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Type Theory and Programming
, 1994
"... This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an im ..."
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This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.
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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Cited by 21 (1 self)
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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 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 [4] and i ..."
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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 HeineBorel co ..."
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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 ...