WORST_CASE(?,O(1)) * Step 1: InnermostRuleRemoval 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: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X)))) All above mentioned rules can be savely removed. * 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 dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(0()) -> 0() 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() dbl#(X) -> c_9() dbl#(0()) -> c_10() first#(X1,X2) -> c_11() first#(0(),X) -> c_12() s#(X) -> c_13() sqr#(0()) -> c_14() terms#(N) -> c_15(sqr#(N),s#(N)) terms#(X) -> c_16() Weak DPs and mark the set of starting terms. * Step 3: UsableRules 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() dbl#(X) -> c_9() dbl#(0()) -> c_10() first#(X1,X2) -> c_11() first#(0(),X) -> c_12() s#(X) -> c_13() sqr#(0()) -> c_14() terms#(N) -> c_15(sqr#(N),s#(N)) terms#(X) -> c_16() - 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 dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(0()) -> 0() 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_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0} - 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: UsableRules + Details: We replace rewrite rules by usable rules: 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() dbl#(X) -> c_9() dbl#(0()) -> c_10() first#(X1,X2) -> c_11() first#(0(),X) -> c_12() s#(X) -> c_13() sqr#(0()) -> c_14() terms#(N) -> c_15(sqr#(N),s#(N)) terms#(X) -> c_16() * Step 4: Trivial 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() dbl#(X) -> c_9() dbl#(0()) -> c_10() first#(X1,X2) -> c_11() first#(0(),X) -> c_12() s#(X) -> c_13() sqr#(0()) -> c_14() terms#(N) -> c_15(sqr#(N),s#(N)) terms#(X) -> c_16() - 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_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0} - 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: Trivial + Details: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S: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:S:activate#(n__dbl(X)) -> c_3(dbl#(X)) -->_1 dbl#(0()) -> c_10():10 -->_1 dbl#(X) -> c_9():9 4:S:activate#(n__first(X1,X2)) -> c_4(first#(X1,X2)) -->_1 first#(0(),X) -> c_12():12 -->_1 first#(X1,X2) -> c_11():11 5:S:activate#(n__s(X)) -> c_5(s#(X)) -->_1 s#(X) -> c_13():13 6:S:activate#(n__terms(X)) -> c_6(terms#(X)) -->_1 terms#(N) -> c_15(sqr#(N),s#(N)):15 -->_1 terms#(X) -> c_16():16 7:S:add#(X1,X2) -> c_7() 8:S:add#(0(),X) -> c_8() 9:S:dbl#(X) -> c_9() 10:S:dbl#(0()) -> c_10() 11:S:first#(X1,X2) -> c_11() 12:S:first#(0(),X) -> c_12() 13:S:s#(X) -> c_13() 14:S:sqr#(0()) -> c_14() 15:S:terms#(N) -> c_15(sqr#(N),s#(N)) -->_1 sqr#(0()) -> c_14():14 -->_2 s#(X) -> c_13():13 16:S:terms#(X) -> c_16() The dependency graph contains no loops, we remove all dependency pairs. * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0} - 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))