WORST_CASE(?,O(n^3)) * Step 1: NaturalMI 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: 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__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] p(a__from) = [1] x1 + [4] p(a__length) = [2] p(a__length1) = [2] p(cons) = [1] x1 + [1] p(from) = [1] x1 + [4] p(length) = [1] p(length1) = [1] p(mark) = [1] x1 + [1] p(nil) = [1] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__from(X) = [1] X + [4] > [1] X + [2] = cons(mark(X),from(s(X))) a__length(X) = [2] > [1] = length(X) a__length(nil()) = [2] > [0] = 0() a__length1(X) = [2] > [1] = length1(X) mark(0()) = [1] > [0] = 0() mark(nil()) = [2] > [1] = nil() Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [4] >= [1] X + [4] = from(X) a__length(cons(X,Y)) = [2] >= [2] = s(a__length1(Y)) a__length1(X) = [2] >= [2] = a__length(X) mark(cons(X1,X2)) = [1] X1 + [2] >= [1] X1 + [2] = cons(mark(X1),X2) mark(from(X)) = [1] X + [5] >= [1] X + [5] = a__from(mark(X)) mark(length(X)) = [2] >= [2] = a__length(X) mark(length1(X)) = [2] >= [2] = a__length1(X) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) * Step 2: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) 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(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(nil()) -> 0() a__length1(X) -> length1(X) mark(0()) -> 0() 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 = 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__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] p(a__from) = [1] x1 + [8] p(a__length) = [4] p(a__length1) = [4] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [8] p(length) = [4] p(length1) = [4] p(mark) = [1] x1 + [8] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: mark(length(X)) = [12] > [4] = a__length(X) mark(length1(X)) = [12] > [4] = a__length1(X) Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [8] >= [1] X + [8] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [8] >= [1] X + [8] = from(X) a__length(X) = [4] >= [4] = length(X) a__length(cons(X,Y)) = [4] >= [4] = s(a__length1(Y)) a__length(nil()) = [4] >= [0] = 0() a__length1(X) = [4] >= [4] = a__length(X) a__length1(X) = [4] >= [4] = length1(X) mark(0()) = [8] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [8] >= [1] X1 + [8] = cons(mark(X1),X2) mark(from(X)) = [1] X + [16] >= [1] X + [16] = a__from(mark(X)) mark(nil()) = [8] >= [0] = nil() mark(s(X)) = [1] X + [8] >= [1] X + [8] = s(mark(X)) * Step 3: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(cons(X,Y)) -> s(a__length1(Y)) 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__length(X) -> length(X) 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 = 3, 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: 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 0] [2] [0 0 0] x1 + [0] [0 0 0] [2] p(a__length) = [0 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(a__length1) = [0 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [0 0 2] [2] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [2] p(from) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] p(length) = [0 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(length1) = [0 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [2 0 0] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: a__length(cons(X,Y)) = [0 0 2] [4] [0 0 0] Y + [0] [0 0 0] [0] > [0 0 2] [3] [0 0 0] Y + [0] [0 0 0] [0] = s(a__length1(Y)) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [2] >= [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [2] >= [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [0] = from(X) a__length(X) = [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] = length(X) a__length(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() a__length1(X) = [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] = a__length(X) a__length1(X) = [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 0] [0 0 2] [2] [0 0 0] X1 + [0 0 0] X2 + [0] [2 0 0] [0 0 4] [4] >= [1 0 0] [0 0 2] [2] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [2] = cons(mark(X1),X2) mark(from(X)) = [1 0 0] [2] [0 0 0] X + [0] [2 0 0] [4] >= [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [2] = a__from(mark(X)) mark(length(X)) = [0 0 2] [0] [0 0 0] X + [0] [0 0 4] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] = a__length(X) mark(length1(X)) = [0 0 2] [0] [0 0 0] X + [0] [0 0 4] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] = a__length1(X) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 0] [3] [0 0 0] X + [0] [2 0 0] [6] >= [1 0 0] [3] [0 0 0] X + [0] [0 0 0] [0] = s(mark(X)) * Step 4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) 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__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 = 3, 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: 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] [2] p(a__from) = [1 0 0] [3] [0 0 0] x1 + [2] [0 0 0] [2] p(a__length) = [0 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [2] p(a__length1) = [0 2 0] [2] [0 1 0] x1 + [0] [0 0 0] [2] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [2] [0 0 0] [0 0 0] [0] p(from) = [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [2] p(length) = [0 2 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(length1) = [0 2 0] [2] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 0] [0] [1 1 2] x1 + [0] [1 0 0] [2] p(nil) = [0] [0] [0] p(s) = [1 0 0] [2] [0 1 0] x1 + [2] [0 0 1] [0] Following rules are strictly oriented: a__length1(X) = [0 2 0] [2] [0 1 0] X + [0] [0 0 0] [2] > [0 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] = a__length(X) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 0] [3] [0 0 0] X + [2] [0 0 0] [2] >= [1 0 0] [2] [0 0 0] X + [2] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 0] [3] [0 0 0] X + [2] [0 0 0] [2] >= [1 0 0] [3] [0 0 0] X + [0] [0 0 0] [2] = from(X) a__length(X) = [0 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] >= [0 2 0] [0] [0 