Alternative Data Definitions for N

Using the above, standard data definition for natural numbers makes it easy to develop all kinds of functions on numbers. Consider, for example, a function that multiplies the first <#62691#><#12923#>n<#12923#><#62691#> numbers. Put differently, it consumes a natural number <#62692#><#12924#>n<#12924#><#62692#> and multiplies all numbers between <#62693#><#12925#>0<#12925#><#62693#> (exclusive) and <#62694#><#12926#>n<#12926#><#62694#> (inclusive). The function is called factorial and has the mathematical notation <#62695#><#12927#>!<#12927#><#62695#>. Its contract is easy to formulate:

<#71002#>;; <#62696#><#12932#>!<#12932#> <#12933#>:<#12933#> <#12934#>N<#12934#> <#12935#><#12935#><#12936#>-;SPMgt;<#12936#><#12937#><#12937#> <#12938#>N<#12938#><#62696#><#71002#>
<#12939#>;; to compute #tex2html_wrap_inline73028#<#12939#> 
<#12940#>(define<#12940#> <#12941#>(!<#12941#> <#12942#>n)<#12942#> <#12943#>...)<#12943#>

It consumes a natural number and produces one. Specifying its input-output relationship is a bit more tricky. We know, of course, what the product of <#62697#><#12947#>1<#12947#><#62697#>, <#62698#><#12948#>2<#12948#><#62698#>, and <#62699#><#12949#>3<#12949#><#62699#> is, so we should have

  <#12954#>(!<#12954#> <#12955#>3)<#12955#>
<#12956#>=<#12956#> <#12957#>6<#12957#> 
<#12958#>;and, similarly, <#12958#> 
  <#12959#>(!<#12959#> <#12960#>10)<#12960#> 
<#12961#>=<#12961#> <#12962#>3628800<#12962#>

The real question is what to do with the input <#62700#><#12966#>0<#12966#><#62700#>. According to the informal description of the task, <#62701#><#12967#>!<#12967#><#62701#> is supposed to produce the product of all numbers between <#62702#><#12968#>0<#12968#><#62702#> (exclusive) and <#62703#><#12969#>n<#12969#><#62703#> (inclusive), the argument. Since <#62704#><#12970#>n<#12970#><#62704#> is <#62705#><#12971#>0<#12971#><#62705#>, this request is rather strange because there are no numbers between <#62706#><#12972#>0<#12972#><#62706#> (exclusive) and <#62707#><#12973#>0<#12973#><#62707#> (inclusive). We solve the problem by following mathematical convention and set that <#62708#><#12974#>(!<#12974#>\ <#12975#>0)<#12975#><#62708#> evaluates to <#62709#><#12976#>1<#12976#><#62709#>. From here, the rest is straightforward. The template for <#62710#><#12977#>!<#12977#><#62710#> is clearly that of a natural number-processing function:

<#12982#>(d<#12982#><#12983#>efine<#12983#> <#12984#>(!<#12984#> <#12985#>n)<#12985#>
  <#12986#>(c<#12986#><#12987#>ond<#12987#> 
    <#12988#>[<#12988#><#12989#>(zero?<#12989#> <#12990#>n)<#12990#> <#12991#>...]<#12991#> 
    <#12992#>[<#12992#><#12993#>else<#12993#> <#12994#>...<#12994#> <#12995#>(!<#12995#> <#12996#>(sub1<#12996#> <#12997#>n))<#12997#> <#12998#>...]<#12998#><#12999#>))<#12999#>

The answer in the first <#62711#><#13003#>cond<#13003#><#62711#>-clause is given: <#62712#><#13004#>1<#13004#><#62712#>. In the second clause, the recursion produces the product of the first <#62713#><#13005#>n<#13005#>\ <#13006#>-<#13006#><#13007#> <#13007#><#13008#>1<#13008#><#62713#> numbers, <#13009#>e.g.<#13009#>, for <#62714#><#13010#>n<#13010#>\ <#13011#>=<#13011#>\ <#13012#>3<#13012#><#62714#>, <#62715#><#13013#>(!<#13013#>\ <#13014#>(sub1<#13014#>\ <#13015#>n))<#13015#><#62715#> evaluates to <#62716#><#13016#>2<#13016#><#62716#>. To get the product of the first <#62717#><#13017#>n<#13017#><#62717#> numbers, we just need to multiply the (value of the) recursion by <#62718#><#13018#>n<#13018#><#62718#>. Figure~#figfact#13019> contains the complete definition of <#62719#><#13020#>!<#13020#><#62719#>.
<#13023#>Exercise 11.4.1<#13023#> Determine the value of <#62720#><#13025#>(!<#13025#>\ <#13026#>2)<#13026#><#62720#> by hand and with DrScheme. Also test <#62721#><#13027#>!<#13027#><#62721#> with <#62722#><#13028#>10<#13028#><#62722#>, <#62723#><#13029#>100<#13029#><#62723#>, and <#62724#><#13030#>1000<#13030#><#62724#>.
<#13031#>Note:<#13031#> The results of these expressions are large numbers, well beyond the native capacities of many other programming languages.

