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.

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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

The Curry--Howard 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 object-oriented structuring mechanisms for verification, we extend the object-model 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 object-oriented proving principles --- including inheritance of proofs, late binding of proofs, and encapsulation of proofs --- as analogues to object-oriented programming principles. We have used Lego, a type-theoretic proof checker, to explore the feasibility of this approach. In particular, we have verified a small hier...

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(...

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

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, A-branching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for power-series 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.