WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> app(x,y)} = mark(app(x,y)) ->^+ a__app(mark(x),mark(y)) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [2] p(a__prefix) = [1] x1 + [2] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [2] p(mark) = [1] x1 + [0] p(nil) = [0] p(prefix) = [1] x1 + [2] p(s) = [1] x1 + [5] p(zWadr) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: a__from(X) = [1] X + [2] > [1] X + [0] = cons(mark(X),from(s(X))) a__prefix(L) = [1] L + [2] > [0] = cons(nil(),zWadr(L,prefix(L))) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [0] >= [1] X + [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [0] >= [1] YS + [0] = mark(YS) a__from(X) = [1] X + [2] >= [1] X + [2] = from(X) a__prefix(X) = [1] X + [2] >= [1] X + [2] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [0] >= [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [0] >= [0] = nil() mark(app(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [2] >= [1] X + [2] = a__from(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(prefix(X)) = [1] X + [2] >= [1] X + [2] = a__prefix(mark(X)) mark(s(X)) = [1] X + [5] >= [1] X + [5] = s(mark(X)) mark(zWadr(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> from(X) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [2] p(a__prefix) = [1] x1 + [10] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [2] p(from) = [1] x1 + [2] p(mark) = [1] x1 + [0] p(nil) = [8] p(prefix) = [1] x1 + [10] p(s) = [1] x1 + [0] p(zWadr) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: a__app(nil(),YS) = [1] YS + [8] > [1] YS + [0] = mark(YS) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [2] >= [1] X + [2] = cons(mark(X),app(XS,YS)) a__from(X) = [1] X + [2] >= [1] X + [2] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [2] >= [1] X + [2] = from(X) a__prefix(L) = [1] L + [10] >= [10] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [10] >= [1] X + [10] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [8] >= [8] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [4] >= [1] X + [1] Y + [4] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [8] >= [8] = nil() mark(app(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [2] >= [1] X1 + [2] = cons(mark(X1),X2) mark(from(X)) = [1] X + [2] >= [1] X + [2] = a__from(mark(X)) mark(nil()) = [8] >= [8] = nil() mark(prefix(X)) = [1] X + [10] >= [1] X + [10] = a__prefix(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> from(X) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = x1 + x2 p(a__from) = 2 + x1 p(a__prefix) = x1 p(a__zWadr) = 2 + x1 + x2 p(app) = x1 + x2 p(cons) = x1 p(from) = 2 + x1 p(mark) = x1 p(nil) = 0 p(prefix) = x1 p(s) = 1 + x1 p(zWadr) = 2 + x1 + x2 Following rules are strictly oriented: a__zWadr(XS,nil()) = 2 + XS > 0 = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = 2 + X + Y > X + Y = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = 2 + YS > 0 = nil() Following rules are (at-least) weakly oriented: a__app(X1,X2) = X1 + X2 >= X1 + X2 = app(X1,X2) a__app(cons(X,XS),YS) = X + YS >= X = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = YS >= YS = mark(YS) a__from(X) = 2 + X >= X = cons(mark(X),from(s(X))) a__from(X) = 2 + X >= 2 + X = from(X) a__prefix(L) = L >= 0 = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = X >= X = prefix(X) a__zWadr(X1,X2) = 2 + X1 + X2 >= 2 + X1 + X2 = zWadr(X1,X2) mark(app(X1,X2)) = X1 + X2 >= X1 + X2 = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(from(X)) = 2 + X >= 2 + X = a__from(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(prefix(X)) = X >= X = a__prefix(mark(X)) mark(s(X)) = 1 + X >= 1 + X = s(mark(X)) mark(zWadr(X1,X2)) = 2 + X1 + X2 >= 2 + X1 + X2 = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> from(X) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [1] p(a__from) = [1] x1 + [0] p(a__prefix) = [1] x1 + [0] p(a__zWadr) = [1] x1 + [1] x2 + [2] p(app) = [1] x1 + [1] x2 + [1] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(prefix) = [1] x1 + [0] p(s) = [1] x1 + [0] p(zWadr) = [1] x1 + [1] x2 + [2] Following rules are strictly oriented: a__app(cons(X,XS),YS) = [1] X + [1] YS + [1] > [1] X + [0] = cons(mark(X),app(XS,YS)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = app(X1,X2) a__app(nil(),YS) = [1] YS + [1] >= [1] YS + [0] = mark(YS) a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__prefix(L) = [1] L + [0] >= [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [0] >= [1] X + [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [2] >= [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [2] >= [1] X + [1] Y + [1] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [2] >= [0] = nil() mark(app(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(prefix(X)) = [1] X + [0] >= [1] X + [0] = a__prefix(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__prefix) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__zWadr) = [1 1 1] [1 1 1] [1] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(zWadr) = [1 1 1] [1 1 1] [1] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] Following rules are strictly oriented: mark(s(X)) = [1 1 0] [1] [0 0 1] X + [1] [0 0 1] [1] > [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 