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(X) -> g(h(f(X)))
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + 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(X) -> g(h(f(X)))
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + 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(X) -> g(h(f(X)))
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = x1    
             p(f) = x1    
             p(g) = 0     
             p(h) = 2 + x1
          p(mark) = 5*x1  
        
        Following rules are strictly oriented:
        mark(h(X)) = 10 + 5*X  
                   > 2 + 5*X   
                   = h(mark(X))
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X) =  X            
                   >= X            
                   =  f(X)         
        
           a__f(X) =  X            
                   >= 0            
                   =  g(h(f(X)))   
        
        mark(f(X)) =  5*X          
                   >= 5*X          
                   =  a__f(mark(X))
        
        mark(g(X)) =  0            
                   >= 0            
                   =  g(X)         
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(X) -> g(h(f(X)))
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
        - Weak TRS:
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = [1] x1 + [1]
             p(f) = [1] x1 + [1]
             p(g) = [1] x1 + [0]
             p(h) = [1] x1 + [0]
          p(mark) = [2] x1 + [1]
        
        Following rules are strictly oriented:
        mark(f(X)) = [2] X + [3]  
                   > [2] X + [2]  
                   = a__f(mark(X))
        
        mark(g(X)) = [2] X + [1]  
                   > [1] X + [0]  
                   = g(X)         
        
        
        Following rules are (at-least) weakly oriented:
           a__f(X) =  [1] X + [1]
                   >= [1] X + [1]
                   =  f(X)       
        
           a__f(X) =  [1] X + [1]
                   >= [1] X + [1]
                   =  g(h(f(X))) 
        
        mark(h(X)) =  [2] X + [1]
                   >= [2] X + [1]
                   =  h(mark(X)) 
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(X) -> g(h(f(X)))
        - Weak TRS:
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {a__f,mark}
        TcT has computed the following interpretation:
          p(a__f) = 10 + x1
             p(f) = 2 + x1 
             p(g) = 0      
             p(h) = x1     
          p(mark) = 8*x1   
        
        Following rules are strictly oriented:
        a__f(X) = 10 + X    
                > 2 + X     
                = f(X)      
        
        a__f(X) = 10 + X    
                > 0         
                = g(h(f(X)))
        
        
        Following rules are (at-least) weakly oriented:
        mark(f(X)) =  16 + 8*X     
                   >= 10 + 8*X     
                   =  a__f(mark(X))
        
        mark(g(X)) =  0            
                   >= 0            
                   =  g(X)         
        
        mark(h(X)) =  8*X          
                   >= 8*X          
                   =  h(mark(X))   
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(X) -> g(h(f(X)))
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> h(mark(X))
        - Signature:
            {a__f/1,mark/1} / {f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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