WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square ,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,7,9,10,12,15} by application of Pre({1,4,7,9,10,12,15}) = {2,3,5,6,8,11,13,14,16}. Here rules are labelled as follows: 1: 2ndsneg#(0(),Z) -> c_1() 2: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 3: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 4: 2ndspos#(0(),Z) -> c_4() 5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) 7: activate#(X) -> c_7() 8: activate#(n__from(X)) -> c_8(from#(X)) 9: from#(X) -> c_9() 10: from#(X) -> c_10() 11: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 12: plus#(0(),Y) -> c_12() 13: plus#(s(X),Y) -> c_13(plus#(X,Y)) 14: square#(X) -> c_14(times#(X,X)) 15: times#(0(),Y) -> c_15() 16: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) * Step 4: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(n__from(X)) -> c_8(from#(X)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_4() activate#(X) -> c_7() from#(X) -> c_9() from#(X) -> c_10() plus#(0(),Y) -> c_12() times#(0(),Y) -> c_15() - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {1,2,3,4}. Here rules are labelled as follows: 1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) 5: activate#(n__from(X)) -> c_8(from#(X)) 6: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 7: plus#(s(X),Y) -> c_13(plus#(X,Y)) 8: square#(X) -> c_14(times#(X,X)) 9: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 10: 2ndsneg#(0(),Z) -> c_1() 11: 2ndspos#(0(),Z) -> c_4() 12: activate#(X) -> c_7() 13: from#(X) -> c_9() 14: from#(X) -> c_10() 15: plus#(0(),Y) -> c_12() 16: times#(0(),Y) -> c_15() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_4() activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() plus#(0(),Y) -> c_12() times#(0(),Y) -> c_15() - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_2 activate#(X) -> c_7():11 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 -->_2 activate#(X) -> c_7():11 -->_1 2ndspos#(0(),Z) -> c_4():10 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_2 activate#(X) -> c_7():11 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_2 activate#(X) -> c_7():11 -->_1 2ndsneg#(0(),Z) -> c_1():9 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) -->_2 from#(X) -> c_10():14 -->_2 from#(X) -> c_9():13 -->_1 2ndspos#(0(),Z) -> c_4():10 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(0(),Y) -> c_12():15 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 times#(0(),Y) -> c_15():16 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(0(),Y) -> c_15():16 -->_1 plus#(0(),Y) -> c_12():15 -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 9:W:2ndsneg#(0(),Z) -> c_1() 10:W:2ndspos#(0(),Z) -> c_4() 11:W:activate#(X) -> c_7() 12:W:activate#(n__from(X)) -> c_8(from#(X)) -->_1 from#(X) -> c_10():14 -->_1 from#(X) -> c_9():13 13:W:from#(X) -> c_9() 14:W:from#(X) -> c_10() 15:W:plus#(0(),Y) -> c_12() 16:W:times#(0(),Y) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: times#(0(),Y) -> c_15() 15: plus#(0(),Y) -> c_12() 10: 2ndspos#(0(),Z) -> c_4() 9: 2ndsneg#(0(),Z) -> c_1() 11: activate#(X) -> c_7() 12: activate#(n__from(X)) -> c_8(from#(X)) 13: from#(X) -> c_9() 14: from#(X) -> c_10() * Step 6: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) * Step 7: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(7,square#(X) -> c_14(times#(X,X)))] * Step 8: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1 ,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} Problem (S) - Strict DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1 ,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} ** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 5:W:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 6:W:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 8:W:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 6: plus#(s(X),Y) -> c_13(plus#(X,Y)) ** Step 8.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) ** Step 8.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) Consider the set of all dependency pairs 1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) 5: pi#(X) -> c_11(2ndspos#(X,from(0()))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 8.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_11) = {1} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [1] p(2ndspos) = [1] x1 + [2] x2 + [1] p(activate) = [3] p(cons) = [1] x1 + [0] p(cons2) = [1] x1 + [1] x2 + [0] p(from) = [9] p(n__from) = [3] p(negrecip) = [1] x1 + [1] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [2] p(rcons) = [1] x2 + [2] p(rnil) = [8] p(s) = [1] x1 + [2] p(square) = [2] x1 + [2] p(times) = [2] x1 + [8] p(2ndsneg#) = [4] x1 + [0] p(2ndspos#) = [4] x1 + [0] p(activate#) = [2] x1 + [0] p(from#) = [1] x1 + [1] p(pi#) = [4] x1 + [0] p(plus#) = [1] x1 + [1] p(square#) = [2] x1 + [0] p(times#) = [1] x1 + [1] x2 + [4] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [7] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [8] p(c_7) = [1] p(c_8) = [1] x1 + [1] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] p(c_15) = [0] p(c_16) = [1] x1 + [2] Following rules are strictly oriented: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [4] N + [8] > [4] N + [7] = c_3(2ndspos#(N,activate(Z))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N),cons(X,Z)) = [4] N + [8] >= [4] N + [8] = c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons(X,Z)) = [4] N + [8] >= [4] N + [8] = c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) = [4] N + [8] >= [4] N + [8] = c_6(2ndsneg#(N,activate(Z))) pi#(X) = [4] X + [0] >= [4] X + [0] = c_11(2ndspos#(X,from(0()))) *** Step 8.