MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,from,length,length1,nil ,s} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__nil()) -> c_4(nil#()) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) length#(n__nil()) -> c_10() length1#(X) -> c_11(length#(activate(X)),activate#(X)) nil#() -> c_12() s#(X) -> c_13() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__nil()) -> c_4(nil#()) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) length#(n__nil()) -> c_10() length1#(X) -> c_11(length#(activate(X)),activate#(X)) nil#() -> c_12() s#(X) -> c_13() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/2,c_3/2,c_4/1,c_5/2,c_6/0,c_7/1,c_8/0 ,c_9/3,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,6,8,10,12,13} by application of Pre({1,6,8,10,12,13}) = {2,3,4,5,7,9,11}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) 3: activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) 4: activate#(n__nil()) -> c_4(nil#()) 5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 6: cons#(X1,X2) -> c_6() 7: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 8: from#(X) -> c_8() 9: length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) 10: length#(n__nil()) -> c_10() 11: length1#(X) -> c_11(length#(activate(X)),activate#(X)) 12: nil#() -> c_12() 13: s#(X) -> c_13() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__nil()) -> c_4(nil#()) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_7(cons#(X,n__from(n__s(X)))) length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() cons#(X1,X2) -> c_6() from#(X) -> c_8() length#(n__nil()) -> c_10() nil#() -> c_12() s#(X) -> c_13() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/2,c_3/2,c_4/1,c_5/2,c_6/0,c_7/1,c_8/0 ,c_9/3,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + 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}) = {1,2,4,6,7}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) 3: activate#(n__nil()) -> c_4(nil#()) 4: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 5: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 6: length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) 7: length1#(X) -> c_11(length#(activate(X)),activate#(X)) 8: activate#(X) -> c_1() 9: cons#(X1,X2) -> c_6() 10: from#(X) -> c_8() 11: length#(n__nil()) -> c_10() 12: nil#() -> c_12() 13: s#(X) -> c_13() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__nil()) -> c_4(nil#()) cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() length#(n__nil()) -> c_10() nil#() -> c_12() s#(X) -> c_13() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/2,c_3/2,c_4/1,c_5/2,c_6/0,c_7/1,c_8/0 ,c_9/3,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__nil()) -> c_4(nil#()):7 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_1 cons#(X1,X2) -> c_6():8 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_1 from#(X) -> c_7(cons#(X,n__from(n__s(X)))):9 -->_2 activate#(n__nil()) -> c_4(nil#()):7 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_8():10 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__nil()) -> c_4(nil#()):7 -->_1 s#(X) -> c_13():13 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 4:S:length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) -->_3 activate#(n__nil()) -> c_4(nil#()):7 -->_2 length1#(X) -> c_11(length#(activate(X)),activate#(X)):5 -->_1 s#(X) -> c_13():13 -->_3 activate#(X) -> c_1():6 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_3 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 5:S:length1#(X) -> c_11(length#(activate(X)),activate#(X)) -->_2 activate#(n__nil()) -> c_4(nil#()):7 -->_1 length#(n__nil()) -> c_10():11 -->_2 activate#(X) -> c_1():6 -->_1 length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)):4 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 6:W:activate#(X) -> c_1() 7:W:activate#(n__nil()) -> c_4(nil#()) -->_1 nil#() -> c_12():12 8:W:cons#(X1,X2) -> c_6() 9:W:from#(X) -> c_7(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_6():8 10:W:from#(X) -> c_8() 11:W:length#(n__nil()) -> c_10() 12:W:nil#() -> c_12() 13:W:s#(X) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: length#(n__nil()) -> c_10() 10: from#(X) -> c_8() 9: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 8: cons#(X1,X2) -> c_6() 6: activate#(X) -> c_1() 13: s#(X) -> c_13() 7: activate#(n__nil()) -> c_4(nil#()) 12: nil#() -> c_12() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/2,c_3/2,c_4/1,c_5/2,c_6/0,c_7/1,c_8/0 ,c_9/3,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 4:S:length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)) -->_2 length1#(X) -> c_11(length#(activate(X)),activate#(X)):5 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_3 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 5:S:length1#(X) -> c_11(length#(activate(X)),activate#(X)) -->_1 length#(n__cons(X,Y)) -> c_9(s#(length1(activate(Y))),length1#(activate(Y)),activate#(Y)):4 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) * Step 6: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) length(n__cons(X,Y)) -> s(length1(activate(Y))) length(n__nil()) -> 0() length1(X) -> length(activate(X)) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) * Step 7: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + 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: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} Problem (S) - Strict DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} ** Step 7.a:1: Failure MAYBE + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 7.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) -->_2 activate#(n__s(X)) -> c_5(activate#(X)):5 -->_2 activate#(n__from(X)) -> c_3(activate#(X)):4 -->_2 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):3 -->_1 length1#(X) -> c_11(length#(activate(X)),activate#(X)):2 2:S:length1#(X) -> c_11(length#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(activate#(X)):5 -->_2 activate#(n__from(X)) -> c_3(activate#(X)):4 -->_2 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):3 -->_1 length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)):1 3:W:activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):5 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):4 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):3 4:W:activate#(n__from(X)) -> c_3(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):5 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):4 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):3 5:W:activate#(n__s(X)) -> c_5(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):5 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):4 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: activate#(n__s(X)) -> c_5(activate#(X)) 4: activate#(n__from(X)) -> c_3(activate#(X)) 3: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) ** Step 7.b:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) length1#(X) -> c_11(length#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)) -->_1 length1#(X) -> c_11(length#(activate(X)),activate#(X)):2 2:S:length1#(X) -> c_11(length#(activate(X)),activate#(X)) -->_1 length#(n__cons(X,Y)) -> c_9(length1#(activate(Y)),activate#(Y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y))) length1#(X) -> c_11(length#(activate(X))) ** Step 7.b:3: Failure MAYBE + Considered Problem: - Strict DPs: length#(n__cons(X,Y)) -> c_9(length1#(activate(Y))) length1#(X) -> c_11(length#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__nil()) -> nil() activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) nil() -> n__nil() s(X) -> n__s(X) - Signature: {activate/1,cons/2,from/1,length/1,length1/1,nil/0,s/1,activate#/1,cons#/2,from#/1,length#/1,length1#/1 ,nil#/0,s#/1} / {0/0,n__cons/2,n__from/1,n__nil/0,n__s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,from#,length#,length1#,nil# ,s#} and constructors {0,n__cons,n__from,n__nil,n__s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE