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

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

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