### Table 1. Encoding of B in Coq and PVS

1999

"... In PAGE 10: ... For instance, the monotonicity of the preconditioning operator is expressed in PVS as MONOTONE_PRE: THEOREM 8 (p:pred[Env],tr1,tr2:Transition): IS_REFINED BY(tr1,tr2) ) IS_REFINED_BY(PRE(p,tr1),PRE(p,tr2)) 3.7 Summary Table1 summarizes the key aspects of the encoding of B in Coq and PVS. This study has led us to the following conclusions: { Dependent types and subtyping allow a simple encoding of abstract ma- chines.... ..."

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### Table 6: Qualitative influences: non-commutative, non-associative operators; RA: right asso- ciative; LA: left associative.

2005

"... In PAGE 15: ...o 0, i.e. S0(Cj, E) holds. square Finally, the results in Table6 for the increasing order operator are proven. Proposition 6 Let B = (G, Pr) be a Bayesian network representing a causal independence model with decomposable interaction function f that is equal to the logical increasing order operator lt;.... In PAGE 15: ... Suppose that this holds for bacteria A and B, then each of these would make the development of infection less likely, even though there could be circumstances where these bacteria turn pathogenic. Now, let C be a bacterium with only pathogenic strains, then the right-associative version of implication ( Table6 ) would model this situation appropriately.... ..."

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### Table 5. Laws for Commuting and Distributing Update Connectives

2006

"... In PAGE 55: ...Schema Variables Table5 . Modi ers for Schema Variables Modi er Applicable to rigid \term A \formula Terms or formulae that can syntactically be identi ed as rigid strict \term A Terms of type A (and not of proper subtypes of A) list \program t Sequences of program entities.... In PAGE 106: ...xample 2. We continue Example 1 and assume the same vocabulary/algebra. a := 1 ; f(a) := 2 a := 1 j f(1) := 2 valS; (a := 1 ; f(a) := 2) = fhai 7! 1; hf; (1)i 7! 2g valS; (a := 1 ; (a := 3 j f(a) := 2)) = fhai 7! 3; hf; (1)i 7! 2g We normalise the update in the second line using the given rewriting rules: a := 1 ; (a := 3 j f(a) := 2) (R45) ! a := 1 j fa := 1g (a := 3 j f(a) := 2) (R48) ! a := 1 j (fa := 1g a := 3 j fa := 1g f(a) := 2) (R47) ! a := 1 j (a := fa := 1g 3 j f(fa := 1g a) := fa := 1g 2) (R2); (R12) ! a := 1 j (a := 3 j f(non-rec(a := 1; a; ())) := 2) (R11) ! a := 1 j (a := 3 j f(if true then 1 else a) := 2) The last expression can be simpli ed further using rules for conditional terms, which are, however, beyond the scope of this paper. Further, using (R54) in Table5 , it is possible to eliminate the assignment a := 1, which is overridden by a := 3. 8 Soundness and Completeness of Update Application The following two lemmas state that the rewriting rules from Sect.... In PAGE 111: ...ewriting rules for update application (than the ones given in Sect. 5). This has been done for the implementation of updates in KeY. Table5 gives, besides others, identities that enable to establish form (1) by turning sequential composition into parallel composition, distributing if and for through parallel composition and commuting if and for. Another impor-... ..."

### Table 9 Signs of additive synergies for the commutative, non-associative opera- tors; right-associative case

2004

"... In PAGE 19: ... a50 Next, the two commutative, non-associative operators are considered. Here, we only supply a proof for the NAND | operator; the results are summarised in Table9 . Note that it is now no longer permitted to only look at the variables I1 and I2.... ..."

### Table 9: Signs of additive synergies for the commutative, non-associative operators; right- associative case.

2005

"... In PAGE 18: ... square Next, the two commutative, non-associative operators are considered. Here, we only supply a proof for the nand | operator; the results are summarised in Table9 . Note that it is now no longer permitted to only look at the variables I1 and I2.... ..."

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### Table 5 Axioms for observational equality (1) P + Q = Q + P

"... In PAGE 8: ... This auxiliary pre x is used in our axiomatization in order to transform the class of message agents in equivalent pre x forms: it is easy to prove that hai apos; a?:0. In Table5 an axiomatic characterization for the observational equality is presented. Axioms (1) and (2) state that the choice composition operator + is commutative and... ..."

### Table 1: Attributes for the top-level ring network. In this example, the scientist again wants to prove a limit on a distributed algorithm;

"... In PAGE 9: ... Since attributes will be carried through to the lower-level model, we will assign attributes rst. Table1 de nes the attributes for each of the top-level connections.... ..."

### Table 2: Commuting distance

"... In PAGE 12: ...5 minutes for workers who do not work from home).23 Further, they have a shorter commuting distance (see Table2 ). The preferred measure of the length of the commute is commuting distance, because commuting time is influenced by the endogenously chosen speed which may differ between employees and self-employed.... In PAGE 14: ... In line with the theoretical model, we find that the excess commute is larger in less urban areas. As can be see in column (3) of Table2 , the elasticity of address density on the excess commute is 0.... In PAGE 24: ...046 (0.008) 17 sectors Included Occupations (83) Included Log Likelihood -10880 N 33902 Note: The explanatory variable Log (commuting distance) is the midpoint of the commuting distance class as reported in Table2... ..."

### Table 4: Infrastructure Damages to Community Ring-Dike Towns

"... In PAGE 27: ... The vertical line at the right of the chart represents the infrastructure damages shown as a variable percentage (X%) of the 1997 flood costs. Table4 lists the 1997 damages for the community ring-dike towns, and the total potential damages after the dyke is breached. The total potential damages are also expressed as a percentage of the 1997 costs, which can then be equated to the vertical line on Figure 10.... In PAGE 43: ... This percentage was added directly to the damages assessed to each land parcel in the flooded area for all municipalities. Total infrastructure damages for the ring diked communities were calculated by adapting the reported and potential damages shown on Table4 into the generic depth-damage relationship shown on Figure 10. For example, The model shows that for a repeat 1826 flood, the St.... In PAGE 51: ...Red River Basin Depth-Damage Curves Update And Preparation of Flood Damage Risk Maps 43 January, 2000 Infrastructure damages estimated for the ring dike communities include an estimate for the Roseau River First Nation community, where reported infrastructure damages for 1997 were not obtained. Table 8: 1997 Flood Simulation Infrastructure Damage Estimate by Municipality The estimated 1997 infrastructure damages for each of the communities with a ring dike are equal to the damages reported for each community as shown on Table4 . The damage estimates are included in the above table under the applicable municipality.... ..."

### Table 7. Signs of additive synergies for the non-commutative, non-associative operators; RA: right-associative; LA: left-associative.

2002

"... In PAGE 4: ... The analysis of the non-commutative and non-associative opera- tors is more difficult, as again a distinction must be made between assuming the operators to be right associative or left associative. Due to lack of space the results are summarised in Table7... ..."

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