WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: goal(xs,ys) -> overlap(xs,ys) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) -> False() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite],notEmpty,overlap ,overlap[Ite][True][Ite]} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() Weak DPs !EQ#(0(),0()) -> c_8() !EQ#(0(),S(y)) -> c_9() !EQ#(S(x),0()) -> c_10() !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_13() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: !EQ#(0(),0()) -> c_8() !EQ#(0(),S(y)) -> c_9() !EQ#(S(x),0()) -> c_10() !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_13() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) goal(xs,ys) -> overlap(xs,ys) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) -> False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() !EQ#(0(),0()) -> c_8() !EQ#(0(),S(y)) -> c_9() !EQ#(S(x),0()) -> c_10() !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_13() notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: !EQ#(0(),0()) -> c_8() !EQ#(0(),S(y)) -> c_9() !EQ#(S(x),0()) -> c_10() !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_13() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5} by application of Pre({4,5}) = {}. Here rules are labelled as follows: 1: goal#(xs,ys) -> c_1(overlap#(xs,ys)) 2: member#(x,Nil()) -> c_2() 3: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) 4: notEmpty#(Cons(x,xs)) -> c_4() 5: notEmpty#(Nil()) -> c_5() 6: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) 7: overlap#(Nil(),ys) -> c_7() 8: !EQ#(0(),0()) -> c_8() 9: !EQ#(0(),S(y)) -> c_9() 10: !EQ#(S(x),0()) -> c_10() 11: !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) 12: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) 13: member[Ite][True][Ite]#(True(),x,xs) -> c_13() 14: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) 15: overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: !EQ#(0(),0()) -> c_8() !EQ#(0(),S(y)) -> c_9() !EQ#(S(x),0()) -> c_10() !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_13() notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:goal#(xs,ys) -> c_1(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):4 -->_1 overlap#(Nil(),ys) -> c_7():5 2:S:member#(x,Nil()) -> c_2() 3:S:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):10 -->_2 !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)):9 -->_1 member[Ite][True][Ite]#(True(),x,xs) -> c_13():11 -->_2 !EQ#(S(x),0()) -> c_10():8 -->_2 !EQ#(0(),S(y)) -> c_9():7 -->_2 !EQ#(0(),0()) -> c_8():6 4:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):14 -->_1 overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15():15 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')):3 -->_2 member#(x,Nil()) -> c_2():2 5:S:overlap#(Nil(),ys) -> c_7() 6:W:!EQ#(0(),0()) -> c_8() 7:W:!EQ#(0(),S(y)) -> c_9() 8:W:!EQ#(S(x),0()) -> c_10() 9:W:!EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) -->_1 !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)):9 -->_1 !EQ#(S(x),0()) -> c_10():8 -->_1 !EQ#(0(),S(y)) -> c_9():7 -->_1 !EQ#(0(),0()) -> c_8():6 10:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')):3 -->_1 member#(x,Nil()) -> c_2():2 11:W:member[Ite][True][Ite]#(True(),x,xs) -> c_13() 12:W:notEmpty#(Cons(x,xs)) -> c_4() 13:W:notEmpty#(Nil()) -> c_5() 14:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():5 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):4 15:W:overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: notEmpty#(Nil()) -> c_5() 12: notEmpty#(Cons(x,xs)) -> c_4() 11: member[Ite][True][Ite]#(True(),x,xs) -> c_13() 9: !EQ#(S(x),S(y)) -> c_11(!EQ#(x,y)) 6: !EQ#(0(),0()) -> c_8() 7: !EQ#(0(),S(y)) -> c_9() 8: !EQ#(S(x),0()) -> c_10() 15: overlap[Ite][True][Ite]#(True(),xs,ys) -> c_15() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:goal#(xs,ys) -> c_1(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):4 -->_1 overlap#(Nil(),ys) -> c_7():5 2:S:member#(x,Nil()) -> c_2() 3:S:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):10 4:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):14 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')):3 -->_2 member#(x,Nil()) -> c_2():2 5:S:overlap#(Nil(),ys) -> c_7() 10:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs)),!EQ#(x,x')):3 -->_1 member#(x,Nil()) -> c_2():2 14:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():5 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) * Step 6: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(overlap#(xs,ys)) member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:goal#(xs,ys) -> c_1(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):4 -->_1 overlap#(Nil(),ys) -> c_7():5 2:S:member#(x,Nil()) -> c_2() 3:S:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):6 4:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):7 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 -->_2 member#(x,Nil()) -> c_2():2 5:S:overlap#(Nil(),ys) -> c_7() 6:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 -->_1 member#(x,Nil()) -> c_2():2 7:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():5 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):4 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). [(1,goal#(xs,ys) -> c_1(overlap#(xs,ys)))] * Step 7: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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: member#(x,Nil()) -> c_2() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1 ,c_12/1,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} Problem (S) - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member#(x,Nil()) -> c_2() member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1 ,c_12/1,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} ** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: member#(x,Nil()) -> c_2() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 2:S:member#(x,Nil()) -> c_2() 3:W:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):6 4:W:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_2 member#(x,Nil()) -> c_2():2 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):7 5:W:overlap#(Nil(),ys) -> c_7() 6:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x,Nil()) -> c_2():2 -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 7:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):4 -->_1 overlap#(Nil(),ys) -> c_7():5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: overlap#(Nil(),ys) -> c_7() ** Step 7.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: member#(x,Nil()) -> c_2() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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: 2: member#(x,Nil()) -> c_2() The strictly oriented rules are moved into the weak component. *** Step 7.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: member#(x,Nil()) -> c_2() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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_3) = {1}, uargs(c_6) = {1,2}, uargs(c_12) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty#,overlap#,overlap[Ite][True][Ite]#} TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [1] p(S) = [0] p(True) = [1] p(goal) = [1] x2 + [8] p(member) = [0] p(member[Ite][True][Ite]) = [6] x1 + [8] p(notEmpty) = [1] p(overlap) = [1] x2 + [2] p(overlap[Ite][True][Ite]) = [2] x1 + [8] x2 + [1] p(!EQ#) = [1] x1 + [2] p(goal#) = [2] x1 + [1] x2 + [0] p(member#) = [1] p(member[Ite][True][Ite]#) = [1] p(notEmpty#) = [1] x1 + [0] p(overlap#) = [12] x1 + [2] x2 + [11] p(overlap[Ite][True][Ite]#) = [12] x2 + [2] x3 + [9] p(c_1) = [2] x1 + [8] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [2] x2 + [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [4] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [1] x1 + [4] p(c_15) = [0] Following rules are strictly oriented: member#(x,Nil()) = [1] > [0] = c_2() Following rules are (at-least) weakly oriented: member#(x',Cons(x,xs)) = [1] >= [1] = c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = [1] >= [1] = c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) = [12] xs + [2] ys + [23] >= [12] xs + [2] ys + [23] = c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [12] xs + [2] ys + [21] >= [12] xs + [2] ys + [15] = c_14(overlap#(xs,ys)) *** Step 7.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: member#(x,Nil()) -> c_2() member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:member#(x,Nil()) -> c_2() 2:W:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):3 3:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):2 -->_1 member#(x,Nil()) -> c_2():1 4:W:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):5 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):2 -->_2 member#(x,Nil()) -> c_2():1 5:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) 5: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) 2: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) 3: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) 1: member#(x,Nil()) -> c_2() *** Step 7.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member#(x,Nil()) -> c_2() member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):5 2:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):6 -->_2 member#(x,Nil()) -> c_2():4 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 3:S:overlap#(Nil(),ys) -> c_7() 4:W:member#(x,Nil()) -> c_2() 5:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x,Nil()) -> c_2():4 -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 6:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():3 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: member#(x,Nil()) -> c_2() ** Step 7.b:2: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1 ,c_12/1,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} Problem (S) - Strict DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1 ,c_12/1,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} *** Step 7.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):5 2:W:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):6 3:W:overlap#(Nil(),ys) -> c_7() 5:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 6:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):2 -->_1 overlap#(Nil(),ys) -> c_7():3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: overlap#(Nil(),ys) -> c_7() *** Step 7.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) Consider the set of all dependency pairs 1: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) 2: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) 5: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) 6: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.b:2.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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_3) = {1}, uargs(c_6) = {1,2}, uargs(c_12) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {!EQ,!