WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(X) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0 ,cons,n__from,n__s,n__take,nil} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(X) sel(0(),cons(X,XS)) -> X take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0 ,cons,n__from,n__s,n__take,nil} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(X) sel(0(),cons(X,XS)) -> X take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + 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(from) = {1}, uargs(s) = {1}, uargs(take) = {1,2}, uargs(from#) = {1}, uargs(head#) = {1}, uargs(s#) = {1}, uargs(take#) = {1,2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(2nd) = [1] x1 + [1] p(activate) = [5] x1 + [1] p(cons) = [1] x2 + [4] p(from) = [1] x1 + [15] p(head) = [4] x1 + [1] p(n__from) = [1] x1 + [4] p(n__s) = [1] x1 + [2] p(n__take) = [1] x1 + [1] x2 + [2] p(nil) = [1] p(s) = [1] x1 + [9] p(sel) = [1] x2 + [1] p(take) = [1] x1 + [1] x2 + [7] p(2nd#) = [7] x1 + [0] p(activate#) = [5] x1 + [6] p(from#) = [1] x1 + [1] p(head#) = [1] x1 + [3] p(s#) = [1] x1 + [0] p(sel#) = [1] x1 + [2] x2 + [10] p(take#) = [1] x1 + [1] x2 + [4] p(c_1) = [1] x1 + [0] p(c_2) = [2] p(c_3) = [1] x1 + [8] p(c_4) = [1] x1 + [9] p(c_5) = [1] x1 + [12] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [8] Following rules are strictly oriented: 2nd#(cons(X,XS)) = [7] XS + [28] > [5] XS + [4] = c_1(head#(activate(XS))) activate#(X) = [5] X + [6] > [2] = c_2() activate#(n__from(X)) = [5] X + [26] > [5] X + [10] = c_3(from#(activate(X))) activate#(n__s(X)) = [5] X + [16] > [5] X + [10] = c_4(s#(activate(X))) from#(X) = [1] X + [1] > [0] = c_6() from#(X) = [1] X + [1] > [0] = c_7() head#(cons(X,XS)) = [1] XS + [7] > [0] = c_8() sel#(0(),cons(X,XS)) = [2] XS + [20] > [0] = c_10() take#(X1,X2) = [1] X1 + [1] X2 + [4] > [1] = c_11() activate(X) = [5] X + [1] > [1] X + [0] = X activate(n__from(X)) = [5] X + [21] > [5] X + [16] = from(activate(X)) activate(n__s(X)) = [5] X + [11] > [5] X + [10] = s(activate(X)) activate(n__take(X1,X2)) = [5] X1 + [5] X2 + [11] > [5] X1 + [5] X2 + [9] = take(activate(X1),activate(X2)) from(X) = [1] X + [15] > [1] X + [10] = cons(X,n__from(n__s(X))) from(X) = [1] X + [15] > [1] X + [4] = n__from(X) s(X) = [1] X + [9] > [1] X + [2] = n__s(X) take(X1,X2) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [2] = n__take(X1,X2) take(0(),XS) = [1] XS + [9] > [1] = nil() Following rules are (at-least) weakly oriented: activate#(n__take(X1,X2)) = [5] X1 + [5] X2 + [16] >= [5] X1 + [5] X2 + [18] = c_5(take#(activate(X1),activate(X2))) s#(X) = [1] X + [0] >= [0] = c_9() take#(0(),XS) = [1] XS + [6] >= [8] = c_12() 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: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) s#(X) -> c_9() take#(0(),XS) -> c_12() - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {1}. Here rules are labelled as follows: 1: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) 2: s#(X) -> c_9() 3: take#(0(),XS) -> c_12() 4: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 5: activate#(X) -> c_2() 6: activate#(n__from(X)) -> c_3(from#(activate(X))) 7: activate#(n__s(X)) -> c_4(s#(activate(X))) 8: from#(X) -> c_6() 9: from#(X) -> c_7() 10: head#(cons(X,XS)) -> c_8() 11: sel#(0(),cons(X,XS)) -> c_10() 12: take#(X1,X2) -> c_11() * Step 6: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) s#(X) -> c_9() - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + 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__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) 2: s#(X) -> c_9() 3: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 4: activate#(X) -> c_2() 5: activate#(n__from(X)) -> c_3(from#(activate(X))) 6: activate#(n__s(X)) -> c_4(s#(activate(X))) 7: from#(X) -> c_6() 8: from#(X) -> c_7() 9: head#(cons(X,XS)) -> c_8() 10: sel#(0(),cons(X,XS)) -> c_10() 11: take#(X1,X2) -> c_11() 12: take#(0(),XS) -> c_12() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_9() - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() sel#(0(),cons(X,XS)) -> c_10() take#(X1,X2) -> c_11() take#(0(),XS) -> c_12() - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:s#(X) -> c_9() 2:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) -->_1 head#(cons(X,XS)) -> c_8():9 3:W:activate#(X) -> c_2() 4:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_7():8 -->_1 from#(X) -> c_6():7 5:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():1 6:W:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) -->_1 take#(0(),XS) -> c_12():12 -->_1 take#(X1,X2) -> c_11():11 7:W:from#(X) -> c_6() 8:W:from#(X) -> c_7() 9:W:head#(cons(X,XS)) -> c_8() 10:W:sel#(0(),cons(X,XS)) -> c_10() 11:W:take#(X1,X2) -> c_11() 12:W:take#(0(),XS) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: sel#(0(),cons(X,XS)) -> c_10() 6: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2))) 11: take#(X1,X2) -> c_11() 12: take#(0(),XS) -> c_12() 4: activate#(n__from(X)) -> c_3(from#(activate(X))) 7: from#(X) -> c_6() 8: from#(X) -> c_7() 3: activate#(X) -> c_2() 2: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 9: head#(cons(X,XS)) -> c_8() * Step 8: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_9() - Weak DPs: activate#(n__s(X)) -> c_4(s#(activate(X))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:s#(X) -> c_9() 5:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():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__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))