WORST_CASE(?,O(n^3)) * Step 1: NaturalPI WORST_CASE(?,O(n^3)) + 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 3 p(a__first) = x1 + x2 p(a__from) = 12 + x1 p(cons) = x1 p(first) = x1 + x2 p(from) = 12 + x1 p(mark) = x1 p(nil) = 0 p(s) = 3 + x1 Following rules are strictly oriented: a__first(0(),X) = 3 + X > 0 = nil() a__first(s(X),cons(Y,Z)) = 3 + X + Y > Y = cons(mark(Y),first(X,Z)) a__from(X) = 12 + X > X = cons(mark(X),from(s(X))) Following rules are (at-least) weakly oriented: a__first(X1,X2) = X1 + X2 >= X1 + X2 = first(X1,X2) a__from(X) = 12 + X >= 12 + X = from(X) mark(0()) = 3 >= 3 = 0() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(first(X1,X2)) = X1 + X2 >= X1 + X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = 12 + X >= 12 + X = a__from(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 3 + X >= 3 + X = s(mark(X)) * Step 2: NaturalMI WORST_CASE(?,O(n^3)) + 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: 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__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] [0] [0] p(a__first) = [1 3 2] [1 0 2] [1] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [2] p(a__from) = [1 0 2] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 2] [1 0 2] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [2] p(from) = [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 2] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(nil) = [0] [1] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__first(X1,X2) = [1 3 2] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] > [1 0 2] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = first(X1,X2) Following rules are (at-least) weakly oriented: a__first(0(),X) = [1 0 2] [1] [0 0 0] X + [1] [0 0 1] [2] >= [0] [1] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 2] [1 0 2] [1] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [2] >= [1 0 2] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 2] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 2] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = from(X) mark(0()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 2] [0] [0 0 0] X1 + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 4] [1 0 4] [4] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] >= [1 0 4] [1 0 4] [4] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 0 4] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 4] [0] [0 0 0] X + [1] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [1] [0] >= [0] [1] [0] = nil() mark(s(X)) = [1 0 2] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) * Step 3: NaturalMI WORST_CASE(?,O(n^3)) + 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: 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__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] [0] [0] p(a__first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [3] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(from) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [3] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [2] Following rules are strictly oriented: mark(from(X)) = [1 0 3] [4] [0 0 0] X + [0] [0 0 1] [3] > [1 0 3] [1] [0 0 0] X + [0] [0 0 1] [3] = a__from(mark(X)) mark(s(X)) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [2] > [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [2] = s(mark(X)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 0] [1 0 1] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [2] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [3] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [3] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [3] = from(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() * Step 4: NaturalMI WORST_CASE(?,O(n^3)) + 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(nil()) -> nil() - 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))) mark(from(X)) -> a__from(mark(X)) 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 = 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__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] [0] [0] p(a__first) = [1 0 0] [1 3 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [0] p(a__from) = [1 2 0] [1] [0 1 0] x1 + [2] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(first) = [1 0 0] [1 3 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(from) = [1 2 0] [0] [0 1 0] x1 + [2] [0 0 0] [0] p(mark) = [1 2 0] [0] [0 1 0] x1 + [0] [1 0 0] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: a__from(X) = [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] > [1 2 0] [0] [0 1 0] X + [2] [0 0 0] [0] = from(X) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 3 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [0] >= [1 0 0] [1 3 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = first(X1,X2) a__first(0(),X) = [1 3 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 0] [1 3 0] [1] [0 1 0] X + [0 1 0] Y + [0] [0 0 0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 2 0] [0] [0 1 0] X1 + [0] [1 0 0] [0] >= [1 2 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 2 0] [1 5 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [1 0 0] [1 3 0] [0] >= [1 2 0] [1 5 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [1 0 0] [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 4 0] [4] [0 1 0] X + [2] [1 2 0] [0] >= [1 4 0] [1] [0 1 0] X + [2] [0 0 0] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 2 0] [1] [0 1 0] X + [0] [1 0 0] [1] >= [1 2 0] [1] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() - 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(from(X)) -> a__from(mark(X)) 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 = 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__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] [0] [0] p(a__first) = [1 0 2] [1 2 0] [3] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [1] p(a__from) = [1 1 3] [0] [0 1 2] x1 + [0] [0 0 0] [0] p(cons) = [1 0 2] [0 0 3] [0] [0 1 1] x1 + [0 0 2] x2 + [0] [0 0 0] [0 0 0] [0] p(first) = [1 0 2] [1 2 0] [3] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [1] p(from) = [1 1 2] [0] [0 1 2] x1 + [0] [0 0 0] [0] p(mark) = [1 1 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 1] [0] [0 1 2] x1 + [2] [0 0 0] [0] Following rules are strictly oriented: mark(first(X1,X2)) = [1 1 3] [1 3 1] [4] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [1] > [1 1 3] [1 3 1] [3] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [1] = a__first(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 2] [1 2 0] [3] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [1] >= [1 0 2] [1 2 0] [3] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [1] = first(X1,X2) a__first(0(),X) = [1 2 0] [3] [0 1 0] X + [0] [0 0 1] [1] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [1 2 4] [0 0 7] [3] [0 1 2] X + [0 1 1] Y + [0 0 2] Z + [2] [0 0 0] [0 0 0] [0 0 0] [1] >= [1 1 3] [0 0 3] [3] [0 1 1] Y + [0 0 2] Z + [2] [0 0 0] [0 0 0] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 1 3] [0] [0 1 2] X + [0] [0 0 0] [0] >= [1 1 3] [0] [0 1 1] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 3] [0] [0 1 2] X + [0] [0 0 0] [0] >= [1 1 2] [0] [0 1 2] X + [0] [0 0 0] [0] = from(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 3] [0 0 5] [0] [0 1 1] X1 + [0 0 2] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 3] [0 0 3] [0] [0 1 1] X1 + [0 0 2] X2 + [0] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 4] [0] [0 1 2] X + [0] [0 0 0] [0] >= [1 2 4] [0] [0 1 2] X + [0] [0 0 0] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 1 3] [2] [0 1 2] X + [2] [0 0 0] [0] >= [1 1 2] [0] [0 1 2] X + [2] [0 0 0] [0] = s(mark(X)) * Step 6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() - 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(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) 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 = 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__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] [0] [2] p(a__first) = [1 0 2] [1 0 2] [0] [2 0 0] x1 + [2 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 2] [0] [2 0 0] x1 + [0] [0 0 1] [0] p(cons) = [1 0 0] [0 2 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(first) = [1 0 2] [1 0 2] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(from) = [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 2] [0] [2 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 1] [3] [0 0 2] x1 + [3] [0 0 1] [1] Following rules are strictly oriented: mark(0()) = [4] [0] [2] > [0] [0] [2] = 0() Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 2] [1 0 2] [0] [2 0 0] X1 + [2 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 2] [4] [2 0 0] X + [0] [0 0 1] [2] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 3] [1 0 2] [0 2 0] [5] [2 0 2] X + [2 0 0] Y + [0 4 0] Z + [6] [0 0 1] [0 0 1] [0 0 0] [1] >= [1 0 2] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 2] [0] [2 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 2] [0] [2 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = from(X) mark(cons(X1,X2)) = [1 0 2] [0 2 0] [0] [2 0 0] X1 + [0 4 