Solution<#62725#><#62725#> Now suppose we wish to design the function <#62726#><#13037#>product-from-20<#13037#><#62726#>, which computes the product from <#62727#><#13038#>20<#13038#><#62727#> (exclusive) to some number <#62728#><#13039#>n<#13039#><#62728#> (inclusive) that is greater or equal to <#62729#><#13040#>20<#13040#><#62729#>. We have several choices here. First, we could define a function that computes <#62730#><#13041#>(!<#13041#>\ <#13042#>n)<#13042#><#62730#> and <#62731#><#13043#>(!<#13043#>\ <#13044#>20)<#13044#><#62731#> and divides the former by the latter. A simple mathematical argument shows that this approach indeed yields the product of all numbers between <#62732#><#13045#>20<#13045#><#62732#> (exclusive) and <#62733#><#13046#>n<#13046#><#62733#> (inclusive):

#displaymath73026#

<#13051#>Exercise 11.4.2<#13051#> Use the idea to define <#62734#><#13053#>product<#13053#><#62734#>, a function that consumes two natural numbers, <#62735#><#13054#>n<#13054#><#62735#> and <#62736#><#13055#>m<#13055#><#62736#>, with <#62737#><#13056#>m<#13056#>\ <#13057#>;SPMgt;<#13057#>\ <#13058#>n<#13058#><#62737#>, and that produces the product of the numbers between <#62738#><#13059#>n<#13059#><#62738#> (exclusive) and <#62739#><#13060#>m<#13060#><#62739#> (inclusive). Solution<#62740#><#62740#>
Second, we can follow our design recipe, starting with a precise characterization of the function's input. Obviously, the inputs belong to the natural numbers, but we know more than that. It belongs the following collection of numbers: <#62741#><#13068#>20<#13068#><#62741#>, <#62742#><#13069#>21<#13069#><#62742#>, <#62743#><#13070#>22<#13070#><#62743#>, ...By now we know how to describe such a set precisely with a data definition:

A <#72202#><#71003#>natural number <#62744#><#13072#>[<#13072#><#13073#>;SPMgt;=<#13073#>\ <#13074#>20]<#13074#><#62744#><#71003#><#72202#> is either

<#62745#><#13076#>20<#13076#><#62745#> or
<#62746#><#13077#>(add1<#13077#>\ <#13078#>n)<#13078#><#62746#> if <#62747#><#13079#>n<#13079#><#62747#> is a natural number <#62748#><#13080#>[<#13080#><#13081#>;SPMgt;=<#13081#>\ <#13082#>20]<#13082#><#62748#>.
Notation: In contracts, we use <#62749#><#13084#>N<#13084#>\ <#13085#>[<#13085#><#13086#>;SPMgt;=<#13086#>\ <#13087#>20]<#13087#><#62749#> instead of ``natural numbers <#62750#><#13088#>[<#13088#><#13089#>;SPMgt;=<#13089#>\ <#13090#>20]<#13090#><#62750#>.''

Using the new data definition, we can formulate a contract for <#62751#><#13092#>product-from-20<#13092#><#62751#>:

<#71004#>;; <#62752#><#13097#>product-from-20:<#13097#> <#13098#>N<#13098#> <#13099#>[<#13099#><#13100#>;SPMgt;=<#13100#> <#13101#>20]<#13101#> <#13102#><#13102#><#13103#>-;SPMgt;<#13103#><#13104#><#13104#> <#13105#>N<#13105#><#62752#><#71004#>
<#13106#>;; to compute #tex2html_wrap_inline73030#<#13106#> 
<#13107#>(define<#13107#> <#13108#>(product-from-20<#13108#> <#13109#>n-above-20)<#13109#> <#13110#>...)<#13110#>

