WORST_CASE(?,O(n^2)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2)) + 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)) -> __(activate(X1),activate(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,n____(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 All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(activate(X1),activate(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,n____(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: 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {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 + [5] 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 + [0] p(n__a) = [5] 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 + [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: __(X1,X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = n____(X1,X2) activate(n__a()) = [5] > [0] = a() activate(n__isNeList(X)) = [1] X + [3] > [1] X + [0] = isNeList(X) and(tt(),X) = [1] X + [5] > [1] X + [0] = activate(X) isQid(n__a()) = [5] > [0] = tt() Following rules are (at-least) weakly oriented: a() = [0] >= [5] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = __(activate(X1),activate(X2)) activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__isList(X)) = [1] X + [0] >= [1] X + [0] = isList(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() e() = [0] >= [0] = 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 + [5] = 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 + [3] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [5] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [0] >= [1] I + [1] P + [5] = and(isQid(activate(I)),n__isPal(activate(P))) 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__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 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isPal(X)) -> isPal(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() 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,n____(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__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Weak TRS: __(X1,X2) -> n____(X1,X2) activate(n__a()) -> a() activate(n__isNeList(X)) -> isNeList(X) and(tt(),X) -> activate(X) isQid(n__a()) -> tt() - 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [5] p(a) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [1] 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 + [0] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [5] p(n__o) = [0] p(n__u) = [4] p(nil) = [0] p(o) = [0] p(tt) = [0] p(u) = [5] Following rules are strictly oriented: activate(n__nil()) = [5] > [0] = nil() e() = [1] > [0] = n__e() isList(n__nil()) = [5] > [0] = tt() isPal(n__nil()) = [5] > [0] = tt() isQid(n__u()) = [4] > [0] = tt() u() = [5] > [4] = n__u() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [0] = 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 + [0] >= [1] X1 + [1] X2 + [5] = __(activate(X1),activate(X2)) activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [0] >= [1] = 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 + [0] >= [1] X + [0] = isPal(X) activate(n__o()) = [0] >= [0] = o() activate(n__u()) = [4] >= [5] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) 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))) 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))) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [0] >= [1] I + [1] P + [0] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1] V + [0] >= [1] V + [0] = isNePal(activate(V)) isPal(X) = [1] X + [0] >= [1] X + [0] = n__isPal(X) isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [0] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() isQid(n__o()) = [0] >= [0] = tt() nil() = [0] >= [5] = n__nil() o() = [0] >= [0] = n__o() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() activate(n__u()) -> u() 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))) 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,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() nil() -> n__nil() o() -> n__o() - Weak TRS: __(X1,X2) -> n____(X1,X2) activate(n__a()) -> a() activate(n__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() and(tt(),X) -> activate(X) e() -> n__e() isList(n__nil()) -> tt() isPal(n__nil()) -> tt() isQid(n__a()) -> 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {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 + [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 + [0] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] 