WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [2] p(a__from) = [1] x1 + [0] p(a__prefix) = [1] x1 + [0] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [0] p(cons) = [1] x1 + [1] p(from) = [0] p(mark) = [5] p(nil) = [0] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [2] > [0] = app(X1,X2) mark(nil()) = [5] > [0] = nil() Following rules are (at-least) weakly oriented: a__app(cons(X,XS),YS) = [1] X + [1] YS + [3] >= [6] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [2] >= [5] = mark(YS) a__from(X) = [1] X + [0] >= [6] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [0] = from(X) a__prefix(L) = [1] L + [0] >= [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [0] >= [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [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 + [2] >= [14] = 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)) = [5] >= [12] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [5] >= [6] = cons(mark(X1),X2) mark(from(X)) = [5] >= [5] = a__from(mark(X)) mark(prefix(X)) = [5] >= [5] = a__prefix(mark(X)) mark(s(X)) = [5] >= [5] = s(mark(X)) mark(zWadr(X1,X2)) = [5] >= [10] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict 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__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(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(X1,X2) -> app(X1,X2) mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [4] p(a__from) = [1] x1 + [0] p(a__prefix) = [1] x1 + [0] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [3] p(nil) = [3] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__app(cons(X,XS),YS) = [1] X + [1] YS + [4] > [3] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [7] > [3] = mark(YS) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [4] >= [0] = app(X1,X2) a__from(X) = [1] X + [0] >= [3] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [0] = from(X) a__prefix(L) = [1] L + [0] >= [3] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [0] >= [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [3] >= [3] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0] >= [10] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [3] >= [3] = nil() mark(app(X1,X2)) = [3] >= [10] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [3] >= [3] = cons(mark(X1),X2) mark(from(X)) = [3] >= [3] = a__from(mark(X)) mark(nil()) = [3] >= [3] = nil() mark(prefix(X)) = [3] >= [3] = a__prefix(mark(X)) mark(s(X)) = [3] >= [3] = s(mark(X)) mark(zWadr(X1,X2)) = [3] >= [6] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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(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(X1,X2) -> app(X1,X2) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__app(nil(),YS) -> mark(YS) mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [4] p(a__from) = [1] x1 + [1] p(a__prefix) = [1] x1 + [0] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [3] p(nil) = [3] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__from(X) = [1] X + [1] > [0] = from(X) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [4] >= [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [4] >= [3] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [7] >= [3] = mark(YS) a__from(X) = [1] X + [1] >= [3] = cons(mark(X),from(s(X))) a__prefix(L) = [1] L + [0] >= [3] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [0] >= [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [3] >= [3] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0] >= [10] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [3] >= [3] = nil() mark(app(X1,X2)) = [3] >= [10] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [3] >= [3] = cons(mark(X1),X2) mark(from(X)) = [3] >= [4] = a__from(mark(X)) mark(nil()) = [3] >= [3] = nil() mark(prefix(X)) = [3] >= [3] = a__prefix(mark(X)) mark(s(X)) = [3] >= [3] = s(mark(X)) mark(zWadr(X1,X2)) = [3] >= [6] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) 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(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) mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [2] p(a__from) = [1] x1 + [1] p(a__prefix) = [1] x1 + [7] p(a__zWadr) = [1] x1 + [1] x2 + [0] p(app) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [1] p(nil) = [1] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__prefix(L) = [1] L + [7] > [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [7] > [0] = prefix(X) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [2] >= [1] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [3] >= [1] = mark(YS) a__from(X) = [1] X + [1] >= [1] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [1] >= [0] = from(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [1] >= [1] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [0] >= [4] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [1] >= [1] = nil() mark(app(X1,X2)) = [1] >= [4] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] >= [1] = cons(mark(X1),X2) mark(from(X)) = [1] >= [2] = a__from(mark(X)) mark(nil()) = [1] >= [1] = nil() mark(prefix(X)) = [1] >= [8] = a__prefix(mark(X)) mark(s(X)) = [1] >= [1] = s(mark(X)) mark(zWadr(X1,X2)) = [1] >= [2] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(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(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(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(L) -> cons(nil(),zWadr(L,prefix(L))) a__prefix(X) -> prefix(X) mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [2] p(a__from) = [1] x1 + [1] p(a__prefix) = [1] x1 + [1] p(a__zWadr) = [1] x1 + [1] x2 + [4] p(app) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [1] p(nil) = [1] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__zWadr(X1,X2) = [1] X1 + [1] X2 + [4] > [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [5] > [1] = nil() a__zWadr(nil(),YS) = [1] YS + [5] > [1] = nil() Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [2] >= [1] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [3] >= [1] = mark(YS) a__from(X) = [1] X + [1] >= [1] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [1] >= [0] = from(X) a__prefix(L) = [1] L + [1] >= [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [1] >= [0] = prefix(X) a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [4] >= [4] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) mark(app(X1,X2)) = [1] >= [4] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] >= [1] = cons(mark(X1),X2) mark(from(X)) = [1] >= [2] = a__from(mark(X)) mark(nil()) = [1] >= [1] = nil() mark(prefix(X)) = [1] >= [2] = a__prefix(mark(X)) mark(s(X)) = [1] >= [1] = s(mark(X)) mark(zWadr(X1,X2)) = [1] >= [6] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) 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(s(X)) -> s(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) -> 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(nil(),YS) -> nil() mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [4] p(a__from) = [1] x1 + [6] p(a__prefix) = [1] x1 + [4] p(a__zWadr) = [1] x1 + [1] x2 + [5] p(app) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [3] p(nil) = [3] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [0] Following rules are strictly oriented: a__from(X) = [1] X + [6] > [3] = cons(mark(X),from(s(X))) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [4] >= [3] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [7] >= [3] = mark(YS) a__from(X) = [1] X + [6] >= [0] = from(X) a__prefix(L) = [1] L + [4] >= [3] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [4] >= [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [5] >= [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [8] >= [3] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [5] >= [10] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1] YS + [8] >= [3] = nil() mark(app(X1,X2)) = [3] >= [10] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [3] >= [3] = cons(mark(X1),X2) mark(from(X)) = [3] >= [9] = a__from(mark(X)) mark(nil()) = [3] >= [3] = nil() mark(prefix(X)) = [3] >= [7] = a__prefix(mark(X)) mark(s(X)) = [3] >= [3] = s(mark(X)) mark(zWadr(X1,X2)) = [3] >= [11] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) 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(s(X)) -> s(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__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(nil(),YS) -> nil() mark(nil()) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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: all TcT has computed the following interpretation: p(a__app) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [0] p(a__prefix) = [1] x1 + [0] p(a__zWadr) = [1] x1 + [1] x2 + [1] p(app) = [0] p(cons) = [1] x1 + [0] p(from) = [0] p(mark) = [0] p(nil) = [0] p(prefix) = [0] p(s) = [1] x1 + [0] p(zWadr) = [1] x2 + [1] Following rules are strictly oriented: a__zWadr(cons(X,XS),cons(Y,YS)) = [1] X + [1] Y + [1] > [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1] X + [1] YS + [0] >= [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1] YS + [0] >= [0] = mark(YS) a__from(X) = [1] X + [0] >= [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [0] = from(X) a__prefix(L) = [1] L + [0] >= [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1] X + [0] >= [0] = prefix(X) a__zWadr(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X2 + [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1] XS + [1] >= [0] = nil() a__zWadr(nil(),YS) = [1] YS + [1] >= [0] = nil() mark(app(X1,X2)) = [0] >= [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(from(X)) = [0] >= [0] = a__from(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(prefix(X)) = [0] >= [0] = a__prefix(mark(X)) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(zWadr(X1,X2)) = [0] >= [1] = a__zWadr(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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(s(X)) -> s(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__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(nil()) -> 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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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 2] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 4] x1 + [6] [0 1] [4] p(a__prefix) = [1 1] x1 + [6] [0 1] [2] p(a__zWadr) = [1 6] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] p(app) = [1 2] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 2] x1 + [0] [0 1] [0] p(from) = [1 4] x1 + [2] [0 1] [4] p(mark) = [1 2] x1 + [0] [0 1] [0] p(nil) = [2] [2] p(prefix) = [1 1] x1 + [4] [0 1] [2] p(s) = [1 2] x1 + [0] [0 1] [0] p(zWadr) = [1 6] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] Following rules are strictly oriented: mark(from(X)) = [1 6] X + [10] [0 1] [4] > [1 6] X + [6] [0 1] [4] = a__from(mark(X)) mark(prefix(X)) = [1 3] X + [8] [0 1] [2] > [1 3] X + [6] [0 1] [2] = a__prefix(mark(X)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 2] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 2] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 4] X + [1 2] YS + [0] [0 1] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 2] YS + [6] [0 1] [2] >= [1 2] YS + [0] [0 1] [0] = mark(YS) a__from(X) = [1 4] X + [6] [0 1] [4] >= [1 4] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 4] X + [6] [0 1] [4] >= [1 4] X + [2] [0 1] [4] = from(X) a__prefix(L) = [1 1] L + [6] [0 1] [2] >= [6] [2] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 1] X + [6] [0 1] [2] >= [1 1] X + [4] [0 1] [2] = prefix(X) a__zWadr(X1,X2) = [1 6] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 6] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 6] XS + [10] [0 1] [2] >= [2] [2] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 8] X + [1 6] Y + [0] [0 1] [0 1] [0] >= [1 8] X + [1 6] Y + [0] [0 1] [0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 4] YS + [14] [0 1] [2] >= [2] [2] = nil() mark(app(X1,X2)) = [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 4] X1 + [0] [0 1] [0] >= [1 4] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(nil()) = [6] [2] >= [2] [2] = nil() mark(s(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = s(mark(X)) mark(zWadr(X1,X2)) = [1 8] X1 + [1 6] X2 + [0] [0 1] [0 1] [0] >= [1 8] X1 + [1 6] X2 + [0] [0 1] [0 1] [0] = a__zWadr(mark(X1),mark(X2)) * Step 9: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(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__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(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(prefix(X)) -> a__prefix(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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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 2] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 2] x1 + [0] [0 1] [0] p(a__prefix) = [1 0] x1 + [4] [0 1] [1] p(a__zWadr) = [1 4] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] p(app) = [1 2] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 0] x1 + [0] [0 1] [0] p(from) = [1 2] x1 + [0] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(prefix) = [1 0] x1 + [4] [0 1] [1] p(s) = [1 0] x1 + [0] [0 1] [1] p(zWadr) = [1 4] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] Following rules are strictly oriented: mark(s(X)) = [1 2] X + [2] [0 1] [1] > [1 2] X + [0] [0 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 2] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 2] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 2] X + [1 2] YS + [0] [0 1] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 