WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(x,xs) -> member(x,xs) 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() - 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() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {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} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs goal#(x,xs) -> c_1(member#(x,xs)) 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() Weak DPs !EQ#(0(),0()) -> c_6() !EQ#(0(),S(y)) -> c_7() !EQ#(S(x),0()) -> c_8() !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_11() and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(x,xs) -> c_1(member#(x,xs)) 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() - Weak DPs: !EQ#(0(),0()) -> c_6() !EQ#(0(),S(y)) -> c_7() !EQ#(S(x),0()) -> c_8() !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_11() - 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(x,xs) -> member(x,xs) 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() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} 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#(x,xs) -> c_1(member#(x,xs)) 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: !EQ#(0(),0()) -> c_6() 7: !EQ#(0(),S(y)) -> c_7() 8: !EQ#(S(x),0()) -> c_8() 9: !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) 10: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) 11: member[Ite][True][Ite]#(True(),x,xs) -> c_11() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(x,xs) -> c_1(member#(x,xs)) 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)) - Weak DPs: !EQ#(0(),0()) -> c_6() !EQ#(0(),S(y)) -> c_7() !EQ#(S(x),0()) -> c_8() !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) member[Ite][True][Ite]#(True(),x,xs) -> c_11() notEmpty#(Cons(x,xs)) -> c_4() notEmpty#(Nil()) -> c_5() - 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(x,xs) -> member(x,xs) 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() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:goal#(x,xs) -> c_1(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 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_10(member#(x',xs)):8 -->_2 !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)):7 -->_1 member[Ite][True][Ite]#(True(),x,xs) -> c_11():9 -->_2 !EQ#(S(x),0()) -> c_8():6 -->_2 !EQ#(0(),S(y)) -> c_7():5 -->_2 !EQ#(0(),0()) -> c_6():4 4:W:!EQ#(0(),0()) -> c_6() 5:W:!EQ#(0(),S(y)) -> c_7() 6:W:!EQ#(S(x),0()) -> c_8() 7:W:!EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) -->_1 !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)):7 -->_1 !EQ#(S(x),0()) -> c_8():6 -->_1 !EQ#(0(),S(y)) -> c_7():5 -->_1 !EQ#(0(),0()) -> c_6():4 8:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(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 9:W:member[Ite][True][Ite]#(True(),x,xs) -> c_11() 10:W:notEmpty#(Cons(x,xs)) -> c_4() 11:W:notEmpty#(Nil()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: notEmpty#(Nil()) -> c_5() 10: notEmpty#(Cons(x,xs)) -> c_4() 9: member[Ite][True][Ite]#(True(),x,xs) -> c_11() 7: !EQ#(S(x),S(y)) -> c_9(!EQ#(x,y)) 4: !EQ#(0(),0()) -> c_6() 5: !EQ#(0(),S(y)) -> c_7() 6: !EQ#(S(x),0()) -> c_8() * Step 4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(x,xs) -> c_1(member#(x,xs)) 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)) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) - 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(x,xs) -> member(x,xs) 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() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:goal#(x,xs) -> c_1(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 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_10(member#(x',xs)):8 8:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(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 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^1)) + Considered Problem: - Strict DPs: goal#(x,xs) -> c_1(member#(x,xs)) member#(x,Nil()) -> c_2() 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_10(member#(x',xs)) - 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(x,xs) -> member(x,xs) 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() - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} 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) goal#(x,xs) -> c_1(member#(x,xs)) 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_10(member#(x',xs)) * Step 6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(x,xs) -> c_1(member#(x,xs)) member#(x,Nil()) -> c_2() 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_10(member#(x',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:goal#(x,xs) -> c_1(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 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_10(member#(x',xs)):4 4:W:member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(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 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#(x,xs) -> c_1(member#(x,xs)))] * Step 7: 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))) - Weak DPs: member[Ite][True][Ite]#(False(),x',Cons(x,xs)) -> c_10(member#(x',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} 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(c_3) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [1] p(0) = [4] p(Cons) = [8] p(False) = [1] p(Nil) = [1] p(S) = [0] p(True) = [1] p(goal) = [8] x1 + [1] x2 + [1] p(member) = [4] x1 + [8] x2 + [0] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [8] x1 + [1] p(!EQ#) = [1] x2 + [1] p(goal#) = [1] x1 + [1] x2 + [1] p(member#) = [11] p(member[Ite][True][Ite]#) = [1] x1 + [12] p(notEmpty#) = [2] x1 + [2] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [4] p(c_5) = [1] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] Following rules are strictly oriented: member#(x,Nil()) = [11] > [1] = c_2() Following rules are (at-least) weakly oriented: member#(x',Cons(x,xs)) = [11] >= [13] = c_3(member[Ite][True][Ite]#(!EQ(x',x),x',Cons(x,xs))) member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = [13] >= [11] = c_10(member#(x',xs)) !EQ(0(),0()) = [1] >= [1] = 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) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: 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_10(member#(x',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} 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(c_3) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [4] p(0) = [0] p(Cons) = [1] x2 + [2] p(False) = [4] p(Nil) = [10] p(S) = [1] p(True) = [4] p(goal) = [8] x1 + [1] x2 + [0] p(member) = [1] x1 + [1] x2 + [2] p(member[Ite][True][Ite]) = [1] x1 + [2] p(notEmpty) = [1] x1 + [0] p(!EQ#) = [4] x1 + [1] p(goal#) = [8] x1 + [4] x2 + [1] p(member#) = [1] x2 + [6] p(member[Ite][True][Ite]#) = [1] x1 + [1] x3 + [1] p(notEmpty#) = [1] x1 + [1] p(c_1) = [2] x1 + [2] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [4] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] Following rules are strictly oriented: member#(x',Cons(x,xs)) = [1] xs + [8] > [1] xs + [7] = c_3(member[Ite][True][Ite]#(!EQ(x',x),x',Cons(x,xs))) Following rules are (at-least) weakly oriented: member#(x,Nil()) = [16] >= [0] = c_2() member[Ite][True][Ite]#(False(),x',Cons(x,xs)) = [1] xs + [7] >= [1] xs + [6] = c_10(member#(x',xs)) !EQ(0(),0()) = [4] >= [4] = True() !EQ(0(),S(y)) = [4] >= [4] = False() !EQ(S(x),0()) = [4] >= [4] = False() !EQ(S(x),S(y)) = [4] >= [4] = !EQ(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: 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_10(member#(x',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) - Signature: {!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,!EQ#/2,goal#/2,member#/2 ,member[Ite][True][Ite]#/3,notEmpty#/1} / {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_6/0,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,goal#,member#,member[Ite][True][Ite]# ,notEmpty#} 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^1))