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) = [2] p(n__u) = [0] p(nil) = [0] p(o) = [3] p(tt) = [0] p(u) = [0] Following rules are strictly oriented: isQid(n__o()) = [2] > [0] = tt() o() = [3] > [2] = n__o() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = 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 + [0] >= [1] X1 + [1] X2 + [0] = __(activate(X1),activate(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 + [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] >= [0] = nil() activate(n__o()) = [2] >= [3] = 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 + [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))) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [0] >= [1] I + [1] P + [0] = and(isQid(activate(I)),n__isPal(activate(P))) 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__u()) = [0] >= [0] = tt() nil() = [0] >= [0] = n__nil() u() = [0] >= [0] = 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^2)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__e()) -> e() activate(n__i()) -> i() activate(n__isList(X)) -> isList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() activate(n__u()) -> u() 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))) 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,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isQid(n__e()) -> tt() isQid(n__i()) -> tt() nil() -> n__nil() - Weak TRS: __(X1,X2) -> n____(X1,X2) activate(n__a()) -> a() activate(n__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() and(tt(),X) -> activate(X) e() -> n__e() isList(n__nil()) -> tt() isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [3] p(a) = [3] p(activate) = [1] x1 + [2] p(and) = [1] x1 + [1] x2 + [1] p(e) = [5] p(i) = [0] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [3] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [2] p(n__a) = [1] p(n__e) = [5] p(n__i) = [0] p(n__isList) = [1] x1 + [1] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [0] p(n__o) = [1] p(n__u) = [2] p(nil) = [0] p(o) = [1] p(tt) = [1] p(u) = [2] Following rules are strictly oriented: a() = [3] > [1] = n__a() activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__e()) = [7] > [5] = e() activate(n__i()) = [2] > [0] = i() activate(n__isList(X)) = [1] X + [3] > [1] X + [2] = isList(X) activate(n__o()) = [3] > [1] = o() activate(n__u()) = [4] > [2] = u() isList(X) = [1] X + [2] > [1] X + [1] = n__isList(X) isNeList(X) = [1] X + [2] > [1] X + [0] = n__isNeList(X) isPal(V) = [1] V + [3] > [1] V + [2] = isNePal(activate(V)) isPal(X) = [1] X + [3] > [1] X + [0] = n__isPal(X) isQid(n__e()) = [5] > [1] = tt() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [2] = n____(X1,X2) activate(n____(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [7] = __(activate(X1),activate(X2)) activate(n__a()) = [3] >= [3] = a() activate(n__isNeList(X)) = [1] X + [2] >= [1] X + [2] = isNeList(X) activate(n__isPal(X)) = [1] X + [2] >= [1] X + [3] = isPal(X) activate(n__nil()) = [2] >= [0] = nil() and(tt(),X) = [1] X + [2] >= [1] X + [2] = activate(X) e() = [5] >= [5] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [2] >= [1] V + [4] = isNeList(activate(V)) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [2] >= [1] = tt() isNeList(V) = [1] V + [2] >= [1] V + [2] = isQid(activate(V)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [7] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [8] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [2] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [4] >= [1] I + [1] P + [5] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(n__nil()) = [3] >= [1] = tt() isQid(n__a()) = [1] >= [1] = tt() isQid(n__i()) = [0] >= [1] = tt() isQid(n__o()) = [1] >= [1] = tt() isQid(n__u()) = [2] >= [1] = tt() nil() = [0] >= [0] = n__nil() o() = [1] >= [1] = n__o() u() = [2] >= [2] = 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^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) i() -> n__i() isList(V) -> isNeList(activate(V)) isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) 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)) isNePal(n____(I,n____(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isQid(n__i()) -> tt() nil() -> n__nil() - Weak 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__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() and(tt(),X) -> activate(X) e() -> n__e() isList(X) -> n__isList(X) isList(n__nil()) -> tt() isNeList(X) -> n__isNeList(X) isPal(V) -> isNePal(activate(V)) isPal(X) -> n__isPal(X) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [4] p(a) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [0] p(i) = [1] p(isList) = [1] x1 + [4] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [5] p(isPal) = [1] x1 + [5] p(isQid) = [1] x1 + [6] p(n____) = [1] x1 + [1] x2 + [3] p(n__a) = [0] p(n__e) = [0] p(n__i) = [1] p(n__isList) = [1] x1 + [4] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [4] p(n__nil) = [2] p(n__o) = [2] p(n__u) = [0] p(nil) = [0] p(o) = [2] p(tt) = [6] p(u) = [0] Following rules are strictly oriented: isList(V) = [1] V + [4] > [1] V + [0] = isNeList(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [11] > [1] I + [1] P + [10] = and(isQid(activate(I)),n__isPal(activate(P))) isQid(n__i()) = [7] > [6] = tt() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [3] = 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 + [3] >= [1] X1 + [1] X2 + [4] = __(activate(X1),activate(X2)) activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [1] >= [1] = i() activate(n__isList(X)) = [1] X + [4] >= [1] X + [4] = isList(X) activate(n__isNeList(X)) = [1] X + [0] >= [1] X + [0] = isNeList(X) activate(n__isPal(X)) = [1] X + [4] >= [1] X + [5] = isPal(X) activate(n__nil()) = [2] >= [0] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [0] >= [0] = u() and(tt(),X) = [1] X + [6] >= [1] X + [0] = activate(X) e() = [0] >= [0] = n__e() i() = [1] >= [1] = n__i() isList(X) = [1] X + [4] >= [1] X + [4] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [7] >= [1] V1 + [1] V2 + [8] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [6] >= [6] = tt() isNeList(V) = [1] V + [0] >= [1] V + [6] = isQid(activate(V)) isNeList(X) = [1] X + [0] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [4] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [5] >= [1] V + [6] = isQid(activate(V)) isPal(V) = [1] V + [5] >= [1] V + [5] = isNePal(activate(V)) isPal(X) = [1] X + [5] >= [1] X + [4] = n__isPal(X) isPal(n__nil()) = [7] >= [6] = tt() isQid(n__a()) = [6] >= [6] = tt() isQid(n__e()) = [6] >= [6] = tt() isQid(n__o()) = [8] >= [6] = tt() isQid(n__u()) = [6] >= [6] = tt() nil() = [0] >= [2] = n__nil() o() = [2] >= [2] = 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^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) i() -> n__i() isList(n____(V1,V2)) -> and(isList(activate(V1)),n__isList(activate(V2))) 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)) nil() -> n__nil() - Weak 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__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() 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__nil()) -> tt() isNeList(X) -> n__isNeList(X) isNePal(n____(I,n____(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() 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [1] p(a) = [2] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [2] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [4] p(isPal) = [1] x1 + [4] p(isQid) = [1] x1 + [2] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [2] p(n__e) = [2] p(n__i) = [0] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [4] p(n__nil) = [4] p(n__o) = [1] p(n__u) = [7] p(nil) = [4] p(o) = [1] p(tt) = [0] p(u) = [7] Following rules are strictly oriented: 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))) isNePal(V) = [1] V + [4] > [1] V + [2] = isQid(activate(V)) Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n____(X1,X2) a() = [2] >= [2] = 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] = __(activate(X1),activate(X2)) activate(n__a()) = [2] >= [2] = a() activate(n__e()) = [2] >= [2] = 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 + [4] >= [1] X + [4] = isPal(X) activate(n__nil()) = [4] >= [4] = nil() activate(n__o()) = [1] >= [1] = o() activate(n__u()) = [7] >= [7] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) e() = [2] >= [2] = 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__nil()) = [4] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [2] = isQid(activate(V)) isNeList(X) = [1] X + [0] >= [1] X + [0] = n__isNeList(X) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [6] >= [1] I + [1] P + [6] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1] V + [4] >= [1] V + [4] = isNePal(activate(V)) isPal(X) = [1] X + [4] >= [1] X + [4] = n__isPal(X) isPal(n__nil()) = [8] >= [0] = tt() isQid(n__a()) = [4] >= [0] = tt() isQid(n__e()) = [4] >= [0] = tt() isQid(n__i()) = [2] >= [0] = tt() isQid(n__o()) = [3] >= [0] = tt() isQid(n__u()) = [9] >= [0] = tt() nil() = [4] >= [4] = n__nil() o() = [1] >= [1] = n__o() u() = [7] >= [7] = 