WORST_CASE(?,O(n^3)) * Step 1: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [1] x2 + [0] p(U62) = [1] x1 + [1] x2 + [0] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [0] p(a__U31) = [1] x1 + [0] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [0] p(a__U52) = [1] x1 + [0] p(a__U61) = [1] x1 + [1] x2 + [3] p(a__U62) = [1] x1 + [1] x2 + [0] p(a__isNat) = [0] p(a__isNatIList) = [0] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [5] Following rules are strictly oriented: a__U61(X1,X2,X3) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1] L + [3] > [1] L + [0] = a__U62(a__isNat(N),L) mark(zeros()) = [5] > [0] = a__zeros() Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [1] X + [0] = U11(X) a__U11(tt()) = [0] >= [0] = tt() a__U21(X) = [1] X + [0] >= [1] X + [0] = U21(X) a__U21(tt()) = [0] >= [0] = tt() a__U31(X) = [1] X + [0] >= [1] X + [0] = U31(X) a__U31(tt()) = [0] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U41(X1,X2) a__U41(tt(),V2) = [0] >= [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [1] X + [0] = U42(X) a__U42(tt()) = [0] >= [0] = tt() a__U51(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U51(X1,X2) a__U51(tt(),V2) = [0] >= [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [0] >= [1] X + [0] = U52(X) a__U52(tt()) = [0] >= [0] = tt() a__U62(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = U62(X1,X2) a__U62(tt(),L) = [1] L + [0] >= [1] L + [0] = s(a__length(mark(L))) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] >= [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0] >= [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] >= [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [0] >= [1] L + [3] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [0] >= [5] = cons(0(),zeros()) a__zeros() = [0] >= [5] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X)) = [1] X + [0] >= [1] X + [0] = a__U11(mark(X)) mark(U21(X)) = [1] X + [0] >= [1] X + [0] = a__U21(mark(X)) mark(U31(X)) = [1] X + [0] >= [1] X + [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__U41(mark(X1),X2) mark(U42(X)) = [1] X + [0] >= [1] X + [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__U51(mark(X1),X2) mark(U52(X)) = [1] X + [0] >= [1] X + [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [3] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [1] X + [0] >= [1] X + [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Weak TRS: a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [0] p(U42) = [1] x1 + [5] p(U51) = [1] x1 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [1] x2 + [5] p(U62) = [1] x1 + [1] x2 + [0] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [0] p(a__U31) = [1] x1 + [0] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [3] p(a__U51) = [1] x1 + [0] p(a__U52) = [1] x1 + [0] p(a__U61) = [1] x1 + [1] x2 + [5] p(a__U62) = [1] x1 + [1] x2 + [5] p(a__isNat) = [0] p(a__isNatIList) = [3] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(cons) = [1] x1 + [1] x2 + [5] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__U42(tt()) = [3] > [0] = tt() a__U62(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = U62(X1,X2) a__U62(tt(),L) = [1] L + [5] > [1] L + [0] = s(a__length(mark(L))) a__isNatIList(V) = [3] > [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [3] > [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [3] > [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [3] > [0] = tt() mark(U42(X)) = [1] X + [5] > [1] X + [3] = a__U42(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [1] X + [0] = U11(X) a__U11(tt()) = [0] >= [0] = tt() a__U21(X) = [1] X + [0] >= [1] X + [0] = U21(X) a__U21(tt()) = [0] >= [0] = tt() a__U31(X) = [1] X + [0] >= [1] X + [0] = U31(X) a__U31(tt()) = [0] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U41(X1,X2) a__U41(tt(),V2) = [0] >= [6] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [3] >= [1] X + [5] = U42(X) a__U51(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U51(X1,X2) a__U51(tt(),V2) = [0] >= [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [0] >= [1] X + [0] = U52(X) a__U52(tt()) = [0] >= [0] = tt() a__U61(X1,X2,X3) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1] L + [5] >= [1] L + [5] = a__U62(a__isNat(N),L) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] >= [0] = a__U21(a__isNat(V1)) a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [5] >= [1] L + [5] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [0] >= [5] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X)) = [1] X + [0] >= [1] X + [0] = a__U11(mark(X)) mark(U21(X)) = [1] X + [0] >= [1] X + [0] = a__U21(mark(X)) mark(U31(X)) = [1] X + [0] >= [1] X + [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__U41(mark(X1),X2) mark(U51(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__U51(mark(X1),X2) mark(U52(X)) = [1] X + [0] >= [1] X + [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [5] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [3] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [1] X + [0] >= [1] X + [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Weak TRS: a__U42(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() mark(U42(X)) -> a__U42(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [1] x1 + [0] p(U31) = [0] p(U41) = [1] x1 + [0] p(U42) = [1] x1 + [0] p(U51) = [1] x1 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [5] p(U62) = [1] x1 + [0] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [0] p(a__U31) = [1] x1 + [1] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [0] p(a__U52) = [1] x1 + [0] p(a__U61) = [1] x1 + [5] p(a__U62) = [1] x1 + [5] p(a__isNat) = [1] p(a__isNatIList) = [1] p(a__isNatList) = [0] p(a__length) = [1] x1 + [6] p(a__zeros) = [0] p(cons) = [1] x1 + [0] p(isNat) = [0] p(isNatIList) = [1] p(isNatList) = [0] p(length) = [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [1] p(zeros) = [0] Following rules are strictly oriented: a__U31(X) = [1] X + [1] > [0] = U31(X) a__U31(tt()) = [2] > [1] = tt() a__U51(tt(),V2) = [1] > [0] = a__U52(a__isNatList(V2)) a__isNat(X) = [1] > [0] = isNat(X) a__isNat(length(V1)) = [1] > [0] = a__U11(a__isNatList(V1)) a__length(X) = [1] X + [6] > [0] = length(X) a__length(cons(N,L)) = [1] N + [6] > [5] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [6] > [0] = 0() Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [0] = U11(X) a__U11(tt()) = [1] >= [1] = tt() a__U21(X) = [1] X + [0] >= [1] X + [0] = U21(X) a__U21(tt()) = [1] >= [1] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U41(X1,X2) a__U41(tt(),V2) = [1] >= [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [1] X + [0] = U42(X) a__U42(tt()) = [1] >= [1] = tt() a__U51(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U51(X1,X2) a__U52(X) = [1] X + [0] >= [1] X + [0] = U52(X) a__U52(tt()) = [1] >= [1] = tt() a__U61(X1,X2,X3) = [1] X1 + [5] >= [1] X1 + [5] = U61(X1,X2,X3) a__U61(tt(),L,N) = [6] >= [6] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [5] >= [1] X1 + [0] = U62(X1,X2) a__U62(tt(),L) = [6] >= [6] = s(a__length(mark(L))) a__isNat(0()) = [1] >= [1] = tt() a__isNat(s(V1)) = [1] >= [1] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1] >= [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1] >= [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] >= [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] >= [1] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [1] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [1] = tt() a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X)) = [0] >= [0] = a__U11(mark(X)) mark(U21(X)) = [0] >= [0] = a__U21(mark(X)) mark(U31(X)) = [0] >= [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0] >= [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0] >= [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0] >= [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0] >= [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0] >= [5] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0] >= [5] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [1] = a__isNat(X) mark(isNatIList(X)) = [0] >= [1] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [6] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [1] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Weak TRS: a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(U42) = [1] x1 + [0] p(U51) = [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [4] p(U62) = [1] x1 + [3] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [7] p(a__U31) = [1] x1 + [4] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [4] p(a__U52) = [1] x1 + [3] p(a__U61) = [1] x1 + [4] p(a__U62) = [1] x1 + [3] p(a__isNat) = [1] p(a__isNatIList) = [5] p(a__isNatList) = [1] p(a__length) = [1] x1 + [1] p(a__zeros) = [2] p(cons) = [1] x1 + [6] p(isNat) = [1] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [1] p(mark) = [2] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__U21(X) = [1] X + [7] > [0] = U21(X) a__U21(tt()) = [7] > [0] = tt() a__U51(X1,X2) = [1] X1 + [4] > [0] = U51(X1,X2) a__U52(X) = [1] X + [3] > [1] X + [0] = U52(X) a__U52(tt()) = [3] > [0] = tt() a__isNat(0()) = [1] > [0] = tt() a__isNatList(X) = [1] > [0] = isNatList(X) a__isNatList(nil()) = [1] > [0] = tt() a__zeros() = [2] > [0] = zeros() mark(0()) = [2] > [1] = 0() mark(isNat(X)) = [2] > [1] = a__isNat(X) mark(isNatList(X)) = [2] > [1] = a__isNatList(X) mark(nil()) = [2] > [0] = nil() mark(tt()) = [2] > [0] = tt() Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [0] = U11(X) a__U11(tt()) = [0] >= [0] = tt() a__U31(X) = [1] X + [4] >= [0] = U31(X) a__U31(tt()) = [4] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [0] = U41(X1,X2) a__U41(tt(),V2) = [0] >= [5] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [1] X + [0] = U42(X) a__U42(tt()) = [0] >= [0] = tt() a__U51(tt(),V2) = [4] >= [4] = a__U52(a__isNatList(V2)) a__U61(X1,X2,X3) = [1] X1 + [4] >= [1] X1 + [4] = U61(X1,X2,X3) a__U61(tt(),L,N) = [4] >= [4] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [3] >= [1] X1 + [3] = U62(X1,X2) a__U62(tt(),L) = [3] >= [3] = s(a__length(mark(L))) a__isNat(X) = [1] >= [1] = isNat(X) a__isNat(length(V1)) = [1] >= [1] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [1] >= [8] = a__U21(a__isNat(V1)) a__isNatIList(V) = [5] >= [5] = a__U31(a__isNatList(V)) a__isNatIList(X) = [5] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [5] >= [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [5] >= [0] = tt() a__isNatList(cons(V1,V2)) = [1] >= [5] = a__U51(a__isNat(V1),V2) a__length(X) = [1] X + [1] >= [1] X + [1] = length(X) a__length(cons(N,L)) = [1] N + [7] >= [5] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] >= [1] = 0() a__zeros() = [2] >= [7] = cons(0(),zeros()) mark(U11(X)) = [2] >= [2] = a__U11(mark(X)) mark(U21(X)) = [2] >= [9] = a__U21(mark(X)) mark(U31(X)) = [2] >= [6] = a__U31(mark(X)) mark(U41(X1,X2)) = [2] >= [2] = a__U41(mark(X1),X2) mark(U42(X)) = [2] >= [2] = a__U42(mark(X)) mark(U51(X1,X2)) = [2] >= [6] = a__U51(mark(X1),X2) mark(U52(X)) = [2] >= [5] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [2] >= [6] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [2] >= [5] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [8] = cons(mark(X1),X2) mark(isNatIList(X)) = [2] >= [5] = a__isNatIList(X) mark(length(X)) = [2] >= [3] = a__length(mark(X)) mark(s(X)) = [2] >= [2] = s(mark(X)) mark(zeros()) = [2] >= [2] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__zeros() -> cons(0(),zeros()) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNatIList(X)) -> a__isNatIList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(U42) = [0] p(U51) = [0] p(U52) = [0] p(U61) = [1] x1 + [0] p(U62) = [1] x1 + [3] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [1] p(a__U31) = [1] x1 + [5] p(a__U41) = [1] x1 + [4] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [6] p(a__U52) = [1] x1 + [6] p(a__U61) = [1] x1 + [4] p(a__U62) = [1] x1 + [3] p(a__isNat) = [1] p(a__isNatIList) = [5] p(a__isNatList) = [0] p(a__length) = [1] x1 + [1] p(a__zeros) = [2] p(cons) = [1] x1 + [4] p(isNat) = [0] p(isNatIList) = [5] p(isNatList) = [0] p(length) = [1] x1 + [1] p(mark) = [2] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [2] Following rules are strictly oriented: a__U41(X1,X2) = [1] X1 + [4] > [0] = U41(X1,X2) Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [0] = U11(X) a__U11(tt()) = [0] >= [0] = tt() a__U21(X) = [1] X + [1] >= [0] = U21(X) a__U21(tt()) = [1] >= [0] = tt() a__U31(X) = [1] X + [5] >= [0] = U31(X) a__U31(tt()) = [5] >= [0] = tt() a__U41(tt(),V2) = [4] >= [5] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [0] = U42(X) a__U42(tt()) = [0] >= [0] = tt() a__U51(X1,X2) = [1] X1 + [6] >= [0] = U51(X1,X2) a__U51(tt(),V2) = [6] >= [6] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [6] >= [0] = U52(X) a__U52(tt()) = [6] >= [0] = tt() a__U61(X1,X2,X3) = [1] X1 + [4] >= [1] X1 + [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [4] >= [4] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [3] >= [1] X1 + [3] = U62(X1,X2) a__U62(tt(),L) = [3] >= [3] = s(a__length(mark(L))) a__isNat(X) = [1] >= [0] = isNat(X) a__isNat(0()) = [1] >= [0] = tt() a__isNat(length(V1)) = [1] >= [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [1] >= [2] = a__U21(a__isNat(V1)) a__isNatIList(V) = [5] >= [5] = a__U31(a__isNatList(V)) a__isNatIList(X) = [5] >= [5] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [5] >= [5] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [5] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [7] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [1] >= [1] X + [1] = length(X) a__length(cons(N,L)) = [1] N + [5] >= [4] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] >= [0] = 0() a__zeros() = [2] >= [4] = cons(0(),zeros()) a__zeros() = [2] >= [2] = zeros() mark(0()) = [2] >= [0] = 0() mark(U11(X)) = [2] >= [2] = a__U11(mark(X)) mark(U21(X)) = [2] >= [3] = a__U21(mark(X)) mark(U31(X)) = [2] >= [7] = a__U31(mark(X)) mark(U41(X1,X2)) = [2] >= [6] = a__U41(mark(X1),X2) mark(U42(X)) = [2] >= [2] = a__U42(mark(X)) mark(U51(X1,X2)) = [2] >= [8] = a__U51(mark(X1),X2) mark(U52(X)) = [2] >= [8] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [2] >= [6] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [2] >= [5] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [6] = cons(mark(X1),X2) mark(isNat(X)) = [2] >= [1] = a__isNat(X) mark(isNatIList(X)) = [2] >= [5] = a__isNatIList(X) mark(isNatList(X)) = [2] >= [0] = a__isNatList(X) mark(length(X)) = [2] >= [3] = a__length(mark(X)) mark(nil()) = [2] >= [0] = nil() mark(s(X)) = [2] >= [2] = s(mark(X)) mark(tt()) = [2] >= [0] = tt() mark(zeros()) = [2] >= [2] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__zeros() -> cons(0(),zeros()) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNatIList(X)) -> a__isNatIList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(U42) = [0] p(U51) = [0] p(U52) = [0] p(U61) = [0] p(U62) = [0] p(a__U11) = [1] x1 + [0] p(a__U21) = [1] x1 + [1] p(a__U31) = [1] x1 + [0] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [5] p(a__U52) = [1] x1 + [0] p(a__U61) = [1] x1 + [5] p(a__U62) = [1] x1 + [4] p(a__isNat) = [1] p(a__isNatIList) = [1] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [2] p(cons) = [1] x1 + [5] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [0] p(mark) = [2] p(nil) = [2] p(s) = [1] x1 + [1] p(tt) = [0] p(zeros) = [1] Following rules are strictly oriented: mark(isNatIList(X)) = [2] > [1] = a__isNatIList(X) Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [0] >= [0] = U11(X) a__U11(tt()) = [0] >= [0] = tt() a__U21(X) = [1] X + [1] >= [0] = U21(X) a__U21(tt()) = [1] >= [0] = tt() a__U31(X) = [1] X + [0] >= [0] = U31(X) a__U31(tt()) = [0] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [0] = U41(X1,X2) a__U41(tt(),V2) = [0] >= [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [0] = U42(X) a__U42(tt()) = [0] >= [0] = tt() a__U51(X1,X2) = [1] X1 + [5] >= [0] = U51(X1,X2) a__U51(tt(),V2) = [5] >= [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [0] >= [0] = U52(X) a__U52(tt()) = [0] >= [0] = tt() a__U61(X1,X2,X3) = [1] X1 + [5] >= [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [5] >= [5] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [4] >= [0] = U62(X1,X2) a__U62(tt(),L) = [4] >= [3] = s(a__length(mark(L))) a__isNat(X) = [1] >= [0] = isNat(X) a__isNat(0()) = [1] >= [0] = tt() a__isNat(length(V1)) = [1] >= [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [1] >= [2] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1] >= [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] >= [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [6] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [5] >= [5] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [2] >= [0] = 0() a__zeros() = [2] >= [5] = cons(0(),zeros()) a__zeros() = [2] >= [1] = zeros() mark(0()) = [2] >= [0] = 0() mark(U11(X)) = [2] >= [2] = a__U11(mark(X)) mark(U21(X)) = [2] >= [3] = a__U21(mark(X)) mark(U31(X)) = [2] >= [2] = a__U31(mark(X)) mark(U41(X1,X2)) = [2] >= [2] = a__U41(mark(X1),X2) mark(U42(X)) = [2] >= [2] = a__U42(mark(X)) mark(U51(X1,X2)) = [2] >= [7] = a__U51(mark(X1),X2) mark(U52(X)) = [2] >= [2] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [2] >= [7] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [2] >= [6] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [7] = cons(mark(X1),X2) mark(isNat(X)) = [2] >= [1] = a__isNat(X) mark(isNatList(X)) = [2] >= [0] = a__isNatList(X) mark(length(X)) = [2] >= [2] = a__length(mark(X)) mark(nil()) = [2] >= [2] = nil() mark(s(X)) = [2] >= [3] = s(mark(X)) mark(tt()) = [2] >= [0] = tt() mark(zeros()) = [2] >= [2] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__zeros() -> cons(0(),zeros()) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [0] p(U21) = [1] x1 + [6] p(U31) = [1] x1 + [1] p(U41) = [1] x1 + [0] p(U42) = [1] x1 + [0] p(U51) = [2] p(U52) = [1] x1 + [1] p(U61) = [1] x1 + [1] p(U62) = [1] x1 + [4] p(a__U11) = [1] x1 + [1] p(a__U21) = [1] x1 + [6] p(a__U31) = [1] x1 + [1] p(a__U41) = [1] x1 + [0] p(a__U42) = [1] x1 + [0] p(a__U51) = [1] x1 + [2] p(a__U52) = [1] x1 + [1] p(a__U61) = [1] x1 + [6] p(a__U62) = [1] x1 + [4] p(a__isNat) = [1] p(a__isNatIList) = [1] p(a__isNatList) = [0] p(a__length) = [1] x1 + [2] p(a__zeros) = [1] p(cons) = [1] x1 + [4] p(isNat) = [1] p(isNatIList) = [1] p(isNatList) = [0] p(length) = [0] p(mark) = [1] p(nil) = [1] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__U11(X) = [1] X + [1] > [1] X + [0] = U11(X) a__U11(tt()) = [1] > [0] = tt() Following rules are (at-least) weakly oriented: a__U21(X) = [1] X + [6] >= [1] X + [6] = U21(X) a__U21(tt()) = [6] >= [0] = tt() a__U31(X) = [1] X + [1] >= [1] X + [1] = U31(X) a__U31(tt()) = [1] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U41(X1,X2) a__U41(tt(),V2) = [0] >= [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1] X + [0] >= [1] X + [0] = U42(X) a__U42(tt()) = [0] >= [0] = tt() a__U51(X1,X2) = [1] X1 + [2] >= [2] = U51(X1,X2) a__U51(tt(),V2) = [2] >= [1] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [1] >= [1] X + [1] = U52(X) a__U52(tt()) = [1] >= [0] = tt() a__U61(X1,X2,X3) = [1] X1 + [6] >= [1] X1 + [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [6] >= [5] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [4] >= [1] X1 + [4] = U62(X1,X2) a__U62(tt(),L) = [4] >= [3] = s(a__length(mark(L))) a__isNat(X) = [1] >= [1] = isNat(X) a__isNat(0()) = [1] >= [0] = tt() a__isNat(length(V1)) = [1] >= [1] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [1] >= [7] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1] >= [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1] >= [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] >= [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [3] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [2] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [6] >= [6] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [3] >= [0] = 0() a__zeros() = [1] >= [4] = cons(0(),zeros()) a__zeros() = [1] >= [0] = zeros() mark(0()) = [1] >= [0] = 0() mark(U11(X)) = [1] >= [2] = a__U11(mark(X)) mark(U21(X)) = [1] >= [7] = a__U21(mark(X)) mark(U31(X)) = [1] >= [2] = a__U31(mark(X)) mark(U41(X1,X2)) = [1] >= [1] = a__U41(mark(X1),X2) mark(U42(X)) = [1] >= [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [1] >= [3] = a__U51(mark(X1),X2) mark(U52(X)) = [1] >= [2] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1] >= [7] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1] >= [5] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1] >= [5] = cons(mark(X1),X2) mark(isNat(X)) = [1] >= [1] = a__isNat(X) mark(isNatIList(X)) = [1] >= [1] = a__isNatIList(X) mark(isNatList(X)) = [1] >= [0] = a__isNatList(X) mark(length(X)) = [1] >= [3] = a__length(mark(X)) mark(nil()) = [1] >= [1] = nil() mark(s(X)) = [1] >= [1] = s(mark(X)) mark(tt()) = [1] >= [0] = tt() mark(zeros()) = [1] >= [1] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__zeros() -> cons(0(),zeros()) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(U11) = [1] x1 + [4] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [0] p(U42) = [1] x1 + [2] p(U51) = [1] x1 + [0] p(U52) = [1] x1 + [0] p(U61) = [1] x1 + [1] x2 + [4] p(U62) = [1] x1 + [1] x2 + [0] p(a__U11) = [1] x1 + [4] p(a__U21) = [1] x1 + [0] p(a__U31) = [1] x1 + [1] p(a__U41) = [1] x1 + [1] x2 + [4] p(a__U42) = [1] x1 + [1] p(a__U51) = [1] x1 + [0] p(a__U52) = [1] x1 + [0] p(a__U61) = [1] x1 + [1] x2 + [4] p(a__U62) = [1] x1 + [1] x2 + [0] p(a__isNat) = [4] p(a__isNatIList) = [1] x1 + [2] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(cons) = [1] x1 + [1] x2 + [6] p(isNat) = [4] p(isNatIList) = [1] x1 + [2] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [6] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__U41(tt(),V2) = [1] V2 + [4] > [1] V2 + [3] = a__U42(a__isNatIList(V2)) Following rules are (at-least) weakly oriented: a__U11(X) = [1] X + [4] >= [1] X + [4] = U11(X) a__U11(tt()) = [4] >= [0] = tt() a__U21(X) = [1] X + [0] >= [1] X + [0] = U21(X) a__U21(tt()) = [0] >= [0] = tt() a__U31(X) = [1] X + [1] >= [1] X + [0] = U31(X) a__U31(tt()) = [1] >= [0] = tt() a__U41(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [0] = U41(X1,X2) a__U42(X) = [1] X + [1] >= [1] X + [2] = U42(X) a__U42(tt()) = [1] >= [0] = tt() a__U51(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U51(X1,X2) a__U51(tt(),V2) = [0] >= [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1] X + [0] >= [1] X + [0] = U52(X) a__U52(tt()) = [0] >= [0] = tt() a__U61(X1,X2,X3) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1] L + [4] >= [1] L + [4] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = U62(X1,X2) a__U62(tt(),L) = [1] L + [0] >= [1] L + [0] = s(a__length(mark(L))) a__isNat(X) = [4] >= [4] = isNat(X) a__isNat(0()) = [4] >= [0] = tt() a__isNat(length(V1)) = [4] >= [4] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [4] >= [4] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1] V + [2] >= [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1] X + [2] >= [1] X + [2] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V2 + [8] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [2] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [4] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [6] >= [1] L + [4] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [6] >= [6] = 0() a__zeros() = [0] >= [12] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [6] >= [6] = 