### Table 2: Laws for the asynchronous -calculus

"... In PAGE 11: ...Table 2: Laws for the asynchronous -calculus We write P v Q if the inequality P v Q is derivable within the system of Table2 . Soundness of the system is straightforward.... ..."

### Table 2: Sequent Calculus of Linear Logic De nition 5.5. A typed expression of l-calculus can be derived using the rules shown in Table 3. If t ` X is derivable we say that the term t has type X. Remark 5.1. Assume that the statement has been derived (x1; : : : ; xn)fj t1; : : : ; tk; a1 b1; : : : ; ar br jg ` A1; : : : ; Ak Its derivation induce a 1 ? 1 correspondence between the proper terms (that are not interactions) and the formulae of the sequent. The correspondence does not depend on the particular derivation of the term. Lemma 5.6. Every term constructed in l-calculus is closed. The demonstration consists only in a routine veri cation on the derivation of the term.

### Table 3: A calculus for -ccp

"... In PAGE 14: ... In fact the operators of the language are modeled in these equations by standard set theoretic notions which in the speci cation logic get replaced by the corresponding logical notions. Table3 contains the rules of the calculus for assertions of the form D:A sat . Rules C0-C6 are given in the usual natural deduction style and are self explaining since they are obtained essentially by a logical reading of equations D0-D6, i.... In PAGE 14: ... Theorem 5.4 (Soundness) If D:A sat can be derived by the inference system in Table3 , then j= D:A sat holds. Completeness of the system is proved in the sense of Cook ([7]): We assume the expressibility of the strongest postcondition of a process D:A, i.... ..."

### Table 1 The sequent calculus for L/MILL with commutativity implicit

"... In PAGE 3: ...ig. 1. Example sequent derivation 2.1 Sequent Calculus Table1 shows the sequent calculus for the Lambek calculus L, first proposed by Lambek (1958). The commutative version, the Lambek-van Benthem calculus LP, is also known as the multiplicative fragment of intuitionistic linear logic MILL.... ..."

### Table 2: The transition system for the -calculus

1998

"... In PAGE 18: ... Also, is the composition of the two substitutions, in which is applied rst; therefore P is (P ) . The operational semantics of the calculus is de ned by the transition rules of Table2 . The silent action P ?! Q has the same meaning as in CCS.... In PAGE 25: ... Normalised replications can be given the simple transition rule rep-nor: :P ?! P0 ! :P ?! P0 j ! : P or, alternatively, the two rules rep-inp: ! a(x): P ab ?! Pfb=xg j ! a(x): P rep-pre: ! : P ?! P j ! : P ; if is not an input Remark 6.2 As an aside, we wish to point out that rule rep-nor (as well as rep-inp and rep-pre) preserves the following pleasant property of -calculus transition system in Table2 , and which we state here very informally: If two inference proofs of transitions P ?! P0 and P ?! P00 consume the same pre x(es) of P, then P0 and P00 are syntactically the same (up to alpha conversion). This is a handy property to have, for instance when examining the set of derivatives of a process, because it makes it easier to reason by structural induction on processes.... ..."

Cited by 53

### Table 2: The transition system for the -calculus

1998

"... In PAGE 18: ... Also, is the composition of the two substitutions, in which is applied rst; therefore P is (P ) . The operational semantics of the calculus is de ned by the transition rules of Table2 . The silent action P ?! Q has the same meaning as in CCS.... In PAGE 24: ... Normalised replications can be given the simple transition rule rep-nor: :P ?! P0 ! :P ?! P0 j ! : P or, alternatively, the two rules rep-inp: ! a(x): P ab ?! Pfb=xg j ! a(x): P rep-pre: ! : P ?! P j ! : P ; if is not an input Remark 6.2 As an aside, we wish to point out that rule rep-nor (as well as rep-inp and rep-pre) preserves the following pleasant property of -calculus transition system in Table2 , and which we state here very informally: If two inference proofs of transitions P ?! P0 and P ?! P00 involve the same pre x(es) of P, then P0 and P00 are syntactically the same (up to alpha conversion). This is a handy property to have, for instance when examining the set of derivatives of a process, because it makes it easier to reason by structural induction on processes.... ..."

Cited by 53

### Table 1: Hypersequent Calculus HIL for Intuitionistic Logic

2003

"... In PAGE 3: ... These are external weak- ening (ew) and external contraction (ec) (see Table 1). As an example, in Table1 one can nd a hypersequent calculus for intuitionistic logic IL which we call HIL. The \hyperlevel quot; of this calculus is in fact redundant, in the sense that a hypersequent 1 ) 1 j : : : j k ) k is derivable if and only if for some i 2 f1; : : : ; kg, already i ) i is derivable.... ..."

Cited by 7

### Table 4. Derived rules.

1992

"... In PAGE 10: ... We also subsume some of the laws for structural congruence (see Table 2) and the convention that [x=x] : P = 1 : P. The axioms of the up-calculus are given in Table 3, and in Table4 we present some derived rules (whose names start with D). De nition 13.... ..."

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### Table 4: Derived rules.

"... In PAGE 13: ... We also subsume some of the laws for structural congruence (see Table 2) and the convention that [x=x] : P = 1 : P . The axioms of the up-calculus are given in Table 3, and in Table4 we present some derived rules (whose names start with D). De nition 9 A substitution agrees with a match sequence ~ M, and ~ M agrees with , if for all x; y which appear in ~ M it holds that (x) = (y) i ~ M ) [x = y].... ..."

### Table 3. Typing Judgements of the Monoidal -calculus The term judgements of are

1999

"... In PAGE 9: ... These let- expressions ensure the context z : A B is isomorphic to the context x : A; y : B as required by the de nition of an L-category. Formally, the typing judgements are of the form ? ` t : A and ? ` f : ?0 and are given in Table3 , while the equality judgements for the calculus are given in Table 4. The -equations are derived from Ghani apos;s adjoint rewriting [9].... ..."

Cited by 1