WORST_CASE(?,O(1)) * Step 1: Sum WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1 ,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1 ,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) activate#(n__dbl(X)) -> c_3(dbl#(X)) activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__terms(X)) -> c_6(terms#(X)) add#(X1,X2) -> c_7() add#(0(),X) -> c_8() add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_10() dbl#(0()) -> c_11() dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_13() first#(0(),X) -> c_14() first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) s#(X) -> c_16() sqr#(0()) -> c_17() sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) terms#(N) -> c_19(sqr#(N),s#(N)) terms#(X) -> c_20() Weak DPs and mark the set of starting terms. * Step 3: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) activate#(n__dbl(X)) -> c_3(dbl#(X)) activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__terms(X)) -> c_6(terms#(X)) add#(X1,X2) -> c_7() add#(0(),X) -> c_8() add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_10() dbl#(0()) -> c_11() dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_13() first#(0(),X) -> c_14() first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) s#(X) -> c_16() sqr#(0()) -> c_17() sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) terms#(N) -> c_19(sqr#(N),s#(N)) terms#(X) -> c_20() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1 ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/0,c_18/5,c_19/2 ,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,7,8,9,10,11,12,13,14,15,16,17,18,20} by application of Pre({1,7,8,9,10,11,12,13,14,15,16,17,18,20}) = {2,3,4,5,6,19}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) 3: activate#(n__dbl(X)) -> c_3(dbl#(X)) 4: activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) 5: activate#(n__s(X)) -> c_5(s#(X)) 6: activate#(n__terms(X)) -> c_6(terms#(X)) 7: add#(X1,X2) -> c_7() 8: add#(0(),X) -> c_8() 9: add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) 10: dbl#(X) -> c_10() 11: dbl#(0()) -> c_11() 12: dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) 13: first#(X1,X2) -> c_13() 14: first#(0(),X) -> c_14() 15: first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) 16: s#(X) -> c_16() 17: sqr#(0()) -> c_17() 18: sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) 19: terms#(N) -> c_19(sqr#(N),s#(N)) 20: terms#(X) -> c_20() * Step 4: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) activate#(n__dbl(X)) -> c_3(dbl#(X)) activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__terms(X)) -> c_6(terms#(X)) terms#(N) -> c_19(sqr#(N),s#(N)) - Weak DPs: activate#(X) -> c_1() add#(X1,X2) -> c_7() add#(0(),X) -> c_8() add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_10() dbl#(0()) -> c_11() dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_13() first#(0(),X) -> c_14() first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) s#(X) -> c_16() sqr#(0()) -> c_17() sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) terms#(X) -> c_20() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1 ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/0,c_18/5,c_19/2 ,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6} by application of Pre({1,2,3,4,6}) = {5}. Here rules are labelled as follows: 1: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) 2: activate#(n__dbl(X)) -> c_3(dbl#(X)) 3: activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) 4: activate#(n__s(X)) -> c_5(s#(X)) 5: activate#(n__terms(X)) -> c_6(terms#(X)) 6: terms#(N) -> c_19(sqr#(N),s#(N)) 7: activate#(X) -> c_1() 8: add#(X1,X2) -> c_7() 9: add#(0(),X) -> c_8() 10: add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) 11: dbl#(X) -> c_10() 12: dbl#(0()) -> c_11() 13: dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) 14: first#(X1,X2) -> c_13() 15: first#(0(),X) -> c_14() 16: first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) 17: s#(X) -> c_16() 18: sqr#(0()) -> c_17() 19: sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) 20: terms#(X) -> c_20() * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__terms(X)) -> c_6(terms#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) activate#(n__dbl(X)) -> c_3(dbl#(X)) activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) activate#(n__s(X)) -> c_5(s#(X)) add#(X1,X2) -> c_7() add#(0(),X) -> c_8() add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_10() dbl#(0()) -> c_11() dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_13() first#(0(),X) -> c_14() first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) s#(X) -> c_16() sqr#(0()) -> c_17() sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) terms#(N) -> c_19(sqr#(N),s#(N)) terms#(X) -> c_20() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1 ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/0,c_18/5,c_19/2 ,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + 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: activate#(n__terms(X)) -> c_6(terms#(X)) 2: activate#(X) -> c_1() 3: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) 4: activate#(n__dbl(X)) -> c_3(dbl#(X)) 5: activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) 6: activate#(n__s(X)) -> c_5(s#(X)) 7: add#(X1,X2) -> c_7() 8: add#(0(),X) -> c_8() 9: add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) 10: dbl#(X) -> c_10() 11: dbl#(0()) -> c_11() 12: dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) 13: first#(X1,X2) -> c_13() 14: first#(0(),X) -> c_14() 15: first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) 16: s#(X) -> c_16() 17: sqr#(0()) -> c_17() 18: sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) 19: terms#(N) -> c_19(sqr#(N),s#(N)) 20: terms#(X) -> c_20() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) activate#(n__dbl(X)) -> c_3(dbl#(X)) activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__terms(X)) -> c_6(terms#(X)) add#(X1,X2) -> c_7() add#(0(),X) -> c_8() add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_10() dbl#(0()) -> c_11() dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_13() first#(0(),X) -> c_14() first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) s#(X) -> c_16() sqr#(0()) -> c_17() sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) terms#(N) -> c_19(sqr#(N),s#(N)) terms#(X) -> c_20() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1 ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/0,c_18/5,c_19/2 ,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(X) -> c_1() 2:W:activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) -->_1 add#(0(),X) -> c_8():8 -->_1 add#(X1,X2) -> c_7():7 3:W:activate#(n__dbl(X)) -> c_3(dbl#(X)) -->_1 dbl#(0()) -> c_11():11 -->_1 dbl#(X) -> c_10():10 4:W:activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) -->_1 first#(0(),X) -> c_14():14 -->_1 first#(X1,X2) -> c_13():13 5:W:activate#(n__s(X)) -> c_5(s#(X)) -->_1 s#(X) -> c_16():16 6:W:activate#(n__terms(X)) -> c_6(terms#(X)) -->_1 terms#(N) -> c_19(sqr#(N),s#(N)):19 -->_1 terms#(X) -> c_20():20 7:W:add#(X1,X2) -> c_7() 8:W:add#(0(),X) -> c_8() 9:W:add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) 10:W:dbl#(X) -> c_10() 11:W:dbl#(0()) -> c_11() 12:W:dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) 13:W:first#(X1,X2) -> c_13() 14:W:first#(0(),X) -> c_14() 15:W:first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) 16:W:s#(X) -> c_16() 17:W:sqr#(0()) -> c_17() 18:W:sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) 19:W:terms#(N) -> c_19(sqr#(N),s#(N)) -->_1 sqr#(0()) -> c_17():17 -->_2 s#(X) -> c_16():16 20:W:terms#(X) -> c_20() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: sqr#(s(X)) -> c_18(s#(n__add(sqr(activate(X)),dbl(activate(X)))) ,sqr#(activate(X)) ,activate#(X) ,dbl#(activate(X)) ,activate#(X)) 15: first#(s(X),cons(Y,Z)) -> c_15(activate#(X),activate#(Z)) 12: dbl#(s(X)) -> c_12(s#(n__s(n__dbl(activate(X)))),activate#(X)) 9: add#(s(X),Y) -> c_9(s#(n__add(activate(X),Y)),activate#(X)) 6: activate#(n__terms(X)) -> c_6(terms#(X)) 20: terms#(X) -> c_20() 19: terms#(N) -> c_19(sqr#(N),s#(N)) 17: sqr#(0()) -> c_17() 5: activate#(n__s(X)) -> c_5(s#(X)) 16: s#(X) -> c_16() 4: activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) 13: first#(X1,X2) -> c_13() 14: first#(0(),X) -> c_14() 3: activate#(n__dbl(X)) -> c_3(dbl#(X)) 10: dbl#(X) -> c_10() 11: dbl#(0()) -> c_11() 2: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)) 7: add#(X1,X2) -> c_7() 8: add#(0(),X) -> c_8() 1: activate#(X) -> c_1() * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__dbl(X)) -> dbl(X) activate(n__first(X1,X2)) -> first(X1,X2) activate(n__s(X)) -> s(X) activate(n__terms(X)) -> terms(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1 ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/0,c_18/5,c_19/2 ,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))