/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__nil,n__o,n__u,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__nil,n__o,n__u,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__nil,n__o,n__u,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: isList(x){x -> n____(x,u)} = isList(n____(x,u)) ->^+ U21(isList(x),activate(u)) = C[isList(x) = isList(x){}] ** Step 1.b:1: NaturalPI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = x1 p(U21) = 1 + x1 + 4*x2 p(U22) = x1 p(U31) = 4*x1 p(U41) = 2 + x1 + 4*x2 p(U42) = x1 p(U51) = 7 + x1 + 4*x2 p(U52) = 6 + x1 p(U61) = x1 p(U71) = 6 + x1 + 2*x2 p(U72) = 3 + x1 p(U81) = 1 + x1 p(__) = 2 + x1 + x2 p(a) = 1 p(activate) = x1 p(e) = 1 p(i) = 5 p(isList) = 1 + 4*x1 p(isNeList) = 4*x1 p(isNePal) = 2*x1 p(isPal) = 1 + 2*x1 p(isQid) = x1 p(n____) = 2 + x1 + x2 p(n__a) = 1 p(n__e) = 1 p(n__i) = 5 p(n__nil) = 2 p(n__o) = 0 p(n__u) = 0 p(nil) = 2 p(o) = 0 p(tt) = 0 p(u) = 0 Following rules are strictly oriented: U41(tt(),V2) = 2 + 4*V2 > 4*V2 = U42(isNeList(activate(V2))) U52(tt()) = 6 > 0 = tt() U71(tt(),P) = 6 + 2*P > 4 + 2*P = U72(isPal(activate(P))) U72(tt()) = 3 > 0 = tt() U81(tt()) = 1 > 0 = tt() __(X,nil()) = 4 + X > X = X __(nil(),X) = 4 + X > X = X isList(V) = 1 + 4*V > 4*V = U11(isNeList(activate(V))) isList(n____(V1,V2)) = 9 + 4*V1 + 4*V2 > 2 + 4*V1 + 4*V2 = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = 9 > 0 = tt() isNeList(n____(V1,V2)) = 8 + 4*V1 + 4*V2 > 3 + 4*V1 + 4*V2 = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = 8 + 4*V1 + 4*V2 > 7 + 4*V1 + 4*V2 = U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) = 8 + 4*I + 2*P > 6 + I + 2*P = U71(isQid(activate(I)),activate(P)) isPal(n__nil()) = 5 > 0 = tt() isQid(n__a()) = 1 > 0 = tt() isQid(n__e()) = 1 > 0 = tt() isQid(n__i()) = 5 > 0 = tt() Following rules are (at-least) weakly oriented: U11(tt()) = 0 >= 0 = tt() U21(tt(),V2) = 1 + 4*V2 >= 1 + 4*V2 = U22(isList(activate(V2))) U22(tt()) = 0 >= 0 = tt() U31(tt()) = 0 >= 0 = tt() U42(tt()) = 0 >= 0 = tt() U51(tt(),V2) = 7 + 4*V2 >= 7 + 4*V2 = U52(isList(activate(V2))) U61(tt()) = 0 >= 0 = tt() __(X1,X2) = 2 + X1 + X2 >= 2 + X1 + X2 = n____(X1,X2) __(__(X,Y),Z) = 4 + X + Y + Z >= 4 + X + Y + Z = __(X,__(Y,Z)) a() = 1 >= 1 = n__a() activate(X) = X >= X = X activate(n____(X1,X2)) = 2 + X1 + X2 >= 2 + X1 + X2 = __(X1,X2) activate(n__a()) = 1 >= 1 = a() activate(n__e()) = 1 >= 1 = e() activate(n__i()) = 5 >= 5 = i() activate(n__nil()) = 2 >= 2 = nil() activate(n__o()) = 0 >= 0 = o() activate(n__u()) = 0 >= 0 = u() e() = 1 >= 1 = n__e() i() = 5 >= 5 = n__i() isNeList(V) = 4*V >= 4*V = U31(isQid(activate(V))) isNePal(V) = 2*V >= V = U61(isQid(activate(V))) isPal(V) = 1 + 2*V >= 1 + 2*V = U81(isNePal(activate(V))) isQid(n__o()) = 0 >= 0 = tt() isQid(n__u()) = 0 >= 0 = tt() nil() = 2 >= 2 = n__nil() o() = 0 >= 0 = n__o() u() = 0 >= 0 = n__u() ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U61(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(V) -> U31(isQid(activate(V))) isNePal(V) -> U61(isQid(activate(V))) isPal(V) -> U81(isNePal(activate(V))) isQid(n__o()) -> tt() isQid(n__u()) -> tt() nil() -> n__nil() o() -> n__o() u() -> n__u() - Weak TRS: U41(tt(),V2) -> U42(isNeList(activate(V2))) U52(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(n__nil()) -> tt() isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [5] p(U22) = [1] x1 + [2] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [5] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [3] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [5] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [2] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [3] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [5] p(n__a) = [0] p(n__e) = [2] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [5] p(n__u) = [2] p(nil) = [0] p(o) = [5] p(tt) = [0] p(u) = [2] Following rules are strictly oriented: U21(tt(),V2) = [1] V2 + [5] > [1] V2 + [2] = U22(isList(activate(V2))) U22(tt()) = [2] > [0] = tt() U51(tt(),V2) = [1] V2 + [5] > [1] V2 + [0] = U52(isList(activate(V2))) isPal(V) = [1] V + [3] > [1] V + [0] = U81(isNePal(activate(V))) isQid(n__o()) = [5] > [0] = tt() isQid(n__u()) = [2] > [0] = tt() Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [0] = tt() U52(tt()) = [0] >= [0] = tt() U61(tt()) = [0] >= [0] = tt() U71(tt(),P) = [1] P + [3] >= [1] P + [3] = U72(isPal(activate(P))) U72(tt()) = [0] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [5] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [10] >= [1] X + [1] Y + [1] Z + [10] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [5] >= [1] X + [0] = X a() = [0] >= [0] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = __(X1,X2) activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [2] >= [2] = e() activate(n__i()) = [0] >= [0] = i() activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [5] >= [5] = o() activate(n__u()) = [2] >= [2] = u() e() = [2] >= [2] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [0] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [0] >= [1] V + [0] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [10] >= [1] I + [1] P + [3] = U71(isQid(activate(I)),activate(P)) isPal(n__nil()) = [3] >= [0] = tt() isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [2] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [5] >= [5] = n__o() u() = [2] >= [2] = n__u() ** Step 1.b:3: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U31(tt()) -> tt() U42(tt()) -> tt() U61(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isNeList(V) -> U31(isQid(activate(V))) isNePal(V) -> U61(isQid(activate(V))) nil() -> n__nil() o() -> n__o() u() -> n__u() - Weak TRS: U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [2] x2 + [6] p(U22) = [1] x1 + [0] p(U31) = [2] x1 + [0] p(U41) = [1] x1 + [2] x2 + [4] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [2] x2 + [6] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [7] p(U72) = [1] x1 + [4] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [5] p(a) = [0] p(activate) = [1] x1 + [1] p(e) = [1] p(i) = [2] p(isList) = [2] x1 + [4] p(isNeList) = [2] x1 + [2] p(isNePal) = [1] x1 + [1] p(isPal) = [1] x1 + [2] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [5] p(n__a) = [0] p(n__e) = [0] p(n__i) = [1] p(n__nil) = [3] p(n__o) = [4] p(n__u) = [0] p(nil) = [4] p(o) = [5] p(tt) = [0] p(u) = [0] Following rules are strictly oriented: activate(X) = [1] X + [1] > [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = __(X1,X2) activate(n__a()) = [1] > [0] = a() activate(n__u()) = [1] > [0] = u() e() = [1] > [0] = n__e() i() = [2] > [1] = n__i() nil() = [4] > [3] = n__nil() o() = [5] > [4] = n__o() Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U21(tt(),V2) = [2] V2 + [6] >= [2] V2 + [6] = U22(isList(activate(V2))) U22(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [2] V2 + [4] >= [2] V2 + [4] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [0] = tt() U51(tt(),V2) = [2] V2 + [6] >= [2] V2 + [6] = U52(isList(activate(V2))) U52(tt()) = [0] >= [0] = tt() U61(tt()) = [0] >= [0] = tt() U71(tt(),P) = [1] P + [7] >= [1] P + [7] = U72(isPal(activate(P))) U72(tt()) = [4] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [9] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [10] >= [1] X + [1] Y + [1] Z + [10] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [9] >= [1] X + [0] = X a() = [0] >= [0] = n__a() activate(n__e()) = [1] >= [1] = e() activate(n__i()) = [2] >= [2] = i() activate(n__nil()) = [4] >= [4] = nil() activate(n__o()) = [5] >= [5] = o() isList(V) = [2] V + [4] >= [2] V + [4] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [2] V1 + [2] V2 + [14] >= [2] V1 + [2] V2 + [14] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [10] >= [0] = tt() isNeList(V) = [2] V + [2] >= [2] V + [2] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [12] >= [2] V1 + [2] V2 + [12] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [12] >= [2] V1 + [2] V2 + [12] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [1] >= [1] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [11] >= [1] I + [1] P + [9] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [2] >= [1] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [5] >= [0] = tt() isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [0] >= [0] = tt() isQid(n__i()) = [1] >= [0] = tt() isQid(n__o()) = [4] >= [0] = tt() isQid(n__u()) = [0] >= [0] = tt() u() = [0] >= [0] = n__u() ** Step 1.b:4: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U31(tt()) -> tt() U42(tt()) -> tt() U61(tt()) -> tt() __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() isNeList(V) -> U31(isQid(activate(V))) isNePal(V) -> U61(isQid(activate(V))) u() -> n__u() - Weak TRS: U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [2] x2 + [0] p(U22) = [1] x1 + [2] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [2] x2 + [0] p(U42) = [1] x1 + [2] p(U51) = [1] x1 + [2] x2 + [0] p(U52) = [1] x1 + [1] p(U61) = [2] x1 + [1] p(U71) = [2] x1 + [4] x2 + [0] p(U72) = [1] x1 + [2] p(U81) = [1] x1 + [1] p(__) = [1] x1 + [1] x2 + [0] p(a) = [1] p(activate) = [1] x1 + [0] p(e) = [1] p(i) = [2] p(isList) = [2] x1 + [0] p(isNeList) = [2] x1 + [0] p(isNePal) = [4] x1 + [1] p(isPal) = [4] x1 + [2] p(isQid) = [2] x1 + [0] p(n____) = [1] x1 + [1] x2 + [0] p(n__a) = [1] p(n__e) = [1] p(n__i) = [2] p(n__nil) = [2] p(n__o) = [1] p(n__u) = [1] p(nil) = [2] p(o) = [1] p(tt) = [2] p(u) = [1] Following rules are strictly oriented: U42(tt()) = [4] > [2] = tt() U61(tt()) = [5] > [2] = tt() Following rules are (at-least) weakly oriented: U11(tt()) = [2] >= [2] = tt() U21(tt(),V2) = [2] V2 + [2] >= [2] V2 + [2] = U22(isList(activate(V2))) U22(tt()) = [4] >= [2] = tt() U31(tt()) = [2] >= [2] = tt() U41(tt(),V2) = [2] V2 + [2] >= [2] V2 + [2] = U42(isNeList(activate(V2))) U51(tt(),V2) = [2] V2 + [2] >= [2] V2 + [1] = U52(isList(activate(V2))) U52(tt()) = [3] >= [2] = tt() U71(tt(),P) = [4] P + [4] >= [4] P + [4] = U72(isPal(activate(P))) U72(tt()) = [4] >= [2] = tt() U81(tt()) = [3] >= [2] = tt() __(X,nil()) = [1] X + [2] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [0] >= [1] X + [1] Y + [1] Z + [0] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [2] >= [1] X + [0] = X a() = [1] >= [1] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = __(X1,X2) activate(n__a()) = [1] >= [1] = a() activate(n__e()) = [1] >= [1] = e() activate(n__i()) = [2] >= [2] = i() activate(n__nil()) = [2] >= [2] = nil() activate(n__o()) = [1] >= [1] = o() activate(n__u()) = [1] >= [1] = u() e() = [1] >= [1] = n__e() i() = [2] >= [2] = n__i() isList(V) = [2] V + [0] >= [2] V + [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [2] = tt() isNeList(V) = [2] V + [0] >= [2] V + [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [4] V + [1] >= [4] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [8] I + [4] P + [1] >= [4] I + [4] P + [0] = U71(isQid(activate(I)),activate(P)) isPal(V) = [4] V + [2] >= [4] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [10] >= [2] = tt() isQid(n__a()) = [2] >= [2] = tt() isQid(n__e()) = [2] >= [2] = tt() isQid(n__i()) = [4] >= [2] = tt() isQid(n__o()) = [2] >= [2] = tt() isQid(n__u()) = [2] >= [2] = tt() nil() = [2] >= [2] = n__nil() o() = [1] >= [1] = n__o() u() = [1] >= [1] = n__u() ** Step 1.b:5: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U31(tt()) -> tt() __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() isNeList(V) -> U31(isQid(activate(V))) isNePal(V) -> U61(isQid(activate(V))) u() -> n__u() - Weak TRS: U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [1] p(U22) = [1] x1 + [1] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [2] p(U42) = [1] x1 + [2] p(U51) = [1] x1 + [1] x2 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [2] x1 + [1] x2 + [4] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [2] p(a) = [1] p(activate) = [1] x1 + [0] p(e) = [1] 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 + [0] p(n____) = [1] x1 + [1] x2 + [2] p(n__a) = [1] p(n__e) = [1] p(n__i) = [0] p(n__nil) = [2] p(n__o) = [2] p(n__u) = [0] p(nil) = [2] p(o) = [2] p(tt) = [0] p(u) = [0] Following rules are strictly oriented: isNePal(V) = [1] V + [4] > [1] V + [0] = U61(isQid(activate(V))) Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U21(tt(),V2) = [1] V2 + [1] >= [1] V2 + [1] = U22(isList(activate(V2))) U22(tt()) = [1] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [1] V2 + [2] >= [1] V2 + [2] = U42(isNeList(activate(V2))) U42(tt()) = [2] >= [0] = tt() U51(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = U52(isList(activate(V2))) U52(tt()) = [0] >= [0] = tt() U61(tt()) = [0] >= [0] = tt() U71(tt(),P) = [1] P + [4] >= [1] P + [4] = U72(isPal(activate(P))) U72(tt()) = [0] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [4] >= [1] X + [0] = X __(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)) __(nil(),X) = [1] X + [4] >= [1] X + [0] = X a() = [1] >= [1] = 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()) = [1] >= [1] = a() activate(n__e()) = [1] >= [1] = e() activate(n__i()) = [0] >= [0] = i() activate(n__nil()) = [2] >= [2] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [0] >= [0] = u() e() = [1] >= [1] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [1] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [2] >= [0] = tt() isNeList(V) = [1] V + [0] >= [1] V + [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [2] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [0] = U51(isNeList(activate(V1)),activate(V2)) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [8] >= [2] I + [1] P + [4] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [4] >= [1] V + [4] = U81(isNePal(activate(V))) isPal(n__nil()) = [6] >= [0] = tt() isQid(n__a()) = [1] >= [0] = tt() isQid(n__e()) = [1] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() isQid(n__o()) = [2] >= [0] = tt() isQid(n__u()) = [0] >= [0] = tt() nil() = [2] >= [2] = n__nil() o() = [2] >= [2] = n__o() u() = [0] >= [0] = n__u() ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U31(tt()) -> tt() __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() isNeList(V) -> U31(isQid(activate(V))) u() -> n__u() - Weak TRS: U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [4] p(U21) = [1] x1 + [2] x2 + [4] p(U22) = [1] x1 + [0] p(U31) = [2] x1 + [0] p(U41) = [1] x1 + [2] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [2] x2 + [4] p(U52) = [1] x1 + [0] p(U61) = [2] x1 + [0] p(U71) = [4] x1 + [2] x2 + [0] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [2] p(a) = [1] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [2] p(isList) = [2] x1 + [4] p(isNeList) = [2] x1 + [0] p(isNePal) = [2] x1 + [0] p(isPal) = [2] x1 + [0] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [2] p(n__a) = [1] p(n__e) = [0] p(n__i) = [2] p(n__nil) = [0] p(n__o) = [4] p(n__u) = [4] p(nil) = [0] p(o) = [4] p(tt) = [0] p(u) = [4] Following rules are strictly oriented: U11(tt()) = [4] > [0] = tt() Following rules are (at-least) weakly oriented: U21(tt(),V2) = [2] V2 + [4] >= [2] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [0] = tt() U51(tt(),V2) = [2] V2 + [4] >= [2] V2 + [4] = U52(isList(activate(V2))) U52(tt()) = [0] >= [0] = tt() U61(tt()) = [0] >= [0] = tt() U71(tt(),P) = [2] P + [0] >= [2] P + [0] = U72(isPal(activate(P))) U72(tt()) = [0] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [2] >= [1] X + [0] = X __(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)) __(nil(),X) = [1] X + [2] >= [1] X + [0] = X a() = [1] >= [1] = 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()) = [1] >= [1] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [2] >= [2] = i() activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [4] >= [4] = o() activate(n__u()) = [4] >= [4] = u() e() = [0] >= [0] = n__e() i() = [2] >= [2] = n__i() isList(V) = [2] V + [4] >= [2] V + [4] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [2] V1 + [2] V2 + [8] >= [2] V1 + [2] V2 + [8] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [0] = tt() isNeList(V) = [2] V + [0] >= [2] V + [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [4] >= [2] V1 + [2] V2 + [4] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [4] >= [2] V1 + [2] V2 + [4] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [2] V + [0] >= [2] V + [0] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [4] I + [2] P + [8] >= [4] I + [2] P + [0] = U71(isQid(activate(I)),activate(P)) isPal(V) = [2] V + [0] >= [2] V + [0] = U81(isNePal(activate(V))) isPal(n__nil()) = [0] >= [0] = tt() isQid(n__a()) = [1] >= [0] = tt() isQid(n__e()) = [0] >= [0] = tt() isQid(n__i()) = [2] >= [0] = tt() isQid(n__o()) = [4] >= [0] = tt() isQid(n__u()) = [4] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [4] >= [4] = n__o() u() = [4] >= [4] = n__u() ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: U31(tt()) -> tt() __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(n__e()) -> e() activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() isNeList(V) -> U31(isQid(activate(V))) u() -> n__u() - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [4] x2 + [4] p(U22) = [1] x1 + [4] p(U31) = [2] x1 + [0] p(U41) = [1] x1 + [4] x2 + [0] p(U42) = [1] x1 + [2] p(U51) = [1] x1 + [4] x2 + [2] p(U52) = [1] x1 + [4] p(U61) = [2] x1 + [0] p(U71) = [2] x1 + [4] x2 + [4] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [6] p(__) = [1] x1 + [1] x2 + [1] p(a) = [6] p(activate) = [1] x1 + [0] p(e) = [1] p(i) = [1] p(isList) = [4] x1 + [0] p(isNeList) = [4] x1 + [0] p(isNePal) = [4] x1 + [0] p(isPal) = [4] x1 + [6] p(isQid) = [2] x1 + [0] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [6] p(n__e) = [1] p(n__i) = [1] p(n__nil) = [1] p(n__o) = [2] p(n__u) = [1] p(nil) = [1] p(o) = [2] p(tt) = [2] p(u) = [1] Following rules are strictly oriented: U31(tt()) = [4] > [2] = tt() Following rules are (at-least) weakly oriented: U11(tt()) = [2] >= [2] = tt() U21(tt(),V2) = [4] V2 + [6] >= [4] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [6] >= [2] = tt() U41(tt(),V2) = [4] V2 + [2] >= [4] V2 + [2] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [2] = tt() U51(tt(),V2) = [4] V2 + [4] >= [4] V2 + [4] = U52(isList(activate(V2))) U52(tt()) = [6] >= [2] = tt() U61(tt()) = [4] >= [2] = tt() U71(tt(),P) = [4] P + [8] >= [4] P + [6] = U72(isPal(activate(P))) U72(tt()) = [2] >= [2] = tt() U81(tt()) = [8] >= [2] = tt() __(X,nil()) = [1] X + [2] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [2] >= [1] X + [1] Y + [1] Z + [2] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [2] >= [1] X + [0] = X a() = [6] >= [6] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = __(X1,X2) activate(n__a()) = [6] >= [6] = a() activate(n__e()) = [1] >= [1] = e() activate(n__i()) = [1] >= [1] = i() activate(n__nil()) = [1] >= [1] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [1] >= [1] = u() e() = [1] >= [1] = n__e() i() = [1] >= [1] = n__i() isList(V) = [4] V + [0] >= [4] V + [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [4] V1 + [4] V2 + [4] >= [4] V1 + [4] V2 + [4] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [2] = tt() isNeList(V) = [4] V + [0] >= [4] V + [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [4] V1 + [4] V2 + [4] >= [4] V1 + [4] V2 + [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [4] V1 + [4] V2 + [4] >= [4] V1 + [4] V2 + [2] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [4] V + [0] >= [4] V + [0] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [8] I + [4] P + [8] >= [4] I + [4] P + [4] = U71(isQid(activate(I)),activate(P)) isPal(V) = [4] V + [6] >= [4] V + [6] = U81(isNePal(activate(V))) isPal(n__nil()) = [10] >= [2] = tt() isQid(n__a()) = [12] >= [2] = tt() isQid(n__e()) = [2] >= [2] = tt() isQid(n__i()) = [2] >= [2] = tt() isQid(n__o()) = [4] >= [2] = tt() isQid(n__u()) = [2] >= [2] = tt() nil() = [1] >= [1] = n__nil() o() = [2] >= [2] = n__o() u() = [1] >= [1] = n__u() ** Step 1.b:8: NaturalMI. 