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()))))
            unsafe(0()) -> 0()
            unsafe(S(x)) -> dbl(unsafe(x),0())
        - Signature:
            {dbl/2,unsafe/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,unsafe} 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()))))
            unsafe(0()) -> 0()
            unsafe(S(x)) -> dbl(unsafe(x),0())
        - Signature:
            {dbl/2,unsafe/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,unsafe} and constructors {0,S}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          unsafe(x){x -> S(x)} =
            unsafe(S(x)) ->^+ dbl(unsafe(x),0())
              = C[unsafe(x) = unsafe(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()))))
            unsafe(0()) -> 0()
            unsafe(S(x)) -> dbl(unsafe(x),0())
        - Signature:
            {dbl/2,unsafe/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,unsafe} 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) = {1}
        
        Following symbols are considered usable:
          {dbl,unsafe}
        TcT has computed the following interpretation:
               p(0) = 0          
               p(S) = 1          
             p(dbl) = 4*x1 + 8*x2
          p(unsafe) = 0          
        
        Following rules are strictly oriented:
        dbl(S(0()),S(0())) = 12             
                           > 1              
                           = S(S(S(S(0()))))
        
        
        Following rules are (at-least) weakly oriented:
          dbl(0(),y) =  8*y               
                     >= y                 
                     =  y                 
        
         unsafe(0()) =  0                 
                     >= 0                 
                     =  0()               
        
        unsafe(S(x)) =  0                 
                     >= 0                 
                     =  dbl(unsafe(x),0())
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            dbl(0(),y) -> y
            unsafe(0()) -> 0()
            unsafe(S(x)) -> dbl(unsafe(x),0())
        - Weak TRS:
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
        - Signature:
            {dbl/2,unsafe/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,unsafe} 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) = {1}
        
        Following symbols are considered usable:
          {dbl,unsafe}
        TcT has computed the following interpretation:
               p(0) = 1            
               p(S) = 1 + x1       
             p(dbl) = 1 + x1 + 4*x2
          p(unsafe) = 8 + 8*x1     
        
        Following rules are strictly oriented:
          dbl(0(),y) = 2 + 4*y           
                     > y                 
                     = y                 
        
         unsafe(0()) = 16                
                     > 1                 
                     = 0()               
        
        unsafe(S(x)) = 16 + 8*x          
                     > 13 + 8*x          
                     = dbl(unsafe(x),0())
        
        
        Following rules are (at-least) weakly oriented:
        dbl(S(0()),S(0())) =  11             
                           >= 5              
                           =  S(S(S(S(0()))))
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            dbl(0(),y) -> y
            dbl(S(0()),S(0())) -> S(S(S(S(0()))))
            unsafe(0()) -> 0()
            unsafe(S(x)) -> dbl(unsafe(x),0())
        - Signature:
            {dbl/2,unsafe/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dbl,unsafe} 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))