WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs 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: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) append#1#(nil(),l2) -> c_3() subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(leaf()) -> c_5() subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) append#1#(nil(),l2) -> c_3() subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(leaf()) -> c_5() subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,5} by application of Pre({3,5}) = {1,4}. Here rules are labelled as follows: 1: append#(l1,l2) -> c_1(append#1#(l1,l2)) 2: append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) 3: append#1#(nil(),l2) -> c_3() 4: subtrees#(t) -> c_4(subtrees#1#(t)) 5: subtrees#1#(leaf()) -> c_5() 6: subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) 7: subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) 8: subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - Weak DPs: append#1#(nil(),l2) -> c_3() subtrees#1#(leaf()) -> c_5() - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_1(append#1#(l1,l2)) -->_1 append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)):2 -->_1 append#1#(nil(),l2) -> c_3():7 2:S:append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) -->_1 append#(l1,l2) -> c_1(append#1#(l1,l2)):1 3:S:subtrees#(t) -> c_4(subtrees#1#(t)) -->_1 subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)):4 -->_1 subtrees#1#(leaf()) -> c_5():8 4:S:subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) -->_1 subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)):5 -->_2 subtrees#(t) -> c_4(subtrees#1#(t)):3 5:S:subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) -->_1 subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)):6 -->_2 subtrees#(t) -> c_4(subtrees#1#(t)):3 6:S:subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) -->_1 append#(l1,l2) -> c_1(append#1#(l1,l2)):1 7:W:append#1#(nil(),l2) -> c_3() 8:W:subtrees#1#(leaf()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: subtrees#1#(leaf()) -> c_5() 7: append#1#(nil(),l2) -> c_3() * Step 4: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) and a lower component append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) Further, following extension rules are added to the lower component. subtrees#(t) -> subtrees#1#(t) subtrees#1#(node(x,t1,t2)) -> subtrees#(t1) subtrees#1#(node(x,t1,t2)) -> subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) -> subtrees#(t2) subtrees#2#(l1,t1,t2,x) -> subtrees#3#(subtrees(t2),l1,t1,t2,x) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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: SimplifyRHS + Details: Consider the dependency graph 1:S:subtrees#(t) -> c_4(subtrees#1#(t)) -->_1 subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)):2 2:S:subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) -->_1 subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)):3 -->_2 subtrees#(t) -> c_4(subtrees#1#(t)):1 3:S:subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#3#(subtrees(t2),l1,t1,t2,x),subtrees#(t2)) -->_2 subtrees#(t) -> c_4(subtrees#1#(t)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#(t2)) ** Step 4.a:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#(t2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/1,c_8/1} - 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 = 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(c_4) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {1} Following symbols are considered usable: {append#,append#1#,subtrees#,subtrees#1#,subtrees#2#,subtrees#3#} TcT has computed the following interpretation: p(append) = [5] x1 + [3] x2 + [0] p(append#1) = [1] x2 + [2] p(cons) = [1] x2 + [2] p(leaf) = [0] p(nil) = [4] p(node) = [1] x1 + [1] x2 + [1] x3 + [2] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [4] x2 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [1] x5 + [2] p(append#) = [2] x1 + [0] p(append#1#) = [1] x1 + [1] x2 + [0] p(subtrees#) = [8] x1 + [0] p(subtrees#1#) = [8] x1 + [0] p(subtrees#2#) = [8] x3 + [2] x4 + [15] p(subtrees#3#) = [1] x1 + [1] x2 + [1] x4 + [1] p(c_1) = [1] x1 + [1] p(c_2) = [0] p(c_3) = [8] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] x2 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [8] Following rules are strictly oriented: subtrees#2#(l1,t1,t2,x) = [8] t2 + [2] x + [15] > [8] t2 + [0] = c_7(subtrees#(t2)) Following rules are (at-least) weakly oriented: subtrees#(t) = [8] t + [0] >= [8] t + [0] = c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) = [8] t1 + [8] t2 + [8] x + [16] >= [8] t1 + [8] t2 + [2] x + [16] = c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) ** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) - Weak DPs: subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#(t2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/1,c_8/1} - 