/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [0] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [0] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [3] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [5] p(n____) = [1] x1 + [1] x2 + [0] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [0] p(n__u) = [0] p(nil) = [0] p(o) = [0] p(tt) = [0] p(u) = [1] Following rules are strictly oriented: isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] > [1] V1 + [1] V2 + [0] = U41(isList(activate(V1)),activate(V2)) isQid(n__a()) = [5] > [0] = tt() isQid(n__e()) = [5] > [0] = tt() isQid(n__i()) = [5] > [0] = tt() isQid(n__o()) = [5] > [0] = tt() isQid(n__u()) = [5] > [0] = tt() u() = [1] > [0] = n__u() Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U21(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = U22(isList(activate(V2))) U22(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [1] V2 + [0] >= [1] V2 + [3] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [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 + [0] >= [1] P + [0] = U72(isPal(activate(P))) U72(tt()) = [0] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [0] >= [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 + [0] >= [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 + [0] >= [1] X1 + [1] X2 + [0] = __(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()) = [0] >= [0] = o() activate(n__u()) = [0] >= [1] = u() e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [0] >= [1] V + [3] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [0] >= [0] = tt() isNeList(V) = [1] V + [3] >= [1] V + [5] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [3] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [0] >= [1] V + [5] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [0] >= [1] I + [1] P + [5] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [0] >= [1] V + [0] = U81(isNePal(activate(V))) isPal(n__nil()) = [0] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [0] >= [0] = n__o() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + 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)) -> 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() nil() -> n__nil() o() -> n__o() - Weak TRS: isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [1] p(U41) = [1] x1 + [1] x2 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [0] p(U72) = [1] x1 + [3] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [0] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [0] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [0] p(isNePal) = [1] x1 + [0] p(isPal) = [1] x1 + [0] p(isQid) = [1] x1 + [6] p(n____) = [1] x1 + [1] x2 + [0] p(n__a) = [0] p(n__e) = [0] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [0] p(n__u) = [0] p(nil) = [0] p(o) = [1] p(tt) = [6] p(u) = [0] Following rules are strictly oriented: U21(tt(),V2) = [1] V2 + [6] > [1] V2 + [0] = U22(isList(activate(V2))) U31(tt()) = [7] > [6] = tt() U41(tt(),V2) = [1] V2 + [6] > [1] V2 + [0] = U42(isNeList(activate(V2))) U51(tt(),V2) = [1] V2 + [6] > [1] V2 + [0] = U52(isList(activate(V2))) U71(tt(),P) = [1] P + [6] > [1] P + [3] = U72(isPal(activate(P))) U72(tt()) = [9] > [6] = tt() o() = [1] > [0] = n__o() Following rules are (at-least) weakly oriented: U11(tt()) = [6] >= [6] = tt() U22(tt()) = [6] >= [6] = tt() U42(tt()) = [6] >= [6] = tt() U52(tt()) = [6] >= [6] = tt() U61(tt()) = [6] >= [6] = tt() U81(tt()) = [6] >= [6] = tt() __(X,nil()) = [1] X + [0] >= [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 + [0] >= [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 + [0] >= [1] X1 + [1] X2 + [0] = __(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()) = [0] >= [1] = o() activate(n__u()) = [0] >= [0] = u() e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(V) = [1] V + [0] >= [1] V + [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [0] >= [6] = tt() isNeList(V) = [1] V + [0] >= [1] V + [7] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [0] >= [1] V + [6] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [0] >= [1] I + [1] P + [6] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [0] >= [1] V + [0] = U81(isNePal(activate(V))) isPal(n__nil()) = [0] >= [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() = [0] >= [0] = n__nil() u() = [0] >= [0] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt()) -> tt() U22(tt()) -> tt() U42(tt()) -> tt() U52(tt()) -> tt() U61(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)) -> 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() nil() -> n__nil() - Weak TRS: U21(tt(),V2) -> U22(isList(activate(V2))) U31(tt()) -> tt() U41(tt(),V2) -> U42(isNeList(activate(V2))) U51(tt(),V2) -> U52(isList(activate(V2))) U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() isNeList(n____(V1,V2)) -> U41(isList(activate(V1)),activate(V2)) isQid(n__a()) -> tt() isQid(n__e()) -> tt() isQid(n__i()) -> tt() isQid(n__o()) -> tt() isQid(n__u()) -> tt() o() -> n__o() u() -> n__u() - Signature: {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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [5] p(U21) = [1] x1 + [1] x2 + [5] p(U22) = [1] x1 + [5] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [1] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [5] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [3] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [2] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [4] p(isPal) = [1] x1 + [5] p(isQid) = [1] x1 + [1] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [2] p(n__e) = [4] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [0] p(n__u) = [4] p(nil) = [0] p(o) = [0] p(tt) = [0] p(u) = [4] Following rules are strictly oriented: U11(tt()) = [5] > [0] = tt() U22(tt()) = [5] > [0] = tt() __(X,nil()) = [1] X + [3] > [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [1] = n____(X1,X2) __(nil(),X) = [1] X + [3] > [1] X + [0] = X activate(n__a()) = [2] > [0] = a() activate(n__e()) = [4] > [0] = e() i() = [2] > [0] = n__i() isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] > [1] V1 + [1] V2 + [1] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [4] > [1] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [8] > [1] I + [1] P + [6] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [5] > [1] V + [4] = U81(isNePal(activate(V))) isPal(n__nil()) = [5] > [0] = tt() Following rules are (at-least) weakly oriented: U21(tt(),V2) = [1] V2 + [5] >= [1] V2 + [5] = U22(isList(activate(V2))) U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [1] V2 + [1] >= [1] V2 + [1] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [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 + [5] >= [1] P + [5] = U72(isPal(activate(P))) U72(tt()) = [0] >= [0] = tt() U81(tt()) = [0] >= [0] = tt() __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [6] >= [1] X + [1] Y + [1] Z + [6] = __(X,__(Y,Z)) a() = [0] >= [2] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [3] = __(X1,X2) activate(n__i()) = [0] >= [2] = i() activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [0] >= [0] = o() activate(n__u()) = [4] >= [4] = u() e() = [0] >= [4] = n__e() isList(V) = [1] V + [0] >= [1] V + [6] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [1] >= [1] V1 + [1] V2 + [5] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [0] >= [0] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [1] = U41(isList(activate(V1)),activate(V2)) isQid(n__a()) = [3] >= [0] = tt() isQid(n__e()) = [5] >= [0] = tt() isQid(n__i()) = [1] >= [0] = tt() isQid(n__o()) = [1] >= [0] = tt() isQid(n__u()) = [5] >= [0] = tt() nil() = [0] >= [0] = n__nil() o() = [0] >= [0] = n__o() u() = [4] >= [4] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U42(tt()) -> tt() U52(tt()) -> tt() U61(tt()) -> tt() U81(tt()) -> tt() __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() 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))) nil() -> n__nil() - 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))) U51(tt(),V2) -> U52(isList(activate(V2))) U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() __(X,nil()) -> X __(X1,X2) -> n____(X1,X2) __(nil(),X) -> X activate(n__a()) -> a() activate(n__e()) -> e() i() -> n__i() 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() 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [1] x2 + [3] p(U22) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [7] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [1] x2 + [1] p(U52) = [1] x1 + [1] p(U61) = [1] x1 + [2] p(U71) = [1] x1 + [1] x2 + [4] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [1] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [5] p(i) = [3] p(isList) = [1] x1 + [0] p(isNeList) = [1] x1 + [7] p(isNePal) = [1] x1 + [2] p(isPal) = [1] x1 + [4] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [1] p(n__e) = [5] p(n__i) = [3] p(n__nil) = [3] p(n__o) = [0] p(n__u) = [0] p(nil) = [4] p(o) = [0] p(tt) = [0] p(u) = [0] Following rules are strictly oriented: U52(tt()) = [1] > [0] = tt() U61(tt()) = [2] > [0] = tt() isList(n__nil()) = [3] > [0] = tt() isNeList(V) = [1] V + [7] > [1] V + [0] = U31(isQid(activate(V))) nil() = [4] > [3] = n__nil() Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = tt() U21(tt(),V2) = [1] V2 + [3] >= [1] V2 + [0] = U22(isList(activate(V2))) U22(tt()) = [0] >= [0] = tt() U31(tt()) = [0] >= [0] = tt() U41(tt(),V2) = [1] V2 + [7] >= [1] V2 + [7] = U42(isNeList(activate(V2))) U42(tt()) = [0] >= [0] = tt() U51(tt(),V2) = [1] V2 + [1] >= [1] V2 + [1] = U52(isList(activate(V2))) 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 + [5] >= [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 + [5] >= [1] X + [0] = X a() = [0] >= [1] = 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()) = [1] >= [0] = a() activate(n__e()) = [5] >= [5] = e() activate(n__i()) = [3] >= [3] = i() activate(n__nil()) = [3] >= [4] = nil() activate(n__o()) = [0] >= [0] = o() activate(n__u()) = [0] >= [0] = u() e() = [5] >= [5] = n__e() i() = [3] >= [3] = n__i() isList(V) = [1] V + [0] >= [1] V + [7] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [1] >= [1] V1 + [1] V2 + [3] = U21(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [7] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = 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 + [4] >= [1] I + [1] P + [4] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [4] >= [1] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [7] >= [0] = tt() isQid(n__a()) = [1] >= [0] = tt() isQid(n__e()) = [5] >= [0] = tt() isQid(n__i()) = [3] >= [0] = tt() isQid(n__o()) = [0] >= [0] = tt() isQid(n__u()) = [0] >= [0] = tt() o() = [0] >= [0] = n__o() u() = [0] >= [0] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U42(tt()) -> tt() U81(tt()) -> tt() __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) - 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))) U51(tt(),V2) -> U52(isList(activate(V2))) U52(tt()) -> tt() U61(tt()) -> tt() U71(tt(),P) -> U72(isPal(activate(P))) U72(tt()) -> tt() __(X,nil()) -> X __(X1,X2) -> n____(X1,X2) __(nil(),X) -> X activate(n__a()) -> a() activate(n__e()) -> e() i() -> n__i() 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1] x1 + [2] p(U21) = [1] x1 + [1] x2 + [5] p(U22) = [1] x1 + [4] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [4] p(U42) = [1] x1 + [3] p(U51) = [1] x1 + [1] x2 + [4] p(U52) = [1] x1 + [5] p(U61) = [1] x1 + [2] p(U71) = [1] x1 + [1] x2 + [7] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [1] p(__) = [1] x1 + [1] x2 + [6] p(a) = [0] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [1] p(isList) = [1] x1 + [0] 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) = [4] p(n__e) = [2] p(n__i) = [1] p(n__nil) = [4] p(n__o) = [2] p(n__u) = [1] p(nil) = [4] p(o) = [2] p(tt) = [1] p(u) = [1] Following rules are strictly oriented: U42(tt()) = [4] > [1] = tt() U81(tt()) = [2] > [1] = tt() Following rules are (at-least) weakly oriented: U11(tt()) = [3] >= [1] = tt() U21(tt(),V2) = [1] V2 + [6] >= [1] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [5] >= [1] = tt() U31(tt()) = [1] >= [1] = tt() U41(tt(),V2) = [1] V2 + [5] >= [1] V2 + [4] = U42(isNeList(activate(V2))) U51(tt(),V2) = [1] V2 + [5] >= [1] V2 + [5] = U52(isList(activate(V2))) U52(tt()) = [6] >= [1] = tt() U61(tt()) = [3] >= [1] = tt() U71(tt(),P) = [1] P + [8] >= [1] P + [4] = U72(isPal(activate(P))) U72(tt()) = [1] >= [1] = tt() __(X,nil()) = [1] X + [10] >= [1] X + [0] = X __(X1,X2) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [4] = n____(X1,X2) __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [12] >= [1] X + [1] Y + [1] Z + [12] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [10] >= [1] X + [0] = X a() = [0] >= [4] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [6] = __(X1,X2) activate(n__a()) = [4] >= [0] = a() activate(n__e()) = [2] >= [0] = e() activate(n__i()) = [1] >= [1] = i() activate(n__nil()) = [4] >= [4] = nil() activate(n__o()) = [2] >= [2] = o() activate(n__u()) = [1] >= [1] = u() e() = [0] >= [2] = n__e() i() = [1] >= [1] = n__i() isList(V) = [1] V + [0] >= [1] V + [3] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [5] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [1] = tt() isNeList(V) = [1] V + [1] >= [1] V + [0] = 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 + [5] = 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 + [12] >= [1] I + [1] P + [7] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [4] >= [1] V + [3] = U81(isNePal(activate(V))) isPal(n__nil()) = [8] >= [1] = tt() isQid(n__a()) = [4] >= [1] = tt() isQid(n__e()) = [2] >= [1] = tt() isQid(n__i()) = [1] >= [1] = tt() isQid(n__o()) = [2] >= [1] = tt() isQid(n__u()) = [1] >= [1] = tt() nil() = [4] >= [4] = n__nil() o() = [2] >= [2] = n__o() u() = [1] >= [1] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() isList(V) -> U11(isNeList(activate(V))) isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) - 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 activate(n__a()) -> a() activate(n__e()) -> e() i() -> n__i() 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [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) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [7] 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) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [6] p(isPal) = [1] x1 + [6] p(isQid) = [1] x1 + [1] p(n____) = [1] x1 + [1] x2 + [1] p(n__a) = [2] p(n__e) = [0] p(n__i) = [0] p(n__nil) = [0] p(n__o) = [4] p(n__u) = [0] p(nil) = [0] p(o) = [4] p(tt) = [1] p(u) = [0] Following rules are strictly oriented: isList(V) = [1] V + [2] > [1] V + [1] = U11(isNeList(activate(V))) Following rules are (at-least) weakly oriented: U11(tt()) = [1] >= [1] = tt() U21(tt(),V2) = [1] V2 + [2] >= [1] V2 + [2] = U22(isList(activate(V2))) U22(tt()) = [1] >= [1] = tt() U31(tt()) = [1] >= [1] = tt() U41(tt(),V2) = [1] V2 + [1] >= [1] V2 + [1] = U42(isNeList(activate(V2))) U42(tt()) = [1] >= [1] = tt() U51(tt(),V2) = [1] V2 + [2] >= [1] V2 + [2] = U52(isList(activate(V2))) U52(tt()) = [1] >= [1] = tt() U61(tt()) = [1] >= [1] = tt() U71(tt(),P) = [1] P + [8] >= [1] P + [7] = U72(isPal(activate(P))) U72(tt()) = [2] >= [1] = tt() U81(tt()) = [1] >= [1] = 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] >= [2] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = __(X1,X2) activate(n__a()) = [2] >= [0] = a() activate(n__e()) = [0] >= [0] = e() activate(n__i()) = [0] >= [0] = i() activate(n__nil()) = [0] >= [0] = nil() activate(n__o()) = [4] >= [4] = o() activate(n__u()) = [0] >= [0] = u() e() = [0] >= [0] = n__e() i() = [0] >= [0] = n__i() isList(n____(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [3] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [2] >= [1] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = 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 + [2] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [6] >= [1] V + [1] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [8] >= [1] I + [1] P + [8] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [6] >= [1] V + [6] = U81(isNePal(activate(V))) isPal(n__nil()) = [6] >= [1] = tt() isQid(n__a()) = [3] >= [1] = tt() isQid(n__e()) = [1] >= [1] = tt() isQid(n__i()) = [1] >= [1] = tt() isQid(n__o()) = [5] >= [1] = tt() isQid(n__u()) = [1] >= [1] = tt() nil() = [0] >= [0] = n__nil() o() = [4] >= [4] = n__o() u() = [0] >= [0] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__e() isList(n____(V1,V2)) -> U21(isList(activate(V1)),activate(V2)) - 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 activate(n__a()) -> a() activate(n__e()) -> e() i() -> n__i() isList(V) -> U11(isNeList(activate(V))) 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [6] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [2] p(U42) = [1] x1 + [2] p(U51) = [1] x1 + [1] x2 + [1] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [0] p(U71) = [1] x1 + [1] x2 + [4] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [3] p(a) = [2] p(activate) = [1] x1 + [0] p(e) = [0] p(i) = [5] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [2] p(isNePal) = [1] x1 + [2] p(isPal) = [1] x1 + [2] p(isQid) = [1] x1 + [2] p(n____) = [1] x1 + [1] x2 + [3] p(n__a) = [6] p(n__e) = [5] p(n__i) = [5] p(n__nil) = [4] p(n__o) = [7] p(n__u) = [4] p(nil) = [4] p(o) = [7] p(tt) = [6] p(u) = [4] Following rules are strictly oriented: isList(n____(V1,V2)) = [1] V1 + [1] V2 + [5] > [1] V1 + [1] V2 + [4] = U21(isList(activate(V1)),activate(V2)) Following rules are (at-least) weakly oriented: U11(tt()) = [6] >= [6] = tt() U21(tt(),V2) = [1] V2 + [8] >= [1] V2 + [8] = U22(isList(activate(V2))) U22(tt()) = [12] >= [6] = tt() U31(tt()) = [6] >= [6] = tt() U41(tt(),V2) = [1] V2 + [8] >= [1] V2 + [4] = U42(isNeList(activate(V2))) U42(tt()) = [8] >= [6] = tt() U51(tt(),V2) = [1] V2 + [7] >= [1] V2 + [2] = U52(isList(activate(V2))) U52(tt()) = [6] >= [6] = tt() U61(tt()) = [6] >= [6] = tt() U71(tt(),P) = [1] P + [10] >= [1] P + [2] = U72(isPal(activate(P))) U72(tt()) = [6] >= [6] = tt() U81(tt()) = [6] >= [6] = tt() __(X,nil()) = [1] X + [7] >= [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 + [7] >= [1] X + [0] = X