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On the proof theory of Coquand's calculus of constructions

Jonathan P. Seldin
1997 Annals of Pure and Applied Logic  
The calculus of constructions is formulated as a natural deduction system in which deductions follow the constructions of the terms to which types are assigned.  ...  The calculus of constructions of Thierry Coquand [5, 7-l l] is a system of typed I-calculus in which the second-order polymorphic typed A-calculus can be interpreted. Furthermore, *  ...  But for the form of the theory of constructions presented by Coquand, it is the most useful approach. This form of the theory of constructions is what is known as a sequent calculus.  ... 
doi:10.1016/s0168-0072(96)00008-5 fatcat:2olfkhtdovgn3pobdsu6tutpay

Page 2745 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
{For the entire collection see MR 98d:03006. } 98e:03019 03B40 03FS0 Seldin, Jonathan P. (3-CONC; Montreal, PQ) On the proof theory of Coquand’s calculus of constructions. (English summary) Ann.  ...  Coquand’scalculus of constructions” is a system of typed 4- calculus in which the second-order polymorphic typed A-calculus can be interpreted.  ... 

Justifying Algorithms for βη-Conversion [chapter]

Healfdene Goguen
2005 Lecture Notes in Computer Science  
Deciding the typing judgement of type theories with dependent types such as the Logical Framework relies on deciding the equality judgement for the same theory.  ...  on established metatheoretic results for the type theory.  ...  Acknowledgments I would like to thank Bob Harper for stimulating my renewed interest in the topic of βη equality for type theories with dependent types.  ... 
doi:10.1007/978-3-540-31982-5_26 fatcat:rfpv4yncgfe27h4zwsscxk5c3e

Page 5700 of Mathematical Reviews Vol. , Issue 89J [page]

1989 Mathematical Reviews  
Thecalculus of constructions”, part of which was the subject of Coquand’s thesis [“Une théorie des constructions”, Thése de 3° cycle, Univ.  ...  The authors give both the formal syntax and the backgrounds of the calculus of constructions.  ... 

Page 5909 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
The authors provide a careful and detailed new proof of SN for Coquand’s calculus of constructions, AC.  ...  Nonetheless, a proof of SN for a categorical-style theory of calculus of constructions as in a paper by J. M. E. Hyland and A. M.  ... 

Page 884 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The proof is based on a reduction of the problem of nonemptiness of proof(a) to the problem of the infiniteness of proof(a).  ...  Curry: es- says on combinatory logic, lambda calculus and formalism, 480—490, Academic Press, London, 1980; MR 82g:03094], this set is equiva- lent to the set of normal form proofs of @ in the natural  ... 

Univalent Foundations of Mathematics [chapter]

Vladimir Voevodsky
2011 Lecture Notes in Computer Science  
The ideas of Martin-Lof found their way into theoretical computer science in part through Coquand's Calculus of Constructions and its extension -Calculus of Inductive Constructions .  ...  Com- bining the ideas of Steve Awodey on the interpretation of the identity types with my ideas on the interpretation of the universes I have constructed the univalent model of the calculus of inductive  ... 
doi:10.1007/978-3-642-20920-8_4 fatcat:3a622h3imfec7oewpp4ma4mmjm

Page 3788 of Mathematical Reviews Vol. , Issue 88g [page]

1988 Mathematical Reviews  
The calculus and its syntactic theory were presented in Coquand’s thesis, and an implementation by the author was used to verify me- chanically a substantial number of proofs demonstrating the power of  ...  Summary: “The calculus of constructions is a higher-order formal- ism for writing constructive proofs in a natural deduction style, inspired from work of de Bruijn, Girard, Martin-Léf and Scott.  ... 

Ensuring streams flow [chapter]

Alastair Telford, David Turner
1997 Lecture Notes in Computer Science  
This analysis is an extension of the check for guardedness that has been used with de nitions over coinductive types in Martin-L of's type theory and in the calculus of constructions.  ...  Thus programmers will have fewer restrictions on their use of in nite streams within a strongly normalizing functional language.  ...  Coquand 2] in Type Theory and Gim enez 5] in the Calculus of Constructions produced syntactic checks upon the de nitions of in nite data structures which they called guardedness.  ... 
doi:10.1007/bfb0000493 fatcat:3rr2gcypm5fffc6hqyrxfcjxmu

Page 7576 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
Tait’s normalization tech- nique played a central role in Martin-L6f’s early papers on type theory; here Tait gives an extension of the usual combinator cal- culus with S and K to a calculus in which the  ...  and philosophy of language (219-238); Wojciech Buszkowski, The Ajdukiewicz calculus, Polish nota- tion and Hilbert-style proofs (241-252); Andrzej Indrzejczak, Jaskowski and Gentzen approaches to natural  ... 

Page 5203 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews  
In the final Section 5 the theory is briefly compared with Martin-Léf’s intu- itionistic theory of types, with de Bruijn’s AUTOMATH and with Coquand’s theory of constructions.  ...  We use some of the ideas behind the design of the Data Encryption Standard for our construction.  ... 

Games and weak-head reduction for classical PCF [chapter]

Hugo Herbelin
1997 Lecture Notes in Computer Science  
The propositional fragment of this calculus is described in 10]. It is a variant of Gentzen's calculus LJ.  ...  In spread E-dialogues both the strategies of the rst player and of the second player are in one-to-one correspondence with B ohm trees.  ...  Acknowledgement This work has bene ted from the discussions with P.-L. Curien, V. Danos and L. Regnier. I specially thank P.-L. Curien for his fruitful feedback on the paper.  ... 
doi:10.1007/3-540-62688-3_38 fatcat:2bctkd5usvcrvpiqzucot26gla

On Choice Rules in Dependent Type Theory [chapter]

Maria Emilia Maietti
2017 Lecture Notes in Computer Science  
Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand's Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative  ...  In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says  ...  We also thank Tatsuji Kawai and Fabio Pasquali very much for their comments on this paper.  ... 
doi:10.1007/978-3-319-55911-7_2 fatcat:vyvvr3w3lfci7hg23ronk6z2kq

An overview of type theories [article]

Nino Guallart
2014 arXiv   pre-print
It is worth exploring some of them in depth, particularly predicative Martin-L\"of's intuitionistic type theory and impredicative Coquand's calculus of constructions.  ...  The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an active role in the development of computational  ...  Coquand's calculus of constructions (CoC in short) is a higher-order lambda calculus theory that combines polimorphism and type construction of Girard's system F ω with dependent types.  ... 
arXiv:1411.1029v2 fatcat:jmsw6oea7nf33jxrfqchq7rxwq

Sets in types, types in sets [chapter]

Benjamin Werner
1997 Lecture Notes in Computer Science  
We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Fr nkel set theory and the opposite way.  ...  Both encodings are quite elementary: type theory is interpreted in set theory through a generalization of Coquand's simple proof-irrelevance interpretation.  ...  The idea of investigating this topic was given to me by Christine Paulin-Mohring and Martin Ho man.  ... 
doi:10.1007/bfb0014566 fatcat:r2wt56ltwbealgf4bwjbqo3a6q
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