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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} 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:
          {+}
        TcT has computed the following interpretation:
          p(+) = 8 + 3*x1 + 8*x2
          p(0) = 0              
          p(s) = x1             
        
        Following rules are strictly oriented:
        +(0(),y) = 8 + 8*y
                 > y      
                 = y      
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),y) =  8 + 3*x + 8*y
                  >= 8 + 3*x + 8*y
                  =  +(x,s(y))    
        
        +(s(x),y) =  8 + 3*x + 8*y
                  >= 8 + 3*x + 8*y
                  =  s(+(x,y))    
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(0(),y) -> y
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} 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:
          {+}
        TcT has computed the following interpretation:
          p(+) = 9*x1 + 8*x2
          p(0) = 0          
          p(s) = 1 + x1     
        
        Following rules are strictly oriented:
        +(s(x),y) = 9 + 9*x + 8*y
                  > 8 + 9*x + 8*y
                  = +(x,s(y))    
        
        +(s(x),y) = 9 + 9*x + 8*y
                  > 1 + 9*x + 8*y
                  = s(+(x,y))    
        
        
        Following rules are (at-least) weakly oriented:
        +(0(),y) =  8*y
                 >= y  
                 =  y  
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} 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^1))