WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X,nil()) -> X __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil(),X) -> X a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() i() -> n__i() isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. __(X,nil()) -> X __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil(),X) -> X isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() i() -> n__i() isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [0] p(a) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [7] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [3] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [5] p(n____) = [1] x1 + [1] x2 + [5] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [3] p(n__isPal) = [1] x1 + [3] p(n__nil) = [0] p(n__o) = [5] p(n__u) = [0] p(nil) = [0] p(o) = [0] p(tt) = [0] p(u) = [1] Following rules are strictly oriented: activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = __(X1,X2) activate(n__isNeList(X)) = [1] X + [3] > [1] X + [1] = isNeList(X) activate(n__isPal(X)) = [1] X + [3] > [1] X + [0] = isPal(X) activate(n__o()) = [5] > [0] = o() and(tt(),X) = [1] X + [7] > [1] X + [0] = activate(X) isList(V) = [1] V + [3] > [1] V + [1] = isNeList(activate(V)) isList(X) = [1] X + [3] > [1] X + [0] = n__isList(X) isList(n__nil()) = [3] > [0] = tt() isQid(n__a()) = [5] > [0] = tt() isQid(n__e()) = [5] > [0] = tt() isQid(n__i()) = [5] > [0] = tt() isQid(n__o()) = [10] > [0] = tt() isQid(n__u()) = [5] > [0] = tt() u() = [1] > [0] = n__u() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [5] = n____(X1,X2) a() = [0] >= [0] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__isList(X)) = [1] X + [0] >= [1] X + [3] = isList(X) activate(n__nil()) = [0] >= [0] = nil() activate(n__u()) = [0] >= [1] = u() e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [10] = and(isList(activate(V1)),n__isList(activate(V2))) isNeList(V) = [1] V + [1] >= [1] V + [5] = isQid(activate(V)) isNeList(X) = [1] X + [1] >= [1] X + [3] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [13] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [8] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [5] = isQid(activate(V)) isPal(V) = [1] V + [0] >= [1] V + [0] = isNePal(activate(V)) isPal(X) = [1] X + [0] >= [1] X + [3] = n__isPal(X) isPal(n__nil()) = [0] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [0] >= [5] = n__o() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() nil() -> n__nil() o() -> n__o() - Weak TRS: activate(n____(X1,X2)) -> __(X1,X2) activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [1] p(a) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [5] p(n__e) = [4] p(n__i) = [1] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [0] p(n__o) = [1] p(n__u) = [1] p(nil) = [1] p(o) = [1] p(tt) = [0] p(u) = [1] Following rules are strictly oriented: activate(n__a()) = [5] > [0] = a() activate(n__e()) = [4] > [0] = e() activate(n__i()) = [1] > [0] = i() isList(n____(V1,V2)) = [1] V1 + [1] V2 + [1] > [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [1] > [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [1] > [1] V1 + [1] V2 + [0] = and(isNeList(activate(V1)),n__isList(activate(V2))) nil() = [1] > [0] = n__nil() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n____(X1,X2) a() = [0] >= [5] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = __(X1,X2) activate(n__isList(X)) = [1] X + [0] >= [1] X + [0] = isList(X) activate(n__isNeList(X)) = [1] X + [0] >= [1] X + [0] = isNeList(X) activate(n__isPal(X)) = [1] X + [0] >= [1] X + [0] = isPal(X) activate(n__nil()) = [0] >= [1] = nil() activate(n__o()) = [1] >= [1] = o() activate(n__u()) = [1] >= [1] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) e() = [0] >= [4] = n__e() i() = [0] >= [1] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = isNeList(activate(V)) isList(X) = [1] X + [0] >= [1] X + [0] = n__isList(X) isList(n__nil()) = [0] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNeList(X) = [1] X + [0] >= [1] X + [0] = n__isNeList(X) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isPal(V) = [1] V + [0] >= [1] V + [0] = isNePal(activate(V)) isPal(X) = [1] X + [0] >= [1] X + [0] = n__isPal(X) isPal(n__nil()) = [0] >= [0] = tt() isQid(n__a()) = [5] >= [0] = tt() isQid(n__e()) = [4] >= [0] = tt() isQid(n__i()) = [1] >= [0] = tt() isQid(n__o()) = [1] >= [0] = tt() isQid(n__u()) = [1] >= [0] = tt() o() = [1] >= [1] = n__o() u() = [1] >= [1] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() o() -> n__o() - Weak