WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append(l1,l2) -> append#1(l1,l2) append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x2 + [8] p(append#1) = [1] x2 + [0] p(cons) = [1] x2 + [3] p(leaf) = [0] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: append(l1,l2) = [1] l2 + [8] > [1] l2 + [0] = append#1(l1,l2) Following rules are (at-least) weakly oriented: append#1(cons(x,xs),l2) = [1] l2 + [0] >= [1] l2 + [11] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 subtrees(t) = [0] >= [0] = subtrees#1(t) subtrees#1(leaf()) = [0] >= [0] = nil() subtrees#1(node(x,t1,t2)) = [0] >= [0] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1] l1 + [0] >= [1] l1 + [0] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [0] >= [1] l2 + [11] = cons(node(x,t1,t2),append(l1,l2)) 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: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Weak TRS: append(l1,l2) -> append#1(l1,l2) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x1 + [1] x2 + [0] p(append#1) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x1 + [1] x2 + [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [1] > [1] l1 + [1] l2 + [0] = cons(node(x,t1,t2),append(l1,l2)) Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = append#1(l1,l2) append#1(cons(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [0] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 subtrees(t) = [0] >= [0] = subtrees#1(t) subtrees#1(leaf()) = [0] >= [0] = nil() subtrees#1(node(x,t1,t2)) = [0] >= [0] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1] l1 + [0] >= [1] l1 + [1] = subtrees#3(subtrees(t2),l1,t1,t2,x) 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: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) - Weak TRS: append(l1,l2) -> append#1(l1,l2) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x2 + [0] p(append#1) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(leaf) = [4] p(nil) = [0] p(node) = [1] x3 + [0] p(subtrees) = [8] p(subtrees#1) = [11] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: subtrees#1(leaf()) = [11] > [0] = nil() subtrees#1(node(x,t1,t2)) = [11] > [8] = subtrees#2(subtrees(t1),t1,t2,x) Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l2 + [0] >= [1] l2 + [0] = append#1(l1,l2) append#1(cons(x,xs),l2) = [1] l2 + [0] >= [1] l2 + [0] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 subtrees(t) = [8] >= [11] = subtrees#1(t) subtrees#2(l1,t1,t2,x) = [1] l1 + [0] >= [1] l1 + [8] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [0] >= [1] l2 + [0] = cons(node(x,t1,t2),append(l1,l2)) 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: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) - Weak TRS: append(l1,l2) -> append#1(l1,l2) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x1 + [1] x2 + [1] p(append#1) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [1] p(leaf) = [0] p(nil) = [1] p(node) = [1] x1 + [1] x2 + [0] p(subtrees) = [0] p(subtrees#1) = [1] p(subtrees#2) = [1] x1 + [1] p(subtrees#3) = [1] x1 + [1] x2 + [2] Following rules are strictly oriented: append#1(nil(),l2) = [1] l2 + [1] > [1] l2 + [0] = l2 Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l1 + [1] l2 + [1] >= [1] l1 + [1] l2 + [0] = append#1(l1,l2) append#1(cons(x,xs),l2) = [1] l2 + [1] xs + [1] >= [1] l2 + [1] xs + [2] = cons(x,append(xs,l2)) subtrees(t) = [0] >= [1] = subtrees#1(t) subtrees#1(leaf()) = [1] >= [1] = nil() subtrees#1(node(x,t1,t2)) = [1] >= [1] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1] l1 + [1] >= [1] l1 + [2] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [2] >= [1] l1 + [1] l2 + [2] = cons(node(x,t1,t2),append(l1,l2)) 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: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) subtrees(t) -> subtrees#1(t) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) - Weak TRS: append(l1,l2) -> append#1(l1,l2) append#1(nil(),l2) -> l2 subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x2 + [0] p(append#1) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(leaf) = [0] p(nil) = [1] p(node) = [1] x2 + [1] p(subtrees) = [0] p(subtrees#1) = [1] p(subtrees#2) = [1] x1 + [1] p(subtrees#3) = [1] x1 + [0] Following rules are strictly oriented: subtrees#2(l1,t1,t2,x) = [1] l1 + [1] > [0] = subtrees#3(subtrees(t2),l1,t1,t2,x) Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l2 + [0] >= [1] l2 + [0] = append#1(l1,l2) append#1(cons(x,xs),l2) = [1] l2 + [0] >= [1] l2 + [0] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 subtrees(t) = [0] >= [1] = subtrees#1(t) subtrees#1(leaf()) = [1] >= [1] = nil() subtrees#1(node(x,t1,t2)) = [1] >= [1] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l2 + [0] >= [1] l2 + [0] = cons(node(x,t1,t2),append(l1,l2)) 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: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) subtrees(t) -> subtrees#1(t) - Weak TRS: append(l1,l2) -> append#1(l1,l2) append#1(nil(),l2) -> l2 subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [1] x2 + [0] p(append#1) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(leaf) = [1] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [3] p(subtrees) = [1] x1 + [1] p(subtrees#1) = [1] x1 + [0] p(subtrees#2) = [1] x1 + [1] x3 + [2] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: subtrees(t) = [1] t + [1] > [1] t + [0] = subtrees#1(t) Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l2 + [0] >= [1] l2 + [0] = append#1(l1,l2) append#1(cons(x,xs),l2) = [1] l2 + [0] >= [1] l2 + [0] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 subtrees#1(leaf()) = [1] >= [0] = nil() subtrees#1(node(x,t1,t2)) = [1] t1 + [1] t2 + [3] >= [1] t1 + [1] t2 + [3] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1] l1 + [1] t2 + [2] >= [1] l1 + [1] t2 + [1] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [0] >= [1] l2 + [0] = cons(node(x,t1,t2),append(l1,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) - Weak TRS: append(l1,l2) -> append#1(l1,l2) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + 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(cons) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3} TcT has computed the following interpretation: p(append) = [0 2] x1 + [1 0] x2 + [0] [0 1] [0 1] [4] p(append#1) = [0 2] x1 + [1 0] x2 + [0] [0 1] [0 1] [4] p(cons) = [1 0] x2 + [0] [0 1] [1] p(leaf) = [3] [0] p(nil) = [4] [0] p(node) = [1 1] x2 + [1 0] x3 + [4] [0 1] [0 1] [5] p(subtrees) = [2 0] x1 + [1] [0 1] [0] p(subtrees#1) = [2 0] x1 + [1] [0 1] [0] p(subtrees#2) = [1 2] x1 + [2 0] x3 + [7] [0 1] [0 1] [5] p(subtrees#3) = [1 0] x1 + [0 2] x2 + [3] [0 1] [0 1] [5] Following rules are strictly oriented: append#1(cons(x,xs),l2) = [1 0] l2 + [0 2] xs + [2] [0 1] [0 1] [5] > [1 0] l2 + [0 2] xs + [0] [0 1] [0 1] [5] = cons(x,append(xs,l2)) Following rules are (at-least) weakly oriented: append(l1,l2) = [0 2] l1 + [1 0] l2 + [0] [0 1] [0 1] [4] >= [0 2] l1 + [1 0] l2 + [0] [0 1] [0 1] [4] = append#1(l1,l2) append#1(nil(),l2) = [1 0] l2 + [0] [0 1] [4] >= [1 0] l2 + [0] [0 1] [0] = l2 subtrees(t) = [2 0] t + [1] [0 1] [0] >= [2 0] t + [1] [0 1] [0] = subtrees#1(t) subtrees#1(leaf()) = [7] [0] >= [4] [0] = nil() subtrees#1(node(x,t1,t2)) = [2 2] t1 + [2 0] t2 + [9] [0 1] [0 1] [5] >= [2 2] t1 + [2 0] t2 + [8] [0 1] [0 1] [5] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1 2] l1 + [2 0] t2 + [7] [0 1] [0 1] [5] >= [0 2] l1 + [2 0] t2 + [4] [0 1] [0 1] [5] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [0 2] l1 + [1 0] l2 + [3] [0 1] [0 1] [5] >= [0 2] l1 + [1 0] l2 + [0] [0 1] [0 1] [5] = cons(node(x,t1,t2),append(l1,l2)) * Step 8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(l1,l2) -> append#1(l1,l2) append#1(cons(x,xs),l2) -> cons(x,append(xs,l2)) append#1(nil(),l2) -> l2 subtrees(t) -> subtrees#1(t) subtrees#1(leaf()) -> nil() subtrees#1(node(x,t1,t2)) -> subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) -> subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) -> cons(node(x,t1,t2),append(l1,l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {cons/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {cons,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))