WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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__sqr,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(n__sqr(activate(X)),n__dbl(activate(X)))) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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__sqr,n__terms,nil,recip} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) activate#(n__terms(X)) -> c_7(terms#(activate(X))) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() dbl#(X) -> c_10() dbl#(0()) -> c_11() first#(X1,X2) -> c_12() first#(0(),X) -> c_13() s#(X) -> c_14() sqr#(X) -> c_15() sqr#(0()) -> c_16() terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() Weak DPs and mark the set of starting terms. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) activate#(n__terms(X)) -> c_7(terms#(activate(X))) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() dbl#(X) -> c_10() dbl#(0()) -> c_11() first#(X1,X2) -> c_12() first#(0(),X) -> c_13() s#(X) -> c_14() sqr#(X) -> c_15() sqr#(0()) -> c_16() terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1}, uargs(add#) = {1,2}, uargs(dbl#) = {1}, uargs(first#) = {1,2}, uargs(sqr#) = {1}, uargs(terms#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_17) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [3] x1 + [1] p(add) = [1] x1 + [1] x2 + [4] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [7] p(first) = [1] x1 + [1] x2 + [4] p(n__add) = [1] x1 + [1] x2 + [2] p(n__dbl) = [1] x1 + [4] p(n__first) = [1] x1 + [1] x2 + [2] p(n__s) = [2] p(n__sqr) = [1] x1 + [2] p(n__terms) = [1] x1 + [2] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [4] p(sqr) = [1] x1 + [3] p(terms) = [1] x1 + [5] p(activate#) = [3] x1 + [0] p(add#) = [1] x1 + [1] x2 + [3] p(dbl#) = [1] x1 + [1] p(first#) = [1] x1 + [1] x2 + [5] p(s#) = [1] p(sqr#) = [1] x1 + [7] p(terms#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [4] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [2] p(c_17) = [1] x1 + [0] p(c_18) = [0] Following rules are strictly oriented: activate#(n__add(X1,X2)) = [3] X1 + [3] X2 + [6] > [3] X1 + [3] X2 + [5] = c_2(add#(activate(X1),activate(X2))) activate#(n__dbl(X)) = [3] X + [12] > [3] X + [6] = c_3(dbl#(activate(X))) activate#(n__s(X)) = [6] > [1] = c_5(s#(X)) activate#(n__terms(X)) = [3] X + [6] > [3] X + [1] = c_7(terms#(activate(X))) add#(X1,X2) = [1] X1 + [1] X2 + [3] > [0] = c_8() add#(0(),X) = [1] X + [4] > [0] = c_9() dbl#(X) = [1] X + [1] > [0] = c_10() dbl#(0()) = [2] > [0] = c_11() first#(X1,X2) = [1] X1 + [1] X2 + [5] > [0] = c_12() first#(0(),X) = [1] X + [6] > [0] = c_13() s#(X) = [1] > [0] = c_14() sqr#(X) = [1] X + [7] > [0] = c_15() sqr#(0()) = [8] > [2] = c_16() activate(X) = [3] X + [1] > [1] X + [0] = X activate(n__add(X1,X2)) = [3] X1 + [3] X2 + [7] > [3] X1 + [3] X2 + [6] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [3] X + [13] > [3] X + [8] = dbl(activate(X)) activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [7] > [3] X1 + [3] X2 + [6] = first(activate(X1),activate(X2)) activate(n__s(X)) = [7] > [4] = s(X) activate(n__sqr(X)) = [3] X + [7] > [3] X + [4] = sqr(activate(X)) activate(n__terms(X)) = [3] X + [7] > [3] X + [6] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [2] = n__add(X1,X2) add(0(),X) = [1] X + [5] > [1] X + [0] = X dbl(X) = [1] X + [7] > [1] X + [4] = n__dbl(X) dbl(0()) = [8] > [1] = 0() first(X1,X2) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [2] = n__first(X1,X2) first(0(),X) = [1] X + [5] > [0] = nil() s(X) = [4] > [2] = n__s(X) sqr(X) = [1] X + [3] > [1] X + [2] = n__sqr(X) sqr(0()) = [4] > [1] = 0() terms(N) = [1] N + [5] > [1] N + [3] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [5] > [1] X + [2] = n__terms(X) Following rules are (at-least) weakly oriented: activate#(X) = [3] X + [0] >= [0] = c_1() activate#(n__first(X1,X2)) = [3] X1 + [3] X2 + [6] >= [3] X1 + [3] X2 + [7] = c_4(first#(activate(X1),activate(X2))) activate#(n__sqr(X)) = [3] X + [6] >= [3] X + [9] = c_6(sqr#(activate(X))) terms#(N) = [1] N + [0] >= [1] N + [7] = c_17(sqr#(N)) terms#(X) = [1] X + [0] >= [0] = c_18() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Weak DPs: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X))) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() dbl#(X) -> c_10() dbl#(0()) -> c_11() first#(X1,X2) -> c_12() first#(0(),X) -> c_13() s#(X) -> c_14() sqr#(X) -> c_15() sqr#(0()) -> c_16() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3} by application of Pre({1,2,3}) = {}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) 3: activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) 4: terms#(N) -> c_17(sqr#(N)) 5: terms#(X) -> c_18() 6: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) 7: activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) 8: activate#(n__s(X)) -> c_5(s#(X)) 9: activate#(n__terms(X)) -> c_7(terms#(activate(X))) 10: add#(X1,X2) -> c_8() 11: add#(0(),X) -> c_9() 12: dbl#(X) -> c_10() 13: dbl#(0()) -> c_11() 14: first#(X1,X2) -> c_12() 15: first#(0(),X) -> c_13() 16: s#(X) -> c_14() 17: sqr#(X) -> c_15() 18: sqr#(0()) -> c_16() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Weak DPs: activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) activate#(n__terms(X)) -> c_7(terms#(activate(X))) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() dbl#(X) -> c_10() dbl#(0()) -> c_11() first#(X1,X2) -> c_12() first#(0(),X) -> c_13() s#(X) -> c_14() sqr#(X) -> c_15() sqr#(0()) -> c_16() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(0()) -> c_16():18 -->_1 sqr#(X) -> c_15():17 2:S:terms#(X) -> c_18() 3:W:activate#(X) -> c_1() 4:W:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) -->_1 add#(0(),X) -> c_9():11 -->_1 add#(X1,X2) -> c_8():10 5:W:activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) -->_1 dbl#(0()) -> c_11():13 -->_1 dbl#(X) -> c_10():12 6:W:activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) -->_1 first#(0(),X) -> c_13():15 -->_1 first#(X1,X2) -> c_12():14 7:W:activate#(n__s(X)) -> c_5(s#(X)) -->_1 s#(X) -> c_14():16 8:W:activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) -->_1 sqr#(0()) -> c_16():18 -->_1 sqr#(X) -> c_15():17 9:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(X) -> c_18():2 -->_1 terms#(N) -> c_17(sqr#(N)):1 10:W:add#(X1,X2) -> c_8() 11:W:add#(0(),X) -> c_9() 12:W:dbl#(X) -> c_10() 13:W:dbl#(0()) -> c_11() 14:W:first#(X1,X2) -> c_12() 15:W:first#(0(),X) -> c_13() 16:W:s#(X) -> c_14() 17:W:sqr#(X) -> c_15() 18:W:sqr#(0()) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: activate#(n__sqr(X)) -> c_6(sqr#(activate(X))) 7: activate#(n__s(X)) -> c_5(s#(X)) 16: s#(X) -> c_14() 6: activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2))) 14: first#(X1,X2) -> c_12() 15: first#(0(),X) -> c_13() 5: activate#(n__dbl(X)) -> c_3(dbl#(activate(X))) 12: dbl#(X) -> c_10() 13: dbl#(0()) -> c_11() 4: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2))) 10: add#(X1,X2) -> c_8() 11: add#(0(),X) -> c_9() 3: activate#(X) -> c_1() 17: sqr#(X) -> c_15() 18: sqr#(0()) -> c_16() * Step 6: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:terms#(N) -> c_17(sqr#(N)) 2:S:terms#(X) -> c_18() 9:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(X) -> c_18():2 -->_1 terms#(N) -> c_17(sqr#(N)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: terms#(N) -> c_17() * Step 7: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_17() terms#(X) -> c_18() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + 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: terms#(N) -> c_17() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) terms#(X) -> c_18() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} Problem (S) - Strict DPs: terms#(X) -> c_18() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) terms#(N) -> c_17() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} ** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_17() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) terms#(X) -> c_18() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:terms#(N) -> c_17() 2:W:terms#(X) -> c_18() 3:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(N) -> c_17():1 -->_1 terms#(X) -> c_18():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: terms#(X) -> c_18() ** Step 7.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_17() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:terms#(N) -> c_17() 3:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(N) -> c_17():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 7.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(X) -> c_18() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) terms#(N) -> c_17() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:terms#(X) -> c_18() 2:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(N) -> c_17():3 -->_1 terms#(X) -> c_18():1 3:W:terms#(N) -> c_17() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: terms#(N) -> c_17() ** Step 7.b:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(X) -> c_18() - Weak DPs: activate#(n__terms(X)) -> c_7(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:terms#(X) -> c_18() 2:W:activate#(n__terms(X)) -> c_7(terms#(activate(X))) -->_1 terms#(X) -> c_18():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 7.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(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(X) -> n__sqr(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__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__sqr/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_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0} - 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__sqr,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))