WORST_CASE(?,O(n^3)) * Step 1: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [11] p(a__from) = [1] x1 + [1] p(a__length) = [0] p(a__length1) = [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(length) = [0] p(length1) = [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__from(X) = [1] X + [1] > [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [1] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: a__length(X) = [0] >= [0] = length(X) a__length(cons(X,Y)) = [0] >= [0] = s(a__length1(Y)) a__length(nil()) = [0] >= [11] = 0() a__length1(X) = [0] >= [0] = a__length(X) a__length1(X) = [0] >= [0] = length1(X) mark(0()) = [11] >= [11] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [1] = a__from(mark(X)) mark(length(X)) = [0] >= [0] = a__length(X) mark(length1(X)) = [0] >= [0] = a__length1(X) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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__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__from) = [1] x1 + [1] p(a__length) = [7] p(a__length1) = [0] p(cons) = [1] x1 + [1] p(from) = [1] x1 + [1] p(length) = [9] p(length1) = [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [9] Following rules are strictly oriented: a__length(nil()) = [7] > [0] = 0() mark(length(X)) = [9] > [7] = a__length(X) Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [1] >= [1] X + [1] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [1] >= [1] X + [1] = from(X) a__length(X) = [7] >= [9] = length(X) a__length(cons(X,Y)) = [7] >= [9] = s(a__length1(Y)) a__length1(X) = [0] >= [7] = a__length(X) a__length1(X) = [0] >= [0] = length1(X) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = cons(mark(X1),X2) mark(from(X)) = [1] X + [1] >= [1] X + [1] = a__from(mark(X)) mark(length1(X)) = [0] >= [0] = a__length1(X) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [1] X + [9] >= [1] X + [9] = 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^3)) + Considered Problem: - Strict TRS: a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(nil()) -> 0() mark(length(X)) -> a__length(X) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(a__from) = [1] x1 + [9] p(a__length) = [6] p(a__length1) = [0] p(cons) = [1] x1 + [4] p(from) = [1] x1 + [9] p(length) = [1] p(length1) = [0] p(mark) = [1] x1 + [5] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__length(X) = [6] > [1] = length(X) a__length(cons(X,Y)) = [6] > [0] = s(a__length1(Y)) mark(0()) = [11] > [6] = 0() mark(length1(X)) = [5] > [0] = a__length1(X) mark(nil()) = [5] > [0] = nil() Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [9] >= [1] X + [9] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [9] >= [1] X + [9] = from(X) a__length(nil()) = [6] >= [6] = 0() a__length1(X) = [0] >= [6] = a__length(X) a__length1(X) = [0] >= [0] = length1(X) mark(cons(X1,X2)) = [1] X1 + [9] >= [1] X1 + [9] = cons(mark(X1),X2) mark(from(X)) = [1] X + [14] >= [1] X + [14] = a__from(mark(X)) mark(length(X)) = [6] >= [6] = a__length(X) mark(s(X)) = [1] X + [5] >= [1] X + [5] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [8] p(a__from) = [1] x1 + [1] p(a__length) = [8] p(a__length1) = [8] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [1] p(length) = [8] p(length1) = [7] p(mark) = [1] x1 + [1] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__length1(X) = [8] > [7] = length1(X) Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [1] >= [1] X + [1] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [1] >= [1] X + [1] = from(X) a__length(X) = [8] >= [8] = length(X) a__length(cons(X,Y)) = [8] >= [8] = s(a__length1(Y)) a__length(nil()) = [8] >= [8] = 0() a__length1(X) = [8] >= [8] = a__length(X) mark(0()) = [9] >= [8] = 0() mark(cons(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = cons(mark(X1),X2) mark(from(X)) = [1] X + [2] >= [1] X + [2] = a__from(mark(X)) mark(length(X)) = [9] >= [8] = a__length(X) mark(length1(X)) = [8] >= [8] = a__length1(X) mark(nil()) = [1] >= [0] = nil() mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__length1(X) -> a__length(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> length1(X) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [4] [1] p(a__from) = [1 0] x1 + [1] [0 0] [1] p(a__length) = [0 4] x1 + [0] [0 5] [0] p(a__length1) = [0 4] x1 + [4] [0 5] [4] p(cons) = [1 0] x1 + [0 1] x2 + [1] [0 0] [0 1] [1] p(from) = [1 0] x1 + [1] [0 0] [0] p(length) = [0 4] x1 + [0] [0 0] [0] p(length1) = [0 4] x1 + [4] [0 0] [4] p(mark) = [1 0] x1 + [0] [2 0] [0] p(nil) = [4] [2] p(s) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: a__length1(X) = [0 4] X + [4] [0 5] [4] > [0 4] X + [0] [0 5] [0] = a__length(X) Following rules are (at-least) weakly oriented: a__from(X) = [1 0] X + [1] [0 0] [1] >= [1 0] X + [1] [0 0] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0] X + [1] [0 0] [1] >= [1 0] X + [1] [0 0] [0] = from(X) a__length(X) = [0 4] X + [0] [0 5] [0] >= [0 4] X + [0] [0 0] [0] = length(X) a__length(cons(X,Y)) = [0 4] Y + [4] [0 5] [5] >= [0 4] Y + [4] [0 5] [4] = s(a__length1(Y)) a__length(nil()) = [8] [10] >= [4] [1] = 0() a__length1(X) = [0 4] X + [4] [0 5] [4] >= [0 4] X + [4] [0 0] [4] = length1(X) mark(0()) = [4] [8] >= [4] [1] = 0() mark(cons(X1,X2)) = [1 0] X1 + [0 1] X2 + [1] [2 0] [0 2] [2] >= [1 0] X1 + [0 1] X2 + [1] [0 0] [0 1] [1] = cons(mark(X1),X2) mark(from(X)) = [1 0] X + [1] [2 0] [2] >= [1 0] X + [1] [0 0] [1] = a__from(mark(X)) mark(length(X)) = [0 4] X + [0] [0 