WORST_CASE(?,O(n^2)) * Step 1: Sum 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: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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) = x1 p(False) = 0 p(Nil) = 0 p(S) = 0 p(True) = 0 p(goal) = 2 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 0 p(overlap) = 0 p(overlap[Ite][True][Ite]) = 2*x1 Following rules are strictly oriented: goal(xs,ys) = 2 > 0 = overlap(xs,ys) Following rules are (at-least) weakly oriented: !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() notEmpty(Cons(x,xs)) = 0 >= 0 = True() notEmpty(Nil()) = 0 >= 0 = False() overlap(Cons(x,xs),ys) = 0 >= 0 = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) = 0 >= 0 = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 0 >= 0 = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 0 >= 0 = True() * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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) goal(xs,ys) -> overlap(xs,ys) 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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 + x1 p(False) = 0 p(Nil) = 0 p(S) = 4 p(True) = 0 p(goal) = 0 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 4 p(overlap) = 0 p(overlap[Ite][True][Ite]) = x1 Following rules are strictly oriented: notEmpty(Cons(x,xs)) = 4 > 0 = True() notEmpty(Nil()) = 4 > 0 = False() Following rules are (at-least) weakly oriented: !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) goal(xs,ys) = 0 >= 0 = overlap(xs,ys) 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() overlap(Cons(x,xs),ys) = 0 >= 0 = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) = 0 >= 0 = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 0 >= 0 = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 0 >= 0 = True() * Step 4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) 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) goal(xs,ys) -> overlap(xs,ys) 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[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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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) = 1 p(Cons) = 1 p(False) = 0 p(Nil) = 2 p(S) = 1 p(True) = 0 p(goal) = 7 + x2 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 3 p(overlap) = 2 p(overlap[Ite][True][Ite]) = x1 + 2*x2 Following rules are strictly oriented: overlap(Nil(),ys) = 2 > 0 = False() Following rules are (at-least) weakly oriented: !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) goal(xs,ys) = 7 + ys >= 2 = overlap(xs,ys) 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() notEmpty(Cons(x,xs)) = 3 >= 0 = True() notEmpty(Nil()) = 3 >= 0 = False() overlap(Cons(x,xs),ys) = 2 >= 2 = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 2 >= 2 = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 2*xs >= 0 = True() * Step 5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,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[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(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} / {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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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) = 1 p(Cons) = 4 + x2 p(False) = 0 p(Nil) = 0 p(S) = 1 + x1 p(True) = 0 p(goal) = 6 + 6*x1 + 2*x2 p(member) = 0 p(member[Ite][True][Ite]) = x1 p(notEmpty) = 2 + x1 p(overlap) = 2 + 2*x1 + x2 p(overlap[Ite][True][Ite]) = 4*x1 + 2*x2 + x3 Following rules are strictly oriented: overlap(Cons(x,xs),ys) = 10 + 2*xs + ys > 8 + 2*xs + ys = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) Following rules are (at-least) weakly oriented: !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) goal(xs,ys) = 6 + 6*xs + 2*ys >= 2 + 2*xs + ys = overlap(xs,ys) 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() notEmpty(Cons(x,xs)) = 6 + xs >= 0 = True() notEmpty(Nil()) = 2 >= 0 = False() overlap(Nil(),ys) = 2 + ys >= 0 = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 8 + 2*xs + ys >= 2 + 2*xs + ys = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 2*xs + ys >= 0 = True() * Step 6: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: member(x,Nil()) -> False() member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(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(xs,ys) -> overlap(xs,ys) 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} / {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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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) = 1 p(Cons) = 2 + x1 + x2 p(False) = 0 p(Nil) = 0 p(S) = 1 p(True) = 0 p(goal) = 3 + 7*x1 + 4*x2 p(member) = 1 p(member[Ite][True][Ite]) = 1 + x1 p(notEmpty) = 2 + 3*x1 p(overlap) = 1 + 7*x1 + 2*x2 p(overlap[Ite][True][Ite]) = x1 + 7*x2 + 2*x3 Following rules are strictly oriented: member(x,Nil()) = 1 > 0 = False() Following rules are (at-least) weakly oriented: !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) goal(xs,ys) = 3 + 7*xs + 4*ys >= 1 + 7*xs + 2*ys = overlap(xs,ys) 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) = 1 >= 0 = True() notEmpty(Cons(x,xs)) = 8 + 3*x + 3*xs >= 0 = True() notEmpty(Nil()) = 2 >= 0 = False() overlap(Cons(x,xs),ys) = 15 + 7*x + 7*xs + 2*ys >= 15 + 7*x + 7*xs + 2*ys = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) = 1 + 2*ys >= 0 = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 14 + 7*x + 7*xs + 2*ys >= 1 + 7*xs + 2*ys = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 7*xs + 2*ys >= 0 = True() * Step 7: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(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(xs,ys) -> overlap(xs,ys) member(x,Nil()) -> False() 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} / {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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1}, uargs(overlap[Ite][True][Ite]) = {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 + x1 + x2 p(False) = 0 p(Nil) = 0 p(S) = x1 p(True) = 0 p(goal) = 5 + 7*x1 + 6*x1*x2 + 4*x1^2 + 6*x2 p(member) = 1 + x1 + 2*x2 p(member[Ite][True][Ite]) = 4*x1 + x2 + 2*x3 p(notEmpty) = 3 + 4*x1^2 p(overlap) = 5 + 2*x1 + 6*x1*x2 + 2*x1^2 + 5*x2 p(overlap[Ite][True][Ite]) = 5 + 2*x1 + 6*x2*x3 + 2*x2^2 Following rules are strictly oriented: member(x',Cons(x,xs)) = 3 + 2*x + x' + 2*xs > 2 + 2*x + x' + 2*xs = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) Following rules are (at-least) weakly oriented: !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) goal(xs,ys) = 5 + 7*xs + 6*xs*ys + 4*xs^2 + 6*ys >= 5 + 2*xs + 6*xs*ys + 2*xs^2 + 5*ys = overlap(xs,ys) member(x,Nil()) = 1 + x >= 0 = False() member[Ite][True][Ite](False(),x',Cons(x,xs)) = 2 + 2*x + x' + 2*xs >= 1 + x' + 2*xs = member(x',xs) member[Ite][True][Ite](True(),x,xs) = x + 2*xs >= 0 = True() notEmpty(Cons(x,xs)) = 7 + 8*x + 8*x*xs + 4*x^2 + 8*xs + 4*xs^2 >= 0 = True() notEmpty(Nil()) = 3 >= 0 = False() overlap(Cons(x,xs),ys) = 9 + 6*x + 4*x*xs + 6*x*ys + 2*x^2 + 6*xs + 6*xs*ys + 2*xs^2 + 11*ys >= 9 + 6*x + 4*x*xs + 6*x*ys + 2*x^2 + 4*xs + 6*xs*ys + 2*xs^2 + 10*ys = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) = 5 + 5*ys >= 0 = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 7 + 4*x + 4*x*xs + 6*x*ys + 2*x^2 + 4*xs + 6*xs*ys + 2*xs^2 + 6*ys >= 5 + 2*xs + 6*xs*ys + 2*xs^2 + 5*ys = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = 5 + 6*xs*ys + 2*xs^2 >= 0 = True() * Step 8: 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) 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} / {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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))