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^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) i() -> n__i() isNeList(V) -> isQid(activate(V)) nil() -> n__nil() - Weak 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__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() 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(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,n____(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() 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [2] p(a) = [6] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [1] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [2] p(n__a) = [6] p(n__e) = [0] p(n__i) = [0] p(n__isList) = [1] x1 + [1] p(n__isNeList) = [1] x1 + [1] 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 + [1] > [1] V + [0] = isQid(activate(V)) Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n____(X1,X2) a() = [6] >= [6] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = __(activate(X1),activate(X2)) activate(n__a()) = [6] >= [6] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__isList(X)) = [1] X + [1] >= [1] X + [1] = isList(X) activate(n__isNeList(X)) = [1] X + [1] >= [1] X + [1] = 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 + [1] >= [1] V + [1] = isNeList(activate(V)) isList(X) = [1] X + [1] >= [1] X + [1] = n__isList(X) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [2] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [1] >= [0] = tt() isNeList(X) = [1] X + [1] >= [1] X + [1] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [2] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [2] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [0] >= [1] V + [0] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [4] >= [1] I + [1] P + [0] = and(isQid(activate(I)),n__isPal(activate(P))) 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()) = [6] >= [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 9: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) i() -> n__i() nil() -> n__nil() - Weak 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__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() 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)) isNePal(n____(I,n____(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() 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [7] p(a) = [3] p(activate) = [1] x1 + [2] p(and) = [1] x1 + [1] x2 + [0] p(e) = [4] p(i) = [0] p(isList) = [1] x1 + [4] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [3] p(isPal) = [1] x1 + [5] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [6] p(n__a) = [3] p(n__e) = [4] p(n__i) = [3] p(n__isList) = [1] x1 + [2] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [6] p(n__o) = [4] p(n__u) = [3] p(nil) = [7] p(o) = [5] p(tt) = [3] p(u) = [3] Following rules are strictly oriented: nil() = [7] > [6] = n__nil() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [6] = n____(X1,X2) a() = [3] >= [3] = n__a() activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [11] = __(activate(X1),activate(X2)) activate(n__a()) = [5] >= [3] = a() activate(n__e()) = [6] >= [4] = e() activate(n__i()) = [5] >= [0] = i() activate(n__isList(X)) = [1] X + [4] >= [1] X + [4] = isList(X) activate(n__isNeList(X)) = [1] X + [2] >= [1] X + [2] = isNeList(X) activate(n__isPal(X)) = [1] X + [2] >= [1] X + [5] = isPal(X) activate(n__nil()) = [8] >= [7] = nil() activate(n__o()) = [6] >= [5] = o() activate(n__u()) = [5] >= [3] = u() and(tt(),X) = [1] X + [3] >= [1] X + [2] = activate(X) e() = [4] >= [4] = n__e() i() = [0] >= [3] = n__i() isList(V) = [1] V + [4] >= [1] V + [4] = isNeList(activate(V)) isList(X) = [1] X + [4] >= [1] X + [2] = 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()) = [10] >= [3] = 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 + [8] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [3] >= [1] V + [2] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [15] >= [1] I + [1] P + [4] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1] V + [5] >= [1] V + [5] = isNePal(activate(V)) isPal(X) = [1] X + [5] >= [1] X + [0] = n__isPal(X) isPal(n__nil()) = [11] >= [3] = tt() isQid(n__a()) = [3] >= [3] = tt() isQid(n__e()) = [4] >= [3] = tt() isQid(n__i()) = [3] >= [3] = tt() isQid(n__o()) = [4] >= [3] = tt() isQid(n__u()) = [3] >= [3] = tt() o() = [5] >= [4] = n__o() u() = [3] >= [3] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) i() -> n__i() - Weak 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__isNeList(X)) -> isNeList(X) activate(n__nil()) -> nil() 