WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0 ,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,s,sqr ,terms} and constructors {0,cons,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(add(X,Y)) dbl(s(X)) -> s(s(dbl(X))) first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) sqr(s(X)) -> s(add(sqr(X),dbl(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__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() half(0()) -> 0() half(dbl(X)) -> X s(X) -> n__s(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,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0 ,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,s,sqr ,terms} and constructors {0,cons,n__first,n__s,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__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) add#(0(),X) -> c_5() dbl#(0()) -> c_6() first#(X1,X2) -> c_7() first#(0(),X) -> c_8() half#(0()) -> c_9() half#(dbl(X)) -> c_10() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) add#(0(),X) -> c_5() dbl#(0()) -> c_6() first#(X1,X2) -> c_7() first#(0(),X) -> c_8() half#(0()) -> c_9() half#(dbl(X)) -> c_10() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() half(0()) -> 0() half(dbl(X)) -> X s(X) -> n__s(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,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) add#(0(),X) -> c_5() dbl#(0()) -> c_6() first#(X1,X2) -> c_7() first#(0(),X) -> c_8() half#(0()) -> c_9() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) add#(0(),X) -> c_5() dbl#(0()) -> c_6() first#(X1,X2) -> c_7() first#(0(),X) -> c_8() half#(0()) -> c_9() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,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(cons) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(s) = {1}, uargs(terms) = {1}, uargs(first#) = {1,2}, uargs(s#) = {1}, uargs(terms#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [5] x1 + [1] p(add) = [0] p(cons) = [1] x1 + [1] p(dbl) = [0] p(first) = [1] x1 + [1] x2 + [5] p(half) = [0] p(n__first) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [1] p(n__terms) = [1] x1 + [1] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [4] p(sqr) = [1] x1 + [1] p(terms) = [1] x1 + [4] p(activate#) = [5] x1 + [6] p(add#) = [0] p(dbl#) = [0] p(first#) = [1] x1 + [1] x2 + [8] p(half#) = [0] p(s#) = [1] x1 + [4] p(sqr#) = [1] x1 + [13] p(terms#) = [1] x1 + [9] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] Following rules are strictly oriented: activate#(X) = [5] X + [6] > [0] = c_1() activate#(n__first(X1,X2)) = [5] X1 + [5] X2 + [26] > [5] X1 + [5] X2 + [10] = c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) = [5] X + [11] > [5] X + [5] = c_3(s#(activate(X))) activate#(n__terms(X)) = [5] X + [11] > [5] X + [10] = c_4(terms#(activate(X))) first#(X1,X2) = [1] X1 + [1] X2 + [8] > [0] = c_7() first#(0(),X) = [1] X + [8] > [0] = c_8() s#(X) = [1] X + [4] > [0] = c_11() sqr#(0()) = [13] > [0] = c_12() terms#(X) = [1] X + [9] > [0] = c_14() activate(X) = [5] X + [1] > [1] X + [0] = X activate(n__first(X1,X2)) = [5] X1 + [5] X2 + [21] > [5] X1 + [5] X2 + [7] = first(activate(X1),activate(X2)) activate(n__s(X)) = [5] X + [6] > [5] X + [5] = s(activate(X)) activate(n__terms(X)) = [5] X + [6] > [5] X + [5] = terms(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = n__first(X1,X2) first(0(),X) = [1] X + [5] > [0] = nil() s(X) = [1] X + [4] > [1] X + [1] = n__s(X) sqr(0()) = [1] > [0] = 0() terms(N) = [1] N + [4] > [1] N + [2] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [4] > [1] X + [1] = n__terms(X) Following rules are (at-least) weakly oriented: add#(0(),X) = [0] >= [0] = c_5() dbl#(0()) = [0] >= [0] = c_6() half#(0()) = [0] >= [0] = c_9() terms#(N) = [1] N + [9] >= [1] N + [13] = c_13(sqr#(N)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: add#(0(),X) -> c_5() dbl#(0()) -> c_6() half#(0()) -> c_9() terms#(N) -> c_13(sqr#(N)) - Weak DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) first#(X1,X2) -> c_7() first#(0(),X) -> c_8() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(X) -> c_14() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,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} by application of Pre({1,2,3}) = {}. Here rules are labelled as follows: 1: add#(0(),X) -> c_5() 2: dbl#(0()) -> c_6() 3: half#(0()) -> c_9() 4: terms#(N) -> c_13(sqr#(N)) 5: activate#(X) -> c_1() 6: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) 7: activate#(n__s(X)) -> c_3(s#(activate(X))) 8: activate#(n__terms(X)) -> c_4(terms#(activate(X))) 9: first#(X1,X2) -> c_7() 10: first#(0(),X) -> c_8() 11: s#(X) -> c_11() 12: sqr#(0()) -> c_12() 13: terms#(X) -> c_14() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_13(sqr#(N)) - Weak DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_3(s#(activate(X))) activate#(n__terms(X)) -> c_4(terms#(activate(X))) add#(0(),X) -> c_5() dbl#(0()) -> c_6() first#(X1,X2) -> c_7() first#(0(),X) -> c_8() half#(0()) -> c_9() s#(X) -> c_11() sqr#(0()) -> c_12() terms#(X) -> c_14() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:terms#(N) -> c_13(sqr#(N)) -->_1 sqr#(0()) -> c_12():12 2:W:activate#(X) -> c_1() 3:W:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) -->_1 first#(0(),X) -> c_8():9 -->_1 first#(X1,X2) -> c_7():8 4:W:activate#(n__s(X)) -> c_3(s#(activate(X))) -->_1 s#(X) -> c_11():11 5:W:activate#(n__terms(X)) -> c_4(terms#(activate(X))) -->_1 terms#(X) -> c_14():13 -->_1 terms#(N) -> c_13(sqr#(N)):1 6:W:add#(0(),X) -> c_5() 7:W:dbl#(0()) -> c_6() 8:W:first#(X1,X2) -> c_7() 9:W:first#(0(),X) -> c_8() 10:W:half#(0()) -> c_9() 11:W:s#(X) -> c_11() 12:W:sqr#(0()) -> c_12() 13:W:terms#(X) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: half#(0()) -> c_9() 7: dbl#(0()) -> c_6() 6: add#(0(),X) -> c_5() 13: terms#(X) -> c_14() 4: activate#(n__s(X)) -> c_3(s#(activate(X))) 11: s#(X) -> c_11() 3: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) 8: first#(X1,X2) -> c_7() 9: first#(0(),X) -> c_8() 2: activate#(X) -> c_1() 12: sqr#(0()) -> c_12() * Step 7: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_13(sqr#(N)) - Weak DPs: activate#(n__terms(X)) -> c_4(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:terms#(N) -> c_13(sqr#(N)) 5:W:activate#(n__terms(X)) -> c_4(terms#(activate(X))) -->_1 terms#(N) -> c_13(sqr#(N)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: terms#(N) -> c_13() * Step 8: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: terms#(N) -> c_13() - Weak DPs: activate#(n__terms(X)) -> c_4(terms#(activate(X))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:terms#(N) -> c_13() 2:W:activate#(n__terms(X)) -> c_4(terms#(activate(X))) -->_1 terms#(N) -> c_13():1 The dependency graph contains no loops, we remove all dependency pairs. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) 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(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,s#/1 ,sqr#/1,terms#/1} / {0/0,cons/2,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_5/0 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,s#,sqr# ,terms#} and constructors {0,cons,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(n^1))