WORST_CASE(?,O(n^3)) * Step 1: Sum WORST_CASE(?,O(n^3)) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalMI WORST_CASE(?,O(n^3)) + 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(__) = {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] x1 + [1] x2 + [2] p(a) = [3] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(e) = [6] p(i) = [2] 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 + [2] p(n__a) = [3] p(n__e) = [6] p(n__i) = [2] p(n__isList) = [1] x1 + [0] p(n__isNeList) = [1] x1 + [0] p(n__isPal) = [1] x1 + [0] p(n__nil) = [1] p(n__o) = [1] p(n__u) = [12] p(nil) = [1] p(o) = [1] p(tt) = [0] p(u) = [12] Following rules are strictly oriented: __(X,nil()) = [1] X + [3] > [1] X + [0] = X __(nil(),X) = [1] X + [3] > [1] X + [0] = X isList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] > [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [1] > [0] = tt() isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] > [1] V1 + [1] V2 + [0] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] > [1] V1 + [1] V2 + [0] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [4] > [1] I + [1] P + [0] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(n__nil()) = [1] > [0] = tt() isQid(n__a()) = [3] > [0] = tt() isQid(n__e()) = [6] > [0] = tt() isQid(n__i()) = [2] > [0] = tt() isQid(n__o()) = [1] > [0] = tt() isQid(n__u()) = [12] > [0] = tt() Following rules are (at-least) weakly oriented: __(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [4] >= [1] X + [1] Y + [1] Z + [4] = __(X,__(Y,Z)) a() = [3] >= [3] = 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] = __(X1,X2) activate(n__a()) = [3] >= [3] = a() activate(n__e()) = [6] >= [6] = e() activate(n__i()) = [2] >= [2] = 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()) = [1] >= [1] = nil() activate(n__o()) = [1] >= [1] = o() activate(n__u()) = [12] >= [12] = u() and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) e() = [6] >= [6] = n__e() i() = [2] >= [2] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = isNeList(activate(V)) isList(X) = [1] X + [0] >= [1] X + [0] = n__isList(X) 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) nil() = [1] >= [1] = n__nil() o() = [1] >= [1] = n__o() u() = [12] >= [12] = n__u() * Step 3: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) 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) 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) nil() -> n__nil() o() -> n__o() u() -> n__u() - Weak TRS: __(X,nil()) -> X __(nil(),X) -> 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(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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(__) = 7 + x1 + x2 p(a) = 1 p(activate) = 1 + x1 p(and) = x1 + x2 p(e) = 5 p(i) = 1 p(isList) = 5 + x1 p(isNeList) = 4 + x1 p(isNePal) = 1 + x1 p(isPal) = 7 + x1 p(isQid) = x1 p(n____) = 7 + x1 + x2 p(n__a) = 1 p(n__e) = 4 p(n__i) = 1 p(n__isList) = 4 + x1 p(n__isNeList) = 4 + x1 p(n__isPal) = 7 + x1 p(n__nil) = 0 p(n__o) = 4 p(n__u) = 2 p(nil) = 1 p(o) = 4 p(tt) = 1 p(u) = 3 Following rules are strictly oriented: activate(X) = 1 + X > X = X activate(n____(X1,X2)) = 8 + X1 + X2 > 7 + X1 + X2 = __(X1,X2) activate(n__a()) = 2 > 1 = a() activate(n__i()) = 2 > 1 = i() activate(n__isNeList(X)) = 5 + X > 4 + X = isNeList(X) activate(n__isPal(X)) = 8 + X > 7 + X = isPal(X) activate(n__o()) = 5 > 4 = o() e() = 5 > 4 = n__e() isList(X) = 5 + X > 4 + X = n__isList(X) isNeList(V) = 4 + V > 1 + V = isQid(activate(V)) isPal(V) = 7 + V > 2 + V = isNePal(activate(V)) nil() = 1 > 0 = n__nil() u() = 3 > 2 = n__u() Following rules are (at-least) weakly oriented: __(X,nil()) = 8 + X >= X = X __(X1,X2) = 7 + X1 + X2 >= 7 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 14 + X + Y + Z >= 14 