*** 1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 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 DP Rules: Weak TRS Rules: !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 basic terms: {!EQ,goal,member,member[Ite][True][Ite],notEmpty}/{0,Cons,False,Nil,S,True} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: 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: {} TcT has computed the following interpretation: p(!EQ) = [3] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [3] p(Nil) = [13] p(S) = [1] x1 + [0] p(True) = [0] p(goal) = [1] x2 + [0] p(member) = [1] x2 + [3] p(member[Ite][True][Ite]) = [1] x1 + [1] x3 + [0] p(notEmpty) = [5] Following rules are strictly oriented: member(x,Nil()) = [16] > [3] = False() notEmpty(Cons(x,xs)) = [5] > [0] = True() notEmpty(Nil()) = [5] > [3] = False() Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [3] >= [0] = True() !EQ(0(),S(y)) = [3] >= [3] = False() !EQ(S(x),0()) = [3] >= [3] = False() !EQ(S(x),S(y)) = [3] >= [3] = !EQ(x,y) goal(x,xs) = [1] xs + [0] >= [1] xs + [3] = member(x,xs) member(x',Cons(x,xs)) = [1] x + [1] xs + [3] >= [1] x + [1] xs + [3] = member[Ite][True][Ite](!EQ(x',x) ,x' ,Cons(x,xs)) member[Ite][True][Ite](False() = [1] x + [1] xs + [3] ,x' ,Cons(x,xs)) >= [1] xs + [3] = member(x',xs) member[Ite][True][Ite](True() = [1] xs + [0] ,x ,xs) >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: goal(x,xs) -> member(x,xs) member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) Weak DP Rules: Weak TRS 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[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 basic terms: {!EQ,goal,member,member[Ite][True][Ite],notEmpty}/{0,Cons,False,Nil,S,True} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: 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: {} TcT has computed the following interpretation: p(!EQ) = [6] p(0) = [2] p(Cons) = [1] x2 + [8] p(False) = [0] p(Nil) = [0] p(S) = [0] p(True) = [6] p(goal) = [4] x1 + [2] x2 + [4] p(member) = [4] x1 + [2] x2 + [2] p(member[Ite][True][Ite]) = [1] x1 + [4] x2 + [2] x3 + [5] p(notEmpty) = [2] x1 + [1] Following rules are strictly oriented: goal(x,xs) = [4] x + [2] xs + [4] > [4] x + [2] xs + [2] = member(x,xs) Following rules are (at-least) weakly oriented: !EQ(0(),0()) = [6] >= [6] = True() !EQ(0(),S(y)) = [6] >= [0] = False() !EQ(S(x),0()) = [6] >= [0] = False() !EQ(S(x),S(y)) = [6] >= [6] = !EQ(x,y) member(x,Nil()) = [4] x + [2] >= [0] = False() member(x',Cons(x,xs)) = [4] x' + [2] xs + [18] >= [4] x' + [2] xs + [27] = member[Ite][True][Ite](!EQ(x',x) ,x' ,Cons(x,xs)) member[Ite][True][Ite](False() = [4] x' + [2] xs + [21] ,x' ,Cons(x,xs)) >= [4] x' + [2] xs + [2] = member(x',xs) member[Ite][True][Ite](True() = [4] x + [2] xs + [11] ,x ,xs) >= [6] = True() notEmpty(Cons(x,xs)) = [2] xs + [17] >= [6] = True() notEmpty(Nil()) = [1] >= [0] = False() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs)) Weak DP Rules: Weak TRS 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) -> 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 basic terms: {!EQ,goal,member,member[Ite][True][Ite],notEmpty}/{0,Cons,False,Nil,S,True} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy} Proof: 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) = [1] x2 + [5] p(False) = [0] p(Nil) = [4] p(S) = [0] p(True) = [0] p(goal) = [4] x2 + [8] p(member) = [4] x2 + [2] p(member[Ite][True][Ite]) = [2] x1 + [4] x3 + [0] p(notEmpty) = [4] x1 + [1] Following rules are strictly oriented: member(x',Cons(x,xs)) = [4] xs + [22] > [4] xs + [20] = 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) = [4] xs + [8] >= [4] xs + [2] = member(x,xs) member(x,Nil()) = [18] >= [0] = False() member[Ite][True][Ite](False() = [4] xs + [20] ,x' ,Cons(x,xs)) >= [4] xs + [2] = member(x',xs) member[Ite][True][Ite](True() = [4] xs + [0] ,x ,xs) >= [0] = True() notEmpty(Cons(x,xs)) = [4] xs + [21] >= [0] = True() notEmpty(Nil()) = [17] >= [0] = False() *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS 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) -> 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 basic terms: {!EQ,goal,member,member[Ite][True][Ite],notEmpty}/{0,Cons,False,Nil,S,True} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).