WORST_CASE(?,O(n^2)) * Step 1: Sum WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/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 {::,leaf,nil,node} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/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 {::,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#(::(@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 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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#(::(@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 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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#(::(@x,@xs),@l2) -> c_2(append#(@xs,@l2)):2 -->_1 append#1#(nil(),@l2) -> c_3():7 2:S:append#1#(::(@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 5: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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#(::(@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 5.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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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 5.a:2: 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)) subtrees#2#(@l1,@t1,@t2,@x) -> c_7(subtrees#(@t2)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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(::) = {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(::) = [1] x2 + [0] p(append) = [1] x2 + [0] p(append#1) = [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#) = [4] p(subtrees#1#) = [0] p(subtrees#2#) = [1] x1 + [2] 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#(@t) = [4] > [0] = c_4(subtrees#1#(@t)) Following rules are (at-least) weakly oriented: subtrees#1#(node(@x,@t1,@t2)) = [0] >= [6] = c_6(subtrees#2#(subtrees(@t1),@t1,@t2,@x),subtrees#(@t1)) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [2] >= [4] = c_7(subtrees#(@t2)) append(@l1,@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = ::(@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] = ::(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.a:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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 DPs: subtrees#(@t) -> c_4(subtrees#1#(@t)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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: {subtrees,subtrees#1,subtrees#2,subtrees#3,append#,append#1#,subtrees#,subtrees#1#,subtrees#2# ,subtrees#3#} TcT has computed the following interpretation: p(::) = [0] p(append) = [2] x2 + [4] p(append#1) = [8] x2 + [2] p(leaf) = [0] p(nil) = [0] p(node) = [1] x1 + [1] x2 + [1] x3 + [2] p(subtrees) = [1] p(subtrees#1) = [1] p(subtrees#2) = [1] p(subtrees#3) = [0] p(append#) = [1] x1 + [8] x2 + [1] p(append#1#) = [1] x1 + [1] p(subtrees#) = [8] x1 + [4] p(subtrees#1#) = [8] x1 + [4] p(subtrees#2#) = [8] x1 + [8] x3 + [8] p(subtrees#3#) = [4] x1 + [2] x4 + [2] x5 + [4] 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 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [2] Following rules are strictly oriented: subtrees#2#(@l1,@t1,@t2,@x) = [8] @l1 + [8] @t2 + [8] > [8] @t2 + [4] = c_7(subtrees#(@t2)) Following rules are (at-least) weakly oriented: subtrees#(@t) = [8] @t + [4] >= [8] @t + [4] = c_4(subtrees#1#(@t)) subtrees#1#(node(@x,@t1,@t2)) = [8] @t1 + [8] @t2 + [8] @x + [20] >= [8] @t1 + [8] @t2 + [20] = c_6(subtrees#2#(subtrees(@t1),@t1,@t2,@x),subtrees#(@t1)) subtrees(@t) = [1] >= [1] = subtrees#1(@t) subtrees#1(leaf()) = [1] >= [0] = nil() subtrees#1(node(@x,@t1,@t2)) = [1] >= [1] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] >= [0] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [0] >= [0] = ::(node(@x,@t1,@t2),append(@l1,@l2)) ** Step 5.