fun isRelatedTo(a, b, []) = false 
   | isRelatedTo(a, b, (h1, h2)::rest) = 
            (a = h1 andalso b = h2) orelse isRelatedTo(a, b, rest);

datatype city = Vancouver | LosAngeles | Mexico | Denver | Omaha |
         KansasCity | Minneapolis | StLouis | Memphis | NewOrleans |
         Chicago | Cincinnati | Pittsburgh | Montreal | NewYork;

datatype river = Missouri | Platte | NPlatte | SPlatte | Arkansas | Canadian | 
         Kansas | Mississippi | Tennessee | Ohio | Allegheny | Monongahela;

datatype Tribe = Asher | Naphtali | Zebulun | Issachar | Dan 
               | Gad | Manasseh | Reuben | Ephraim | Benjamin 
               | Judah | Simeon;


datatype WaterBody = Mediterranean | Dead | Jordan 
                    | Chinnereth | Arnon | Jabbok;


val waterImmedWestOf = 
 [(Mediterranean, Chinnereth), 
  (Mediterranean, Jordan), (Mediterranean, Dead),
  (Chinnereth, Jabbok), (Chinnereth, Arnon), 
  (Jordan, Jabbok), (Jordan, Arnon), 
  (Dead, Jabbok), (Dead, Arnon)];

val waterWestOf = 
 [(Mediterranean, Chinnereth), (Mediterranean, Jordan), (Mediterranean, Dead),
  (Mediterranean, Jabbok), (Mediterranean, Arnon), (Chinnereth, Jabbok), 
  (Chinnereth, Arnon), (Jordan, Jabbok), (Jordan, Arnon), (Dead, Jabbok), 
  (Dead, Arnon)];

val flowsInto = [(Platte, Missouri), (Kansas, Missouri),(Missouri, Mississippi),
                 (Canadian, Arkansas), (Arkansas, Mississippi),
                 (Allegheny, Ohio), (Monongahela, Ohio), (Tennessee, Ohio),
                 (Ohio, Mississippi), (NPlatte, Platte), (SPlatte, Platte)];

val isDownstreamFrom = 
    [(Omaha, Denver), (KansasCity, Omaha), (KansasCity, Denver),
     (StLouis, KansasCity), (StLouis, Omaha), (StLouis, Denver),
     (StLouis, Minneapolis), (Cincinnati, Pittsburgh),
     (Memphis, StLouis), (Memphis, KansasCity), (Memphis, Omaha),
     (Memphis, Denver), (Memphis, Minneapolis), (Memphis, Cincinnati),
     (Memphis, Pittsburgh), (NewOrleans, Memphis),
     (NewOrleans, StLouis), (NewOrleans, KansasCity), (NewOrleans, Omaha),
     (NewOrleans, Denver), (NewOrleans, Minneapolis), 
     (NewOrleans,Cincinnati), (NewOrleans, Pittsburgh)];

val inSameTimeZone = 
    [(Montreal, NewYork), (NewYork, Montreal), 
     (NewYork, Pittsburgh), (Pittsburgh, NewYork), (Montreal, Pittsburgh),
     (Pittsburgh, Montreal), (Chicago, NewOrleans), (NewOrleans, Chicago),
     (Chicago, Mexico), (Mexico, Chicago), (NewOrleans, Mexico),
     (Mexico, NewOrleans), (LosAngeles, Vancouver),
     (Vancouver, LosAngeles)];

val waterVerticalAlign =
 [(Chinnereth, Jordan), (Chinnereth, Dead), (Jordan, Chinnereth),(Jordan, Dead),
  (Dead, Chinnereth), (Dead, Jordan), (Jabbok, Arnon), (Arnon, Jabbok)];

fun testOnePair((a, b), []) = true
  | testOnePair((a, b), (c, d)::rest) =
    ((not (b = c)) orelse isRelatedTo(a, d, relation))
    andalso testOnePair((a,b), rest);



fun test([]) = true
  | test((a,b)::rest) = 
    testOnePair((a,b), relation) andalso test(rest);

