Regnier, L. (1992). -Calcul et r'eseaux. PhD thesis, Universit'e Paris VII.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... things) the sign of F (i.e. oe(F ) or O if F occurs positively, oe(F ) Gamma if F occurs negatively in the sequent Gamma ) Delta (i.e. in the formula V Gamma W Delta) or either ffi or ffl (in case a (sub)formula disappears in a cut) Using the terminology of Regnier(1992), we will call a formula in terminal if it either is a cutformula in or a formula in the conclusion. Then, to be precise, we define T (F ) for (sub)formulas F of terminal formulas in ; in case F is a cutformula we put T (F ) T 0 (F ) where 0 denotes the subderivation of ....

..... Gamma ) A 1 . Gamma ) A . Pi; Delta; A; A ) B Pi; Gamma; Delta; A ) B Pi; Gamma; Gamma; Delta ) B . Pi; Gamma; Delta ) B 10 The concepts of chain and path are similar to that of trace , which plays an important role in the study of proofnets, cf. Regnier(1992). If i OE is boxed and moreover is element of an adequate chain Sigma in , then we can show the converse, i.e. there exists a series of reductions (a reduction strategy) starting from , that eventually either will erase or duplicate i . An important step towards a proof of this is the ....

Regnier, L. (1992). -Calcul et r'eseaux. PhD thesis, Universit'e Paris VII.

....contextformulas of the final sequent of the transported subderivation back down along the tree. At branchings originally due to explicit contractions on A, the contractions now are inherited by these contextformulas. cf. the complete exponential reduction of proof nets, as described e.g. in Regnier(1992). This procedure of course forms a radical departure from the usual one up reduction steps that are considered in most proofs of eliminability of cut. 6 Weak equivalence A well known, sometimes execrated, property of sequent calculus, is the high number of possible variations on a given ....

....in D Gamma ( this subderivation is an exponential box with, say, D(A) as principal door, and the decorated cut is an exponential cut. Reduction of this cut in D Gamma ( consists in moving it up the principal door s exponential tree (this again is the complete exponential reduction of Regnier(1992)) possibly) duplicating and or erasing the box, then cutting it on the derelictions introducing the outermost exponential of ancestors of ( D(A) These derelictions follow introductions in either an axiom or a logical rule. In case of an axiom we proceed by a promotion dereliction ....

Regnier, L. (1992). -Calcul et r'eseaux. PhD thesis, Universit'e Paris VII.

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