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__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() isNeList(V) -> U31(isQid(activate(V))) u() -> n__u() - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [4] x2 + [4] p(U22) = [1] x1 + [0] p(U31) = [4] x1 + [0] p(U41) = [1] x1 + [4] x2 + [4] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [4] x2 + [4] p(U52) = [1] x1 + [0] p(U61) = [4] x1 + [1] p(U71) = [4] x1 + [4] x2 + [3] p(U72) = [1] x1 + [1] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [1] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [0] p(isList) = [4] x1 + [4] p(isNeList) = [4] x1 + [4] p(isNePal) = [4] x1 + [2] p(isPal) = [4] x1 + [2] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [2] p(n__u) = [1] p(nil) = [0] p(o) = [2] p(tt) = [0] p(u) = [1] Following rules are strictly oriented: isNeList(V) = [4] V + [4] > [4] V + [0] = U31(isQid(activate(V))) Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U21(tt(),V2) = [4] V2 + [4] >= [4] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [4] V2 + [4] >= [4] V2 + [4] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [0] = tt() U51(tt(),V2) = [4] V2 + [4] >= [4] V2 + [4] = U52(isList(activate(V2))) U52(tt()) = [0] >= [0] = tt() U61(tt()) = [1] >= [0] = tt() U71(tt(),P) = [4] P + [3] >= [4] P + [3] = U72(isPal(activate(P))) U72(tt()) = [1] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [1] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [2] >= [1] X + [1] Y + [1] Z + [2] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [1] >= [1] X + [0] = X a() = [0] >= [0] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = __(X1,X2) activate(n__a()) = [0] >= [0] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [1] >= [1] = u() e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(V) = [4] V + [4] >= [4] V + [4] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [0] = tt() isNeList(n____(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [4] V + [2] >= [4] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [8] I + [4] P + [10] >= [4] I + [4] P + [3] = U71(isQid(activate(I)),activate(P)) isPal(V) = [4] V + [2] >= [4] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [2] >= [0] = tt() isQid(n__a()) = [0] >= [0] = tt() isQid(n__e()) = [0] >= [0] = tt() isQid(n__i()) = [0] >= [0] = tt() isQid(n__o()) = [2] >= [0] = tt() isQid(n__u()) = [1] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [2] >= [2] = n__o() u() = [1] >= [1] = n__u() ** Step 1.b:9: NaturalMI. 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__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() u() -> n__u() - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [4] p(U22) = [1] x1 + [2] p(U31) = [1] x1 + [4] p(U41) = [1] x1 + [1] x2 + [3] p(U42) = [1] x1 + [1] p(U51) = [1] x1 + [1] x2 + [5] p(U52) = [1] x1 + [1] p(U61) = [1] x1 + [0] p(U71) = [2] x1 + [1] x2 + [3] p(U72) = [1] x1 + [6] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [7] p(a) = [6] p(activate) = [1] x1 + [1] p(e) = [7] p(i) = [6] p(isList) = [1] x1 + [7] p(isNeList) = [1] x1 + [6] p(isNePal) = [1] x1 + [1] p(isPal) = [1] x1 + [3] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [7] p(n__a) = [6] p(n__e) = [6] p(n__i) = [6] p(n__nil) = [4] p(n__o) = [6] p(n__u) = [6] p(nil) = [4] p(o) = [6] p(tt) = [6] p(u) = [7] Following rules are strictly oriented: activate(n__i()) = [7] > [6] = i() activate(n__nil()) = [5] > [4] = nil() activate(n__o()) = [7] > [6] = o() u() = [7] > [6] = n__u() Following rules are (at-least) weakly oriented: U11(tt()) = [6] >= [6] = tt() U21(tt(),V2) = [1] V2 + [10] >= [1] V2 + [10] = U22(isList(activate(V2))) U22(tt()) = [8] >= [6] = tt() U31(tt()) = [10] >= [6] = tt() U41(tt(),V2) = [1] V2 + [9] >= [1] V2 + [8] = U42(isNeList(activate(V2))) U42(tt()) = [7] >= [6] = tt() U51(tt(),V2) = [1] V2 + [11] >= [1] V2 + [9] = U52(isList(activate(V2))) U52(tt()) = [7] >= [6] = tt() U61(tt()) = [6] >= [6] = tt() U71(tt(),P) = [1] P + [15] >= [1] P + [10] = U72(isPal(activate(P))) U72(tt()) = [12] >= [6] = tt() U81(tt()) = [6] >= [6] = tt() __(X,nil()) = [1] X + [11] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [14] >= [1] X + [1] Y + [1] Z + [14] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [11] >= [1] X + [0] = X a() = [6] >= [6] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [7] = __(X1,X2) activate(n__a()) = [7] >= [6] = a() activate(n__e()) = [7] >= [7] = e() activate(n__u()) = [7] >= [7] = u() e() = [7] >= [6] = n__e() i() = [6] >= [6] = n__i() isList(V) = [1] V + [7] >= [1] V + [7] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [14] >= [1] V1 + [1] V2 + [13] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [11] >= [6] = tt() isNeList(V) = [1] V + [6] >= [1] V + [5] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [13] >= [1] V1 + [1] V2 + [12] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [13] >= [1] V1 + [1] V2 + [13] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [1] >= [1] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [15] >= [2] I + [1] P + [6] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [3] >= [1] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [7] >= [6] = tt() isQid(n__a()) = [6] >= [6] = tt() isQid(n__e()) = [6] >= [6] = tt() isQid(n__i()) = [6] >= [6] = tt() isQid(n__o()) = [6] >= [6] = tt() isQid(n__u()) = [6] >= [6] = tt() nil() = [4] >= [4] = n__nil() o() = [6] >= [6] = n__o() ** Step 1.b:10: NaturalMI. 