1 0] [1 1 0] [0] [0 0 1] X + [0 0 1] YS + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 1 0] [0] [0 0 1] YS + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = from(X) a__prefix(L) = [1 1 0] [0] [0 0 1] L + [0] [0 0 1] [0] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [1] [0 0 1] XS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 1 1] [1 1 1] [1] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [1] [0 0 1] YS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() mark(app(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(prefix(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__prefix(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 0 1] [1 0 1] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [1] p(a__from) = [1 0 1] [1] [0 1 1] x1 + [0] [0 0 1] [0] p(a__prefix) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__zWadr) = [1 1 1] [1 1 1] [1] [0 1 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(app) = [1 0 1] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [1] [0 1 1] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(s) = [1 1 1] [1] [0 1 0] x1 + [1] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 1 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] Following rules are strictly oriented: a__app(X1,X2) = [1 0 1] [1 0 1] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] > [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] = app(X1,X2) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [1] [0 1 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] > [1 1 1] [1 1 1] [0] [0 1 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = zWadr(X1,X2) Following rules are (at-least) weakly oriented: a__app(cons(X,XS),YS) = [1 0 1] [1 0 1] [1] [0 0 1] X + [0 1 0] YS + [1] [0 0 1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 0 1] [1] [0 1 0] YS + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 0 1] [1] [0 1 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [1] [0 1 1] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 1 1] X + [0] [0 0 1] [0] = from(X) a__prefix(L) = [1 0 0] [0] [0 0 1] L + [0] [0 0 1] [0] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] = prefix(X) a__zWadr(XS,nil()) = [1 1 1] [1] [0 1 1] XS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 0 2] [1 0 2] [1] [0 0 2] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 2] [1] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() mark(app(X1,X2)) = [1 0 2] [1 0 2] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 2] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [1] [0 1 1] X + [0] [0 0 1] [0] >= [1 0 2] [1] [0 1 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(prefix(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__prefix(mark(X)) mark(s(X)) = [1 1 2] [1] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 2] [1] [0 1 0] X + [1] [0 0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 1 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 2] [1 1 2] [1] [0 1 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__prefix(X) -> prefix(X) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__prefix) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [1] [1] p(prefix) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(s) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] Following rules are strictly oriented: mark(nil()) = [1] [1] [1] > [0] [1] [1] = nil() Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 1 0] [1 1 0] [0] [0 0 1] X + [0 0 1] YS + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 1 0] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = from(X) a__prefix(L) = [1 0 0] [0] [0 1 0] L + [1] [0 0 1] [1] >= [0] [1] [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [2] [0 0 1] XS + [1] [0 0 1] [1] >= [0] [1] [1] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 1 1] [1 1 1] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [2] [0 0 1] YS + [1] [0 0 1] [1] >= [0] [1] [1] = nil() mark(app(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(prefix(X)) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = a__prefix(mark(X)) mark(s(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__prefix(X) -> prefix(X) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(prefix(X)) -> a__prefix(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 0 1] [1 0 1] [1] [0 0 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(a__prefix) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__zWadr) = [1 1 1] [1 1 1] [1] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(app) = [1 0 1] [1 0 1] [1] [0 0 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [1] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] Following rules are strictly oriented: mark(from(X)) = [1 0 2] [1] [0 1 1] X + [0] [0 0 1] [1] > [1 0 2] [0] [0 1 1] X + [0] [0 0 1] [1] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 0 1] [1 0 1] [1] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 1] [1] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 0 1] [1 0 1] [1] [0 0 1] X + [0 1 0] YS + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 0 1] [1] [0 1 0] YS + [0] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 0 1] [0] [0 1 1] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 