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 8 + X >= X = X a() = 1 >= 1 = n__a() activate(n__e()) = 5 >= 5 = e() activate(n__isList(X)) = 5 + X >= 5 + X = isList(X) activate(n__nil()) = 1 >= 1 = nil() activate(n__u()) = 3 >= 3 = u() and(tt(),X) = 1 + X >= 1 + X = activate(X) i() = 1 >= 1 = n__i() isList(V) = 5 + V >= 5 + V = isNeList(activate(V)) isList(n____(V1,V2)) = 12 + V1 + V2 >= 11 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 5 >= 1 = tt() isNeList(X) = 4 + X >= 4 + X = n__isNeList(X) isNeList(n____(V1,V2)) = 11 + V1 + V2 >= 11 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 11 + V1 + V2 >= 10 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = 1 + V >= 1 + V = isQid(activate(V)) isNePal(n____(I,__(P,I))) = 15 + 2*I + P >= 9 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(X) = 7 + X >= 7 + X = n__isPal(X) isPal(n__nil()) = 7 >= 1 = tt() isQid(n__a()) = 1 >= 1 = tt() isQid(n__e()) = 4 >= 1 = tt() isQid(n__i()) = 1 >= 1 = tt() isQid(n__o()) = 4 >= 1 = tt() isQid(n__u()) = 2 >= 1 = tt() o() = 4 >= 4 = n__o() * Step 4: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() and(tt(),X) -> activate(X) i() -> n__i() isList(V) -> isNeList(activate(V)) isNeList(X) -> n__isNeList(X) isNePal(V) -> isQid(activate(V)) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__i()) -> i() activate(n__isNeList(X)) -> isNeList(X) activate(n__isPal(X)) -> isPal(X) activate(n__o()) -> o() e() -> n__e() 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(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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(__) = 4 + x1 + x2 p(a) = 3 p(activate) = x1 p(and) = 4 + x1 + x2 p(e) = 0 p(i) = 0 p(isList) = x1 p(isNeList) = x1 p(isNePal) = 7 + x1 p(isPal) = 7 + x1 p(isQid) = x1 p(n____) = 4 + x1 + x2 p(n__a) = 3 p(n__e) = 0 p(n__i) = 0 p(n__isList) = x1 p(n__isNeList) = x1 p(n__isPal) = 7 + x1 p(n__nil) = 6 p(n__o) = 0 p(n__u) = 0 p(nil) = 6 p(o) = 0 p(tt) = 0 p(u) = 0 Following rules are strictly oriented: and(tt(),X) = 4 + X > X = activate(X) isNePal(V) = 7 + V > V = isQid(activate(V)) Following rules are (at-least) weakly oriented: __(X,nil()) = 10 + X >= X = X __(X1,X2) = 4 + X1 + X2 >= 4 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 8 + X + Y + Z >= 8 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 10 + X >= X = X a() = 3 >= 3 = n__a() activate(X) = X >= X = X activate(n____(X1,X2)) = 4 + X1 + X2 >= 4 + X1 + X2 = __(X1,X2) activate(n__a()) = 3 >= 3 = a() activate(n__e()) = 0 >= 0 = e() activate(n__i()) = 0 >= 0 = i() activate(n__isList(X)) = X >= X = isList(X) activate(n__isNeList(X)) = X >= X = isNeList(X) activate(n__isPal(X)) = 7 + X >= 7 + X = isPal(X) activate(n__nil()) = 6 >= 6 = 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) = V >= V = isNeList(activate(V)) isList(X) = X >= X = n__isList(X) isList(n____(V1,V2)) = 4 + V1 + V2 >= 4 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 6 >= 0 = tt() isNeList(V) = V >= V = isQid(activate(V)) isNeList(X) = X >= X = n__isNeList(X) isNeList(n____(V1,V2)) = 4 + V1 + V2 >= 4 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 4 + V1 + V2 >= 4 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(n____(I,__(P,I))) = 15 + 2*I + P >= 11 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = 7 + V >= 7 + V = isNePal(activate(V)) isPal(X) = 7 + X >= 7 + X = n__isPal(X) isPal(n__nil()) = 13 >= 0 = tt() isQid(n__a()) = 3 >= 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() = 6 >= 6 = n__nil() o() = 0 >= 0 = n__o() u() = 0 >= 0 = n__u() * Step 5: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() i() -> n__i() isList(V) -> isNeList(activate(V)) isNeList(X) -> n__isNeList(X) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() 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) e() -> n__e() 