WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4() first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) from#(X) -> c_7() from#(X) -> c_8() sel#(0(),cons(X,Z)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4() first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) from#(X) -> c_7() from#(X) -> c_8() sel#(0(),cons(X,Z)) -> c_9() sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,8,9} by application of Pre({1,4,5,7,8,9}) = {2,3,6,10}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 3: activate#(n__from(X)) -> c_3(from#(X)) 4: first#(X1,X2) -> c_4() 5: first#(0(),Z) -> c_5() 6: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) 7: from#(X) -> c_7() 8: from#(X) -> c_8() 9: sel#(0(),cons(X,Z)) -> c_9() 10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(X) -> c_1() first#(X1,X2) -> c_4() first#(0(),Z) -> c_5() from#(X) -> c_7() from#(X) -> c_8() sel#(0(),cons(X,Z)) -> c_9() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {3,4}. Here rules are labelled as follows: 1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 2: activate#(n__from(X)) -> c_3(from#(X)) 3: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) 4: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) 5: activate#(X) -> c_1() 6: first#(X1,X2) -> c_4() 7: first#(0(),Z) -> c_5() 8: from#(X) -> c_7() 9: from#(X) -> c_8() 10: sel#(0(),cons(X,Z)) -> c_9() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4() first#(0(),Z) -> c_5() from#(X) -> c_7() from#(X) -> c_8() sel#(0(),cons(X,Z)) -> c_9() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)):2 -->_1 first#(0(),Z) -> c_5():7 -->_1 first#(X1,X2) -> c_4():6 2:S:first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) -->_1 activate#(n__from(X)) -> c_3(from#(X)):5 -->_1 activate#(X) -> c_1():4 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 3:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_2 activate#(n__from(X)) -> c_3(from#(X)):5 -->_1 sel#(0(),cons(X,Z)) -> c_9():10 -->_2 activate#(X) -> c_1():4 -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):3 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 4:W:activate#(X) -> c_1() 5:W:activate#(n__from(X)) -> c_3(from#(X)) -->_1 from#(X) -> c_8():9 -->_1 from#(X) -> c_7():8 6:W:first#(X1,X2) -> c_4() 7:W:first#(0(),Z) -> c_5() 8:W:from#(X) -> c_7() 9:W:from#(X) -> c_8() 10:W:sel#(0(),cons(X,Z)) -> c_9() 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,Z)) -> c_9() 6: first#(X1,X2) -> c_4() 7: first#(0(),Z) -> c_5() 4: activate#(X) -> c_1() 5: activate#(n__from(X)) -> c_3(from#(X)) 8: from#(X) -> c_7() 9: from#(X) -> c_8() * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) and a lower component activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) Further, following extension rules are added to the lower component. sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)) -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) ** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + 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) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(first) = [1] x2 + [0] p(from) = [0] p(n__first) = [1] x2 + [0] p(n__from) = [0] p(nil) = [0] p(s) = [1] x1 + [1] p(sel) = [0] p(activate#) = [0] p(first#) = [0] p(from#) = [0] p(sel#) = [1] x1 + [1] x2 + [10] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: sel#(s(X),cons(Y,Z)) = [1] X + [1] Z + [11] > [1] X + [1] Z + [10] = c_10(sel#(X,activate(Z))) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = first(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) first(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] Z + [0] >= [1] Z + [0] = cons(Y,n__first(X,activate(Z))) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) - Weak DPs: sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + 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) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [4] p(n__first) = [1] x1 + [1] x2 + [0] p(n__from) = [4] p(nil) = [3] p(s) = [1] x1 + [3] p(sel) = [0] p(activate#) = [0] p(first#) = [1] p(from#) = [0] p(sel#) = [4] x1 + [1] x2 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [5] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: first#(s(X),cons(Y,Z)) = [1] > [0] = c_6(activate#(Z)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [0] >= [6] = c_2(first#(X1,X2)) sel#(s(X),cons(Y,Z)) = [4] X + [1] Z + [13] >= [0] = activate#(Z) sel#(s(X),cons(Y,Z)) = [4] X + [1] Z + [13] >= [4] X + [1] Z + [1] = sel#(X,activate(Z)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) activate(n__from(X)) = [4] >= [4] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [3] >= [3] = nil() first(s(X),cons(Y,Z)) = [1] X + [1] Z + [3] >= [1] X + [1] Z + [0] = cons(Y,n__first(X,activate(Z))) from(X) = [4] >= [4] = cons(X,n__from(s(X))) from(X) = [4] >= [4] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + 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) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(first) = [1] x1 + [1] x2 + [4] p(from) = [0] p(n__first) = [1] x1 + [1] x2 + [4] p(n__from) = [0] p(nil) = [1] p(s) = [1] x1 + [1] p(sel) = [1] x1 + [1] p(activate#) = [1] x1 + [0] p(first#) = [1] x2 + [1] p(from#) = [2] p(sel#) = [1] x1 + [1] x2 + [3] p(c_1) = [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [2] x1 + [2] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [4] > [1] X2 + [2] = c_2(first#(X1,X2)) Following rules are (at-least) weakly oriented: first#(s(X),cons(Y,Z)) = [1] Z + [1] >= [1] Z + [1] = c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) = [1] X + [1] Z + [4] >= [1] Z + [0] = activate#(Z) sel#(s(X),cons(Y,Z)) = [1] X + [1] Z + [4] >= [1] X + [1] Z + [3] = sel#(X,activate(Z)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = first(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n__first(X1,X2) first(0(),Z) = [1] Z + [9] >= [1] = nil() first(s(X),cons(Y,Z)) = [1] X + [1] Z + [5] >= [1] X + [1] Z + [4] = cons(Y,n__first(X,activate(Z))) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))