WORST_CASE(?,O(n^2)) * Step 1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__first,a__from,mark} TcT has computed the following interpretation: p(0) = [2] p(a__first) = [1] x1 + [1] x2 + [4] p(a__from) = [1] x1 + [2] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [4] p(from) = [1] x1 + [2] p(mark) = [1] x1 + [0] p(nil) = [2] p(s) = [1] x1 + [3] Following rules are strictly oriented: a__first(0(),X) = [1] X + [6] > [2] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [1] Y + [7] > [1] Y + [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1] X + [2] > [1] X + [0] = cons(mark(X),from(s(X))) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = first(X1,X2) a__from(X) = [1] X + [2] >= [1] X + [2] = from(X) mark(0()) = [2] >= [2] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1] X + [2] >= [1] X + [2] = a__from(mark(X)) mark(nil()) = [2] >= [2] = nil() mark(s(X)) = [1] X + [3] >= [1] X + [3] = s(mark(X)) * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__first) = [1] x1 + [1] x2 + [15] p(a__from) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [0] p(from) = [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [15] > [1] X1 + [0] = first(X1,X2) Following rules are (at-least) weakly oriented: a__first(0(),X) = [1] X + [15] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [1] Y + [15] >= [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1] X + [0] >= [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [0] = from(X) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [0] >= [15] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__first) = [1] x1 + [1] x2 + [15] p(a__from) = [1] x1 + [5] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [0] p(from) = [0] p(mark) = [5] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__from(X) = [1] X + [5] > [0] = from(X) mark(0()) = [5] > [0] = 0() mark(nil()) = [5] > [0] = nil() Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [15] >= [1] X1 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [15] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [1] Y + [15] >= [5] = cons(mark(Y),first(X,Z)) a__from(X) = [1] X + [5] >= [5] = cons(mark(X),from(s(X))) mark(cons(X1,X2)) = [5] >= [5] = cons(mark(X1),X2) mark(first(X1,X2)) = [5] >= [25] = a__first(mark(X1),mark(X2)) mark(from(X)) = [5] >= [10] = a__from(mark(X)) mark(s(X)) = [5] >= [5] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(nil()) -> nil() - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__first,a__from,mark} TcT has computed the following interpretation: p(0) = [1] [0] p(a__first) = [1 0] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 1] x1 + [0] [0 1] [1] p(cons) = [1 0] x1 + [0] [0 1] [0] p(first) = [1 0] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(from) = [1 1] x1 + [0] [0 1] [1] p(mark) = [1 1] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(from(X)) = [1 2] X + [1] [0 1] [1] > [1 2] X + [0] [0 1] [1] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0] X1 + [1 1] X2 + [0] [0 1] [0 1] [0] >= [1 0] X1 + [1 1] X2 + [0] [0 1] [0 1] [0] = first(X1,X2) a__first(0(),X) = [1 1] X + [1] [0 1] [0] >= [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0] X + [1 1] Y + [0] [0 1] [0 1] [0] >= [1 1] Y + [0] [0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 1] X + [0] [0 1] [1] >= [1 1] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1] X + [0] [0 1] [1] >= [1 1] X + [0] [0 1] [1] = from(X) mark(0()) = [1] [0] >= [1] [0] = 0() mark(cons(X1,X2)) = [1 1] X1 + [0] [0 1] [0] >= [1 1] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] >= [0] [0] = nil() mark(s(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = s(mark(X)) * Step 5: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__first,a__from,mark} TcT has computed the following interpretation: p(0) = [2] [3] p(a__first) = [1 0] x1 + [1 7] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 4] x1 + [0] [0 1] [3] p(cons) = [1 0] x1 + [0] [0 1] [2] p(first) = [1 0] x1 + [1 7] x2 + [0] [0 1] [0 1] [0] p(from) = [1 4] x1 + [0] [0 1] [3] p(mark) = [1 4] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 