MAYBE

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

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x,y) -> gr#(x,0())
   cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
   cond2#(true(),x,y) -> p#(x)
   cond2#(true(),x,y) -> add#(x,y)
   cond2#(true(),x,y) -> gr#(add(x,y),0())
   cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
   cond2#(false(),x,y) -> gr#(y,0())
   cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
   cond3#(true(),x,y) -> p#(y)
   cond3#(true(),x,y) -> add#(x,y)
   cond3#(true(),x,y) -> gr#(add(x,y),0())
   cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
   cond3#(false(),x,y) -> add#(x,y)
   cond3#(false(),x,y) -> gr#(add(x,y),0())
   cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
   gr#(s(x),s(y)) -> gr#(x,y)
   add#(s(x),y) -> add#(x,y)
  TRS:
   cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
   cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
   cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
   cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
   cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   add(0(),x) -> x
   add(s(x),y) -> s(add(x,y))
   p(0()) -> 0()
   p(s(x)) -> x
  Usable Rule Processor:
   DPs:
    cond1#(true(),x,y) -> gr#(x,0())
    cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
    cond2#(true(),x,y) -> p#(x)
    cond2#(true(),x,y) -> add#(x,y)
    cond2#(true(),x,y) -> gr#(add(x,y),0())
    cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
    cond2#(false(),x,y) -> gr#(y,0())
    cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
    cond3#(true(),x,y) -> p#(y)
    cond3#(true(),x,y) -> add#(x,y)
    cond3#(true(),x,y) -> gr#(add(x,y),0())
    cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
    cond3#(false(),x,y) -> add#(x,y)
    cond3#(false(),x,y) -> gr#(add(x,y),0())
    cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
    gr#(s(x),s(y)) -> gr#(x,y)
    add#(s(x),y) -> add#(x,y)
   TRS:
    f16(x,y) -> x
    f16(x,y) -> y
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    add(0(),x) -> x
    add(s(x),y) -> s(add(x,y))
    p(0()) -> 0()
    p(s(x)) -> x
   CDG Processor:
    DPs:
     cond1#(true(),x,y) -> gr#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
     cond2#(true(),x,y) -> p#(x)
     cond2#(true(),x,y) -> add#(x,y)
     cond2#(true(),x,y) -> gr#(add(x,y),0())
     cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
     cond2#(false(),x,y) -> gr#(y,0())
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
     cond3#(true(),x,y) -> p#(y)
     cond3#(true(),x,y) -> add#(x,y)
     cond3#(true(),x,y) -> gr#(add(x,y),0())
     cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
     cond3#(false(),x,y) -> add#(x,y)
     cond3#(false(),x,y) -> gr#(add(x,y),0())
     cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
     gr#(s(x),s(y)) -> gr#(x,y)
     add#(s(x),y) -> add#(x,y)
    TRS:
     f16(x,y) -> x
     f16(x,y) -> y
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     add(0(),x) -> x
     add(s(x),y) -> s(add(x,y))
     p(0()) -> 0()
     p(s(x)) -> x
    graph:
     cond3#(false(),x,y) -> add#(x,y) ->
     add#(s(x),y) -> add#(x,y)
     cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y) ->
     cond1#(true(),x,y) -> gr#(x,0())
     cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y) ->
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
     cond3#(true(),x,y) -> add#(x,y) ->
     add#(s(x),y) -> add#(x,y)
     cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y)) ->
     cond1#(true(),x,y) -> gr#(x,0())
     cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y)) ->
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
     add#(s(x),y) -> add#(x,y) ->
     add#(s(x),y) -> add#(x,y)
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(true(),x,y) -> p#(y)
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(true(),x,y) -> add#(x,y)
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(true(),x,y) -> gr#(add(x,y),0())
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(false(),x,y) -> add#(x,y)
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(false(),x,y) -> gr#(add(x,y),0())
     cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y) ->
     cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
     cond2#(true(),x,y) -> add#(x,y) ->
     add#(s(x),y) -> add#(x,y)
     cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y) ->
     cond1#(true(),x,y) -> gr#(x,0())
     cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y) ->
     cond1#(true(),x,y) -> cond2#(gr(x,0()),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,0()),x,y) ->
     cond2#(true(),x,y) -> p#(x)
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y) ->
     cond2#(true(),x,y) -> add#(x,y)
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y) ->
     cond2#(true(),x,y) -> gr#(add(x,y),0())
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y) ->
     cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y) ->
     cond2#(false(),x,y) -> gr#(y,0())
     cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y) -> cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
    Restore Modifier:
     DPs:
      cond1#(true(),x,y) -> gr#(x,0())
      cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
      cond2#(true(),x,y) -> p#(x)
      cond2#(true(),x,y) -> add#(x,y)
      cond2#(true(),x,y) -> gr#(add(x,y),0())
      cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
      cond2#(false(),x,y) -> gr#(y,0())
      cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
      cond3#(true(),x,y) -> p#(y)
      cond3#(true(),x,y) -> add#(x,y)
      cond3#(true(),x,y) -> gr#(add(x,y),0())
      cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
      cond3#(false(),x,y) -> add#(x,y)
      cond3#(false(),x,y) -> gr#(add(x,y),0())
      cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
      gr#(s(x),s(y)) -> gr#(x,y)
      add#(s(x),y) -> add#(x,y)
     TRS:
      cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
      cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
      cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
      cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
      cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      add(0(),x) -> x
      add(s(x),y) -> s(add(x,y))
      p(0()) -> 0()
      p(s(x)) -> x
     SCC Processor:
      #sccs: 3
      #rules: 7
      #arcs: 24/289
      DPs:
       gr#(s(x),s(y)) -> gr#(x,y)
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
       cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
       cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
       cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
       cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       add(0(),x) -> x
       add(s(x),y) -> s(add(x,y))
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gr#](x0, x1) = x1 + 1,
        
