### Table 5. Box rewrite rules

"... In PAGE 9: ... Table 7 gives the usual cut-elimination steps, whereas Table 6 gives the extra cut-elimination steps for the \unordered n-ary quot; treat- ment we give for the contraction links. Table5 lists additional rewrites for storage boxes needed for the categorical semantics. In (Blute et al 1992) a considerable e ort was spent in making the rewiring of thinning links as \local quot; as possible (see the discussion there of rules of surgery).... In PAGE 36: ... the reductions and expansions of Tables 4, 6, 7, are just those of (Danos 1990), and so form a con uent system for which we have strong normalization. The \box-expansion quot; rules of Table5... In PAGE 40: ... A terminal storage link may be handled by a method that depends on the type of thinning link involved. For an exponential thinning, the box rewrite rule in Table5 allows us to move the thinning link outside the box. For a unit thinning, the thinning link and the empire of the unit lie either completely inside or completely outside the storage box|in either case the induction assumption is easily applied.... ..."

### Table 5: Rewrite rules for BPA

1994

"... In PAGE 8: ... x (y z + z) ?! x(y z + z) if x + y ?!! y 3.6 The entire TRS The entire TRS is given once again in Table5 . It is easy to see that all rules can be deduced from BPA .... In PAGE 10: ... Theorem 3.8 The TRS R in Table5... In PAGE 13: ... Suppose that q is a normal form of a term rs. Each rule in Table5 that applies to a term of the form tu or t u, reduces it to one of either forms again, and so q must be in one of either forms. But Rules 3,4,7 and 8 reduce p(tu) and p (tu) and p(t u) and p (t u) respectively.... ..."

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### Table 1. The rewriting system for the -calculus

"... In PAGE 3: ... Thus, in addition to the substitution of the free occurrences of the index 1 by the corresponding term, free occurrences of indices should be actualized (decreased) because of the elimination of the abstractor. Table1 includes the rewriting system of the -calculus augmented with an Eta rule for the -reduction, as presented in [6].... In PAGE 4: ... And when b is restricted as well to be in -normal form in principle no problem appears, but without these restrictions subject reduction may be violated. In fact, assuming the de nition of Eta in Table1 and without any restriction on terms, observe that (2 1 ) ! 2 [ (1 1 ):1 : quot;], since 2 = 2 [ (1 1 ):1 : quot;] [ quot;], where 2 abbreviates 1 [ quot;], because (1 [ quot;])[ (1 1 ):1 : quot;] !Clos 1 [ quot; ( (1 1 ):1 : quot;)] !ShiftCons 1 [1 : quot;] !V arCons 1. But the term (1 1 ) is not typable in the simply typed -calculus.... In PAGE 4: ... Consequently, in order to have a constructive and implementable de nition of Eta one needs to explicitly de ne how the condition of the rule should be decided. The de nition of Eta for the -calculus in Table1 is inherited from the usual (non constructive) de nition of -reduction given in the literature for the -calculus ` a la de Bruijn: (a 1) ! b if b+ = a, where b+ denotes the lifting of b, operator which increases by one the free indices... ..."

### Table 4: Normalization rewriting rules

1998

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### Table 2. Rewriting TM predicates

1994

"... In PAGE 12: ...Examples of predicates that may or may not be rewritten into the desired format are listed in Table2 . Predicates above the separation line are predicates that may occur in SQL (being a subset of TM); predicates below the separation line are specific TM pred-... In PAGE 13: ...a modification of the regular join. Consider the following query: SELECT x FROM X x WHERE x:a SELECT y:a (P1) FROM Y y WHERE x:b = y:b ^ y:c SELECT z:c (P2) FROM Z z WHERE y:d = z:d Predicates P1 and P2 between query blocks do require grouping (see Table2 ), so we may have the following execution strategy: (1) A nest join with operands Y and Z on join predicate y:d = z:d. Each element of Z satisfying the join predicate is projected on the c attribute.... ..."

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### Table 3: Rewrite of Fig. 4

1993

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### Table 2: Term Rewriting Process

1998

Cited by 4