WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [10] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: s(X) = [1] X + [1] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [10] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [10] X1 + [0] >= [10] X1 + [0] = cons(activate(X1),X2) activate(n__from(X)) = [10] X + [0] >= [10] X + [0] = from(activate(X)) activate(n__s(X)) = [10] X + [0] >= [10] X + [1] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__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 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) - Weak TRS: s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [10] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [1] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: from(X) = [1] X + [1] > [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [1] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [10] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [10] X1 + [0] >= [10] X1 + [0] = cons(activate(X1),X2) activate(n__from(X)) = [10] X + [0] >= [10] X + [1] = from(activate(X)) activate(n__s(X)) = [10] X + [0] >= [10] X + [0] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) - Weak TRS: from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [5] p(from) = [1] x1 + [5] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [5] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(n__from(X)) = [2] X + [10] > [2] X + [5] = from(activate(X)) cons(X1,X2) = [1] X1 + [5] > [1] X1 + [0] = n__cons(X1,X2) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [0] >= [2] X1 + [5] = cons(activate(X1),X2) activate(n__s(X)) = [2] X + [0] >= [2] X + [0] = s(activate(X)) from(X) = [1] X + [5] >= [1] X + [5] = cons(X,n__from(n__s(X))) from(X) = [1] X + [5] >= [1] X + [5] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__s(X)) -> s(activate(X)) - Weak TRS: activate(n__from(X)) -> from(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {2nd,activate,cons,from,s} TcT has computed the following interpretation: p(2nd) = [2] p(activate) = [8] x1 + [2] p(cons) = [1] x1 + [3] p(from) = [1] x1 + [8] p(n__cons) = [1] x1 + [2] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [8] X + [2] > [1] X + [0] = X activate(n__cons(X1,X2)) = [8] X1 + [18] > [8] X1 + [5] = cons(activate(X1),X2) Following rules are (at-least) weakly oriented: activate(n__from(X)) = [8] X + [10] >= [8] X + [10] = from(activate(X)) activate(n__s(X)) = [8] X + [2] >= [8] X + [2] = s(activate(X)) cons(X1,X2) = [1] X1 + [3] >= [1] X1 + [2] = n__cons(X1,X2) from(X) = [1] X + [8] >= [1] X + [3] = cons(X,n__from(n__s(X))) from(X) = [1] X + [8] >= [1] X + [1] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) * Step 6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__s(X)) -> s(activate(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {2nd,activate,cons,from,s} TcT has computed the following interpretation: p(2nd) = [1] x1 + [1] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [6] p(from) = [1] x1 + [6] p(n__cons) = [1] x1 + [4] p(n__from) = [1] x1 + [5] p(n__s) = [1] x1 + [1] p(s) = [1] x1 + [1] Following rules are strictly oriented: activate(n__s(X)) = [2] X + [2] > [2] X + [1] = s(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [8] >= [2] X1 + [6] = cons(activate(X1),X2) activate(n__from(X)) = [2] X + [10] >= [2] X + [6] = from(activate(X)) cons(X1,X2) = [1] X1 + [6] >= [1] X1 + [4] = n__cons(X1,X2) from(X) = [1] X + [6] >= [1] X + [6] = cons(X,n__from(n__s(X))) from(X) = [1] X + [6] >= [1] X + [5] = n__from(X) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))