WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> cons(x,y)} = mark(cons(x,y)) ->^+ cons(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalMI 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: 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 1.b: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 0 0] [1 2 2] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 2 1] [0] p(a__from) = [1 2 2] [0] [0 1 0] x1 + [0] [0 2 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 2 2] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(from) = [1 2 2] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 2] [0] [0 0 0] x1 + [0] [0 1 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: mark(s(X)) = [1 0 2] [2] [0 0 0] X + [0] [0 1 1] [1] > [1 0 2] [0] [0 0 0] X + [0] [0 1 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 2 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 2 1] [0] >= [1 0 0] [1 2 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 2 2] [0] [0 1 0] X + [0] [0 2 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 0] [1 2 2] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 2 1] [1] >= [1 0 2] [0] [0 0 0] Y + [0] [0 1 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 2 2] [0] [0 1 0] X + [0] [0 2 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 1 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2 2] [0] [0 1 0] X + [0] [0 2 1] [0] >= [1 2 2] [0] [0 1 0] X + [0] [0 0 1] [0] = from(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 2] [0] [0 0 0] X1 + [0] [0 1 1] [0] >= [1 0 2] [0] [0 0 0] X1 + [0] [0 1 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 2] [1 2 4] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 1 1] [0 1 1] [0] >= [1 0 2] [1 2 4] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 1 1] [0 1 1] [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 2 4] [0] [0 0 0] X + [0] [0 1 1] [0] >= [1 2 4] [0] [0 0 0] X + [0] [0 1 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() ** Step 1.b:3: 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() - 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))) 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 1 2] [0] [0 1 0] x1 + [0 0 2] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 2] [3] [0 0 2] x1 + [0] [0 0 1] [2] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 1 2] [0] [0 1 0] x1 + [0 0 2] x2 + [0] [0 0 1] [0 0 1] [0] p(from) = [1 1 2] [0] [0 0 2] x1 + [0] [0 0 1] [2] p(mark) = [1 1 2] [0] [0 0 2] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [2] [0 1 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__from(X) = [1 1 2] [3] [0 0 2] X + [0] [0 0 1] [2] > [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [2] = from(X) mark(from(X)) = [1 1 6] [4] [0 0 2] X + [4] [0 0 1] [2] > [1 1 6] [3] [0 0 2] X + [0] [0 0 1] [2] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 1 2] [0] [0 1 0] X1 + [0 0 2] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 1 2] [0] [0 1 0] X1 + [0 0 2] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 0] [1 1 2] [2] [0 1 0] X + [0 0 2] Y + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [0] [0 0 2] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 1 2] [3] [0 0 2] X + [0] [0 0 1] [2] >= [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 2] [0] [0 0 2] X1 + [0] [0 0 1] [0] >= [1 1 2] [0] [0 0 2] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 1 2] [1 1 6] [0] [0 0 2] X1 + [0 0 2] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 6] [0] [0 0 2] X1 + [0 0 2] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 1 2] [2] [0 0 2] X + [0] [0 0 1] [0] >= [1 1 2] [2] [0 0 2] X + [0] [0 0 1] [0] = s(mark(X)) ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__first(X1,X2) -> first(X1,X2) 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(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 1 0] [1 0 1] [3] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [2] p(a__from) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [2] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [2] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [2] [0 1 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__first(X1,X2) = [1 1 0] [1 0 1] [3] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] > [1 0 0] [1 0 1] [2] [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 1] [3] [0 0 0] X + [1] [0 0 1] [2] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 1 0] [1 0 1] [5] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [2] >= [1 0 1] [0] [0 0 0] Y + [1] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 1] [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 1] [0] [0 0 0] X1 + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [1] [0 0 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 1] [1 0 2] [4] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] >= [1 0 1] [1 0 2] [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 2] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [1] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [1] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 1] [2] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] X + [1] [0 0 1] [0] = s(mark(X)) ** Step 