Here is a first example for <#62753#><#13114#>product-from-20<#13114#><#62753#>'s input-output specification:

  <#13119#>(product-from-20<#13119#> <#13120#>21)<#13120#>
<#13121#>=<#13121#> <#13122#>21<#13122#>

Since the function multiplies all numbers between <#62754#><#13126#>20<#13126#><#62754#> (exclusively) and its input, <#62755#><#13127#>(product-from-20<#13127#>\ <#13128#>21)<#13128#><#62755#> must produce <#62756#><#13129#>21<#13129#><#62756#>, which is the only number in the interval. Similarly,

  <#13134#>(product-from-20<#13134#> <#13135#>22)<#13135#>
<#13136#>=<#13136#> <#13137#>462<#13137#>

for the same reason. Finally, we again follow mathematical convention and agree that

  <#13145#>(product-from-20<#13145#> <#13146#>20)<#13146#>
<#13147#>=<#13147#> <#13148#>1<#13148#>

The template for <#62757#><#13152#>product-from-20<#13152#><#62757#> is a straightforward adaptation of the template for <#62758#><#13153#>!<#13153#><#62758#>, or any natural number-processing function:

<#13158#>(d<#13158#><#13159#>efine<#13159#> <#13160#>(product-from-20<#13160#> <#13161#>n-above-20)<#13161#>
  <#13162#>(c<#13162#><#13163#>ond<#13163#> 
    <#13164#>[<#13164#><#13165#>(=<#13165#> <#13166#>n-above-20<#13166#> <#13167#>20)<#13167#> <#13168#>...]<#13168#> 
    <#13169#>[<#13169#><#13170#>else<#13170#> <#13171#>...<#13171#> <#13172#>(product-from-20<#13172#> <#13173#>(sub1<#13173#> <#13174#>n-above-20))<#13174#> <#13175#>...]<#13175#><#13176#>))<#13176#>

The input <#62759#><#13180#>n-above-20<#13180#><#62759#> is either <#62760#><#13181#>20<#13181#><#62760#> or larger. If it is <#62761#><#13182#>20<#13182#><#62761#>, it does not have any components according to the data definition. Otherwise, it is the result of adding <#62762#><#13183#>1<#13183#><#62762#> to a natural number <#62763#><#13184#>[<#13184#><#13185#>;SPMgt;=<#13185#>\ <#13186#>20]<#13186#><#62763#>, and we can recover this ``component'' by subtracting <#62764#><#13187#>1<#13187#><#62764#>. The value of this selector expression belongs to the same class of data as the input and is thus a candidate for natural recursion. Completing the template is equally straightforward. As specified, the result of <#62765#><#13188#>(product-from-20<#13188#>\ <#13189#>20)<#13189#><#62765#> is <#62766#><#13190#>1<#13190#><#62766#>, which determines the answer for the first <#62767#><#13191#>cond<#13191#><#62767#>-clauses. Otherwise, <#62768#><#13192#>(product-from-20<#13192#>\ <#13193#>(sub1<#13193#>\ <#13194#>n-above-20))<#13194#><#62768#> already produces the product of all the numbers between <#62769#><#13195#>20<#13195#><#62769#> (exclusive) and <#62770#><#13196#>n-above-20<#13196#>\ <#13197#>-<#13197#>\ <#13198#>1<#13198#><#62770#>. The only number not included in this range is <#62771#><#13199#>n-above-20<#13199#><#62771#>. Hence, <#62772#><#13200#>(*<#13200#>\ <#13201#>n-above-20<#13201#>\ <#13202#>(product-from-20<#13202#>\ <#13203#>(sub1<#13203#>\ <#13204#>n-above-20)))<#13204#><#62772#> is the correct answer in the second clause. Figure~#figfact#13205> contains the complete definition of <#62773#><#13206#>product-from-20<#13206#><#62773#>.
<#13209#>Exercise 11.4.3<#13209#> Develop <#62774#><#13211#>product-from-minus-11<#13211#><#62774#>. The function consumes an integer <#62775#><#13212#>n<#13212#><#62775#> greater or equal to <#62776#><#13213#>-11<#13213#><#62776#> and produces the product of all the integers between <#62777#><#13214#>-11<#13214#><#62777#> (exclusive) and <#62778#><#13215#>n<#13215#><#62778#> (inclusive).

