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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + 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(a__f) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
             p(a) = 0      
          p(a__f) = 14 + x1
             p(f) = 14 + x1
             p(g) = 0      
          p(mark) = 3 + x1 
        
        Following rules are strictly oriented:
        a__f(f(a())) = 28             
                     > 14             
                     = a__f(g(f(a())))
        
           mark(a()) = 3              
                     > 0              
                     = a()            
        
          mark(g(X)) = 3              
                     > 0              
                     = g(X)           
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X) =  14 + X       
                   >= 14 + X       
                   =  f(X)         
        
        mark(f(X)) =  17 + X       
                   >= 17 + X       
                   =  a__f(mark(X))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            mark(f(X)) -> a__f(mark(X))
        - Weak TRS:
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + 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(a__f) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
             p(a) = 3     
          p(a__f) = 2 + x1
             p(f) = 1 + x1
             p(g) = 2     
          p(mark) = 4*x1  
        
        Following rules are strictly oriented:
           a__f(X) = 2 + X        
                   > 1 + X        
                   = f(X)         
        
        mark(f(X)) = 4 + 4*X      
                   > 2 + 4*X      
                   = a__f(mark(X))
        
        
        Following rules are (at-least) weakly oriented:
        a__f(f(a())) =  6              
                     >= 4              
                     =  a__f(g(f(a())))
        
           mark(a()) =  12             
                     >= 3              
                     =  a()            
        
          mark(g(X)) =  8              
                     >= 2              
                     =  g(X)           
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
            mark(a()) -> a()
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Signature:
            {a__f/1,mark/1} / {a/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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