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() - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [2] x2 + [1] p(U22) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [2] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [2] x2 + [2] p(U52) = [1] x1 + [1] p(U61) = [1] x1 + [4] p(U71) = [2] x1 + [1] x2 + [1] p(U72) = [1] x1 + [1] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [3] p(a) = [5] p(activate) = [1] x1 + [1] p(e) = [5] p(i) = [5] p(isList) = [2] x1 + [3] p(isNeList) = [2] x1 + [1] p(isNePal) = [1] x1 + [5] p(isPal) = [1] x1 + [7] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [3] p(n__a) = [4] p(n__e) = [4] p(n__i) = [5] p(n__nil) = [2] p(n__o) = [5] p(n__u) = [5] p(nil) = [3] p(o) = [5] p(tt) = [4] p(u) = [5] Following rules are strictly oriented: a() = [5] > [4] = n__a() Following rules are (at-least) weakly oriented: U11(tt()) = [4] >= [4] = tt() U21(tt(),V2) = [2] V2 + [5] >= [2] V2 + [5] = U22(isList(activate(V2))) U22(tt()) = [4] >= [4] = tt() U31(tt()) = [4] >= [4] = tt() U41(tt(),V2) = [2] V2 + [4] >= [2] V2 + [3] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [4] = tt() U51(tt(),V2) = [2] V2 + [6] >= [2] V2 + [6] = U52(isList(activate(V2))) U52(tt()) = [5] >= [4] = tt() U61(tt()) = [8] >= [4] = tt() U71(tt(),P) = [1] P + [9] >= [1] P + [9] = U72(isPal(activate(P))) U72(tt()) = [5] >= [4] = tt() U81(tt()) = [4] >= [4] = tt() __(X,nil()) = [1] X + [6] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [6] >= [1] X + [1] Y + [1] Z + [6] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [6] >= [1] X + [0] = X activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [3] = __(X1,X2) activate(n__a()) = [5] >= [5] = a() activate(n__e()) = [5] >= [5] = e() activate(n__i()) = [6] >= [5] = i() activate(n__nil()) = [3] >= [3] = nil() activate(n__o()) = [6] >= [5] = o() activate(n__u()) = [6] >= [5] = u() e() = [5] >= [4] = n__e() i() = [5] >= [5] = n__i() isList(V) = [2] V + [3] >= [2] V + [3] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [2] V1 + [2] V2 + [9] >= [2] V1 + [2] V2 + [8] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [7] >= [4] = tt() isNeList(V) = [2] V + [1] >= [1] V + [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [7] >= [2] V1 + [2] V2 + [7] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [2] V1 + [2] V2 + [7] >= [2] V1 + [2] V2 + [7] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [5] >= [1] V + [5] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [11] >= [2] I + [1] P + [4] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [7] >= [1] V + [6] = U81(isNePal(activate(V))) isPal(n__nil()) = [9] >= [4] = tt() isQid(n__a()) = [4] >= [4] = tt() isQid(n__e()) = [4] >= [4] = tt() isQid(n__i()) = [5] >= [4] = tt() isQid(n__o()) = [5] >= [4] = tt() isQid(n__u()) = [5] >= [4] = tt() nil() = [3] >= [2] = n__nil() o() = [5] >= [5] = n__o() u() = [5] >= [5] = n__u() ** Step 1.b:11: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) activate(n__e()) -> e() - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__a()) -> a() activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [0] p(U22) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [1] p(U52) = [1] x1 + [0] p(U61) = [2] x1 + [1] p(U71) = [1] x1 + [2] x2 + [7] p(U72) = [1] x1 + [3] p(U81) = [1] x1 + [1] p(__) = [1] x1 + [1] x2 + [3] p(a) = [6] p(activate) = [1] x1 + [1] p(e) = [4] p(i) = [7] p(isList) = [1] x1 + [3] p(isNeList) = [1] x1 + [2] p(isNePal) = [2] x1 + [3] p(isPal) = [2] x1 + [6] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [3] p(n__a) = [6] p(n__e) = [4] p(n__i) = [7] p(n__nil) = [1] p(n__o) = [5] p(n__u) = [4] p(nil) = [2] p(o) = [6] p(tt) = [4] p(u) = [4] Following rules are strictly oriented: activate(n__e()) = [5] > [4] = e() Following rules are (at-least) weakly oriented: U11(tt()) = [4] >= [4] = tt() U21(tt(),V2) = [1] V2 + [4] >= [1] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [4] >= [4] = tt() U31(tt()) = [4] >= [4] = tt() U41(tt(),V2) = [1] V2 + [4] >= [1] V2 + [3] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [4] = tt() U51(tt(),V2) = [1] V2 + [5] >= [1] V2 + [4] = U52(isList(activate(V2))) U52(tt()) = [4] >= [4] = tt() U61(tt()) = [9] >= [4] = tt() U71(tt(),P) = [2] P + [11] >= [2] P + [11] = U72(isPal(activate(P))) U72(tt()) = [7] >= [4] = tt() U81(tt()) = [5] >= [4] = tt() __(X,nil()) = [1] X + [5] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [6] >= [1] X + [1] Y + [1] Z + [6] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [5] >= [1] X + [0] = X a() = [6] >= [6] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [3] = __(X1,X2) activate(n__a()) = [7] >= [6] = a() activate(n__i()) = [8] >= [7] = i() activate(n__nil()) = [2] >= [2] = nil() activate(n__o()) = [6] >= [6] = o() activate(n__u()) = [5] >= [4] = u() e() = [4] >= [4] = n__e() i() = [7] >= [7] = n__i() isList(V) = [1] V + [3] >= [1] V + [3] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [5] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [4] = tt() isNeList(V) = [1] V + [2] >= [1] V + [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [2] V + [3] >= [2] V + [3] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [4] I + [2] P + [15] >= [1] I + [2] P + [10] = U71(isQid(activate(I)),activate(P)) isPal(V) = [2] V + [6] >= [2] V + [6] = U81(isNePal(activate(V))) isPal(n__nil()) = [8] >= [4] = tt() isQid(n__a()) = [6] >= [4] = tt() isQid(n__e()) = [4] >= [4] = tt() isQid(n__i()) = [7] >= [4] = tt() isQid(n__o()) = [5] >= [4] = tt() isQid(n__u()) = [4] >= [4] = tt() nil() = [2] >= [1] = n__nil() o() = [6] >= [5] = n__o() u() = [4] >= [4] = n__u() ** Step 1.b:12: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(X1,X2) -> n____(X1,X2) __(__(X,Y),Z) -> __(X,__(Y,Z)) - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [2] p(U22) = [1] x1 + [3] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [1] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [1] p(U71) = [1] x1 + [1] x2 + [1] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [5] p(a) = [5] p(activate) = [1] x1 + [1] p(e) = [5] p(i) = [5] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [2] p(isPal) = [1] x1 + [4] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [4] p(n__a) = [5] p(n__e) = [4] p(n__i) = [5] p(n__nil) = [2] p(n__o) = [4] p(n__u) = [4] p(nil) = [2] p(o) = [5] p(tt) = [4] p(u) = [4] Following rules are strictly oriented: __(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = n____(X1,X2) Following rules are (at-least) weakly oriented: U11(tt()) = [4] >= [4] = tt() U21(tt(),V2) = [1] V2 + [6] >= [1] V2 + [6] = U22(isList(activate(V2))) U22(tt()) = [7] >= [4] = tt() U31(tt()) = [4] >= [4] = tt() U41(tt(),V2) = [1] V2 + [4] >= [1] V2 + [2] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [4] = tt() U51(tt(),V2) = [1] V2 + [5] >= [1] V2 + [3] = U52(isList(activate(V2))) U52(tt()) = [4] >= [4] = tt() U61(tt()) = [5] >= [4] = tt() U71(tt(),P) = [1] P + [5] >= [1] P + [5] = U72(isPal(activate(P))) U72(tt()) = [4] >= [4] = tt() U81(tt()) = [4] >= [4] = tt() __(X,nil()) = [1] X + [7] >= [1] X + [0] = X __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [10] >= [1] X + [1] Y + [1] Z + [10] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [7] >= [1] X + [0] = X a() = [5] >= [5] = n__a() activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = __(X1,X2) activate(n__a()) = [6] >= [5] = a() activate(n__e()) = [5] >= [5] = e() activate(n__i()) = [6] >= [5] = i() activate(n__nil()) = [3] >= [2] = nil() activate(n__o()) = [5] >= [5] = o() activate(n__u()) = [5] >= [4] = u() e() = [5] >= [4] = n__e() i() = [5] >= [5] = n__i() isList(V) = [1] V + [2] >= [1] V + [2] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [6] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [4] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [2] >= [1] V + [2] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [11] >= [1] I + [1] P + [3] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [4] >= [1] V + [3] = U81(isNePal(activate(V))) isPal(n__nil()) = [6] >= [4] = tt() isQid(n__a()) = [5] >= [4] = tt() isQid(n__e()) = [4] >= [4] = tt() isQid(n__i()) = [5] >= [4] = tt() isQid(n__o()) = [4] >= [4] = tt() isQid(n__u()) = [4] >= [4] = tt() nil() = [2] >= [2] = n__nil() o() = [5] >= [4] = n__o() u() = [4] >= [4] = n__u() ** Step 1.b:13: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,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(U11) = {1}, uargs(U21) = {1,2}, uargs(U22) = {1}, uargs(U31) = {1}, uargs(U41) = {1,2}, uargs(U42) = {1}, uargs(U51) = {1,2}, uargs(U52) = {1}, uargs(U61) = {1}, uargs(U71) = {1,2}, uargs(U72) = {1}, uargs(U81) = {1}, uargs(__) = {2}, uargs(activate) = {1}, uargs(isList) = {1}, uargs(isNeList) = {1}, uargs(isNePal) = {1}, uargs(isPal) = {1}, uargs(isQid) = {1}, uargs(n____) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U21) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [1] p(U22) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] p(U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(U41) = [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 1 0] [0] p(U42) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(U51) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(U61) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 0] [0] p(U71) = [1 1 1] [1 1 0] [0] [1 0 0] x1 + [0 0 1] x2 + [1] [0 1 0] [0 0 1] [0] p(U72) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(U81) = [1 0 0] [0] [0 0 0] x1 + [1] [0 1 0] [0] p(__) = [1 0 1] [1 0 0] [1] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a) = [1] [0] [0] p(activate) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(e) = [1] [0] [1] p(i) = [1] [1] [0] p(isList) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(isNeList) = [1 0 0] [0] [0 0 0] x1 + [0] [0 1 0] [1] p(isNePal) = [1 1 0] [1] [1 1 0] x1 + [1] [1 1 0] [0] p(isPal) = [1 1 0] [1] [0 0 1] x1 + [1] [1 1 1] [1] p(isQid) = [1 0 0] [0] [1 1 0] x1 + [1] [0 0 1] [1] p(n____) = [1 0 1] [1 0 