WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
            a__c() -> c()
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {c,f,g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
            a__c() -> c()
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {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:
          none
        
        Following symbols are considered usable:
          {a__c,a__f,mark}
        TcT has computed the following interpretation:
          p(a__c) = [12]
          p(a__f) = [12]
             p(c) = [10]
             p(f) = [12]
             p(g) = [12]
          p(mark) = [12]
        
        Following rules are strictly oriented:
        a__c() = [12]
               > [10]
               = c() 
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [12]        
                   >= [12]        
                   =  a__f(g(c()))
        
           a__f(X) =  [12]        
                   >= [12]        
                   =  f(X)        
        
        a__f(g(X)) =  [12]        
                   >= [12]        
                   =  g(X)        
        
         mark(c()) =  [12]        
                   >= [12]        
                   =  a__c()      
        
        mark(f(X)) =  [12]        
                   >= [12]        
                   =  a__f(X)     
        
        mark(g(X)) =  [12]        
                   >= [12]        
                   =  g(X)        
        
* Step 3: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Weak TRS:
            a__c() -> c()
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {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:
          none
        
        Following symbols are considered usable:
          {a__c,a__f,mark}
        TcT has computed the following interpretation:
          p(a__c) = [8]
          p(a__f) = [8]
             p(c) = [8]
             p(f) = [6]
             p(g) = [8]
          p(mark) = [8]
        
        Following rules are strictly oriented:
        a__f(X) = [8] 
                > [6] 
                = f(X)
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [8]         
                   >= [8]         
                   =  a__f(g(c()))
        
            a__c() =  [8]         
                   >= [8]         
                   =  c()         
        
        a__f(g(X)) =  [8]         
                   >= [8]         
                   =  g(X)        
        
         mark(c()) =  [8]         
                   >= [8]         
                   =  a__c()      
        
        mark(f(X)) =  [8]         
                   >= [8]         
                   =  a__f(X)     
        
        mark(g(X)) =  [8]         
                   >= [8]         
                   =  g(X)        
        
* Step 4: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Weak TRS:
            a__c() -> c()
            a__f(X) -> f(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {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:
          none
        
        Following symbols are considered usable:
          {a__c,a__f,mark}
        TcT has computed the following interpretation:
          p(a__c) = [1]
          p(a__f) = [1]
             p(c) = [1]
             p(f) = [0]
             p(g) = [0]
          p(mark) = [1]
        
        Following rules are strictly oriented:
        a__f(g(X)) = [1] 
                   > [0] 
                   = g(X)
        
        mark(g(X)) = [1] 
                   > [0] 
                   = g(X)
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [1]         
                   >= [1]         
                   =  a__f(g(c()))
        
            a__c() =  [1]         
                   >= [1]         
                   =  c()         
        
           a__f(X) =  [1]         
                   >= [0]         
                   =  f(X)        
        
         mark(c()) =  [1]         
                   >= [1]         
                   =  a__c()      
        
        mark(f(X)) =  [1]         
                   >= [1]         
                   =  a__f(X)     
        
* Step 5: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
        - Weak TRS:
            a__c() -> c()
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {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:
          none
        
        Following symbols are considered usable:
          {a__c,a__f,mark}
        TcT has computed the following interpretation:
          p(a__c) = [0]
          p(a__f) = [0]
             p(c) = [0]
             p(f) = [0]
             p(g) = [0]
          p(mark) = [1]
        
        Following rules are strictly oriented:
         mark(c()) = [1]    
                   > [0]    
                   = a__c() 
        
        mark(f(X)) = [1]    
                   > [0]    
                   = a__f(X)
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [0]         
                   >= [0]         
                   =  a__f(g(c()))
        
            a__c() =  [0]         
                   >= [0]         
                   =  c()         
        
           a__f(X) =  [0]         
                   >= [0]         
                   =  f(X)        
        
        a__f(g(X)) =  [0]         
                   >= [0]         
                   =  g(X)        
        
        mark(g(X)) =  [1]         
                   >= [0]         
                   =  g(X)        
        
* Step 6: NaturalPI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> a__f(g(c()))
        - Weak TRS:
            a__c() -> c()
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {c,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:
          none
        
        Following symbols are considered usable:
          {a__c,a__f,mark}
        TcT has computed the following interpretation:
          p(a__c) = 9 
          p(a__f) = 0 
             p(c) = 0 
             p(f) = 0 
             p(g) = 0 
          p(mark) = 12
        
        Following rules are strictly oriented:
        a__c() = 9           
               > 0           
               = a__f(g(c()))
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  9      
                   >= 0      
                   =  c()    
        
           a__f(X) =  0      
                   >= 0      
                   =  f(X)   
        
        a__f(g(X)) =  0      
                   >= 0      
                   =  g(X)   
        
         mark(c()) =  12     
                   >= 9      
                   =  a__c() 
        
        mark(f(X)) =  12     
                   >= 0      
                   =  a__f(X)
        
        mark(g(X)) =  12     
                   >= 0      
                   =  g(X)   
        
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__c() -> a__f(g(c()))
            a__c() -> c()
            a__f(X) -> f(X)
            a__f(g(X)) -> g(X)
            mark(c()) -> a__c()
            mark(f(X)) -> a__f(X)
            mark(g(X)) -> g(X)
        - Signature:
            {a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {c,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))