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 constant 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}, uargs(subtrees#2#) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {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) = [0] p(node) = [1] x1 + [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] p(append#) = [0] p(append#1#) = [0] p(subtrees#) = [0] p(subtrees#1#) = [3] p(subtrees#2#) = [1] x1 + [0] p(subtrees#3#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] x2 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] Following rules are strictly oriented: subtrees#1#(node(x,t1,t2)) = [3] > [0] = c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) Following rules are (at-least) weakly oriented: subtrees#(t) = [0] >= [3] = c_4(subtrees#1#(t)) subtrees#2#(l1,t1,t2,x) = [1] l1 + [0] >= [0] = c_7(subtrees#(t2)) 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] >= [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 + [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.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: subtrees#(t) -> c_4(subtrees#1#(t)) - Weak DPs: subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#(t2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/1,c_8/1} - 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 constant 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}, uargs(subtrees#2#) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {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) = [0] p(node) = [1] x1 + [1] x2 + [1] x3 + [4] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] p(append#) = [0] p(append#1#) = [0] p(subtrees#) = [2] x1 + [2] p(subtrees#1#) = [2] x1 + [1] p(subtrees#2#) = [1] x1 + [2] x3 + [2] x4 + [4] p(subtrees#3#) = [1] x4 + [0] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] x2 + [3] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: subtrees#(t) = [2] t + [2] > [2] t + [1] = c_4(subtrees#1#(t)) Following rules are (at-least) weakly oriented: subtrees#1#(node(x,t1,t2)) = [2] t1 + [2] t2 + [2] x + [9] >= [2] t1 + [2] t2 + [2] x + [9] = c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) = [1] l1 + [2] t2 + [2] x + [4] >= [2] t2 + [2] = c_7(subtrees#(t2)) 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] >= [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 + [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.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: subtrees#(t) -> c_4(subtrees#1#(t)) subtrees#1#(node(x,t1,t2)) -> c_6(subtrees#2#(subtrees(t1),t1,t2,x),subtrees#(t1)) subtrees#2#(l1,t1,t2,x) -> c_7(subtrees#(t2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/1,c_8/1} - 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). ** Step 4.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - Weak DPs: subtrees#(t) -> subtrees#1#(t) subtrees#1#(node(x,t1,t2)) -> subtrees#(t1) subtrees#1#(node(x,t1,t2)) -> subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) -> subtrees#(t2) 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(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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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 constant 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}, uargs(subtrees#2#) = {1}, uargs(subtrees#3#) = {1}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_8) = {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) = [0] p(node) = [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] p(append#) = [0] p(append#1#) = [7] p(subtrees#) = [0] p(subtrees#1#) = [0] p(subtrees#2#) = [1] x1 + [0] p(subtrees#3#) = [1] x1 + [1] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [2] x1 + [2] x2 + [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: append#1#(cons(x,xs),l2) = [7] > [0] = c_2(append#(xs,l2)) Following rules are (at-least) weakly oriented: append#(l1,l2) = [0] >= [7] = c_1(append#1#(l1,l2)) subtrees#(t) = [0] >= [0] = subtrees#1#(t) subtrees#1#(node(x,t1,t2)) = [0] >= [0] = subtrees#(t1) subtrees#1#(node(x,t1,t2)) = [0] >= [0] = subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) = [1] l1 + [0] >= [0] = subtrees#(t2) 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] >= [0] = c_8(append#(l1,l2)) 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] >= [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 + [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.