*** 1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} Obligation: Full basic terms: {2nd,activate,cons,from}/{n__cons,n__from,s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following weak dependency pairs: Strict DPs 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Weak DPs and mark the set of starting terms. *** 1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Strict TRS Rules: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1} Obligation: Full basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) *** 1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1} Obligation: Full basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [0] p(cons) = [0] p(from) = [0] p(n__cons) = [0] p(n__from) = [0] p(s) = [0] p(2nd#) = [0] p(activate#) = [1] p(cons#) = [0] p(from#) = [5] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] Following rules are strictly oriented: activate#(X) = [1] > [0] = c_2(X) activate#(n__cons(X1,X2)) = [1] > [0] = c_3(cons#(X1,X2)) from#(X) = [5] > [0] = c_6(cons#(X,n__from(s(X)))) from#(X) = [5] > [0] = c_7(X) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [1] >= [5] = c_4(from#(X)) cons#(X1,X2) = [0] >= [0] = c_5(X1,X2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) Strict TRS Rules: Weak DP Rules: activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1} Obligation: Full basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [0] p(cons) = [0] p(from) = [0] p(n__cons) = [0] p(n__from) = [0] p(s) = [0] p(2nd#) = [0] p(activate#) = [11] p(cons#) = [11] p(from#) = [11] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] Following rules are strictly oriented: cons#(X1,X2) = [11] > [0] = c_5(X1,X2) Following rules are (at-least) weakly oriented: activate#(X) = [11] >= [0] = c_2(X) activate#(n__cons(X1,X2)) = [11] >= [11] = c_3(cons#(X1,X2)) activate#(n__from(X)) = [11] >= [11] = c_4(from#(X)) from#(X) = [11] >= [11] = c_6(cons#(X,n__from(s(X)))) from#(X) = [11] >= [1] = c_7(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_4(from#(X)) Strict TRS Rules: Weak DP Rules: activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1} Obligation: Full basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [0] p(cons) = [0] p(from) = [0] p(n__cons) = [0] p(n__from) = [2] p(s) = [0] p(2nd#) = [0] p(activate#) = [1] x1 + [11] p(cons#) = [0] p(from#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [11] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] Following rules are strictly oriented: activate#(n__from(X)) = [13] > [0] = c_4(from#(X)) Following rules are (at-least) weakly oriented: activate#(X) = [1] X + [11] >= [0] = c_2(X) activate#(n__cons(X1,X2)) = [11] >= [11] = c_3(cons#(X1,X2)) cons#(X1,X2) = [0] >= [0] = c_5(X1,X2) from#(X) = [0] >= [0] = c_6(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_7(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: activate#(X) -> c_2(X) activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)) activate#(n__from(X)) -> c_4(from#(X)) cons#(X1,X2) -> c_5(X1,X2) from#(X) -> c_6(cons#(X,n__from(s(X)))) from#(X) -> c_7(X) Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1} Obligation: Full basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).