0() mark(U11(X)) = [1] X + [4] >= [1] X + [4] = a__U11(mark(X)) mark(U21(X)) = [1] X + [0] >= [1] X + [0] = a__U21(mark(X)) mark(U31(X)) = [1] X + [0] >= [1] X + [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [4] = a__U41(mark(X1),X2) mark(U42(X)) = [1] X + [2] >= [1] X + [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__U51(mark(X1),X2) mark(U52(X)) = [1] X + [0] >= [1] X + [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [6] = cons(mark(X1),X2) mark(isNat(X)) = [4] >= [4] = a__isNat(X) mark(isNatIList(X)) = [1] X + [2] >= [1] X + [2] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [1] X + [0] >= [1] X + [0] = a__length(mark(X)) mark(nil()) = [6] >= [6] = nil() mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__zeros() -> cons(0(),zeros()) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U21) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U31) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U41) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U42) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U51) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U61) = [1 1 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(U62) = [1 1 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__U41) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U51) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U61) = [1 1 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(a__U62) = [1 1 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(a__isNat) = [1] [1] [0] p(a__isNatIList) = [1] [1] [0] p(a__isNatList) = [0 0 0] [1] [0 0 1] x1 + [0] [0 0 0] [0] p(a__length) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(a__zeros) = [1] [0] [1] p(cons) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 0] [0 0 0] [1] p(isNat) = [0] [1] [0] p(isNatIList) = [0] [1] [0] p(isNatList) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(length) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 0] [1] p(nil) = [1] [1] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(tt) = [1] [1] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__zeros() = [1] [0] [1] > [0] [0] [1] = cons(0(),zeros()) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U11(X) a__U11(tt()) = [1] [1] [0] >= [1] [1] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U21(X) a__U21(tt()) = [1] [1] [0] >= [1] [1] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U31(X) a__U31(tt()) = [1] [1] [0] >= [1] [1] [0] = tt() a__U41(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] = U41(X1,X2) a__U41(tt(),V2) = [1] [1] [0] >= [1] [1] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U42(X) a__U42(tt()) = [1] [1] [0] >= [1] [1] [0] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [1] [1] [0] >= [1] [1] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U52(X) a__U52(tt()) = [1] [1] [0] >= [1] [1] [0] = tt() a__U61(X1,X2,X3) = [1 1 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1 0 0] [1 0 0] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [2] [0 0 0] L + [0] [0 0 0] [0] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = U62(X1,X2) a__U62(tt(),L) = [1 0 0] [2] [0 0 0] L + [0] [0 0 0] [0] >= [1 0 0] [2] [0 0 0] L + [0] [0 0 0] [0] = s(a__length(mark(L))) a__isNat(X) = [1] [1] [0] >= [0] [1] [0] = isNat(X) a__isNat(0()) = [1] [1] [0] >= [1] [1] [0] = tt() a__isNat(length(V1)) = [1] [1] [0] >= [1] [1] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [1] [1] [0] >= [1] [1] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1] [1] [0] >= [1] [1] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1] [1] [0] >= [0] [1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] [1] [0] >= [1] [1] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] [1] [0] >= [1] [1] [0] = tt() a__isNatList(X) = [0 0 0] [1] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [1] [1] [0] >= [1] [1] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [1] [1] [0] >= [1] [1] [0] = tt() a__length(X) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = length(X) a__length(cons(N,L)) = [1 0 1] [1 0 0] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 0] [0 0 0] [0] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [2] [0] [0] >= [0] [0] [0] = 0() a__zeros() = [1] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [1] [0] [1] >= [0] [0] [0] = 0() mark(U11(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] = a__U11(mark(X)) mark(U21(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] = a__U21(mark(X)) mark(U31(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [1 0 0] [1] [0 0 0] X1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X1 + [1] [0 0 0] [1] = a__U41(mark(X1),X2) mark(U42(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [1 0 0] [1] [0 0 0] X1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X1 + [1] [0 0 0] [1] = a__U51(mark(X1),X2) mark(U52(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1 1 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [1] >= [1 1 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1 1 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [1] >= [1 1 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 0] [0 0 0] [1] >= [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 0] [0 0 0] [1] = cons(mark(X1),X2) mark(isNat(X)) = [1] [1] [1] >= [1] [1] [0] = a__isNat(X) mark(isNatIList(X)) = [1] [1] [1] >= [1] [1] [0] = a__isNatIList(X) mark(isNatList(X)) = [0 0 0] [1] [0 0 1] X + [0] [0 0 0] [1] >= [0 0 0] [1] [0 0 1] X + [0] [0 0 0] [0] = a__isNatList(X) mark(length(X)) = [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] X + [0] [0 0 0] [0] = a__length(mark(X)) mark(nil()) = [2] [1] [1] >= [1] [1] [1] = nil() mark(s(X)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = s(mark(X)) mark(tt()) = [2] [1] [1] >= [1] [1] [0] = tt() mark(zeros()) = [1] [0] [1] >= [1] [0] [1] = a__zeros() * Step 10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U41) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U42) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U51) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(U62) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U41) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__U62) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__isNat) = [0] [0] [1] p(a__isNatIList) = [0] [0] [1] p(a__isNatList) = [0] [0] [1] p(a__length) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [0] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0] [0] [1] p(isNatIList) = [0] [0] [1] p(isNatList) = [0] [0] [1] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [1] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(length(X)) = [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] > [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U31(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U31(X) a__U31(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U41(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U41(X1,X2) a__U41(tt(),V2) = [0] [0] [1] >= [0] [0] [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0] [0] [1] >= [0] [0] [1] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = U62(X1,X2) a__U62(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0] [0] [1] >= [0] [0] [1] = isNat(X) a__isNat(0()) = [0] [0] [1] >= [0] [0] [1] = tt() a__isNat(length(V1)) = [0] [0] [1] >= [0] [0] [1] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] [0] [1] >= [0] [0] [1] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0] [0] [1] >= [0] [0] [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0] [0] [1] >= [0] [0] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] [0] [1] >= [0] [0] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [1] >= [0] [0] [1] = tt() a__isNatList(X) = [0] [0] [1] >= [0] [0] [1] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [1] >= [0] [0] [1] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [1] >= [0] [0] [1] = tt() a__length(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(U11(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U11(mark(X)) mark(U21(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U21(mark(X)) mark(U31(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [1] [0] [1] >= [0] [0] [1] = a__isNat(X) mark(isNatIList(X)) = [1] [0] [1] >= [0] [0] [1] = a__isNatIList(X) mark(isNatList(X)) = [1] [0] [1] >= [0] [0] [1] = a__isNatList(X) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(tt()) = [1] [0] [1] >= [0] [0] [1] = tt() mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 11: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U41) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U42) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U51) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U52) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [0 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(U62) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U41) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U52) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0] [0] [0] p(a__isNatIList) = [0] [0] [0] p(a__isNatList) = [0] [0] [0] p(a__length) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [0] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0] [0] [0] p(isNatIList) = [0] [0] [0] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(U61(X1,X2,X3)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U62(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U31(X) a__U31(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U41(X1,X2) a__U41(tt(),V2) = [0] [0] [0] >= [0] [0] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 1 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0] [0] [0] >= [0] [0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0] [0] [0] >= [0] [0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0] [0] [0] >= [0] [0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0] [0] [0] >= [0] [0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNatList(X) = [0] [0] [0] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(U11(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U11(mark(X)) mark(U21(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U21(mark(X)) mark(U31(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U52(mark(X)) mark(cons(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] [0] [0] >= [0] [0] [0] = a__isNat(X) mark(isNatIList(X)) = [0] [0] [0] >= [0] [0] [0] = a__isNatIList(X) mark(isNatList(X)) = [0] [0] [0] >= [0] [0] [0] = a__isNatList(X) mark(length(X)) = [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 12: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(U41) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(U42) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U51) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [1 0 0] [0 0 0] [0 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(U62) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__U41) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U42) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 0 1] [0 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [0 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(a__isNatIList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__isNatList) = [0] [0] [0] p(a__length) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [1] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(isNatIList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [0 0 1] [0] [1 1 1] x1 + [1] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(U31(X)) = [0 0 1] [1] [0 0 1] X + [2] [0 0 1] [1] > [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 0 1] [0 0 1] [1] [1 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [1] > [0 0 1] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U41(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [1] [0] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = U31(X) a__U31(tt()) = [0] [0] [1] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U41(X1,X2) a__U41(tt(),V2) = [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0] [0] [0] >= [0] [0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 0 1] [0 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 0] [0 0 0] [0 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 0 1] [0 0 0] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0 0 0] [0] [0 0 1] V1 + [1] [0 0 0] [0] >= [0] [1] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 0] [0] [0 0 1] V1 + [0] [0 0 0] [0] >= [0] [0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 0] [0] [0 0 0] V + [0] [0 0 1] [1] >= [0] [0] [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 0] [0 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [1] >= [0] [0] [0] = tt() a__isNatList(X) = [0] [0] [0] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 0] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [0] [2] >= [0] [0] [0] = 0() a__zeros() = [0] [1] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [1] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(U11(X)) = [0 0 1] [0] [0 0 1] X + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] = a__U11(mark(X)) mark(U21(X)) = [0 0 1] [0] [0 0 1] X + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U21(mark(X)) mark(U42(X)) = [0 0 1] [0] [0 0 1] X + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 0 1] [0] [0 0 1] X1 + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 0 1] [0] [1 0 1] X + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 0 1] [0 0 1] [0 0 1] [1] [1 0 1] X1 + [0 0 1] X2 + [0 0 1] X3 + [3] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 1] [0 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 0 1] [0 0 1] [0] [1 0 1] X1 + [0 0 3] X2 + [1] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 0] [0] [0 0 1] X + [1] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 0 1] [1] [0 0 1] X + [2] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = a__isNatIList(X) mark(isNatList(X)) = [0] [1] [0] >= [0] [0] [0] = a__isNatList(X) mark(length(X)) = [0 0 1] [1] [0 0 1] X + [3] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [1] [2] [1] >= [0] [0] [1] = nil() mark(s(X)) = [0 0 1] [0] [1 0 1] X + [1] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(tt()) = [0] [1] [0] >= [0] [0] [0] = tt() mark(zeros()) = [0] [1] [0] >= [0] [1] [0] = a__zeros() * Step 13: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U41) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(U42) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U51) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U52) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(U62) = [0 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U41) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 0 1] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0] [0] [0] p(a__isNatIList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__isNatList) = [0] [0] [0] p(a__length) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__zeros) = [0] [0] [1] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(isNat) = [0] [0] [0] p(isNatIList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(nil) = [1] [0] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] > [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U31(X) a__U31(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = U41(X1,X2) a__U41(tt(),V2) = [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0] [0] [0] >= [0] [0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 0 1] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 0 1] [0 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0] [0] [0] >= [0] [0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 0] [0] [0 0 0] V + [0] [0 0 1] [0] >= [0] [0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 0] [0 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNatList(X) = [0] [0] [0] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [1] >= [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(U11(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U11(mark(X)) mark(U21(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U21(mark(X)) mark(U31(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U41(mark(X1),X2) mark(U42(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [1] = a__U51(mark(X1),X2) mark(U52(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 0 1] [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [2] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U62(mark(X1),X2) mark(isNat(X)) = [0] [0] [1] >= [0] [0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = a__isNatIList(X) mark(isNatList(X)) = [0] [0] [1] >= [0] [0] [0] = a__isNatList(X) mark(length(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [1] [0] [2] >= [1] [0] [1] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(tt()) = [0] [0] [1] >= [0] [0] [0] = tt() mark(zeros()) = [0] [0] [1] >= [0] [0] [1] = a__zeros() * Step 14: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__U42(X) -> U42(X) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U41) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(U42) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(U51) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U52) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(U62) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U41) = [1 1 0] [0 0 1] [1] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U42) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__U51) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0] [0] [0] p(a__isNatIList) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__isNatList) = [0] [0] [0] p(a__length) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__zeros) = [0] [0] [1] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(isNat) = [0] [0] [0] p(isNatIList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(nil) = [1] [0] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__U42(X) = [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] > [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = U42(X) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U31(X) a__U31(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 1 0] [0 0 1] [1] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U41(X1,X2) a__U41(tt(),V2) = [0 0 1] [1] [0 0 0] V2 + [0] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] V2 + [0] [0 0 1] [1] = a__U42(a__isNatIList(V2)) a__U42(tt()) = [1] [0] [1] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X1 + [0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0] [0] [0] >= [0] [0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0] [0] [0] >= [0] [0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] [0] [0] >= [0] [0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 1] [0] [0 0 0] V + [0] [0 0 1] [0] >= [0] [0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 1] [0 0 1] [1] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] V2 + [0] [0 0 1] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNatList(X) = [0] [0] [0] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [1] >= [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(U11(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U11(mark(X)) mark(U21(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U21(mark(X)) mark(U31(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U41(mark(X1),X2) mark(U42(X)) = [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [2] >= [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [2] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [1] = a__U51(mark(X1),X2) mark(U52(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = cons(mark(X1),X2) mark(isNat(X)) = [0] [0] [1] >= [0] [0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__isNatIList(X) mark(isNatList(X)) = [0] [0] [1] >= [0] [0] [0] = a__isNatList(X) mark(length(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [1] [0] [2] >= [1] [0] [1] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(tt()) = [0] [0] [1] >= [0] [0] [0] = tt() mark(zeros()) = [0] [0] [1] >= [0] [0] [1] = a__zeros() * Step 15: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U31) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(U41) = [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(U42) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(U51) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(U61) = [1 0 0] [1 1 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(U62) = [1 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U21) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U41) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__U51) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [1 1 1] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [1 1 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(a__isNatIList) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(a__isNatList) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(a__length) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__zeros) = [0] [0] [1] p(cons) = [1 0 1] [1 1 1] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(isNat) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(isNatIList) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(isNatList) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(length) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__isNatList(cons(V1,V2)) = [0 0 1] [0 0 1] [1] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 0] [0 0 0] [0] > [0 0 1] [0 0 1] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 0] [0 0 0] [0] = a__U51(a__isNat(V1),V2) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U11(X) a__U11(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 1 0] [1] [0 0 0] X + [1] [0 0 1] [0] >= [1 1 0] [1] [0 0 0] X + [1] [0 0 1] [0] = U31(X) a__U31(tt()) = [1] [1] [0] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] = U41(X1,X2) a__U41(tt(),V2) = [1 0 1] [1] [0 0 0] V2 + [1] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] V2 + [1] [0 0 1] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [1] [0] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 1] [0] [0 0 0] V2 + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] V2 + [0] [0 0 0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [1 1 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 1 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1 1 1] [0 0 1] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [1 1 1] [0 0 1] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 0] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [1 1 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [1 1 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [1 1 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0 0 1] [0] [0 0 0] V1 + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] V1 + [0] [0 0 0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 1] [0] [0 0 0] V1 + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] V1 + [0] [0 0 0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [1 0 1] [1] [0 0 0] V + [1] [0 0 1] [0] >= [0 0 1] [1] [0 0 0] V + [1] [0 0 0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [1 0 1] [1] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1 0 2] [1 1 2] [2] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 1] [0 0 1] [1] >= [0 0 1] [1 0 1] [1] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 0] [0 0 1] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] [1] [0] >= [0] [0] [0] = tt() a__isNatList(X) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = isNatList(X) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 