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: subtrees#1#(node(@x,@t1,@t2)) -> c_6(subtrees#2#(subtrees(@t1),@t1,@t2,@x),subtrees#(@t1)) - Weak DPs: subtrees#(@t) -> c_4(subtrees#1#(@t)) subtrees#2#(@l1,@t1,@t2,@x) -> c_7(subtrees#(@t2)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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(::) = {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(::) = [1] x2 + [0] p(append) = [1] x2 + [0] p(append#1) = [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [3] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [0] p(append#) = [0] p(append#1#) = [2] x1 + [0] p(subtrees#) = [3] x1 + [6] p(subtrees#1#) = [3] x1 + [6] p(subtrees#2#) = [1] x1 + [3] x3 + [7] p(subtrees#3#) = [0] p(c_1) = [0] p(c_2) = [2] x1 + [1] p(c_3) = [4] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [1] x2 + [1] p(c_7) = [1] x1 + [1] p(c_8) = [1] Following rules are strictly oriented: subtrees#1#(node(@x,@t1,@t2)) = [3] @t1 + [3] @t2 + [15] > [3] @t1 + [3] @t2 + [14] = c_6(subtrees#2#(subtrees(@t1),@t1,@t2,@x),subtrees#(@t1)) Following rules are (at-least) weakly oriented: subtrees#(@t) = [3] @t + [6] >= [3] @t + [6] = c_4(subtrees#1#(@t)) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [3] @t2 + [7] >= [3] @t2 + [7] = c_7(subtrees#(@t2)) append(@l1,@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = ::(@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] >= [0] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l2 + [0] >= [1] @l2 + [0] = ::(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.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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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(::) = {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(::) = [1] x2 + [0] p(append) = [1] x2 + [0] p(append#1) = [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#) = [1] x1 + [1] x2 + [0] p(append#1#) = [1] x1 + [1] x2 + [0] p(subtrees#) = [2] p(subtrees#1#) = [2] p(subtrees#2#) = [1] x1 + [2] p(subtrees#3#) = [1] x1 + [1] x2 + [2] 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) = [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: subtrees#3#(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [2] > [1] @l1 + [1] @l2 + [0] = c_8(append#(@l1,@l2)) Following rules are (at-least) weakly oriented: append#(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = c_1(append#1#(@l1,@l2)) append#1#(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [0] >= [1] @l2 + [1] @xs + [0] = c_2(append#(@xs,@l2)) subtrees#(@t) = [2] >= [2] = subtrees#1#(@t) subtrees#1#(node(@x,@t1,@t2)) = [2] >= [2] = subtrees#(@t1) subtrees#1#(node(@x,@t1,@t2)) = [2] >= [2] = subtrees#2#(subtrees(@t1),@t1,@t2,@x) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [2] >= [2] = subtrees#(@t2) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [2] >= [1] @l1 + [2] = subtrees#3#(subtrees(@t2),@l1,@t1,@t2,@x) append(@l1,@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = ::(@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] = ::(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.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@x,@xs),@l2) -> c_2(append#(@xs,@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) subtrees#3#(@l2,@l1,@t1,@t2,@x) -> c_8(append#(@l1,@l2)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,leaf,nil,node} + 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(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(::) = 1 + x2 p(append) = x1 + x2 p(append#1) = x1 + x2 p(leaf) = 4 p(nil) = 0 p(node) = 4 + x2 + x3 p(subtrees) = x1 p(subtrees#1) = x1 p(subtrees#2) = 4 + x1 + x3 p(subtrees#3) = 1 + x1 + x2 p(append#) = 2 + 2*x1 p(append#1#) = 1 + 2*x1 p(subtrees#) = 2*x1 p(subtrees#1#) = 2*x1 p(subtrees#2#) = 5 + 2*x1 + 2*x3 p(subtrees#3#) = 4 + 2*x2 p(c_1) = x1 p(c_2) = 1 + x1 p(c_3) = 1 p(c_4) = 0 p(c_5) = 0 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 Following rules are strictly oriented: append#(@l1,@l2) = 2 + 2*@l1 > 1 + 2*@l1 = c_1(append#1#(@l1,@l2)) Following rules are (at-least) weakly oriented: append#1#(::(@x,@xs),@l2) = 3 + 2*@xs >= 3 + 