Polymorphic extensions of simple type structures. With an application to a bar recursive minimization
Annals of Pure and Applied Logic
The technical contribution of this paper is threefold. First we show how to encode ftmctionals in a 'flat' applicative structure by adding oracles to untyped I-calculus and mimicking the applicative behaviour of the functionals with an impredicatively defined reduction relation. The main achievement here is a Church-Rosser result for the extended reduction relation. Second, by combining the previous result with the model construction based on partial equivalence relations, we show how to extend
... a i-closed simple type structure to a model of the polymorphic I-calculus. Third, we specialize the previous result to a counter model against a simple minimization. This minimization is realized by a bar recursive functional, which contrasts the results of Spector and Girard which imply that the bar recursive functions are exactly those that are definable in the polymorphic &calculus. As a spin-off, we obtain directly the non-conservativity of the extensions of Godel's T with bar recursion, fan functional, and Luckhardt's minimization functional, respectively. For the latter two extensions these results are new. SSDIOl68-0072(95)00025-9 E. Barendsen, M. Bezem I Annals of Pure and Applied Logic 79 (1996) 221-280 Lemma 1.2.21. M 237 Proof. Set A4 3 MI, M 3 A42. Distinguish cases as to the relative positions of A i and AZ. Case 1: A, and A2 are disjoint. This case is trivial. Case 2: Ai and A2 coincide. Then we are done by the Non-ambiguity Lemma 1.2.10.