fun isTransitive(relation) =
  let fun testOnePair((a, b), []) = true
        | testOnePair((a, b), (c, d)::rest) =
                 ((not (b = c)) orelse isRelatedTo(a, d, relation))
                 andalso testOnePair((a,b), rest);
      fun test([]) = true
        | test((a,b)::rest) = 
                 testOnePair((a,b), relation) andalso test(rest);
  in
    test(relation)
  end;

fun counterTransitive(relation) =
 let fun testOnePair((a, b), []) = []
        | testOnePair((a, b), (c,d)::rest) = 
                      (if ((not (b=c)) orelse isRelatedTo(a, d, relation)) 
                               then [] else [(a, d)])
                       @ testOnePair((a,b), rest);
      fun test([]) = []
       | test((a,b)::rest) = testOnePair((a,b), relation) @ test(rest)
 in
    test(relation)
 end;

val waterVerticalAlignCorrected =
 [(Chinnereth, Jordan), (Chinnereth, Dead), (Jordan, Chinnereth),(Jordan, Dead),
  (Dead, Chinnereth), (Dead, Jordan), (Jabbok, Arnon), (Arnon, Jabbok),
  (Chinnereth,Chinnereth),(Jordan,Jordan),
   (Dead,Dead),(Jabbok,Jabbok),(Arnon,Arnon)];

(* Ex 5.6.1, isSymmetric *)
fun isSymmetric(relation) = 
    let fun testOnePair((a,b), [] ) = ??
          | testOnePair((a,b), (c,d)::rest) = ??;
        fun test([]) = ??
           | test((a,b)::rest) = ??
    in
        test(relation)
    end;

(* Ex 5.6.3, counterSymmetric *)
fun counterSymmetric(relation) = 
        let fun testOnePair((a,b), [] ) = ??
          | testOnePair((a,b), (c,d)::rest) = ??;
        fun test([]) = ??
           | test((a,b)::rest) = ??
    in
        test(relation)
        end;

val flowsInto = [(Canadian,Arkansas), (NPlatte,Platte), (SPlatte,Platte),
                 (Platte,Missouri), (Kansas,Missouri), (Missouri,Mississippi),
                 (Allegheny,Ohio), (Monongahela,Ohio), (Tennessee,Ohio),
                 (Ohio,Mississippi)];

fun contains(x, []) = false
  | contains(x, y::rest) = if x = y then true else contains(x, rest);

fun makeNoRepeats([]) = []
  | makeNoRepeats(x::rest) =
       if contains(x, rest) then makeNoRepeats(rest)
                            else x::makeNoRepeats(rest);

fun transitiveClosure(relation) =
    relation @ makeNoRepeats(counterTransitive(relation));



fun transitiveClosure(relation) =
  if isTransitive(relation) 
            then relation 
            else transitiveClosure(makeNoRepeats(counterTransitive(relation))
                                   @ relation);

val flowsInto = [(Canadian,Arkansas), (NPlatte,Platte), (SPlatte,Platte),
                 (Platte,Missouri), (Kansas,Missouri), (Missouri,Mississippi),
                 (Allegheny,Ohio), (Monongahela,Ohio), (Tennessee,Ohio),
                 (Ohio,Mississippi)];

fun flowsInto(Canadian, Arkansas) = true
  | flowsInto(NPlatte, Platte) = true 
  |  flowsInto(SPlatte, Platte) = true
  | flowsInto(Platte, Missouri) = true
  | flowsInto(Kansas, Missouri) = true 
  | flowsInto(Missouri, Mississippi) = true
  | flowsInto(Allegheny, Ohio) = true
  | flowsInto(Monongahela, Ohio) = true 
  |  flowsInto(Tennessee, Ohio) = true
  | flowsInto(Ohio, MIssissippi) = true
  | flowsInto(x, y) = false;

(* Ex 5.7.5; symmetricClosure *)
fun symmetricClosure(relation) = 
    ??