/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> 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,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons,n__from ,n__take,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> 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,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons,n__from ,n__take,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> 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,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons,n__from ,n__take,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(v){v -> n__take(s(x),cons(z,v))} = activate(n__take(s(x),cons(z,v))) ->^+ cons(z,n__take(x,activate(v))) = C[activate(v) = activate(v){}] ** Step 1.b:1: 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(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> 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,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons,n__from ,n__take,nil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following weak dependency pairs: Strict DPs 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> 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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) 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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {11} by application of Pre({11}) = {2,4,5,6,7,8,10,12}. Here rules are labelled as follows: 1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 2: activate#(X) -> c_2(X) 3: activate#(n__from(X)) -> c_3(from#(X)) 4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) 5: from#(X) -> c_5(X,X) 6: from#(X) -> c_6(X) 7: head#(cons(X,XS)) -> c_7(X) 8: sel#(0(),cons(X,XS)) -> c_8(X) 9: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) 10: take#(X1,X2) -> c_10(X1,X2) 11: take#(0(),XS) -> c_11() 12: take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) ** Step 1.b:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: take#(0(),XS) -> c_11() - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [1] p(from) = [0] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x2 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(sel) = [0] p(take) = [1] x2 + [0] p(2nd#) = [5] x1 + [0] p(activate#) = [1] x1 + [0] p(from#) = [0] p(head#) = [1] x1 + [0] p(sel#) = [1] x2 + [0] p(take#) = [1] x2 + [0] p(c_1) = [1] x1 + [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) = [1] x1 + [0] p(c_10) = [1] x2 + [0] p(c_11) = [0] p(c_12) = [1] x3 + [0] Following rules are strictly oriented: 2nd#(cons(X,XS)) = [5] XS + [5] > [1] XS + [0] = c_1(head#(activate(XS))) head#(cons(X,XS)) = [1] XS + [1] > [0] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [1] > [0] = c_8(X) sel#(s(N),cons(X,XS)) = [1] XS + [1] > [1] XS + [0] = c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) = [1] XS + [1] > [1] XS + [0] = c_12(X,N,activate#(XS)) Following rules are (at-least) weakly oriented: activate#(X) = [1] X + [0] >= [1] X + [0] = c_2(X) activate#(n__from(X)) = [0] >= [0] = c_3(from#(X)) activate#(n__take(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = c_4(take#(X1,X2)) from#(X) = [0] >= [0] = c_5(X,X) from#(X) = [0] >= [0] = c_6(X) take#(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = c_10(X1,X2) take#(0(),XS) = [1] XS + [0] >= [0] = c_11() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) activate(n__take(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = take(X1,X2) from(X) = [0] >= [1] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [0] = nil() take(s(N),cons(X,XS)) = [1] XS + [1] >= [1] XS + [1] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:5: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) take#(X1,X2) -> c_10(X1,X2) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [6] p(from) = [7] p(head) = [1] p(n__from) = [0] p(n__take) = [1] x2 + [0] p(nil) = [0] p(s) = [0] p(sel) = [0] p(take) = [1] x2 + [0] p(2nd#) = [1] x1 + [2] p(activate#) = [2] x1 + [5] p(from#) = [0] p(head#) = [1] x1 + [1] p(sel#) = [1] x2 + [0] p(take#) = [2] x2 + [3] p(c_1) = [1] x1 + [7] p(c_2) = [2] x1 + [4] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [7] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [2] x2 + [0] p(c_11) = [2] p(c_12) = [1] x3 + [5] Following rules are strictly oriented: activate#(X) = [2] X + [5] > [2] X + [4] = c_2(X) activate#(n__from(X)) = [5] > [0] = c_3(from#(X)) activate#(n__take(X1,X2)) = [2] X2 + [5] > [2] X2 + [3] = c_4(take#(X1,X2)) take#(X1,X2) = [2] X2 + [3] > [2] X2 + [0] = c_10(X1,X2) from(X) = [7] > [6] = cons(X,n__from(s(X))) from(X) = [7] > [0] = n__from(X) Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [1] XS + [8] >= [1] XS + [8] = c_1(head#(activate(XS))) from#(X) = [0] >= [0] = c_5(X,X) from#(X) = [0] >= [0] = c_6(X) head#(cons(X,XS)) = [1] XS + [7] >= [7] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [6] >= [0] = c_8(X) sel#(s(N),cons(X,XS)) = [1] XS + [6] >= [1] XS + [0] = c_9(sel#(N,activate(XS))) take#(0(),XS) = [2] XS + [3] >= [2] = c_11() take#(s(N),cons(X,XS)) = [2] XS + [15] >= [2] XS + [10] = c_12(X,N,activate#(XS)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [7] = from(X) activate(n__take(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = take(X1,X2) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [0] = nil() take(s(N),cons(X,XS)) = [1] XS + [6] >= [1] XS + [6] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:6: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: from#(X) -> c_5(X,X) from#(X) -> c_6(X) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Weak TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [1] x1 + [3] p(cons) = [1] x2 + [4] p(from) = [4] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x2 + [4] p(nil) = [0] p(s) = [0] p(sel) = [0] p(take) = [1] x2 + [0] p(2nd#) = [2] x1 + [0] p(activate#) = [0] p(from#) = [0] p(head#) = [1] x1 + [2] p(sel#) = [1] x2 + [6] p(take#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [2] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] x1 + [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x3 + [0] Following rules are strictly oriented: activate(X) = [1] X + [3] > [1] X + [0] = X activate(n__take(X1,X2)) = [1] X2 + [7] > [1] X2 + [0] = take(X1,X2) Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [2] XS + [8] >= [1] XS + [5] = c_1(head#(activate(XS))) activate#(X) = [0] >= [0] = c_2(X) activate#(n__from(X)) = [0] >= [0] = c_3(from#(X)) activate#(n__take(X1,X2)) = [0] >= [0] = c_4(take#(X1,X2)) from#(X) = [0] >= [0] = c_5(X,X) from#(X) = [0] >= [2] = c_6(X) head#(cons(X,XS)) = [1] XS + [6] >= [0] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [10] >= [1] = c_8(X) sel#(s(N),cons(X,XS)) = [1] XS + [10] >= [1] XS + [10] = c_9(sel#(N,activate(XS))) take#(X1,X2) = [0] >= [0] = c_10(X1,X2) take#(0(),XS) = [0] >= [0] = c_11() take#(s(N),cons(X,XS)) = [0] >= [0] = c_12(X,N,activate#(XS)) activate(n__from(X)) = [3] >= [4] = from(X) from(X) = [4] >= [4] = cons(X,n__from(s(X))) from(X) = [4] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [4] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [0] = nil() take(s(N),cons(X,XS)) = [1] XS + [4] >= [1] XS + [11] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:7: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: from#(X) -> c_5(X,X) from#(X) -> c_6(X) - Strict TRS: activate(n__from(X)) -> from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [1] p(from) = [1] p(head) = [1] x1 + [2] p(n__from) = [0] p(n__take) = [1] x2 + [2] p(nil) = [1] p(s) = [1] x1 + [2] p(sel) = [1] x1 + [0] p(take) = [1] x2 + [0] p(2nd#) = [1] x1 + [6] p(activate#) = [4] p(from#) = [0] p(head#) = [1] x1 + [2] p(sel#) = [1] x1 + [1] x2 + [4] p(take#) = [4] p(c_1) = [1] x1 + [3] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [0] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [4] p(c_12) = [1] x3 + [0] Following rules are strictly oriented: activate(n__from(X)) = [2] > [1] = from(X) Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [1] XS + [7] >= [1] XS + [7] = c_1(head#(activate(XS))) activate#(X) = [4] >= [1] = c_2(X) activate#(n__from(X)) = [4] >= [0] = c_3(from#(X)) activate#(n__take(X1,X2)) = [4] >= [4] = c_4(take#(X1,X2)) from#(X) = [0] >= [1] = c_5(X,X) from#(X) = [0] >= [1] = c_6(X) head#(cons(X,XS)) = [1] XS + [3] >= [0] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [5] >= [4] = c_8(X) sel#(s(N),cons(X,XS)) = [1] N + [1] XS + [7] >= [1] N + [1] XS + [6] = c_9(sel#(N,activate(XS))) take#(X1,X2) = [4] >= [0] = c_10(X1,X2) take#(0(),XS) = [4] >= [4] = c_11() take#(s(N),cons(X,XS)) = [4] >= [4] = c_12(X,N,activate#(XS)) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__take(X1,X2)) = [1] X2 + [4] >= [1] X2 + [0] = take(X1,X2) from(X) = [1] >= [1] = cons(X,n__from(s(X))) from(X) = [1] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [2] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [1] = nil() take(s(N),cons(X,XS)) = [1] XS + [1] >= [1] XS + [5] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:8: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: from#(X) -> c_5(X,X) from#(X) -> c_6(X) - Strict TRS: take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [4] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [0] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x2 + [3] p(nil) = [0] p(s) = [0] p(sel) = [1] x1 + [4] x2 + [1] p(take) = [1] x2 + [3] p(2nd#) = [2] x1 + [0] p(activate#) = [2] x1 + [1] p(from#) = [1] p(head#) = [1] x1 + [0] p(sel#) = [1] x2 + [0] p(take#) = [2] x2 + [7] p(c_1) = [1] x1 + [0] p(c_2) = [2] x1 + [1] 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) = [1] x1 + [0] p(c_10) = [2] x2 + [7] p(c_11) = [7] p(c_12) = [1] x3 + [6] Following rules are strictly oriented: from#(X) = [1] > [0] = c_5(X,X) from#(X) = [1] > [0] = c_6(X) take(0(),XS) = [1] XS + [3] > [0] = nil() Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [2] XS + [0] >= [1] XS + [0] = c_1(head#(activate(XS))) activate#(X) = [2] X + [1] >= [2] X + [1] = c_2(X) activate#(n__from(X)) = [1] >= [1] = c_3(from#(X)) activate#(n__take(X1,X2)) = [2] X2 + [7] >= [2] X2 + [7] = c_4(take#(X1,X2)) head#(cons(X,XS)) = [1] XS + [0] >= [0] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [0] >= [0] = c_8(X) sel#(s(N),cons(X,XS)) = [1] XS + [0] >= [1] XS + [0] = c_9(sel#(N,activate(XS))) take#(X1,X2) = [2] X2 + [7] >= [2] X2 + [7] = c_10(X1,X2) take#(0(),XS) = [2] XS + [7] >= [7] = c_11() take#(s(N),cons(X,XS)) = [2] XS + [7] >= [2] XS + [7] = c_12(X,N,activate#(XS)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) activate(n__take(X1,X2)) = [1] X2 + [3] >= [1] X2 + [3] = take(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [3] >= [1] X2 + [3] = n__take(X1,X2) take(s(N),cons(X,XS)) = [1] XS + [3] >= [1] XS + [3] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:9: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: take(X1,X2) -> n__take(X1,X2) take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(activate) = {1}, uargs(cons) = {2}, uargs(n__take) = {2}, uargs(take) = {2}, uargs(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(2nd) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [0] p(from) = [2] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x1 + [1] x2 + [3] p(nil) = [0] p(s) = [1] x1 + [2] p(sel) = [1] x2 + [1] p(take) = [1] x1 + [1] x2 + [4] p(2nd#) = [2] x1 + [5] p(activate#) = [4] x1 + [2] p(from#) = [0] p(head#) = [1] x1 + [2] p(sel#) = [4] x1 + [1] x2 + [0] p(take#) = [1] x1 + [4] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [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) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x3 + [0] Following rules are strictly oriented: take(X1,X2) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [3] = n__take(X1,X2) take(s(N),cons(X,XS)) = [1] N + [1] XS + [6] > [1] N + [1] XS + [5] = cons(X,n__take(N,activate(XS))) Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [2] XS + [5] >= [1] XS + [4] = c_1(head#(activate(XS))) activate#(X) = [4] X + [2] >= [0] = c_2(X) activate#(n__from(X)) = [2] >= [0] = c_3(from#(X)) activate#(n__take(X1,X2)) = [4] X1 + [4] X2 + [14] >= [1] X1 + [4] X2 + [0] = c_4(take#(X1,X2)) from#(X) = [0] >= [0] = c_5(X,X) from#(X) = [0] >= [0] = c_6(X) head#(cons(X,XS)) = [1] XS + [2] >= [0] = c_7(X) sel#(0(),cons(X,XS)) = [1] XS + [4] >= [1] = c_8(X) sel#(s(N),cons(X,XS)) = [4] N + [1] XS + [8] >= [4] N + [1] XS + [2] = c_9(sel#(N,activate(XS))) take#(X1,X2) = [1] X1 + [4] X2 + [0] >= [0] = c_10(X1,X2) take#(0(),XS) = [4] XS + [1] >= [0] = c_11() take#(s(N),cons(X,XS)) = [1] N + [4] XS + [2] >= [4] XS + [2] = c_12(X,N,activate#(XS)) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__from(X)) = [2] >= [2] = from(X) activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [4] = take(X1,X2) from(X) = [2] >= [0] = cons(X,n__from(s(X))) from(X) = [2] >= [0] = n__from(X) take(0(),XS) = [1] XS + [5] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2(X) activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5(X,X) from#(X) -> c_6(X) head#(cons(X,XS)) -> c_7(X) sel#(0(),cons(X,XS)) -> c_8(X) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10(X1,X2) take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(X,N,activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) 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,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/0 ,c_12/3} - Obligation: runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel#,take#} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))