8] [0] >= [0 4] X + [0] [0 5] [0] = a__length(X) mark(length1(X)) = [0 4] X + [4] [0 8] [8] >= [0 4] X + [4] [0 5] [4] = a__length1(X) mark(nil()) = [4] [8] >= [4] [2] = nil() mark(s(X)) = [1 0] X + [0] [2 0] [0] >= [1 0] X + [0] [2 0] [0] = s(mark(X)) * Step 6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__from) = [1 2 0] [1] [0 1 0] x1 + [2] [0 0 2] [0] p(a__length) = [0] [0] [0] p(a__length1) = [0] [0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(from) = [1 2 0] [1] [0 1 0] x1 + [2] [0 0 0] [0] p(length) = [0] [0] [0] p(length1) = [0] [0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [1] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(from(X)) = [1 3 0] [3] [0 1 0] X + [2] [0 0 0] [0] > [1 3 0] [1] [0 1 0] X + [2] [0 0 0] [0] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 2 0] [1] [0 1 0] X + [2] [0 0 2] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2 0] [1] [0 1 0] X + [2] [0 0 2] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = from(X) a__length(X) = [0] [0] [0] >= [0] [0] [0] = length(X) a__length(cons(X,Y)) = [0] [0] [0] >= [0] [0] [0] = s(a__length1(Y)) a__length(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() a__length1(X) = [0] [0] [0] >= [0] [0] [0] = a__length(X) a__length1(X) = [0] [0] [0] >= [0] [0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(length(X)) = [0] [0] [0] >= [0] [0] [0] = a__length(X) mark(length1(X)) = [0] [0] [0] >= [0] [0] [0] = a__length1(X) mark(nil()) = [1] [0] [0] >= [1] [0] [0] = nil() mark(s(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(mark(X)) * Step 7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__from) = [1 0 1] [0] [2 0 1] x1 + [0] [0 0 1] [1] p(a__length) = [0] [0] [0] p(a__length1) = [0 0 0] [0] [0 0 2] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(from) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(length) = [0] [0] [0] p(length1) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(mark) = [1 0 1] [0] [2 2 0] x1 + [0] [0 0 1] [0] p(nil) = [2] [0] [0] p(s) = [1 0 0] [0] [0 0 2] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 0 1] [1] [2 0 0] X1 + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 1] [0] [2 0 1] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [2 0 1] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] = from(X) a__length(X) = [0] [0] [0] >= [0] [0] [0] = length(X) a__length(cons(X,Y)) = [0] [0] [0] >= [0] [0] [0] = s(a__length1(Y)) a__length(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() a__length1(X) = [0 0 0] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0] [0] [0] = a__length(X) a__length1(X) = [0 0 0] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(from(X)) = [1 0 2] [1] [2 0 4] X + [0] [0 0 1] [1] >= [1 0 2] [0] [2 0 3] X + [0] [0 0 1] [1] = a__from(mark(X)) mark(length(X)) = [0] [0] [0] >= [0] [0] [0] = a__length(X) mark(length1(X)) = [0 0 0] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 2] X + [0] [0 0 0] [0] = a__length1(X) mark(nil()) = [2] [4] [0] >= [2] [0] [0] = nil() mark(s(X)) = [1 0 1] [0] [2 0 4] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 2] X + [0] [0 0 1] [0] = s(mark(X)) * Step 8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(a__from) = [1 0 1 0] [1] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [1] p(a__length) = [0 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(a__length1) = [0 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(cons) = [1 0 0 0] [0 0 0 1] [1] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] p(from) = [1 0 1 0] [1] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(length) = [0 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(length1) = [0 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(mark) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [1 0 0 0] [0] p(nil) = [1] [0] [0] [0] p(s) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: mark(s(X)) = [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [1] [1 0 0 0] [0] > [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = s(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [1] >= [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [1] >= [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = from(X) a__length(X) = [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = length(X) a__length(cons(X,Y)) = [0 0 0 0] [0] [0 0 0 0] Y + [0] [0 0 0 1] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] Y + [0] [0 0 0 1] [1] [0 0 0 0] [0] = s(a__length1(Y)) a__length(nil()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = 0() a__length1(X) = [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = a__length(X) a__length1(X) = [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = length1(X) mark(0()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 1 0] [0 0 0 1] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 0 0] [0] [1 0 0 0] [0 0 0 1] [1] >= [1 0 1 0] [0 0 0 1] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] = cons(mark(X1),X2) mark(from(X)) = [1 0 2 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [1 0 1 0] [1] >= [1 0 2 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [1] = a__from(mark(X)) mark(length(X)) = [0 0 0 1] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = a__length(X) mark(length1(X)) = [0 0 0 1] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = a__length1(X) mark(nil()) = [1] [0] [0] [1] >= [1] [0] [0] [0] = nil() * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))