a() = [2] >= [6] = n__a() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n____(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = __(X1,X2) activate(n__a()) = [6] >= [2] = a() activate(n__e()) = [5] >= [0] = e() activate(n__i()) = [5] >= [5] = i() activate(n__nil()) = [4] >= [4] = nil() activate(n__o()) = [7] >= [7] = o() activate(n__u()) = [4] >= [4] = u() e() = [0] >= [5] = n__e() i() = [5] >= [5] = n__i() isList(V) = [1] V + [2] >= [1] V + [2] = U11(isNeList(activate(V))) isList(n__nil()) = [6] >= [6] = tt() isNeList(V) = [1] V + [2] >= [1] V + [2] = 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 + [3] = 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 + [8] >= [1] I + [1] P + [6] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [2] >= [1] V + [2] = U81(isNePal(activate(V))) isPal(n__nil()) = [6] >= [6] = tt() isQid(n__a()) = [8] >= [6] = tt() isQid(n__e()) = [7] >= [6] = tt() isQid(n__i()) = [7] >= [6] = tt() isQid(n__o()) = [9] >= [6] = tt() isQid(n__u()) = [6] >= [6] = tt() nil() = [4] >= [4] = n__nil() o() = [7] >= [7] = n__o() u() = [4] >= [4] = n__u() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) a() -> n__a() activate(X) -> X activate(n____(X1,X2)) -> __(X1,X2) activate(n__i()) -> i() activate(n__nil()) -> nil() activate(n__o()) -> o() activate(n__u()) -> u() e() -> n__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 __(X1,X2) -> n____(X1,X2) __(nil(),X) -> X activate(n__a()) -> a() activate(n__e()) -> 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [0] p(U42) = [1] x1 + [1] p(U51) = [1] x1 + [1] x2 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [2] p(U71) = [1] x1 + [1] x2 + [2] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [3] p(a) = [5] p(activate) = [1] x1 + [1] p(e) = [5] p(i) = [7] p(isList) = [1] x1 + [2] p(isNeList) = [1] x1 + [1] p(isNePal) = [1] x1 + [3] p(isPal) = [1] x1 + [4] p(isQid) = [1] x1 + [0] p(n____) = [1] x1 + [1] x2 + [3] p(n__a) = [4] p(n__e) = [4] p(n__i) = [7] p(n__nil) = [4] p(n__o) = [4] p(n__u) = [4] p(nil) = [4] p(o) = [5] p(tt) = [3] p(u) = [4] Following rules are strictly oriented: a() = [5] > [4] = 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__i()) = [8] > [7] = i() activate(n__nil()) = [5] > [4] = nil() activate(n__u()) = [5] > [4] = u() e() = [5] > [4] = n__e() Following rules are (at-least) weakly oriented: U11(tt()) = [3] >= [3] = tt() U21(tt(),V2) = [1] V2 + [4] >= [1] V2 + [4] = U22(isList(activate(V2))) U22(tt()) = [4] >= [3] = tt() U31(tt()) = [3] >= [3] = tt() U41(tt(),V2) = [1] V2 + [3] >= [1] V2 + [3] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [3] = tt() U51(tt(),V2) = [1] V2 + [3] >= [1] V2 + [3] = U52(isList(activate(V2))) U52(tt()) = [3] >= [3] = tt() U61(tt()) = [5] >= [3] = tt() U71(tt(),P) = [1] P + [5] >= [1] P + [5] = U72(isPal(activate(P))) U72(tt()) = [3] >= [3] = tt() U81(tt()) = [3] >= [3] = tt() __(X,nil()) = [1] X + [7] >= [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 + [7] >= [1] X + [0] = X activate(n__a()) = [5] >= [5] = a() activate(n__e()) = [5] >= [5] = e() activate(n__o()) = [5] >= [5] = o() i() = [7] >= [7] = n__i() isList(V) = [1] V + [2] >= [1] V + [2] = 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()) = [6] >= [3] = tt() isNeList(V) = [1] V + [1] >= [1] V + [1] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [3] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1] V + [3] >= [1] V + [3] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2] I + [1] P + [9] >= [1] 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()) = [8] >= [3] = tt() isQid(n__a()) = [4] >= [3] = tt() isQid(n__e()) = [4] >= [3] = tt() isQid(n__i()) = [7] >= [3] = tt() isQid(n__o()) = [4] >= [3] = tt() isQid(n__u()) = [4] >= [3] = tt() nil() = [4] >= [4] = n__nil() o() = [5] >= [4] = n__o() u() = [4] >= [4] = n__u() * Step 