MAYBE

Problem:
 cond1(true(),x,y) -> cond2(gr(x,y),x,y)
 cond2(true(),x,y) -> cond1(gr(x,0()),y,y)
 cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 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) -> cond1#(gr(x,0()),y,y)
   cond2#(false(),x,y) -> p#(x)
   cond2#(false(),x,y) -> gr#(x,0())
   cond2#(false(),x,y) -> cond1#(gr(x,0()),p(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) -> cond1(gr(x,0()),y,y)
   cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   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) -> cond1#(gr(x,0()),y,y)
    cond2#(false(),x,y) -> p#(x)
    cond2#(false(),x,y) -> gr#(x,0())
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(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) -> cond1(gr(x,0()),y,y)
    cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    p(0()) -> 0()
    p(s(x)) -> x
   graph:
    cond2#(false(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
    cond1#(true(),x,y) -> gr#(x,y)
    cond2#(true(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y) ->
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y) ->
    cond1#(true(),x,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) -> cond1#(gr(x,0()),p(x),y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(false(),x,y) -> gr#(x,0())
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(false(),x,y) -> p#(x)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
    cond2#(true(),x,y) -> cond1#(gr(x,0()),y,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)
   EDG 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) -> cond1#(gr(x,0()),y,y)
     cond2#(false(),x,y) -> p#(x)
     cond2#(false(),x,y) -> gr#(x,0())
     cond2#(false(),x,y) -> cond1#(gr(x,0()),p(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) -> cond1(gr(x,0()),y,y)
     cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     p(0()) -> 0()
     p(s(x)) -> x
    graph:
     cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
     cond1#(true(),x,y) -> gr#(x,y)
     cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y) ->
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
     cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y) ->
     cond1#(true(),x,y) -> gr#(x,y)
     cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y) ->
     cond1#(true(),x,y) -> cond2#(gr(x,y),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#(true(),x,y) -> gr#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y)
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(false(),x,y) -> p#(x)
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(false(),x,y) -> gr#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
     cond1#(true(),x,y) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
    SCC Processor:
     #sccs: 2
     #rules: 4
     #arcs: 11/64
     DPs:
      cond2#(false(),x,y) -> cond1#(gr(x,0()),p(x),y)
      cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
      cond2#(true(),x,y) -> cond1#(gr(x,0()),y,y)
     TRS:
      cond1(true(),x,y) -> cond2(gr(x,y),x,y)
      cond2(true(),x,y) -> cond1(gr(x,0()),y,y)
      cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      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) -> cond1(gr(x,0()),y,y)
      cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
     KBO Processor:
      argument filtering:
       pi(true) = []
       pi(cond1) = 2
       pi(gr) = 0
       pi(cond2) = 2
       pi(0) = []
       pi(false) = []
       pi(p) = [0]
       pi(s) = [0]
       pi(gr#) = 1
      weight function:
       w0 = 1
       w(gr#) = w(s) = w(p) = w(false) = w(0) = w(cond2) = w(gr) = w(cond1) = w(true) = 1
      precedence:
       gr# ~ s ~ p ~ 0 ~ cond2 ~ cond1 ~ true > false ~ gr
      problem:
       DPs:
        
       TRS:
        cond1(true(),x,y) -> cond2(gr(x,y),x,y)
        cond2(true(),x,y) -> cond1(gr(x,0()),y,y)
        cond2(false(),x,y) -> cond1(gr(x,0()),p(x),y)
        gr(0(),x) -> false()
        gr(s(x),0()) -> true()
        gr(s(x),s(y)) -> gr(x,y)
        p(0()) -> 0()
        p(s(x)) -> x
      Qed