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 2] [0 2 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 4] [1 0 4] [0] [2 0 4] X1 + [2 0 4] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 4] [1 0 4] [0] [2 0 4] X1 + [2 0 4] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 0 4] [0] [2 0 4] X + [0] [0 0 1] [0] >= [1 0 4] [0] [2 0 4] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 3] [5] [2 0 2] X + [6] [0 0 1] [1] >= [1 0 3] [3] [0 0 2] X + [3] [0 0 1] [1] = s(mark(X)) * Step 7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() - 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(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) 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 = 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__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] [3] [0] p(a__first) = [1 1 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [2] p(a__from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(first) = [1 1 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [1] p(from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 2 2] [0] p(nil) = [0] [2] [0] p(s) = [1 3 0] [3] [0 1 0] x1 + [0] [0 0 1] [2] Following rules are strictly oriented: mark(nil()) = [2] [2] [4] > [0] [2] [0] = nil() Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 1 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [2] >= [1 1 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [1] = first(X1,X2) a__first(0(),X) = [1 1 0] [3] [0 1 0] X + [3] [0 0 0] [2] >= [0] [2] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 4 0] [1 1 0] [3] [0 1 0] X + [0 1 0] Y + [0] [0 0 0] [0 0 0] [2] >= [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(mark(Y),first(X,Z)) a__from(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] = cons(mark(X),from(s(X))) a__from(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] = from(X) mark(0()) = [3] [3] [6] >= [0] [3] [0] = 0() mark(cons(X1,X2)) = [1 1 0] [0] [0 1 0] X1 + [0] [0 2 0] [0] >= [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 2 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 2 0] [0 2 0] [2] >= [1 2 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [2] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 2 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__from(mark(X)) mark(s(X)) = [1 4 0] [3] [0 1 0] X + [0] [0 2 2] [4] >= [1 4 0] [3] [0 1 0] X + [0] [0 2 2] [2] = s(mark(X)) * Step 8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) - 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(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 = 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__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] [0] [1] p(a__first) = [1 0 0] [1 0 2] [0] [0 1 0] x1 + [0 1 0] x2 + [2] [0 0 1] [0 0 1] [2] p(a__from) = [1 2 2] [3] [0 0 1] x1 + [2] [0 0 1] [2] p(cons) = [1 2 0] [0] [0 0 1] x1 + [2] [0 0 1] [2] p(first) = [1 0 0] [1 0 2] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [2] p(from) = [1 2 2] [3] [0 0 1] x1 + [0] [0 0 1] [2] p(mark) = [1 2 0] [2] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [3] p(s) = [1 2 2] [0] [0 0 1] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 2 2] [6] [0 0 1] X1 + [2] [0 0 1] [2] > [1 2 2] [2] [0 0 1] X1 + [2] [0 0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 0 2] [0] [0 1 0] X1 + [0 1 0] X2 + [2] [0 0 1] [0 0 1] [2] >= [1 0 0] [1 0 2] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [2] = first(X1,X2) a__first(0(),X) = [1 0 2] [2] [0 1 0] X + [2] [0 0 1] [3] >= [0] [0] [3] = nil() a__first(s(X),cons(Y,Z)) = [1 2 2] [1 2 2] [4] [0 0 1] X + [0 0 1] Y + [4] [0 0 1] [0 0 1] [4] >= [1 2 2] [2] [0 0 1] Y + [2] [0 0 1] [2] = cons(mark(Y),first(X,Z)) a__from(X) = [1 2 2] [3] [0 0 1] X + [2] [0 0 1] [2] >= [1 2 2] [2] [0 0 1] X + [2] [0 0 1] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 2 2] [3] [0 0 1] X + [2] [0 0 1] [2] >= [1 2 2] [3] [0 0 1] X + [0] [0 0 1] [2] = from(X) mark(0()) = [4] [1] [1] >= [2] [0] [1] = 0() mark(first(X1,X2)) = [1 2 0] [1 2 2] [4] [0 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [2] >= [1 2 0] [1 2 2] [4] [0 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [2] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 2 4] [5] [0 0 1] X + [2] [0 0 1] [2] >= [1 2 4] [5] [0 0 1] X + [2] [0 0 1] [2] = a__from(mark(X)) mark(nil()) = [2] [3] [3] >= [0] [0] [3] = nil() mark(s(X)) = [1 2 4] [2] [0 0 1] X + [0] [0 0 1] [0] >= [1 2 4] [2] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) * Step 9: 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^3))