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak DPs: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 8.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 2:W:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 3:W:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 4:W:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1 5:W:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: pi#(X) -> c_11(2ndspos#(X,from(0()))) 1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) 3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) *** Step 8.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: pi#(X) -> c_11(2ndspos#(X,from(0()))) 2: plus#(s(X),Y) -> c_13(plus#(X,Y)) 3: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 4: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 5: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 6: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 7: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) ** Step 8.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 2:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):2 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 3:W:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):4 4:W:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):5 5:W:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6 6:W:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):4 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):3 7:W:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: pi#(X) -> c_11(2ndspos#(X,from(0()))) 3: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) 5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 4: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) ** Step 8.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1 ,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} Problem (S) - Strict DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1 ,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} *** Step 8.b:3.a:1: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) *** Step 8.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: plus#(s(X),Y) -> c_13(plus#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 8.b:3.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_13) = {1}, uargs(c_16) = {1,2} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = 1 p(2ndsneg) = 1 + 4*x1^2 + x2 p(2ndspos) = 1 + 4*x1 + x1^2 p(activate) = 1 p(cons) = 1 p(cons2) = x2 p(from) = 4 + 4*x1 p(n__from) = 0 p(negrecip) = 0 p(pi) = 1 p(plus) = 3*x1 + 4*x1*x2 + 2*x1^2 + 4*x2 p(posrecip) = 0 p(rcons) = 0 p(rnil) = 1 p(s) = 2 + x1 p(square) = 4 + 4*x1^2 p(times) = x1 + x1*x2 + x2 + 2*x2^2 p(2ndsneg#) = 2 + 2*x1 + 2*x1^2 + x2 + 2*x2^2 p(2ndspos#) = 4*x1*x2 + x2^2 p(activate#) = 0 p(from#) = 2 + x1 + 2*x1^2 p(pi#) = x1 p(plus#) = 4*x1 p(square#) = 0 p(times#) = 2 + 2*x1*x2 + 2*x2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 1 + x1 p(c_4) = 2 p(c_5) = 1 p(c_6) = 1 + x1 p(c_7) = 1 p(c_8) = 4 p(c_9) = 0 p(c_10) = 1 p(c_11) = 1 p(c_12) = 0 p(c_13) = x1 p(c_14) = 1 p(c_15) = 0 p(c_16) = x1 + x2 Following rules are strictly oriented: plus#(s(X),Y) = 8 + 4*X > 4*X = c_13(plus#(X,Y)) Following rules are (at-least) weakly oriented: times#(s(X),Y) = 2 + 2*X*Y + 6*Y >= 2 + 2*X*Y + 6*Y = c_16(plus#(Y,times(X,Y)),times#(X,Y)) **** Step 8.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 8.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 2:W:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):2 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 1: plus#(s(X),Y) -> c_13(plus#(X,Y)) **** Step 8.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 8.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):2 -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):1 2:W:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: plus#(s(X),Y) -> c_13(plus#(X,Y)) *** Step 8.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: times#(s(X),Y) -> c_16(times#(X,Y)) *** Step 8.b:3.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: times#(s(X),Y) -> c_16(times#(X,Y)) *** Step 8.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: times#(s(X),Y) -> c_16(times#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 8.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_16) = {1} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [2] p(2ndsneg) = [2] x1 + [1] x2 + [0] p(2ndspos) = [1] x1 + [1] x2 + [0] p(activate) = [2] x1 + [1] p(cons) = [1] x1 + [1] x2 + [1] p(cons2) = [1] x1 + [1] x2 + [0] p(from) = [0] p(n__from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [0] p(times) = [0] p(2ndsneg#) = [0] p(2ndspos#) = [0] p(activate#) = [1] p(from#) = [1] p(pi#) = [8] x1 + [0] p(plus#) = [1] p(square#) = [2] x1 + [0] p(times#) = [4] x1 + [4] p(c_1) = [8] p(c_2) = [8] x1 + [8] p(c_3) = [8] p(c_4) = [8] p(c_5) = [8] p(c_6) = [2] p(c_7) = [1] p(c_8) = [2] x1 + [0] p(c_9) = [2] p(c_10) = [4] p(c_11) = [1] x1 + [1] p(c_12) = [4] p(c_13) = [1] p(c_14) = [2] x1 + [1] p(c_15) = [4] p(c_16) = [1] x1 + [2] Following rules are strictly oriented: times#(s(X),Y) = [4] X + [8] > [4] X + [6] = c_16(times#(X,Y)) Following rules are (at-least) weakly oriented: **** Step 8.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 8.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:times#(s(X),Y) -> c_16(times#(X,Y)) -->_1 times#(s(X),Y) -> c_16(times#(X,Y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: times#(s(X),Y) -> c_16(times#(X,Y)) **** Step 8.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))