constructions as a domain-free pure type system [36]. This is becau... |

65 |
Metamathematical investigations on a calculus of constructions
- Coquand
- 1990
(Show Context)
Citation Context ... 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 calcul... |

52 |
Foundations of Mathematical Logic
- Curry
- 1963
(Show Context)
Citation Context ...urry's theory of functionality (his name for type assignment), which he considered a part of illative combinatory logic, and the proof-theoretic tools of Gentzen [7], especially as developed by Curry =-=[8,9]-=-, and Prawitz [10]. Seldin also wanted to allow for the possibility of assumptions other than those assigning types to variables; he thought that such assumptions might be useful, for example in deali... |

44 | Definitions by rewriting in the calculus of constructions - Blanqui |

41 |
Une The'orie Des Constructions
- Coquand
- 1985
(Show Context)
Citation Context ...rted 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... |

40 |
Natural Deduction, Almqvist & Wiksell
- Prawitz
- 1965
(Show Context)
Citation Context ...nctionality (his name for type assignment), which he considered a part of illative combinatory logic, and the proof-theoretic tools of Gentzen [7], especially as developed by Curry [8,9], and Prawitz =-=[10]-=-. Seldin also wanted to allow for the possibility of assumptions other than those assigning types to variables; he thought that such assumptions might be useful, for example in dealing with the possib... |

35 | Domain-free pure type systems
- Barthe, Sørensen
- 2000
(Show Context)
Citation Context ...ctive types [25,26], the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction [34,35], and the calculus of constructions as a domain-free pure type system =-=[36]-=-. This is because the di#erent variants we deal with here could be defined for all of these systems, and there are so many of them that considering them all would make the paper much, much longer. Fur... |

33 | Constructions, Inductive Types and Strong Normalization
- Altenkirch
- 1993
(Show Context)
Citation Context ...here are di#erent kinds of extensions of the calculus of constructions that we do not consider here: the extended calculus of constructions [23,24], the calculus of constructions with inductive types =-=[25,26]-=-, the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction [34,35], and the calculus of constructions as a domain-free pure type system [36]. This is becau... |

29 |
Een nadere bewijstheoretische analyse van GSTT’s
- Terlouw
- 1989
(Show Context)
Citation Context ...ever, Seldin has been asked on several occasions about the exact relationship between his versions and the other ones. After Coquand introduced the calculus of constructions, Berardi [17] and Terlouw =-=[18]-=- introduced the concept of a PTS as a generalization of the GTSs of the Barendregt cube (see [19]). A comparison of the definition of a PTS to the definitions of the original versions of the calculus ... |

29 | Checking Algorithms for Pure Type Systems
- Jutting, S, et al.
- 1993
(Show Context)
Citation Context ...." If #(S) is a regular PTS with specification S, then Severi calls the corresponding PTS without the #-condition # # (S). Similar variants are considered by van Benthem Jutting, McKinna, and Pol=-=lack [40]-=-, in particular with respect to the conditions under which they are equivalent to the corresponding ordinary PTSs. Definition 30 The system TOC2A is obtained from the system TOC2P by replacing rule (A... |

27 |
Towards a mathematical analysis of the coquandhuet calculus of constructions and the other systems in barendregt’s cube
- Berardi
- 1988
(Show Context)
Citation Context ...6]. Recently, however, Seldin has been asked on several occasions about the exact relationship between his versions and the other ones. After Coquand introduced the calculus of constructions, Berardi =-=[17]-=- and Terlouw [18] introduced the concept of a PTS as a generalization of the GTSs of the Barendregt cube (see [19]). A comparison of the definition of a PTS to the definitions of the original versions... |

22 | Combining first and higher order rewrite systems with type assignment systems - Barbanera, Fernández - 1993 |

16 | Modularity of termination and confluence in combinations of rewrite systems with λω - Barbanera, Fernández - 1993 |

