WORST_CASE(?,O(n^2))
* Step 1: Sum WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            loop(Cons(x,xs),Nil(),pp,ss) -> False()
            loop(Cons(x',xs'),Cons(x,xs),pp,ss) -> loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
            loop(Nil(),s,pp,ss) -> True()
            match1(p,s) -> loop(p,s,p,s)
        - Weak TRS:
            !EQ(0(),0()) -> True()
            !EQ(0(),S(y)) -> False()
            !EQ(S(x),0()) -> False()
            !EQ(S(x),S(y)) -> !EQ(x,y)
            loop[Ite](False(),p,s,pp,Cons(x,xs)) -> loop(pp,xs,pp,xs)
            loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) -> loop(xs',xs,pp,ss)
        - Signature:
            {!EQ/2,loop/4,loop[Ite]/5,match1/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {!EQ,loop,loop[Ite],match1} and constructors {0,Cons,False
            ,Nil,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            loop(Cons(x,xs),Nil(),pp,ss) -> False()
            loop(Cons(x',xs'),Cons(x,xs),pp,ss) -> loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
            loop(Nil(),s,pp,ss) -> True()
            match1(p,s) -> loop(p,s,p,s)
        - Weak TRS:
            !EQ(0(),0()) -> True()
            !EQ(0(),S(y)) -> False()
            !EQ(S(x),0()) -> False()
            !EQ(S(x),S(y)) -> !EQ(x,y)
            loop[Ite](False(),p,s,pp,Cons(x,xs)) -> loop(pp,xs,pp,xs)
            loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) -> loop(xs',xs,pp,ss)
        - Signature:
            {!EQ/2,loop/4,loop[Ite]/5,match1/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {!EQ,loop,loop[Ite],match1} and constructors {0,Cons,False
            ,Nil,S,True}
    + 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(loop[Ite]) = {1}
        
        Following symbols are considered usable:
          {!EQ,loop,loop[Ite],match1}
        TcT has computed the following interpretation:
                p(!EQ) = 0       
                  p(0) = 0       
               p(Cons) = 1       
              p(False) = 0       
                p(Nil) = 1       
                  p(S) = 1       
               p(True) = 0       
               p(loop) = 2       
          p(loop[Ite]) = 2 + 2*x1
             p(match1) = 2       
        
        Following rules are strictly oriented:
        loop(Cons(x,xs),Nil(),pp,ss) = 2      
                                     > 0      
                                     = False()
        
                 loop(Nil(),s,pp,ss) = 2      
                                     > 0      
                                     = True() 
        
        
        Following rules are (at-least) weakly oriented:
                                           !EQ(0(),0()) =  0                                                 
                                                        >= 0                                                 
                                                        =  True()                                            
        
                                          !EQ(0(),S(y)) =  0                                                 
                                                        >= 0                                                 
                                                        =  False()                                           
        
                                          !EQ(S(x),0()) =  0                                                 
                                                        >= 0                                                 
                                                        =  False()                                           
        
                                         !EQ(S(x),S(y)) =  0                                                 
                                                        >= 0                                                 
                                                        =  !EQ(x,y)                                          
        
                    loop(Cons(x',xs'),Cons(x,xs),pp,ss) =  2                                                 
                                                        >= 2                                                 
                                                        =  loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
        
                   loop[Ite](False(),p,s,pp,Cons(x,xs)) =  2                                                 
                                                        >= 2                                                 
                                                        =  loop(pp,xs,pp,xs)                                 
        
        loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) =  2                                                 
                                                        >= 2                                                 
                                                        =  loop(xs',xs,pp,ss)                                
        
                                            match1(p,s) =  2                                                 
                                                        >= 2                                                 
                                                        =  loop(p,s,p,s)                                     
        
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            loop(Cons(x',xs'),Cons(x,xs),pp,ss) -> loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
            match1(p,s) -> loop(p,s,p,s)
        - Weak TRS:
            !EQ(0(),0()) -> True()
            !EQ(0(),S(y)) -> False()
            !EQ(S(x),0()) -> False()
            !EQ(S(x),S(y)) -> !EQ(x,y)
            loop(Cons(x,xs),Nil(),pp,ss) -> False()
            loop(Nil(),s,pp,ss) -> True()
            loop[Ite](False(),p,s,pp,Cons(x,xs)) -> loop(pp,xs,pp,xs)
            loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) -> loop(xs',xs,pp,ss)
        - Signature:
            {!EQ/2,loop/4,loop[Ite]/5,match1/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {!EQ,loop,loop[Ite],match1} and constructors {0,Cons,False
            ,Nil,S,True}
    + 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(loop[Ite]) = {1}
        
        Following symbols are considered usable:
          {!EQ,loop,loop[Ite],match1}
        TcT has computed the following interpretation:
                p(!EQ) = 0        
                  p(0) = 0        
               p(Cons) = x2       
              p(False) = 0        
                p(Nil) = 0        
                  p(S) = 1        
               p(True) = 0        
               p(loop) = 2*x4     
          p(loop[Ite]) = x1 + 2*x5
             p(match1) = 4 + 2*x2 
        
        Following rules are strictly oriented:
        match1(p,s) = 4 + 2*s      
                    > 2*s          
                    = loop(p,s,p,s)
        
        
        Following rules are (at-least) weakly oriented:
                                           !EQ(0(),0()) =  0                                                 
                                                        >= 0                                                 
                                                        =  True()                                            
        
                                          !EQ(0(),S(y)) =  0                                                 
                                                        >= 0                                                 
                                                        =  False()                                           
        
