@MISC{Sellink96onthe, author = {M. P. A. Sellink}, title = {On the Conservativity of Leibniz Equality}, year = {1996} }

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Abstract

We embed a first order theory with equality in the Pure Type System L that is a subsystem of the well-known type system PRED2. The embedding is based on the Curry-Howard isomorphism, i.e. \Gamma\Gamma\Gamma? and 8 coincide with ! and \Pi. Formulas of the form t1 s = t2 are treated as Leibniz equalities. That is, t1 s = t2 is identified with the second order formula 8P : P (t1 )\Gamma\Gamma\Gamma?P (t2 ), which contains only \Gamma\Gamma\Gamma?'s and 8's and can hence be embedded straightforwardly. We give a syntactic proof for the equivalence between derivability in the logic and inhabitance in L. The idea of the proof is to introduce extra reduction steps, that reduce those proofterms that do not correspond to derivations in the logic to ones that do correspond to derivations in the logic. Introduction Many logics can be interpreted in type systems. For instance Implicational Propositional Logic can be interpreted in ! . In [Ber89], Berardi designed a type system (PRED!) in w...

...x and C j &. Both cases are ruled out by the Generation Lemma (Lemma 2.2 on page 8). 2 Lemma 2.5. \Gamma ` M :: A =) A 2 S. Proof. Suppose \Gamma ` M : oe and \Gamma ` oe : A. By Correctness of types =-=[Bar92]-=- there exists a sort & 2 S such that \Gamma ` oe : & or oe 2 S. In the first case we know, by Uniqueness of Types [Bar92], that A = fi & and hence A j & by Lemma 2.4, so A 2 S and we are done. In the ...

...Minimal Higher Order Many Sorted Intuitionistic Predicate Logic (PRED!) can be interpreted, as worked out ingreat detail in [TF92]. All these interpretations are based on the Curry-Howard-isomorphism =-=[How80]-=-. In this paper we focus on the interpretation in type theory of a first order theory with equality. We interpret this theory, which is a subsystem of second order predicate logic (PRED2), in a Pure T...

...ty of type systems. In [Sel96] we give the criterion (for PTS's) which should be fulfilled in order to apply the ideas of the proof. This criterion is fulfilled for all type systems of the logic cube =-=[Geu93]-=-. The rest of this paper is organised as follows: in Section 1 we introduce the first order theory and its accompanying proof theory. Then, in Section 2, the Pure Type System L is introduced. We assum...

... b used for the encoding of equality, and it is (almost) the weakest type system (PTS) with this property. We expect, however, that the ideas of the proof apply to a large variety of type systems. In =-=[Sel96]-=- we give the criterion (for PTS's) which should be fulfilled in order to apply the ideas of the proof. This criterion is fulfilled for all type systems of the logic cube [Geu93]. The rest of this pape...

...erpreted ins! . In [Ber89], Berardi designed a type system (PRED!) in which Minimal Higher Order Many Sorted Intuitionistic Predicate Logic (PRED!) can be interpreted, as worked out ingreat detail in =-=[TF92]-=-. All these interpretations are based on the Curry-Howard-isomorphism [How80]. In this paper we focus on the interpretation in type theory of a first order theory with equality. We interpret this theo...