Recall the class of <#69993#><#56967#>posn<#56967#><#69993#> structures from part~#partbasic#56968>.
A <#69994#><#56969#>posn<#56969#><#69994#> combines two numbers; its fields are called <#69995#><#56970#>x<#56970#><#69995#> and
<#69996#><#56971#>y<#56971#><#69996#>. Here are two examples:
<#56976#>(make-posn<#56976#> <#56977#>3<#56977#> <#56978#>4)<#56978#>
<#56984#>(make-posn<#56984#> <#56985#>8<#56985#> <#56986#>6)<#56986#>
They are obviously distinct. In contrast, the following two
<#56994#>(make-posn<#56994#> <#56995#>12<#56995#> <#56996#>1)<#56996#>
<#57002#>(make-posn<#57002#> <#57003#>12<#57003#> <#57004#>1)<#57004#>
are equal. They both contain <#69997#><#57008#>12<#57008#><#69997#> in the <#69998#><#57009#>x<#57009#><#69998#>-field
and <#69999#><#57010#>1<#57010#><#69999#> in the <#70000#><#57011#>y<#57011#><#70000#>-field.
More generally, we consider two structures as equal if they contain equal
components. This assumes that we know how to compare the components, but
that's not surprising. It just reminds us that processing structures
follows the data definition that comes with the structure
definition. Philosophers refer to this notion of equality as
<#70001#><#57012#>EXTENSIONAL EQUALITY<#57012#><#70001#>.
Section~#secequaltest#57013> introduced extensional equality and discussed
its use for building tests. As a reminder, let's consider a function for
determining the extensional equality of <#70002#><#57014#>posn<#57014#><#70002#> structures:
<#72045#>;; <#70003#><#57019#>equal-posn?<#57019#> <#57020#>:<#57020#> <#57021#>posn<#57021#> <#57022#>posn<#57022#> <#57023#><#57023#><#57024#>-;SPMgt;<#57024#><#57025#><#57025#> <#57026#>boolean<#57026#><#70003#><#72045#>
<#72046#>;; to determine whether two <#70004#><#57027#>posn<#57027#><#70004#>s are extensionally equal <#72046#>
<#57028#>(d<#57028#><#57029#>efine<#57029#> <#57030#>(equal-posn<#57030#> <#57031#>p1<#57031#> <#57032#>p2)<#57032#>
<#57033#>(and<#57033#> <#57034#>(=<#57034#> <#57035#>(posn-x<#57035#> <#57036#>p1)<#57036#> <#57037#>(posn-x<#57037#> <#57038#>p2))<#57038#>
<#57039#>(=<#57039#> <#57040#>(posn-y<#57040#> <#57041#>p1)<#57041#> <#57042#>(posn-y<#57042#> <#57043#>p2))))<#57043#>
The function consumes two <#70005#><#57047#>posn<#57047#><#70005#> structures, extracts their field
values, and then compares the corresponding field values using <#70006#><#57048#>=<#57048#><#70006#>,
the predicate for comparing numbers. Its organization matches that of the
data definition for <#70007#><#57049#>posn<#57049#><#70007#> structures; its design is standard. This
implies that for recursive classes of data, we naturally need recursive
equality functions.
<#57052#>Exercise 42.1.1<#57052#>
Develop an extensional equality function for the class of <#70008#><#57054#>child<#57054#><#70008#>
structures from exercise~#exftss1#57055>. If <#70009#><#57056#>ft1<#57056#><#70009#> and <#70010#><#57057#>ft2<#57057#><#70010#>
are family tree nodes, how long is the maximal abstract running time of the
function?~ 
Solution<#70011#><#70011#>
<#57063#>Exercise 42.1.2<#57063#>
Use exercise~#exeqextensional1#57065> to abstract <#70012#><#57066#>equal-posn<#57066#><#70012#> so
that its instances can test the intensional equality of any given class of
structures.~ Solution<#70013#><#70013#>