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The splitting rule for universal quantification is simpler than that for summation. The difference is that conjunction is idempotent whereas addition is not. When splitting the range in a universal quantification it does not matter whether some elements of the range are repeated in the two conjuncts. When splitting the range in a summation it does matter whether elements of the range are repeated. This additional flexibility allows the range in the splitting rule to be generalized from a disjunction P v Q of two predicates on the dummy to an arbitrary disjunction of predicates on the dummy. That is, we replace an 'or' by an existential quantification: [Splitting] (Vj:tf:{Vk:S:T = (Vfe : ( 3 j : R : S ) : T) . (11.26) (The side condition on this rule, when used from right to left, demands that '&' is not free in R.) Trading terms in the range is the same as summation, with the appropriate replacements for the operators and constants. In particular, 0 (the unit of summation) is replaced by true (the unit of conjunction). But, since if p ~ r n - n p truefi is the same as P=>T, trading with the term part can be simplified. [Trading] [Trading] (V k&K : P A Q : T > - ( V k e { k K \ P } : Q : T ) . (Vk:PAQ:T) = <Vk:Q:P=>T) . (11.27) (11.28)



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Note that the items output from a GroupBy() call are of type IGrouping<TKey, TElement> which has a property for the key that the query is grouping on (employeeDepartmentId) However, it does not have a property for the items within the group Rather, IGrouping<TKey, TElement> derives from IEnumerable<T>, allowing for enumeration of the items within the group using a foreach statement or for aggregating the data into something such as a count of items (employeeGroupCount())





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Implementing a One-to-Many Relationship with GroupJoin() Listing 1421 and Listing 1422 are virtually identical Either Join() call could have produced the same output just by changing the anonymous type definition When trying to create a list of employees, Listing 1421 provides the correct result department ends up as a property of each anonymous type representing the joined employee However, Listing 1422 is not optimal Given support for collections, a preferable representation of a department would have a collection of employees rather than a single anonymous type record for each department-employee relationship Listing 1424 demonstrates; Output 1411 shows the preferred output

The final rules govern the term part. The distributMty law is just one example of a distributMty property governing universal quantification. We see shortly that there are several more distributMty laws. [Rearranging] <Vk : R : T0 ATi) - (Vk:R:T0) A ( V k : R : T i ) . [DistributMty] < Vk :R:pvT) = p v (Vk:R: T) . (11.29) (11.30)

Template logic isn t tied to C#; if you re more comfortable with Visual Basic, then you can use that for the control logic in your templates

Listing 1424: Creating a Child Collection with SystemLinqEnumerable GroupJoin()

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using System; using SystemLinq; // Department[] departments = CorporateDataDepartments; Employee[] employees = CorporateDataEmployees; var items = departmentsGroupJoin( employees, department => departmentId, employee => employeeDepartmentId, (department, departmentEmployees) => new { departmentId, departmentName,

These are the rules for existential quantification. Not surprisingly, they are entirely dual to the rules for universal quantification. (In the rule (11.39), if P "Tn -ip - false fi has been simplified to P A T.) Once again, the side condition that free variables may not be captured, and bound variables may not be released, applies to all rules. [Dummy Renaming] ( 3 j : R : T ) = (3k:R[j := k]:T[j := k]> . [Nesting] <3 jjs : R AS : T) = ( 3 j : R : ( 3 j s : S : T ) ) . [Rearranging] ( 3 j , k : R : T ) = ( 3 k , j : R : T ) . [Empty Range] ( 3k: false :T> = false . [One-Point] (3k:k = e : T ) = T[k := e] . (11.31) (11.32) (11.33) (11.34) (11.35)

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