        [s](x0) = x0 + 1,
        
        [cond3](x0, x1, x2) = 0,
        
        [false] = 0,
        
        [p](x0) = x0,
        
        [add](x0, x1) = x0 + x1 + 1,
        
        [cond2](x0, x1, x2) = 0,
        
        [gr](x0, x1) = 0,
        
        [0] = 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,0()),x,y)
        
        cond2(true(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),p(x),y)
        
        cond2(false(),x,y) = 0 >= 0 = cond3(gr(y,0()),x,y)
        
        cond3(true(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),x,p(y))
        
        cond3(false(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),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)
        
        add(0(),x) = x + 1 >= x = x
        
        add(s(x),y) = x + y + 2 >= x + y + 2 = s(add(x,y))
        
        p(0()) = 0 >= 0 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
         cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
         cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
         cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
         cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         add(0(),x) -> x
         add(s(x),y) -> s(add(x,y))
         p(0()) -> 0()
         p(s(x)) -> x
       Qed
      
      DPs:
       cond3#(false(),x,y) -> cond1#(gr(add(x,y),0()),x,y)
       cond1#(true(),x,y) -> cond2#(gr(x,0()),x,y)
       cond2#(false(),x,y) -> cond3#(gr(y,0()),x,y)
       cond3#(true(),x,y) -> cond1#(gr(add(x,y),0()),x,p(y))
       cond2#(true(),x,y) -> cond1#(gr(add(x,y),0()),p(x),y)
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
       cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
       cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
       cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
       cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       add(0(),x) -> x
       add(s(x),y) -> s(add(x,y))
       p(0()) -> 0()
       p(s(x)) -> x
      Open
      
      DPs:
       add#(s(x),y) -> add#(x,y)
      TRS:
       cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
       cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
       cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
       cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
       cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       add(0(),x) -> x
       add(s(x),y) -> s(add(x,y))
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [add#](x0, x1) = x0 + 1,
        
        [s](x0) = x0 + 1,
        
        [cond3](x0, x1, x2) = 0,
        
        [false] = 0,
        
        [p](x0) = x0,
        
        [add](x0, x1) = x0 + x1 + 1,
        
        [cond2](x0, x1, x2) = 0,
        
        [gr](x0, x1) = 0,
        
        [0] = 0,
        
        [cond1](x0, x1, x2) = 0,
        
        [true] = 0
       orientation:
        add#(s(x),y) = x + 2 >= x + 1 = add#(x,y)
        
        cond1(true(),x,y) = 0 >= 0 = cond2(gr(x,0()),x,y)
        
        cond2(true(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),p(x),y)
        
        cond2(false(),x,y) = 0 >= 0 = cond3(gr(y,0()),x,y)
        
        cond3(true(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),x,p(y))
        
        cond3(false(),x,y) = 0 >= 0 = cond1(gr(add(x,y),0()),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)
        
        add(0(),x) = x + 1 >= x = x
        
        add(s(x),y) = x + y + 2 >= x + y + 2 = s(add(x,y))
        
        p(0()) = 0 >= 0 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x,y) -> cond2(gr(x,0()),x,y)
         cond2(true(),x,y) -> cond1(gr(add(x,y),0()),p(x),y)
         cond2(false(),x,y) -> cond3(gr(y,0()),x,y)
         cond3(true(),x,y) -> cond1(gr(add(x,y),0()),x,p(y))
         cond3(false(),x,y) -> cond1(gr(add(x,y),0()),x,y)
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         add(0(),x) -> x
         add(s(x),y) -> s(add(x,y))
         p(0()) -> 0()
         p(s(x)) -> x
       Qed