1.b: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 2 0] [1 1 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 2] [1] [0 0 1] x1 + [1] [0 0 1] [3] p(cons) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(first) = [1 2 0] [1 1 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(from) = [1 0 2] [1] [0 0 1] x1 + [1] [0 0 1] [3] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 1] [2] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 0 2] [1] [0 0 1] X1 + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 1] X1 + [0] [0 0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 2 0] [1 1 0] [0] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 2 0] [1 1 0] [0] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [1 0 2] [2] [0 0 0] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [3] >= [1 0 2] [0] [0 0 1] Y + [0] [0 0 1] [1] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [3] >= [1 0 2] [0] [0 0 1] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [3] >= [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [3] = from(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(first(X1,X2)) = [1 2 1] [1 1 1] [0] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 2 1] [1 1 1] [0] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 0 3] [4] [0 0 1] X + [1] [0 0 1] [3] >= [1 0 3] [1] [0 0 1] X + [1] [0 0 1] [3] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 2] [4] [0 0 0] X + [0] [0 0 1] [2] >= [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [2] = s(mark(X)) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(0()) -> 0() 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(cons(X1,X2)) -> cons(mark(X1),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] [0] p(a__first) = [1 0 0] [1 2 2] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 2 1] [2] p(a__from) = [1 3 3] [2] [0 2 0] x1 + [0] [0 2 1] [0] p(cons) = [1 2 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 2 2] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [2] p(from) = [1 3 3] [2] [0 1 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [2] [0 0 0] x1 + [0] [0 1 1] [0] p(nil) = [0] [0] [0] p(s) = [1 1 1] [2] [0 1 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(0()) = [2] [0] [0] > [0] [0] [0] = 0() mark(nil()) = [2] [0] [0] > [0] [0] [0] = nil() Following rules are (at-least) weakly oriented: a__first(X1,X2) = [1 0 0] [1 2 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 2 1] [2] >= [1 0 0] [1 2 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [2] = first(X1,X2) a__first(0(),X) = [1 2 2] [0] [0 1 0] X + [0] [0 2 1] [2] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 1 1] [1 4 3] [2] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 2 1] [2] >= [1 1 2] [2] [0 0 0] Y + [0] [0 1 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 3 3] [2] [0 2 0] X + [0] [0 2 1] [0] >= [1 1 2] [2] [0 0 0] X + [0] [0 1 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 3 3] [2] [0 2 0] X + [0] [0 2 1] [0] >= [1 3 3] [2] [0 1 0] X + [0] [0 0 1] [0] = from(X) mark(cons(X1,X2)) = [1 2 2] [2] [0 0 0] X1 + [0] [0 1 1] [0] >= [1 1 2] [2] [0 0 0] X1 + [0] [0 1 1] [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1 0 1] [1 2 3] [4] [0 0 0] X1 + [0 0 0] X2 + [0] [0 1 1] [0 1 1] [2] >= [1 0 1] [1 2 3] [4] [0 0 0] X1 + [0 0 0] X2 + [0] [0 1 1] [0 1 1] [2] = a__first(mark(X1),mark(X2)) mark(from(X)) = [1 3 4] [4] [0 0 0] X + [0] [0 1 1] [0] >= [1 3 4] [4] [0 0 0] X + [0] [0 1 1] [0] = a__from(mark(X)) mark(s(X)) = [1 1 2] [4] [0 0 0] X + [0] [0 1 1] [0] >= [1 1 2] [4] [0 0 0] X + [0] [0 1 1] [0] = s(mark(X)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(first(X1,X2)) -> a__first(mark(X1),mark(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(cons(X1,X2)) -> cons(mark(X1),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) = [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] [2] p(a__from) = [1 0 2] [3] [0 2 0] x1 + [0] [0 0 1] [0] p(cons) = [1 0 0] [2] [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] [2] p(from) = [1 0 2] [3] [0 0 0] x1 + [0] [0 0 1] [0] 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 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: mark(first(X1,X2)) = [1 0 1] [1 0 2] [2] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] > [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__first(mark(X1),mark(X2)) 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] [2] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [2] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 0] [1 0 1] [2] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [3] >= [1 0 1] [2] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__from(X) = [1 0 2] [3] [0 2 0] X + [0] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 2] [3] [0 2 0] X + [0] [0 0 1] [0] >= [1 0 2] [3] [0 0 0] X + [0] [0 0 1] [0] = from(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [2] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 3] [3] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 3] [3] [0 0 0] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = s(mark(X)) ** Step 1.b: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(Omega(n^1),O(n^3))