1 1] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 1 1] X + [0] [0 0 1] [1] = from(X) a__prefix(L) = [1 0 0] [0] [0 0 0] L + [0] [0 0 1] [0] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [1] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [1] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [1] [0 1 1] XS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 0 2] [1 0 2] [1] [0 0 2] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [1] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [1] [0 1 0] YS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() mark(app(X1,X2)) = [1 0 2] [1 0 2] [1] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [1] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(prefix(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__prefix(mark(X)) mark(s(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] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 2] [1] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__prefix(X) -> prefix(X) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(prefix(X)) -> a__prefix(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 0 1] [1 0 1] [0] [0 0 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__prefix) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(app) = [1 0 1] [1 0 1] [0] [0 0 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(from) = [1 1 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] Following rules are strictly oriented: a__prefix(X) = [1 1 0] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] = prefix(X) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 0 1] [1 0 1] [0] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 1] [0] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 0 1] [1 0 1] [0] [0 0 1] X + [0 1 0] YS + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = from(X) a__prefix(L) = [1 1 0] [1] [0 0 0] L + [0] [0 0 1] [1] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [0] [0 1 0] XS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 0 2] [1 0 2] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [0] [0 1 0] YS + [0] [0 0 1] [0] >= [0] [0] [0] = nil() mark(app(X1,X2)) = [1 0 2] [1 0 2] [0] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [0] [0 0 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 2] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 2] [0] [0 0 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(prefix(X)) = [1 1 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] X + [0] [0 0 1] [1] = a__prefix(mark(X)) mark(s(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(prefix(X)) -> a__prefix(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 1 0] [1 1 0] [1] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(a__prefix) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [1] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(app) = [1 1 0] [1 1 0] [1] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(s) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] Following rules are strictly oriented: mark(app(X1,X2)) = [1 1 1] [1 1 1] [2] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] > [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__app(mark(X1),mark(X2)) mark(prefix(X)) = [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] > [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] = a__prefix(mark(X)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 1 0] [1 1 0] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 1 0] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = app(X1,X2) a__app(cons(X,XS),YS) = [1 1 0] [1 1 0] [1] [0 0 1] X + [0 0 1] YS + [1] [0 0 1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 1 0] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = from(X) a__prefix(L) = [1 1 0] [0] [0 0 1] L + [1] [0 0 1] [1] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [1] [0 0 1] XS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 1 1] [1 1 1] [1] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [1] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:11: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 0 1] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__from) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__prefix) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(app) = [1 0 1] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [1] [0] p(prefix) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] Following rules are strictly oriented: mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] > [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__zWadr(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = app(X1,X2) a__app(cons(X,XS),YS) = [1 0 1] [1 0 1] [0] [0 0 1] X + [0 1 0] YS + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 0 1] [0] [0 1 0] YS + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = from(X) a__prefix(L) = [1 0 0] [1] [0 0 0] L + [1] [0 0 1] [1] >= [0] [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0 0] [1] [0 0 0] X + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [1] [0 0 1] XS + [1] [0 0 1] [1] >= [0] [1] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 0 2] [1 0 2] [0] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 