4] X1 + [8] [0 1] [2] > [1 4] X1 + [0] [0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0] X1 + [1 7] X2 + [0] [0 1] [0 1] [0] >= [1 0] X1 + [1 7] X2 + [0] [0 1] [0 1] [0] = first(X1,X2) a__first(0(),X) = [1 7] X + [2] [0 1] [3] >= [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0] X + [1 7] Y + [14] [0 1] [0 1] [2] >= [1 4] Y + [0] [0 1] [2] = cons(mark(Y),first(X,Z)) a__from(X) = [1 4] X + [0] [0 1] [3] >= [1 4] X + [0] [0 1] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 4] X + [0] [0 1] [3] >= [1 4] X + [0] [0 1] [3] = from(X) mark(0()) = [14] [3] >= [2] [3] = 0() mark(first(X1,X2)) = [1 4] X1 + [1 11] X2 + [0] [0 1] [0 1] [0] >= [1 4] X1 + [1 11] X2 + [0] [0 1] [0 1] [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 8] X + [12] [0 1] [3] >= [1 8] X + [0] [0 1] [3] = a__from(mark(X)) mark(nil()) = [0] [0] >= [0] [0] = nil() mark(s(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = s(mark(X)) * Step 6: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__first,a__from,mark} TcT has computed the following interpretation: p(0) = [1] [2] p(a__first) = [1 2] x1 + [1 4] x2 + [1] [0 1] [0 1] [2] p(a__from) = [1 4] x1 + [6] [0 1] [1] p(cons) = [1 0] x1 + [0] [0 1] [1] p(first) = [1 2] x1 + [1 4] x2 + [0] [0 1] [0 1] [2] p(from) = [1 4] x1 + [2] [0 1] [1] p(mark) = [1 4] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(s) = [1 4] x1 + [4] [0 1] [0] Following rules are strictly oriented: mark(first(X1,X2)) = [1 6] X1 + [1 8] X2 + [8] [0 1] [0 1] [2] > [1 6] X1 + [1 8] X2 + [1] [0 1] [0 1] [2] = a__first(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 2] X1 + [1 4] X2 + [1] [0 1] [0 1] [2] >= [1 2] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] = first(X1,X2) a__first(0(),X) = [1 4] X + [6] [0 1] [4] >= [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 6] X + [1 4] Y + [9] [0 1] [0 1] [3] >= [1 4] Y + [0] [0 1] [1] = cons(mark(Y),first(X,Z)) a__from(X) = [1 4] X + [6] [0 1] [1] >= [1 4] X + [0] [0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 4] X + [6] [0 1] [1] >= [1 4] X + [2] [0 1] [1] = from(X) mark(0()) = [9] [2] >= [1] [2] = 0() mark(cons(X1,X2)) = [1 4] X1 + [4] [0 1] [1] >= [1 4] X1 + [0] [0 1] [1] = cons(mark(X1),X2) mark(from(X)) = [1 8] X + [6] [0 1] [1] >= [1 8] X + [6] [0 1] [1] = a__from(mark(X)) mark(nil()) = [0] [0] >= [0] [0] = nil() mark(s(X)) = [1 8] X + [4] [0 1] [0] >= [1 8] X + [4] [0 1] [0] = s(mark(X)) * Step 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(s(X)) -> s(mark(X)) - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(a__first) = {1,2}, uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__first,a__from,mark} TcT has computed the following interpretation: p(0) = [0] [6] p(a__first) = [1 1] x1 + [1 4] x2 + [0] [0 1] [0 1] [5] p(a__from) = [1 2] x1 + [7] [0 1] [0] p(cons) = [1 0] x1 + [0] [0 1] [0] p(first) = [1 1] x1 + [1 4] x2 + [0] [0 1] [0 1] [5] p(from) = [1 2] x1 + [7] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [0] p(nil) = [2] [0] p(s) = [1 2] x1 + [0] [0 1] [1] Following rules are strictly oriented: mark(s(X)) = [1 4] X + [2] [0 1] [1] > [1 4] X + [0] [0 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 1] X1 + [1 4] X2 + [0] [0 1] [0 1] [5] >= [1 1] X1 + [1 4] X2 + [0] [0 1] [0 1] [5] = first(X1,X2) a__first(0(),X) = [1 4] X + [6] [0 1] [11] >= [2] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 3] X + [1 4] Y + [1] [0 1] [0 1] [6] >= [1 2] Y + [0] [0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 2] X + [7] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2] X + [7] [0 1] [0] >= [1 2] X + [7] [0 1] [0] = from(X) mark(0()) = [12] [6] >= [0] [6] = 0() mark(cons(X1,X2)) = [1 2] X1 + [0] [0 1] [0] >= [1 2] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 3] X1 + [1 6] X2 + [10] [0 1] [0 1] [5] >= [1 3] X1 + [1 6] X2 + [0] [0 1] [0 1] [5] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 4] X + [7] [0 1] [0] >= [1 4] X + [7] [0 1] [0] = a__from(mark(X)) mark(nil()) = [2] [0] >= [2] [0] = nil() * Step 8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))