WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
            app_xs#1(Nil(),x6) -> x6
            main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
        - Signature:
            {app_xs#1/2,main/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil}
    + 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) = {2},
            uargs(app_xs#1) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [0]
                 p(Nil) = [0]         
            p(app_xs#1) = [1] x2 + [0]
                p(main) = [2] x2 + [1]
          
          Following rules are strictly oriented:
          main(x4,x2) = [2] x2 + [1]                
                      > [1] x2 + [0]                
                      = app_xs#1(x4,app_xs#1(x4,x2))
          
          
          Following rules are (at-least) weakly oriented:
          app_xs#1(Cons(x7,x8),x10) =  [1] x10 + [0]            
                                    >= [1] x10 + [0]            
                                    =  Cons(x7,app_xs#1(x8,x10))
          
                 app_xs#1(Nil(),x6) =  [1] x6 + [0]             
                                    >= [1] x6 + [0]             
                                    =  x6                       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
            app_xs#1(Nil(),x6) -> x6
        - Weak TRS:
            main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
        - Signature:
            {app_xs#1/2,main/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil}
    + 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) = {2},
            uargs(app_xs#1) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(Cons) = [1] x1 + [1] x2 + [0]
                 p(Nil) = [7]                  
            p(app_xs#1) = [1] x1 + [1] x2 + [0]
                p(main) = [2] x1 + [2] x2 + [0]
          
          Following rules are strictly oriented:
          app_xs#1(Nil(),x6) = [1] x6 + [7]
                             > [1] x6 + [0]
                             = x6          
          
          
          Following rules are (at-least) weakly oriented:
          app_xs#1(Cons(x7,x8),x10) =  [1] x10 + [1] x7 + [1] x8 + [0]
                                    >= [1] x10 + [1] x7 + [1] x8 + [0]
                                    =  Cons(x7,app_xs#1(x8,x10))      
          
                        main(x4,x2) =  [2] x2 + [2] x4 + [0]          
                                    >= [1] x2 + [2] x4 + [0]          
                                    =  app_xs#1(x4,app_xs#1(x4,x2))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
        - Weak TRS:
            app_xs#1(Nil(),x6) -> x6
            main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
        - Signature:
            {app_xs#1/2,main/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil}
    + 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) = {2},
            uargs(app_xs#1) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [1]           
                 p(Nil) = [2]                    
            p(app_xs#1) = [6] x1 + [1] x2 + [7]  
                p(main) = [12] x1 + [2] x2 + [14]
          
          Following rules are strictly oriented:
          app_xs#1(Cons(x7,x8),x10) = [1] x10 + [6] x8 + [13]  
                                    > [1] x10 + [6] x8 + [8]   
                                    = Cons(x7,app_xs#1(x8,x10))
          
          
          Following rules are (at-least) weakly oriented:
          app_xs#1(Nil(),x6) =  [1] x6 + [19]               
                             >= [1] x6 + [0]                
                             =  x6                          
          
                 main(x4,x2) =  [2] x2 + [12] x4 + [14]     
                             >= [1] x2 + [12] x4 + [14]     
                             =  app_xs#1(x4,app_xs#1(x4,x2))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10))
            app_xs#1(Nil(),x6) -> x6
            main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2))
        - Signature:
            {app_xs#1/2,main/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))