MAYBE

Problem:
 +(X,0()) -> X
 +(X,s(Y)) -> s(+(X,Y))
 f(0(),s(0()),X) -> f(X,+(X,X),X)
 g(X,Y) -> X
 g(X,Y) -> Y

Proof:
 DP Processor:
  DPs:
   +#(X,s(Y)) -> +#(X,Y)
   f#(0(),s(0()),X) -> +#(X,X)
   f#(0(),s(0()),X) -> f#(X,+(X,X),X)
  TRS:
   +(X,0()) -> X
   +(X,s(Y)) -> s(+(X,Y))
   f(0(),s(0()),X) -> f(X,+(X,X),X)
   g(X,Y) -> X
   g(X,Y) -> Y
  Usable Rule Processor:
   DPs:
    +#(X,s(Y)) -> +#(X,Y)
    f#(0(),s(0()),X) -> +#(X,X)
    f#(0(),s(0()),X) -> f#(X,+(X,X),X)
   TRS:
    +(X,0()) -> X
    +(X,s(Y)) -> s(+(X,Y))
   EDG Processor:
    DPs:
     +#(X,s(Y)) -> +#(X,Y)
     f#(0(),s(0()),X) -> +#(X,X)
     f#(0(),s(0()),X) -> f#(X,+(X,X),X)
    TRS:
     +(X,0()) -> X
     +(X,s(Y)) -> s(+(X,Y))
    graph:
     f#(0(),s(0()),X) -> f#(X,+(X,X),X) ->
     f#(0(),s(0()),X) -> +#(X,X)
     f#(0(),s(0()),X) -> f#(X,+(X,X),X) ->
     f#(0(),s(0()),X) -> f#(X,+(X,X),X)
     f#(0(),s(0()),X) -> +#(X,X) -> +#(X,s(Y)) -> +#(X,Y)
     +#(X,s(Y)) -> +#(X,Y) -> +#(X,s(Y)) -> +#(X,Y)
    Restore Modifier:
     DPs:
      +#(X,s(Y)) -> +#(X,Y)
      f#(0(),s(0()),X) -> +#(X,X)
      f#(0(),s(0()),X) -> f#(X,+(X,X),X)
     TRS:
      +(X,0()) -> X
      +(X,s(Y)) -> s(+(X,Y))
      f(0(),s(0()),X) -> f(X,+(X,X),X)
      g(X,Y) -> X
      g(X,Y) -> Y
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 4/9
      DPs:
       f#(0(),s(0()),X) -> f#(X,+(X,X),X)
      TRS:
       +(X,0()) -> X
       +(X,s(Y)) -> s(+(X,Y))
       f(0(),s(0()),X) -> f(X,+(X,X),X)
       g(X,Y) -> X
       g(X,Y) -> Y
      Open
      
      DPs:
       +#(X,s(Y)) -> +#(X,Y)
      TRS:
       +(X,0()) -> X
       +(X,s(Y)) -> s(+(X,Y))
       f(0(),s(0()),X) -> f(X,+(X,X),X)
       g(X,Y) -> X
       g(X,Y) -> Y
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [+#](x0, x1) = x1 + 1,
        
        [g](x0, x1) = x0 + x1,
        
        [f](x0, x1, x2) = 0,
        
        [s](x0) = x0 + 1,
        
        [+](x0, x1) = x0 + x1 + 1,
        
        [0] = 0
       orientation:
        +#(X,s(Y)) = Y + 2 >= Y + 1 = +#(X,Y)
        
        +(X,0()) = X + 1 >= X = X
        
        +(X,s(Y)) = X + Y + 2 >= X + Y + 2 = s(+(X,Y))
        
        f(0(),s(0()),X) = 0 >= 0 = f(X,+(X,X),X)
        
        g(X,Y) = X + Y >= X = X
        
        g(X,Y) = X + Y >= Y = Y
       problem:
        DPs:
         
        TRS:
         +(X,0()) -> X
         +(X,s(Y)) -> s(+(X,Y))
         f(0(),s(0()),X) -> f(X,+(X,X),X)
         g(X,Y) -> X
         g(X,Y) -> Y
       Qed