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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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(*) = {2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {*,-,exp}
        TcT has computed the following interpretation:
            p(*) = 1 + x2       
            p(+) = x2           
            p(-) = x1 + 9*x2    
            p(0) = 1            
          p(exp) = 8 + x1 + 2*x2
            p(s) = 2 + x1       
        
        Following rules are strictly oriented:
            -(x,0()) = 9 + x        
                     > x            
                     = x            
        
        -(s(x),s(y)) = 20 + x + 9*y 
                     > x + 9*y      
                     = -(x,y)       
        
          exp(x,0()) = 10 + x       
                     > 3            
                     = s(0())       
        
         exp(x,s(y)) = 12 + x + 2*y 
                     > 9 + x + 2*y  
                     = *(x,exp(x,y))
        
        
        Following rules are (at-least) weakly oriented:
         *(0(),y) =  1 + y      
                  >= 1          
                  =  0()        
        
        *(s(x),y) =  1 + y      
                  >= 1 + y      
                  =  +(y,*(x,y))
        
         -(0(),y) =  1 + 9*y    
                  >= 1          
                  =  0()        
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(0(),y) -> 0()
        - Weak TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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(*) = {2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {*,-,exp}
        TcT has computed the following interpretation:
            p(*) = x2      
            p(+) = x2      
            p(-) = 7 + 2*x1
            p(0) = 0       
          p(exp) = 12      
            p(s) = 10 + x1 
        
        Following rules are strictly oriented:
        -(0(),y) = 7  
                 > 0  
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
            *(0(),y) =  y            
                     >= 0            
                     =  0()          
        
           *(s(x),y) =  y            
                     >= y            
                     =  +(y,*(x,y))  
        
            -(x,0()) =  7 + 2*x      
                     >= x            
                     =  x            
        
        -(s(x),s(y)) =  27 + 2*x     
                     >= 7 + 2*x      
                     =  -(x,y)       
        
          exp(x,0()) =  12           
                     >= 10           
                     =  s(0())       
        
         exp(x,s(y)) =  12           
                     >= 12           
                     =  *(x,exp(x,y))
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
        - Weak TRS:
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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(*) = {2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {*,-,exp}
        TcT has computed the following interpretation:
            p(*) = 9 + x2         
            p(+) = x2             
            p(-) = 8*x1           
            p(0) = 0              
          p(exp) = 4 + 2*x1 + 8*x2
            p(s) = 3 + x1         
        
        Following rules are strictly oriented:
        *(0(),y) = 9 + y
                 > 0    
                 = 0()  
        
        
        Following rules are (at-least) weakly oriented:
           *(s(x),y) =  9 + y         
                     >= 9 + y         
                     =  +(y,*(x,y))   
        
            -(x,0()) =  8*x           
                     >= x             
                     =  x             
        
            -(0(),y) =  0             
                     >= 0             
                     =  0()           
        
        -(s(x),s(y)) =  24 + 8*x      
                     >= 8*x           
                     =  -(x,y)        
        
          exp(x,0()) =  4 + 2*x       
                     >= 3             
                     =  s(0())        
        
         exp(x,s(y)) =  28 + 2*x + 8*y
                     >= 13 + 2*x + 8*y
                     =  *(x,exp(x,y)) 
        
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(s(x),y) -> +(y,*(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(*) = {2},
          uargs(+) = {2}
        
        Following symbols are considered usable:
          {*,-,exp}
        TcT has computed the following interpretation:
            p(*) = 2 + x1 + x2        
            p(+) = x2                 
            p(-) = x1*x2 + 2*x1^2 + x2
            p(0) = 2                  
          p(exp) = x1*x2 + 2*x2       
            p(s) = 1 + x1             
        
        Following rules are strictly oriented:
        *(s(x),y) = 3 + x + y  
                  > 2 + x + y  
                  = +(y,*(x,y))
        
        
        Following rules are (at-least) weakly oriented:
            *(0(),y) =  4 + y                      
                     >= 2                          
                     =  0()                        
        
            -(x,0()) =  2 + 2*x + 2*x^2            
                     >= x                          
                     =  x                          
        
            -(0(),y) =  8 + 3*y                    
                     >= 2                          
                     =  0()                        
        
        -(s(x),s(y)) =  4 + 5*x + x*y + 2*x^2 + 2*y
                     >= x*y + 2*x^2 + y            
                     =  -(x,y)                     
        
          exp(x,0()) =  4 + 2*x                    
                     >= 3                          
                     =  s(0())                     
        
         exp(x,s(y)) =  2 + x + x*y + 2*y          
                     >= 2 + x + x*y + 2*y          
                     =  *(x,exp(x,y))              
        
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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^2))