### Table 1: Sequent rules for LF

2002

"... In PAGE 6: ... However, it turns out that once we establish the appropriate left and right introduction rules for the two way rules above, in particular for the typing rules, these mixed cut rules will be derivable, and so need not be added to the sequent calculus. In Table1 we list the sequent rules for this logic LF, using the convention that when a rule applies to either sequent calculus, we use a variable such as x to represent either of S or T. We have given the left and right introduction rules corresponding to the two way rules discussed above.... In PAGE 6: ...ules. We shall illustrate this with the following pair of lemmas. Lemma 1.2 In LF (as presented in Table1 ), the mixed cut rules are derivable rules. Proof.... In PAGE 8: ... 2 Lemma 1.3 In LF (as presented in Table1 ), the two way tensor and the two way typing rules are derivable rules. Proof.... ..."

Cited by 4

### Table 3: the sequent calculus BCT

1995

"... In PAGE 10: ... This should motivate the rules of the sequent calculus BCT, the \By Cases Theory quot; given in the next de nition. In Table3 we use the vector-notation described earlier: in rule ( !), the terms ~ A are of types such that (f ~ A) has product type; and in rule ByCases, the terms ~ P are of types such that (h ~ P ) has sum type. De nition 5.... ..."

Cited by 10

### Table 1: Inference rules for sequents.

2000

"... In PAGE 12: ...2 (Sequents) Let R = h ; E; L; Ri be a rewrite theory. We say that R entails a sequent [s] ) [t], written R ` [s] ) [t], if and only if [s] ) [t] can be obtained by a nite number of applications of the inference rules in Table1 , where t(~ w=~x) denotes the simultaneous substitution of wi for xi in t. A rewrite theory is just a static description of `what a system can do apos;; the behaviour of the theory is instead given by the rewrite relation induced by the rules of deduction.... In PAGE 12: ... A rewrite theory is just a static description of `what a system can do apos;; the behaviour of the theory is instead given by the rewrite relation induced by the rules of deduction. The deduction system in Table1 was introduced in [54], and it is only one of the possible, equivalent ways to entail the same class of sequents. It has, however, the advantage of being rather intuitive.... In PAGE 17: ...Extending the paradigm to non-cartesian structures The deduction rules presented in Table1 make clear that the underlying idea of the rewriting logic paradigm is that the rewrite relation has to be built in- ductively, lifting to computations the structure of terms. Such an intuition can be exploited to describe suitable notions of computation also over structures other than terms: In particular, over elements of gs-monoidal theories, as for the deduction system presented in this section.... In PAGE 17: ... Of course, the deduction system we just presented is also valid for rewriting over monoidal theories: Since we are not interested in the eventual structure of proof terms, we just need to change the premise of the re exivity rule, re- stricting the attention to terms in ME( ). The system in Table 2 induces over terms the same rewrite relation as the one de ned in Table1 for alge- braic sequents, since algebraic theories are just gs-monoidal theories plus the naturality axioms En, that is, AE( ) = GSE[En( ). The correspondence re- sult between the two deduction systems is explicitly given by the following proposition, stated here only for rewrite theories with an empty set of axioms.... ..."

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### Table 1. Sequent and anti-sequent left rules - Introduction into antecedent

### Table 1: Sequent calculus I.

2000

Cited by 23

### Table 1: A sequent system K for PL.

1995

"... In PAGE 5: ..., range over sequences of formulae of LP. A sequent system K for the logic is given in Table1 . In the sequel `K means that the sequent ` can be derived in K.... ..."

Cited by 2