1 0] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] X + [1] [0 0 1] [0] = length(X) a__length(cons(N,L)) = [1 1 2] [1 0 1] [0] [0 0 0] L + [0 0 0] N + [1] [0 0 1] [0 0 1] [1] >= [1 1 2] [0 0 1] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [1] [0] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [1] >= [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(U11(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U11(mark(X)) mark(U21(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U21(mark(X)) mark(U31(X)) = [1 1 1] [1] [0 0 0] X + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] X + [1] [0 0 1] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [1] = a__U41(mark(X1),X2) mark(U42(X)) = [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [1 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [1] >= [1 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [1] = a__U51(mark(X1),X2) mark(U52(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1 0 1] [1 1 1] [0 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [2] >= [1 0 1] [1 1 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [2] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1 0 1] [1 1 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [1 0 1] [1 1 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1 0 2] [1 1 2] [1] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [2] >= [1 0 2] [1 1 1] [1] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [2] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [1 0 1] [1] [0 0 0] X + [1] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [1] [0 0 1] [0] = a__isNatIList(X) mark(isNatList(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = a__isNatList(X) mark(length(X)) = [1 1 1] [0] [0 0 0] X + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 0] X + [1] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [0] [0] [1] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(tt()) = [0] [0] [1] >= [0] [0] [0] = tt() mark(zeros()) = [0] [0] [1] >= [0] [0] [1] = a__zeros() * Step 16: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U21) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(U31) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U41) = [1 0 0] [0 1 1] [0] [0 0 0] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [1] p(U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(U51) = [1 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(U52) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(U61) = [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(U62) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__U11) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__U21) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__U31) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__U41) = [1 0 0] [0 1 1] [0] [0 0 0] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [1] p(a__U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(a__U51) = [1 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__U52) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__U61) = [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(a__U62) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__isNat) = [0 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__isNatIList) = [0 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(a__isNatList) = [0 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(a__length) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__zeros) = [1] [0] [1] p(cons) = [1 1 0] [1 1 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 0] [0 0 0] [1] p(isNat) = [0 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(isNatIList) = [0 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(isNatList) = [0 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(length) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 1 0] x1 + [0] [1 0 1] [0] p(nil) = [0] [0] [1] p(s) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(tt) = [0] [1] [0] p(zeros) = [1] [0] [0] Following rules are strictly oriented: a__isNat(s(V1)) = [0 1 0] [1] [0 0 0] V1 + [1] [0 0 0] [0] > [0 1 0] [0] [0 0 0] V1 + [1] [0 0 0] [0] = a__U21(a__isNat(V1)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U11(X) a__U11(tt()) = [0] [1] [0] >= [0] [1] [0] = tt() a__U21(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = U21(X) a__U21(tt()) = [0] [1] [0] >= [0] [1] [0] = tt() a__U31(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U31(X) a__U31(tt()) = [0] [1] [0] >= [0] [1] [0] = tt() a__U41(X1,X2) = [1 0 0] [0 1 1] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [1] >= [1 0 0] [0 1 1] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [1] = U41(X1,X2) a__U41(tt(),V2) = [0 1 1] [0] [0 1 1] V2 + [1] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] V2 + [1] [0 0 0] [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] = U42(X) a__U42(tt()) = [0] [1] [1] >= [0] [1] [0] = tt() a__U51(X1,X2) = [1 0 0] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 1 0] [0] [0 0 0] V2 + [1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] V2 + [1] [0 0 0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = U52(X) a__U52(tt()) = [0] [1] [0] >= [0] [1] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [1 0 0] [0 1 0] [0] [0 1 0] L + [0 1 0] N + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [0 1 0] [0] [0 1 0] L + [0 0 0] N + [1] [0 0 0] [0 0 0] [0] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = U62(X1,X2) a__U62(tt(),L) = [1 0 0] [0] [0 1 0] L + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] L + [1] [0 0 0] [0] = s(a__length(mark(L))) a__isNat(X) = [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [1] [0] >= [0] [1] [0] = tt() a__isNat(length(V1)) = [0 1 0] [0] [0 0 0] V1 + [1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] V1 + [1] [0 0 0] [0] = a__U11(a__isNatList(V1)) a__isNatIList(V) = [0 1 0] [0] [0 1 0] V + [1] [0 0 0] [1] >= [0 1 0] [0] [0 0 0] V + [1] [0 0 0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 1 0] [0 1 1] [0] [0 1 0] V1 + [0 1 1] V2 + [1] [0 0 0] [0 0 0] [1] >= [0 1 0] [0 1 1] [0] [0 0 0] V1 + [0 1 1] V2 + [1] [0 0 0] [0 0 0] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [1] [1] >= [0] [1] [0] = tt() a__isNatList(X) = [0 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] >= [0 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 1 0] [0 1 1] [0] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [0] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 0] [0 0 0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [1] [0] >= [0] [1] [0] = tt() a__length(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = length(X) a__length(cons(N,L)) = [1 1 0] [1 1 0] [0] [0 1 1] L + [0 1 0] N + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0 1 0] [0] [0 1 1] L + [0 1 0] N + [0] [0 0 0] [0 0 0] [0] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() a__zeros() = [1] [0] [1] >= [1] [0] [1] = cons(0(),zeros()) a__zeros() = [1] [0] [1] >= [1] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(U11(X)) = [1 0 0] [0] [0 0 0] X + [1] [1 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = a__U11(mark(X)) mark(U21(X)) = [1 0 0] [0] [0 1 0] X + [0] [1 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__U21(mark(X)) mark(U31(X)) = [1 0 0] [0] [0 0 0] X + [1] [1 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [1 0 0] [0 1 1] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [1 0 0] [0 1 1] [1] >= [1 0 0] [0 1 1] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [1] = a__U41(mark(X1),X2) mark(U42(X)) = [1 0 0] [0] [0 1 0] X + [0] [1 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [1 0 0] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [1] [1 0 0] [0 1 0] [0] >= [1 0 0] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [1 0 0] [0] [0 0 0] X + [1] [1 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [1 0 0] [1 0 0] [0 1 0] [0] >= [1 0 0] [1 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [1] [1 0 0] [1 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [1 1 0] [1 1 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [1 1 0] [1 1 0] [1] >= [1 1 0] [1 1 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 0] [0 0 0] [1] = cons(mark(X1),X2) mark(isNat(X)) = [0 1 0] [0] [0 0 0] X + [1] [0 1 0] [0] >= [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 1 0] [0] [0 1 0] X + [1] [0 1 0] [1] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 1 0] [0] [0 0 1] X + [0] [0 1 0] [0] >= [0 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] = a__isNatList(X) mark(length(X)) = [1 0 0] [0] [0 1 0] X + [0] [1 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__length(mark(X)) mark(nil()) = [0] [0] [1] >= [0] [0] [1] = nil() mark(s(X)) = [1 0 0] [0] [0 1 0] X + [1] [1 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = s(mark(X)) mark(tt()) = [0] [1] [0] >= [0] [1] [0] = tt() mark(zeros()) = [1] [0] [1] >= [1] [0] [1] = a__zeros() * Step 17: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(U21) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U31) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U41) = [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [1] p(U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U51) = [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(U52) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U61) = [0 0 0] [0 0 1] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] p(U62) = [0 0 0] [0 1 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(a__U21) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U31) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U41) = [1 0 0] [0 0 1] [1] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 0] [0 1 1] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] p(a__U62) = [1 0 0] [0 1 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__isNat) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__isNatIList) = [0 0 1] [1] [0 1 0] x1 + [0] [0 0 1] [1] p(a__isNatList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__length) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [0] [0] p(cons) = [1 0 0] [0 1 1] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(isNatIList) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(isNatList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(length) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(mark) = [0 1 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(U11(X)) = [0 1 1] [1] [0 1 0] X + [0] [0 0 1] [1] > [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [1] = a__U11(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = U11(X) a__U11(tt()) = [0] [0] [1] >= [0] [0] [0] = tt() a__U21(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U21(X) a__U21(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U31(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U31(X) a__U31(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U41(X1,X2) = [1 0 0] [0 0 1] [1] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [1] = U41(X1,X2) a__U41(tt(),V2) = [0 0 1] [1] [0 1 1] V2 + [0] [0 0 1] [1] >= [0 0 1] [1] [0 1 0] V2 + [0] [0 0 1] [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U51(X1,X2) = [1 0 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0] [0 1 1] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 1] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 1 1] [0 0 0] [0] [0 1 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 0 0] [0] [0 1 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 1 1] [0] [0 1 0] L + [0] [0 0 1] [1] >= [0 1 1] [0] [0 1 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0 0 0] [0] [0 0 0] V1 + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] V1 + [0] [0 0 1] [1] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 