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The Groupoid Interpretation of Type Theory
 In Venice Festschrift
, 1996
"... ion and application Suppose that M 2 Tm(B). We define its abstraction A;B (M) 2 Tm(\Pi LF (A; B)) on objects by A;B (M)(fl)(a) = M(fl; a) A;B (M)(fl)(q) = M(id fl ; q) If p : fl ! fl 0 then we need a natural transformation A;B (M)(p) : p \Delta A;B (M)(fl) ! A;B (M)(fl 0 ) At object a ..."
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ion and application Suppose that M 2 Tm(B). We define its abstraction A;B (M) 2 Tm(\Pi LF (A; B)) on objects by A;B (M)(fl)(a) = M(fl; a) A;B (M)(fl)(q) = M(id fl ; q) If p : fl ! fl 0 then we need a natural transformation A;B (M)(p) : p \Delta A;B (M)(fl) ! A;B (M)(fl 0 ) At object a 2 A(fl 0 ) it is given by M(p; id a ). Conversely, if M 2 Tm(\Pi(A; B)) we define a dependent object \Gamma1 A;B 2 Tm(B). Its object part is given by \Gamma1 A;B (M)(fl; a) = M(fl)(a) For the morphism part assume p : fl ! fl 0 and q : p \Delta a ! a 0 . We define \Gamma1 A;B (M)(p; q) = M(fl 0 )(q) ffi (id fl 0 ; q) \Delta M(p) p \Delta a We claim that \Gamma1 A;B (M)(p; q) : (p; q) \Delta \Gamma1 A;B (M)(fl; a) ! \Gamma1 A;B (M)(fl 0 ; a 0 ) as required. To see this, first note that M(fl 0 )(q) : (id fl 0 ; q) \Delta M 0 (fl 0 )(p \Delta a) !M 0 (fl 0 )(a 0 ) because q : p \Delta a ! a 0 . On the other hand M(p) : p \Delta M(fl) !M(fl 0 )...
A Pointfree approach to Constructive Analysis in Type Theory
, 1997
"... The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from ..."
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The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the HeineBorel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the HahnBanach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional MartinLöf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...
A Machine Assisted Proof of the HahnBanach Theorem
, 1997
"... We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with ..."
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We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with [4] and it was a help in the development of the informal proofs. 1 Introduction We present a machine assisted formalisation of pointfree topology in MartinLof's type theory. The continuum and the basic definitions needed in a pointfree approach to functional analysis are given and in this setting we describe implementations of localic formulations of the Alaoglu and the HahnBanach theorems. The classical HahnBanach theorem says that, if M is a subspace of a normed linear space A and f is a bounded linear functional on M , then f can be extended to a linear functional F on A so that kFk = kfk. (In our proof we use the equivalent formulation: if kfk 1 then f can be extended to F so that kFk 1.) A...
An Implementation of the HeineBorel Covering Theorem in Type Theory
"... . We describe an implementation, in type theory, of a proof of a pointfree formulation of the HeineBorel covering theorem for intervals with rational endpoints. 1 Introduction The proof presented here is a complete formalisation of the proof presented in "A constructive proof of the HeineBor ..."
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. We describe an implementation, in type theory, of a proof of a pointfree formulation of the HeineBorel covering theorem for intervals with rational endpoints. 1 Introduction The proof presented here is a complete formalisation of the proof presented in "A constructive proof of the HeineBorel covering theorem for formal reals" [CN]. We describe an implementation, in type theory, of a proof of a pointfree formulation of the HeineBorel covering theorem for intervals with rational endpoints. The implementations also contain a definition of formal spaces as a type, and definitions of the continuum and the closed rational interval as instances of that type. The paper is organised as follows: in section 2 we describe the proofchecker Half, in which the implementation has been done, and the type theory it is based on. The rest of the paper is devoted to formal definitions and the proof of the HeineBorel covering theorem. In section 3 some general definitions are given. In section 4 we ...
An Implementation of the HeineBorel Covering Theorem in Type Theory
"... Abstract. We describe an implementation, in type theory, of a proof of apointfree formulation of the HeineBorel covering theorem for intervals with rational endpoints. 1 ..."
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Abstract. We describe an implementation, in type theory, of a proof of apointfree formulation of the HeineBorel covering theorem for intervals with rational endpoints. 1