MAYBE

Problem:
 digits() -> d(0())
 d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
 if(true(),x) -> cons(x,d(s(x)))
 if(false(),x) -> nil()
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)

Proof:
 DP Processor:
  DPs:
   digits#() -> d#(0())
   d#(x) -> le#(x,s(s(s(s(s(s(s(s(s(0()))))))))))
   d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
   if#(true(),x) -> d#(s(x))
   le#(s(x),s(y)) -> le#(x,y)
  TRS:
   digits() -> d(0())
   d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
   if(true(),x) -> cons(x,d(s(x)))
   if(false(),x) -> nil()
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
  TDG Processor:
   DPs:
    digits#() -> d#(0())
    d#(x) -> le#(x,s(s(s(s(s(s(s(s(s(0()))))))))))
    d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
    if#(true(),x) -> d#(s(x))
    le#(s(x),s(y)) -> le#(x,y)
   TRS:
    digits() -> d(0())
    d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
    if(true(),x) -> cons(x,d(s(x)))
    if(false(),x) -> nil()
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
   graph:
    if#(true(),x) -> d#(s(x)) ->
    d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
    if#(true(),x) -> d#(s(x)) ->
    d#(x) -> le#(x,s(s(s(s(s(s(s(s(s(0()))))))))))
    le#(s(x),s(y)) -> le#(x,y) ->
    le#(s(x),s(y)) -> le#(x,y)
    d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x) ->
    if#(true(),x) -> d#(s(x))
    d#(x) -> le#(x,s(s(s(s(s(s(s(s(s(0())))))))))) ->
    le#(s(x),s(y)) -> le#(x,y)
    digits#() -> d#(0()) ->
    d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
    digits#() -> d#(0()) -> d#(x) -> le#(x,s(s(s(s(s(s(s(s(s(0()))))))))))
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 7/25
    DPs:
     if#(true(),x) -> d#(s(x))
     d#(x) -> if#(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
    TRS:
     digits() -> d(0())
     d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
     if(true(),x) -> cons(x,d(s(x)))
     if(false(),x) -> nil()
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
    Open
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     digits() -> d(0())
     d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
     if(true(),x) -> cons(x,d(s(x)))
     if(false(),x) -> nil()
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
    KBO Processor:
     argument filtering:
      pi(digits) = []
      pi(0) = []
      pi(d) = 0
      pi(s) = [0]
      pi(le) = 0
      pi(if) = 0
      pi(true) = []
      pi(cons) = []
      pi(false) = []
      pi(nil) = []
      pi(le#) = 0
     weight function:
      w0 = 1
      w(le#) = w(nil) = w(false) = w(cons) = w(true) = w(if) = w(le) = w(
      s) = w(d) = w(0) = w(digits) = 1
     precedence:
      le# ~ false ~ s ~ digits > 0 > true > nil ~ cons ~ if ~ le ~ d
     problem:
      DPs:
       
      TRS:
       digits() -> d(0())
       d(x) -> if(le(x,s(s(s(s(s(s(s(s(s(0())))))))))),x)
       if(true(),x) -> cons(x,d(s(x)))
       if(false(),x) -> nil()
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
     Qed