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Unifying Constructive and Nonstandard Analysis
 Bull. Symbolic Logic
, 1999
"... This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions bet ..."
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This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions between nonstandard versions of standard sets. We also briefly indicate how to make the Loeb measure construction over hyperfinite sets. Section 4 discusses the relation between nonstandard real numbers and the canonical real numbers of N . In the final section we exemplify the use of the model to prove results in the calculus of several variables, e.g. the Implicit Function Theorem.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Inheritance of Proofs
, 1996
"... The CurryHoward isomorphism, a fundamental property shared by many type theories, establishes a direct correspondence between programs and proofs. This suggests that the same structuring principles that ease programming be used to simplify proving as well. To exploit objectoriented structuring me ..."
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Cited by 4 (0 self)
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The CurryHoward isomorphism, a fundamental property shared by many type theories, establishes a direct correspondence between programs and proofs. This suggests that the same structuring principles that ease programming be used to simplify proving as well. To exploit objectoriented structuring mechanisms for verification, we extend the objectmodel of Pierce and Turner, based on the higher order typed calculus F ! , with a proof component. By enriching the (functional) signature of objects with a specification, the methods and their correctness proofs are packed together in the objects. The uniform treatment of methods and proofs gives rise in a natural way to objectoriented proving principles  including inheritance of proofs, late binding of proofs, and encapsulation of proofs  as analogues to objectoriented programming principles. We have used Lego, a typetheoretic proof checker, to explore the feasibility of this approach. In particular, we have verified a small hier...
A construction of Type:Type in MartinLöf's partial type theory with one universe
"... ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x ..."
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ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x)R 1 (x; p(c))i; for c 2 PAR, and let e := fix((c)f(c)). Hence e 2 PAR is a fixed point of f , e = f(e). The right summand of f corresponds to the rules (2). We now interpret Type:Type. The universe (U 1 ; T 1 ) is defined by letting U 1 := T (p(e)) and T 1 (a) := T (Ap(q(e); a)); for a 2 U 1 . Thus the rules (1) are verified. Using the equality e = f(e) and the commutation of T with \Sigma, \Pi and + we get U 1 = T (p(e)) = T (p(f(e))) (4) = T (oe(p(e); (x)Ap(q(e); x) ! p(e))) + T (n 1 ) = (\Sigmax 2 T (p(e)))[T (Ap(q(e); x)) \Gamma! T (p(e))] +N 1 = (\Sigmax 2 U 1 )[T 1 (x) \Gamma! U 1 ] +N 1 and hence j(0 1 ) 2 U 1 . Furthermore we have T 1 (j(0 1 )) = T (Ap(q(...
Infinite Objects In Type Theory
, 1997
"... Contents 1. Introduction 1 2. Lazy Evaluation and Infinite Objects 3 3. Mathematics of Infinity 10 3.1. The extension of type theory with an external infinite object 11 4. Models of Type Theory 18 4.1. The Structure of a Model for Type Theory 19 4.2. The Comma Model 24 4.3. Necessity 26 5. Mathemat ..."
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Contents 1. Introduction 1 2. Lazy Evaluation and Infinite Objects 3 3. Mathematics of Infinity 10 3.1. The extension of type theory with an external infinite object 11 4. Models of Type Theory 18 4.1. The Structure of a Model for Type Theory 19 4.2. The Comma Model 24 4.3. Necessity 26 5. Mathematics of Infinity Formalized 27 5.1. The term model 27 5.2. T ff 30 5.3. The standard model 33 5.4. Nonstandard truth 39 6. Examples 43 6.1. Infinity 43 6.2. Linear Search 45 7. Conclusions 50 References 52 1. Introduction It is one of the purposes of this work to explore type theory as a programming language where programs come out correct by construction. This has already been done to some extent following the correspondence types/specifications/propositions and objects /programs/proofs. In our case we will be interested in a particular analysis of streams and other infinite objects as they occur in lazy functio
M4M 2007 Continuous Functions on Final Coalgebras
"... It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, Abranching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on fin ..."
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It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, Abranching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for powerseries functors on the category of sets and functions. While our main technical contribution is the characterisation of all continuous functions, defined on a final coalgebra and taking values in a discrete space by means of inductive types, a methodological point is that these inductive types are most conveniently formulated in a framework of dependent type theory.
A note on Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis
, 2011
"... A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer princi ..."
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A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer principle is weak. In this paper it is shown that this principle can be strengthened by employing Brouwerian continuity axioms familiar from intuitionistic systems. We end by commenting on the relation between the transfer principle and Ishihara’s boundedness principle. 1
Wander Types A Formalization of CoinductionRecursion ∗
"... Wander types are a coinductive version of inductiverecursive definitions. They are defined by simultaneously specifying the constructors of the type and a function on the type itself. The types of the constructors can refer to the function component and the function itself is given by pattern match ..."
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Wander types are a coinductive version of inductiverecursive definitions. They are defined by simultaneously specifying the constructors of the type and a function on the type itself. The types of the constructors can refer to the function component and the function itself is given by pattern matching on the constructors. Wander types are different from inductiverecursive types in two ways: the structure of the elements is not required to be wellfounded, so infinite applications of the constructors are allowed; and the recursive calls in the definition of the function are not required to be on structurally smaller arguments. Wander types generalize several known type formers. We can use the functional component to control the way the data branch. This allows not only the implementation of coinduction, but also of induction, by imposing wellfoundedness through an appropriate function definition. Special instances of wander types are: plain inductive and coinductive types, inductiverecursive types, mixed inductivecoinductive types, continuous stream processors. 1
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"... A note on Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis ERIK PALMGREN 1 Abstract: A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultrapower construction. A constructive approach due to Schmieden and ..."
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A note on Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis ERIK PALMGREN 1 Abstract: A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultrapower construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer principle is weak. In this paper it is shown that this principle can be strengthened by employing Brouwerian continuity axioms familiar from intuitionistic systems. We end by commenting on the relation between the transfer principle and Ishihara’s boundedness principle.