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty#,overlap#,overlap[Ite][True][Ite]#} TcT has computed the following interpretation: p(!EQ) = x1^2 + x2 p(0) = 1 p(Cons) = 2 + x2 p(False) = 2 p(Nil) = 0 p(S) = 2 + x1 p(True) = 2 p(goal) = x1^2 p(member) = x1 + 2*x1*x2 p(member[Ite][True][Ite]) = x1*x2 + x1*x3 + 2*x2*x3 + 2*x2^2 p(notEmpty) = 1 p(overlap) = 1 + x1 + x1*x2 + 2*x2^2 p(overlap[Ite][True][Ite]) = 2 + x1*x2 + 2*x1*x3 + x1^2 + x2*x3 + 2*x2^2 + x3^2 p(!EQ#) = 0 p(goal#) = x1*x2 p(member#) = 1 + 2*x2 p(member[Ite][True][Ite]#) = 2*x3 p(notEmpty#) = 1 p(overlap#) = 2 + x1 + x1*x2 + 3*x2 p(overlap[Ite][True][Ite]#) = x2 + x2*x3 + x3 p(c_1) = x1 p(c_2) = 2 p(c_3) = x1 p(c_4) = 1 p(c_5) = 0 p(c_6) = x1 + x2 p(c_7) = 2 p(c_8) = 1 p(c_9) = 1 p(c_10) = 0 p(c_11) = 1 + x1 p(c_12) = x1 p(c_13) = 2 p(c_14) = x1 p(c_15) = 0 Following rules are strictly oriented: member#(x',Cons(x,xs)) = 5 + 2*xs > 4 + 2*xs = c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) Following rules are (at-least) weakly oriented: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = 4 + 2*xs >= 1 + 2*xs = c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) = 4 + xs + xs*ys + 5*ys >= 3 + xs + xs*ys + 5*ys = c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = 2 + xs + xs*ys + 3*ys >= 2 + xs + xs*ys + 3*ys = c_14(overlap#(xs,ys)) !EQ(0(),0()) = 2 >= 2 = True() !EQ(0(),S(y)) = 3 + y >= 2 = False() !EQ(S(x),0()) = 5 + 4*x + x^2 >= 2 = False() !EQ(S(x),S(y)) = 6 + 4*x + x^2 + y >= x^2 + y = !EQ(x,y) **** Step 7.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):2 2:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 3:W:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):4 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):1 4:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) ,member#(x,ys)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) 4: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) 1: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) 2: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) **** Step 7.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):5 -->_2 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 2:S:overlap#(Nil(),ys) -> c_7() 3:W:member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) -->_1 member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)):4 4:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) -->_1 member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))):3 5:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():2 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: member#(x',Cons(x,xs)) -> c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) 4: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) *** Step 7.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):5 2:S:overlap#(Nil(),ys) -> c_7() 5:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():2 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys),member#(x,ys)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) *** Step 7.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) 2: overlap#(Nil(),ys) -> c_7() Consider the set of all dependency pairs 1: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) 2: overlap#(Nil(),ys) -> c_7() 3: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1,2} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) overlap#(Nil(),ys) -> c_7() - Weak DPs: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + 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}, uargs(c_14) = {1} Following symbols are considered usable: {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty#,overlap#,overlap[Ite][True][Ite]#} TcT has computed the following interpretation: p(!EQ) = [4] x1 + [6] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [2] p(False) = [2] p(Nil) = [0] p(S) = [6] p(True) = [4] p(goal) = [8] x2 + [8] p(member) = [1] x1 + [0] p(member[Ite][True][Ite]) = [3] x1 + [4] x3 + [2] p(notEmpty) = [2] p(overlap) = [1] p(overlap[Ite][True][Ite]) = [2] x2 + [0] p(!EQ#) = [2] x2 + [0] p(goal#) = [1] x1 + [8] p(member#) = [8] x2 + [1] p(member[Ite][True][Ite]#) = [8] x1 + [0] p(notEmpty#) = [1] x1 + [0] p(overlap#) = [8] x1 + [8] p(overlap[Ite][True][Ite]#) = [8] x2 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [0] p(c_5) = [8] p(c_6) = [1] x1 + [0] p(c_7) = [6] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [2] p(c_12) = [1] p(c_13) = [2] p(c_14) = [1] x1 + [8] p(c_15) = [0] Following rules are strictly oriented: overlap#(Cons(x,xs),ys) = [8] x + [8] xs + [24] > [8] x + [8] xs + [16] = c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) overlap#(Nil(),ys) = [8] > [6] = c_7() Following rules are (at-least) weakly oriented: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [8] x + [8] xs + [16] >= [8] xs + [16] = c_14(overlap#(xs,ys)) **** Step 7.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) overlap#(Nil(),ys) -> c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) -->_1 overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)):3 2:W:overlap#(Nil(),ys) -> c_7() 3:W:overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) -->_1 overlap#(Nil(),ys) -> c_7():2 -->_1 overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: overlap#(Cons(x,xs),ys) -> c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) 3: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> c_14(overlap#(xs,ys)) 2: overlap#(Nil(),ys) -> c_7() **** Step 7.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs) member[Ite][True][Ite](True(),x,xs) -> True() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3,!EQ#/2 ,goal#/2,member#/2,member[Ite][True][Ite]#/3,notEmpty#/1,overlap#/2,overlap[Ite][True][Ite]#/3} / {0/0 ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]#,notEmpty# ,overlap#,overlap[Ite][True][Ite]#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))