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_ = WIDP} + Details: We add the following weak innermost dependency pairs: 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: UsableRules 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)) - 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,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))) 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)) * Step 3: 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)) - 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 4: 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() - 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 5: 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)) - 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 6: Decompose 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - 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)) -> 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} Problem (S) - Strict DPs: 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#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) - 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} ** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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)) -> 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) The strictly oriented rules are moved into the weak component. *** Step 6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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)) -> 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {revapp#,select#,selects#} TcT has computed the following interpretation: p(Cons) = 1 + x1 + x2 p(Nil) = 1 p(revapp) = 2 p(select) = 0 p(selects) = 1 + x1 + x1^2 + 2*x2^2 p(revapp#) = 2*x1 p(select#) = 1 p(selects#) = 2 + 4*x1 + 2*x1*x2 + 5*x1*x3 + 3*x2 + 5*x2*x3 + 3*x3^2 p(c_1) = x1 p(c_2) = 1 p(c_3) = 2 p(c_4) = 0 p(c_5) = 2 + x1 p(c_6) = x1 + x2 Following rules are strictly oriented: revapp#(Cons(x,xs),rest) = 2 + 2*x + 2*xs > 2*xs = c_1(revapp#(xs,Cons(x,rest))) Following rules are (at-least) weakly oriented: selects#(x,revprefix,Nil()) = 5 + 8*revprefix + 2*revprefix*x + 9*x >= 2 + 2*revprefix = c_5(revapp#(revprefix,Nil())) selects#(x',revprefix,Cons(x,xs)) = 5 + 8*revprefix + 5*revprefix*x + 2*revprefix*x' + 5*revprefix*xs + 6*x + 5*x*x' + 6*x*xs + 3*x^2 + 9*x' + 5*x'*xs + 6*xs + 3*xs^2 >= 5 + 5*revprefix + 2*revprefix*x + 5*revprefix*xs + 6*x + 2*x*x' + 5*x*xs + 3*x' + 5*x'*xs + 5*xs + 3*xs^2 = c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) *** Step 6.a:1.a:2: Assumption 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)) -> 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.a:1.b:1: RemoveWeakSuffixes 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)) -> 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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:W:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 3:W: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)):3 -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):2 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)) 2: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) 1: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) *** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). ** Step 6.b:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) - 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 {1} by application of Pre({1}) = {2}. Here rules are labelled as follows: 1: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) 2: selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs)) 3: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) ** Step 6.b:2: RemoveWeakSuffixes 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)) - Weak DPs: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) - 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:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)) -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3 -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):2 -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)) ,selects#(x,Cons(x',revprefix),xs)):1 2:W: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))):2 3:W:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())) 2: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))) ** Step 6.b:3: 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.b:4: PredecessorEstimationCP 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) The strictly oriented rules are moved into the weak component. *** Step 6.b:4.a:1: NaturalMI 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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: {revapp#,select#,selects#} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [4] p(Nil) = [1] p(revapp) = [1] x2 + [2] p(select) = [2] p(selects) = [1] x1 + [4] x2 + [1] x3 + [0] p(revapp#) = [1] x1 + [4] p(select#) = [1] p(selects#) = [4] x1 + [4] x3 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x1 + [15] Following rules are strictly oriented: selects#(x',revprefix,Cons(x,xs)) = [4] x + [4] x' + [4] xs + [16] > [4] x + [4] xs + [15] = c_6(selects#(x,Cons(x',revprefix),xs)) Following rules are (at-least) weakly oriented: *** Step 6.b:4.a:2: Assumption 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.b:4.b:1: RemoveWeakSuffixes 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) -->_1 selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs)) *** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). WORST_CASE(?,O(n^2))