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Extensional Set Equality in the Calculus of Constructions

2001
*
Journal of Logic and Computation
*

*The*original representation

*of*set theory

*in*

*the*

*calculus*

*of*

*constructions*, has met an objection on

*the*grounds that

*the*axiom

*of*exstensionality is incompatible with

*the*assumptions for arithmetic. ...

*In*this paper, it is shown that there is really no problem with

*the*axiom

*of*

*extensionality*. ... Acknowledgements This work was supported

*in*part by grant RGP-23391-98 from

*the*Natural Sciences and Engineering Research Council

*of*Canada, and was presented at

*the*Festival Workshop on

*the*Foundations ...

##
###
Internalizing Relational Parametricity in the Extensional Calculus of Constructions

2013
*
Annual Conference for Computer Science Logic
*

We give

doi:10.4230/lipics.csl.2013.432
dblp:conf/csl/KrishnaswamiD13
fatcat:ekzodbdhxjd7xfpc2iylif3aia
*the*first relationally parametric model*of**the**extensional**calculus**of**constructions*. ... Our model remains as simple as traditional PER models*of*types, but unlike them, it additionally permits*the*relating*of*terms that implement abstract types*in*different ways. ... Almost all*of**the*typing rules are standard for*the**calculus**of**constructions*, and we discuss only our variations*in*detail. Our treatment*of*equality follows*extensional*type theory. ...##
###
Deduction System for TIL-2010

2012
*
Recent Advances in Slavonic Natural Languages Processing
*

Then we describe

dblp:conf/raslan/DuziMV12
fatcat:ma2bklnidrexvhocladriyelzy
*the*adjustments*of**the**calculus*so that it be applicable to hyperintensions within*the*ramified hierarchy*of*types. ... Tichý defined a sequent*calculus*for pre-1988 TIL, that is TIL based on*the*simple theory*of*types. Thus we first briefly recapitulate*the*rules*of*this simple-type*calculus*. ...*of*Semantics' and also by*the*internal grant agency*of*VSB-Technical University Ostrava, Project SP2012/26, "An utilization*of*artificial intelligence*in*knowledge mining from software processes". ...##
###
Realizability Models for Type Theories

1999
*
Electronical Notes in Theoretical Computer Science
*

These range from simply typed -

doi:10.1016/s1571-0661(04)00108-2
fatcat:7v24nb5g6nfifec3xmtg7swqum
*calculus*over second-order polymorphic -*calculus*to*the**Calculus**of**Constructions*as an example*of*dependent t ype theory. ... Finally, b y considering complete*extensional*pers, an approach to bridge*the*gap from type theory to*constructive*domain theory is discussed. Reus (see e.g. 7]). ... Thanks also go to Thorsten Altenkirch, Alexander Knapp, and several workshop participants who gave v aluable feedback on di erent preliminary drafts*of*this survey, and to*the*editors,*in*particular Pino ...##
###
Page 12 of Mathematical Reviews Vol. , Issue 81G
[page]

1981
*
Mathematical Reviews
*

*The*author then extends

*the*

*construction*

*of*this model

*in*two ways.

*The*first is to obtain a model

*of*

*the*ASK-

*calculus*without

*the*w-rule. ... Any set-theoretic models

*of*

*the*A-

*calculus*which are natural (

*in*

*the*above sense) have to be

*constructed*

*in*set theories without this axiom. ...

##
###
On a Question of H. Friedman

1996
*
Information and Computation
*

*In*this paper we answer a question

*of*Friedman, providing an ω-separable model M

*of*

*the*λβη-

*calculus*. There therefore exists an α-separable model for any α ≥ 0. ...

*The*open term model embeds

*in*M: by way

*of*contrast we provide a model which cannot embed

*in*any non-trivial model separating all pairs

*of*distinct elements. ...

*In*Section 2 we answer Friedman's question positively, giving a term model

*construction*

*of*an ω-separable model

*of*

*the*λβη-

*calculus*. ...

##
###
Towards an Implementation in LambdaProlog of the Two Level Minimalist Foundation (short paper)

2018
*
International Conference on Intelligent Computer Mathematics
*

Linking

dblp:conf/mkm/FioriC18
fatcat:hm66eh7eqbby5eaclj5ve64oei
*the*two level there is an interpretation*of**the**extensional*level*in*a dependent setoid model build upon*the*intensional level where*extensionality*is modeled by intensional propositions. ...*In*this work we will present an implementation*of**the*Minimalist Foundation [MS05; Mai09] using*the*Prolog [NM88] implementation ELPI [Dun+15] . ... To reasonably implement*the**extensional**calculus*and*the*interpretation we need a way to*construct*intensional proofs from*extensional*proof and a way to contain*the*undecidability*of**the*strong elimination ...##
###
Hot: A concurrent automated theorem prover based on higher-order tableaux
[chapter]

1998
*
Lecture Notes in Computer Science
*

*The*rst part

*of*this paper introduces an improved variant

*of*

*the*

*calculus*which closely corresponds to

*the*proof procedure implemented

*in*Hot. ... Hot is an automated higher-order theorem prover based on HT E, an

*extensional*higher-order tableaux

*calculus*. ... o px ) py

*The*following tableau

*construction*rules complement ETAB with regard to

*extensionality*

*in*Henkin model semantics: Soundness and Completeness Soundness and completeness proofs for

*extensional*...

