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: 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)
        - 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:
        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) = x1
          p(mark) = 6 
        
        Following rules are strictly oriented:
        mark(g(X)) = 6   
                   > 0   
                   = g(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)) =  6            
                   >= 6            
                   =  a__f(mark(X))
        
        mark(h(X)) =  6            
                   >= 6            
                   =  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)))
            mark(f(X)) -> a__f(mark(X))
        - Weak TRS:
            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) = 7 + x1
             p(f) = 4 + x1
             p(g) = 0     
             p(h) = x1    
          p(mark) = 4*x1  
        
        Following rules are strictly oriented:
           a__f(X) = 7 + X        
                   > 4 + X        
                   = f(X)         
        
           a__f(X) = 7 + X        
                   > 0            
                   = g(h(f(X)))   
        
        mark(f(X)) = 16 + 4*X     
                   > 7 + 4*X      
                   = a__f(mark(X))
        
        
        Following rules are (at-least) weakly oriented:
        mark(g(X)) =  0         
                   >= 0         
                   =  g(X)      
        
        mark(h(X)) =  4*X       
                   >= 4*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))