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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            dbl(0(),y) -> y
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
            save(0()) -> 0()
            save(S(x)) -> dbl(0(),save(x))
        - Signature:
            {dbl/2,save/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,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(dbl) = {2}
        
        Following symbols are considered usable:
          {dbl,save}
        TcT has computed the following interpretation:
             p(0) = 0   
             p(S) = 1   
           p(dbl) = 2*x2
          p(save) = 0   
        
        Following rules are strictly oriented:
        dbl(S(0()),S(0())) = 2              
                           > 1              
                           = S(S(S(S(0()))))
        
        
        Following rules are (at-least) weakly oriented:
        dbl(0(),y) =  2*y             
                   >= y               
                   =  y               
        
         save(0()) =  0               
                   >= 0               
                   =  0()             
        
        save(S(x)) =  0               
                   >= 0               
                   =  dbl(0(),save(x))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            dbl(0(),y) -> y
            save(0()) -> 0()
            save(S(x)) -> dbl(0(),save(x))
        - Weak TRS:
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
        - Signature:
            {dbl/2,save/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,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(dbl) = {2}
        
        Following symbols are considered usable:
          {dbl,save}
        TcT has computed the following interpretation:
             p(0) = 0 
             p(S) = 4 
           p(dbl) = x2
          p(save) = 9 
        
        Following rules are strictly oriented:
        save(0()) = 9  
                  > 0  
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
                dbl(0(),y) =  y               
                           >= y               
                           =  y               
        
        dbl(S(0()),S(0())) =  4               
                           >= 4               
                           =  S(S(S(S(0())))) 
        
                save(S(x)) =  9               
                           >= 9               
                           =  dbl(0(),save(x))
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            dbl(0(),y) -> y
            save(S(x)) -> dbl(0(),save(x))
        - Weak TRS:
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
            save(0()) -> 0()
        - Signature:
            {dbl/2,save/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,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(dbl) = {2}
        
        Following symbols are considered usable:
          {dbl,save}
        TcT has computed the following interpretation:
             p(0) = 6          
             p(S) = 2 + x1     
           p(dbl) = 1 + x1 + x2
          p(save) = 4*x1       
        
        Following rules are strictly oriented:
        dbl(0(),y) = 7 + y           
                   > y               
                   = y               
        
        save(S(x)) = 8 + 4*x         
                   > 7 + 4*x         
                   = dbl(0(),save(x))
        
        
        Following rules are (at-least) weakly oriented:
        dbl(S(0()),S(0())) =  17             
                           >= 14             
                           =  S(S(S(S(0()))))
        
                 save(0()) =  24             
                           >= 6              
                           =  0()            
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            dbl(0(),y) -> y
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
            save(0()) -> 0()
            save(S(x)) -> dbl(0(),save(x))
        - Signature:
            {dbl/2,save/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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