15 |
The Church-Rosser property for βη-reduction in typed λ -calculi
- Geuvers
- 1992
(Show Context)
Citation Context ...ded calculus of constructions [23,24], the calculus of constructions with inductive types [25,26], the calculus of constructions with rewriting [27–33], the calculus of constructions with η-reduction =-=[34,35]-=-, and the calculus of constructions as a domain-free pure type system [36]. This is because the different variants we deal with here could be defined for all of these systems, and there are so many of... |

14 |
On the church-rosser property for expressive type systems and its consequences for their metatheoretic study
- Geuvers, Werner
- 1994
(Show Context)
Citation Context ...ed calculus of constructions [23,24], the calculus of constructions with inductive types [25,26], the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction =-=[34,35]-=-, and the calculus of constructions as a domain-free pure type system [36]. This is because the di#erent variants we deal with here could be defined for all of these systems, and there are so many of ... |

10 | Termination of rewriting in the Calculus of Constructions - Walukiewicz-Chrzaszcz |

9 | Adding algebraic rewriting to the calculus of constructions: Strong normalization preserved - Barbanera - 1990 |

8 |
Concepts Mathématiques et Informatiques Formalisés dans le Calcul des Constructions.” Presented at the Colloque de Logique d’Orsay
- Coquand, Huet
- 1985
(Show Context)
Citation Context ...rted 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... |

7 |
On the proof theory of Coquand's calculus of constructions
- Seldin
- 1997
(Show Context)
Citation Context ...tems significantly more general than the original formulation of Coquand and Huet. Originally, Seldin did not think these di#erences were terribly important, and he continued to use these versions in =-=[6,13,12,14--16]-=-. Recently, however, Seldin has been asked on several occasions about the exact relationship between his versions and the other ones. After Coquand introduced the calculus of constructions, Berardi [1... |

7 | Coquand's calculus of constructions: A mathematical foundation for a proof development system - Seldin - 1992 |

7 | Symbolic Logic, The Ronald - Fitch - 1952 |

5 | Combining term rewriting and type assignment systems - Barbanera - 1990 |

5 |
the church-rosser property for ##-reduction in typed #-calculi
- Geuvers
- 1992
(Show Context)
Citation Context ...ed calculus of constructions [23,24], the calculus of constructions with inductive types [25,26], the calculus of constructions with rewriting [27--33], the calculus of constructions with #-reduction =-=[34,35]-=-, and the calculus of constructions as a domain-free pure type system [36]. This is because the di#erent variants we deal with here could be defined for all of these systems, and there are so many of ... |

5 |
Progress report on generalized functionality
- Seldin
- 1979
(Show Context)
Citation Context ...ith this restriction, TOC2A is equivalent to TOC2P. We can now prove the equivalence of TOC2A and TOC0A. For this proof, # # 2 E will mean that # # E is provable in TOC2A, and # # 0 E will mean 6 See =-=[44]-=-. 19 that # # E is provable in TOC0A. The other conventions from the proofs of Theorems 27 and 28 will remain unchanged. Theorem 35 If # # 2 E (7) then # # 0 E. (8) PROOF. The proof is the same as tha... |

4 | A sequent calculus for type assignment - Seldin - 1977 |

4 | A Gentzen-style sequent calculus of constructions with expansion rules - Seldin - 2000 |

4 |
Equivalences between pure type systems and systems of illative combinatory logics”, Notre Dame
- Bunder, Dekkers
(Show Context)
Citation Context ...ows that the latter is a special case of the former. More recently, Bunder and Dekkers have been studying variants of PTSs for the purpose of comparing them with systems of illative combinatory logic =-=[20]-=- (see also [21] and [22]), and as a result the exact di#erences between these di#erent formulations now seem more important. The purpose of this paper is to study these di#erent formulations of the ca... |

