WORST_CASE(?,O(n^1))
* Step 1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold#3(Cons(x4,x2)) -> plus#2(x4,fold#3(x2))
            fold#3(Nil()) -> 0()
            main(x1) -> fold#3(x1)
            plus#2(0(),x12) -> x12
            plus#2(S(x4),x2) -> S(plus#2(x4,x2))
        - Signature:
            {fold#3/1,main/1,plus#2/2} / {0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#3,main,plus#2} and constructors {0,Cons,Nil,S}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(S) = {1},
          uargs(plus#2) = {2}
        
        Following symbols are considered usable:
          {fold#3,main,plus#2}
        TcT has computed the following interpretation:
               p(0) = 0     
            p(Cons) = 2     
             p(Nil) = 2     
               p(S) = x1    
          p(fold#3) = 0     
            p(main) = 2 + x1
          p(plus#2) = 4*x2  
        
        Following rules are strictly oriented:
        main(x1) = 2 + x1    
                 > 0         
                 = fold#3(x1)
        
        
        Following rules are (at-least) weakly oriented:
        fold#3(Cons(x4,x2)) =  0                    
                            >= 0                    
                            =  plus#2(x4,fold#3(x2))
        
              fold#3(Nil()) =  0                    
                            >= 0                    
                            =  0()                  
        
            plus#2(0(),x12) =  4*x12                
                            >= x12                  
                            =  x12                  
        
           plus#2(S(x4),x2) =  4*x2                 
                            >= 4*x2                 
                            =  S(plus#2(x4,x2))     
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold#3(Cons(x4,x2)) -> plus#2(x4,fold#3(x2))
            fold#3(Nil()) -> 0()
            plus#2(0(),x12) -> x12
            plus#2(S(x4),x2) -> S(plus#2(x4,x2))
        - Weak TRS:
            main(x1) -> fold#3(x1)
        - Signature:
            {fold#3/1,main/1,plus#2/2} / {0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#3,main,plus#2} and constructors {0,Cons,Nil,S}
    + 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(S) = {1},
          uargs(plus#2) = {2}
        
        Following symbols are considered usable:
          {fold#3,main,plus#2}
        TcT has computed the following interpretation:
               p(0) = [0]                  
            p(Cons) = [1] x1 + [1] x2 + [4]
             p(Nil) = [0]                  
               p(S) = [1] x1 + [0]         
          p(fold#3) = [4] x1 + [0]         
            p(main) = [8] x1 + [3]         
          p(plus#2) = [1] x2 + [2]         
        
        Following rules are strictly oriented:
        fold#3(Cons(x4,x2)) = [4] x2 + [4] x4 + [16]
                            > [4] x2 + [2]          
                            = plus#2(x4,fold#3(x2)) 
        
            plus#2(0(),x12) = [1] x12 + [2]         
                            > [1] x12 + [0]         
                            = x12                   
        
        
        Following rules are (at-least) weakly oriented:
           fold#3(Nil()) =  [0]             
                         >= [0]             
                         =  0()             
        
                main(x1) =  [8] x1 + [3]    
                         >= [4] x1 + [0]    
                         =  fold#3(x1)      
        
        plus#2(S(x4),x2) =  [1] x2 + [2]    
                         >= [1] x2 + [2]    
                         =  S(plus#2(x4,x2))
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold#3(Nil()) -> 0()
            plus#2(S(x4),x2) -> S(plus#2(x4,x2))
        - Weak TRS:
            fold#3(Cons(x4,x2)) -> plus#2(x4,fold#3(x2))
            main(x1) -> fold#3(x1)
            plus#2(0(),x12) -> x12
        - Signature:
            {fold#3/1,main/1,plus#2/2} / {0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#3,main,plus#2} and constructors {0,Cons,Nil,S}
    + 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(S) = {1},
          uargs(plus#2) = {2}
        
        Following symbols are considered usable:
          {fold#3,main,plus#2}
        TcT has computed the following interpretation:
               p(0) = [0]         
            p(Cons) = [1] x1 + [4]
             p(Nil) = [1]         
               p(S) = [1] x1 + [0]
          p(fold#3) = [1]         
            p(main) = [4] x1 + [8]
          p(plus#2) = [1] x2 + [0]
        
        Following rules are strictly oriented:
        fold#3(Nil()) = [1]
                      > [0]
                      = 0()
        
        
        Following rules are (at-least) weakly oriented:
        fold#3(Cons(x4,x2)) =  [1]                  
                            >= [1]                  
                            =  plus#2(x4,fold#3(x2))
        
                   main(x1) =  [4] x1 + [8]         
                            >= [1]                  
                            =  fold#3(x1)           
        
            plus#2(0(),x12) =  [1] x12 + [0]        
                            >= [1] x12 + [0]        
                            =  x12                  
        
           plus#2(S(x4),x2) =  [1] x2 + [0]         
                            >= [1] x2 + [0]         
                            =  S(plus#2(x4,x2))     
        
* Step 4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            plus#2(S(x4),x2) -> S(plus#2(x4,x2))
        - Weak TRS:
            fold#3(Cons(x4,x2)) -> plus#2(x4,fold#3(x2))
            fold#3(Nil()) -> 0()
            main(x1) -> fold#3(x1)
            plus#2(0(),x12) -> x12
        - Signature:
            {fold#3/1,main/1,plus#2/2} / {0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#3,main,plus#2} and constructors {0,Cons,Nil,S}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(S) = {1},
          uargs(plus#2) = {2}
        
        Following symbols are considered usable:
          {fold#3,main,plus#2}
        TcT has computed the following interpretation:
               p(0) = 5            
            p(Cons) = 2 + x1 + x2  
             p(Nil) = 0            
               p(S) = 1 + x1       
          p(fold#3) = 14 + 8*x1    
            p(main) = 15 + 10*x1   
          p(plus#2) = 4 + 2*x1 + x2
        
        Following rules are strictly oriented:
        plus#2(S(x4),x2) = 6 + x2 + 2*x4   
                         > 5 + x2 + 2*x4   
                         = S(plus#2(x4,x2))
        
        
        Following rules are (at-least) weakly oriented:
        fold#3(Cons(x4,x2)) =  30 + 8*x2 + 8*x4     
                            >= 18 + 8*x2 + 2*x4     
                            =  plus#2(x4,fold#3(x2))
        
              fold#3(Nil()) =  14                   
                            >= 5                    
                            =  0()                  
        
                   main(x1) =  15 + 10*x1           
                            >= 14 + 8*x1            
                            =  fold#3(x1)           
        
            plus#2(0(),x12) =  14 + x12             
                            >= x12                  
                            =  x12                  
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            fold#3(Cons(x4,x2)) -> plus#2(x4,fold#3(x2))
            fold#3(Nil()) -> 0()
            main(x1) -> fold#3(x1)
            plus#2(0(),x12) -> x12
            plus#2(S(x4),x2) -> S(plus#2(x4,x2))
        - Signature:
            {fold#3/1,main/1,plus#2/2} / {0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#3,main,plus#2} and constructors {0,Cons,Nil,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))