2] YS + [0] [0 1] [0] >= [1 2] YS + [0] [0 1] [0] = mark(YS) a__from(X) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = from(X) a__prefix(L) = [1 0] L + [4] [0 1] [1] >= [0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0] X + [4] [0 1] [1] >= [1 0] X + [4] [0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 4] XS + [0] [0 1] [0] >= [0] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [1 4] Y + [0] [0 1] [0 1] [0] >= [1 4] X + [1 4] Y + [0] [0 1] [0 1] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 4] YS + [0] [0 1] [0] >= [0] [0] = nil() mark(app(X1,X2)) = [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 2] X1 + [0] [0 1] [0] >= [1 2] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = a__from(mark(X)) mark(nil()) = [0] [0] >= [0] [0] = nil() mark(prefix(X)) = [1 2] X + [6] [0 1] [1] >= [1 2] X + [4] [0 1] [1] = a__prefix(mark(X)) mark(zWadr(X1,X2)) = [1 6] X1 + [1 6] X2 + [0] [0 1] [0 1] [0] >= [1 6] X1 + [1 6] X2 + [0] [0 1] [0 1] [0] = a__zWadr(mark(X1),mark(X2)) * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) 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__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(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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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 2] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 2] x1 + [0] [0 1] [0] p(a__prefix) = [1 4] x1 + [3] [0 1] [0] p(a__zWadr) = [1 2] x1 + [1 6] x2 + [4] [0 1] [0 1] [1] p(app) = [1 2] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 0] x1 + [0 1] x2 + [0] [0 1] [0 0] [0] p(from) = [1 2] x1 + [0] [0 1] [0] p(mark) = [1 1] x1 + [0] [0 1] [0] p(nil) = [1] [0] p(prefix) = [1 4] x1 + [3] [0 1] [0] p(s) = [1 0] x1 + [1] [0 1] [0] p(zWadr) = [1 2] x1 + [1 6] x2 + [4] [0 1] [0 1] [1] Following rules are strictly oriented: mark(zWadr(X1,X2)) = [1 3] X1 + [1 7] X2 + [5] [0 1] [0 1] [1] > [1 3] X1 + [1 7] X2 + [4] [0 1] [0 1] [1] = a__zWadr(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 2] X1 + [1 1] X2 + [0] [0 1] [0 1] [0] >= [1 2] X1 + [1 1] X2 + [0] [0 1] [0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 2] X + [0 1] XS + [1 1] YS + [0] [0 1] [0 0] [0 1] [0] >= [1 1] X + [0 1] XS + [0 1] YS + [0] [0 1] [0 0] [0 0] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 1] YS + [1] [0 1] [0] >= [1 1] YS + [0] [0 1] [0] = mark(YS) a__from(X) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = from(X) a__prefix(L) = [1 4] L + [3] [0 1] [0] >= [0 2] L + [2] [0 0] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 4] X + [3] [0 1] [0] >= [1 4] X + [3] [0 1] [0] = prefix(X) a__zWadr(X1,X2) = [1 2] X1 + [1 6] X2 + [4] [0 1] [0 1] [1] >= [1 2] X1 + [1 6] X2 + [4] [0 1] [0 1] [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 2] XS + [5] [0 1] [1] >= [1] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 2] X + [0 1] XS + [1 6] Y + [0 1] YS + [4] [0 1] [0 0] [0 1] [0 0] [1] >= [1 2] X + [0 1] XS + [1 3] Y + [0 1] YS + [1] [0 1] [0 0] [0 1] [0 0] [0] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 6] YS + [5] [0 1] [1] >= [1] [0] = nil() mark(app(X1,X2)) = [1 3] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 3] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = a__app(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1] X1 + [0 1] X2 + [0] [0 1] [0 0] [0] >= [1 1] X1 + [0 1] X2 + [0] [0 1] [0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 3] X + [0] [0 1] [0] >= [1 3] X + [0] [0 1] [0] = a__from(mark(X)) mark(nil()) = [1] [0] >= [1] [0] = nil() mark(prefix(X)) = [1 5] X + [3] [0 1] [0] >= [1 5] X + [3] [0 1] [0] = a__prefix(mark(X)) mark(s(X)) = [1 1] X + [1] [0 1] [0] >= [1 1] X + [1] [0 1] [0] = s(mark(X)) * Step 11: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) 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(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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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 3] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(a__from) = [1 2] x1 + [3] [0 1] [1] p(a__prefix) = [1 1] x1 + [3] [0 1] [2] p(a__zWadr) = [1 4] x1 + [1 5] x2 + [2] [0 1] [0 1] [1] p(app) = [1 3] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 0] x1 + [1] [0 1] [1] p(from) = [1 2] x1 + [1] [0 1] [1] p(mark) = [1 2] x1 + [0] [0 1] [0] p(nil) = [2] [0] p(prefix) = [1 1] x1 + [0] [0 1] [2] p(s) = [1 4] x1 + [0] [0 1] [6] p(zWadr) = [1 4] x1 + [1 5] x2 + [0] [0 1] [0 1] [1] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 