WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest select(Cons(x,xs)) -> selects(x,Nil(),xs) select(Nil()) -> Nil() selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil()) selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs))) ,selects(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {revapp,select,selects} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) revapp#(Nil(),rest) -> c_2() select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) select#(Nil()) -> c_4() selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) revapp#(Nil(),rest) -> c_2() select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) select#(Nil()) -> c_4() selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Weak TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest select(Cons(x,xs)) -> selects(x,Nil(),xs) select(Nil()) -> Nil() selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil()) selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs))) ,selects(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4} by application of Pre({2,4}) = {1,5,6}. Here rules are labelled as follows: 1: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) 2: revapp#(Nil(),rest) -> c_2() 3: select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) 4: select#(Nil()) -> c_4() 5: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) 6: selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Weak DPs: revapp#(Nil(),rest) -> c_2() select#(Nil()) -> c_4() - Weak TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest select(Cons(x,xs)) -> selects(x,Nil(),xs) select(Nil()) -> Nil() selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil()) selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs))) ,selects(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) -->_1 revapp#(Nil(),rest) -> c_2():5 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 2:S:select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) -->_1 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):4 -->_1 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3 3:S:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) -->_1 revapp#(Nil(),rest) -> c_2():5 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 4:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)) -->_1 revapp#(Nil(),rest) -> c_2():5 -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):4 -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 5:W:revapp#(Nil(),rest) -> c_2() 6:W:select#(Nil()) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: select#(Nil()) -> c_4() 5: revapp#(Nil(),rest) -> c_2() * Step 4: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Weak TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest select(Cons(x,xs)) -> selects(x,Nil(),xs) select(Nil()) -> Nil() selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil()) selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs))) ,selects(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 2:S:select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)) -->_1 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):4 -->_1 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3 3:S:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 4:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)) -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):4 -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(2,select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)))] * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Weak TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest select(Cons(x,xs)) -> selects(x,Nil(),xs) select(Nil()) -> Nil() selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil()) selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs))) ,selects(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + 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 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) and a lower component revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) Further, following extension rules are added to the lower component. selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)) -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) ** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + 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(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [2] p(Nil) = [0] p(revapp) = [0] p(select) = [0] p(selects) = [0] p(revapp#) = [0] p(select#) = [4] p(selects#) = [12] x1 + [12] x3 + [1] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [0] p(c_5) = [2] x1 + [1] p(c_6) = [1] x1 + [1] Following rules are strictly oriented: selects#(x',revprefix,Cons(x,xs)) = [12] x + [12] x' + [12] xs + [25] > [12] x + [12] xs + [2] = c_6(selects#(x,Cons(x',revprefix),xs)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) - Weak DPs: selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + 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(c_1) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x1 + [0] p(Nil) = [0] p(revapp) = [0] p(select) = [0] p(selects) = [0] p(revapp#) = [0] p(select#) = [0] p(selects#) = [1] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] Following rules are strictly oriented: selects#(x,revprefix,Nil()) = [1] > [0] = c_5(revapp#(revprefix,Nil())) Following rules are (at-least) weakly oriented: revapp#(Cons(x,xs),rest) = [0] >= [0] = c_1(revapp#(xs,Cons(x,rest))) selects#(x',revprefix,Cons(x,xs)) = [1] >= [0] = revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) = [1] >= [1] = selects#(x,Cons(x',revprefix),xs) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) - Weak DPs: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + 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(c_1) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x2 + [2] p(Nil) = [2] p(revapp) = [1] x1 + [0] p(select) = [1] p(selects) = [1] x1 + [1] x3 + [1] p(revapp#) = [12] x1 + [10] x2 + [5] p(select#) = [1] x1 + [0] p(selects#) = [12] x2 + [12] x3 + [4] p(c_1) = [1] x1 + [3] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [8] p(c_5) = [1] x1 + [0] p(c_6) = [4] x2 + [1] Following rules are strictly oriented: revapp#(Cons(x,xs),rest) = [10] rest + [12] xs + [29] > [10] rest + [12] xs + [28] = c_1(revapp#(xs,Cons(x,rest))) Following rules are (at-least) weakly oriented: selects#(x,revprefix,Nil()) = [12] revprefix + [28] >= [12] revprefix + [25] = c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) = [12] revprefix + [12] xs + [28] >= [12] revprefix + [10] xs + [25] = revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) = [12] revprefix + [12] xs + [28] >= [12] revprefix + [12] xs + [28] = selects#(x,Cons(x',revprefix),xs) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs)) selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs) - Signature: {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1 ,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))