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)) isNePal(n____(I,n____(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: 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [7] p(a) = [2] p(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [0] p(e) = [4] p(i) = [3] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [3] p(isPal) = [1] x1 + [5] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [4] p(n__a) = [1] p(n__e) = [4] p(n__i) = [2] p(n__isList) = [1] x1 + [1] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [2] p(n__o) = [1] p(n__u) = [2] p(nil) = [3] p(o) = [1] p(tt) = [1] p(u) = [3] Following rules are strictly oriented: i() = [3] > [2] = n__i() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [4] = n____(X1,X2) a() = [2] >= [1] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [9] = __(activate(X1),activate(X2)) activate(n__a()) = [2] >= [2] = a() activate(n__e()) = [5] >= [4] = e() activate(n__i()) = [3] >= [3] = i() activate(n__isList(X)) = [1] X + [2] >= [1] X + [2] = isList(X) activate(n__isNeList(X)) = [1] X + [1] >= [1] X + [1] = isNeList(X) activate(n__isPal(X)) = [1] X + [1] >= [1] X + [5] = isPal(X) activate(n__nil()) = [3] >= [3] = nil() activate(n__o()) = [2] >= [1] = o() activate(n__u()) = [3] >= [3] = u() and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) e() = [4] >= [4] = n__e() isList(V) = [1] V + [2] >= [1] V + [2] = isNeList(activate(V)) isList(X) = [1] X + [2] >= [1] X + [1] = 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()) = [4] >= [1] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = isQid(activate(V)) isNeList(X) = [1] X + [1] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = 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 + [3] >= [1] V + [1] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [11] >= [1] I + [1] P + [2] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1] V + [5] >= [1] V + [4] = isNePal(activate(V)) isPal(X) = [1] X + [5] >= [1] X + [0] = n__isPal(X) isPal(n__nil()) = [7] >= [1] = tt() isQid(n__a()) = [1] >= [1] = tt() isQid(n__e()) = [4] >= [1] = tt() isQid(n__i()) = [2] >= [1] = tt() isQid(n__o()) = [1] >= [1] = tt() isQid(n__u()) = [2] >= [1] = tt() nil() = [3] >= [2] = n__nil() o() = [1] >= [1] = n__o() u() = [3] >= [2] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 11: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) activate(n__isPal(X)) -> isPal(X) - Weak 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__isNeList(X)) -> isNeList(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,n____(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: 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(__) = [1] x1 + [1] x2 + [4] p(a) = [4] p(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [0] p(e) = [2] p(i) = [5] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [1] p(isPal) = [1] x1 + [4] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [4] p(n__a) = [4] p(n__e) = [2] p(n__i) = [4] p(n__isList) = [1] x1 + [2] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [4] p(n__nil) = [0] p(n__o) = [1] p(n__u) = [2] p(nil) = [1] p(o) = [2] p(tt) = [1] p(u) = [2] Following rules are strictly oriented: activate(n__isPal(X)) = [1] X + [5] > [1] X + [4] = isPal(X) Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n____(X1,X2) a() = [4] >= [4] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [6] = __(activate(X1),activate(X2)) activate(n__a()) = [5] >= [4] = a() activate(n__e()) = [3] >= [2] = e() activate(n__i()) = [5] >= [5] = i() activate(n__isList(X)) = [1] X + [3] >= [1] X + [2] = isList(X) activate(n__isNeList(X)) = [1] X + [1] >= [1] X + [1] = isNeList(X) activate(n__nil()) = [1] >= [1] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [3] >= [2] = u() and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) e() = [2] >= [2] = n__e() i() = [5] >= [4] = 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 + [6] >= [1] V1 + [1] V2 + [6] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [2] >= [1] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = isQid(activate(V)) isNeList(X) = [1] X + [1] >= [1] X + [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1] V + [1] >= [1] V + [1] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2] I + [1] P + [9] >= [1] I + [1] P + [6] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1] V + [4] >= [1] V + [2] = isNePal(activate(V)) isPal(X) = [1] X + [4] >= [1] X + [4] = n__isPal(X) isPal(n__nil()) = [4] >= [1] = tt() isQid(n__a()) = [4] >= [1] = tt() isQid(n__e()) = [2] >= [1] = tt() isQid(n__i()) = [4] >= [1] = tt() isQid(n__o()) = [1] >= [1] = tt() isQid(n__u()) = [2] >= [1] = tt() nil() = [1] >= [0] = n__nil() o() = [2] >= [1] = n__o() u() = [2] >= [2] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 