2] X1 + [3] [0 1] [1] > [1 2] X1 + [1] [0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 3] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 3] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = app(X1,X2) a__app(cons(X,XS),YS) = [1 3] X + [1 2] YS + [4] [0 1] [0 1] [1] >= [1 2] X + [1] [0 1] [1] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 2] YS + [2] [0 1] [0] >= [1 2] YS + [0] [0 1] [0] = mark(YS) a__from(X) = [1 2] X + [3] [0 1] [1] >= [1 2] X + [1] [0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 2] X + [3] [0 1] [1] >= [1 2] X + [1] [0 1] [1] = from(X) a__prefix(L) = [1 1] L + [3] [0 1] [2] >= [3] [1] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 1] X + [3] [0 1] [2] >= [1 1] X + [0] [0 1] [2] = prefix(X) a__zWadr(X1,X2) = [1 4] X1 + [1 5] X2 + [2] [0 1] [0 1] [1] >= [1 4] X1 + [1 5] X2 + [0] [0 1] [0 1] [1] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 4] XS + [4] [0 1] [1] >= [2] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [1 5] Y + [13] [0 1] [0 1] [3] >= [1 4] X + [1 5] Y + [4] [0 1] [0 1] [2] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 5] YS + [4] [0 1] [1] >= [2] [0] = nil() mark(app(X1,X2)) = [1 5] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 5] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = a__app(mark(X1),mark(X2)) mark(from(X)) = [1 4] X + [3] [0 1] [1] >= [1 4] X + [3] [0 1] [1] = a__from(mark(X)) mark(nil()) = [2] [0] >= [2] [0] = nil() mark(prefix(X)) = [1 3] X + [4] [0 1] [2] >= [1 3] X + [3] [0 1] [2] = a__prefix(mark(X)) mark(s(X)) = [1 6] X + [12] [0 1] [6] >= [1 6] X + [0] [0 1] [6] = s(mark(X)) mark(zWadr(X1,X2)) = [1 6] X1 + [1 7] X2 + [2] [0 1] [0 1] [1] >= [1 6] X1 + [1 7] X2 + [2] [0 1] [0 1] [1] = a__zWadr(mark(X1),mark(X2)) * Step 12: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(app(X1,X2)) -> a__app(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__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(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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__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 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [1] p(a__from) = [1 2] x1 + [5] [0 1] [0] p(a__prefix) = [1 0] x1 + [2] [0 1] [1] p(a__zWadr) = [1 4] x1 + [1 4] x2 + [4] [0 1] [0 1] [2] p(app) = [1 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [1] p(cons) = [1 0] x1 + [0] [0 1] [0] p(from) = [1 2] x1 + [5] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [0] p(nil) = [2] [0] p(prefix) = [1 0] x1 + [0] [0 1] [1] p(s) = [1 0] x1 + [4] [0 1] [1] p(zWadr) = [1 4] x1 + [1 4] x2 + [0] [0 1] [0 1] [2] Following rules are strictly oriented: mark(app(X1,X2)) = [1 4] X1 + [1 4] X2 + [3] [0 1] [0 1] [1] > [1 4] X1 + [1 4] X2 + [1] [0 1] [0 1] [1] = a__app(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__app(X1,X2) = [1 2] X1 + [1 2] X2 + [1] [0 1] [0 1] [1] >= [1 2] X1 + [1 2] X2 + [1] [0 1] [0 1] [1] = app(X1,X2) a__app(cons(X,XS),YS) = [1 2] X + [1 2] YS + [1] [0 1] [0 1] [1] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),app(XS,YS)) a__app(nil(),YS) = [1 2] YS + [3] [0 1] [1] >= [1 2] YS + [0] [0 1] [0] = mark(YS) a__from(X) = [1 2] X + [5] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 2] X + [5] [0 1] [0] >= [1 2] X + [5] [0 1] [0] = from(X) a__prefix(L) = [1 0] L + [2] [0 1] [1] >= [2] [0] = cons(nil(),zWadr(L,prefix(L))) a__prefix(X) = [1 0] X + [2] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = prefix(X) a__zWadr(X1,X2) = [1 4] X1 + [1 4] X2 + [4] [0 1] [0 1] [2] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] = zWadr(X1,X2) a__zWadr(XS,nil()) = [1 4] XS + [6] [0 1] [2] >= [2] [0] = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = [1 4] X + [1 4] Y + [4] [0 1] [0 1] [2] >= [1 4] X + [1 4] Y + [1] [0 1] [0 1] [1] = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__zWadr(nil(),YS) = [1 4] YS + [6] [0 1] [2] >= [2] [0] = nil() mark(cons(X1,X2)) = [1 2] X1 + [0] [0 1] [0] >= [1 2] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 4] X + [5] [0 1] [0] >= [1 4] X + [5] [0 1] [0] = a__from(mark(X)) mark(nil()) = [2] [0] >= [2] [0] = nil() mark(prefix(X)) = [1 2] X + [2] [0 1] [1] >= [1 2] X + [2] [0 1] [1] = a__prefix(mark(X)) mark(s(X)) = [1 2] X + [6] [0 1] [1] >= [1 2] X + [4] [0 1] [1] = s(mark(X)) mark(zWadr(X1,X2)) = [1 6] X1 + [1 6] X2 + [4] [0 1] [0 1] [2] >= [1 6] X1 + [1 6] X2 + [4] [0 1] [0 1] [2] = a__zWadr(mark(X1),mark(X2)) * Step 13: 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(?,O(n^2))