(STRATEGY
    INNERMOST)

(VAR
    m n x y z)
(DATATYPES
    A = µX.< true, false, s(X), 0, nil, cons(X, X), app(X, X), error, b, c >)
(SIGNATURES
    average :: [A x A] -> A
    if :: [A x A x A] -> A
    averIter :: [A x A x A] -> A
    ifIter :: [A x A x A x A] -> A
    append :: [A x A] -> A
    low :: [A x A] -> A
    if_low :: [A x A x A] -> A
    high :: [A x A] -> A
    if_high :: [A x A x A] -> A
    quicksort :: [A] -> A
    ifquick :: [A x A] -> A
    plus :: [A x A] -> A
    isempty :: [A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    ge :: [A x A] -> A
    a :: [] -> A)
(RULES
    average(x,y) -> if(ge(x,y),x,y)
    if(true(),x,y) -> averIter(y
                              ,x
                              ,y)
    if(false(),x,y) -> averIter(x
                               ,y
                               ,x)
    averIter(x,y,z) -> ifIter(ge(x
                                ,y)
                             ,x
                             ,y
                             ,z)
    ifIter(true(),x,y,z) -> z
    ifIter(false(),x,y,z) ->
      averIter(plus(x,s(s(s(0()))))
              ,plus(y,s(0()))
              ,plus(z,s(0())))
    append(nil(),y) -> y
    append(cons(n,x),y) -> cons(n
                               ,app(x,y))
    low(n,nil()) -> nil()
    low(n,cons(m,x)) -> if_low(ge(m
                                 ,n)
                              ,n
                              ,cons(m,x))
    if_low(false(),n,cons(m,x)) ->
      cons(m,low(n,x))
    if_low(true(),n,cons(m,x)) ->
      low(n,x)
    high(n,nil()) -> nil()
    high(n,cons(m,x)) ->
      if_high(ge(m,n),n,cons(m,x))
    if_high(false(),n,cons(m,x)) ->
      high(n,x)
    if_high(true(),n,cons(m,x)) ->
      cons(average(m,m),high(n,x))
    quicksort(x) ->
      ifquick(isempty(x),x)
    ifquick(true(),x) -> nil()
    ifquick(false(),x) ->
      append(quicksort(low(head(x)
                          ,tail(x)))
            ,cons(tail(x)
                 ,quicksort(high(head(x)
                                ,tail(x)))))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    isempty(nil()) -> true()
    isempty(cons(n,x)) -> false()
    head(nil()) -> error()
    head(cons(n,x)) -> n
    tail(nil()) -> nil()
    tail(cons(n,x)) -> x
    ge(x,0()) -> true()
    ge(0(),s(y)) -> false()
    ge(s(x),s(y)) -> ge(x,y)
    a() -> b()
    a() -> c())