0] [0] [0 0 0] V1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] V1 + [0] [0 0 1] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 1] [1] [0 1 0] V + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] V + [0] [0 0 1] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 1] [1] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 1] [0 0 1] [1] [0 1 0] V1 + [0 1 1] V2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 1] [1] [0 0 0] V1 + [0 1 1] V2 + [0] [0 0 1] [0 0 1] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] [0] [1] >= [0] [0] [0] = tt() a__isNatList(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0] [0 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 1 1] [1 0 0] [0] [0 1 1] L + [0 1 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 0 0] [0] [0 1 1] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(U21(X)) = [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U21(mark(X)) mark(U31(X)) = [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 1 1] [0 1 2] [1] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 0 1] [1] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [1] = a__U41(mark(X1),X2) mark(U42(X)) = [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 1 1] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 1 1] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 1 1] [0 1 1] [0 0 1] [1] [0 1 1] X1 + [0 1 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 1 1] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 1 1] [0 1 1] [1] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 1 1] [0 1 2] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 1 1] [0 1 1] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = a__isNat(X) mark(isNatIList(X)) = [0 1 1] [1] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 1] [1] [0 1 0] X + [0] [0 0 1] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = a__isNatList(X) mark(length(X)) = [0 1 1] [1] [0 1 0] X + [0] [0 0 1] [1] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 18: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(U21) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(U31) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(U41) = [0 0 0] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U51) = [1 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [0] p(U52) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(U61) = [0 0 0] [0 1 0] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0 0 1] x3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] p(U62) = [0 0 0] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__U11) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(a__U21) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(a__U31) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(a__U41) = [1 0 0] [0 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__U42) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U51) = [1 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [0] p(a__U52) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__U61) = [1 0 1] [0 1 0] [0 0 1] [0] [0 1 1] x1 + [0 1 0] x2 + [0 0 1] x3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] p(a__U62) = [1 0 1] [0 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__isNat) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__isNatIList) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(a__isNatList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__length) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(a__zeros) = [0] [0] [0] p(cons) = [1 0 1] [0 1 1] [0] [0 1 1] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(isNatIList) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(isNatList) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(length) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(tt) = [0] [0] [1] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(s(X)) = [0 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = s(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] = U11(X) a__U11(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U21(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] = U21(X) a__U21(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U31(X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [0 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = U31(X) a__U31(tt()) = [0] [1] [1] >= [0] [0] [1] = tt() a__U41(X1,X2) = [1 0 0] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = U41(X1,X2) a__U41(tt(),V2) = [0 0 0] [0] [0 1 0] V2 + [1] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] V2 + [1] [0 0 1] [1] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U42(X) a__U42(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U51(X1,X2) = [1 0 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [0] >= [1 0 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = U52(X) a__U52(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() a__U61(X1,X2,X3) = [1 0 1] [0 1 0] [0 0 1] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 1] X3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] >= [0 0 0] [0 1 0] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 1] X3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 1 0] [0 0 1] [1] [0 1 0] L + [0 0 1] N + [2] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 0 1] [1] [0 1 0] L + [0 0 1] N + [2] [0 0 0] [0 0 0] [0] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 1] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = U62(X1,X2) a__U62(tt(),L) = [0 1 0] [1] [0 1 0] L + [2] [0 0 0] [0] >= [0 1 0] [0] [0 1 0] L + [2] [0 0 0] [0] = s(a__length(mark(L))) a__isNat(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = isNat(X) a__isNat(0()) = [0] [0] [1] >= [0] [0] [1] = tt() a__isNat(length(V1)) = [0] [0] [1] >= [0] [0] [1] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0] [0] [1] >= [0] [0] [1] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 0] [0] [0 1 0] V + [1] [0 0 1] [1] >= [0] [1] [1] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 0] [0 0 0] [0] [0 1 1] V1 + [0 1 1] V2 + [1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 1] V1 + [0 1 0] V2 + [1] [0 0 1] [0 0 1] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [1] [1] >= [0] [0] [1] = tt() a__isNatList(X) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0] [0 0 0] [0] [0 0 0] V1 + [0 0 0] V2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] V2 + [0] [0 0 1] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [1] >= [0] [0] [1] = tt() a__length(X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = length(X) a__length(cons(N,L)) = [0 1 1] [1 0 1] [0] [0 1 1] L + [0 1 1] N + [1] [0 0 0] [0 0 0] [0] >= [0 1 1] [0 0 1] [0] [0 1 1] L + [0 0 1] N + [1] [0 0 0] [0 0 0] [0] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [0] [1] [0] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(U11(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = a__U11(mark(X)) mark(U21(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = a__U21(mark(X)) mark(U31(X)) = [0 1 0] [1] [0 1 0] X + [1] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 1 1] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 1 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [0] >= [0 1 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 1 1] [0 1 0] [0 0 1] [1] [0 1 1] X1 + [0 1 0] X2 + [0 0 1] X3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] >= [0 1 1] [0 1 0] [0 0 1] [0] [0 1 1] X1 + [0 1 0] X2 + [0 0 1] X3 + [1] [0 0 0] [0 0 0] [0 0 0] [0] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 1 1] [0 1 0] [1] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [0 1 1] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 1 1] [0 1 1] [0] [0 1 1] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 1 1] [0 1 1] [0] [0 1 1] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = a__isNat(X) mark(isNatIList(X)) = [0 1 0] [1] [0 1 0] X + [1] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = a__isNatList(X) mark(length(X)) = [0 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = a__length(mark(X)) mark(nil()) = [0] [0] [1] >= [0] [0] [1] = nil() mark(tt()) = [0] [0] [1] >= [0] [0] [1] = tt() mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 19: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(U21(X)) -> a__U21(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, 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 (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U21) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U31) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U41) = [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(U42) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U51) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 0 1 1] x2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(U52) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U61) = [0 0 0 0] [0 1 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0 1 1 0] x3 + [0] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] p(U62) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 1 0] x2 + [1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__U11) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U21) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U31) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(a__U41) = [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__U42) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U51) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 0 1 1] x2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__U52) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U61) = [1 0 1 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0 1 1 0] x3 + [0] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] p(a__U62) = [1 0 0 0] [0 1 1 0] [1] [0 1 0 0] x1 + [0 1 1 0] x2 + [1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__isNat) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__isNatIList) = [0 0 0 0] [0] [0 1 1 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] p(a__isNatList) = [0 0 0 0] [0] [0 0 1 0] x1 + [1] [0 0 0 1] [0] [0 0 0 0] [0] p(a__length) = [1 0 1 0] [0] [0 1 1 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(a__zeros) = [1] [1] [0] [1] p(cons) = [1 0 0 0] [0 1 0 0] [1] [0 1 0 0] x1 + [0 1 1 0] x2 + [1] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [1] p(isNat) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(isNatIList) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(isNatList) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(length) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(mark) = [0 1 0 0] [1] [0 1 0 0] x1 + [1] [0 0 1 0] [0] [1 0 0 0] [0] p(nil) = [1] [0] [0] [1] p(s) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(tt) = [0] [0] [1] [0] p(zeros) = [1] [0] [0] [0] Following rules are strictly oriented: mark(U51(X1,X2)) = [0 1 0 0] [0 0 1 1] [2] [0 1 0 0] X1 + [0 0 1 1] X2 + [2] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] > [0 1 0 0] [0 0 0 0] [1] [0 1 0 0] X1 + [0 0 1 1] X2 + [2] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U51(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U11(X) a__U11(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U21(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U21(X) a__U21(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U31(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U31(X) a__U31(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U41(X1,X2) = [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = U41(X1,X2) a__U41(tt(),V2) = [0 0 0 0] [0] [0 1 1 0] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 1 0] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U42(X) a__U42(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U51(X1,X2) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 