WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} 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(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,sum}
        TcT has computed the following interpretation:
            p(+) = x1
            p(0) = 2 
            p(s) = x1
          p(sum) = 10
        
        Following rules are strictly oriented:
        sum(0()) = 10 
                 > 2  
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  x             
                  >= x             
                  =  x             
        
        +(x,s(y)) =  x             
                  >= x             
                  =  s(+(x,y))     
        
        sum(s(x)) =  10            
                  >= 10            
                  =  +(sum(x),s(x))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(s(x)) -> +(sum(x),s(x))
        - Weak TRS:
            sum(0()) -> 0()
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} 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(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,sum}
        TcT has computed the following interpretation:
            p(+) = x1 + x2    
            p(0) = 2          
            p(s) = 2 + x1     
          p(sum) = 2*x1 + x1^2
        
        Following rules are strictly oriented:
         +(x,0()) = 2 + x         
                  > x             
                  = x             
        
        sum(s(x)) = 8 + 6*x + x^2 
                  > 2 + 3*x + x^2 
                  = +(sum(x),s(x))
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  2 + x + y
                  >= 2 + x + y
                  =  s(+(x,y))
        
         sum(0()) =  8        
                  >= 2        
                  =  0()      
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} 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(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,sum}
        TcT has computed the following interpretation:
            p(+) = 2 + x1 + 2*x2
            p(0) = 0            
            p(s) = 2 + x1       
          p(sum) = 1 + 3*x1^2   
        
        Following rules are strictly oriented:
        +(x,s(y)) = 6 + x + 2*y
                  > 4 + x + 2*y
                  = s(+(x,y))  
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  2 + x            
                  >= x                
                  =  x                
        
         sum(0()) =  1                
                  >= 0                
                  =  0()              
        
        sum(s(x)) =  13 + 12*x + 3*x^2
                  >= 7 + 2*x + 3*x^2  
                  =  +(sum(x),s(x))   
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))