MAYBE

Problem:
 cond(true(),x,y) -> cond(gr(x,y),x,add(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))

Proof:
 DP Processor:
  DPs:
   cond#(true(),x,y) -> add#(x,y)
   cond#(true(),x,y) -> gr#(x,y)
   cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
   gr#(s(x),s(y)) -> gr#(x,y)
   add#(s(x),y) -> add#(x,y)
  TRS:
   cond(true(),x,y) -> cond(gr(x,y),x,add(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))
  Usable Rule Processor:
   DPs:
    cond#(true(),x,y) -> add#(x,y)
    cond#(true(),x,y) -> gr#(x,y)
    cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
    gr#(s(x),s(y)) -> gr#(x,y)
    add#(s(x),y) -> add#(x,y)
   TRS:
    f10(x,y) -> x
    f10(x,y) -> y
    add(0(),x) -> x
    add(s(x),y) -> s(add(x,y))
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
   ADG Processor:
    DPs:
     cond#(true(),x,y) -> add#(x,y)
     cond#(true(),x,y) -> gr#(x,y)
     cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
     gr#(s(x),s(y)) -> gr#(x,y)
     add#(s(x),y) -> add#(x,y)
    TRS:
     f10(x,y) -> x
     f10(x,y) -> y
     add(0(),x) -> x
     add(s(x),y) -> s(add(x,y))
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
    graph:
     gr#(s(x),s(y)) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
     add#(s(x),y) -> add#(x,y) -> add#(s(x),y) -> add#(x,y)
     cond#(true(),x,y) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
     cond#(true(),x,y) -> add#(x,y) ->
     add#(s(x),y) -> add#(x,y)
     cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y)) ->
     cond#(true(),x,y) -> add#(x,y)
     cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y)) ->
     cond#(true(),x,y) -> gr#(x,y)
     cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y)) ->
     cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
    Restore Modifier:
     DPs:
      cond#(true(),x,y) -> add#(x,y)
      cond#(true(),x,y) -> gr#(x,y)
      cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
      gr#(s(x),s(y)) -> gr#(x,y)
      add#(s(x),y) -> add#(x,y)
     TRS:
      cond(true(),x,y) -> cond(gr(x,y),x,add(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))
     SCC Processor:
      #sccs: 3
      #rules: 3
      #arcs: 7/25
      DPs:
       cond#(true(),x,y) -> cond#(gr(x,y),x,add(x,y))
      TRS:
       cond(true(),x,y) -> cond(gr(x,y),x,add(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))
      Open
      
      DPs:
       add#(s(x),y) -> add#(x,y)
      TRS:
       cond(true(),x,y) -> cond(gr(x,y),x,add(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))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [add#](x0, x1) = x0,
        
        [s](x0) = x0 + 1,
        
        [false] = 0,
        
        [0] = 0,
        
        [add](x0, x1) = x0 + x1 + 1,
        
        [gr](x0, x1) = 0,
        
        [cond](x0, x1, x2) = x1,
        
        [true] = 0
       orientation:
        add#(s(x),y) = x + 1 >= x = add#(x,y)
        
        cond(true(),x,y) = x >= x = cond(gr(x,y),x,add(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))
       problem:
        DPs:
         
        TRS:
         cond(true(),x,y) -> cond(gr(x,y),x,add(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))
       Qed
      
      DPs:
       gr#(s(x),s(y)) -> gr#(x,y)
      TRS:
       cond(true(),x,y) -> cond(gr(x,y),x,add(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))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gr#](x0, x1) = x0 + x1,
        
        [s](x0) = x0 + 1,
        
        [false] = 0,
        
        [0] = 0,
        
        [add](x0, x1) = x0 + x1 + 1,
        
        [gr](x0, x1) = 0,
        
        [cond](x0, x1, x2) = 0,
        
        [true] = 0
       orientation:
        gr#(s(x),s(y)) = x + y + 2 >= x + y = gr#(x,y)
        
        cond(true(),x,y) = 0 >= 0 = cond(gr(x,y),x,add(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))
       problem:
        DPs:
         
        TRS:
         cond(true(),x,y) -> cond(gr(x,y),x,add(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))
       Qed