0 0] [0] [0 0 1 1] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U52(X) a__U52(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U61(X1,X2,X3) = [1 0 1 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0 1 1 0] X3 + [0] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 1 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0 1 1 0] X3 + [0] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 1 1 0] [0 0 0 0] [1] [0 1 1 0] L + [0 1 1 0] N + [1] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 1 1 0] [0 0 0 0] [1] [0 1 1 0] L + [0 0 1 0] N + [1] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0 0] [0 1 1 0] [1] [0 1 0 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = U62(X1,X2) a__U62(tt(),L) = [0 1 1 0] [1] [0 1 1 0] L + [1] [0 0 1 0] [1] [0 0 0 0] [0] >= [0 1 1 0] [1] [0 1 1 0] L + [1] [0 0 1 0] [1] [0 0 0 0] [0] = s(a__length(mark(L))) a__isNat(X) = [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__isNat(length(V1)) = [0 0 0 0] [0] [0 0 1 0] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 0 0] [0] [0 0 1 0] V1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 0 0] [0] [0 1 1 0] V + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 0 0] [0] [0 1 1 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] V1 + [0 1 2 1] V2 + [2] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] V1 + [0 1 1 0] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [1] [1] [0] >= [0] [0] [1] [0] = tt() a__isNatList(X) = [0 0 0 0] [0] [0 0 1 0] X + [1] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] V1 + [0 0 1 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] V1 + [0 0 1 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [1] [1] [0] >= [0] [0] [1] [0] = tt() a__length(X) = [1 0 1 0] [0] [0 1 1 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = length(X) a__length(cons(N,L)) = [0 1 1 1] [1 0 1 0] [1] [0 1 2 1] L + [0 1 1 0] N + [1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 1 1 1] [0 0 0 0] [0] [0 1 2 1] L + [0 1 1 0] N + [1] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [0] [1] [0] >= [0] [0] [0] [0] = 0() a__zeros() = [1] [1] [0] [1] >= [1] [1] [0] [1] = cons(0(),zeros()) a__zeros() = [1] [1] [0] [1] >= [1] [0] [0] [0] = zeros() mark(0()) = [1] [1] [0] [0] >= [0] [0] [0] [0] = 0() mark(U11(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U11(mark(X)) mark(U21(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U21(mark(X)) mark(U31(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [1 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [1 0 0 0] [0] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 1 1 0] [0 1 1 0] [1] [0 1 1 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 0] [0 0 0 0] [1] [1 0 0 0] [0 0 0 0] [0] >= [0 1 0 0] [0 0 0 0] [1] [0 1 1 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U42(mark(X)) mark(U52(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 1 1 0] [0 1 1 0] [0 1 1 0] [1] [0 1 1 0] X1 + [0 1 1 0] X2 + [0 1 1 0] X3 + [1] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 1 0 0] [0 0 0 0] [0] >= [0 1 1 0] [0 1 1 0] [0 0 0 0] [1] [0 1 1 0] X1 + [0 1 1 0] X2 + [0 1 1 0] X3 + [1] [0 0 0 0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 1 0 0] [0 1 1 0] [2] [0 1 0 0] X1 + [0 1 1 0] X2 + [2] [0 0 0 0] [0 0 1 0] [1] [1 0 0 0] [0 0 0 0] [0] >= [0 1 0 0] [0 1 1 0] [2] [0 1 0 0] X1 + [0 1 1 0] X2 + [2] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 1 0 0] [0 1 1 0] [2] [0 1 0 0] X1 + [0 1 1 0] X2 + [2] [0 0 1 0] [0 0 1 1] [0] [1 0 0 0] [0 1 0 0] [1] >= [0 1 0 0] [0 1 0 0] [2] [0 1 0 0] X1 + [0 1 1 0] X2 + [2] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [1] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 1 0] [1] [0 0 1 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 1 1 0] [1] [0 1 1 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__isNatIList(X) mark(isNatList(X)) = [0 0 1 0] [1] [0 0 1 0] X + [1] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [1] [0 0 0 1] [0] [0 0 0 0] [0] = a__isNatList(X) mark(length(X)) = [0 1 1 0] [1] [0 1 1 0] X + [1] [0 0 1 0] [1] [1 0 0 0] [0] >= [0 1 1 0] [1] [0 1 1 0] X + [1] [0 0 1 0] [1] [0 0 0 0] [0] = a__length(mark(X)) mark(nil()) = [1] [1] [0] [1] >= [1] [0] [0] [1] = nil() mark(s(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 1 0] [0] [1 0 0 0] [0] >= [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 1 0] [0] [0 0 0 0] [0] = s(mark(X)) mark(tt()) = [1] [1] [1] [0] >= [0] [0] [1] [0] = tt() mark(zeros()) = [1] [1] [0] [1] >= [1] [1] [0] [1] = a__zeros() * Step 20: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(U21(X)) -> a__U21(mark(X)) mark(U52(X)) -> a__U52(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, 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 (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U21) = [0 0 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] p(U31) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U41) = [0 0 0 0] [0 0 1 1] [0] [0 1 0 0] x1 + [0 0 1 1] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(U42) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U51) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 0 1 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(U52) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(U61) = [1 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0 0 1 0] x3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] p(U62) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] p(a__U11) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U21) = [1 0 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U31) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(a__U41) = [1 0 0 0] [0 0 1 1] [0] [0 1 0 0] x1 + [0 0 1 1] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__U42) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U51) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 0 1 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(a__U52) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__U61) = [1 0 0 0] [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0 0 1 0] x3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [0 0 0 0] [0] p(a__U62) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] p(a__isNat) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__isNatIList) = [0 0 1 1] [0] [0 0 1 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] p(a__isNatList) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(a__length) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(a__zeros) = [1] [1] [0] [1] p(cons) = [1 0 0 0] [0 1 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [1] p(isNat) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(isNatIList) = [0 0 1 1] [0] [0 0 1 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] p(isNatList) = [0 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(length) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(mark) = [0 1 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [1] p(nil) = [1] [1] [0] [1] p(s) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(tt) = [0] [0] [1] [0] p(zeros) = [0] [1] [0] [0] Following rules are strictly oriented: mark(U21(X)) = [0 1 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [0 1 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U21(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U11(X) a__U11(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U21(X) = [1 0 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = U21(X) a__U21(tt()) = [0] [1] [1] [0] >= [0] [0] [1] [0] = tt() a__U31(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U31(X) a__U31(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U41(X1,X2) = [1 0 0 0] [0 0 1 1] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 1 1] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = U41(X1,X2) a__U41(tt(),V2) = [0 0 1 1] [0] [0 0 1 1] V2 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 1 1] [0] [0 0 1 1] V2 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U42(X) a__U42(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U51(X1,X2) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 0 0] [0] [0 0 1 0] V2 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V2 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = U52(X) a__U52(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__U61(X1,X2,X3) = [1 0 0 0] [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] L + [0 0 1 0] N + [0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] L + [0 0 1 0] N + [0] [0 0 1 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] = U62(X1,X2) a__U62(tt(),L) = [0 1 0 0] [0] [0 1 0 0] L + [0] [0 0 1 0] [1] [0 0 0 0] [0] >= [0 1 0 0] [0] [0 1 0 0] L + [0] [0 0 1 0] [1] [0 0 0 0] [0] = s(a__length(mark(L))) a__isNat(X) = [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__isNat(length(V1)) = [0 0 0 0] [0] [0 0 1 0] V1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 0 0] [0] [0 0 1 0] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 0 1 1] [0] [0 0 1 1] V + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 1 0] V + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 0 1 1] [0] [0 0 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 1 1] [0] [0 0 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 1 0] [0 0 1 1] [1] [0 0 1 0] V1 + [0 0 1 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 0] [0 0 1 1] [0] [0 0 1 0] V1 + [0 0 1 1] V2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [0] [0] [1] [1] >= [0] [0] [1] [0] = tt() a__isNatList(X) = [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] V1 + [0 0 1 1] V2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] V1 + [0 0 1 0] V2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() a__length(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = length(X) a__length(cons(N,L)) = [0 1 0 0] [1 0 0 0] [0] [0 1 1 0] L + [0 1 1 0] N + [0] [0 0 1 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [0 1 0 0] [0 0 0 0] [0] [0 1 1 0] L + [0 0 1 0] N + [0] [0 0 1 1] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [1] [0] [0] >= [0] [0] [0] [0] = 0() a__zeros() = [1] [1] [0] [1] >= [1] [1] [0] [1] = cons(0(),zeros()) a__zeros() = [1] [1] [0] [1] >= [0] [1] [0] [0] = zeros() mark(0()) = [0] [0] [0] [1] >= [0] [0] [0] [0] = 0() mark(U11(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U11(mark(X)) mark(U31(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 1 0 0] [0 0 1 1] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0 1 0 0] [0 0 1 1] [0] [0 1 0 0] X1 + [0 0 1 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U41(mark(X1),X2) mark(U42(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 1 0 0] [0 0 1 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U51(mark(X1),X2) mark(U52(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__U52(mark(X)) mark(U61(X1,X2,X3)) = [0 1 0 0] [0 1 0 0] [0 0 1 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] >= [0 1 0 0] [0 1 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 1 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [0 1 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 1 1 0] [0 1 1 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [1] >= [0 1 0 0] [0 1 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [1] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 1 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 0 1 1] [0] [0 0 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 1 1] [0] [0 0 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 1 0] [0] [0 0 1 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 1 0] X + [0] [0 0 0 1] [0] [0 0 0 