WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(Cons) = {1,2},
            uargs(S) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [1]                  
                p(Cons) = [1] x1 + [1] x2 + [6]
                 p(Nil) = [10]                 
                   p(S) = [1] x1 + [1]         
                p(main) = [2] x1 + [13]        
               p(map#2) = [1] x1 + [2] x2 + [5]
              p(plus_x) = [4]                  
            p(plus_x#1) = [2] x2 + [4]         
          
          Following rules are strictly oriented:
                           main(x5,x12) = [2] x5 + [13]                             
                                        > [2] x5 + [9]                              
                                        = map#2(plus_x(x12),x5)                     
          
                map#2(plus_x(x2),Nil()) = [29]                                      
                                        > [10]                                      
                                        = Nil()                                     
          
          map#2(plus_x(x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [21]                    
                                        > [2] x2 + [2] x4 + [19]                    
                                        = Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
          
                       plus_x#1(0(),x8) = [2] x8 + [4]                              
                                        > [1] x8 + [0]                              
                                        = x8                                        
          
          
          Following rules are (at-least) weakly oriented:
          plus_x#1(S(x12),x14) =  [2] x14 + [4]       
                               >= [2] x14 + [5]       
                               =  S(plus_x#1(x12,x14))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
        - Signature:
            {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S
            ,plus_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) = {1,2},
          uargs(S) = {1}
        
        Following symbols are considered usable:
          {main,map#2,plus_x#1}
        TcT has computed the following interpretation:
                 p(0) = 0                        
              p(Cons) = 2 + x1 + x2              
               p(Nil) = 3                        
                 p(S) = 2 + x1                   
              p(main) = 2 + 2*x1 + 5*x1*x2 + x2^2
             p(map#2) = 5*x1*x2 + x1^2 + 2*x2    
            p(plus_x) = x1                       
          p(plus_x#1) = 4*x1 + x1*x2 + 2*x2      
        
        Following rules are strictly oriented:
        plus_x#1(S(x12),x14) = 8 + 4*x12 + x12*x14 + 4*x14
                             > 2 + 4*x12 + x12*x14 + 2*x14
                             = S(plus_x#1(x12,x14))       
        
        
        Following rules are (at-least) weakly oriented:
                         main(x5,x12) =  2 + 5*x12*x5 + x12^2 + 2*x5                       
                                      >= 5*x12*x5 + x12^2 + 2*x5                           
                                      =  map#2(plus_x(x12),x5)                             
        
              map#2(plus_x(x2),Nil()) =  6 + 15*x2 + x2^2                                  
                                      >= 3                                                 
                                      =  Nil()                                             
        
        map#2(plus_x(x6),Cons(x4,x2)) =  4 + 2*x2 + 5*x2*x6 + 2*x4 + 5*x4*x6 + 10*x6 + x6^2
                                      >= 2 + 2*x2 + 5*x2*x6 + 2*x4 + x4*x6 + 4*x6 + x6^2   
                                      =  Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))        
        
                     plus_x#1(0(),x8) =  2*x8                                              
                                      >= x8                                                
                                      =  x8                                                
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))