9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) activate(n__o()) -> o() - 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__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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: 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 + [0] p(U31) = [1] x1 + [2] 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 + [4] p(U71) = [1] x1 + [1] x2 + [6] p(U72) = [1] x1 + [0] p(U81) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [3] p(a) = [6] p(activate) = [1] x1 + [1] p(e) = [4] p(i) = [4] p(isList) = [1] x1 + [4] p(isNeList) = [1] x1 + [3] p(isNePal) = [1] x1 + [5] p(isPal) = [1] 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) = [4] p(n__nil) = [0] p(n__o) = [5] p(n__u) = [4] p(nil) = [0] p(o) = [5] p(tt) = [4] p(u) = [5] Following rules are strictly oriented: activate(n__o()) = [6] > [5] = o() Following rules are (at-least) weakly oriented: U11(tt()) = [4] >= [4] = tt() U21(tt(),V2) = [1] V2 + [5] >= [1] V2 + [5] = U22(isList(activate(V2))) U22(tt()) = [4] >= [4] = tt() U31(tt()) = [6] >= [4] = tt() U41(tt(),V2) = [1] V2 + [4] >= [1] V2 + [4] = U42(isNeList(activate(V2))) U42(tt()) = [4] >= [4] = tt() U51(tt(),V2) = [1] V2 + [5] >= [1] V2 + [5] = U52(isList(activate(V2))) U52(tt()) = [4] >= [4] = tt() U61(tt()) = [8] >= [4] = tt() U71(tt(),P) = [1] P + [10] >= [1] P + [7] = U72(isPal(activate(P))) U72(tt()) = [4] >= [4] = tt() U81(tt()) = [4] >= [4] = tt() __(X,nil()) = [1] X + [3] >= [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 + [3] >= [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__e()) = [5] >= [4] = e() activate(n__i()) = [5] >= [4] = i() activate(n__nil()) = [1] >= [0] = nil() activate(n__u()) = [5] >= [5] = u() e() = [4] >= [4] = n__e() i() = [4] >= [4] = n__i() isList(V) = [1] V + [4] >= [1] V + [4] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1] V1 + [1] V2 + [7] >= [1] V1 + [1] V2 + [7] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [4] >= [4] = tt() isNeList(V) = [1] V + [3] >= [1] V + [3] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [6] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [6] = 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] >= [1] I + [1] P + [8] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1] V + [6] >= [1] V + [6] = U81(isNePal(activate(V))) isPal(n__nil()) = [6] >= [4] = tt() isQid(n__a()) = [6] >= [4] = tt() isQid(n__e()) = [4] >= [4] = tt() isQid(n__i()) = [4] >= [4] = tt() isQid(n__o()) = [5] >= [4] = tt() isQid(n__u()) = [4] >= [4] = tt() nil() = [0] >= [0] = n__nil() o() = [5] >= [5] = n__o() u() = [5] >= [4] = n__u() * Step 10: NaturalMI. WORST_CASE(?,O(n^2)) + 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [1 0] x1 + [0] [0 0] [0] p(U21) = [1 0] x1 + [1 0] x2 + [1] [0 0] [0 0] [2] p(U22) = [1 0] x1 + [0] [0 0] [0] p(U31) = [1 0] x1 + [0] [0 0] [0] p(U41) = [1 0] x1 + [1 0] x2 + [1] [0 0] [0 0] [0] p(U42) = [1 0] x1 + [0] [0 1] [0] p(U51) = [1 0] x1 + [1 0] x2 + [1] [0 0] [0 0] [0] p(U52) = [1 0] x1 + [0] [0 0] [0] p(U61) = [1 0] x1 + [0] [0 0] [0] p(U71) = [1 1] x1 + [1 0] x2 + [2] [0 0] [0 0] [1] p(U72) = [1 0] x1 + [0] [0 0] [1] p(U81) = [1 0] x1 + [0] [0 0] [3] p(__) = [1 1] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] p(a) = [3] [1] p(activate) = [1 0] x1 + [0] [0 1] [0] p(e) = [0] [2] p(i) = [0] [0] p(isList) = [1 0] x1 + [3] [0 1] [0] p(isNeList) = [1 0] x1 + [3] [0 0] [0] p(isNePal) = [1 0] x1 + [3] [0 0] [1] p(isPal) = [1 0] x1 + [3] [0 0] [3] p(isQid) = [1 0] x1 + [2] [0 1] [0] p(n____) = [1 1] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] p(n__a) = [3] [1] p(n__e) = [0] [2] p(n__i) = [0] [0] p(n__nil) = [0] [0] p(n__o) = [0] [0] p(n__u) = [0] [0] p(nil) = [0] [0] p(o) = [0] [0] p(tt) = [2] [0] p(u) = [0] [0] Following rules are strictly oriented: __(__(X,Y),Z) = [1 2] X + [1 1] Y + [1 0] Z + [4] [0 1] [0 1] [0 1] [4] > [1 1] X + [1 1] Y + [1 0] Z + [2] [0 1] [0 1] [0 1] [4] = __(X,__(Y,Z)) Following rules are (at-least) weakly oriented: U11(tt()) = [2] [0] >= [2] [0] = tt() U21(tt(),V2) = [1 0] V2 + [3] [0 0] [2] >= [1 0] V2 + [3] [0 0] [0] = U22(isList(activate(V2))) U22(tt()) = [2] [0] >= [2] [0] = tt() U31(tt()) = [2] [0] >= [2] [0] = tt() U41(tt(),V2) = [1 0] V2 + [3] [0 0] [0] >= [1 0] V2 + [3] [0 0] [0] = U42(isNeList(activate(V2))) U42(tt()) = [2] [0] >= [2] [0] = tt() U51(tt(),V2) = [1 0] V2 + [3] [0 0] [0] >= [1 0] V2 + [3] [0 0] [0] = U52(isList(activate(V2))) U52(tt()) = [2] [0] >= [2] [0] = tt() U61(tt()) = [2] [0] >= [2] [0] = tt() U71(tt(),P) = [1 0] P + [4] [0 0] [1] >= [1 0] P + [3] [0 0] [1] = U72(isPal(activate(P))) U72(tt()) = [2] [1] >= [2] [0] = tt() U81(tt()) = [2] [3] >= [2] [0] = tt() __(X,nil()) = [1 1] X + [1] [0 1] [2] >= [1 0] X + [0] [0 1] [0] = X __(X1,X2) = [1 1] X1 + [1 0] X2 + [1] [0 1] [0 1] [2] >= [1 1] X1 + [1 0] X2 + [1] [0 1] [0 1] [2] = n____(X1,X2) __(nil(),X) = [1 0] X + [1] [0 1] [2] >= [1 0] X + [0] [0 1] [0] = X a() = [3] [1] >= [3] [1] = n__a() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n____(X1,X2)) = [1 1] X1 + [1 0] X2 + [1] [0 1] [0 1] [2] >= [1 1] X1 + [1 0] X2 + [1] [0 1] [0 1] [2] = __(X1,X2) activate(n__a()) = [3] [1] >= [3] [1] = a() activate(n__e()) = [0] [2] >= [0] [2] = e() activate(n__i()) = [0] [0] >= [0] [0] = i() activate(n__nil()) = [0] [0] >= [0] [0] = nil() activate(n__o()) = [0] [0] >= [0] [0] = o() activate(n__u()) = [0] [0] >= [0] [0] = u() e() = [0] [2] >= [0] [2] = n__e() i() = [0] [0] >= [0] [0] = n__i() isList(V) = [1 0] V + [3] [0 1] [0] >= [1 0] V + [3] [0 0] [0] = U11(isNeList(activate(V))) isList(n____(V1,V2)) = [1 1] V1 + [1 0] V2 + [4] [0 1] [0 1] [2] >= [1 0] V1 + [1 0] V2 + [4] [0 0] [0 0] [2] = U21(isList(activate(V1)),activate(V2)) isList(n__nil()) = [3] [0] >= [2] [0] = tt() isNeList(V) = [1 0] V + [3] [0 0] [0] >= [1 0] V + [2] [0 0] [0] = U31(isQid(activate(V))) isNeList(n____(V1,V2)) = [1 1] V1 + [1 0] V2 + [4] [0 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [4] [0 0] [0 0] [0] = U41(isList(activate(V1)),activate(V2)) isNeList(n____(V1,V2)) = [1 1] V1 + [1 0] V2 + [4] [0 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [4] [0 0] [0 0] [0] = U51(isNeList(activate(V1)),activate(V2)) isNePal(V) = [1 0] V + [3] [0 0] [1] >= [1 0] V + [2] [0 0] [0] = U61(isQid(activate(V))) isNePal(n____(I,__(P,I))) = [2 1] I + [1 1] P + [5] [0 0] [0 0] [1] >= [1 1] I + [1 0] P + [4] [0 0] [0 0] [1] = U71(isQid(activate(I)),activate(P)) isPal(V) = [1 0] V + [3] [0 0] [3] >= [1 0] V + [3] [0 0] [3] = U81(isNePal(activate(V))) isPal(n__nil()) = [3] [3] >= [2] [0] = tt() isQid(n__a()) = [5] [1] >= [2] [0] = tt() isQid(n__e()) = [2] [2] >= [2] [0] = tt() isQid(n__i()) = [2] [0] >= [2] [0] = tt() isQid(n__o()) = [2] [0] >= [2] [0] = tt() isQid(n__u()) = [2] [0] >= [2] [0] = tt() nil() = [0] [0] >= [0] [0] = n__nil() o() = [0] [0] >= [0] [0] = n__o() u() = [0] [0] >= [0] [0] = n__u() * Step 11: 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: innermost derivational complexity wrt. signature {U11,U21,U22,U31,U41,U42,U51,U52,U61,U71,U72,U81,__,a ,activate,e,i,isList,isNeList,isNePal,isPal,isQid,n____,n__a,n__e,n__i,n__nil,n__o,n__u,nil,o,tt,u} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))