12: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n____(X1,X2)) -> __(activate(X1),activate(X2)) - Weak 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__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,n____(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: NaturalMI {miDimension = 2, miDegree = 2, 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(__) = {1,2}, uargs(and) = {1,2}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isQid) = {1}, uargs(n__isList) = {1}, uargs(n__isNeList) = {1}, uargs(n__isPal) = {1} Following symbols are considered usable: {__,a,activate,and,e,i,isList,isNeList,isNePal,isPal,isQid,nil,o,u} TcT has computed the following interpretation: p(__) = [1 2] x1 + [1 2] x2 + [4] [0 1] [0 1] [1] p(a) = [2] [6] p(activate) = [1 1] x1 + [0] [0 1] [0] p(and) = [1 0] x1 + [1 1] x2 + [0] [0 0] [0 4] [0] p(e) = [0] [0] p(i) = [4] [5] p(isList) = [1 2] x1 + [0] [0 0] [0] p(isNeList) = [1 1] x1 + [0] [0 0] [0] p(isNePal) = [1 1] x1 + [2] [0 0] [0] p(isPal) = [1 2] x1 + [2] [0 0] [0] p(isQid) = [1 0] x1 + [0] [0 0] [0] p(n____) = [1 2] x1 + [1 2] x2 + [4] [0 1] [0 1] [1] p(n__a) = [2] [6] p(n__e) = [0] [0] p(n__i) = [4] [5] p(n__isList) = [1 2] x1 + [0] [0 0] [0] p(n__isNeList) = [1 1] x1 + [0] [0 0] [0] p(n__isPal) = [1 2] x1 + [2] [0 0] [0] p(n__nil) = [0] [0] p(n__o) = [4] [0] p(n__u) = [0] [1] p(nil) = [0] [0] p(o) = [4] [0] p(tt) = [0] [0] p(u) = [1] [1] Following rules are strictly oriented: activate(n____(X1,X2)) = [1 3] X1 + [1 3] X2 + [5] [0 1] [0 1] [1] > [1 3] X1 + [1 3] X2 + [4] [0 1] [0 1] [1] = __(activate(X1),activate(X2)) Following rules are (at-least) weakly oriented: __(X1,X2) = [1 2] X1 + [1 2] X2 + [4] [0 1] [0 1] [1] >= [1 2] X1 + [1 2] X2 + [4] [0 1] [0 1] [1] = n____(X1,X2) a() = [2] [6] >= [2] [6] = n__a() activate(X) = [1 1] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__a()) = [8] [6] >= [2] [6] = a() activate(n__e()) = [0] [0] >= [0] [0] = e() activate(n__i()) = [9] [5] >= [4] [5] = i() activate(n__isList(X)) = [1 2] X + [0] [0 0] [0] >= [1 2] X + [0] [0 0] [0] = isList(X) activate(n__isNeList(X)) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = isNeList(X) activate(n__isPal(X)) = [1 2] X + [2] [0 0] [0] >= [1 2] X + [2] [0 0] [0] = isPal(X) activate(n__nil()) = [0] [0] >= [0] [0] = nil() activate(n__o()) = [4] [0] >= [4] [0] = o() activate(n__u()) = [1] [1] >= [1] [1] = u() and(tt(),X) = [1 1] X + [0] [0 4] [0] >= [1 1] X + [0] [0 1] [0] = activate(X) e() = [0] [0] >= [0] [0] = n__e() i() = [4] [5] >= [4] [5] = n__i() isList(V) = [1 2] V + [0] [0 0] [0] >= [1 2] V + [0] [0 0] [0] = isNeList(activate(V)) isList(X) = [1 2] X + [0] [0 0] [0] >= [1 2] X + [0] [0 0] [0] = n__isList(X) isList(n____(V1,V2)) = [1 4] V1 + [1 4] V2 + [6] [0 0] [0 0] [0] >= [1 3] V1 + [1 3] V2 + [0] [0 0] [0 0] [0] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [0] [0] >= [0] [0] = tt() isNeList(V) = [1 1] V + [0] [0 0] [0] >= [1 1] V + [0] [0 0] [0] = isQid(activate(V)) isNeList(X) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1 3] V1 + [1 3] V2 + [5] [0 0] [0 0] [0] >= [1 3] V1 + [1 2] V2 + [0] [0 0] [0 0] [0] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1 3] V1 + [1 3] V2 + [5] [0 0] [0 0] [0] >= [1 2] V1 + [1 3] V2 + [0] [0 0] [0 0] [0] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1 1] V + [2] [0 0] [0] >= [1 1] V + [0] [0 0] [0] = isQid(activate(V)) isNePal(n____(I,n____(P,I))) = [2 8] I + [1 5] P + [14] [0 0] [0 0] [0] >= [1 1] I + [1 3] P + [2] [0 0] [0 0] [0] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1 2] V + [2] [0 0] [0] >= [1 2] V + [2] [0 0] [0] = isNePal(activate(V)) isPal(X) = [1 2] X + [2] [0 0] [0] >= [1 2] X + [2] [0 0] [0] = n__isPal(X) isPal(n__nil()) = [2] [0] >= [0] [0] = tt() isQid(n__a()) = [2] [0] >= [0] [0] = tt() isQid(n__e()) = [0] [0] >= [0] [0] = tt() isQid(n__i()) = [4] [0] >= [0] [0] = tt() isQid(n__o()) = [4] [0] >= [0] [0] = tt() isQid(n__u()) = [0] [0] >= [0] [0] = tt() nil() = [0] [0] >= [0] [0] = n__nil() o() = [4] [0] >= [4] [0] = n__o() u() = [1] [1] >= [0] [1] = n__u() * Step 13: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: __(X1,X2) -> n____(X1,X2) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(activate(X1),activate(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,n____(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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))