2*@xs = c_2(append#(@xs,@l2)) subtrees#(@t) = 2*@t >= 2*@t = subtrees#1#(@t) subtrees#1#(node(@x,@t1,@t2)) = 8 + 2*@t1 + 2*@t2 >= 2*@t1 = subtrees#(@t1) subtrees#1#(node(@x,@t1,@t2)) = 8 + 2*@t1 + 2*@t2 >= 5 + 2*@t1 + 2*@t2 = subtrees#2#(subtrees(@t1),@t1,@t2,@x) subtrees#2#(@l1,@t1,@t2,@x) = 5 + 2*@l1 + 2*@t2 >= 2*@t2 = subtrees#(@t2) subtrees#2#(@l1,@t1,@t2,@x) = 5 + 2*@l1 + 2*@t2 >= 4 + 2*@l1 = subtrees#3#(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3#(@l2,@l1,@t1,@t2,@x) = 4 + 2*@l1 >= 2 + 2*@l1 = c_8(append#(@l1,@l2)) append(@l1,@l2) = @l1 + @l2 >= @l1 + @l2 = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = 1 + @l2 + @xs >= 1 + @l2 + @xs = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = @l2 >= @l2 = @l2 subtrees(@t) = @t >= @t = subtrees#1(@t) subtrees#1(leaf()) = 4 >= 0 = nil() subtrees#1(node(@x,@t1,@t2)) = 4 + @t1 + @t2 >= 4 + @t1 + @t2 = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = 4 + @l1 + @t2 >= 1 + @l1 + @t2 = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = 1 + @l1 + @l2 >= 1 + @l1 + @l2 = ::(node(@x,@t1,@t2),append(@l1,@l2)) ** Step 5.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#1#(::(@x,@xs),@l2) -> c_2(append#(@xs,@l2)) - Weak DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,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(::) = {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(::) = [1] x2 + [1] p(append) = [1] x1 + [1] x2 + [0] p(append#1) = [1] x1 + [1] x2 + [0] p(leaf) = [7] p(nil) = [7] p(node) = [1] x2 + [1] x3 + [4] p(subtrees) = [1] x1 + [1] p(subtrees#1) = [1] x1 + [0] p(subtrees#2) = [1] x1 + [1] x3 + [3] p(subtrees#3) = [1] x1 + [1] x2 + [1] p(append#) = [1] x1 + [2] p(append#1#) = [1] x1 + [2] p(subtrees#) = [2] x1 + [4] p(subtrees#1#) = [2] x1 + [0] p(subtrees#2#) = [1] x1 + [1] x2 + [2] x3 + [7] p(subtrees#3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [5] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [1] x1 + [1] x2 + [4] p(c_7) = [1] x1 + [1] x2 + [1] p(c_8) = [1] x1 + [3] Following rules are strictly oriented: append#1#(::(@x,@xs),@l2) = [1] @xs + [3] > [1] @xs + [2] = c_2(append#(@xs,@l2)) Following rules are (at-least) weakly oriented: append#(@l1,@l2) = [1] @l1 + [2] >= [1] @l1 + [2] = c_1(append#1#(@l1,@l2)) subtrees#(@t) = [2] @t + [4] >= [2] @t + [0] = subtrees#1#(@t) subtrees#1#(node(@x,@t1,@t2)) = [2] @t1 + [2] @t2 + [8] >= [2] @t1 + [4] = subtrees#(@t1) subtrees#1#(node(@x,@t1,@t2)) = [2] @t1 + [2] @t2 + [8] >= [2] @t1 + [2] @t2 + [8] = subtrees#2#(subtrees(@t1),@t1,@t2,@x) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t1 + [2] @t2 + [7] >= [2] @t2 + [4] = subtrees#(@t2) subtrees#2#(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t1 + [2] @t2 + [7] >= [1] @l1 + [1] @t1 + [2] @t2 + [6] = subtrees#3#(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3#(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [1] @t1 + [1] @t2 + [5] >= [1] @l1 + [5] = c_8(append#(@l1,@l2)) append(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [1] >= [1] @l2 + [1] @xs + [1] = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = [1] @l2 + [7] >= [1] @l2 + [0] = @l2 subtrees(@t) = [1] @t + [1] >= [1] @t + [0] = subtrees#1(@t) subtrees#1(leaf()) = [7] >= [7] = nil() subtrees#1(node(@x,@t1,@t2)) = [1] @t1 + [1] @t2 + [4] >= [1] @t1 + [1] @t2 + [4] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t2 + [3] >= [1] @l1 + [1] @t2 + [2] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [1] >= [1] @l1 + [1] @l2 + [1] = ::(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.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(@l1,@l2) -> c_1(append#1#(@l1,@l2)) append#1#(::(@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(::(@x,@xs),@l2) -> ::(@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) -> ::(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} / {::/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 {::,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))