The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...

We give two nite axiomatizations of indexed inductive-recursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductive-recursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductive-recursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.

Introduction We present our implementation AGDA of type theory. We limit ourselves in this presentation to a rather primitive form of type theory (dependent product with a simple notion of sorts) that we extend to structure facility we find in most programming language: let expressions (local definition) and a package mechanism. We call this language Structured Type Theory. The first part describes the syntax of the language and an informal description of the type-checking. The second part contains a detailed description of a core language, which is used to implement Strutured Type Theory. We give a realisability semantics, and type-checking rules are proved correct with respect to this semantics. The notion of meta-variables is explained at this level. The third part explains how to interpret Structured Type Theory in this core language. The main contributions are: ffl use of explicit substitution to simplify and make

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We prove normalization for a dependently typed lambda-calculus extended with first-order data types and computation schemata for first-order size-change terminating recursive functions. Size-change termination, introduced by C.S. Lee, N.D. Jones and A.M. Ben-Amram, can be seen as a generalized form of structural induction, which allows inductive computations and proofs to be defined in a straight-forward manner. The language can be used as a proof system—an extension of Martin-Löf’s Logical Framework.

form of the knight's tour problem \Is it possible for the k-th player to win?" Solution. We have to nd a boolean function which associates to the k-th player true if and only if there exists the game tree and it has a node labelled with a state which is winning for the k-th player. The problem can be expressed by the following type: (8k < numP layers) fx 2 Boolj (x = true) $ (9t 2 ExpandedT ree(State; gameGraph; initState)) HasWinning(t; k)g where HasWinning(t; k) (9a 2 State) (a InT ree t) & (winning(a; k) = true) 60 The abstract solution A general solution of this problem is then given by the function k:find((expand(gameGraph)) m (leaf(initState)); (s) winning(s; k)) since, supposing t is a tree and b a condition, it is possible to prove that find(t; b) 2 fx 2 Boolj (x = true) $ (9a 2 State) (a InT ree t) & (b(a) = true)g 61 What should be avoided If you want to avoid classical logic we should not add ... Extensional power-set U(x) prop [x : S] fUg 2 P(S) x"U $ x"V [x : S] fUg = fV g 2 P(S) Extensional nite power-set U(x) prop [x : S] Fin(U) fUg 2 P!(S) x"U $ x"V [x : S] fUg = fV g 2 P!(S) where Fin(U) (9n 2 Nat)(9f : Nn ! S) U Im[f ] Extensional quotient set a 2 A [a] R 2 A=R [a] R = [b] R 2 A=R R(a; b) true Extensional couple set W 1 ; W 2 2 fU; V g x"W 1 $ x"W 2 [x : S] W 1 = W 2 2 fU; V g 62 The Power-set Constructor The notion of Subset Let S be a set, then U is a subset of S if U(x) prop [x : S] or equivalently U : (x : S) prop Thus . . .

We present a denotational semantics of a type system with dependent types, where types are interpreted as finitary projections. We prove then the correctness of a type-checking algorithm w.r.t. this semantics. In this way, we can justify some simple optimisation in this algorithm. We then sketch how to extend this semantics to allow a simple record mechanism with manifest fields.