                                          !EQ(S(x),0()) =  0                                                 
                                                        >= 0                                                 
                                                        =  False()                                           
        
                                         !EQ(S(x),S(y)) =  0                                                 
                                                        >= 0                                                 
                                                        =  !EQ(x,y)                                          
        
                           loop(Cons(x,xs),Nil(),pp,ss) =  2*ss                                              
                                                        >= 0                                                 
                                                        =  False()                                           
        
                    loop(Cons(x',xs'),Cons(x,xs),pp,ss) =  2*ss                                              
                                                        >= 2*ss                                              
                                                        =  loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
        
                                    loop(Nil(),s,pp,ss) =  2*ss                                              
                                                        >= 0                                                 
                                                        =  True()                                            
        
                   loop[Ite](False(),p,s,pp,Cons(x,xs)) =  2*xs                                              
                                                        >= 2*xs                                              
                                                        =  loop(pp,xs,pp,xs)                                 
        
        loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) =  2*ss                                              
                                                        >= 2*ss                                              
                                                        =  loop(xs',xs,pp,ss)                                
        
* Step 4: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            loop(Cons(x',xs'),Cons(x,xs),pp,ss) -> loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
        - Weak TRS:
            !EQ(0(),0()) -> True()
            !EQ(0(),S(y)) -> False()
            !EQ(S(x),0()) -> False()
            !EQ(S(x),S(y)) -> !EQ(x,y)
            loop(Cons(x,xs),Nil(),pp,ss) -> False()
            loop(Nil(),s,pp,ss) -> True()
            loop[Ite](False(),p,s,pp,Cons(x,xs)) -> loop(pp,xs,pp,xs)
            loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) -> loop(xs',xs,pp,ss)
            match1(p,s) -> loop(p,s,p,s)
        - Signature:
            {!EQ/2,loop/4,loop[Ite]/5,match1/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {!EQ,loop,loop[Ite],match1} and constructors {0,Cons,False
            ,Nil,S,True}
    + 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(loop[Ite]) = {1}
        
        Following symbols are considered usable:
          {!EQ,loop,loop[Ite],match1}
        TcT has computed the following interpretation:
                p(!EQ) = 0                                       
                  p(0) = 0                                       
               p(Cons) = 1 + x2                                  
              p(False) = 0                                       
                p(Nil) = 0                                       
                  p(S) = 0                                       
               p(True) = 0                                       
               p(loop) = 1 + x1 + 2*x3*x4 + x4^2                 
          p(loop[Ite]) = x1 + x1*x4 + x1*x5 + x2 + 2*x4*x5 + x5^2
             p(match1) = 1 + 2*x1 + 3*x1*x2 + 3*x1^2 + 2*x2^2    
        
        Following rules are strictly oriented:
        loop(Cons(x',xs'),Cons(x,xs),pp,ss) = 2 + 2*pp*ss + ss^2 + xs'                          
                                            > 1 + 2*pp*ss + ss^2 + xs'                          
                                            = loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
        
        
        Following rules are (at-least) weakly oriented:
                                           !EQ(0(),0()) =  0                                   
                                                        >= 0                                   
                                                        =  True()                              
        
                                          !EQ(0(),S(y)) =  0                                   
                                                        >= 0                                   
                                                        =  False()                             
        
                                          !EQ(S(x),0()) =  0                                   
                                                        >= 0                                   
                                                        =  False()                             
        
                                         !EQ(S(x),S(y)) =  0                                   
                                                        >= 0                                   
                                                        =  !EQ(x,y)                            
        
                           loop(Cons(x,xs),Nil(),pp,ss) =  2 + 2*pp*ss + ss^2 + xs             
                                                        >= 0                                   
                                                        =  False()                             
        
                                    loop(Nil(),s,pp,ss) =  1 + 2*pp*ss + ss^2                  
                                                        >= 0                                   
                                                        =  True()                              
        
                   loop[Ite](False(),p,s,pp,Cons(x,xs)) =  1 + p + 2*pp + 2*pp*xs + 2*xs + xs^2
                                                        >= 1 + pp + 2*pp*xs + xs^2             
                                                        =  loop(pp,xs,pp,xs)                   
        
        loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) =  1 + 2*pp*ss + ss^2 + xs'            
                                                        >= 1 + 2*pp*ss + ss^2 + xs'            
                                                        =  loop(xs',xs,pp,ss)                  
        
                                            match1(p,s) =  1 + 2*p + 3*p*s + 3*p^2 + 2*s^2     
                                                        >= 1 + p + 2*p*s + s^2                 
                                                        =  loop(p,s,p,s)                       
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            !EQ(0(),0()) -> True()
            !EQ(0(),S(y)) -> False()
            !EQ(S(x),0()) -> False()
            !EQ(S(x),S(y)) -> !EQ(x,y)
            loop(Cons(x,xs),Nil(),pp,ss) -> False()
            loop(Cons(x',xs'),Cons(x,xs),pp,ss) -> loop[Ite](!EQ(x',x),Cons(x',xs'),Cons(x,xs),pp,ss)
            loop(Nil(),s,pp,ss) -> True()
            loop[Ite](False(),p,s,pp,Cons(x,xs)) -> loop(pp,xs,pp,xs)
            loop[Ite](True(),Cons(x',xs'),Cons(x,xs),pp,ss) -> loop(xs',xs,pp,ss)
            match1(p,s) -> loop(p,s,p,s)
        - Signature:
            {!EQ/2,loop/4,loop[Ite]/5,match1/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {!EQ,loop,loop[Ite],match1} and constructors {0,Cons,False
            ,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))