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())) -> c(f(g(f(a()))))
            mark(a()) -> a()
            mark(c(X)) -> c(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/1,mark/1} / {a/0,c/1,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,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())) -> c(f(g(f(a()))))
            mark(a()) -> a()
            mark(c(X)) -> c(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/1,mark/1} / {a/0,c/1,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,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: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> c(f(g(f(a()))))
            mark(a()) -> a()
            mark(c(X)) -> c(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/1,mark/1} / {a/0,c/1,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,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) = [0]         
          p(a__f) = [1] x1 + [6]
             p(c) = [4]         
             p(f) = [1] x1 + [2]
             p(g) = [1] x1 + [0]
          p(mark) = [4] x1 + [0]
        
        Following rules are strictly oriented:
             a__f(X) = [1] X + [6]    
                     > [1] X + [2]    
                     = f(X)           
        
        a__f(f(a())) = [8]            
                     > [4]            
                     = c(f(g(f(a()))))
        
          mark(c(X)) = [16]           
                     > [4]            
                     = c(X)           
        
          mark(f(X)) = [4] X + [8]    
                     > [4] X + [6]    
                     = a__f(mark(X))  
        
        
        Following rules are (at-least) weakly oriented:
         mark(a()) =  [0]        
                   >= [0]        
                   =  a()        
        
        mark(g(X)) =  [4] X + [0]
                   >= [4] X + [0]
                   =  g(mark(X)) 
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            mark(a()) -> a()
            mark(g(X)) -> g(mark(X))
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> c(f(g(f(a()))))
            mark(c(X)) -> c(X)
            mark(f(X)) -> a__f(mark(X))
        - Signature:
            {a__f/1,mark/1} / {a/0,c/1,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,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) = [0]         
          p(a__f) = [1] x1 + [8]
             p(c) = [8]         
             p(f) = [1] x1 + [6]
             p(g) = [1] x1 + [5]
          p(mark) = [2] x1 + [6]
        
        Following rules are strictly oriented:
         mark(a()) = [6]         
                   > [0]         
                   = a()         
        
        mark(g(X)) = [2] X + [16]
                   > [2] X + [11]
                   = g(mark(X))  
        
        
        Following rules are (at-least) weakly oriented:
             a__f(X) =  [1] X + [8]    
                     >= [1] X + [6]    
                     =  f(X)           
        
        a__f(f(a())) =  [14]           
                     >= [8]            
                     =  c(f(g(f(a()))))
        
          mark(c(X)) =  [22]           
                     >= [8]            
                     =  c(X)           
        
          mark(f(X)) =  [2] X + [18]   
                     >= [2] X + [14]   
                     =  a__f(mark(X))  
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(f(a())) -> c(f(g(f(a()))))
            mark(a()) -> a()
            mark(c(X)) -> c(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(mark(X))
        - Signature:
            {a__f/1,mark/1} / {a/0,c/1,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,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))