WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          walk#1(x){x -> Node(x,y)} =
            walk#1(Node(x,y)) ->^+ comp_f_g(walk#1(x),walk#1(y))
              = C[walk#1(x) = walk#1(x){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(Cons) = {2},
          uargs(comp_f_g) = {1,2},
          uargs(comp_f_g#1) = {1,2,3}
        
        Following symbols are considered usable:
          {comp_f_g#1,main,walk#1}
        TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [4]                  
                p(Leaf) = [2]                           
                 p(Nil) = [8]                           
                p(Node) = [1] x1 + [1] x2 + [2]         
            p(comp_f_g) = [1] x1 + [1] x2 + [0]         
          p(comp_f_g#1) = [4] x1 + [4] x2 + [1] x3 + [0]
              p(cons_x) = [3]                           
                p(main) = [8] x1 + [8]                  
              p(walk#1) = [2] x1 + [0]                  
        
        Following rules are strictly oriented:
        comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) = [1] x4 + [4] x7 + [4] x9 + [12]        
                                                  > [1] x4 + [4] x7 + [4] x9 + [4]         
                                                  = comp_f_g#1(x7,x9,Cons(x2,x4))          
        
        comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) = [1] x3 + [4] x5 + [4] x7 + [12]        
                                                  > [1] x3 + [4] x5 + [4] x7 + [4]         
                                                  = Cons(x2,comp_f_g#1(x5,x7,x3))          
        
             comp_f_g#1(cons_x(x5),cons_x(x2),x4) = [1] x4 + [24]                          
                                                  > [1] x4 + [8]                           
                                                  = Cons(x5,Cons(x2,x4))                   
        
                                   main(Leaf(x4)) = [24]                                   
                                                  > [12]                                   
                                                  = Cons(x4,Nil())                         
        
                                main(Node(x9,x5)) = [8] x5 + [8] x9 + [24]                 
                                                  > [8] x5 + [8] x9 + [8]                  
                                                  = comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
                                 walk#1(Leaf(x2)) = [4]                                    
                                                  > [3]                                    
                                                  = cons_x(x2)                             
        
                              walk#1(Node(x5,x3)) = [2] x3 + [2] x5 + [4]                  
                                                  > [2] x3 + [2] x5 + [0]                  
                                                  = comp_f_g(walk#1(x5),walk#1(x3))        
        
        
        Following rules are (at-least) weakly oriented:
        comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) =  [1] x1 + [4] x2 + [4] x3 + [4] x4 + [4] x5 + [0]
                                                       >= [1] x1 + [4] x2 + [4] x3 + [4] x4 + [4] x5 + [0]
                                                       =  comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))          
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
        - Weak TRS:
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(Cons) = {2},
          uargs(comp_f_g) = {1,2},
          uargs(comp_f_g#1) = {1,2,3}
        
        Following symbols are considered usable:
          {comp_f_g#1,main,walk#1}
        TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [0]                  
                p(Leaf) = [1]                           
                 p(Nil) = [15]                          
                p(Node) = [1] x1 + [1] x2 + [2]         
            p(comp_f_g) = [1] x1 + [1] x2 + [4]         
          p(comp_f_g#1) = [1] x1 + [1] x2 + [1] x3 + [0]
              p(cons_x) = [0]                           
                p(main) = [14] x1 + [2]                 
              p(walk#1) = [4] x1 + [0]                  
        
        Following rules are strictly oriented:
        comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [8]
                                                       > [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0]
                                                       = comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))          
        
        
        Following rules are (at-least) weakly oriented:
        comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) =  [1] x4 + [1] x7 + [1] x9 + [4]         
                                                  >= [1] x4 + [1] x7 + [1] x9 + [0]         
                                                  =  comp_f_g#1(x7,x9,Cons(x2,x4))          
        
        comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) =  [1] x3 + [1] x5 + [1] x7 + [4]         
                                                  >= [1] x3 + [1] x5 + [1] x7 + [0]         
                                                  =  Cons(x2,comp_f_g#1(x5,x7,x3))          
        
             comp_f_g#1(cons_x(x5),cons_x(x2),x4) =  [1] x4 + [0]                           
                                                  >= [1] x4 + [0]                           
                                                  =  Cons(x5,Cons(x2,x4))                   
        
                                   main(Leaf(x4)) =  [16]                                   
                                                  >= [15]                                   
                                                  =  Cons(x4,Nil())                         
        
                                main(Node(x9,x5)) =  [14] x5 + [14] x9 + [30]               
                                                  >= [4] x5 + [4] x9 + [15]                 
                                                  =  comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
                                 walk#1(Leaf(x2)) =  [4]                                    
                                                  >= [0]                                    
                                                  =  cons_x(x2)                             
        
                              walk#1(Node(x5,x3)) =  [4] x3 + [4] x5 + [8]                  
                                                  >= [4] x3 + [4] x5 + [4]                  
                                                  =  comp_f_g(walk#1(x5),walk#1(x3))        
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1))
            comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4))
            comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3))
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - Signature:
            {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil
            ,Node,comp_f_g,cons_x}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))