WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys goal(xs) -> shuffle(xs) reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs))) shuffle(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {append,goal,reverse,shuffle} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(Nil(),ys) -> c_2() goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) reverse#(Nil()) -> c_5() shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) shuffle#(Nil()) -> c_7() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(Nil(),ys) -> c_2() goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) reverse#(Nil()) -> c_5() shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) shuffle#(Nil()) -> c_7() - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys goal(xs) -> shuffle(xs) reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() shuffle(Cons(x,xs)) -> Cons(x,shuffle(reverse(xs))) shuffle(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(Nil(),ys) -> c_2() goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) reverse#(Nil()) -> c_5() shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) shuffle#(Nil()) -> c_7() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(Nil(),ys) -> c_2() goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) reverse#(Nil()) -> c_5() shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) shuffle#(Nil()) -> c_7() - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,5,7} by application of Pre({2,5,7}) = {1,3,4,6}. Here rules are labelled as follows: 1: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) 2: append#(Nil(),ys) -> c_2() 3: goal#(xs) -> c_3(shuffle#(xs)) 4: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) 5: reverse#(Nil()) -> c_5() 6: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) 7: shuffle#(Nil()) -> c_7() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak DPs: append#(Nil(),ys) -> c_2() reverse#(Nil()) -> c_5() shuffle#(Nil()) -> c_7() - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(Nil(),ys) -> c_2():5 -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:goal#(xs) -> c_3(shuffle#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4 -->_1 shuffle#(Nil()) -> c_7():7 3:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) -->_2 reverse#(Nil()) -> c_5():6 -->_1 append#(Nil(),ys) -> c_2():5 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3 -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 4:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Nil()) -> c_7():7 -->_2 reverse#(Nil()) -> c_5():6 -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3 5:W:append#(Nil(),ys) -> c_2() 6:W:reverse#(Nil()) -> c_5() 7:W:shuffle#(Nil()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: reverse#(Nil()) -> c_5() 7: shuffle#(Nil()) -> c_7() 5: append#(Nil(),ys) -> c_2() * Step 5: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) goal#(xs) -> c_3(shuffle#(xs)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:goal#(xs) -> c_3(shuffle#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4 3:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3 -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 4:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):4 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):3 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,goal#(xs) -> c_3(shuffle#(xs)))] * Step 6: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} Problem (S) - Strict DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} ** Step 6.a:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) and a lower component append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) Further, following extension rules are added to the lower component. shuffle#(Cons(x,xs)) -> reverse#(xs) shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) *** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: {append,reverse,append#,goal#,reverse#,shuffle#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [1] p(Nil) = [0] p(append) = [1] x1 + [1] x2 + [0] p(goal) = [8] x1 + [1] p(reverse) = [1] x1 + [0] p(shuffle) = [8] x1 + [2] p(append#) = [1] x1 + [1] x2 + [1] p(goal#) = [8] x1 + [0] p(reverse#) = [1] x1 + [0] p(shuffle#) = [4] x1 + [0] p(c_1) = [2] x1 + [2] p(c_2) = [1] p(c_3) = [4] x1 + [8] p(c_4) = [2] x1 + [1] x2 + [1] p(c_5) = [1] p(c_6) = [1] x1 + [3] p(c_7) = [0] Following rules are strictly oriented: shuffle#(Cons(x,xs)) = [4] xs + [4] > [4] xs + [3] = c_6(shuffle#(reverse(xs)),reverse#(xs)) Following rules are (at-least) weakly oriented: append(Cons(x,xs),ys) = [1] xs + [1] ys + [1] >= [1] xs + [1] ys + [1] = Cons(x,append(xs,ys)) append(Nil(),ys) = [1] ys + [0] >= [1] ys + [0] = ys reverse(Cons(x,xs)) = [1] xs + [1] >= [1] xs + [1] = append(reverse(xs),Cons(x,Nil())) reverse(Nil()) = [0] >= [0] = Nil() **** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) **** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> reverse#(xs) shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> reverse#(xs) shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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_4) = {1,2} Following symbols are considered usable: {append,reverse,append#,goal#,reverse#,shuffle#} TcT has computed the following interpretation: p(Cons) = 1 + x1 + x2 p(Nil) = 0 p(append) = x1 + x2 p(goal) = 4 + 4*x1^2 p(reverse) = x1 p(shuffle) = 2 + x1 + 2*x1^2 p(append#) = x1 + x2^2 p(goal#) = 1 + 2*x1 p(reverse#) = 2 + 4*x1 + x1^2 p(shuffle#) = 6 + 2*x1^2 p(c_1) = x1 p(c_2) = 2 p(c_3) = 4 + x1 p(c_4) = x1 + x2 p(c_5) = 1 p(c_6) = 1 p(c_7) = 0 Following rules are strictly oriented: append#(Cons(x,xs),ys) = 1 + x + xs + ys^2 > xs + ys^2 = c_1(append#(xs,ys)) Following rules are (at-least) weakly oriented: reverse#(Cons(x,xs)) = 7 + 6*x + 2*x*xs + x^2 + 6*xs + xs^2 >= 3 + 2*x + x^2 + 5*xs + xs^2 = c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) = 8 + 4*x + 4*x*xs + 