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On the proof theory of Coquand's calculus of constructions

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*. ...

##
###
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’s*“*calculus**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]

2005
*
Lecture Notes in Computer Science
*

Deciding

doi:10.1007/978-3-540-31982-5_26
fatcat:rfpv4yncgfe27h4zwsscxk5c3e
*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. ...##
###
Page 5700 of Mathematical Reviews Vol. , Issue 89J
[page]

1989
*
Mathematical Reviews
*

*The*“

*calculus*

*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]

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 ...

##
###
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]

1997
*
Lecture Notes in Computer Science
*

This analysis is an extension

doi:10.1007/bfb0000493
fatcat:3rr2gcypm5fffc6hqyrxfcjxmu
*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. ...##
###
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]

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. ...

##
###
On Choice Rules in Dependent Type Theory
[chapter]

2017
*
Lecture Notes in Computer Science
*

Here we show that

doi:10.1007/978-3-319-55911-7_2
fatcat:vyvvr3w3lfci7hg23ronk6z2kq
*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. ...##
###
An overview of type theories
[article]

2014
*
arXiv
*
pre-print

It is worth exploring some

arXiv:1411.1029v2
fatcat:jmsw6oea7nf33jxrfqchq7rxwq
*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. ...##
###
Sets in types, types in sets
[chapter]

1997
*
Lecture Notes in Computer Science
*

We present two mutual encodings, respectively

doi:10.1007/bfb0014566
fatcat:r2wt56ltwbealgf4bwjbqo3a6q
*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. ...
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