0] [1] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(n__a) = [1] [0] [0] p(n__e) = [1] [0] [0] p(n__i) = [1] [1] [0] p(n__nil) = [1] [0] [0] p(n__o) = [1] [0] [0] p(n__u) = [1] [1] [1] p(nil) = [1] [0] [0] p(o) = [1] [0] [1] p(tt) = [1] [0] [0] p(u) = [1] [1] [1] Following rules are strictly oriented: __(__(X,Y),Z) = [1 0 2] [1 0 1] [1 0 0] [3] [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] [2] [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: U11(tt()) = [1] [1] [0] >= [1] [0] [0] = tt() U21(tt(),V2) = [1 0 0] [1] [0 0 0] V2 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] V2 + [0] [0 0 0] [1] = U22(isList(activate(V2))) U22(tt()) = [2] [0] [1] >= [1] [0] [0] = tt() U31(tt()) = [1] [0] [1] >= [1] [0] [0] = tt() U41(tt(),V2) = [1 0 0] [2] [0 0 0] V2 + [0] [0 1 0] [0] >= [1 0 0] [1] [0 0 0] V2 + [0] [0 0 0] [0] = U42(isNeList(activate(V2))) U42(tt()) = [2] [0] [0] >= [1] [0] [0] = tt() U51(tt(),V2) = [1 0 0] [1] [0 0 0] V2 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] V2 + [0] [0 0 0] [0] = U52(isList(activate(V2))) U52(tt()) = [1] [0] [0] >= [1] [0] [0] = tt() U61(tt()) = [2] [0] [0] >= [1] [0] [0] = tt() U71(tt(),P) = [1 1 0] [1] [0 0 1] P + [2] [0 0 1] [0] >= [1 1 0] [1] [0 0 1] P + [2] [0 0 0] [0] = U72(isPal(activate(P))) U72(tt()) = [1] [0] [0] >= [1] [0] [0] = tt() U81(tt()) = [1] [1] [0] >= [1] [0] [0] = tt() __(X,nil()) = [1 0 1] [2] [0 1 0] X + [0] [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] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = n____(X1,X2) __(nil(),X) = [1 0 0] [2] [0 1 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X a() = [1] [0] [0] >= [1] [0] [0] = n__a() activate(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X activate(n____(X1,X2)) = [1 0 1] [1 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = __(X1,X2) activate(n__a()) = [1] [0] [1] >= [1] [0] [0] = a() activate(n__e()) = [1] [0] [1] >= [1] [0] [1] = e() activate(n__i()) = [1] [1] [1] >= [1] [1] [0] = i() activate(n__nil()) = [1] [0] [1] >= [1] [0] [0] = nil() activate(n__o()) = [1] [0] [1] >= [1] [0] [1] = o() activate(n__u()) = [1] [1] [2] >= [1] [1] [1] = u() e() = [1] [0] [1] >= [1] [0] [0] = n__e() i() = [1] [1] [0] >= [1] [1] [0] = n__i() isList(V) = [1 0 0] [0] [0 0 1] V + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] V + [1] [0 0 0] [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1 0 1] [1 0 0] [1] [0 0 1] V1 + [0 0 1] V2 + [1] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 0] [0 0 0] [1] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [1] [1] [1] >= [1] [0] [0] = tt() isNeList(V) = [1 0 0] [0] [0 0 0] V + [0] [0 1 0] [1] >= [1 0 0] [0] [0 0 0] V + [0] [0 0 0] [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1 0 1] [1 0 0] [1] [0 0 0] V1 + [0 0 0] V2 + [0] [0 1 0] [0 1 0] [1] >= [1 0 0] [1 0 0] [1] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 0] [0 1 0] [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1 0 1] [1 0 0] [1] [0 0 0] V1 + [0 0 0] V2 + [0] [0 1 0] [0 1 0] [1] >= [1 0 0] [1 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 0] [0 0 0] [0] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1 1 0] [1] [1 1 0] V + [1] [1 1 0] [0] >= [1 0 0] [1] [1 1 0] V + [1] [0 0 0] [0] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2 2 1] [1 1 1] [3] [2 2 1] I + [1 1 1] P + [3] [2 2 1] [1 1 1] [2] >= [2 1 1] [1 1 0] [3] [1 0 0] I + [0 0 1] P + [2] [1 1 0] [0 0 1] [2] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1 1 0] [1] [0 0 1] V + [1] [1 1 1] [1] >= [1 1 0] [1] [0 0 0] V + [1] [1 1 0] [1] = U81(isNePal(activate(V))) isPal(n__nil()) = [2] [1] [2] >= [1] [0] [0] = tt() isQid(n__a()) = [1] [2] [1] >= [1] [0] [0] = tt() isQid(n__e()) = [1] [2] [1] >= [1] [0] [0] = tt() isQid(n__i()) = [1] [3] [1] >= [1] [0] [0] = tt() isQid(n__o()) = [1] [2] [1] >= [1] [0] [0] = tt() isQid(n__u()) = [1] [3] [2] >= [1] [0] [0] = tt() nil() = [1] [0] [0] >= [1] [0] [0] = n__nil() o() = [1] [0] [1] >= [1] [0] [0] = n__o() u() = [1] [1] [1] >= [1] [1] [1] = n__u() ** Step 1.b:14: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt()) -> tt() U21(tt(),V2) -> U22(isList(activate(V2))) U22(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() U81(tt()) -> tt() __(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__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) isList(n__nil()) -> tt() isNeList(V) -> U31(isQid(activate(V))) isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) -> U51(isNeList(activate(V1)),activate(V2)) isNePal(V) -> U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) -> U71(isQid(activate(I)),activate(P)) isPal(V) -> U81(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() o() -> n__o() u() -> n__u() - Signature: {U11/1,U21/2,U22/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/1,U71/2,U72/1,U81/1,__/2,a/0,activate/1,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__nil/0 ,n__o/0,n__u/0,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a,activate,e,i ,isList,isNeList,isNePal,isPal,isQid,nil,o,u} and constructors {n____,n__a,n__e,n__i,n__nil,n__o,n__u,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))