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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(+(x,y)) -> *(f(x),f(y))
            f(+(x,s(0()))) -> +(s(s(0())),f(x))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {f/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} 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(*) = {1,2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(*) = x1 + x2    
          p(+) = 4 + x1 + x2
          p(0) = 0          
          p(f) = x1         
          p(s) = 0          
        
        Following rules are strictly oriented:
        f(+(x,y)) = 4 + x + y   
                  > x + y       
                  = *(f(x),f(y))
        
        
        Following rules are (at-least) weakly oriented:
        f(+(x,s(0()))) =  4 + x              
                       >= 4 + x              
                       =  +(s(s(0())),f(x))  
        
                f(0()) =  0                  
                       >= 0                  
                       =  s(0())             
        
             f(s(0())) =  0                  
                       >= 0                  
                       =  *(s(s(0())),f(0()))
        
             f(s(0())) =  0                  
                       >= 0                  
                       =  s(s(0()))          
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(+(x,s(0()))) -> +(s(s(0())),f(x))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Weak TRS:
            f(+(x,y)) -> *(f(x),f(y))
        - Signature:
            {f/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} 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(*) = {1,2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(*) = x1 + x2    
          p(+) = 1 + x1 + x2
          p(0) = 0          
          p(f) = 8*x1       
          p(s) = 0          
        
        Following rules are strictly oriented:
        f(+(x,s(0()))) = 8 + 8*x          
                       > 1 + 8*x          
                       = +(s(s(0())),f(x))
        
        
        Following rules are (at-least) weakly oriented:
        f(+(x,y)) =  8 + 8*x + 8*y      
                  >= 8*x + 8*y          
                  =  *(f(x),f(y))       
        
           f(0()) =  0                  
                  >= 0                  
                  =  s(0())             
        
        f(s(0())) =  0                  
                  >= 0                  
                  =  *(s(s(0())),f(0()))
        
        f(s(0())) =  0                  
                  >= 0                  
                  =  s(s(0()))          
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Weak TRS:
            f(+(x,y)) -> *(f(x),f(y))
            f(+(x,s(0()))) -> +(s(s(0())),f(x))
        - Signature:
            {f/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} 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(*) = {1,2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(*) = 2 + x1 + x2
          p(+) = 1 + x1 + x2
          p(0) = 0          
          p(f) = 4 + 8*x1   
          p(s) = 2          
        
        Following rules are strictly oriented:
           f(0()) = 4                  
                  > 2                  
                  = s(0())             
        
        f(s(0())) = 20                 
                  > 8                  
                  = *(s(s(0())),f(0()))
        
        f(s(0())) = 20                 
                  > 2                  
                  = s(s(0()))          
        
        
        Following rules are (at-least) weakly oriented:
             f(+(x,y)) =  12 + 8*x + 8*y   
                       >= 10 + 8*x + 8*y   
                       =  *(f(x),f(y))     
        
        f(+(x,s(0()))) =  28 + 8*x         
                       >= 7 + 8*x          
                       =  +(s(s(0())),f(x))
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(+(x,y)) -> *(f(x),f(y))
            f(+(x,s(0()))) -> +(s(s(0())),f(x))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {f/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} 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))