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)) isNePal(n____(I,__(P,I))) -> and(isQid(activate(I)),n__isPal(activate(P))) 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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 + x1 + x2 p(a) = 6 p(activate) = x1 p(and) = x1 + x2 p(e) = 1 p(i) = 1 p(isList) = 1 + x1 p(isNeList) = x1 p(isNePal) = x1 p(isPal) = 2 + x1 p(isQid) = x1 p(n____) = 1 + x1 + x2 p(n__a) = 6 p(n__e) = 1 p(n__i) = 1 p(n__isList) = 1 + x1 p(n__isNeList) = x1 p(n__isPal) = 2 + x1 p(n__nil) = 5 p(n__o) = 0 p(n__u) = 1 p(nil) = 5 p(o) = 0 p(tt) = 0 p(u) = 1 Following rules are strictly oriented: isList(V) = 1 + V > V = isNeList(activate(V)) Following rules are (at-least) weakly oriented: __(X,nil()) = 6 + X >= X = X __(X1,X2) = 1 + X1 + X2 >= 1 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 2 + X + Y + Z >= 2 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 6 + X >= X = X a() = 6 >= 6 = n__a() activate(X) = X >= X = X activate(n____(X1,X2)) = 1 + X1 + X2 >= 1 + X1 + X2 = __(X1,X2) activate(n__a()) = 6 >= 6 = a() activate(n__e()) = 1 >= 1 = e() activate(n__i()) = 1 >= 1 = i() activate(n__isList(X)) = 1 + X >= 1 + X = isList(X) activate(n__isNeList(X)) = X >= X = isNeList(X) activate(n__isPal(X)) = 2 + X >= 2 + X = isPal(X) activate(n__nil()) = 5 >= 5 = nil() activate(n__o()) = 0 >= 0 = o() activate(n__u()) = 1 >= 1 = u() and(tt(),X) = X >= X = activate(X) e() = 1 >= 1 = n__e() i() = 1 >= 1 = n__i() isList(X) = 1 + X >= 1 + X = n__isList(X) isList(n____(V1,V2)) = 2 + V1 + V2 >= 2 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 6 >= 0 = tt() isNeList(V) = V >= V = isQid(activate(V)) isNeList(X) = X >= X = n__isNeList(X) isNeList(n____(V1,V2)) = 1 + V1 + V2 >= 1 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 1 + V1 + V2 >= 1 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = V >= V = isQid(activate(V)) isNePal(n____(I,__(P,I))) = 2 + 2*I + P >= 2 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = 2 + V >= V = isNePal(activate(V)) isPal(X) = 2 + X >= 2 + X = n__isPal(X) isPal(n__nil()) = 7 >= 0 = tt() isQid(n__a()) = 6 >= 0 = tt() isQid(n__e()) = 1 >= 0 = tt() isQid(n__i()) = 1 >= 0 = tt() isQid(n__o()) = 0 >= 0 = tt() isQid(n__u()) = 1 >= 0 = tt() nil() = 5 >= 5 = n__nil() o() = 0 >= 0 = n__o() u() = 1 >= 1 = n__u() * Step 6: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() activate(n__u()) -> u() i() -> n__i() isNeList(X) -> n__isNeList(X) isPal(X) -> n__isPal(X) o() -> n__o() - Weak TRS: __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() 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) 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(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(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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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(__) = 4 + x1 + x2 p(a) = 2 p(activate) = 1 + x1 p(and) = x1 + x2 p(e) = 3 p(i) = 1 p(isList) = 3 + x1 p(isNeList) = 1 + x1 p(isNePal) = 2 + x1 p(isPal) = 3 + x1 p(isQid) = x1 p(n____) = 4 + x1 + x2 p(n__a) = 1 p(n__e) = 3 p(n__i) = 1 p(n__isList) = 2 + x1 p(n__isNeList) = x1 p(n__isPal) = 2 + x1 p(n__nil) = 3 p(n__o) = 3 p(n__u) = 4 p(nil) = 4 p(o) = 4 p(tt) = 1 p(u) = 4 Following rules are strictly oriented: a() = 2 > 1 = n__a() activate(n__e()) = 4 > 3 = e() activate(n__u()) = 5 > 4 = u() isNeList(X) = 1 + X > X = n__isNeList(X) isPal(X) = 3 + X > 2 + X = n__isPal(X) o() = 4 > 3 = n__o() Following rules are (at-least) weakly oriented: __(X,nil()) = 8 + X >= X = X __(X1,X2) = 4 + X1 + X2 >= 4 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 8 + X + Y + Z >= 8 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 8 + X >= X = X activate(X) = 