##
###
A Calculus of Coroutines
[chapter]

2004
*
Lecture Notes in Computer Science
*

*extensional*

*in*

*the*presence

*of*

*of*errors;

*in*particular adding errors to PCF allows

*the*order

*of*evaluation

*of*functions to be observed. ...

*In*particular, Laird showed that

*the*bistable functions (a new notion) gave rise to a universal model

*of*

*the*λ-

*calculus*with two constants, and that CPS-translations allowed this to be extended to programming ... Moreover,

*the*

*extensional*understanding

*of*computation

*in*games models afforded by this work allowed

*the*

*construction*

*of*new models

*of*nondeterminism: by completing

*the*model with meets or joins, one obtains ...

##
###
Extensional Universal Types for Call-by-Value
[chapter]

2008
*
Lecture Notes in Computer Science
*

To demonstrate validity

doi:10.1007/978-3-540-89330-1_9
fatcat:icvmldilfzgm7jmgpo3thtwxv4
*of**the**calculus*, we*construct*concrete models for*the**calculus**in*a generic manner, exploiting "relevant" parametricity. ...*In**the*present paper we propose a second-order polymorphic call-by-value lambda*calculus*with*extensional*universal types. ...*The*λc-*calculus*λ c 2 η -*calculus*, to demonstrate how many effects are consistent with*extensional*universal types. ...##
###
Page 1375 of Mathematical Reviews Vol. , Issue 82d
[page]

1982
*
Mathematical Reviews
*

(

*The*author’s*construction**of*C; is*in*fact Cy, with M@*the*closed term model*of*7.) If there are enough arrows 1— U (1 is*the*terminal object), then I2(C, U) is (weakly)*extensional*. ... Given a domain U (and*the*retraction maps)*in*a Ccc C, one can extend*the*mentioned correspondence to type-free A-*calculus*and*construct*a A-theory T(C, U). ...##
###
Programs as Data Structures in λSF-Calculus

2016
*
Electronical Notes in Theoretical Computer Science
*

*In*turn, all closed normal forms are data structures,

*in*

*the*sense that their internal structure is accessible through queries defined

*in*

*the*

*calculus*, even to

*the*point

*of*

*constructing*

*the*Goedel number ... First, N is

*extensionally*equivalent to M where

*extensional*equivalence is defined

*in*terms

*of*eta-reduction. ...

*In*conclusion, λSF -

*calculus*adds intensionality to

*the*

*extensional*nature

*of*λ

*calculus*, so that one can query

*the*internal structure

*of*arbitrary closed normal forms, and treat programs as data structures ...

##
###
Page 304 of The Philosophical Review Vol. 52, Issue 3
[page]

1943
*
The Philosophical Review
*

For

*the*intuitionistic rejection*of*certain principles*of*classical mathematics (e.g.,*the*principle*of*excluded middle) refers, not to*the*abstract*construction**of*a*calculus*, but to*the**construction*...*of*a*calculus*whose interpretation is at least partly given. ...##
###
Extensional Models of Untyped Lambda-mu Calculus

2012
*
Electronic Proceedings in Theoretical Computer Science
*

*The*other is called

*the*stream combinatory algebra, which is an extension

*of*

*the*combinatory algebra, and it is proved that

*the*

*extensional*equality

*of*

*the*Lambda-mu

*calculus*is equivalent to equality ... This paper proposes new mathematical models

*of*

*the*untyped Lambda-mu

*calculus*. ... One natural direction

*of*study is to analyze

*the*local structure

*of*

*the*domain-theoretic

*extensional*stream models

*constructed*from

*the*solutions

*of*(2)

*in*Section 3.2. ...

##
###
Dr. Zilsel on the Concept of Physical Law

1943
*
Philosophical Review
*

For

doi:10.2307/2180924
fatcat:rg24t7e4drbufn5uv3gxignt74
*the*intuitionistic rejection*of*certain principles*of*classical mathematics (e.g.,*the*principle*of*excluded middle) refers, not to*the*abstract*construction**of*a*calculus*, but to*the**construction*...*of*a*calculus*whose interpretation is at least partly given. ...
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