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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(half(x))))
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} 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(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {half,log}
        TcT has computed the following interpretation:
             p(0) = 0       
          p(half) = 0       
           p(log) = 1 + 4*x1
             p(s) = x1      
        
        Following rules are strictly oriented:
        log(s(0())) = 1  
                    > 0  
                    = 0()
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  0                 
                      >= 0                 
                      =  0()               
        
        half(s(s(x))) =  0                 
                      >= 0                 
                      =  s(half(x))        
        
         log(s(s(x))) =  1 + 4*x           
                      >= 1                 
                      =  s(log(s(half(x))))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(s(x))) -> s(log(s(half(x))))
        - Weak TRS:
            log(s(0())) -> 0()
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} 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(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {half,log}
        TcT has computed the following interpretation:
             p(0) = 4        
          p(half) = x1       
           p(log) = 15 + 2*x1
             p(s) = 4 + x1   
        
        Following rules are strictly oriented:
        half(s(s(x))) = 8 + x             
                      > 4 + x             
                      = s(half(x))        
        
         log(s(s(x))) = 31 + 2*x          
                      > 27 + 2*x          
                      = s(log(s(half(x))))
        
        
        Following rules are (at-least) weakly oriented:
          half(0()) =  4  
                    >= 4  
                    =  0()
        
        log(s(0())) =  31 
                    >= 4  
                    =  0()
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
        - Weak TRS:
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(half(x))))
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} 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(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {half,log}
        TcT has computed the following interpretation:
             p(0) = 0       
          p(half) = 1 + x1  
           p(log) = 2 + 2*x1
             p(s) = 2 + x1  
        
        Following rules are strictly oriented:
        half(0()) = 1  
                  > 0  
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
        half(s(s(x))) =  5 + x             
                      >= 3 + x             
                      =  s(half(x))        
        
          log(s(0())) =  6                 
                      >= 0                 
                      =  0()               
        
         log(s(s(x))) =  10 + 2*x          
                      >= 10 + 2*x          
                      =  s(log(s(half(x))))
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            half(0()) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(half(x))))
        - Signature:
            {half/1,log/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {half,log} 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))