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) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/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) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/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) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/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(+) = 15 + 8*x2
          p(-) = x1       
          p(0) = 8        
          p(s) = x1       
        
        Following rules are strictly oriented:
        +(0(),y) = 15 + 8*y
                 > y       
                 = y       
        
        
        Following rules are (at-least) weakly oriented:
           +(s(x),y) =  15 + 8*y 
                     >= 15 + 8*y 
                     =  s(+(x,y))
        
            -(x,0()) =  x        
                     >= x        
                     =  x        
        
            -(0(),y) =  8        
                     >= 8        
                     =  0()      
        
        -(s(x),s(y)) =  x        
                     >= x        
                     =  -(x,y)   
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Weak TRS:
            +(0(),y) -> y
        - Signature:
            {+/2,-/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(+) = 4 + x1 + 8*x2
          p(-) = 1 + 2*x1 + x2
          p(0) = 1            
          p(s) = 8 + x1       
        
        Following rules are strictly oriented:
            -(x,0()) = 2 + 2*x     
                     > x           
                     = x           
        
            -(0(),y) = 3 + y       
                     > 1           
                     = 0()         
        
        -(s(x),s(y)) = 25 + 2*x + y
                     > 1 + 2*x + y 
                     = -(x,y)      
        
        
        Following rules are (at-least) weakly oriented:
         +(0(),y) =  5 + 8*y     
                  >= y           
                  =  y           
        
        +(s(x),y) =  12 + x + 8*y
                  >= 12 + x + 8*y
                  =  s(+(x,y))   
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(0(),y) -> y
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/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(+) = 4 + 3*x1 + 4*x2
          p(-) = 3 + 4*x1       
          p(0) = 0              
          p(s) = 4 + x1         
        
        Following rules are strictly oriented:
        +(s(x),y) = 16 + 3*x + 4*y
                  > 8 + 3*x + 4*y 
                  = s(+(x,y))     
        
        
        Following rules are (at-least) weakly oriented:
            +(0(),y) =  4 + 4*y 
                     >= y       
                     =  y       
        
            -(x,0()) =  3 + 4*x 
                     >= x       
                     =  x       
        
            -(0(),y) =  3       
                     >= 0       
                     =  0()     
        
        -(s(x),s(y)) =  19 + 4*x
                     >= 3 + 4*x 
                     =  -(x,y)  
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/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))