MAYBE

Problem:
 cond1(true(),x,y) -> cond2(gr(x,y),x,y)
 cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
 cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
 cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
 cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
 cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
 cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 and(true(),true()) -> true()
 and(false(),x) -> false()
 and(x,false()) -> false()
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x,y) -> gr#(x,y)
   cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
   cond2#(true(),x,y) -> gr#(x,0())
   cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y)
   cond2#(false(),x,y) -> gr#(y,0())
   cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y)
   cond3#(true(),x,y) -> p#(x)
   cond3#(true(),x,y) -> gr#(x,0())
   cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y)
   cond3#(false(),x,y) -> gr#(y,0())
   cond3#(false(),x,y) -> gr#(x,0())
   cond3#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
   cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
   cond4#(true(),x,y) -> p#(y)
   cond4#(true(),x,y) -> gr#(y,0())
   cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y))
   cond4#(false(),x,y) -> gr#(y,0())
   cond4#(false(),x,y) -> gr#(x,0())
   cond4#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
   cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
   gr#(s(x),s(y)) -> gr#(x,y)
  TRS:
   cond1(true(),x,y) -> cond2(gr(x,y),x,y)
   cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
   cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
   cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
   cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
   cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
   cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   and(true(),true()) -> true()
   and(false(),x) -> false()
   and(x,false()) -> false()
   p(0()) -> 0()
   p(s(x)) -> x
  TDG Processor:
   DPs:
    cond1#(true(),x,y) -> gr#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond2#(true(),x,y) -> gr#(x,0())
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y)
    cond2#(false(),x,y) -> gr#(y,0())
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y)
    cond3#(true(),x,y) -> p#(x)
    cond3#(true(),x,y) -> gr#(x,0())
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y)
    cond3#(false(),x,y) -> gr#(y,0())
    cond3#(false(),x,y) -> gr#(x,0())
    cond3#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    cond4#(true(),x,y) -> p#(y)
    cond4#(true(),x,y) -> gr#(y,0())
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y))
    cond4#(false(),x,y) -> gr#(y,0())
    cond4#(false(),x,y) -> gr#(x,0())
    cond4#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    gr#(s(x),s(y)) -> gr#(x,y)
   TRS:
    cond1(true(),x,y) -> cond2(gr(x,y),x,y)
    cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
    cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
    cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
    cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
    cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
    cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    and(true(),true()) -> true()
    and(false(),x) -> false()
    and(x,false()) -> false()
    p(0()) -> 0()
    p(s(x)) -> x
   graph:
    cond4#(false(),x,y) -> gr#(y,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond4#(false(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y) ->
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y) ->
    cond1#(true(),x,y) -> gr#(x,y)
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(false(),x,y) -> gr#(x,0())
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(false(),x,y) -> gr#(y,0())
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y))
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(true(),x,y) -> gr#(y,0())
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y)) ->
    cond4#(true(),x,y) -> p#(y)
    cond4#(true(),x,y) -> gr#(y,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
    cond3#(false(),x,y) -> gr#(y,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond3#(false(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y) ->
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y) ->
    cond1#(true(),x,y) -> gr#(x,y)
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(false(),x,y) -> gr#(x,0())
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(false(),x,y) -> gr#(y,0())
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y)
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(true(),x,y) -> gr#(x,0())
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y) ->
    cond3#(true(),x,y) -> p#(x)
    cond3#(true(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(false(),x,y) -> gr#(x,0())
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(false(),x,y) -> gr#(y,0())
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y))
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(true(),x,y) -> gr#(y,0())
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y) ->
    cond4#(true(),x,y) -> p#(y)
    cond2#(false(),x,y) -> gr#(y,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(false(),x,y) -> and#(gr(x,0()),gr(y,0()))
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(false(),x,y) -> gr#(x,0())
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(false(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y)
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(true(),x,y) -> gr#(x,0())
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y) ->
    cond3#(true(),x,y) -> p#(x)
    cond2#(true(),x,y) -> gr#(x,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
    gr#(s(x),s(y)) -> gr#(x,y) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(false(),x,y) -> gr#(y,0())
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(true(),x,y) -> gr#(x,0())
    cond1#(true(),x,y) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
   SCC Processor:
    #sccs: 2
    #rules: 8
    #arcs: 46/441
    DPs:
     cond4#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
     cond2#(true(),x,y) -> cond3#(gr(x,0()),x,y)
     cond3#(true(),x,y) -> cond3#(gr(x,0()),p(x),y)
     cond3#(false(),x,y) -> cond1#(and(gr(x,0()),gr(y,0())),x,y)
     cond2#(false(),x,y) -> cond4#(gr(y,0()),x,y)
     cond4#(true(),x,y) -> cond4#(gr(y,0()),x,p(y))
    TRS:
     cond1(true(),x,y) -> cond2(gr(x,y),x,y)
     cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
     cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
     cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
     cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
     cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
     cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     and(true(),true()) -> true()
     and(false(),x) -> false()
     and(x,false()) -> false()
     p(0()) -> 0()
     p(s(x)) -> x
    Open
    
    DPs:
     gr#(s(x),s(y)) -> gr#(x,y)
    TRS:
     cond1(true(),x,y) -> cond2(gr(x,y),x,y)
     cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
     cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
     cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
     cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
     cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
     cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     and(true(),true()) -> true()
     and(false(),x) -> false()
     and(x,false()) -> false()
     p(0()) -> 0()
     p(s(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(gr#) = 1
     problem:
      DPs:
       
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,y),x,y)
       cond2(true(),x,y) -> cond3(gr(x,0()),x,y)
       cond2(false(),x,y) -> cond4(gr(y,0()),x,y)
       cond3(true(),x,y) -> cond3(gr(x,0()),p(x),y)
       cond3(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
       cond4(true(),x,y) -> cond4(gr(y,0()),x,p(y))
       cond4(false(),x,y) -> cond1(and(gr(x,0()),gr(y,0())),x,y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       and(true(),true()) -> true()
       and(false(),x) -> false()
       and(x,false()) -> false()
       p(0()) -> 0()
       p(s(x)) -> x
     Qed