WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__f(X1,X2) -> f(X1,X2)
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {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(X1,X2) -> f(X1,X2)
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mark(x){x -> f(x,y)} =
            mark(f(x,y)) ->^+ a__f(mark(x),y)
              = C[mark(x) = mark(x){}]

** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__f(X1,X2) -> f(X1,X2)
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {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(a__f) = {1},
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = [1] x1 + [8]
             p(f) = [1] x1 + [8]
             p(g) = [1] x1 + [4]
          p(mark) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        a__f(g(X),Y) = [1] X + [12]           
                     > [1] X + [8]            
                     = a__f(mark(X),f(g(X),Y))
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X1,X2) =  [1] X1 + [8]     
                       >= [1] X1 + [8]     
                       =  f(X1,X2)         
        
        mark(f(X1,X2)) =  [1] X1 + [8]     
                       >= [1] X1 + [8]     
                       =  a__f(mark(X1),X2)
        
            mark(g(X)) =  [1] X + [4]      
                       >= [1] X + [4]      
                       =  g(mark(X))       
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__f(X1,X2) -> f(X1,X2)
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Weak TRS:
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {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(a__f) = {1},
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = [1 0 1]      [0]
                    [0 0 0] x1 + [2]
                    [0 0 1]      [1]
             p(f) = [1 0 1]      [0]
                    [0 0 0] x1 + [0]
                    [0 0 1]      [1]
             p(g) = [1 1 3]      [0]
                    [0 0 0] x1 + [0]
                    [0 0 1]      [4]
          p(mark) = [1 0 2]      [4]
                    [0 0 0] x1 + [2]
                    [0 0 1]      [0]
        
        Following rules are strictly oriented:
        mark(f(X1,X2)) = [1 0 3]      [6] 
                         [0 0 0] X1 + [2] 
                         [0 0 1]      [1] 
                       > [1 0 3]      [4] 
                         [0 0 0] X1 + [2] 
                         [0 0 1]      [1] 
                       = a__f(mark(X1),X2)
        
            mark(g(X)) = [1 1 5]     [12] 
                         [0 0 0] X + [2]  
                         [0 0 1]     [4]  
                       > [1 0 5]     [6]  
                         [0 0 0] X + [0]  
                         [0 0 1]     [4]  
                       = g(mark(X))       
        
        
        Following rules are (at-least) weakly oriented:
         a__f(X1,X2) =  [1 0 1]      [0]       
                        [0 0 0] X1 + [2]       
                        [0 0 1]      [1]       
                     >= [1 0 1]      [0]       
                        [0 0 0] X1 + [0]       
                        [0 0 1]      [1]       
                     =  f(X1,X2)               
        
        a__f(g(X),Y) =  [1 1 4]     [4]        
                        [0 0 0] X + [2]        
                        [0 0 1]     [5]        
                     >= [1 0 3]     [4]        
                        [0 0 0] X + [2]        
                        [0 0 1]     [1]        
                     =  a__f(mark(X),f(g(X),Y))
        
** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            a__f(X1,X2) -> f(X1,X2)
        - Weak TRS:
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {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(a__f) = {1},
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = [1 0 6]      [1]
                    [0 0 2] x1 + [7]
                    [0 0 1]      [4]
             p(f) = [1 0 6]      [0]
                    [0 0 2] x1 + [2]
                    [0 0 1]      [4]
             p(g) = [1 0 2]      [1]
                    [0 1 0] x1 + [1]
                    [0 0 1]      [1]
          p(mark) = [1 0 2]      [7]
                    [1 1 0] x1 + [6]
                    [0 0 1]      [0]
        
        Following rules are strictly oriented:
        a__f(X1,X2) = [1 0 6]      [1]
                      [0 0 2] X1 + [7]
                      [0 0 1]      [4]
                    > [1 0 6]      [0]
                      [0 0 2] X1 + [2]
                      [0 0 1]      [4]
                    = f(X1,X2)        
        
        
        Following rules are (at-least) weakly oriented:
          a__f(g(X),Y) =  [1 0 8]     [8]        
                          [0 0 2] X + [9]        
                          [0 0 1]     [5]        
                       >= [1 0 8]     [8]        
                          [0 0 2] X + [7]        
                          [0 0 1]     [4]        
                       =  a__f(mark(X),f(g(X),Y))
        
        mark(f(X1,X2)) =  [1 0 8]      [15]      
                          [1 0 8] X1 + [8]       
                          [0 0 1]      [4]       
                       >= [1 0 8]      [8]       
                          [0 0 2] X1 + [7]       
                          [0 0 1]      [4]       
                       =  a__f(mark(X1),X2)      
        
            mark(g(X)) =  [1 0 4]     [10]       
                          [1 1 2] X + [8]        
                          [0 0 1]     [1]        
                       >= [1 0 4]     [8]        
                          [1 1 0] X + [7]        
                          [0 0 1]     [1]        
                       =  g(mark(X))             
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X1,X2) -> f(X1,X2)
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/2,mark/1} / {f/2,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))