(STRATEGY
    INNERMOST)

(VAR
    x y)
(DATATYPES
    A = µX.< s(X), 0, true, false >)
(SIGNATURES
    g :: [A x A] -> A
    n :: [A x A] -> A
    m :: [A x A] -> A
    k :: [A x A] -> A
    t :: [A] -> A
    p :: [A x A] -> A
    minus :: [A x A] -> A
    id :: [A] -> A
    if :: [A x A x A] -> A
    not :: [A] -> A
    and :: [A x A] -> A
    f :: [A] -> A
    h :: [A] -> A
    gt :: [A x A] -> A)
(RULES
    g(s(x),s(y)) -> if(and(f(s(x))
                          ,f(s(y)))
                      ,t(g(k(minus(m(x,y),n(x,y))
                            ,s(s(0())))
                          ,k(n(s(x),s(y)),s(s(0())))))
                      ,g(minus(m(x,y),n(x,y))
                        ,n(s(x),s(y))))
    n(0(),y) -> 0()
    n(x,0()) -> 0()
    n(s(x),s(y)) -> s(n(x,y))
    m(0(),y) -> y
    m(x,0()) -> x
    m(s(x),s(y)) -> s(m(x,y))
    k(0(),s(y)) -> 0()
    k(s(x),s(y)) -> s(k(minus(x,y)
                       ,s(y)))
    t(x) -> p(x,x)
    p(s(x),s(y)) -> s(s(p(if(gt(x,y)
                            ,x
                            ,y)
                         ,if(not(gt(x,y)),id(x),id(y)))))
    p(s(x),x) -> p(if(gt(x,x)
                     ,id(x)
                     ,id(x))
                  ,s(x))
    p(0(),y) -> y
    p(id(x),s(y)) -> s(p(x
                        ,if(gt(s(y),y),y,s(y))))
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    id(x) -> x
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    not(x) -> if(x,false(),true())
    and(x,false()) -> false()
    and(true(),true()) -> true()
    f(0()) -> true()
    f(s(x)) -> h(x)
    h(0()) -> false()
    h(s(x)) -> f(x)
    gt(s(x),0()) -> true()
    gt(0(),y) -> false()
    gt(s(x),s(y)) -> gt(x,y))