WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          sum(x){x -> s(x)} =
            sum(s(x)) ->^+ +(sum(x),s(x))
              = C[sum(x) = sum(x){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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,sum1}
        TcT has computed the following interpretation:
             p(+) = x1      
             p(0) = 3       
             p(s) = x1      
           p(sum) = 5 + 4*x1
          p(sum1) = 1 + 4*x1
        
        Following rules are strictly oriented:
         sum(0()) = 17 
                  > 3  
                  = 0()
        
        sum1(0()) = 13 
                  > 3  
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
         sum(s(x)) =  5 + 4*x             
                   >= 5 + 4*x             
                   =  +(sum(x),s(x))      
        
        sum1(s(x)) =  1 + 4*x             
                   >= 1 + 4*x             
                   =  s(+(sum1(x),+(x,x)))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(s(x)) -> +(sum(x),s(x))
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Weak TRS:
            sum(0()) -> 0()
            sum1(0()) -> 0()
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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,sum1}
        TcT has computed the following interpretation:
             p(+) = 8 + x1   
             p(0) = 4        
             p(s) = 8 + x1   
           p(sum) = 2*x1     
          p(sum1) = 15 + 2*x1
        
        Following rules are strictly oriented:
        sum(s(x)) = 16 + 2*x      
                  > 8 + 2*x       
                  = +(sum(x),s(x))
        
        
        Following rules are (at-least) weakly oriented:
          sum(0()) =  8                   
                   >= 4                   
                   =  0()                 
        
         sum1(0()) =  23                  
                   >= 4                   
                   =  0()                 
        
        sum1(s(x)) =  31 + 2*x            
                   >= 31 + 2*x            
                   =  s(+(sum1(x),+(x,x)))
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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,sum1}
        TcT has computed the following interpretation:
             p(+) = 4 + x1
             p(0) = 8     
             p(s) = 8 + x1
           p(sum) = 2*x1  
          p(sum1) = 2*x1  
        
        Following rules are strictly oriented:
        sum1(s(x)) = 16 + 2*x            
                   > 12 + 2*x            
                   = s(+(sum1(x),+(x,x)))
        
        
        Following rules are (at-least) weakly oriented:
         sum(0()) =  16            
                  >= 8             
                  =  0()           
        
        sum(s(x)) =  16 + 2*x      
                  >= 4 + 2*x       
                  =  +(sum(x),s(x))
        
        sum1(0()) =  16            
                  >= 8             
                  =  0()           
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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))