WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          fold#3(y){y -> Cons(x,y)} =
            fold#3(Cons(x,y)) ->^+ plus#2(x,fold#3(y))
              = C[fold#3(y) = fold#3(y){}]

** Step 1.b: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 1.b:2: 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()
            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:
        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) = 14      
            p(Cons) = 2 + x2  
             p(Nil) = 5       
               p(S) = x1      
          p(fold#3) = 4 + 2*x1
            p(main) = 4 + 2*x1
          p(plus#2) = x2      
        
        Following rules are strictly oriented:
        fold#3(Cons(x4,x2)) = 8 + 2*x2             
                            > 4 + 2*x2             
                            = plus#2(x4,fold#3(x2))
        
        
        Following rules are (at-least) weakly oriented:
           fold#3(Nil()) =  14              
                         >= 14              
                         =  0()             
        
                main(x1) =  4 + 2*x1        
                         >= 4 + 2*x1        
                         =  fold#3(x1)      
        
         plus#2(0(),x12) =  x12             
                         >= x12             
                         =  x12             
        
        plus#2(S(x4),x2) =  x2              
                         >= x2              
                         =  S(plus#2(x4,x2))
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold#3(Nil()) -> 0()
            plus#2(0(),x12) -> x12
            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)
        - 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) = 3 
            p(Cons) = 0 
             p(Nil) = 0 
               p(S) = x1
          p(fold#3) = 5 
            p(main) = 5 
          p(plus#2) = x2
        
        Following rules are strictly oriented:
        fold#3(Nil()) = 5  
                      > 3  
                      = 0()
        
        
        Following rules are (at-least) weakly oriented:
        fold#3(Cons(x4,x2)) =  5                    
                            >= 5                    
                            =  plus#2(x4,fold#3(x2))
        
                   main(x1) =  5                    
                            >= 5                    
                            =  fold#3(x1)           
        
            plus#2(0(),x12) =  x12                  
                            >= x12                  
                            =  x12                  
        
           plus#2(S(x4),x2) =  x2                   
                            >= x2                   
                            =  S(plus#2(x4,x2))     
        
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            plus#2(0(),x12) -> x12
            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)
        - 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) = 8       
            p(Cons) = x1 + x2 
             p(Nil) = 5       
               p(S) = 7 + x1  
          p(fold#3) = 1 + 4*x1
            p(main) = 8 + 8*x1
          p(plus#2) = x1 + x2 
        
        Following rules are strictly oriented:
        plus#2(0(),x12) = 8 + x12
                        > x12    
                        = x12    
        
        
        Following rules are (at-least) weakly oriented:
        fold#3(Cons(x4,x2)) =  1 + 4*x2 + 4*x4      
                            >= 1 + 4*x2 + x4        
                            =  plus#2(x4,fold#3(x2))
        
              fold#3(Nil()) =  21                   
                            >= 8                    
                            =  0()                  
        
                   main(x1) =  8 + 8*x1             
                            >= 1 + 4*x1             
                            =  fold#3(x1)           
        
           plus#2(S(x4),x2) =  7 + x2 + x4          
                            >= 7 + x2 + x4          
                            =  S(plus#2(x4,x2))     
        
** Step 1.b:5: 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 1.b:6: 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(Omega(n^1),O(n^1))