Solution<#62779#><#62779#> <#13221#>Exercise 11.4.4<#13221#> In exercise~#extabulatef#13223>, we developed a function that tabulates some mathematical function <#62780#><#13224#>f<#13224#><#62780#> for an interval of the shape (0,n]. Develop the function <#62781#><#13225#>tabulate-f20<#13225#><#62781#>, which tabulates the values of <#62782#><#13226#>f<#13226#><#62782#> for natural numbers greater than <#62783#><#13227#>20<#13227#><#62783#>. Specifically, the function consumes a natural number <#62784#><#13228#>n<#13228#><#62784#> greater or equal to <#62785#><#13229#>20<#13229#><#62785#> and produces a list of <#62786#><#13230#>posn<#13230#><#62786#>s, each of which has the shape <#62787#><#13231#>(make-posn<#13231#>\ <#13232#>n<#13232#>\ <#13233#>(f<#13233#>\ <#13234#>n))<#13234#><#62787#> for some <#62788#><#13235#>n<#13235#><#62788#> between <#62789#><#13236#>20<#13236#><#62789#> (exclusive) and <#62790#><#13237#>n<#13237#><#62790#> (inclusive).

Solution<#62791#><#62791#>

<#71005#>;; <#62792#><#13249#>!<#13249#> <#13250#>:<#13250#> <#13251#>N<#13251#> <#13252#><#13252#><#13253#>-;SPMgt;<#13253#><#13254#><#13254#> <#13255#>N<#13255#><#62792#><#71005#>
<#13256#>;; to compute #tex2html_wrap_inline73034#<#13256#> 
<#13257#>(d<#13257#><#13258#>efine<#13258#> <#13259#>(!<#13259#> <#13260#>n)<#13260#> 
  <#13261#>(c<#13261#><#13262#>ond<#13262#> 
    <#13263#>[<#13263#><#13264#>(zero?<#13264#> <#13265#>n)<#13265#> <#13266#>1]<#13266#> 
    <#13267#>[<#13267#><#13268#>else<#13268#> <#13269#>(*<#13269#> <#13270#>n<#13270#> <#13271#>(!<#13271#> <#13272#>(sub1<#13272#> <#13273#>n)))]<#13273#><#13274#>))<#13274#> 
<#71006#>;; <#62793#><#13275#>product-from-20:<#13275#> <#13276#>N<#13276#> <#13277#>[<#13277#><#13278#>;SPMgt;=<#13278#> <#13279#>20]<#13279#> <#13280#><#13280#><#13281#>-;SPMgt;<#13281#><#13282#><#13282#> <#13283#>N<#13283#><#62793#><#71006#> 
<#13284#>;; to compute #tex2html_wrap_inline73036#<#13284#> 
<#13285#>(d<#13285#><#13286#>efine<#13286#> <#13287#>(product-from-20<#13287#> <#13288#>n-above-20)<#13288#> 
  <#13289#>(c<#13289#><#13290#>ond<#13290#> 
    <#13291#>[<#13291#><#13292#>(=<#13292#> <#13293#>n-above-20<#13293#> <#13294#>20)<#13294#> <#13295#>1]<#13295#> 
    <#13296#>[<#13296#><#13297#>else<#13297#> <#13298#>(*<#13298#> <#13299#>n-above-20<#13299#> <#13300#>(product-from-20<#13300#> <#13301#>(sub1<#13301#> <#13302#>n-above-20)))]<#13302#><#13303#>))<#13303#> 
<#71007#>;; <#62794#><#13304#>product:<#13304#> <#13305#>N<#13305#><#13306#>[<#13306#><#13307#>limit]<#13307#> <#13308#>N<#13308#><#13309#>[<#13309#><#13310#>;SPMgt;=<#13310#> <#13311#>limit]<#13311#> <#13312#><#13312#><#13313#>-;SPMgt;<#13313#><#13314#><#13314#> <#13315#>N<#13315#><#62794#><#71007#> 
<#62795#>;; to compute #tex2html_wrap_inline73038#<#62795#> 
<#13317#>(d<#13317#><#13318#>efine<#13318#> <#13319#>(product<#13319#> <#13320#>limit<#13320#> <#13321#>n)<#13321#> 
  <#13322#>(c<#13322#><#13323#>ond<#13323#> 
    <#13324#>[<#13324#><#13325#>(=<#13325#> <#13326#>n<#13326#> <#13327#>limit)<#13327#> <#13328#>1]<#13328#> 
    <#13329#>[<#13329#><#13330#>else<#13330#> <#13331#>(*<#13331#> <#13332#>n<#13332#> <#13333#>(product<#13333#> <#13334#>limit<#13334#> <#13335#>(sub1<#13335#> <#13336#>n)))]<#13336#><#13337#>))<#13337#>