4 |
Strong normalization for terms of the theory of constructions
- Pottinger
- 1987
(Show Context)
Citation Context ...re similar to the original formulation of Coquand [1], or equivalently, [5], (which is equivalent to the system called TOC2P below), and a formulation of this kind was called TOC2 by Garrel Pottinger =-=[37]. Hence, in this pap-=-er, systems of this kind will be called "TOC2-like". (TOC stands for "Theory of Constructions.") In others judgments are of the form # # E where # is a set of formulas and E is a f... |

4 |
Type inference for pure type systems
- Severi
- 1998
(Show Context)
Citation Context ...conversion rules are essentially those of a PTS; 3 This is equivalent to # # Prop : Type. 4 the addition of "A" will indicate a system with a modified abstraction rule, such as the systems #=-= # (S) of [38], and the addition o-=-f "C" will denote a system with a modified rule of conversion between types. If both the abstraction and conversion rules are modified, "AC" will be added to the name. The original... |

4 | Studies in illative combinatory logic - Seldin - 1968 |

3 |
MATHESIS: The mathematical foundation for ULYSSES, Interim Report RADC-TR-87-223, Rome Air Development Center, Air force Systems Command, Griffiss Air Force Base
- Seldin
- 1988
(Show Context)
Citation Context ...that company, had hired Seldin to work on a project that involved using some version of typed #-calculus with the propositions-as-types notion to develop a very general system for formal verification =-=[6]-=-, and the first task he assigned Seldin was to choose the best version of typed #-calculus. As part of the process of choosing, Platek and Seldin visited Carnegie-Mellon University in early 1986 to se... |

2 | On lists and other abstract data types in the calculus of constructions - Seldin |

2 | Extensional set equality in the calculus of constructions - Seldin - 2000 |

2 |
Pure type systems with more liberal rules
- Bunder, Dekkers
(Show Context)
Citation Context ...tter is a special case of the former. More recently, Bunder and Dekkers have been studying variants of PTSs for the purpose of comparing them with systems of illative combinatory logic [20] (see also =-=[21]-=- and [22]), and as a result the exact di#erences between these di#erent formulations now seem more important. The purpose of this paper is to study these di#erent formulations of the calculus of const... |

1 |
A Theory of Formal Deducibility, Notre Dame Mathematical Lectures Number 6
- Curry
- 1950
(Show Context)
Citation Context ...urry's theory of functionality (his name for type assignment), which he considered a part of illative combinatory logic, and the proof-theoretic tools of Gentzen [7], especially as developed by Curry =-=[8,9]-=-, and Prawitz [10]. Seldin also wanted to allow for the possibility of assumptions other than those assigning types to variables; he thought that such assumptions might be useful, for example in deali... |

1 | Equivalences between illative combinatory logics and pure type systems - Bunder, Dekkers, et al. - 1997 |

1 | Rewriting modulo in deduction modulo, in: Rewriting Techniques and Applications - Blanqui |

1 |
Set theory based on combinatory logic
- Bunder
- 1969
(Show Context)
Citation Context ... the only 5 Bunder acknowledges in a footnote an appearance of essentially the same theorem from di#erent assumptions that appeared in Seldin [42, Theorem 4C1, p. 111], which actually appeared before =-=[41]-=-. However, Seldin had seen Bunder present his version of this theorem in a seminar in Amsterdam in early November, 1967, whereas Seldin did not begin his own work on this result until May, 1968. Furth... |

1 |
Some inconsistencies in illative combinatory logic, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20
- Bunder
- 1974
(Show Context)
Citation Context ...of this theorem in a seminar in Amsterdam in early November, 1967, whereas Seldin did not begin his own work on this result until May, 1968. Furthermore, Seldin's system was later proved inconsistent =-=[43]-=-. 18 restriction on the form of the rule #, Xx # Y x # # # #XY, which is known to be inconsistent, applies to the antecedent. The corresponding rule for Curry's theory of functionality is #, Xx # Y (Z... |