Troelstra, A. S., and D. van Dalen: "Constructivism in mathematics: an introduction (vol 2)", North Holland, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....induction on the associated proof terms supported by inductions on measures of the strong normalisability. The proof is complex: there is more than one cut rule to consider. The calculus is of special interest because its cut free derivations are in natural 1 1 correspondence with the dp normal [28] natural deduction proofs of first order intuitionistic logic. The main purpose, and the novelty, of the present paper is to illustrate the use of the recursive path ordering (r.p.o. theorem of Dershowitz [7] by giving a simple proof of Herbelin s cut elimination theorem. We begin with a routine ....

Troelstra, A. S., and D. van Dalen: "Constructivism in mathematics: an introduction (vol 2)", North Holland, 1988.

....induction on the associated proof terms supported by inductions on measures of the strong normalisability. The proof is complex: there is more than one cut rule to consider. The calculus is of special interest because its cut free derivations are in natural 1 1 correspondence with the dp normal [28] natural deduction proofs of first order intuitionistic logic. The main purpose, and the novelty, of the present paper is to illustrate the use of the recursive path ordering (r.p.o. theorem of Dershowitz [7] by giving a simple proof of Herbelin s cut elimination theorem. We begin with a routine ....

Troelstra, A. S., and D. van Dalen: "Constructivism in mathematics: an introduction (vol 2)", North Holland, 1988.

....of Prawitz [32, 33] are not applicable: i.e. no instance of disjunction elimination occurs at the root of the major premiss of an elimination rule, and similarly for existential elimination and absurdity elimination. Normality has to be carefully defined; in the terminology of [37], normal deductions are dp normal , i.e. they admit no detour or absurdity reductions or permutation reductions but they may allow some immediate simplifications. 3. The correspondence between MJ and NJ (untyped case) Given variable sets U (for individuals) and V (for proofs) with individual ....

....q(Ms) ae apq(T , Ms) spl(U.V.M) N : an(A) lV.N i(N) j(N) wn(A,V.N ,V.N) pr(N , N) efq(A) lU.N prq(T , N) ee(A,U.V. N) A : var(V) ap(A,N) fst(A) snd(A) apn(A,T) This definition of normal lambda terms is designed, as noted above, to capture exactly the dp normal terms of [37]. For example, the choice of efq as a constructor of N terms rather than of A terms ensures that no conclusion of an absurdity elimination rule can be the major premiss of an elimination, since the first argument of each constructor ( wn, efq, ee, ap, fst, snd, apn) for an elimination step is ....

Troelstra, A. S., and D. van Dalen: "Constructivism in mathematics: an introduction (vol 2)", North Holland, (1988).

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