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

WORST_CASE(?,O(n^2))