WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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(a__h) = {1}
        
        Following symbols are considered usable:
          {a__c,a__f,a__h,mark}
        TcT has computed the following interpretation:
          p(a__c) = 0 
          p(a__f) = x1
          p(a__h) = x1
             p(c) = 0 
             p(d) = 0 
             p(f) = x1
             p(g) = 0 
             p(h) = x1
          p(mark) = 2 
        
        Following rules are strictly oriented:
        mark(c(X)) = 2      
                   > 0      
                   = a__c(X)
        
        mark(d(X)) = 2      
                   > 0      
                   = d(X)   
        
        mark(g(X)) = 2      
                   > 0      
                   = g(X)   
        
        
        Following rules are (at-least) weakly oriented:
           a__c(X) =  0               
                   >= 0               
                   =  c(X)            
        
           a__c(X) =  0               
                   >= 0               
                   =  d(X)            
        
           a__f(X) =  X               
                   >= X               
                   =  f(X)            
        
        a__f(f(X)) =  X               
                   >= 0               
                   =  a__c(f(g(f(X))))
        
           a__h(X) =  X               
                   >= 0               
                   =  a__c(d(X))      
        
           a__h(X) =  X               
                   >= X               
                   =  h(X)            
        
        mark(f(X)) =  2               
                   >= 2               
                   =  a__f(mark(X))   
        
        mark(h(X)) =  2               
                   >= 2               
                   =  a__h(mark(X))   
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(f(X)) -> a__f(mark(X))
            mark(h(X)) -> a__h(mark(X))
        - Weak TRS:
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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(a__h) = {1}
        
        Following symbols are considered usable:
          {a__c,a__f,a__h,mark}
        TcT has computed the following interpretation:
          p(a__c) = 0       
          p(a__f) = 1 + x1  
          p(a__h) = 6 + x1  
             p(c) = 0       
             p(d) = 0       
             p(f) = 1 + x1  
             p(g) = x1      
             p(h) = 1 + x1  
          p(mark) = 1 + 8*x1
        
        Following rules are strictly oriented:
        a__f(f(X)) = 2 + X           
                   > 0               
                   = a__c(f(g(f(X))))
        
           a__h(X) = 6 + X           
                   > 0               
                   = a__c(d(X))      
        
           a__h(X) = 6 + X           
                   > 1 + X           
                   = h(X)            
        
        mark(f(X)) = 9 + 8*X         
                   > 2 + 8*X         
                   = a__f(mark(X))   
        
        mark(h(X)) = 9 + 8*X         
                   > 7 + 8*X         
                   = a__h(mark(X))   
        
        
        Following rules are (at-least) weakly oriented:
           a__c(X) =  0      
                   >= 0      
                   =  c(X)   
        
           a__c(X) =  0      
                   >= 0      
                   =  d(X)   
        
           a__f(X) =  1 + X  
                   >= 1 + X  
                   =  f(X)   
        
        mark(c(X)) =  1      
                   >= 0      
                   =  a__c(X)
        
        mark(d(X)) =  1      
                   >= 0      
                   =  d(X)   
        
        mark(g(X)) =  1 + 8*X
                   >= X      
                   =  g(X)   
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
        - Weak TRS:
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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(a__h) = {1}
        
        Following symbols are considered usable:
          {a__c,a__f,a__h,mark}
        TcT has computed the following interpretation:
          p(a__c) = 4       
          p(a__f) = 3 + x1  
          p(a__h) = 7 + x1  
             p(c) = 4       
             p(d) = 2       
             p(f) = 1 + x1  
             p(g) = 3 + x1  
             p(h) = 4 + x1  
          p(mark) = 2 + 4*x1
        
        Following rules are strictly oriented:
        a__c(X) = 4    
                > 2    
                = d(X) 
        
        a__f(X) = 3 + X
                > 1 + X
                = f(X) 
        
        
        Following rules are (at-least) weakly oriented:
           a__c(X) =  4               
                   >= 4               
                   =  c(X)            
        
        a__f(f(X)) =  4 + X           
                   >= 4               
                   =  a__c(f(g(f(X))))
        
           a__h(X) =  7 + X           
                   >= 4               
                   =  a__c(d(X))      
        
           a__h(X) =  7 + X           
                   >= 4 + X           
                   =  h(X)            
        
        mark(c(X)) =  18              
                   >= 4               
                   =  a__c(X)         
        
        mark(d(X)) =  10              
                   >= 2               
                   =  d(X)            
        
        mark(f(X)) =  6 + 4*X         
                   >= 5 + 4*X         
                   =  a__f(mark(X))   
        
        mark(g(X)) =  14 + 4*X        
                   >= 3 + X           
                   =  g(X)            
        
        mark(h(X)) =  18 + 4*X        
                   >= 9 + 4*X         
                   =  a__h(mark(X))   
        
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c(X) -> c(X)
        - Weak TRS:
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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(a__h) = {1}
        
        Following symbols are considered usable:
          {a__c,a__f,a__h,mark}
        TcT has computed the following interpretation:
          p(a__c) = 15      
          p(a__f) = 11 + x1 
          p(a__h) = 15 + x1 
             p(c) = 4       
             p(d) = 0       
             p(f) = 8 + x1  
             p(g) = 4       
             p(h) = 8 + x1  
          p(mark) = 6 + 3*x1
        
        Following rules are strictly oriented:
        a__c(X) = 15  
                > 4   
                = c(X)
        
        
        Following rules are (at-least) weakly oriented:
           a__c(X) =  15              
                   >= 0               
                   =  d(X)            
        
           a__f(X) =  11 + X          
                   >= 8 + X           
                   =  f(X)            
        
        a__f(f(X)) =  19 + X          
                   >= 15              
                   =  a__c(f(g(f(X))))
        
           a__h(X) =  15 + X          
                   >= 15              
                   =  a__c(d(X))      
        
           a__h(X) =  15 + X          
                   >= 8 + X           
                   =  h(X)            
        
        mark(c(X)) =  18              
                   >= 15              
                   =  a__c(X)         
        
        mark(d(X)) =  6               
                   >= 0               
                   =  d(X)            
        
        mark(f(X)) =  30 + 3*X        
                   >= 17 + 3*X        
                   =  a__f(mark(X))   
        
        mark(g(X)) =  18              
                   >= 4               
                   =  g(X)            
        
        mark(h(X)) =  30 + 3*X        
                   >= 21 + 3*X        
                   =  a__h(mark(X))   
        
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__c(X) -> c(X)
            a__c(X) -> d(X)
            a__f(X) -> f(X)
            a__f(f(X)) -> a__c(f(g(f(X))))
            a__h(X) -> a__c(d(X))
            a__h(X) -> h(X)
            mark(c(X)) -> a__c(X)
            mark(d(X)) -> d(X)
            mark(f(X)) -> a__f(mark(X))
            mark(g(X)) -> g(X)
            mark(h(X)) -> a__h(mark(X))
        - Signature:
            {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,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))