WORST_CASE(?,O(n^2)) * Step 1: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [4] p(0) = [0] p(Cons) = [2] p(False) = [0] p(Nil) = [1] p(S) = [0] p(True) = [0] p(goal) = [5] x1 + [2] x2 + [1] p(member) = [1] p(member[Ite][True][Ite]) = [1] x1 + [1] x3 + [4] p(notEmpty) = [2] x1 + [0] p(overlap) = [2] x2 + [0] p(overlap[Ite][True][Ite]) = [1] x1 + [2] x3 + [0] Following rules are strictly oriented: goal(xs,ys) = [5] xs + [2] ys + [1] > [2] ys + [0] = overlap(xs,ys) member(x,Nil()) = [1] > [0] = False() notEmpty(Cons(x,xs)) = [4] > [0] = True() notEmpty(Nil()) = [2] > [0] = False() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [4] >= [0] = True() !EQ(0(),S(y)) = [4] >= [0] = False() !EQ(S(x),0()) = [4] >= [0] = False() !EQ(S(x),S(y)) = [4] >= [4] = !EQ(x,y) member(x',Cons(x,xs)) = [1] >= [10] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [6] >= [1] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [1] xs + [4] >= [0] = True() overlap(Cons(x,xs),ys) = [2] ys + [0] >= [2] ys + [1] = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap(Nil(),ys) = [2] ys + [0] >= [0] = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [2] ys + [0] >= [2] ys + [0] = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = [2] ys + [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap 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)) 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(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[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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [6] p(0) = [0] p(Cons) = [1] x2 + [4] p(False) = [1] p(Nil) = [7] p(S) = [1] p(True) = [0] p(goal) = [4] x1 + [4] x2 + [4] p(member) = [1] p(member[Ite][True][Ite]) = [1] x1 + [0] p(notEmpty) = [1] x1 + [1] p(overlap) = [2] p(overlap[Ite][True][Ite]) = [1] x1 + [4] Following rules are strictly oriented: overlap(Nil(),ys) = [2] > [1] = False() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [6] >= [0] = True() !EQ(0(),S(y)) = [6] >= [1] = False() !EQ(S(x),0()) = [6] >= [1] = False() !EQ(S(x),S(y)) = [6] >= [6] = !EQ(x,y) goal(xs,ys) = [4] xs + [4] ys + [4] >= [2] = overlap(xs,ys) member(x,Nil()) = [1] >= [1] = False() member(x',Cons(x,xs)) = [1] >= [6] = 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() notEmpty(Cons(x,xs)) = [1] xs + [5] >= [0] = True() notEmpty(Nil()) = [8] >= [1] = False() overlap(Cons(x,xs),ys) = [2] >= [5] = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [5] >= [2] = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = [4] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap 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)) 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(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(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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [1] p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [1] p(False) = [0] p(Nil) = [1] p(S) = [2] p(True) = [0] p(goal) = [1] x1 + [1] x2 + [5] p(member) = [0] p(member[Ite][True][Ite]) = [1] x1 + [2] p(notEmpty) = [4] x1 + [2] p(overlap) = [1] x1 + [2] p(overlap[Ite][True][Ite]) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: overlap(Cons(x,xs),ys) = [1] x + [1] xs + [3] > [1] x + [1] xs + [2] = overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys) Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [1] >= [0] = True() !EQ(0(),S(y)) = [1] >= [0] = False() !EQ(S(x),0()) = [1] >= [0] = False() !EQ(S(x),S(y)) = [1] >= [1] = !EQ(x,y) goal(xs,ys) = [1] xs + [1] ys + [5] >= [1] xs + [2] = overlap(xs,ys) member(x,Nil()) = [0] >= [0] = False() member(x',Cons(x,xs)) = [0] >= [3] = member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [2] >= [0] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [2] >= [0] = True() notEmpty(Cons(x,xs)) = [4] x + [4] xs + [6] >= [0] = True() notEmpty(Nil()) = [6] >= [0] = False() overlap(Nil(),ys) = [3] >= [0] = False() overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [1] x + [1] xs + [2] >= [1] xs + [2] = overlap(xs,ys) overlap[Ite][True][Ite](True(),xs,ys) = [1] xs + [1] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: 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 5: 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))