WORST_CASE(?,O(n^1)) * Step 1: WeightGap 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: 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [10] p(0) = [0] p(Cons) = [1] x2 + [0] p(False) = [9] p(Nil) = [0] p(S) = [2] p(True) = [0] p(goal) = [10] x2 + [13] p(member) = [2] x2 + [9] p(member[Ite][True][Ite]) = [1] x1 + [2] x3 + [0] p(notEmpty) = [5] Following rules are strictly oriented: goal(x,xs) = [10] xs + [13] > [2] xs + [9] = member(x,xs) notEmpty(Cons(x,xs)) = [5] > [0] = True() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [10] >= [0] = True() !EQ(0(),S(y)) = [10] >= [9] = False() !EQ(S(x),0()) = [10] >= [9] = False() !EQ(S(x),S(y)) = [10] >= [10] = !EQ(x,y) member(x,Nil()) = [9] >= [9] = False() member(x',Cons(x,xs)) = [2] xs + [9] >= [2] xs + [10] = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [2] xs + [9] >= [2] xs + [9] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [2] xs + [0] >= [0] = True() notEmpty(Nil()) = [5] >= [9] = False() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + 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(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) goal(x,xs) -> member(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() - 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [0] p(False) = [0] p(Nil) = [0] p(S) = [0] p(True) = [0] p(goal) = [2] p(member) = [2] p(member[Ite][True][Ite]) = [4] x1 + [2] p(notEmpty) = [8] Following rules are strictly oriented: member(x,Nil()) = [2] > [0] = False() notEmpty(Nil()) = [8] > [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(x,xs) = [2] >= [2] = member(x,xs) member(x',Cons(x,xs)) = [2] >= [2] = member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) member[Ite][True][Ite](False(),x',Cons(x,xs)) = [2] >= [2] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [2] >= [0] = True() notEmpty(Cons(x,xs)) = [8] >= [0] = True() * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + 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(x,xs) -> member(x,xs) 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() - 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(member[Ite][True][Ite]) = {1} Following symbols are considered usable: {!EQ,goal,member,member[Ite][True][Ite],notEmpty} TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [8] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [2] p(S) = [1] x1 + [1] p(True) = [0] p(goal) = [8] x1 + [5] x2 + [14] p(member) = [3] x1 + [4] x2 + [14] p(member[Ite][True][Ite]) = [8] x1 + [3] x2 + [4] x3 + [12] p(notEmpty) = [6] x1 + [2] Following rules are strictly oriented: member(x',Cons(x,xs)) = [3] x' + [4] xs + [18] > [3] x' + [4] xs + [16] = 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(x,xs) = [8] x + [5] xs + [14] >= [3] x + [4] xs + [14] = member(x,xs) member(x,Nil()) = [3] x + [22] >= [0] = False() member[Ite][True][Ite](False(),x',Cons(x,xs)) = [3] x' + [4] xs + [16] >= [3] x' + [4] xs + [14] = member(x',xs) member[Ite][True][Ite](True(),x,xs) = [3] x + [4] xs + [12] >= [0] = True() notEmpty(Cons(x,xs)) = [6] xs + [8] >= [0] = True() notEmpty(Nil()) = [14] >= [0] = False() * Step 4: 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(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} / {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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))