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - Weak DPs: append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#(t) -> subtrees#1#(t) subtrees#1#(node(x,t1,t2)) -> subtrees#(t1) subtrees#1#(node(x,t1,t2)) -> subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) -> subtrees#(t2) 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(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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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 constant 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}, uargs(subtrees#2#) = {1}, uargs(subtrees#3#) = {1}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_8) = {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) = [1] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [4] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] p(append#) = [1] x2 + [0] p(append#1#) = [1] x2 + [3] p(subtrees#) = [2] x1 + [0] p(subtrees#1#) = [2] x1 + [0] p(subtrees#2#) = [1] x1 + [2] x2 + [2] x3 + [6] p(subtrees#3#) = [1] x1 + [1] x3 + [2] x4 + [2] p(c_1) = [1] x1 + [3] p(c_2) = [1] x1 + [2] p(c_3) = [0] p(c_4) = [1] x1 + [2] p(c_5) = [4] p(c_6) = [4] p(c_7) = [4] x1 + [4] x2 + [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: subtrees#3#(l2,l1,t1,t2,x) = [1] l2 + [1] t1 + [2] t2 + [2] > [1] l2 + [0] = c_8(append#(l1,l2)) Following rules are (at-least) weakly oriented: append#(l1,l2) = [1] l2 + [0] >= [1] l2 + [6] = c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) = [1] l2 + [3] >= [1] l2 + [2] = c_2(append#(xs,l2)) subtrees#(t) = [2] t + [0] >= [2] t + [0] = subtrees#1#(t) subtrees#1#(node(x,t1,t2)) = [2] t1 + [2] t2 + [8] >= [2] t1 + [0] = subtrees#(t1) subtrees#1#(node(x,t1,t2)) = [2] t1 + [2] t2 + [8] >= [2] t1 + [2] t2 + [6] = subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) = [1] l1 + [2] t1 + [2] t2 + [6] >= [2] t2 + [0] = subtrees#(t2) subtrees#2#(l1,t1,t2,x) = [1] l1 + [2] t1 + [2] t2 + [6] >= [1] t1 + [2] t2 + [2] = subtrees#3#(subtrees(t2),l1,t1,t2,x) 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 + [0] = subtrees#3(subtrees(t2),l1,t1,t2,x) subtrees#3(l2,l1,t1,t2,x) = [1] l1 + [1] l2 + [0] >= [1] l1 + [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.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) - Weak DPs: append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#(t) -> subtrees#1#(t) subtrees#1#(node(x,t1,t2)) -> subtrees#(t1) subtrees#1#(node(x,t1,t2)) -> subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) -> subtrees#(t2) subtrees#2#(l1,t1,t2,x) -> subtrees#3#(subtrees(t2),l1,t1,t2,x) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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 = 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(c_1) = {1}, uargs(c_2) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3,append#,append#1#,subtrees#,subtrees#1# ,subtrees#2#,subtrees#3#} 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 + [2] p(leaf) = [4] p(nil) = [4] p(node) = [1] x2 + [1] x3 + [2] p(subtrees) = [1] x1 + [0] p(subtrees#1) = [1] x1 + [0] p(subtrees#2) = [1] x1 + [1] x3 + [2] p(subtrees#3) = [1] x1 + [1] x2 + [2] p(append#) = [2] x1 + [1] p(append#1#) = [2] x1 + [0] p(subtrees#) = [10] x1 + [0] p(subtrees#1#) = [10] x1 + [0] p(subtrees#2#) = [8] x1 + [10] x3 + [6] p(subtrees#3#) = [8] x2 + [1] x4 + [6] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [2] x2 + [0] p(c_8) = [2] x1 + [4] Following rules are strictly oriented: append#(l1,l2) = [2] l1 + [1] > [2] l1 + [0] = c_1(append#1#(l1,l2)) Following rules are (at-least) weakly oriented: append#1#(cons(x,xs),l2) = [2] xs + [4] >= [2] xs + [2] = c_2(append#(xs,l2)) subtrees#(t) = [10] t + [0] >= [10] t + [0] = subtrees#1#(t) subtrees#1#(node(x,t1,t2)) = [10] t1 + [10] t2 + [20] >= [10] t1 + [0] = subtrees#(t1) subtrees#1#(node(x,t1,t2)) = [10] t1 + [10] t2 + [20] >= [8] t1 + [10] t2 + [6] = subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) = [8] l1 + [10] t2 + [6] >= [10] t2 + [0] = subtrees#(t2) subtrees#2#(l1,t1,t2,x) = [8] l1 + [10] t2 + [6] >= [8] l1 + [1] t2 + [6] = subtrees#3#(subtrees(t2),l1,t1,t2,x) subtrees#3#(l2,l1,t1,t2,x) = [8] l1 + [1] t2 + [6] >= [4] l1 + [6] = c_8(append#(l1,l2)) 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 + [2] >= [1] l2 + [1] xs + [2] = cons(x,append(xs,l2)) append#1(nil(),l2) = [1] l2 + [4] >= [1] l2 + [0] = l2 subtrees(t) = [1] t + [0] >= [1] t + [0] = subtrees#1(t) subtrees#1(leaf()) = [4] >= [4] = nil() subtrees#1(node(x,t1,t2)) = [1] t1 + [1] t2 + [2] >= [1] t1 + [1] t2 + [2] = subtrees#2(subtrees(t1),t1,t2,x) subtrees#2(l1,t1,t2,x) = [1] l1 + [1] t2 + [2] >= [1] l1 + [1] t2 + [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)) ** Step 4.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(l1,l2) -> c_1(append#1#(l1,l2)) append#1#(cons(x,xs),l2) -> c_2(append#(xs,l2)) subtrees#(t) -> subtrees#1#(t) subtrees#1#(node(x,t1,t2)) -> subtrees#(t1) subtrees#1#(node(x,t1,t2)) -> subtrees#2#(subtrees(t1),t1,t2,x) subtrees#2#(l1,t1,t2,x) -> subtrees#(t2) subtrees#2#(l1,t1,t2,x) -> subtrees#3#(subtrees(t2),l1,t1,t2,x) subtrees#3#(l2,l1,t1,t2,x) -> c_8(append#(l1,l2)) - 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,append#/2,append#1#/2,subtrees#/1 ,subtrees#1#/1,subtrees#2#/4,subtrees#3#/5} / {cons/2,leaf/0,nil/0,node/3,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/2,c_8/1} - 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))