MAYBE

Problem:
 app(f(),app(f(),x)) -> app(g(),app(f(),x))
 app(g(),app(g(),x)) -> app(f(),x)
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 DP Processor:
  DPs:
   app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
   app#(g(),app(g(),x)) -> app#(f(),x)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
   app#(app(map(),fun),app(app(cons(),x),xs)) ->
   app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) ->
   app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
  TRS:
   app(f(),app(f(),x)) -> app(g(),app(f(),x))
   app(g(),app(g(),x)) -> app(f(),x)
   app(app(map(),fun),nil()) -> nil()
   app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app(app(filter(),fun),nil()) -> nil()
   app(app(filter(),fun),app(app(cons(),x),xs)) ->
   app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
   app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
  EDG Processor:
   DPs:
    app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
    app#(g(),app(g(),x)) -> app#(f(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
   TRS:
    app(f(),app(f(),x)) -> app(g(),app(f(),x))
    app(g(),app(g(),x)) -> app(f(),x)
    app(app(map(),fun),nil()) -> nil()
    app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app(app(filter(),fun),nil()) -> nil()
    app(app(filter(),fun),app(app(cons(),x),xs)) ->
    app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
    app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
   graph:
    app#(g(),app(g(),x)) -> app#(f(),x) ->
    app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(g(),app(g(),x)) -> app#(f(),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(g(),app(g(),x)) -> app#(f(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(f(),app(f(),x)) -> app#(g(),app(f(),x)) -> app#(g(),app(g(),x)) -> app#(f(),x)
   SCC Processor:
    #sccs: 2
    #rules: 7
    #arcs: 50/289
    DPs:
     app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
     app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
     app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
     app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
     app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    TRS:
     app(f(),app(f(),x)) -> app(g(),app(f(),x))
     app(g(),app(g(),x)) -> app(f(),x)
     app(app(map(),fun),nil()) -> nil()
     app(app(map(),fun),app(app(cons(),x),xs)) ->
     app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
     app(app(filter(),fun),nil()) -> nil()
     app(app(filter(),fun),app(app(cons(),x),xs)) ->
     app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
     app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
     app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
    Open
    
    DPs:
     app#(g(),app(g(),x)) -> app#(f(),x)
     app#(f(),app(f(),x)) -> app#(g(),app(f(),x))
    TRS:
     app(f(),app(f(),x)) -> app(g(),app(f(),x))
     app(g(),app(g(),x)) -> app(f(),x)
     app(app(map(),fun),nil()) -> nil()
     app(app(map(),fun),app(app(cons(),x),xs)) ->
     app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
     app(app(filter(),fun),nil()) -> nil()
     app(app(filter(),fun),app(app(cons(),x),xs)) ->
     app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
     app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
     app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
    Matrix Interpretation Processor: dim=2
     
     interpretation:
      [app#](x0, x1) = [0 1]x0 + [0 2]x1,
      
                [0]
      [false] = [0],
      
               [0]
      [true] = [0],
      
                  [0]
      [filter2] = [0],
      
                 [0]
      [filter] = [0],
      
               [0]
      [cons] = [0],
      
              [0]
      [nil] = [0],
      
              [0]
      [map] = [0],
      
            [1]
      [g] = [2],
      
                      [1 0]     [0 0]  
      [app](x0, x1) = [1 0]x0 + [0 1]x1,
      
            [1]
      [f] = [3]
     orientation:
      app#(g(),app(g(),x)) = [0 2]x + [4] >= [0 2]x + [3] = app#(f(),x)
      
      app#(f(),app(f(),x)) = [0 2]x + [5] >= [0 2]x + [4] = app#(g(),app(f(),x))
      
                            [0 0]    [1]    [0 0]    [1]                      
      app(f(),app(f(),x)) = [0 1]x + [2] >= [0 1]x + [2] = app(g(),app(f(),x))
      
                            [0 0]    [1]    [0 0]    [1]             
      app(g(),app(g(),x)) = [0 1]x + [2] >= [0 1]x + [1] = app(f(),x)
      
                                  [0]    [0]        
      app(app(map(),fun),nil()) = [0] >= [0] = nil()
      
                                                  [0 0]      [0 0]                                                       
      app(app(map(),fun),app(app(cons(),x),xs)) = [0 1]xs >= [0 1]xs = app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
      
                                     [0]    [0]        
      app(app(filter(),fun),nil()) = [0] >= [0] = nil()
      
                                                     [0 0]      [0 0]                                                      
      app(app(filter(),fun),app(app(cons(),x),xs)) = [0 1]xs >= [0 1]xs = app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
      
                                                      [0 0]      [0 0]                                                 
      app(app(app(app(filter2(),true()),fun),x),xs) = [0 1]xs >= [0 1]xs = app(app(cons(),x),app(app(filter(),fun),xs))
      
                                                       [0 0]      [0 0]                              
      app(app(app(app(filter2(),false()),fun),x),xs) = [0 1]xs >= [0 1]xs = app(app(filter(),fun),xs)
     problem:
      DPs:
       
      TRS:
       app(f(),app(f(),x)) -> app(g(),app(f(),x))
       app(g(),app(g(),x)) -> app(f(),x)
       app(app(map(),fun),nil()) -> nil()
       app(app(map(),fun),app(app(cons(),x),xs)) ->
       app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
       app(app(filter(),fun),nil()) -> nil()
       app(app(filter(),fun),app(app(cons(),x),xs)) ->
       app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
       app(app(app(app(filter2(),true()),fun),x),xs) ->
       app(app(cons(),x),app(app(filter(),fun),xs))
       app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
     Qed