(STRATEGY
    INNERMOST)

(VAR
    b x y)
(DATATYPES
    A = µX.< 0, true, s(X), false, log_undefined >)
(SIGNATURES
    le :: [A x A] -> A
    inc :: [A] -> A
    minus :: [A x A] -> A
    quot :: [A x A] -> A
    log :: [A] -> A
    log2 :: [A x A] -> A
    if :: [A x A x A x A] -> A
    if2 :: [A x A x A] -> A)
(RULES
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    minus(0(),y) -> 0()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) ->
      s(quot(minus(x,y),s(y)))
    log(x) -> log2(x,0())
    log2(x,y) -> if(le(x,0())
                   ,le(x,s(0()))
                   ,x
                   ,inc(y))
    if(true(),b,x,y) ->
      log_undefined()
    if(false(),b,x,y) -> if2(b,x,y)
    if2(true(),x,s(y)) -> y
    if2(false(),x,y) -> log2(quot(x
                                 ,s(s(0())))
                            ,y))