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: 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) 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: 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 3: 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) 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: 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 4: 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) 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: 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 5: 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))) 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) 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_3/1,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: 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() 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))) 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)) * 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: DecomposeDG 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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component 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)) and a lower component 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#(Nil(),ys) -> c_7() Further, following extension rules are added to the lower component. overlap#(Cons(x,xs),ys) -> member#(x,ys) overlap#(Cons(x,xs),ys) -> overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> overlap#(xs,ys) ** Step 7.a:1: 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[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)):2 2:S: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)):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.a:2: WeightGap 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[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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]#) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [1] p(0) = [4] p(Cons) = [1] x2 + [0] p(False) = [1] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(goal) = [0] p(member) = [1] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [0] p(overlap) = [0] p(overlap[Ite][True][Ite]) = [0] p(!EQ#) = [0] p(goal#) = [0] p(member#) = [0] p(member[Ite][True][Ite]#) = [0] p(notEmpty#) = [2] x1 + [1] p(overlap#) = [0] p(overlap[Ite][True][Ite]#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [4] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] Following rules are strictly oriented: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [1] > [0] = c_14(overlap#(xs,ys)) Following rules are (at-least) weakly oriented: overlap#(Cons(x,xs),ys) = [0] >= [1] = c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) !EQ(0(),0()) = [1] >= [0] = True() !EQ(0(),S(y)) = [1] >= [1] = False() !EQ(S(x),0()) = [1] >= [1] = False() !EQ(S(x),S(y)) = [1] >= [1] = !EQ(x,y) member(x,Nil()) = [1] >= [1] = False() member(x',Cons(x,xs)) = [1] >= [1] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [1] >= [1] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:3: WeightGap 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)) - 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]#) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [1] x2 + [4] p(False) = [0] p(Nil) = [1] p(S) = [1] x1 + [4] p(True) = [0] p(goal) = [1] x2 + [0] p(member) = [0] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [1] p(overlap) = [1] x1 + [1] p(overlap[Ite][True][Ite]) = [2] x1 + [1] x2 + [1] p(!EQ#) = [2] x1 + [1] p(goal#) = [1] x1 + [1] p(member#) = [4] x1 + [1] x2 + [0] p(member[Ite][True][Ite]#) = [1] x3 + [1] p(notEmpty#) = [2] x1 + [1] p(overlap#) = [3] x1 + [1] x2 + [3] p(overlap[Ite][True][Ite]#) = [1] x1 + [3] x2 + [1] x3 + [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [4] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [4] p(c_12) = [4] p(c_13) = [1] p(c_14) = [1] x1 + [0] p(c_15) = [0] Following rules are strictly oriented: overlap#(Cons(x,xs),ys) = [3] xs + [1] ys + [15] > [3] xs + [1] ys + [13] = c_6(overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys)) Following rules are (at-least) weakly oriented: overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [3] xs + [1] ys + [13] >= [3] xs + [1] ys + [3] = c_14(overlap#(xs,ys)) !EQ(0(),0()) = [0] >= [0] = True() !EQ(0(),S(y)) = [0] >= [0] = False() !EQ(S(x),0()) = [0] >= [0] = False() !EQ(S(x),S(y)) = [0] >= [0] = !EQ(x,y) member(x,Nil()) = [0] >= [0] = False() member(x',Cons(x,xs)) = [0] >= [0] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [0] >= [0] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:4: EmptyProcessor 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[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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: WeightGap WORST_CASE(?,O(n^1)) + 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#(Nil(),ys) -> c_7() - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> member#(x,ys) overlap#(Cons(x,xs),ys) -> overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(member[Ite][True][Ite]#) = {1}, uargs(overlap[Ite][True][Ite]#) = {1}, uargs(c_3) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(goal) = [0] p(member) = [0] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [0] p(overlap) = [0] p(overlap[Ite][True][Ite]) = [1] p(!EQ#) = [0] p(goal#) = [0] p(member#) = [3] p(member[Ite][True][Ite]#) = [1] x1 + [3] p(notEmpty#) = [0] p(overlap#) = [3] p(overlap[Ite][True][Ite]#) = [1] x1 + [3] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] Following rules are strictly oriented: member#(x,Nil()) = [3] > [0] = c_2() overlap#(Nil(),ys) = [3] > [0] = c_7() Following rules are (at-least) weakly oriented: member#(x',Cons(x,xs)) = [3] >= [3] = c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = [3] >= [3] = c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) = [3] >= [3] = member#(x,ys) overlap#(Cons(x,xs),ys) = [3] >= [3] = overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [3] >= [3] = overlap#(xs,ys) !EQ(0(),0()) = [0] >= [0] = True() !EQ(0(),S(y)) = [0] >= [0] = False() !EQ(S(x),0()) = [0] >= [0] = False() !EQ(S(x),S(y)) = [0] >= [0] = !EQ(x,y) member(x,Nil()) = [0] >= [0] = False() member(x',Cons(x,xs)) = [0] >= [0] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [0] >= [0] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.b:2: WeightGap WORST_CASE(?,O(n^1)) + 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#(x,Nil()) -> c_2() member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) -> member#(x,ys) overlap#(Cons(x,xs),ys) -> 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) -> 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(member[Ite][True][Ite]#) = {1}, uargs(overlap[Ite][True][Ite]#) = {1}, uargs(c_3) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all 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) = [1] x1 + [4] p(True) = [0] p(goal) = [2] x1 + [4] x2 + [0] p(member) = [0] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [1] x1 + [0] p(overlap) = [1] x1 + [4] p(overlap[Ite][True][Ite]) = [2] x2 + [1] x3 + [0] p(!EQ#) = [0] p(goal#) = [4] x1 + [1] p(member#) = [1] x2 + [1] p(member[Ite][True][Ite]#) = [1] x1 + [1] x3 + [0] p(notEmpty#) = [4] x1 + [2] p(overlap#) = [1] x2 + [2] p(overlap[Ite][True][Ite]#) = [1] x1 + [1] x3 + [2] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [4] p(c_6) = [1] x1 + [1] x2 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [2] p(c_14) = [4] p(c_15) = [4] Following rules are strictly oriented: member#(x',Cons(x,xs)) = [1] xs + [2] > [1] xs + [1] = c_3(member[Ite][True][Ite]#(!EQ(x,x'),x',Cons(x,xs))) Following rules are (at-least) weakly oriented: member#(x,Nil()) = [2] >= [0] = c_2() member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = [1] xs + [1] >= [1] xs + [1] = c_12(member#(x',xs)) overlap#(Cons(x,xs),ys) = [1] ys + [2] >= [1] ys + [1] = member#(x,ys) overlap#(Cons(x,xs),ys) = [1] ys + [2] >= [1] ys + [2] = overlap[Ite][True][Ite]#(member(x,ys),Cons(x,xs),ys) overlap#(Nil(),ys) = [1] ys + [2] >= [0] = c_7() overlap[Ite][True][Ite]#(False(),Cons(x,xs),ys) = [1] ys + [2] >= [1] ys + [2] = overlap#(xs,ys) !EQ(0(),0()) = [0] >= [0] = True() !EQ(0(),S(y)) = [0] >= [0] = False() !EQ(S(x),0()) = [0] >= [0] = False() !EQ(S(x),S(y)) = [0] >= [0] = !EQ(x,y) member(x,Nil()) = [0] >= [0] = False() member(x',Cons(x,xs)) = [0] >= [0] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [0] >= [0] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.b:3: EmptyProcessor 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) -> member#(x,ys) overlap#(Cons(x,xs),ys) -> 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) -> 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))