### Table 5 : Achievable recycling rates for household packaging waste (%) High density Low density % Kerb. Bring Kerb. Bring Plastics PET bottles 59-69 22-32 70-80 35-45

2003

"... In PAGE 18: ...able 4 : scenarios relating to industrial amp; commercial packaging waste.....................................38 Table5 : Achievable recycling rates for household packaging waste (%) .... ..."

### Table 2. The initial data relation for induction.

"... In PAGE 5: ...hich is {M.S., M.A., Ph.D.}. Then the data about graduates can be retrieved and projected on relevant attributes Name, Major, Birth_Place, and GPA, which results in an initial data relation on which induc- tion can be performed. Table2 reflects the result of this preprocessing and a special attribute vote is attached to each tuple with its initial value set to 1. Such a prepro- cessed data relation is called an initial relation.... ..."

### Table 2. The initial data relation for induction.

"... In PAGE 5: ...hich is {M.S., M.A., Ph.D.}. Then the data about graduates can be retrieved and projected on relevant attributes Name, Major, Birth_Place, and GPA, which results in an initial data relation on which induc- tion can be performed. Table2 reflects the result of this preprocessing and a special attribute vote is attached to each tuple with its initial value set to 1. Such a prepro- cessed data relation is called an initial relation.... ..."

### Table 1: Representation of ZC formulae in HOL

2001

"... In PAGE 7: ...ollows from the membership of type string in this class, see Section 1.4. Since type term is simply an instantiation of datatype dbterm, the technical frame- work of Section 1 applies. The syntactic representation of ZC logical constants on top of type term is straightforward, see Table1 . In fact, there is nothing speci c to ZC about these constants - they are simply a minimal set of constants for rst-order predicate logic.... ..."

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### Table 2. Comparison of approaches for reasoning about static spatial relations.

"... In PAGE 6: ... Many of such qual- itative spatial calculi have been developed during the past decades, mainly for topological or positional reasoning; however, they are often not fully specified, and mostly no implementation is made available [14]. Table2 shows a compar- ison of popular qualitative spatial calculi which are classified according to the four relation types presented above. Many of them incorporate the spatial con-... ..."

### TABLE 3 Composition of Mutual Fund Expenses, 1997

in Administrative Costs and The Organization of Individual Account Systems: A Comparative Perspective

1999

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### Table 1. Sample proofs whose solution requires meta-reasoning about failures.

"... In PAGE 13: ... There- fore, we tested the bene t in three domains, the - -proofs from the analysis textbook [1], the residue class domain, and inductive proofs. Table1 gives sam- ple problems from all three domains and the failure-reasoning they require. The numbered colons denote (i) case split introduction, (ii) unblock constraint solv- ing, (iii) unblock by lemma speculation, (iv) analyze variable dependencies.... In PAGE 13: ... Note that x ! a and x ! a+ denote the left-hand limit and the right-hand limit, respectively. The relevance of failure reasoning is not only demonstrated by Table1 . Its gures alone are underestimating because many similar problems can be formu- lated.... In PAGE 14: ...xperiments). Some representative examples occur in Table 1. Inductive Proofs So far, we did not apply Multi to inductive proofs. The induc- tive theorems in Table1 are taken from [9], which describes failure reasoning by so-called critics in the proof planner CLaM. Since the critics employed in CLaM are a special case bound to a particular method (see related work in section 7), our general failure reasoning rules for case-split introduction and lemma spec- ulation are applicable for inductive proofs as well.... ..."

### Table 2: Representation of ZC terms in HOL

2001

"... In PAGE 8: ... This is provided by the following primitive recursive function: rep typ :: zty ! term rep typ NatT = Const \NatT\ rep typ (SetT T) = App \SetT\ [rep typ T] rep typ (PrdT U V ) = App \PrdT\ [rep typ U; rep typ V ] rep typ (RcdT S) = App \RcdT\ (rcd term (map rcd rep typ S)) Injectivity of function rep typ can be proven by structural induction on the datatype zty. Table2 shows the representation of ZC terms as elements of type term. As an example, here is the translation of the term fx 2 N j suc (x) = 0g to HOL: COMPREH ( quot;x quot;,NatT) NAT SET (SUC (Free( quot;x quot;,NatT)) EQ ZERO) A fundamental property of the HOL representation of ZC syntax is its faith- fulness - di erent ZC terms are mapped to di erent HOL terms.... ..."

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### Table 2: Inductive de nition of FOR

"... In PAGE 9: ... (EVAL C1 = EVAL C2) == gt; (EVAL (for (LVAR i) (VAL 1) (VAL m) C1) = EVAL (for (LVAR i) (VAL 1) (VAL m) C2)) for LEMMA1 is a loop termination theorem, for LEMMA2 says that assigning the index variable before a for loop has no e ect, for LEMMA3 unrolls a loop from the bottom, for LEMMA4 unrolls a loop from the top and for LEMMA5 is a congruence theorem. To reason about individual commands in a for command body we have de ned the operator FOR as an inductive de nition (See Table2 ) and proved an additional lemma: FOR_LEMMA = |- !m i C. EVAL (for (LVAR i) (VAL 1) (VAL m) C) = FOR (LVAR i) (VAL 1) (VAL m) (EVAL C) To reason about individual commands inside the scope of a local declaration we... ..."

### Table A.1: Overview about each package.

2001

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