0] [0] = a__isNatList(X) mark(length(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = a__length(mark(X)) mark(nil()) = [1] [1] [0] [1] >= [1] [1] [0] [1] = nil() mark(s(X)) = [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = s(mark(X)) mark(tt()) = [0] [0] [1] [1] >= [0] [0] [1] [0] = tt() mark(zeros()) = [1] [1] [0] [1] >= [1] [1] [0] [1] = a__zeros() * Step 21: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(U52(X)) -> a__U52(mark(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, 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 (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U21) = {1}, uargs(a__U31) = {1}, uargs(a__U41) = {1}, uargs(a__U42) = {1}, uargs(a__U51) = {1}, uargs(a__U52) = {1}, uargs(a__U61) = {1}, uargs(a__U62) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61,a__U62,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(U21) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(U31) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(U41) = [0 0 0 0] [0 1 1 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] p(U42) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(U51) = [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 0 0 1] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [0] p(U52) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(U61) = [0 0 0 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0 0 0 1] x3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] p(U62) = [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] x1 + [0 1 1 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__U11) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(a__U21) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(a__U31) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(a__U41) = [1 0 0 0] [0 1 1 0] [0] [0 1 1 0] x1 + [0 1 1 1] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__U42) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(a__U51) = [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 0 0 1] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [0] p(a__U52) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [1] [0 0 0 1] [0] p(a__U61) = [1 0 0 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 1] x2 + [0 0 0 1] x3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] p(a__U62) = [1 0 0 0] [0 1 1 0] [0] [0 1 1 1] x1 + [0 1 1 1] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__isNat) = [0 0 0 0] [1] [0 0 0 1] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(a__isNatIList) = [0 1 0 0] [1] [0 1 0 1] x1 + [1] [0 0 0 0] [1] [0 0 0 1] [0] p(a__isNatList) = [0 0 0 0] [1] [0 0 0 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(a__length) = [1 0 0 0] [1] [0 1 1 0] x1 + [1] [0 0 0 0] [0] [0 0 0 1] [0] p(a__zeros) = [0] [0] [0] [1] p(cons) = [1 0 0 0] [0 1 1 0] [0] [0 1 1 1] x1 + [0 1 1 1] x2 + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [1] p(isNat) = [0 0 0 0] [1] [0 0 0 1] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(isNatIList) = [0 0 0 0] [1] [0 1 0 1] x1 + [1] [0 0 0 0] [1] [0 0 0 1] [0] p(isNatList) = [0 0 0 0] [1] [0 0 0 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(length) = [0 0 0 0] [0] [0 1 1 0] x1 + [1] [0 0 0 0] [0] [0 0 0 1] [0] p(mark) = [0 1 1 0] [0] [0 1 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [1] p(nil) = [0] [0] [0] [1] p(s) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(tt) = [1] [1] [0] [0] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: mark(U52(X)) = [0 1 1 0] [1] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 1] [1] > [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 1] [1] = a__U52(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] = U11(X) a__U11(tt()) = [1] [1] [0] [0] >= [1] [1] [0] [0] = tt() a__U21(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] = U21(X) a__U21(tt()) = [1] [1] [0] [0] >= [1] [1] [0] [0] = tt() a__U31(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] = U31(X) a__U31(tt()) = [1] [1] [1] [0] >= [1] [1] [0] [0] = tt() a__U41(X1,X2) = [1 0 0 0] [0 1 1 0] [0] [0 1 1 0] X1 + [0 1 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] >= [0 0 0 0] [0 1 1 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] = U41(X1,X2) a__U41(tt(),V2) = [0 1 1 0] [1] [0 1 1 1] V2 + [2] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 0 0] [1] [0 1 0 1] V2 + [2] [0 0 0 0] [0] [0 0 0 1] [0] = a__U42(a__isNatIList(V2)) a__U42(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] = U42(X) a__U42(tt()) = [1] [1] [0] [0] >= [1] [1] [0] [0] = tt() a__U51(X1,X2) = [1 0 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 0 0 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 0 0 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [0] = U51(X1,X2) a__U51(tt(),V2) = [0 0 0 0] [1] [0 0 0 1] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [1] [0 0 0 1] V2 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U52(a__isNatList(V2)) a__U52(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 0 0] [1] [0 0 0 1] [0] = U52(X) a__U52(tt()) = [1] [1] [1] [0] >= [1] [1] [0] [0] = tt() a__U61(X1,X2,X3) = [1 0 0 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 1] X2 + [0 0 0 1] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] >= [0 0 0 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0 0 0 1] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] = U61(X1,X2,X3) a__U61(tt(),L,N) = [0 1 1 0] [0 0 0 0] [1] [0 1 1 1] L + [0 0 0 1] N + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [1] >= [0 1 1 0] [0 0 0 0] [1] [0 1 1 1] L + [0 0 0 1] N + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [1] = a__U62(a__isNat(N),L) a__U62(X1,X2) = [1 0 0 0] [0 1 1 0] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] X1 + [0 1 1 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] = U62(X1,X2) a__U62(tt(),L) = [0 1 1 0] [1] [0 1 1 1] L + [1] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [1] [0 1 1 1] L + [1] [0 0 0 0] [0] [0 0 0 1] [1] = s(a__length(mark(L))) a__isNat(X) = [0 0 0 0] [1] [0 0 0 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [0 0 0 0] [1] [0 0 0 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = isNat(X) a__isNat(0()) = [1] [1] [0] [0] >= [1] [1] [0] [0] = tt() a__isNat(length(V1)) = [0 0 0 0] [1] [0 0 0 1] V1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [0 0 0 0] [1] [0 0 0 1] V1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = [0 0 0 0] [1] [0 0 0 1] V1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [0 0 0 0] [1] [0 0 0 1] V1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = a__U21(a__isNat(V1)) a__isNatIList(V) = [0 1 0 0] [1] [0 1 0 1] V + [1] [0 0 0 0] [1] [0 0 0 1] [0] >= [0 0 0 0] [1] [0 0 0 1] V + [1] [0 0 0 0] [1] [0 0 0 0] [0] = a__U31(a__isNatList(V)) a__isNatIList(X) = [0 1 0 0] [1] [0 1 0 1] X + [1] [0 0 0 0] [1] [0 0 0 1] [0] >= [0 0 0 0] [1] [0 1 0 1] X + [1] [0 0 0 0] [1] [0 0 0 1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 1 1 1] [0 1 1 1] [1] [0 1 1 2] V1 + [0 1 1 2] V2 + [2] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 1] [1] >= [0 0 0 0] [0 1 1 0] [1] [0 0 0 1] V1 + [0 1 1 1] V2 + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] = a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) = [1] [1] [1] [0] >= [1] [1] [0] [0] = tt() a__isNatList(X) = [0 0 0 0] [1] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [0 0 0 0] [1] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] V1 + [0 0 0 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] V1 + [0 0 0 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = a__U51(a__isNat(V1),V2) a__isNatList(nil()) = [1] [1] [1] [0] >= [1] [1] [0] [0] = tt() a__length(X) = [1 0 0 0] [1] [0 1 1 0] X + [1] [0 0 0 0] [0] [0 0 0 1] [0] >= [0 0 0 0] [0] [0 1 1 0] X + [1] [0 0 0 0] [0] [0 0 0 1] [0] = length(X) a__length(cons(N,L)) = [0 1 1 0] [1 0 0 0] [1] [0 1 1 2] L + [0 1 1 1] N + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] >= [0 1 1 0] [0 0 0 0] [1] [0 1 1 2] L + [0 0 0 1] N + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [1] = a__U61(a__isNatList(L),L,N) a__length(nil()) = [1] [1] [0] [1] >= [0] [0] [0] [0] = 0() a__zeros() = [0] [0] [0] [1] >= [0] [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [0] [1] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] [1] >= [0] [0] [0] [0] = 0() mark(U11(X)) = [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = a__U11(mark(X)) mark(U21(X)) = [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = a__U21(mark(X)) mark(U31(X)) = [0 1 1 0] [1] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 1] [1] >= [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 1] [1] = a__U31(mark(X)) mark(U41(X1,X2)) = [0 1 1 0] [0 1 1 0] [1] [0 1 1 1] X1 + [0 1 1 1] X2 + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [2] >= [0 1 1 0] [0 1 1 0] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [2] = a__U41(mark(X1),X2) mark(U42(X)) = [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = a__U42(mark(X)) mark(U51(X1,X2)) = [0 1 1 0] [0 0 0 1] [1] [0 1 1 1] X1 + [0 0 0 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [1] >= [0 1 1 0] [0 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 1] X2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [1] = a__U51(mark(X1),X2) mark(U61(X1,X2,X3)) = [0 1 1 0] [0 1 1 0] [0 0 0 1] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [0 0 0 1] X3 + [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [2] >= [0 1 1 0] [0 1 1 0] [0 0 0 0] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [0 0 0 1] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0 0 0 0] [2] = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = [0 1 1 1] [0 1 1 0] [0] [0 1 1 2] X1 + [0 1 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [2] >= [0 1 1 0] [0 1 1 0] [0] [0 1 1 2] X1 + [0 1 1 1] X2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [2] = a__U62(mark(X1),X2) mark(cons(X1,X2)) = [0 1 1 1] [0 1 1 2] [0] [0 1 1 2] X1 + [0 1 1 2] X2 + [1] [0 0 0 0] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [2] >= [0 1 1 0] [0 1 1 0] [0] [0 1 1 2] X1 + [0 1 1 1] X2 + [1] [0 0 0 0] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [2] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 0 1] [1] [0 0 0 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [1] >= [0 0 0 0] [1] [0 0 0 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 1 0 1] [2] [0 1 0 2] X + [1] [0 0 0 0] [1] [0 0 0 1] [1] >= [0 1 0 0] [1] [0 1 0 1] X + [1] [0 0 0 0] [1] [0 0 0 1] [0] = a__isNatIList(X) mark(isNatList(X)) = [0 0 0 1] [1] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 0] [1] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = a__isNatList(X) mark(length(X)) = [0 1 1 0] [1] [0 1 1 1] X + [1] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [1] [0 1 1 1] X + [1] [0 0 0 0] [0] [0 0 0 1] [1] = a__length(mark(X)) mark(nil()) = [0] [1] [0] [2] >= [0] [0] [0] [1] = nil() mark(s(X)) = [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [0 1 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = s(mark(X)) mark(tt()) = [1] [1] [0] [1] >= [1] [1] [0] [0] = tt() mark(zeros()) = [0] [0] [0] [1] >= [0] [0] [0] [1] = a__zeros() * Step 22: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X) -> U31(X) a__U31(tt()) -> tt() a__U41(X1,X2) -> U41(X1,X2) a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(X) -> U42(X) a__U42(tt()) -> tt() a__U51(X1,X2) -> U51(X1,X2) a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(X) -> U52(X) a__U52(tt()) -> tt() a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(X1,X2) -> U62(X1,X2) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/1,a__U21/1,a__U31/1,a__U41/2,a__U42/1,a__U51/2,a__U52/1,a__U61/3,a__U62/2,a__isNat/1,a__isNatIList/1 ,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__U31,a__U41,a__U42,a__U51,a__U52,a__U61 ,a__U62,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} and constructors {0,U11,U21,U31,U41,U42 ,U51,U52,U61,U62,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))