WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x){x -> s(x)} =
            f(s(x)) ->^+ f(x)
              = C[f(x) = f(x){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {f,g,h}
        TcT has computed the following interpretation:
             p(0) = 0       
          p(cons) = 4       
             p(f) = 10      
             p(g) = 0       
             p(h) = 8 + 4*x1
             p(s) = 0       
        
        Following rules are strictly oriented:
        h(cons(X,Y)) = 24             
                     > 8              
                     = h(g(cons(X,Y)))
        
        
        Following rules are (at-least) weakly oriented:
                f(s(X)) =  10  
                        >= 10  
                        =  f(X)
        
         g(cons(0(),Y)) =  0   
                        >= 0   
                        =  g(Y)
        
        g(cons(s(X),Y)) =  0   
                        >= 0   
                        =  s(X)
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
        - Weak TRS:
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {f,g,h}
        TcT has computed the following interpretation:
             p(0) = 0     
          p(cons) = 2     
             p(f) = 8     
             p(g) = 1     
             p(h) = 3 + x1
             p(s) = 0     
        
        Following rules are strictly oriented:
        g(cons(s(X),Y)) = 1   
                        > 0   
                        = s(X)
        
        
        Following rules are (at-least) weakly oriented:
               f(s(X)) =  8              
                       >= 8              
                       =  f(X)           
        
        g(cons(0(),Y)) =  1              
                       >= 1              
                       =  g(Y)           
        
          h(cons(X,Y)) =  5              
                       >= 4              
                       =  h(g(cons(X,Y)))
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
        - Weak TRS:
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {f,g,h}
        TcT has computed the following interpretation:
             p(0) = 1      
          p(cons) = 2 + x2 
             p(f) = 1      
             p(g) = x1     
             p(h) = 15 + x1
             p(s) = 0      
        
        Following rules are strictly oriented:
        g(cons(0(),Y)) = 2 + Y
                       > Y    
                       = g(Y) 
        
        
        Following rules are (at-least) weakly oriented:
                f(s(X)) =  1              
                        >= 1              
                        =  f(X)           
        
        g(cons(s(X),Y)) =  2 + Y          
                        >= 0              
                        =  s(X)           
        
           h(cons(X,Y)) =  17 + Y         
                        >= 17 + Y         
                        =  h(g(cons(X,Y)))
        
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(X)) -> f(X)
        - Weak TRS:
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + 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(h) = {1}
        
        Following symbols are considered usable:
          {f,g,h}
        TcT has computed the following interpretation:
             p(0) = 0      
          p(cons) = x1 + x2
             p(f) = x1     
             p(g) = x1     
             p(h) = x1     
             p(s) = 4 + x1 
        
        Following rules are strictly oriented:
        f(s(X)) = 4 + X
                > X    
                = f(X) 
        
        
        Following rules are (at-least) weakly oriented:
         g(cons(0(),Y)) =  Y              
                        >= Y              
                        =  g(Y)           
        
        g(cons(s(X),Y)) =  4 + X + Y      
                        >= 4 + X          
                        =  s(X)           
        
           h(cons(X,Y)) =  X + Y          
                        >= X + Y          
                        =  h(g(cons(X,Y)))
        
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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