WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from ,n__s} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) 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__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: sel(0(),cons(X,Y)) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(activate(X)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) from(X) = [1] X + [0] >= [1] X + [1] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = 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 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Weak TRS: sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [9] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: activate(X) = [1] X + [9] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(n__from(X)) = [1] X + [9] >= [1] X + [9] = from(activate(X)) activate(n__s(X)) = [1] X + [9] >= [1] X + [9] = s(activate(X)) 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) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sel(0(),cons(X,Y)) = [1] X + [0] >= [1] X + [0] = 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(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Weak TRS: activate(X) -> X sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [9] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [3] p(sel) = [1] x2 + [0] Following rules are strictly oriented: s(X) = [1] X + [3] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [9] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [9] >= [1] X + [9] = from(activate(X)) activate(n__s(X)) = [1] X + [9] >= [1] X + [12] = s(activate(X)) 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) sel(0(),cons(X,Y)) = [1] X + [0] >= [1] X + [0] = 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(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) - Weak TRS: activate(X) -> X s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,from,s,sel} TcT has computed the following interpretation: p(0) = [0] p(activate) = [8] x1 + [0] p(cons) = [1] x1 + [2] p(from) = [1] x1 + [2] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [2] p(s) = [1] x1 + [2] p(sel) = [1] x1 + [4] x2 + [10] Following rules are strictly oriented: activate(n__from(X)) = [8] X + [16] > [8] X + [2] = from(activate(X)) activate(n__s(X)) = [8] X + [16] > [8] X + [2] = s(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [8] X + [0] >= [1] X + [0] = X from(X) = [1] X + [2] >= [1] X + [2] = cons(X,n__from(n__s(X))) from(X) = [1] X + [2] >= [1] X + [2] = n__from(X) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) sel(0(),cons(X,Y)) = [4] X + [18] >= [1] X + [0] = X * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [3] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [3] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: from(X) = [1] X + [3] > [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [3] > [1] X + [1] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [3] X + [3] >= [3] X + [3] = from(activate(X)) activate(n__s(X)) = [3] X + [0] >= [3] X + [0] = s(activate(X)) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sel(0(),cons(X,Y)) = [1] X + [0] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,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))