<#13341#>Figure: Computing factorial, product-from-20, and product<#13341#>

A comparison of <#62796#><#13343#>!<#13343#><#62796#> and <#62797#><#13344#>product-from-20<#13344#><#62797#> suggests the natural question of how to design a function that multiplies <#13345#>all<#13345#> natural numbers in some range. Roughly speaking, <#62798#><#13346#>product<#13346#><#62798#> is like <#62799#><#13347#>product-from-20<#13347#><#62799#> except that the limit is not a part of the function definition. Instead, it is another input, which suggests the following contract:

<#71008#>;; <#62800#><#13352#>product:<#13352#> <#13353#>N<#13353#> <#13354#>N<#13354#> <#13355#><#13355#><#13356#>-;SPMgt;<#13356#><#13357#><#13357#> <#13358#>N<#13358#><#62800#><#71008#>
<#62801#>;; to compute #tex2html_wrap_inline73040#<#62801#> 
<#13360#>(define<#13360#> <#13361#>(product<#13361#> <#13362#>limit<#13362#> <#13363#>n)<#13363#> <#13364#>...)<#13364#>

The intention is that <#62802#><#13368#>product<#13368#><#62802#>, like <#62803#><#13369#>product-from-20<#13369#><#62803#>, computes the product from <#62804#><#13370#>limit<#13370#><#62804#> (exclusive) to some number <#62805#><#13371#>n<#13371#><#62805#> (inclusive) that is greater or equal to <#62806#><#13372#>limit<#13372#><#62806#>. Unfortunately, <#62807#><#13373#>product<#13373#><#62807#>'s contract, in contrast <#62808#><#13374#>product-from-20<#13374#><#62808#>'s, is rather imprecise. In particular, it does not describe the collection of numbers that <#62809#><#13375#>product<#13375#><#62809#> consumes as the second input. Given its first input, <#62810#><#13376#>limit<#13376#><#62810#>, we know that the second input belongs to <#62811#><#13377#>limit<#13377#><#62811#>, <#62812#><#13378#>(add1<#13378#>\ <#13379#>limit)<#13379#><#62812#>, <#62813#><#13380#>(add1<#13380#>\ <#13381#>(add1<#13381#>\ <#13382#>limit))<#13382#><#62813#>, <#13383#>etc<#13383#>. While it is easy to enumerate the possible second inputs, it also shows that the description of the collection <#13384#>depends on the first input<#13384#>---an unusal situation that we have not encountered before. Still, if we assume limit is some number, the data description for the second input is nearly obvious:

Let <#62814#><#13386#>limit<#13386#><#62814#> be a natural number. A <#72203#><#71009#>natural number <#62815#><#13387#>[<#13387#><#13388#>;SPMgt;=<#13388#>\ <#13389#>limit]<#13389#><#62815#><#71009#><#72203#> (<#62816#><#13390#>N<#13390#><#13391#>[<#13391#><#13392#>;SPMgt;=limit]<#13392#><#62816#>) is either

<#62817#><#13394#>limit<#13394#><#62817#> or
<#62818#><#13395#>(add1<#13395#>\ <#13396#>n)<#13396#><#62818#> if <#62819#><#13397#>n<#13397#><#62819#> is a natural number <#62820#><#13398#>[<#13398#><#13399#>;SPMgt;=<#13399#>\ <#13400#>limit]<#13400#><#62820#>.

In other words, the data definition is like that for natural numbers <#62821#><#13403#>[<#13403#><#13404#>;SPMgt;=<#13404#>\ <#13405#>limit]<#13405#><#62821#> with <#62822#><#13406#>20<#13406#><#62822#> replaced by a variable <#62823#><#13407#>limit<#13407#><#62823#>. Of course, in high school, we refer to <#62824#><#13408#>N<#13408#><#13409#>[<#13409#><#13410#>;SPMgt;=0]<#13410#><#62824#> as <#13411#>the<#13411#> natural numbers, and <#62825#><#13412#>N<#13412#><#13413#>[<#13413#><#13414#>;SPMgt;=1]<#13414#><#62825#> as <#13415#>the<#13415#> positive integers. With this new data definition, we specify the contract for <#62826#><#13416#>product<#13416#><#62826#> as follows:

<#71010#>;; <#62827#><#13421#>product:<#13421#> <#13422#>N<#13422#><#13423#>[<#13423#><#13424#>limit]<#13424#> <#13425#>N<#13425#> <#13426#>[<#13426#><#13427#>;SPMgt;=<#13427#> <#13428#>limit]<#13428#> <#13429#><#13429#><#13430#>-;SPMgt;<#13430#><#13431#><#13431#> <#13432#>N<#13432#><#62827#><#71010#>
<#62828#>;; to compute #tex2html_wrap_inline73042#<#62828#> 
<#13434#>(define<#13434#> <#13435#>(product<#13435#> <#13436#>limit<#13436#> <#13437#>n)<#13437#> <#13438#>...)<#13438#>

That is, we name the first input, a natural number, with the notation <#62829#><#13442#>[<#13442#><#13443#>limit]<#13443#><#62829#> and specify the second input using the name for the first one. The rest of the program development is straightforward. It is basically the same as that for <#62830#><#13444#>product-from-20<#13444#><#62830#> with <#62831#><#13445#>20<#13445#><#62831#> replaced by <#62832#><#13446#>limit<#13446#><#62832#> throughout. The only modification concerns the natural recusion in the function template. Since the function consumes a <#62833#><#13447#>limit<#13447#><#62833#> and a <#62834#><#13448#>N<#13448#>\ <#13449#>[<#13449#><#13450#>;SPMgt;=<#13450#>\ <#13451#>limit]<#13451#><#62834#>, the template must contain an application of <#62835#><#13452#>product<#13452#><#62835#> to <#62836#><#13453#>limit<#13453#><#62836#> and <#62837#><#13454#>(sub1<#13454#>\ <#13455#>n)<#13455#><#62837#>:

<#13460#>(d<#13460#><#13461#>efine<#13461#> <#13462#>(product<#13462#> <#13463#>limit<#13463#> <#13464#>n)<#13464#>
  <#13465#>(c<#13465#><#13466#>ond<#13466#> 
    <#13467#>[<#13467#><#13468#>(=<#13468#> <#13469#>n<#13469#> <#13470#>limit)<#13470#> <#13471#>...]<#13471#> 
    <#13472#>[<#13472#><#13473#>else<#13473#> <#13474#>...<#13474#> <#13475#>(product<#13475#> <#13476#>limit<#13476#> <#13477#>(sub1<#13477#> <#13478#>n))<#13478#> <#13479#>...]<#13479#><#13480#>))<#13480#>

Otherwise things remain the same. The full function definition is contained in figure~#figfact#13484>.
<#13487#>Exercise 11.4.5<#13487#> In exercises~#extabulatef#13489> and #extabulatef20#13490>, we developed functions that tabulate <#62838#><#13491#>f<#13491#><#62838#> from some natural number or natural number <#62839#><#13492#>[<#13492#><#13493#>;SPMgt;=<#13493#>\ <#13494#>20]<#13494#><#62839#> down to <#62840#><#13495#>0<#13495#><#62840#> or <#62841#><#13496#>20<#13496#><#62841#> (exclusive), respectively. Develop the function <#62842#><#13497#>tabulate-f-lim<#13497#><#62842#>, which tabulates the values of <#62843#><#13498#>f<#13498#><#62843#> in an analoguos manner from some natural number <#62844#><#13499#>n<#13499#><#62844#> down to some other natural number <#62845#><#13500#>limit<#13500#><#62845#>.

Solution<#62846#><#62846#> <#13506#>Exercise 11.4.6<#13506#> In exercises~#extabulatef#13508>, #extabulatef20#13509>, and~#extabulateflimit#13510>, we developed functions that tabulate the mathematical function <#62847#><#13511#>f<#13511#><#62847#> in various ranges. In both cases, the final function produced a list of <#62848#><#13512#>posn<#13512#><#62848#>s that was ordered in <#13513#>descending<#13513#> order. That is, an expression like <#62849#><#13514#>(tabulate-f<#13514#>\ <#13515#>3)<#13515#><#62849#> yields