TRS: activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [4] p(a) = [1] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [4] p(e) = [1] p(i) = [1] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [5] p(isQid) = [1] x1 + [2] p(n____) = [1] x1 + [1] x2 + [4] p(n__a) = [1] p(n__e) = [1] p(n__i) = [1] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [5] p(n__nil) = [1] p(n__o) = [3] p(n__u) = [7] p(nil) = [1] p(o) = [0] p(tt) = [0] p(u) = [7] Following rules are strictly oriented: isPal(V) = [1] V + [5] > [1] V + [0] = isNePal(activate(V)) isPal(n__nil()) = [6] > [0] = tt() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n____(X1,X2) a() = [1] >= [1] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = __(X1,X2) activate(n__a()) = [1] >= [1] = a() activate(n__e()) = [1] >= [1] = e() activate(n__i()) = [1] >= [1] = i() activate(n__isList(X)) = [1] X + [0] >= [1] X + [0] = isList(X) activate(n__isNeList(X)) = [1] X + [0] >= [1] X + [0] = isNeList(X) activate(n__isPal(X)) = [1] X + [5] >= [1] X + [5] = isPal(X) activate(n__nil()) = [1] >= [1] = nil() activate(n__o()) = [3] >= [0] = o() activate(n__u()) = [7] >= [7] = u() and(tt(),X) = [1] X + [4] >= [1] X + [0] = activate(X) e() = [1] >= [1] = n__e() i() = [1] >= [1] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = isNeList(activate(V)) isList(X) = [1] X + [0] >= [1] X + [0] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [1] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [2] = isQid(activate(V)) isNeList(X) = [1] X + [0] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [2] = isQid(activate(V)) isPal(X) = [1] X + [5] >= [1] X + [5] = n__isPal(X) isQid(n__a()) = [3] >= [0] = tt() isQid(n__e()) = [3] >= [0] = tt() isQid(n__i()) = [3] >= [0] = tt() isQid(n__o()) = [5] >= [0] = tt() isQid(n__u()) = [9] >= [0] = tt() nil() = [1] >= [1] = n__nil() o() = [0] >= [3] = n__o() u() = [7] >= [7] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNePal(V) -> isQid(activate(V)) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isPal(V) -> isNePal(activate(V)) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [0] p(a) = [2] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [6] p(isPal) = [1] x1 + [6] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [0] p(n__a) = [3] p(n__e) = [6] p(n__i) = [0] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [6] p(n__nil) = [0] p(n__o) = [2] p(n__u) = [3] p(nil) = [1] p(o) = [0] p(tt) = [0] p(u) = [3] Following rules are strictly oriented: isNePal(V) = [1] V + [6] > [1] V + [0] = isQid(activate(V)) Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n____(X1,X2) a() = [2] >= [3] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = __(X1,X2) activate(n__a()) = [3] >= [2] = a() activate(n__e()) = [6] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__isList(X)) = [1] X + [0] >= [1] X + [0] = isList(X) activate(n__isNeList(X)) = [1] X + [0] >= [1] X + [0] = isNeList(X) activate(n__isPal(X)) = [1] X + [6] >= [1] X + [6] = isPal(X) activate(n__nil()) = [0] >= [1] = nil() activate(n__o()) = [2] >= [0] = o() activate(n__u()) = [3] >= [3] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) e() = [0] >= [6] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = isNeList(activate(V)) isList(X) = [1] X + [0] >= [1] X + [0] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [0] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNeList(X) = [1] X + [0] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = and(isNeList(activate(V1)),n__isList(activate(V2))) isPal(V) = [1] V + [6] >= [1] V + [6] = isNePal(activate(V)) isPal(X) = [1] X + [6] >= [1] X + [6] = n__isPal(X) isPal(n__nil()) = [6] >= [0] = tt() isQid(n__a()) = [3] >= [0] = tt() isQid(n__e()) = [6] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() isQid(n__o()) = [2] >= [0] = tt() isQid(n__u()) = [3] >= [0] = tt() nil() = [1] >= [0] = n__nil() o() = [0] >= [2] = n__o() u() = [3] >= [3] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [7] p(a) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [7] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__isList) = [1] x1 + [2] p(n__isNeList) = [1] x1 + [2] p(n__isPal) = [1] x1 + [0] p(n__nil) = [0] p(n__o) = [0] p(n__u) = [0] p(nil) = [0] p(o) = [0] p(tt) = [0] p(u) = [0] Following rules are strictly oriented: isNeList(V) = [1] V + [2] > [1] V + [0] = isQid(activate(V)) Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = n____(X1,X2) a() = [0] >= [0] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = __(X1,X2) activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__isList(X)) = [1] X + [2] >= [1] X + [2] = isList(X) activate(n__isNeList(X)) = [1] X + [2] >= [1] X + [2] = isNeList(X) activate(n__isPal(X)) = [1] X + [0] >= [1] X + [0] = isPal(X) activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [0] >= [0] = o() activate(n__u()) = [0] >= [0] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [2] >= [1] V + [2] = isNeList(activate(V)) isList(X) = [1] X + [2] >= [1] X + [2] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [9] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [2] >= [0] = tt() isNeList(X) = [1] X + [2] >= [1] X + [2] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [9] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [9] >= [1] V1 + [1] V2 + [4] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isPal(V) = [1] V + [0] >= [1] V + [0] = isNePal(activate(V)) isPal(X) = [1] X + [0] >= [1] X + [0] = n__isPal(X) isPal(n__nil()) = [0] >= [0] = tt() isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [0] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() isQid(n__o()) = [0] >= [0] = tt() isQid(n__u()) = [0] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [0] >= [0] = n__o() u() = [0] >= [0] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(X) -> n__isNeList(X) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() and(tt(),X) -> activate(X) isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [0] p(a) = [1] p(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [1] p(e) = [4] p(i) = [0] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [2] p(isPal) = [1] x1 + [3] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [4] p(n__a) = [0] p(n__e) = [3] p(n__i) = [1] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [2] p(n__nil) = [0] p(n__o) = [2] p(n__u) = [1] p(nil) = [2] p(o) = [3] p(tt) = [0] p(u) = [1] Following rules are strictly oriented: a() = [1] > [0] = n__a() activate(X) = [1] X + [1] > [1] X + [0] = X activate(n__u()) = [2] > [1] = u() e() = [4] > [3] = n__e() isNeList(X) = [1] X + [1] > [1] X + [0] = n__isNeList(X) isPal(X) = [1] X + [3] > [1] X + [2] = n__isPal(X) o() = [3] > [2] = n__o() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [4] = n____(X1,X2) activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [0] = __(X1,X2) activate(n__a()) = [1] >= [1] = a() activate(n__e()) = [4] >= [4] = e() activate(n__i()) = [2] >= [0] = i() activate(n__isList(X)) = [1] X + [1] >= [1] X + [2] = isList(X) activate(n__isNeList(X)) = [1] X + [1] >= [1] X + [1] = isNeList(X) activate(n__isPal(X)) = [1] X + [3] >= [1] X + [3] = isPal(X) activate(n__nil()) = [1] >= [2] = nil() activate(n__o()) = [3] >= [3] = o() and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) i() = [0] >= [1] = n__i() isList(V) = [1] V + [2] >= [1] V + [2] = isNeList(activate(V)) isList(X) = [1] X + [2] >= [1] X + [0] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [5] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [2] >= [0] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = isQid(activate(V)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [2] >= [1] V + [1] = isQid(activate(V)) isPal(V) = [1] V + [3] >= [1] V + [3] = isNePal(activate(V)) isPal(n__nil()) = [3] >= [0] = tt() isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [3] >= [0] = tt() isQid(n__i()) = [1] >= [0] = tt() isQid(n__o()) = [2] >= [0] = tt() isQid(n__u()) = [1] >= [0] = tt() nil() = [2] >= [0] = n__nil() u() = [1] >= [1] = n__u() 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 TRS: __(X1,X2) -> n____(X1,X2) activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() i() -> n__i() - Weak TRS: a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [0] p(a) = [3] p(activate) = [1] x1 + [2] p(and) = [1] x1 + [1] x2 + [0] p(e) = [4] p(i) = [5] p(isList) = [1] x1 + [4] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [4] p(isPal) = [1] x1 + [6] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [6] p(n__a) = [2] p(n__e) = [4] p(n__i) = [4] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [4] p(n__nil) = [0] p(n__o) = [2] p(n__u) = [4] p(nil) = [0] p(o) = [2] p(tt) = [2] p(u) = [5] Following rules are strictly oriented: activate(n__nil()) = [2] > [0] = nil() i() = [5] > [4] = n__i() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [6] = n____(X1,X2) a() = [3] >= [2] = n__a() activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [0] = __(X1,X2) activate(n__a()) = [4] >= [3] = a() activate(n__e()) = [6] >= [4] = e() activate(n__i()) = [6] >= [5] = i() activate(n__isList(X)) = [1] X + [2] >= [1] X + [4] = isList(X) activate(n__isNeList(X)) = [1] X + [2] >= [1] X + [2] = isNeList(X) activate(n__isPal(X)) = [1] X + [6] >= [1] X + [6] = isPal(X) activate(n__o()) = [4] >= [2] = o() activate(n__u()) = [6] >= [5] = u() and(tt(),X) = [1] X + [2] >= [1] X + [2] = activate(X) e() = [4] >= [4] = n__e() isList(V) = [1] V + [4] >= [1] V + [4] = isNeList(activate(V)) isList(X) = [1] X + [4] >= [1] X + [0] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [10] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [4] >= [2] = tt() isNeList(V) = [1] V + [2] >= [1] V + [2] = isQid(activate(V)) isNeList(X) = [1] X + [2] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [6] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [4] >= [1] V + [2] = isQid(activate(V)) isPal(V) = [1] V + [6] >= [1] V + [6] = isNePal(activate(V)) isPal(X) = [1] X + [6] >= [1] X + [4] = n__isPal(X) isPal(n__nil()) = [6] >= [2] = tt() isQid(n__a()) = [2] >= [2] = tt() isQid(n__e()) = [4] >= [2] = tt() isQid(n__i()) = [4] >= [2] = tt() isQid(n__o()) = [2] >= [2] = tt() isQid(n__u()) = [4] >= [2] = tt() nil() = [0] >= [0] = n__nil() o() = [2] >= [2] = n__o() u() = [5] >= [4] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) activate(n__isList(X)) -> isList(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() i() -> n__i() isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + 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(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [7] p(a) = [7] p(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [0] p(e) = [1] p(i) = [4] p(isList) = [1] x1 + [4] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [1] p(isPal) = [1] x1 + [2] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [6] p(n__a) = [7] p(n__e) = [1] p(n__i) = [4] p(n__isList) = [1] x1 + [4] p(n__isNeList) = [1] x1 + [2] p(n__isPal) = [1] x1 + [2] p(n__nil) = [0] p(n__o) = [3] p(n__u) = [1] p(nil) = [0] p(o) = [3] p(tt) = [1] p(u) = [2] Following rules are strictly oriented: __(X1,X2) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [6] = n____(X1,X2) activate(n__isList(X)) = [1] X + [5] > [1] X + [4] = isList(X) Following rules are (at-least) weakly oriented: a() = [7] >= [7] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = __(X1,X2) activate(n__a()) = [8] >= [7] = a() activate(n__e()) = [2] >= [1] = e() activate(n__i()) = [5] >= [4] = i() activate(n__isNeList(X)) = [1] X + [3] >= [1] X + [2] = isNeList(X) activate(n__isPal(X)) = [1] X + [3] >= [1] X + [2] = isPal(X) activate(n__nil()) = [1] >= [0] = nil() activate(n__o()) = [4] >= [3] = o() activate(n__u()) = [2] >= [2] = u() and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) e() = [1] >= [1] = n__e() i() = [4] >= [4] = n__i() isList(V) = [1] V + [4] >= [1] V + [3] = isNeList(activate(V)) isList(X) = [1] X + [4] >= [1] X + [4] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [10] >= [1] V1 + [1] V2 + [10] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [4] >= [1] = tt() isNeList(V) = [1] V + [2] >= [1] V + [1] = isQid(activate(V)) isNeList(X) = [1] X + [2] >= [1] X + [2] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [1] >= [1] V + [1] = isQid(activate(V)) isPal(V) = [1] V + [2] >= [1] V + [2] = isNePal(activate(V)) isPal(X) = [1] X + [2] >= [1] X + [2] = n__isPal(X) isPal(n__nil()) = [2] >= [1] = tt() isQid(n__a()) = [7] >= [1] = tt() isQid(n__e()) = [1] >= [1] = tt() isQid(n__i()) = [4] >= [1] = tt() isQid(n__o()) = [3] >= [1] = tt() isQid(n__u()) = [1] >= [1] = tt() nil() = [0] >= [0] = n__nil() o() = [3] >= [3] = n__o() u() = [2] >= [1] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() i() -> n__i() isList(V) -> isNeList(activate(V)) isList(X) -> n__isList(X) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) -> tt() isNeList(V) -> isQid(activate(V)) isNeList(X) -> n__isNeList(X) isNeList(n____(V1,V2)) -> and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) -> and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) -> isQid(activate(V)) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Signature: {__/2,a/0,activate/1,and/2,e/0,i/0,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,nil/0,o/0,u/0} / {n____/2 ,n__a/0,n__e/0,n__i/0,n__isList/1,n__isNeList/1,n__isPal/1,n__nil/0,n__o/0,n__u/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid ,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__isList,n__isNeList,n__isPal,n__nil,n__o,n__u,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))