### Table 3: Context and structural rules for XML

in Abstract

1992

"... In PAGE 33: ... = ! : U 1 . Therefore, using the type equality rule from Table3 , wemay giveany term with type type ! , and vice versa. This allows us to giveanyuntyped lambda term type , including untyped terms with no normal form.... ..."

### Table 3: The modi ed computational transform, T . The next transform we study is the call-by-value version of the Fischer- Reynolds CPS transform. The de nition we use (as well as the overline notation) is taken from [Plo75] (see Table 4). Analogous to the CPS transform, used to give a denotational semantics of programs with mutable store, instead of control operators, is the state passing style transform (SPS). In the de nition, given in Table 5, we used the pairing constructs as abbreviations. Namely, (let hx1; x2i=M in N) 5-7

1998

"... In PAGE 7: ...3 Transforms In this work we concentrate on three transforms mapping lambda terms to lambda terms. First we study the modi ed computational transform, T , mapping pure lambda terms to lambda terms extended with two constants E and R (see Table3 ). The transform T can be viewed as an abstract trans- form which captures, for our purposes, important properties of the CPS and the SPS transforms.... ..."

Cited by 1

### Table 3: The modi ed computational transform, T . The next transform we study is the call-by-value version of the Fischer- Reynolds CPS transform. The de nition we use (as well as the overline notation) is taken from [Plo75] (see Table 4). Analogous to the CPS transform, used to give a denotational semantics of programs with mutable store, instead of control operators, is the state passing style transform (SPS). In the de nition, given in Table 5, we used the pairing constructs as abbreviations. Namely, (let hx1; x2i=M in N) 5-7

1998

"... In PAGE 7: ...3 Transforms In this work we concentrate on three transforms mapping lambda terms to lambda terms. First we study the modi ed computational transform, T , mapping pure lambda terms to lambda terms extended with two constants E and R (see Table3 ). The transform T can be viewed as an abstract trans- form which captures, for our purposes, important properties of the CPS and the SPS transforms.... ..."

Cited by 1

### Table 1. Simply Typed DCC: Typing Rules.

"... In PAGE 3: ... We write turnstileleft s, and say that s is a theorem, when turnstileleft s is derivable by the rules of Table 2. Equivalently, s is a theorem when there is a term e such that turnstileleft e : s is derivable by the rules of Table1 . In this case, we say that e inhabits s, and e represents a proof of s.... ..."

### Table 1: Syntax and equational rules for the simply-typed -calculus.

1992

"... In PAGE 3: ...Of course, there is nothing in the syntax that forces this choice, and some of the theorems below will be for languages in which the base type is interpreted to be lists or sets of natural numbers. The set of simply-typed terms is given by the forma- tion rules of Table1 . We assume that , a signature, is a countable set of typed constants.... In PAGE 3: ... The usual de nitions of free and bound vari- ables apply to this set of terms, and terms are identi- ed up to renaming of bound variables [2]. The equa- tional axioms and rules of the simply-typed -calculus also appear in Table1 ; M[x := N] denotes substitu- tion of N for x in M, where the bound variables of M are renamed to avoid the capture of the free variables of N [2]. We write (M = N) if M and N are prov- ably equivalent in this system.... ..."

Cited by 1

### Table 5. Equational Rules for the Simply-Typed -Calculus

1994

"... In PAGE 23: ...3 Type frames. Although we have given a way to associate a `meaning apos; [[H B M : t]] to a triple H; M; t such that H ` M : t and demonstrated that our assignment of meaning preserves the required equations from Table5 , we did not ac- tually provide a rigorous description of the ground rules for saying when such an assignment really is a model of the simply-typed -calculus. In fact, there is more than one way to do this, depending on what one consid- ers important about the model.... In PAGE 74: ... More convincing programming examples can be given, but this shows that the phenomenon arises quite naturally. The equational rules for the pure polymorphic -calculus are those in Table5 together with the rules that appear in Table 22 modulo the theory T that appears on the left-hand sides of the turnstiles.1 The new rules fTypeCongg and fType g assert that type application and type abstrac- tion are congruences.... ..."

Cited by 3

### Table 1: Syntax and equational rules for the simply-typed -calculus.

"... In PAGE 5: ... The usual de nitions of free and bound variables apply to this set of terms, and terms are identi ed up to renaming of bound variables [1]. The equational axioms and rules of the simply-typed -calculus also appear in Table1 ; M[x := N] denotes substitution of N for x in M, where the bound variables of M are renamed to avoid the capture of the free variables of N [1]. We write (M = N) if M and N are provably equivalent in this system.... In PAGE 7: ... An algebraic equation t = t0 is just an equation between algebraic terms t and t0. These equations may be easily added to the equational theory of the -calculus by rst converting the algebraic terms on either side of the equation to simply-typed terms as outlined above, and then adding these equations to the axioms of Table1 . Given a set of algebraic equations E and algebraic terms t and t0, we say that t =E t0 if the equation holds using the rules of Table 2.... In PAGE 7: ... Given a set of algebraic equations E and algebraic terms t and t0, we say that t =E t0 if the equation holds using the rules of Table 2. It is important to note that any algebraic equation provable in this system is provable in the simply-typed -calculus from the equations E and the axioms and rules of Table1 . It is obvious that all but (sub) is a rule of Table 1; the (sub) rule, on the other hand, can be derived from ( ), ( ), and (trans).... ..."

### Table 1. Processes, types and typing rules of the simply typed AP-calculus

2004

"... In PAGE 3: ...ype a priori. We write DC BM CC to mean that the name DC has type CC . A judgment CO C8 says that C8 is a well-typed process, and CO DA BM CC says that DA is a well-typed value of type CC . The syntax of types and processes as well as the typing rules are shown in Table1 . We use the usual constructors of monadic AP-calculus.... In PAGE 4: ... We assign a level, which is a natural number, to each channel name and incorporate it into the type of the name. Now the syntax of link type takes the new form: C4 ::= CLD2CE link types D2 ::= BDBN BEBN A1 A1 A1 levels The typing rules in Table1 are still valid (by obvious adjustments for link types), with the exception of rule T-rep, which takes the new form: T-rep CO CP BM CLD2CC DC BM CC CO C8 BKCQ BE D3D7B4C8B5BND0DAB4CQB5 BO D2 COAXCPB4DCB5BMC8 where D3D7B4C8 B5 is a set collecting all names in C8 which appear as subjects of those outputs that are not underneath any replicated input (we say this kind of outputs are active). The function D0DAB4CQB5 calculates the level of channel CQ from its type.... In PAGE 6: ... We write AMCQDA BB CPB4DCB5 if one of the two cases holds: (i) D1 BO D2; (ii) D1 BP D2BN CB BP CC BP C6CPD8 and DA BO DC. By substituting the following rule for T-rep in Table1 , we get the extended type system CC BC. The second condition in the definition of BB allows us to include some recursive inputs and gives us the difference from CC .... In PAGE 9: ...here the input pattern has length BD, i.e., it is composed of just one input prefix. We extend the definition of weight to input patterns by taking account of the levels of input subjects: DBD8B4CPBDB4DCBDB5BM A1 A1 A1 BMCPD2B4DCD2B5B5 BP BCCZBD B7 A1 A1 A1 B7 BCCZD2 where D0DAB4CPCXB5 BP CZCX. The typing rule T-rep in Table1 is replaced by the following one. T-rep CO AKBMC8 DBD8B4AKB5 AV DBD8B4C8B5 COAXAKBMC8 Intuitively, this rule means that we consume more than what we produce.... ..."

Cited by 5

### Table 2.4: Processes, types and typing rules of the simply typed pi-calculus

2005