YES(?,O(n^2))
9.09/3.68	YES(?,O(n^2))
9.09/3.68	
9.09/3.68	We are left with following problem, upon which TcT provides the
9.09/3.68	certificate YES(?,O(n^2)).
9.09/3.68	
9.09/3.68	Strict Trs:
9.09/3.68	  { from(X) -> cons(X, n__from(s(X)))
9.09/3.68	  , from(X) -> n__from(X)
9.09/3.68	  , cons(X1, X2) -> n__cons(X1, X2)
9.09/3.68	  , 2ndspos(s(N), cons(X, n__cons(Y, Z))) ->
9.09/3.68	    rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
9.09/3.68	  , 2ndspos(0(), Z) -> rnil()
9.09/3.68	  , activate(X) -> X
9.09/3.68	  , activate(n__from(X)) -> from(X)
9.09/3.68	  , activate(n__cons(X1, X2)) -> cons(X1, X2)
9.09/3.68	  , 2ndsneg(s(N), cons(X, n__cons(Y, Z))) ->
9.09/3.68	    rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
9.09/3.68	  , 2ndsneg(0(), Z) -> rnil()
9.09/3.68	  , pi(X) -> 2ndspos(X, from(0()))
9.09/3.68	  , plus(s(X), Y) -> s(plus(X, Y))
9.09/3.68	  , plus(0(), Y) -> Y
9.09/3.68	  , times(s(X), Y) -> plus(Y, times(X, Y))
9.09/3.68	  , times(0(), Y) -> 0()
9.09/3.68	  , square(X) -> times(X, X) }
9.09/3.68	Obligation:
9.09/3.68	  innermost runtime complexity
9.09/3.68	Answer:
9.09/3.68	  YES(?,O(n^2))
9.09/3.68	
9.09/3.68	Arguments of following rules are not normal-forms:
9.09/3.68	
9.09/3.68	{ 2ndspos(s(N), cons(X, n__cons(Y, Z))) ->
9.09/3.68	  rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
9.09/3.68	, 2ndsneg(s(N), cons(X, n__cons(Y, Z))) ->
9.09/3.69	  rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) }
9.09/3.69	
9.09/3.69	All above mentioned rules can be savely removed.
9.09/3.69	
9.09/3.69	We are left with following problem, upon which TcT provides the
9.09/3.69	certificate YES(?,O(n^2)).
9.09/3.69	
9.09/3.69	Strict Trs:
9.09/3.69	  { from(X) -> cons(X, n__from(s(X)))
9.09/3.69	  , from(X) -> n__from(X)
9.09/3.69	  , cons(X1, X2) -> n__cons(X1, X2)
9.09/3.69	  , 2ndspos(0(), Z) -> rnil()
9.09/3.69	  , activate(X) -> X
9.09/3.69	  , activate(n__from(X)) -> from(X)
9.09/3.69	  , activate(n__cons(X1, X2)) -> cons(X1, X2)
9.09/3.69	  , 2ndsneg(0(), Z) -> rnil()
9.09/3.69	  , pi(X) -> 2ndspos(X, from(0()))
9.09/3.69	  , plus(s(X), Y) -> s(plus(X, Y))
9.09/3.69	  , plus(0(), Y) -> Y
9.09/3.69	  , times(s(X), Y) -> plus(Y, times(X, Y))
9.09/3.69	  , times(0(), Y) -> 0()
9.09/3.69	  , square(X) -> times(X, X) }
9.09/3.69	Obligation:
9.09/3.69	  innermost runtime complexity
9.09/3.69	Answer:
9.09/3.69	  YES(?,O(n^2))
9.09/3.69	
9.09/3.69	The input was oriented with the instance of 'Small Polynomial Path
9.09/3.69	Order (PS,2-bounded)' as induced by the safe mapping
9.09/3.69	
9.09/3.69	 safe(from) = {1}, safe(cons) = {1, 2}, safe(n__from) = {1},
9.09/3.69	 safe(s) = {1}, safe(2ndspos) = {1, 2}, safe(0) = {},
9.09/3.69	 safe(rnil) = {}, safe(n__cons) = {1, 2}, safe(activate) = {1},
9.09/3.69	 safe(2ndsneg) = {}, safe(pi) = {}, safe(plus) = {2},
9.09/3.69	 safe(times) = {}, safe(square) = {}
9.09/3.69	
9.09/3.69	and precedence
9.09/3.69	
9.09/3.69	 from > cons, from > plus, activate > from, activate > cons,
9.09/3.69	 activate > 2ndspos, activate > plus, 2ndsneg > from,
9.09/3.69	 2ndsneg > cons, 2ndsneg > plus, 2ndsneg > times, pi > from,
9.09/3.69	 pi > cons, pi > 2ndspos, pi > plus, times > cons, times > plus,
9.09/3.69	 square > from, square > cons, square > 2ndspos, square > 2ndsneg,
9.09/3.69	 square > plus, square > times, cons ~ plus, activate ~ pi .
9.09/3.69	
9.09/3.69	Following symbols are considered recursive:
9.09/3.69	
9.09/3.69	 {cons, plus, times}
9.09/3.69	
9.09/3.69	The recursion depth is 2.
9.09/3.69	
9.09/3.69	For your convenience, here are the satisfied ordering constraints:
9.09/3.69	
9.09/3.69	                       from(; X) > cons(; X,  n__from(; s(; X)))
9.09/3.69	                                                                
9.09/3.69	                       from(; X) > n__from(; X)                 
9.09/3.69	                                                                
9.09/3.69	                 cons(; X1,  X2) > n__cons(; X1,  X2)           
9.09/3.69	                                                                
9.09/3.69	              2ndspos(; 0(),  Z) > rnil()                       
9.09/3.69	                                                                
9.09/3.69	                   activate(; X) > X                            
9.09/3.69	                                                                
9.09/3.69	        activate(; n__from(; X)) > from(; X)                    
9.09/3.69	                                                                
9.09/3.69	  activate(; n__cons(; X1,  X2)) > cons(; X1,  X2)              
9.09/3.69	                                                                
9.09/3.69	               2ndsneg(0(),  Z;) > rnil()                       
9.09/3.69	                                                                
9.09/3.69	                          pi(X;) > 2ndspos(; X,  from(; 0()))   
9.09/3.69	                                                                
9.09/3.69	                 plus(s(; X); Y) > s(; plus(X; Y))              
9.09/3.69	                                                                
9.09/3.69	                    plus(0(); Y) > Y                            
9.09/3.69	                                                                
9.09/3.69	              times(s(; X),  Y;) > plus(Y; times(X,  Y;))       
9.09/3.69	                                                                
9.09/3.69	                 times(0(),  Y;) > 0()                          
9.09/3.69	                                                                
9.09/3.69	                      square(X;) > times(X,  X;)                
9.09/3.69	                                                                
9.09/3.69	
9.09/3.69	Hurray, we answered YES(?,O(n^2))
9.25/3.70	EOF