<#13520#>(c<#13520#><#13521#>ons<#13521#> <#13522#>(make-posn<#13522#> <#13523#>3<#13523#> <#13524#>2.4)<#13524#>
  <#13525#>(c<#13525#><#13526#>ons<#13526#> <#13527#>(make-posn<#13527#> <#13528#>2<#13528#> <#13529#>3.4)<#13529#> 
    <#13530#>(c<#13530#><#13531#>ons<#13531#> <#13532#>(make-posn<#13532#> <#13533#>1<#13533#> <#13534#>3.6)<#13534#> 
      <#13535#>(c<#13535#><#13536#>ons<#13536#> <#13537#>(make-posn<#13537#> <#13538#>0<#13538#> <#13539#>3.0)<#13539#> 
        <#13540#>empty))))<#13540#>

If we prefer a list of <#62850#><#13544#>posn<#13544#><#62850#>s in <#13545#>ascending<#13545#> order, we must look at a different data collection, natural numbers up to a certain point in the chain:

A <#72204#><#71011#>natural number <#62851#><#13547#>[<#13547#><#13548#>;SPMlt;=<#13548#>\ <#13549#>20]<#13549#><#62851#><#71011#><#72204#> (<#62852#><#13550#>N<#13550#><#13551#>[<#13551#><#13552#>;SPMlt;=20]<#13552#><#62852#>) is either

<#62853#><#13554#>20<#13554#><#62853#> or
<#62854#><#13555#>(sub1<#13555#>\ <#13556#>n)<#13556#><#62854#> if <#62855#><#13557#>n<#13557#><#62855#> is a natural number <#62856#><#13558#>[<#13558#><#13559#>;SPMlt;=<#13559#>\ <#13560#>20]<#13560#><#62856#>.

Of course, in high school, we refer to <#62857#><#13563#>N<#13563#><#13564#>[<#13564#><#13565#>;SPMlt;=-1]<#13565#><#62857#> as <#13566#>the<#13566#> negative integers. Develop the function

<#71012#>;; <#62858#><#13571#>tabulate-f-up-to-20<#13571#> <#13572#>:<#13572#> <#13573#>N<#13573#> <#13574#>[<#13574#><#13575#>;SPMlt;=<#13575#> <#13576#>20]<#13576#> <#13577#><#13577#><#13578#>-;SPMgt;<#13578#><#13579#><#13579#> <#13580#>N<#13580#><#62858#><#71012#>
<#13581#>(define<#13581#> <#13582#>(tabulate-f-up-to-20<#13582#> <#13583#>n-above-20)<#13583#> <#13584#>...)<#13584#>

which tabulates the values of <#62859#><#13588#>f<#13588#><#62859#> for natural numbers less than <#62860#><#13589#>20<#13589#><#62860#>. Specifically, it consumes a natural number <#62861#><#13590#>n<#13590#><#62861#> less or equal to <#62862#><#13591#>20<#13591#><#62862#> and produces a list of <#62863#><#13592#>posn<#13592#><#62863#>s, each of which has the shape <#62864#><#13593#>(make-posn<#13593#>\ <#13594#>n<#13594#>\ <#13595#>(f<#13595#>\ <#13596#>n))<#13596#><#62864#> for some <#62865#><#13597#>n<#13597#><#62865#> between <#62866#><#13598#>0<#13598#><#62866#> and <#62867#><#13599#>n<#13599#><#62867#> (inclusively).

Solution<#62868#><#62868#> <#13605#>Exercise 11.4.7<#13605#> Develop the function <#62869#><#13607#>is-not-divisible-by;SPMlt;=i<#13607#><#62869#>. It consumes a natural number <#62870#><#13608#>[<#13608#><#13609#>;SPMgt;=<#13609#>\ <#13610#>1]<#13610#><#62870#>, <#62871#><#13611#>i<#13611#><#62871#>, and a natural number <#62872#><#13612#>m<#13612#><#62872#>, with <#62873#><#13613#>i<#13613#>\ <#13614#>;SPMlt;<#13614#>\ <#13615#>m<#13615#><#62873#>. If <#62874#><#13616#>m<#13616#><#62874#> is not divisible by any number between <#62875#><#13617#>1<#13617#><#62875#> (exclusive) and <#62876#><#13618#>i<#13618#><#62876#> (inclusive), the function produces <#62877#><#13619#>true<#13619#><#62877#>; otherwise, its output is <#62878#><#13620#>false<#13620#><#62878#>. Use <#62879#><#13621#>is-not-divisible-by;SPMlt;=i<#13621#><#62879#> to define <#62880#><#13622#>prime?<#13622#><#62880#>, which consumes a natural number and determines whether or not it is prime.

Solution<#62881#><#62881#>