2*x^2 + 4*xs + 2*xs^2 >= 2 + 4*xs + xs^2 = reverse#(xs) shuffle#(Cons(x,xs)) = 8 + 4*x + 4*x*xs + 2*x^2 + 4*xs + 2*xs^2 >= 6 + 2*xs^2 = shuffle#(reverse(xs)) append(Cons(x,xs),ys) = 1 + x + xs + ys >= 1 + x + xs + ys = Cons(x,append(xs,ys)) append(Nil(),ys) = ys >= ys = ys reverse(Cons(x,xs)) = 1 + x + xs >= 1 + x + xs = append(reverse(xs),Cons(x,Nil())) reverse(Nil()) = 0 >= 0 = Nil() **** Step 6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> reverse#(xs) shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> reverse#(xs) shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 2:W:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):2 -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):1 3:W:shuffle#(Cons(x,xs)) -> reverse#(xs) -->_1 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):2 4:W:shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) -->_1 shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)):4 -->_1 shuffle#(Cons(x,xs)) -> reverse#(xs):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: shuffle#(Cons(x,xs)) -> shuffle#(reverse(xs)) 3: shuffle#(Cons(x,xs)) -> reverse#(xs) 2: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) 1: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) **** Step 6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak DPs: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):3 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1 2:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1 3:W:append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: append#(Cons(x,xs),ys) -> c_1(append#(xs,ys)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/2,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)) -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1 2:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2 -->_2 reverse#(Cons(x,xs)) -> c_4(append#(reverse(xs),Cons(x,Nil())),reverse#(xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) ** Step 6.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} Problem (S) - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} *** Step 6.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) The strictly oriented rules are moved into the weak component. **** Step 6.b:3.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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_4) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {append,reverse,append#,goal#,reverse#,shuffle#} TcT has computed the following interpretation: p(Cons) = 1 + x2 p(Nil) = 0 p(append) = x1 + x2 p(goal) = x1 p(reverse) = x1 p(shuffle) = x1 + x1^2 p(append#) = x1 p(goal#) = 4 p(reverse#) = 2*x1 p(shuffle#) = 4 + 4*x1^2 p(c_1) = 1 p(c_2) = 2 p(c_3) = 0 p(c_4) = x1 p(c_5) = 4 p(c_6) = 2 + x1 + x2 p(c_7) = 1 Following rules are strictly oriented: reverse#(Cons(x,xs)) = 2 + 2*xs > 2*xs = c_4(reverse#(xs)) Following rules are (at-least) weakly oriented: shuffle#(Cons(x,xs)) = 8 + 8*xs + 4*xs^2 >= 6 + 2*xs + 4*xs^2 = c_6(shuffle#(reverse(xs)),reverse#(xs)) append(Cons(x,xs),ys) = 1 + xs + ys >= 1 + xs + ys = Cons(x,append(xs,ys)) append(Nil(),ys) = ys >= ys = ys reverse(Cons(x,xs)) = 1 + xs >= 1 + xs = append(reverse(xs),Cons(x,Nil())) reverse(Nil()) = 0 >= 0 = Nil() **** Step 6.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) -->_1 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):1 2:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):2 -->_2 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) 1: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) **** Step 6.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak DPs: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_2 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):2 -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1 2:W:reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) -->_1 reverse#(Cons(x,xs)) -> c_4(reverse#(xs)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: reverse#(Cons(x,xs)) -> c_4(reverse#(xs)) *** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs)),reverse#(xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) *** Step 6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) The strictly oriented rules are moved into the weak component. **** Step 6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} 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: {append,reverse,append#,goal#,reverse#,shuffle#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [2] p(Nil) = [0] p(append) = [1] x1 + [1] x2 + [0] p(goal) = [2] x1 + [2] p(reverse) = [1] x1 + [0] p(shuffle) = [1] x1 + [1] p(append#) = [1] x2 + [0] p(goal#) = [0] p(reverse#) = [1] x1 + [8] p(shuffle#) = [8] x1 + [0] p(c_1) = [2] x1 + [0] p(c_2) = [2] p(c_3) = [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [14] p(c_7) = [0] Following rules are strictly oriented: shuffle#(Cons(x,xs)) = [8] xs + [16] > [8] xs + [14] = c_6(shuffle#(reverse(xs))) Following rules are (at-least) weakly oriented: append(Cons(x,xs),ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = Cons(x,append(xs,ys)) append(Nil(),ys) = [1] ys + [0] >= [1] ys + [0] = ys reverse(Cons(x,xs)) = [1] xs + [2] >= [1] xs + [2] = append(reverse(xs),Cons(x,Nil())) reverse(Nil()) = [0] >= [0] = Nil() **** Step 6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) -->_1 shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: shuffle#(Cons(x,xs)) -> c_6(shuffle#(reverse(xs))) **** Step 6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(Cons(x,xs),ys) -> Cons(x,append(xs,ys)) append(Nil(),ys) -> ys reverse(Cons(x,xs)) -> append(reverse(xs),Cons(x,Nil())) reverse(Nil()) -> Nil() - Signature: {append/2,goal/1,reverse/1,shuffle/1,append#/2,goal#/1,reverse#/1,shuffle#/1} / {Cons/2,Nil/0,c_1/1,c_2/0 ,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,goal#,reverse#,shuffle#} and constructors {Cons ,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))