/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: 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: runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons,n__first,n__from ,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + 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: runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons,n__first,n__from ,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + 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: runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons,n__first,n__from ,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(v){v -> n__first(s(x),cons(z,v))} = activate(n__first(s(x),cons(z,v))) ->^+ cons(z,n__first(x,activate(v))) = C[activate(v) = activate(v){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + 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: 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 weak dependency pairs: Strict DPs activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - 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,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1 ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - 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) - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 {5} by application of Pre({5}) = {1,2,4,6,7,8,9}. Here rules are labelled as follows: 1: activate#(X) -> c_1(X) 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(X1,X2) 5: first#(0(),Z) -> c_5() 6: first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) 7: from#(X) -> c_7(X,X) 8: from#(X) -> c_8(X) 9: sel#(0(),cons(X,Z)) -> c_9(X) 10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) ** Step 1.b:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - 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) - Weak DPs: first#(0(),Z) -> c_5() - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, 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 + [3] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [7] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(sel) = [0] p(activate#) = [0] p(first#) = [0] p(from#) = [0] p(sel#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x3 + [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#(0(),cons(X,Z)) = [1] Z + [3] > [0] = c_9(X) sel#(s(X),cons(Y,Z)) = [1] Z + [3] > [1] Z + [0] = c_10(sel#(X,activate(Z))) activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [0] = first(X1,X2) Following rules are (at-least) weakly oriented: activate#(X) = [0] >= [0] = c_1(X) activate#(n__first(X1,X2)) = [0] >= [0] = c_2(first#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_3(from#(X)) first#(X1,X2) = [0] >= [0] = c_4(X1,X2) first#(0(),Z) = [0] >= [0] = c_5() first#(s(X),cons(Y,Z)) = [0] >= [0] = c_6(Y,X,activate#(Z)) from#(X) = [0] >= [0] = c_7(X,X) from#(X) = [0] >= [0] = c_8(X) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [7] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] X + [1] Z + [3] >= [1] X + [1] Z + [10] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [0] >= [1] X + [3] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) 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_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) - Strict TRS: activate(X) -> X 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) - Weak DPs: first#(0(),Z) -> c_5() sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(n__first(X1,X2)) -> first(X1,X2) - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, 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] x1 + [1] x2 + [3] p(from) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [3] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(sel) = [0] p(activate#) = [0] p(first#) = [0] p(from#) = [1] p(sel#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x3 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: from#(X) = [1] > [0] = c_7(X,X) from#(X) = [1] > [0] = c_8(X) first(0(),Z) = [1] Z + [3] > [0] = nil() Following rules are (at-least) weakly oriented: activate#(X) = [0] >= [0] = c_1(X) activate#(n__first(X1,X2)) = [0] >= [0] = c_2(first#(X1,X2)) activate#(n__from(X)) = [0] >= [1] = c_3(from#(X)) first#(X1,X2) = [0] >= [0] = c_4(X1,X2) first#(0(),Z) = [0] >= [0] = c_5() first#(s(X),cons(Y,Z)) = [0] >= [0] = c_6(Y,X,activate#(Z)) sel#(0(),cons(X,Z)) = [1] Z + [0] >= [0] = c_9(X) sel#(s(X),cons(Y,Z)) = [1] Z + [0] >= [1] Z + [0] = c_10(sel#(X,activate(Z))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = first(X1,X2) activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__first(X1,X2) first(s(X),cons(Y,Z)) = [1] X + [1] Z + [3] >= [1] X + [1] Z + [3] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) 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: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) 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) - Weak DPs: first#(0(),Z) -> c_5() from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(n__first(X1,X2)) -> first(X1,X2) first(0(),Z) -> nil() - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, 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] x1 + [1] x2 + [0] p(from) = [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__from) = [0] p(nil) = [0] p(s) = [1] x1 + [1] p(sel) = [0] p(activate#) = [1] x1 + [0] p(first#) = [1] x1 + [1] x2 + [0] p(from#) = [0] p(sel#) = [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 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x2 + [1] x3 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: first#(s(X),cons(Y,Z)) = [1] X + [1] Z + [1] > [1] X + [1] Z + [0] = c_6(Y,X,activate#(Z)) first(s(X),cons(Y,Z)) = [1] X + [1] Z + [1] > [1] X + [1] Z + [0] = cons(Y,n__first(X,activate(Z))) Following rules are (at-least) weakly oriented: activate#(X) = [1] X + [0] >= [1] X + [0] = c_1(X) activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_2(first#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_3(from#(X)) first#(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_4(X1,X2) first#(0(),Z) = [1] Z + [0] >= [0] = c_5() from#(X) = [0] >= [0] = c_7(X,X) from#(X) = [0] >= [0] = c_8(X) sel#(0(),cons(X,Z)) = [1] Z + [0] >= [0] = c_9(X) sel#(s(X),cons(Y,Z)) = [1] Z + [0] >= [1] Z + [0] = c_10(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)) = [0] >= [0] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() 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 1.b:7: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z))) - Weak TRS: activate(n__first(X1,X2)) -> first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [1] p(cons) = [1] x2 + [3] p(first) = [1] x2 + [2] p(from) = [1] x1 + [0] p(n__first) = [1] x2 + [1] p(n__from) = [1] x1 + [7] p(nil) = [1] p(s) = [1] x1 + [1] p(sel) = [2] x1 + [2] p(activate#) = [1] p(first#) = [5] p(from#) = [0] p(sel#) = [2] x1 + [1] x2 + [6] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [5] p(c_6) = [1] x3 + [4] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [4] Following rules are strictly oriented: activate#(X) = [1] > [0] = c_1(X) activate#(n__from(X)) = [1] > [0] = c_3(from#(X)) first#(X1,X2) = [5] > [0] = c_4(X1,X2) activate(X) = [1] X + [1] > [1] X + [0] = X activate(n__from(X)) = [1] X + [8] > [1] X + [0] = from(X) first(X1,X2) = [1] X2 + [2] > [1] X2 + [1] = n__first(X1,X2) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [1] >= [6] = c_2(first#(X1,X2)) first#(0(),Z) = [5] >= [5] = c_5() first#(s(X),cons(Y,Z)) = [5] >= [5] = c_6(Y,X,activate#(Z)) from#(X) = [0] >= [0] = c_7(X,X) from#(X) = [0] >= [0] = c_8(X) sel#(0(),cons(X,Z)) = [1] Z + [9] >= [0] = c_9(X) sel#(s(X),cons(Y,Z)) = [2] X + [1] Z + [11] >= [2] X + [1] Z + [11] = c_10(sel#(X,activate(Z))) activate(n__first(X1,X2)) = [1] X2 + [2] >= [1] X2 + [2] = first(X1,X2) first(0(),Z) = [1] Z + [2] >= [1] = nil() first(s(X),cons(Y,Z)) = [1] Z + [5] >= [1] Z + [5] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [0] >= [1] X + [11] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [7] = n__from(X) 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: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) - Strict TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) 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))) - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, 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 + [4] p(first) = [1] x2 + [5] p(from) = [0] p(n__first) = [1] x2 + [5] p(n__from) = [0] p(nil) = [5] p(s) = [1] x1 + [0] p(sel) = [0] p(activate#) = [1] x1 + [2] p(first#) = [1] x2 + [0] p(from#) = [0] p(sel#) = [1] x2 + [0] p(c_1) = [1] x1 + [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [2] p(c_4) = [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x3 + [2] p(c_7) = [0] p(c_8) = [0] p(c_9) = [4] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [1] X2 + [7] > [1] X2 + [0] = c_2(first#(X1,X2)) Following rules are (at-least) weakly oriented: activate#(X) = [1] X + [2] >= [1] X + [2] = c_1(X) activate#(n__from(X)) = [2] >= [2] = c_3(from#(X)) first#(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = c_4(X1,X2) first#(0(),Z) = [1] Z + [0] >= [0] = c_5() first#(s(X),cons(Y,Z)) = [1] Z + [4] >= [1] Z + [4] = c_6(Y,X,activate#(Z)) from#(X) = [0] >= [0] = c_7(X,X) from#(X) = [0] >= [0] = c_8(X) sel#(0(),cons(X,Z)) = [1] Z + [4] >= [4] = c_9(X) sel#(s(X),cons(Y,Z)) = [1] Z + [4] >= [1] Z + [0] = c_10(sel#(X,activate(Z))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [5] >= [1] X2 + [5] = first(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) first(X1,X2) = [1] X2 + [5] >= [1] X2 + [5] = n__first(X1,X2) first(0(),Z) = [1] Z + [5] >= [5] = nil() first(s(X),cons(Y,Z)) = [1] Z + [9] >= [1] Z + [9] = cons(Y,n__first(X,activate(Z))) from(X) = [0] >= [4] = 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 1.b:9: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) 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))) - 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [1] p(first) = [1] x2 + [5] p(from) = [1] x1 + [1] p(n__first) = [1] x2 + [4] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [6] p(sel) = [1] x2 + [0] p(activate#) = [5] p(first#) = [5] p(from#) = [1] p(sel#) = [1] x2 + [5] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x3 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [6] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: from(X) = [1] X + [1] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: activate#(X) = [5] >= [0] = c_1(X) activate#(n__first(X1,X2)) = [5] >= [5] = c_2(first#(X1,X2)) activate#(n__from(X)) = [5] >= [1] = c_3(from#(X)) first#(X1,X2) = [5] >= [0] = c_4(X1,X2) first#(0(),Z) = [5] >= [0] = c_5() first#(s(X),cons(Y,Z)) = [5] >= [5] = c_6(Y,X,activate#(Z)) from#(X) = [1] >= [1] = c_7(X,X) from#(X) = [1] >= [1] = c_8(X) sel#(0(),cons(X,Z)) = [1] X + [1] Z + [6] >= [6] = c_9(X) sel#(s(X),cons(Y,Z)) = [1] Y + [1] Z + [6] >= [1] Z + [6] = c_10(sel#(X,activate(Z))) activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [5] >= [1] X2 + [5] = first(X1,X2) activate(n__from(X)) = [1] X + [1] >= [1] X + [1] = from(X) first(X1,X2) = [1] X2 + [5] >= [1] X2 + [4] = n__first(X1,X2) first(0(),Z) = [1] Z + [5] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] Y + [1] Z + [6] >= [1] Y + [1] Z + [6] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [1] >= [1] X + [7] = cons(X,n__from(s(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(s(X))) - Weak DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) 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) -> 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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 = 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(first) = {2}, uargs(n__first) = {2}, uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {3}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [3] p(cons) = [1] x2 + [2] p(first) = [1] x1 + [1] x2 + [4] p(from) = [3] p(n__first) = [1] x1 + [1] x2 + [4] p(n__from) = [0] p(nil) = [0] p(s) = [1] x1 + [4] p(sel) = [0] p(activate#) = [0] p(first#) = [0] p(from#) = [0] p(sel#) = [3] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x3 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [2] Following rules are strictly oriented: from(X) = [3] > [2] = cons(X,n__from(s(X))) Following rules are (at-least) weakly oriented: activate#(X) = [0] >= [0] = c_1(X) activate#(n__first(X1,X2)) = [0] >= [0] = c_2(first#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_3(from#(X)) first#(X1,X2) = [0] >= [0] = c_4(X1,X2) first#(0(),Z) = [0] >= [0] = c_5() first#(s(X),cons(Y,Z)) = [0] >= [0] = c_6(Y,X,activate#(Z)) from#(X) = [0] >= [0] = c_7(X,X) from#(X) = [0] >= [0] = c_8(X) sel#(0(),cons(X,Z)) = [1] Z + [2] >= [0] = c_9(X) sel#(s(X),cons(Y,Z)) = [3] X + [1] Z + [14] >= [3] X + [1] Z + [5] = c_10(sel#(X,activate(Z))) activate(X) = [1] X + [3] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [4] = first(X1,X2) activate(n__from(X)) = [3] >= [3] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n__first(X1,X2) first(0(),Z) = [1] Z + [4] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] X + [1] Z + [10] >= [1] X + [1] Z + [9] = cons(Y,n__first(X,activate(Z))) from(X) = [3] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:11: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_1(X) activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) first#(X1,X2) -> c_4(X1,X2) first#(0(),Z) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z)) from#(X) -> c_7(X,X) from#(X) -> c_8(X) sel#(0(),cons(X,Z)) -> c_9(X) 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_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1} - Obligation: 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(Omega(n^1),O(n^1))