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))
  TDG 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))
   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) -> cond#(gr(x,y),x,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) -> 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))
    KBO Processor:
     argument filtering:
      pi(true) = []
      pi(cond) = 1
      pi(gr) = [1]
      pi(add) = [0,1]
      pi(0) = []
      pi(false) = []
      pi(s) = [0]
      pi(add#) = 0
     weight function:
      w0 = 1
      w(add#) = w(s) = w(false) = w(0) = w(gr) = w(cond) = w(true) = 1
      w(add) = 0
     precedence:
      false ~ 0 ~ gr ~ cond ~ true > add > add# ~ s
     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))
    KBO Processor:
     argument filtering:
      pi(true) = []
      pi(cond) = 1
      pi(gr) = []
      pi(add) = [0,1]
      pi(0) = []
      pi(false) = []
      pi(s) = [0]
      pi(gr#) = 1
     weight function:
      w0 = 1
      w(gr#) = w(s) = w(false) = w(0) = w(add) = w(gr) = w(cond) = w(true) = 1
     precedence:
      gr# ~ add ~ gr ~ cond > s ~ false ~ 0 ~ true
     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