1 + X >= X = X activate(n____(X1,X2)) = 5 + X1 + X2 >= 4 + X1 + X2 = __(X1,X2) activate(n__a()) = 2 >= 2 = a() activate(n__i()) = 2 >= 1 = i() activate(n__isList(X)) = 3 + X >= 3 + X = isList(X) activate(n__isNeList(X)) = 1 + X >= 1 + X = isNeList(X) activate(n__isPal(X)) = 3 + X >= 3 + X = isPal(X) activate(n__nil()) = 4 >= 4 = nil() activate(n__o()) = 4 >= 4 = o() and(tt(),X) = 1 + X >= 1 + X = activate(X) e() = 3 >= 3 = n__e() i() = 1 >= 1 = n__i() isList(V) = 3 + V >= 2 + V = isNeList(activate(V)) isList(X) = 3 + X >= 2 + X = n__isList(X) isList(n____(V1,V2)) = 7 + V1 + V2 >= 7 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 6 >= 1 = tt() isNeList(V) = 1 + V >= 1 + V = isQid(activate(V)) isNeList(n____(V1,V2)) = 5 + V1 + V2 >= 5 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 5 + V1 + V2 >= 5 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = 2 + V >= 1 + V = isQid(activate(V)) isNePal(n____(I,__(P,I))) = 10 + 2*I + P >= 4 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = 3 + V >= 3 + V = isNePal(activate(V)) isPal(n__nil()) = 6 >= 1 = tt() isQid(n__a()) = 1 >= 1 = tt() isQid(n__e()) = 3 >= 1 = tt() isQid(n__i()) = 1 >= 1 = tt() isQid(n__o()) = 3 >= 1 = tt() isQid(n__u()) = 4 >= 1 = tt() nil() = 4 >= 3 = n__nil() u() = 4 >= 4 = n__u() * Step 7: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) activate(n__isList(X)) -> isList(X) activate(n__nil()) -> nil() i() -> n__i() - Weak TRS: __(X,nil()) -> X __(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__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)) 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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(__) = 4 + x1 + x2 p(a) = 6 p(activate) = 1 + x1 p(and) = x1 + x2 p(e) = 6 p(i) = 3 p(isList) = 2 + x1 p(isNeList) = 1 + x1 p(isNePal) = 3 + x1 p(isPal) = 4 + x1 p(isQid) = x1 p(n____) = 4 + x1 + x2 p(n__a) = 5 p(n__e) = 6 p(n__i) = 2 p(n__isList) = 2 + x1 p(n__isNeList) = 1 + x1 p(n__isPal) = 3 + x1 p(n__nil) = 2 p(n__o) = 5 p(n__u) = 7 p(nil) = 2 p(o) = 6 p(tt) = 2 p(u) = 7 Following rules are strictly oriented: activate(n__isList(X)) = 3 + X > 2 + X = isList(X) activate(n__nil()) = 3 > 2 = nil() i() = 3 > 2 = n__i() Following rules are (at-least) weakly oriented: __(X,nil()) = 6 + X >= X = X __(X1,X2) = 4 + X1 + X2 >= 4 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 8 + X + Y + Z >= 8 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 6 + X >= X = X a() = 6 >= 5 = n__a() activate(X) = 1 + X >= X = X activate(n____(X1,X2)) = 5 + X1 + X2 >= 4 + X1 + X2 = __(X1,X2) activate(n__a()) = 6 >= 6 = a() activate(n__e()) = 7 >= 6 = e() activate(n__i()) = 3 >= 3 = i() activate(n__isNeList(X)) = 2 + X >= 1 + X = isNeList(X) activate(n__isPal(X)) = 4 + X >= 4 + X = isPal(X) activate(n__o()) = 6 >= 6 = o() activate(n__u()) = 8 >= 7 = u() and(tt(),X) = 2 + X >= 1 + X = activate(X) e() = 6 >= 6 = n__e() isList(V) = 2 + V >= 2 + V = isNeList(activate(V)) isList(X) = 2 + X >= 2 + X = n__isList(X) isList(n____(V1,V2)) = 6 + V1 + V2 >= 6 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 4 >= 2 = tt() isNeList(V) = 1 + V >= 1 + V = isQid(activate(V)) isNeList(X) = 1 + X >= 1 + X = n__isNeList(X) isNeList(n____(V1,V2)) = 5 + V1 + V2 >= 5 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 5 + V1 + V2 >= 5 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = 3 + V >= 1 + V = isQid(activate(V)) isNePal(n____(I,__(P,I))) = 11 + 2*I + P >= 5 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = 4 + V >= 4 + V = isNePal(activate(V)) isPal(X) = 4 + X >= 3 + X = n__isPal(X) isPal(n__nil()) = 6 >= 2 = tt() isQid(n__a()) = 5 >= 2 = tt() isQid(n__e()) = 6 >= 2 = tt() isQid(n__i()) = 2 >= 2 = tt() isQid(n__o()) = 5 >= 2 = tt() isQid(n__u()) = 7 >= 2 = tt() nil() = 2 >= 2 = n__nil() o() = 6 >= 5 = n__o() u() = 7 >= 7 = n__u() * Step 8: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) - Weak TRS: __(X,nil()) -> X __(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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(__) = {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(__) = 7 + x1 + x2 p(a) = 0 p(activate) = 1 + x1 p(and) = 2 + x1 + x2 p(e) = 4 p(i) = 0 p(isList) = 2 + x1 p(isNeList) = 1 + x1 p(isNePal) = 1 + x1 p(isPal) = 4 + x1 p(isQid) = x1 p(n____) = 6 + x1 + x2 p(n__a) = 0 p(n__e) = 3 p(n__i) = 0 p(n__isList) = 1 + x1 p(n__isNeList) = x1 p(n__isPal) = 3 + x1 p(n__nil) = 0 p(n__o) = 0 p(n__u) = 0 p(nil) = 0 p(o) = 1 p(tt) = 0 p(u) = 0 Following rules are strictly oriented: __(X1,X2) = 7 + X1 + X2 > 6 + X1 + X2 = n____(X1,X2) Following rules are (at-least) weakly oriented: __(X,nil()) = 7 + X >= X = X __(__(X,Y),Z) = 14 + X + Y + Z >= 14 + X + Y + Z = __(X,__(Y,Z)) __(nil(),X) = 7 + X >= X = X a() = 0 >= 0 = n__a() activate(X) = 1 + X >= X = X activate(n____(X1,X2)) = 7 + X1 + X2 >= 7 + X1 + X2 = __(X1,X2) activate(n__a()) = 1 >= 0 = a() activate(n__e()) = 4 >= 4 = e() activate(n__i()) = 1 >= 0 = i() activate(n__isList(X)) = 2 + X >= 2 + X = isList(X) activate(n__isNeList(X)) = 1 + X >= 1 + X = isNeList(X) activate(n__isPal(X)) = 4 + X >= 4 + X = isPal(X) activate(n__nil()) = 1 >= 0 = nil() activate(n__o()) = 1 >= 1 = o() activate(n__u()) = 1 >= 0 = u() and(tt(),X) = 2 + X >= 1 + X = activate(X) e() = 4 >= 3 = n__e() i() = 0 >= 0 = n__i() isList(V) = 2 + V >= 2 + V = isNeList(activate(V)) isList(X) = 2 + X >= 1 + X = n__isList(X) isList(n____(V1,V2)) = 8 + V1 + V2 >= 7 + V1 + V2 = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = 2 >= 0 = tt() isNeList(V) = 1 + V >= 1 + V = isQid(activate(V)) isNeList(X) = 1 + X >= X = n__isNeList(X) isNeList(n____(V1,V2)) = 7 + V1 + V2 >= 6 + V1 + V2 = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = 7 + V1 + V2 >= 6 + V1 + V2 = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = 1 + V >= 1 + V = isQid(activate(V)) isNePal(n____(I,__(P,I))) = 14 + 2*I + P >= 7 + I + P = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = 4 + V >= 2 + V = isNePal(activate(V)) isPal(X) = 4 + X >= 3 + X = n__isPal(X) isPal(n__nil()) = 4 >= 0 = tt() isQid(n__a()) = 0 >= 0 = tt() isQid(n__e()) = 3 >= 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() = 1 >= 0 = n__o() u() = 0 >= 0 = n__u() * Step 9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) - Weak TRS: __(X,nil()) -> X __(X1,X2) -> n____(X1,X2) __(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: NaturalMI {miDimension = 3, miDegree = 3, 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(__) = {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 0 1] [1 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a) = [0] [1] [0] p(activate) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(and) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 0] [0 0 1] [0] p(e) = [0] [1] [0] p(i) = [0] [1] [0] p(isList) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(isNeList) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(isNePal) = [1 0 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(isPal) = [1 0 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(isQid) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(n____) = [1 0 1] [1 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(n__a) = [0] [1] [0] p(n__e) = [0] [1] [0] p(n__i) = [0] [1] [0] p(n__isList) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(n__isNeList) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(n__isPal) = [1 0 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(n__nil) = [0] [1] [0] p(n__o) = [0] [1] [0] p(n__u) = [0] [1] [0] p(nil) = [0] [1] [0] p(o) = [0] [1] [0] p(tt) = [0] [1] [0] p(u) = [0] [1] [0] Following