0 0] X + [0] [0 0 0] [0] = length(X) a__length(cons(X,Y)) = [0 2 0] [4] [0 1 0] Y + [2] [0 0 0] [2] >= [0 2 0] [4] [0 1 0] Y + [2] [0 0 0] [2] = s(a__length1(Y)) a__length(nil()) = [0] [0] [2] >= [0] [0] [2] = 0() a__length1(X) = [0 2 0] [2] [0 1 0] X + [0] [0 0 0] [2] >= [0 2 0] [2] [0 1 0] X + [0] [0 0 0] [0] = length1(X) mark(0()) = [0] [4] [2] >= [0] [0] [2] = 0() mark(cons(X1,X2)) = [1 0 0] [0 0 1] [0] [1 0 0] X1 + [0 1 1] X2 + [2] [1 0 0] [0 0 1] [2] >= [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [2] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 0] [3] [1 0 0] X + [7] [1 0 0] [5] >= [1 0 0] [3] [0 0 0] X + [2] [0 0 0] [2] = a__from(mark(X)) mark(length(X)) = [0 2 0] [0] [0 2 0] X + [0] [0 2 0] [2] >= [0 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] = a__length(X) mark(length1(X)) = [0 2 0] [2] [0 3 0] X + [2] [0 2 0] [4] >= [0 2 0] [2] [0 1 0] X + [0] [0 0 0] [2] = a__length1(X) mark(nil()) = [0] [0] [2] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 0] [2] [1 1 2] X + [4] [1 0 0] [4] >= [1 0 0] [2] [1 1 2] X + [2] [1 0 0] [2] = s(mark(X)) * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(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__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 = 3, 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) = [0] [0] [0] p(a__from) = [1 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [2] p(a__length) = [0 0 2] [0] [0 0 2] x1 + [0] [0 0 0] [0] p(a__length1) = [0 0 2] [0] [0 0 2] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [1] p(from) = [1 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(length) = [0 0 0] [0] [0 0 2] x1 + [0] [0 0 0] [0] p(length1) = [0 0 1] [0] [0 0 2] x1 + [0] [0 0 0] [0] p(mark) = [1 2 0] [0] [0 1 0] x1 + [0] [0 0 2] [0] p(nil) = [0] [2] [2] p(s) = [1 0 0] [2] [0 1 0] x1 + [2] [0 0 0] [0] Following rules are strictly oriented: mark(s(X)) = [1 2 0] [6] [0 1 0] X + [2] [0 0 0] [0] > [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [0] = s(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [2] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [1] = from(X) a__length(X) = [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 2] X + [0] [0 0 0] [0] = length(X) a__length(cons(X,Y)) = [0 0 2] [2] [0 0 2] Y + [2] [0 0 0] [0] >= [0 0 2] [2] [0 0 2] Y + [2] [0 0 0] [0] = s(a__length1(Y)) a__length(nil()) = [4] [4] [0] >= [0] [0] [0] = 0() a__length1(X) = [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] = a__length(X) a__length1(X) = [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 2] X + [0] [0 0 0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 2 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 2] [2] >= [1 2 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [1] = cons(mark(X1),X2) mark(from(X)) = [1 4 0] [0] [0 1 0] X + [0] [0 0 0] [2] >= [1 4 0] [0] [0 1 0] X + [0] [0 0 0] [2] = a__from(mark(X)) mark(length(X)) = [0 0 4] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] = a__length(X) mark(length1(X)) = [0 0 5] [0] [0 0 2] X + [0] [0 0 0] [0] >= [0 0 2] [0] [0 0 2] X + [0] [0 0 0] [0] = a__length1(X) mark(nil()) = [4] [2] [4] >= [0] [2] [2] = nil() * Step 6: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(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() 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: NaturalMI {miDimension = 3, 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: 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] [1] [0 2 0] x1 + [0] [0 0 1] [2] p(a__length) = [0] [0] [0] p(a__length1) = [0] [0] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [2] p(length) = [0] [0] [0] p(length1) = [0] [0] [0] p(mark) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [3] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__from(X) = [1 0 1] [1] [0 2 0] X + [0] [0 0 1] [2] > [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [2] = from(X) mark(from(X)) = [1 0 2] [3] [0 0 0] X + [0] [0 0 1] [2] > [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [2] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 1] [1] [0 2 0] X + [0] [0 0 1] [2] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(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()) = [1] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(length(X)) = [1] [0] [0] >= [0] [0] [0] = a__length(X) mark(length1(X)) = [1] [0] [0] >= [0] [0] [0] = a__length1(X) mark(nil()) = [4] [0] [0] >= [3] [0] [0] = nil() mark(s(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) * Step 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) - 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() 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: NaturalMI {miDimension = 3, 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: 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 1] x1 + [2] [0 0 3] [0] p(a__length) = [0] [0] [0] p(a__length1) = [0] [0] [0] p(cons) = [1 0 3] [0 0 1] [0] [0 1 0] x1 + [0 0 1] x2 + [2] [0 0 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 2 0] [1] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [3] [0] [0] p(s) = [1 1 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 2 3] [0 0 3] [5] [0 1 0] X1 + [0 0 1] X2 + [2] [0 0 0] [0 0 0] [0] > [1 2 0] [0 0 1] [1] [0 1 0] X1 + [0 0 1] X2 + [2] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__from(X) = [1 2 0] [1] [0 1 1] X + [2] [0 0 3] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2 0] [1] [0 1 1] X + [2] [0 0 3] [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()) = [1] [0] [0] >= [0] [0] [0] = 0() mark(from(X)) = [1 4 0] [6] [0 1 0] X + [2] [0 0 0] [0] >= [1 4 0] [2] [0 1 0] X + [2] [0 0 0] [0] = a__from(mark(X)) mark(length(X)) = [1] [0] [0] >= [0] [0] [0] = a__length(X) mark(length1(X)) = [1] [0] [0] >= [0] [0] [0] = a__length1(X) mark(nil()) = [4] [0] [0] >= [3] [0] [0] = nil() mark(s(X)) = [1 3 1] [1] [0 1 0] X + [0] [0 0 0] [0] >= [1 3 0] [1] [0 1 0] X + [0] [0 0 0] [0] = s(mark(X)) * Step 8: 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))