WORST_CASE(?,O(n^1))
* Step 1: NaturalPI 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:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        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) = x2          
                p(Leaf) = 0           
                 p(Nil) = 0           
                p(Node) = 2 + x1 + x2 
            p(comp_f_g) = 2 + x1 + x2 
          p(comp_f_g#1) = x1 + x2 + x3
              p(cons_x) = 0           
                p(main) = x1          
              p(walk#1) = x1          
        
        Following rules are strictly oriented:
        comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) = 4 + x1 + x2 + x3 + x4 + x5             
                                                       > x1 + x2 + x3 + x4 + x5                 
                                                       = 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) = 2 + x4 + x7 + x9                       
                                                       > x4 + x7 + x9                           
                                                       = comp_f_g#1(x7,x9,Cons(x2,x4))          
        
             comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) = 2 + x3 + x5 + x7                       
                                                       > x3 + x5 + x7                           
                                                       = Cons(x2,comp_f_g#1(x5,x7,x3))          
        
                                     main(Node(x9,x5)) = 2 + x5 + x9                            
                                                       > x5 + x9                                
                                                       = comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
        
        Following rules are (at-least) weakly oriented:
        comp_f_g#1(cons_x(x5),cons_x(x2),x4) =  x4                             
                                             >= x4                             
                                             =  Cons(x5,Cons(x2,x4))           
        
                              main(Leaf(x4)) =  0                              
                                             >= 0                              
                                             =  Cons(x4,Nil())                 
        
                            walk#1(Leaf(x2)) =  0                              
                                             >= 0                              
                                             =  cons_x(x2)                     
        
                         walk#1(Node(x5,x3)) =  2 + x3 + x5                    
                                             >= 2 + x3 + x5                    
                                             =  comp_f_g(walk#1(x5),walk#1(x3))
        
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            main(Leaf(x4)) -> Cons(x4,Nil())
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - 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))
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        - 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:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        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) = x2          
                p(Leaf) = 2           
                 p(Nil) = 0           
                p(Node) = 0           
            p(comp_f_g) = x1 + x2     
          p(comp_f_g#1) = x1 + x2 + x3
              p(cons_x) = 0           
                p(main) = 2           
              p(walk#1) = 0           
        
        Following rules are strictly oriented:
        main(Leaf(x4)) = 2             
                       > 0             
                       = Cons(x4,Nil())
        
        
        Following rules are (at-least) weakly oriented:
        comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) =  x1 + x2 + x3 + x4 + x5                 
                                                       >= x1 + x2 + x3 + x4 + x5                 
                                                       =  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) =  x4 + x7 + x9                           
                                                       >= x4 + x7 + x9                           
                                                       =  comp_f_g#1(x7,x9,Cons(x2,x4))          
        
             comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) =  x3 + x5 + x7                           
                                                       >= x3 + x5 + x7                           
                                                       =  Cons(x2,comp_f_g#1(x5,x7,x3))          
        
                  comp_f_g#1(cons_x(x5),cons_x(x2),x4) =  x4                                     
                                                       >= x4                                     
                                                       =  Cons(x5,Cons(x2,x4))                   
        
                                     main(Node(x9,x5)) =  2                                      
                                                       >= 0                                      
                                                       =  comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
                                      walk#1(Leaf(x2)) =  0                                      
                                                       >= 0                                      
                                                       =  cons_x(x2)                             
        
                                   walk#1(Node(x5,x3)) =  0                                      
                                                       >= 0                                      
                                                       =  comp_f_g(walk#1(x5),walk#1(x3))        
        
* Step 3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
            walk#1(Leaf(x2)) -> cons_x(x2)
            walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3))
        - 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))
            main(Leaf(x4)) -> Cons(x4,Nil())
            main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        - 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:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        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) = x2          
                p(Leaf) = 1 + x1      
                 p(Nil) = 0           
                p(Node) = 2 + x1 + x2 
            p(comp_f_g) = x1 + x2     
          p(comp_f_g#1) = x1 + x2 + x3
              p(cons_x) = 0           
                p(main) = 2 + 2*x1    
              p(walk#1) = 1 + 2*x1    
        
        Following rules are strictly oriented:
           walk#1(Leaf(x2)) = 3 + 2*x2                       
                            > 0                              
                            = cons_x(x2)                     
        
        walk#1(Node(x5,x3)) = 5 + 2*x3 + 2*x5                
                            > 2 + 2*x3 + 2*x5                
                            = 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) =  x1 + x2 + x3 + x4 + x5                 
                                                       >= x1 + x2 + x3 + x4 + x5                 
                                                       =  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) =  x4 + x7 + x9                           
                                                       >= x4 + x7 + x9                           
                                                       =  comp_f_g#1(x7,x9,Cons(x2,x4))          
        
             comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) =  x3 + x5 + x7                           
                                                       >= x3 + x5 + x7                           
                                                       =  Cons(x2,comp_f_g#1(x5,x7,x3))          
        
                  comp_f_g#1(cons_x(x5),cons_x(x2),x4) =  x4                                     
                                                       >= x4                                     
                                                       =  Cons(x5,Cons(x2,x4))                   
        
                                        main(Leaf(x4)) =  4 + 2*x4                               
                                                       >= 0                                      
                                                       =  Cons(x4,Nil())                         
        
                                     main(Node(x9,x5)) =  6 + 2*x5 + 2*x9                        
                                                       >= 2 + 2*x5 + 2*x9                        
                                                       =  comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
* Step 4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4))
        - 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))
            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:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        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) = x2          
                p(Leaf) = 2           
                 p(Nil) = 0           
                p(Node) = 1 + x1 + x2 
            p(comp_f_g) = 1 + x1 + x2 
          p(comp_f_g#1) = x1 + x2 + x3
              p(cons_x) = 2           
                p(main) = 2 + 2*x1    
              p(walk#1) = x1          
        
        Following rules are strictly oriented:
        comp_f_g#1(cons_x(x5),cons_x(x2),x4) = 4 + x4              
                                             > x4                  
                                             = Cons(x5,Cons(x2,x4))
        
        
        Following rules are (at-least) weakly oriented:
        comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) =  2 + x1 + x2 + x3 + x4 + x5             
                                                       >= x1 + x2 + x3 + x4 + x5                 
                                                       =  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) =  3 + x4 + x7 + x9                       
                                                       >= x4 + x7 + x9                           
                                                       =  comp_f_g#1(x7,x9,Cons(x2,x4))          
        
             comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) =  3 + x3 + x5 + x7                       
                                                       >= x3 + x5 + x7                           
                                                       =  Cons(x2,comp_f_g#1(x5,x7,x3))          
        
                                        main(Leaf(x4)) =  6                                      
                                                       >= 0                                      
                                                       =  Cons(x4,Nil())                         
        
                                     main(Node(x9,x5)) =  4 + 2*x5 + 2*x9                        
                                                       >= x5 + x9                                
                                                       =  comp_f_g#1(walk#1(x9),walk#1(x5),Nil())
        
                                      walk#1(Leaf(x2)) =  2                                      
                                                       >= 2                                      
                                                       =  cons_x(x2)                             
        
                                   walk#1(Node(x5,x3)) =  1 + x3 + x5                            
                                                       >= 1 + x3 + x5                            
                                                       =  comp_f_g(walk#1(x5),walk#1(x3))        
        
* Step 5: 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(?,O(n^1))