rules are strictly oriented: __(__(X,Y),Z) = [1 0 2] [1 0 1] [1 0 0] [1] [0 1 0] X + [0 1 0] Y + [0 1 0] Z + [0] [0 0 1] [0 0 1] [0 0 1] [2] > [1 0 1] [1 0 1] [1 0 0] [0] [0 1 0] X + [0 1 0] Y + [0 1 0] Z + [0] [0 0 1] [0 0 1] [0 0 1] [2] = __(X,__(Y,Z)) Following rules are (at-least) weakly oriented: __(X,nil()) = [1 0 1] [0] [0 1 0] X + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X __(X1,X2) = [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = n____(X1,X2) __(nil(),X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X a() = [0] [1] [0] >= [0] [1] [0] = n__a() activate(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X activate(n____(X1,X2)) = [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = __(X1,X2) activate(n__a()) = [0] [1] [0] >= [0] [1] [0] = a() activate(n__e()) = [0] [1] [0] >= [0] [1] [0] = e() activate(n__i()) = [0] [1] [0] >= [0] [1] [0] = i() activate(n__isList(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] = isList(X) activate(n__isNeList(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] = isNeList(X) activate(n__isPal(X)) = [1 0 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 0] [0] = isPal(X) activate(n__nil()) = [0] [1] [0] >= [0] [1] [0] = nil() activate(n__o()) = [0] [1] [0] >= [0] [1] [0] = o() activate(n__u()) = [0] [1] [0] >= [0] [1] [0] = u() and(tt(),X) = [1 0 0] [0] [0 1 1] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = activate(X) e() = [0] [1] [0] >= [0] [1] [0] = n__e() i() = [0] [1] [0] >= [0] [1] [0] = n__i() isList(V) = [1 0 1] [0] [0 1 0] V + [0] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] V + [0] [0 0 0] [0] = isNeList(activate(V)) isList(X) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] = n__isList(X) isList(n____(V1,V2)) = [1 0 2] [1 0 1] [1] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 1] [0] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] = and(isList(activate(V1)),n__isList(activate(V2))) isList(n__nil()) = [0] [1] [0] >= [0] [1] [0] = tt() isNeList(V) = [1 0 1] [0] [0 1 0] V + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] V + [0] [0 0 0] [0] = isQid(activate(V)) isNeList(X) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] = n__isNeList(X) isNeList(n____(V1,V2)) = [1 0 2] [1 0 1] [1] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 1] [0] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] = and(isList(activate(V1)),n__isNeList(activate(V2))) isNeList(n____(V1,V2)) = [1 0 2] [1 0 1] [1] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 1] [0] [0 1 0] V1 + [0 1 0] V2 + [0] [0 0 0] [0 0 0] [0] = and(isNeList(activate(V1)),n__isList(activate(V2))) isNePal(V) = [1 0 0] [0] [0 1 1] V + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] V + [0] [0 0 0] [0] = isQid(activate(V)) isNePal(n____(I,__(P,I))) = [2 0 1] [1 0 1] [0] [0 2 2] I + [0 1 1] P + [3] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 1] [0] [0 1 0] I + [0 1 1] P + [1] [0 0 0] [0 0 0] [0] = and(isQid(activate(I)),n__isPal(activate(P))) isPal(V) = [1 0 1] [0] [0 1 1] V + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 1] V + [1] [0 0 0] [0] = isNePal(activate(V)) isPal(X) = [1 0 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 0] [0] = n__isPal(X) isPal(n__nil()) = [0] [2] [0] >= [0] [1] [0] = tt() isQid(n__a()) = [0] [1] [0] >= [0] [1] [0] = tt() isQid(n__e()) = [0] [1] [0] >= [0] [1] [0] = tt() isQid(n__i()) = [0] [1] [0] >= [0] [1] [0] = tt() isQid(n__o()) = [0] [1] [0] >= [0] [1] [0] = tt() isQid(n__u()) = [0] [1] [0] >= [0] [1] [0] = tt() nil() = [0] [1] [0] >= [0] [1] [0] = n__nil() o() = [0] [1] [0] >= [0] [1] [0] = n__o() u() = [0] [1] [0] >= [0] [1] [0] = n__u() * Step 10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))