MAYBE

Problem:
 cond1(true(),x,y) -> cond2(gr(x,y),x,y)
 cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
 cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 neq(0(),0()) -> false()
 neq(0(),s(x)) -> true()
 neq(s(x),0()) -> true()
 neq(s(x),s(y)) -> neq(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) -> neq#(x,0())
   cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y)
   cond2#(false(),x,y) -> p#(x)
   cond2#(false(),x,y) -> neq#(x,0())
   cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
   gr#(s(x),s(y)) -> gr#(x,y)
   neq#(s(x),s(y)) -> neq#(x,y)
  TRS:
   cond1(true(),x,y) -> cond2(gr(x,y),x,y)
   cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
   cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   neq(0(),0()) -> false()
   neq(0(),s(x)) -> true()
   neq(s(x),0()) -> true()
   neq(s(x),s(y)) -> neq(x,y)
   p(0()) -> 0()
   p(s(x)) -> x
  Usable Rule Processor:
   DPs:
    cond1#(true(),x,y) -> gr#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
    cond2#(true(),x,y) -> neq#(x,0())
    cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y)
    cond2#(false(),x,y) -> p#(x)
    cond2#(false(),x,y) -> neq#(x,0())
    cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
    gr#(s(x),s(y)) -> gr#(x,y)
    neq#(s(x),s(y)) -> neq#(x,y)
   TRS:
    f14(x,y) -> x
    f14(x,y) -> y
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    neq(0(),0()) -> false()
    neq(s(x),0()) -> true()
    p(0()) -> 0()
    p(s(x)) -> x
   EDG Processor:
    DPs:
     cond1#(true(),x,y) -> gr#(x,y)
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
     cond2#(true(),x,y) -> neq#(x,0())
     cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y)
     cond2#(false(),x,y) -> p#(x)
     cond2#(false(),x,y) -> neq#(x,0())
     cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
     gr#(s(x),s(y)) -> gr#(x,y)
     neq#(s(x),s(y)) -> neq#(x,y)
    TRS:
     f14(x,y) -> x
     f14(x,y) -> y
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     neq(0(),0()) -> false()
     neq(s(x),0()) -> true()
     p(0()) -> 0()
     p(s(x)) -> x
    graph:
     neq#(s(x),s(y)) -> neq#(x,y) ->
     neq#(s(x),s(y)) -> neq#(x,y)
     cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y) ->
     cond1#(true(),x,y) -> gr#(x,y)
     cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y) ->
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
     cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y) ->
     cond1#(true(),x,y) -> gr#(x,y)
     cond2#(true(),x,y) -> cond1#(neq(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) -> neq#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(true(),x,y) -> cond1#(neq(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) -> neq#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(x,y),x,y) ->
     cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
     cond1#(true(),x,y) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
    Restore Modifier:
     DPs:
      cond1#(true(),x,y) -> gr#(x,y)
      cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
      cond2#(true(),x,y) -> neq#(x,0())
      cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y)
      cond2#(false(),x,y) -> p#(x)
      cond2#(false(),x,y) -> neq#(x,0())
      cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
      gr#(s(x),s(y)) -> gr#(x,y)
      neq#(s(x),s(y)) -> neq#(x,y)
     TRS:
      cond1(true(),x,y) -> cond2(gr(x,y),x,y)
      cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
      cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      neq(0(),0()) -> false()
      neq(0(),s(x)) -> true()
      neq(s(x),0()) -> true()
      neq(s(x),s(y)) -> neq(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
     SCC Processor:
      #sccs: 3
      #rules: 5
      #arcs: 12/81
      DPs:
       cond2#(false(),x,y) -> cond1#(neq(x,0()),p(x),y)
       cond1#(true(),x,y) -> cond2#(gr(x,y),x,y)
       cond2#(true(),x,y) -> cond1#(neq(x,0()),y,y)
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,y),x,y)
       cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
       cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y)) -> neq(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(neq(x,0()),y,y)
       cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y)) -> neq(x,y)
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gr#](x0, x1) = x1 + 1,
        
        [s](x0) = x0 + 1,
        
        [p](x0) = x0,
        
        [false] = 0,
        
        [neq](x0, x1) = 0,
        
        [0] = 0,
        
        [cond2](x0, x1, x2) = 0,
        
        [gr](x0, x1) = 0,
        
        [cond1](x0, x1, x2) = 0,
        
        [true] = 0
       orientation:
        gr#(s(x),s(y)) = y + 2 >= y + 1 = gr#(x,y)
        
        cond1(true(),x,y) = 0 >= 0 = cond2(gr(x,y),x,y)
        
        cond2(true(),x,y) = 0 >= 0 = cond1(neq(x,0()),y,y)
        
        cond2(false(),x,y) = 0 >= 0 = cond1(neq(x,0()),p(x),y)
        
        gr(0(),x) = 0 >= 0 = false()
        
        gr(s(x),0()) = 0 >= 0 = true()
        
        gr(s(x),s(y)) = 0 >= 0 = gr(x,y)
        
        neq(0(),0()) = 0 >= 0 = false()
        
        neq(0(),s(x)) = 0 >= 0 = true()
        
        neq(s(x),0()) = 0 >= 0 = true()
        
        neq(s(x),s(y)) = 0 >= 0 = neq(x,y)
        
        p(0()) = 0 >= 0 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x,y) -> cond2(gr(x,y),x,y)
         cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
         cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         neq(0(),0()) -> false()
         neq(0(),s(x)) -> true()
         neq(s(x),0()) -> true()
         neq(s(x),s(y)) -> neq(x,y)
         p(0()) -> 0()
         p(s(x)) -> x
       Qed
      
      DPs:
       neq#(s(x),s(y)) -> neq#(x,y)
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,y),x,y)
       cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
       cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y)) -> neq(x,y)
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [neq#](x0, x1) = x1 + 1,
        
        [s](x0) = x0 + 1,
        
        [p](x0) = x0,
        
        [false] = 0,
        
        [neq](x0, x1) = 0,
        
        [0] = 0,
        
        [cond2](x0, x1, x2) = 0,
        
        [gr](x0, x1) = 0,
        
        [cond1](x0, x1, x2) = 0,
        
        [true] = 0
       orientation:
        neq#(s(x),s(y)) = y + 2 >= y + 1 = neq#(x,y)
        
        cond1(true(),x,y) = 0 >= 0 = cond2(gr(x,y),x,y)
        
        cond2(true(),x,y) = 0 >= 0 = cond1(neq(x,0()),y,y)
        
        cond2(false(),x,y) = 0 >= 0 = cond1(neq(x,0()),p(x),y)
        
        gr(0(),x) = 0 >= 0 = false()
        
        gr(s(x),0()) = 0 >= 0 = true()
        
        gr(s(x),s(y)) = 0 >= 0 = gr(x,y)
        
        neq(0(),0()) = 0 >= 0 = false()
        
        neq(0(),s(x)) = 0 >= 0 = true()
        
        neq(s(x),0()) = 0 >= 0 = true()
        
        neq(s(x),s(y)) = 0 >= 0 = neq(x,y)
        
        p(0()) = 0 >= 0 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x,y) -> cond2(gr(x,y),x,y)
         cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
         cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         neq(0(),0()) -> false()
         neq(0(),s(x)) -> true()
         neq(s(x),0()) -> true()
         neq(s(x),s(y)) -> neq(x,y)
         p(0()) -> 0()
         p(s(x)) -> x
       Qed