WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            main(x,y) -> add(x,y)
        - Signature:
            {add/2,main/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,main} and constructors {0,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}
        
        Following symbols are considered usable:
          {add,main}
        TcT has computed the following interpretation:
             p(0) = [0]          
           p(add) = [2] x2 + [10]
          p(main) = [8] x2 + [10]
             p(s) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
        add(0(),y) = [2] y + [10]
                   > [1] y + [0] 
                   = y           
        
        
        Following rules are (at-least) weakly oriented:
        add(s(x),y) =  [2] y + [10]
                    >= [2] y + [10]
                    =  s(add(x,y)) 
        
          main(x,y) =  [8] y + [10]
                    >= [2] y + [10]
                    =  add(x,y)    
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add(s(x),y) -> s(add(x,y))
            main(x,y) -> add(x,y)
        - Weak TRS:
            add(0(),y) -> y
        - Signature:
            {add/2,main/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,main} and constructors {0,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}
        
        Following symbols are considered usable:
          {add,main}
        TcT has computed the following interpretation:
             p(0) = [0]                   
           p(add) = [2] x2 + [0]          
          p(main) = [8] x1 + [8] x2 + [10]
             p(s) = [1] x1 + [0]          
        
        Following rules are strictly oriented:
        main(x,y) = [8] x + [8] y + [10]
                  > [2] y + [0]         
                  = add(x,y)            
        
        
        Following rules are (at-least) weakly oriented:
         add(0(),y) =  [2] y + [0]
                    >= [1] y + [0]
                    =  y          
        
        add(s(x),y) =  [2] y + [0]
                    >= [2] y + [0]
                    =  s(add(x,y))
        
* Step 3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add(s(x),y) -> s(add(x,y))
        - Weak TRS:
            add(0(),y) -> y
            main(x,y) -> add(x,y)
        - Signature:
            {add/2,main/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,main} and constructors {0,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}
        
        Following symbols are considered usable:
          {add,main}
        TcT has computed the following interpretation:
             p(0) = 2              
           p(add) = 4*x1 + 8*x2    
          p(main) = 8 + 5*x1 + 9*x2
             p(s) = 4 + x1         
        
        Following rules are strictly oriented:
        add(s(x),y) = 16 + 4*x + 8*y
                    > 4 + 4*x + 8*y 
                    = s(add(x,y))   
        
        
        Following rules are (at-least) weakly oriented:
        add(0(),y) =  8 + 8*y      
                   >= y            
                   =  y            
        
         main(x,y) =  8 + 5*x + 9*y
                   >= 4*x + 8*y    
                   =  add(x,y)     
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            main(x,y) -> add(x,y)
        - Signature:
            {add/2,main/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,main} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))