2] [0] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [1] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [0] [1] [0] = nil() mark(app(X1,X2)) = [1 0 2] [1 0 2] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [1] [0] >= [0] [1] [0] = nil() mark(prefix(X)) = [1 0 1] [1] [0 0 0] X + [1] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [1] [0 0 1] [1] = a__prefix(mark(X)) mark(s(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 1] [0] = s(mark(X)) ** Step 1.b:12: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [1] [0 0 1] x1 + [1] [0 0 1] [1] p(a__prefix) = [1 1 0] [1] [0 0 1] x1 + [0] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(app) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [1] [0] [1] p(prefix) = [1 1 0] [1] [0 0 1] x1 + [0] [0 0 1] [1] p(s) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 1] [0] p(zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] Following rules are strictly oriented: a__from(X) = [1 1 0] [1] [0 0 1] X + [1] [0 0 1] [1] > [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = from(X) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 1 0] [1 1 0] [0] [0 0 1] X + [0 0 1] YS + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 1 0] [1] [0 0 1] YS + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 1 0] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__prefix(L) = [1 1 0] [1] [0 0 1] L + [0] [0 0 1] [1] >= [1] [0] [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 1 0] [1] [0 0 1] X + [0] [0 0 1] [1] >= [1 1 0] [1] [0 0 1] X + [0] [0 0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [2] [0 0 1] XS + [1] [0 0 1] [1] >= [1] [0] [1] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 1 1] [1 1 1] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X + [0 0 1] Y + [0] [0 0 1] [0 0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [2] [0 0 1] YS + [1] [0 0 1] [1] >= [1] [0] [1] = nil() mark(app(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] = a__from(mark(X)) mark(nil()) = [1] [1] [1] >= [1] [0] [1] = nil() mark(prefix(X)) = [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 1] X + [0] [0 0 1] [1] = a__prefix(mark(X)) mark(s(X)) = [1 1 0] [1] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [1] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:13: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + 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__app) = {1,2}, uargs(a__from) = {1}, uargs(a__prefix) = {1}, uargs(a__zWadr) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__app,a__from,a__prefix,a__zWadr,mark} TcT has computed the following interpretation: p(a__app) = [1 0 1] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__from) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(a__prefix) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [1] p(app) = [1 0 1] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(from) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(prefix) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(zWadr) = [1 1 1] [1 1 1] [0] [0 0 1] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [1] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 0 1] [1] [0 0 1] X1 + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = app(X1,X2) a__app(cons(X,XS),YS) = [1 0 1] [1 0 1] [1] [0 0 1] X + [0 1 0] YS + [0] [0 0 1] [0 0 1] [2] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [1] >= [1 0 1] [0] [0 1 0] YS + [0] [0 0 1] [0] = mark(YS) a__from(X) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] = from(X) a__prefix(L) = [1 0 0] [1] [0 0 0] L + [0] [0 0 1] [1] >= [0] [0] [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 1 1] [0] [0 0 1] XS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 0 2] [1 0 2] [2] [0 0 1] X + [0 0 1] Y + [2] [0 0 1] [0 0 1] [3] >= [1 0 2] [1 0 2] [1] [0 0 1] X + [0 0 1] Y + [2] [0 0 1] [0 0 1] [3] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 1 1] [0] [0 1 0] YS + [1] [0 0 1] [1] >= [0] [0] [0] = nil() mark(app(X1,X2)) = [1 0 2] [1 0 2] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 2] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__app(mark(X1),mark(X2)) mark(from(X)) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 2] [0] [0 0 1] X + [1] [0 0 1] [1] = a__from(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(prefix(X)) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = a__prefix(mark(X)) 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)) mark(zWadr(X1,X2)) = [1 1 2] [1 1 2] [1] [0 0 1] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 2] [1 1 2] [0] [0 0 1] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [1] = a__zWadr(mark(X1),mark(X2)) ** Step 1.b:14: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__app(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) -> nil() mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(mark(X)) mark(s(X)) -> s(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) - Signature: {a__app/2,a__from/1,a__prefix/1,a__zWadr/2,mark/1} / {app/2,cons/2,from/1,nil/0,prefix/1,s/1,zWadr/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__app,a__from,a__prefix,a__zWadr ,mark} and constructors {app,cons,from,nil,prefix,s,zWadr} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))