MAYBE

Problem:
 cond1(true(),x,y,z) -> cond2(gr(y,z),x,y,z)
 cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
 cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
 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,z) -> gr#(y,z)
   cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z)
   cond2#(true(),x,y,z) -> p#(y)
   cond2#(true(),x,y,z) -> gr#(y,z)
   cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z)
   cond2#(false(),x,y,z) -> p#(x)
   cond2#(false(),x,y,z) -> gr#(x,z)
   cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z)
   gr#(s(x),s(y)) -> gr#(x,y)
  TRS:
   cond1(true(),x,y,z) -> cond2(gr(y,z),x,y,z)
   cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
   cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
   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,z) -> gr#(y,z)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z)
    cond2#(true(),x,y,z) -> p#(y)
    cond2#(true(),x,y,z) -> gr#(y,z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z)
    cond2#(false(),x,y,z) -> p#(x)
    cond2#(false(),x,y,z) -> gr#(x,z)
    cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z)
    gr#(s(x),s(y)) -> gr#(x,y)
   TRS:
    cond1(true(),x,y,z) -> cond2(gr(y,z),x,y,z)
    cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
    cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
    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,z) -> gr#(x,z) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z) ->
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z)
    cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z) ->
    cond1#(true(),x,y,z) -> gr#(y,z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(false(),x,y,z) -> gr#(x,z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(false(),x,y,z) -> p#(x)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(true(),x,y,z) -> gr#(y,z)
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z) ->
    cond2#(true(),x,y,z) -> p#(y)
    cond2#(true(),x,y,z) -> gr#(y,z) -> 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,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(false(),x,y,z) -> gr#(x,z)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(false(),x,y,z) -> p#(x)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(true(),x,y,z) -> gr#(y,z)
    cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z) ->
    cond2#(true(),x,y,z) -> p#(y)
    cond1#(true(),x,y,z) -> gr#(y,z) -> gr#(s(x),s(y)) -> gr#(x,y)
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 18/81
    DPs:
     cond2#(false(),x,y,z) -> cond1#(gr(x,z),p(x),y,z)
     cond1#(true(),x,y,z) -> cond2#(gr(y,z),x,y,z)
     cond2#(true(),x,y,z) -> cond2#(gr(y,z),x,p(y),z)
    TRS:
     cond1(true(),x,y,z) -> cond2(gr(y,z),x,y,z)
     cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
     cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
     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,z) -> cond2(gr(y,z),x,y,z)
     cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
     cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
     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) = 3
      pi(gr) = 0
      pi(cond2) = 3
      pi(p) = [0]
      pi(false) = []
      pi(0) = []
      pi(s) = [0]
      pi(gr#) = 1
     weight function:
      w0 = 1
      w(gr#) = w(s) = w(0) = w(false) = w(cond2) = w(gr) = w(cond1) = w(true) = 1
      w(p) = 0
     precedence:
      cond1 > gr# ~ s ~ p ~ cond2 ~ true > 0 > false ~ gr
     problem:
      DPs:
       
      TRS:
       cond1(true(),x,y,z) -> cond2(gr(y,z),x,y,z)
       cond2(true(),x,y,z) -> cond2(gr(y,z),x,p(y),z)
       cond2(false(),x,y,z) -> cond1(gr(x,z),p(x),y,z)
       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