MAYBE

Problem:
 f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
 f(x,f(y,z)) -> f(f(x,y),z)

Proof:
 DP Processor:
  DPs:
   f#(f(a(),a()),x) -> f#(a(),x)
   f#(f(a(),a()),x) -> f#(b(),f(a(),x))
   f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
   f#(x,f(y,z)) -> f#(x,y)
   f#(x,f(y,z)) -> f#(f(x,y),z)
  TRS:
   f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
   f(x,f(y,z)) -> f(f(x,y),z)
  ADG Processor:
   DPs:
    f#(f(a(),a()),x) -> f#(a(),x)
    f#(f(a(),a()),x) -> f#(b(),f(a(),x))
    f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
    f#(x,f(y,z)) -> f#(x,y)
    f#(x,f(y,z)) -> f#(f(x,y),z)
   TRS:
    f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
    f(x,f(y,z)) -> f(f(x,y),z)
   graph:
    f#(f(a(),a()),x) -> f#(b(),f(a(),x)) ->
    f#(x,f(y,z)) -> f#(x,y)
    f#(f(a(),a()),x) -> f#(b(),f(a(),x)) ->
    f#(x,f(y,z)) -> f#(f(x,y),z)
    f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x))) ->
    f#(x,f(y,z)) -> f#(x,y)
    f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x))) ->
    f#(x,f(y,z)) -> f#(f(x,y),z)
    f#(f(a(),a()),x) -> f#(a(),x) -> f#(x,f(y,z)) -> f#(x,y)
    f#(f(a(),a()),x) -> f#(a(),x) -> f#(x,f(y,z)) -> f#(f(x,y),z)
    f#(x,f(y,z)) -> f#(f(x,y),z) -> f#(f(a(),a()),x) -> f#(a(),x)
    f#(x,f(y,z)) -> f#(f(x,y),z) ->
    f#(f(a(),a()),x) -> f#(b(),f(a(),x))
    f#(x,f(y,z)) -> f#(f(x,y),z) ->
    f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
    f#(x,f(y,z)) -> f#(f(x,y),z) -> f#(x,f(y,z)) -> f#(x,y)
    f#(x,f(y,z)) -> f#(f(x,y),z) -> f#(x,f(y,z)) -> f#(f(x,y),z)
    f#(x,f(y,z)) -> f#(x,y) -> f#(f(a(),a()),x) -> f#(a(),x)
    f#(x,f(y,z)) -> f#(x,y) -> f#(f(a(),a()),x) -> f#(b(),f(a(),x))
    f#(x,f(y,z)) -> f#(x,y) -> f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
    f#(x,f(y,z)) -> f#(x,y) -> f#(x,f(y,z)) -> f#(x,y)
    f#(x,f(y,z)) -> f#(x,y) -> f#(x,f(y,z)) -> f#(f(x,y),z)
   Restore Modifier:
    DPs:
     f#(f(a(),a()),x) -> f#(a(),x)
     f#(f(a(),a()),x) -> f#(b(),f(a(),x))
     f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
     f#(x,f(y,z)) -> f#(x,y)
     f#(x,f(y,z)) -> f#(f(x,y),z)
    TRS:
     f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
     f(x,f(y,z)) -> f(f(x,y),z)
    SCC Processor:
     #sccs: 1
     #rules: 5
     #arcs: 16/25
     DPs:
      f#(f(a(),a()),x) -> f#(b(),f(a(),x))
      f#(x,f(y,z)) -> f#(f(x,y),z)
      f#(x,f(y,z)) -> f#(x,y)
      f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
      f#(f(a(),a()),x) -> f#(a(),x)
     TRS:
      f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
      f(x,f(y,z)) -> f(f(x,y),z)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0, x1) = x0,
       
       [b] = 0,
       
       [f](x0, x1) = x0,
       
       [a] = 1
      orientation:
       f#(f(a(),a()),x) = 1 >= 0 = f#(b(),f(a(),x))
       
       f#(x,f(y,z)) = x >= x = f#(f(x,y),z)
       
       f#(x,f(y,z)) = x >= x = f#(x,y)
       
       f#(f(a(),a()),x) = 1 >= 1 = f#(a(),f(b(),f(a(),x)))
       
       f#(f(a(),a()),x) = 1 >= 1 = f#(a(),x)
       
       f(f(a(),a()),x) = 1 >= 1 = f(a(),f(b(),f(a(),x)))
       
       f(x,f(y,z)) = x >= x = f(f(x,y),z)
      problem:
       DPs:
        f#(x,f(y,z)) -> f#(f(x,y),z)
        f#(x,f(y,z)) -> f#(x,y)
        f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
        f#(f(a(),a()),x) -> f#(a(),x)
       TRS:
        f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
        f(x,f(y,z)) -> f(f(x,y),z)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1) = x0 + x1 + 1,
        
        [b] = 0,
        
        [f](x0, x1) = x0 + x1,
        
        [a] = 1
       orientation:
        f#(x,f(y,z)) = x + y + z + 1 >= x + y + z + 1 = f#(f(x,y),z)
        
        f#(x,f(y,z)) = x + y + z + 1 >= x + y + 1 = f#(x,y)
        
        f#(f(a(),a()),x) = x + 3 >= x + 3 = f#(a(),f(b(),f(a(),x)))
        
        f#(f(a(),a()),x) = x + 3 >= x + 2 = f#(a(),x)
        
        f(f(a(),a()),x) = x + 2 >= x + 2 = f(a(),f(b(),f(a(),x)))
        
        f(x,f(y,z)) = x + y + z >= x + y + z = f(f(x,y),z)
       problem:
        DPs:
         f#(x,f(y,z)) -> f#(f(x,y),z)
         f#(x,f(y,z)) -> f#(x,y)
         f#(f(a(),a()),x) -> f#(a(),f(b(),f(a(),x)))
        TRS:
         f(f(a(),a()),x) -> f(a(),f(b(),f(a(),x)))
         f(x,f(y,z)) -> f(f(x,y),z)
       Open