WORST_CASE(?,O(n^3))
* Step 1: NaturalPI 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:
        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},
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = x1    
             p(f) = x1    
             p(g) = 2 + x1
          p(mark) = x1    
        
        Following rules are strictly oriented:
        a__f(g(X),Y) = 2 + X                  
                     > X                      
                     = a__f(mark(X),f(g(X),Y))
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X1,X2) =  X1               
                       >= X1               
                       =  f(X1,X2)         
        
        mark(f(X1,X2)) =  X1               
                       >= X1               
                       =  a__f(mark(X1),X2)
        
            mark(g(X)) =  2 + X            
                       >= 2 + X            
                       =  g(mark(X))       
        
* Step 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 2 6]      [0]
                    [0 1 4] x1 + [0]
                    [0 0 1]      [2]
             p(f) = [1 2 0]      [0]
                    [0 1 4] x1 + [0]
                    [0 0 1]      [2]
             p(g) = [1 2 1]      [0]
                    [0 1 4] x1 + [2]
                    [0 0 1]      [0]
          p(mark) = [1 2 0]      [2]
                    [0 1 1] x1 + [0]
                    [0 0 1]      [0]
        
        Following rules are strictly oriented:
        mark(g(X)) = [1 4 9]     [6]
                     [0 1 5] X + [2]
                     [0 0 1]     [0]
                   > [1 4 3]     [2]
                     [0 1 5] X + [2]
                     [0 0 1]     [0]
                   = g(mark(X))     
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X1,X2) =  [1 2 6]      [0]       
                          [0 1 4] X1 + [0]       
                          [0 0 1]      [2]       
                       >= [1 2 0]      [0]       
                          [0 1 4] X1 + [0]       
                          [0 0 1]      [2]       
                       =  f(X1,X2)               
        
          a__f(g(X),Y) =  [1 4 15]     [4]       
                          [0 1  8] X + [2]       
                          [0 0  1]     [2]       
                       >= [1 4 8]     [2]        
                          [0 1 5] X + [0]        
                          [0 0 1]     [2]        
                       =  a__f(mark(X),f(g(X),Y))
        
        mark(f(X1,X2)) =  [1 4 8]      [2]       
                          [0 1 5] X1 + [2]       
                          [0 0 1]      [2]       
                       >= [1 4 8]      [2]       
                          [0 1 5] X1 + [0]       
                          [0 0 1]      [2]       
                       =  a__f(mark(X1),X2)      
        
* Step 3: 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)
        - Weak TRS:
            a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
            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 2]      [0 0 1]      [3]
                    [0 1 0] x1 + [0 0 0] x2 + [2]
                    [0 0 1]      [0 0 0]      [2]
             p(f) = [1 0 2]      [0 0 1]      [0]
                    [0 1 0] x1 + [0 0 0] x2 + [2]
                    [0 0 1]      [0 0 0]      [2]
             p(g) = [1 4 1]      [2]             
                    [0 0 1] x1 + [1]             
                    [0 0 1]      [5]             
          p(mark) = [1 4 0]      [0]             
                    [0 0 1] x1 + [0]             
                    [0 0 1]      [2]             
        
        Following rules are strictly oriented:
           a__f(X1,X2) = [1 0 2]      [0 0 1]      [3]
                         [0 1 0] X1 + [0 0 0] X2 + [2]
                         [0 0 1]      [0 0 0]      [2]
                       > [1 0 2]      [0 0 1]      [0]
                         [0 1 0] X1 + [0 0 0] X2 + [2]
                         [0 0 1]      [0 0 0]      [2]
                       = f(X1,X2)                     
        
        mark(f(X1,X2)) = [1 4 2]      [0 0 1]      [8]
                         [0 0 1] X1 + [0 0 0] X2 + [2]
                         [0 0 1]      [0 0 0]      [4]
                       > [1 4 2]      [0 0 1]      [7]
                         [0 0 1] X1 + [0 0 0] X2 + [2]
                         [0 0 1]      [0 0 0]      [4]
                       = a__f(mark(X1),X2)            
        
        
        Following rules are (at-least) weakly oriented:
        a__f(g(X),Y) =  [1 4 3]     [0 0 1]     [15]
                        [0 0 1] X + [0 0 0] Y + [3] 
                        [0 0 1]     [0 0 0]     [7] 
                     >= [1 4 3]     [14]            
                        [0 0 1] X + [2]             
                        [0 0 1]     [4]             
                     =  a__f(mark(X),f(g(X),Y))     
        
          mark(g(X)) =  [1 4 5]     [6]             
                        [0 0 1] X + [5]             
                        [0 0 1]     [7]             
                     >= [1 4 5]     [4]             
                        [0 0 1] X + [3]             
                        [0 0 1]     [7]             
                     =  g(mark(X))                  
        
* Step 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(?,O(n^3))