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(X)
            mark(g(X)) -> g(mark(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(X)
            mark(g(X)) -> g(mark(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 -> g(x)} =
            mark(g(x)) ->^+ g(mark(x))
              = C[mark(x) = mark(x){}]

** Step 1.b:1: NaturalMI 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(X)
            mark(g(X)) -> g(mark(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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
             p(a) = [4]         
          p(a__f) = [8]         
             p(f) = [8]         
             p(g) = [1] x1 + [8]
          p(mark) = [2] x1 + [4]
        
        Following rules are strictly oriented:
         mark(a()) = [12]        
                   > [4]         
                   = a()         
        
        mark(f(X)) = [20]        
                   > [8]         
                   = a__f(X)     
        
        mark(g(X)) = [2] X + [20]
                   > [2] X + [12]
                   = g(mark(X))  
        
        
        Following rules are (at-least) weakly oriented:
             a__f(X) =  [8]            
                     >= [8]            
                     =  f(X)           
        
        a__f(f(a())) =  [8]            
                     >= [8]            
                     =  a__f(g(f(a())))
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> a__f(g(f(a())))
        - Weak TRS:
            mark(a()) -> a()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(mark(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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
             p(a) = [7]         
          p(a__f) = [7]         
             p(f) = [5]         
             p(g) = [1] x1 + [0]
          p(mark) = [7]         
        
        Following rules are strictly oriented:
        a__f(X) = [7] 
                > [5] 
                = f(X)
        
        
        Following rules are (at-least) weakly oriented:
        a__f(f(a())) =  [7]            
                     >= [7]            
                     =  a__f(g(f(a())))
        
           mark(a()) =  [7]            
                     >= [7]            
                     =  a()            
        
          mark(f(X)) =  [7]            
                     >= [7]            
                     =  a__f(X)        
        
          mark(g(X)) =  [7]            
                     >= [7]            
                     =  g(mark(X))     
        
** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(f(a())) -> a__f(g(f(a())))
        - Weak TRS:
            a__f(X) -> f(X)
            mark(a()) -> a()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(mark(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:
        NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
             p(a) = [0]             
                    [0]             
                    [0]             
          p(a__f) = [0 0 4]      [0]
                    [0 0 1] x1 + [2]
                    [0 0 0]      [4]
             p(f) = [0 0 2]      [0]
                    [0 0 1] x1 + [0]
                    [0 0 0]      [3]
             p(g) = [1 4 0]      [1]
                    [0 0 3] x1 + [1]
                    [0 0 0]      [1]
          p(mark) = [2 4 4]      [1]
                    [0 1 2] x1 + [1]
                    [0 0 1]      [1]
        
        Following rules are strictly oriented:
        a__f(f(a())) = [12]           
                       [5]            
                       [4]            
                     > [4]            
                       [3]            
                       [4]            
                     = a__f(g(f(a())))
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X) =  [0 0 4]     [0]  
                      [0 0 1] X + [2]  
                      [0 0 0]     [4]  
                   >= [0 0 2]     [0]  
                      [0 0 1] X + [0]  
                      [0 0 0]     [3]  
                   =  f(X)             
        
         mark(a()) =  [1]              
                      [1]              
                      [1]              
                   >= [0]              
                      [0]              
                      [0]              
                   =  a()              
        
        mark(f(X)) =  [0 0 8]     [13] 
                      [0 0 1] X + [7]  
                      [0 0 0]     [4]  
                   >= [0 0 4]     [0]  
                      [0 0 1] X + [2]  
                      [0 0 0]     [4]  
                   =  a__f(X)          
        
        mark(g(X)) =  [2 8 12]     [11]
                      [0 0  3] X + [4] 
                      [0 0  0]     [2] 
                   >= [2 8 12]     [6] 
                      [0 0  3] X + [4] 
                      [0 0  0]     [1] 
                   =  g(mark(X))       
        
** Step 1.b:4: 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(X)
            mark(g(X)) -> g(mark(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))