MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [1] x2 + [0] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [13] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: mark(0()) = [13] > [0] = 0() mark(nil()) = [13] > [0] = nil() mark(tt()) = [13] > [0] = tt() mark(zeros()) = [13] > [0] = a__zeros() Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(tt(),X) = [1] X + [0] >= [1] X + [13] = mark(X) a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [0] >= [1] L + [13] = s(a__length(mark(L))) a__length(nil()) = [0] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__take(s(M),cons(N,IL)) = [1] IL + [1] M + [1] N + [0] >= [1] IL + [1] M + [1] N + [13] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(and(X1,X2)) = [1] X1 + [1] X2 + [13] >= [1] X1 + [1] X2 + [13] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [13] >= [1] X1 + [1] X2 + [13] = cons(mark(X1),X2) mark(length(X)) = [1] X + [13] >= [1] X + [13] = a__length(mark(X)) mark(s(X)) = [1] X + [13] >= [1] X + [13] = s(mark(X)) mark(take(X1,X2)) = [1] X1 + [1] X2 + [13] >= [1] X1 + [1] X2 + [26] = a__take(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Weak TRS: mark(0()) -> 0() mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [1] x2 + [0] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [9] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: mark(take(X1,X2)) = [1] X1 + [1] X2 + [9] > [1] X1 + [1] X2 + [0] = a__take(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(tt(),X) = [1] X + [0] >= [1] X + [0] = mark(X) a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [0] >= [1] L + [0] = s(a__length(mark(L))) a__length(nil()) = [0] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [9] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__take(s(M),cons(N,IL)) = [1] IL + [1] M + [1] N + [0] >= [1] IL + [1] M + [1] N + [9] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(mark(X1),X2) 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 MAYBE + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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: mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [0] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [2] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [2] p(take) = [1] x1 + [1] x2 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__take(s(M),cons(N,IL)) = [1] M + [1] N + [4] > [2] = cons(mark(N),take(M,IL)) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] N + [2] >= [2] = s(a__length(mark(L))) a__length(nil()) = [0] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__zeros() = [0] >= [2] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [2] = cons(mark(X1),X2) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [2] = s(mark(X)) mark(take(X1,X2)) = [0] >= [0] = a__take(mark(X1),mark(X2)) 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 4: WeightGap MAYBE + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [5] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [0] p(cons) = [1] x1 + [5] p(length) = [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [5] p(take) = [1] x2 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__and(X1,X2) = [1] X1 + [5] > [0] = and(X1,X2) a__and(tt(),X) = [5] > [0] = mark(X) Following rules are (at-least) weakly oriented: a__length(X) = [1] X + [0] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [5] >= [5] = s(a__length(mark(L))) a__length(nil()) = [0] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X2 + [0] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__take(s(M),cons(N,IL)) = [1] M + [1] N + [10] >= [5] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [5] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [0] >= [5] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [5] = cons(mark(X1),X2) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [5] = s(mark(X)) mark(take(X1,X2)) = [0] >= [0] = a__take(mark(X1),mark(X2)) 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 5: WeightGap MAYBE + Considered Problem: - Strict TRS: a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [0] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [6] p(length) = [1] x1 + [1] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [2] p(take) = [1] x1 + [4] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__length(cons(N,L)) = [1] N + [6] > [2] = s(a__length(mark(L))) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__length(X) = [1] X + [0] >= [1] X + [1] = length(X) a__length(nil()) = [0] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [4] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__take(s(M),cons(N,IL)) = [1] M + [1] N + [8] >= [6] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [6] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [6] = cons(mark(X1),X2) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [2] = s(mark(X)) mark(take(X1,X2)) = [0] >= [0] = a__take(mark(X1),mark(X2)) 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 6: WeightGap MAYBE + Considered Problem: - Strict TRS: a__length(X) -> length(X) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__and) = [1] x1 + [0] p(a__length) = [1] x1 + [4] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [5] p(length) = [1] x1 + [1] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [1] p(take) = [1] x2 + [0] p(tt) = [0] p(zeros) = [1] Following rules are strictly oriented: a__length(X) = [1] X + [4] > [1] X + [1] = length(X) a__length(nil()) = [4] > [0] = 0() Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__length(cons(N,L)) = [1] N + [9] >= [5] = s(a__length(mark(L))) a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X2 + [0] = take(X1,X2) a__take(0(),IL) = [1] IL + [0] >= [0] = nil() a__take(s(M),cons(N,IL)) = [1] M + [1] N + [6] >= [5] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [5] = cons(0(),zeros()) a__zeros() = [0] >= [1] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [5] = cons(mark(X1),X2) mark(length(X)) = [0] >= [4] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [1] = s(mark(X)) mark(take(X1,X2)) = [0] >= [0] = a__take(mark(X1),mark(X2)) 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 7: WeightGap MAYBE + Considered Problem: - Strict TRS: a__take(X1,X2) -> take(X1,X2) a__take(0(),IL) -> nil() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(a__and) = [1] x1 + [1] x2 + [1] p(a__length) = [1] x1 + [1] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [2] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [5] p(s) = [1] x1 + [2] p(take) = [1] x1 + [1] x2 + [1] p(tt) = [4] p(zeros) = [0] Following rules are strictly oriented: a__take(0(),IL) = [1] IL + [6] > [5] = nil() Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(tt(),X) = [1] X + [5] >= [1] X + [0] = mark(X) a__length(X) = [1] X + [1] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [3] >= [1] L + [3] = s(a__length(mark(L))) a__length(nil()) = [6] >= [6] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = take(X1,X2) a__take(s(M),cons(N,IL)) = [1] IL + [1] M + [1] N + [4] >= [1] IL + [1] M + [1] N + [3] = cons(mark(N),take(M,IL)) a__zeros() = [0] >= [8] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [6] >= [6] = 0() mark(and(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = cons(mark(X1),X2) mark(length(X)) = [1] X + [0] >= [1] X + [1] = a__length(mark(X)) mark(nil()) = [5] >= [5] = nil() mark(s(X)) = [1] X + [2] >= [1] X + [2] = s(mark(X)) mark(take(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [0] = a__take(mark(X1),mark(X2)) mark(tt()) = [4] >= [4] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap MAYBE + Considered Problem: - Strict TRS: a__take(X1,X2) -> take(X1,X2) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(and(X1,X2)) -> a__and(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__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__and) = [1] x1 + [1] x2 + [0] p(a__length) = [1] x1 + [2] p(a__take) = [1] x1 + [1] x2 + [0] p(a__zeros) = [2] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [4] p(length) = [1] x1 + [2] p(mark) = [1] x1 + [1] p(nil) = [2] p(s) = [1] x1 + [3] p(take) = [1] x1 + [1] x2 + [2] p(tt) = [2] p(zeros) = [1] Following rules are strictly oriented: a__zeros() = [2] > [1] = zeros() Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(tt(),X) = [1] X + [2] >= [1] X + [1] = mark(X) a__length(X) = [1] X + [2] >= [1] X + [2] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [6] >= [1] L + [6] = s(a__length(mark(L))) a__length(nil()) = [4] >= [4] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [2] = take(X1,X2) a__take(0(),IL) = [1] IL + [4] >= [2] = nil() a__take(s(M),cons(N,IL)) = [1] IL + [1] M + [1] N + [7] >= [1] IL + [1] M + [1] N + [7] = cons(mark(N),take(M,IL)) a__zeros() = [2] >= [9] = cons(0(),zeros()) mark(0()) = [5] >= [4] = 0() mark(and(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = cons(mark(X1),X2) mark(length(X)) = [1] X + [3] >= [1] X + [3] = a__length(mark(X)) mark(nil()) = [3] >= [2] = nil() mark(s(X)) = [1] X + [4] >= [1] X + [4] = s(mark(X)) mark(take(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [2] = a__take(mark(X1),mark(X2)) mark(tt()) = [3] >= [2] = tt() mark(zeros()) = [2] >= [2] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__take(X1,X2) -> take(X1,X2) a__zeros() -> cons(0(),zeros()) mark(and(X1,X2)) -> a__and(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__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) a__zeros() -> zeros() mark(0()) -> 0() mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,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__and) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__and,a__length,a__take,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(a__and) = [1 0 0 1] [1 0 0 1] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__length) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] p(a__take) = [1 0 1 0] [1 0 1 0] [0] [0 1 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] p(a__zeros) = [0] [0] [0] [0] p(and) = [1 0 0 1] [1 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] p(cons) = [1 0 0 0] [1 0 0 1] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 1] [0 0 1 1] [0] [0 0 0 1] [0 0 0 1] [0] p(length) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] p(mark) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] p(nil) = [0] [0] [0] [0] p(s) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 1] [0] [0 0 0 1] [0] p(take) = [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] p(tt) = [0] [0] [1] [0] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: mark(and(X1,X2)) = [1 0 0 2] [1 0 0 1] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] > [1 0 0 2] [1 0 0 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] = a__and(mark(X1),X2) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1 0 0 1] [1 0 0 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] >= [1 0 0 1] [1 0 0 0] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] = and(X1,X2) a__and(tt(),X) = [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [0] = mark(X) a__length(X) = [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [0] >= [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [0] = length(X) a__length(cons(N,L)) = [1 0 0 2] [1 0 0 1] [0] [0 0 0 0] L + [0 0 0 0] N + [0] [0 0 1 1] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 0 2] [0] [0 0 0 0] L + [0] [0 0 1 1] [0] [0 0 0 1] [0] = s(a__length(mark(L))) a__length(nil()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = 0() a__take(X1,X2) = [1 0 1 0] [1 0 1 0] [0] [0 1 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] = take(X1,X2) a__take(0(),IL) = [1 0 1 0] [0] [0 0 0 0] IL + [0] [0 0 1 0] [0] [0 0 0 1] [0] >= [0] [0] [0] [0] = nil() a__take(s(M),cons(N,IL)) = [1 0 1 2] [1 0 1 1] [1 0 0 1] [0] [0 0 0 0] IL + [0 0 0 0] M + [0 0 0 0] N + [0] [0 0 1 1] [0 0 1 1] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [0 0 0 1] [0] >= [1 0 1 1] [1 0 1 1] [1 0 0 1] [0] [0 0 0 0] IL + [0 0 0 0] M + [0 0 0 0] N + [0] [0 0 1 1] [0 0 1 1] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [0 0 0 1] [0] = cons(mark(N),take(M,IL)) a__zeros() = [0] [0] [0] [0] >= [0] [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] [0] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 0 1] [1 0 0 2] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 1] [0 0 1 1] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 0 1] [1 0 0 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 0 1] [0 0 1 1] [0] [0 0 0 1] [0 0 0 1] [0] = cons(mark(X1),X2) mark(length(X)) = [1 0 0 2] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [0] >= [1 0 0 2] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [0] = a__length(mark(X)) mark(nil()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = nil() mark(s(X)) = [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 1] [0] [0 0 0 1] [0] >= [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 1 1] [0] [0 0 0 1] [0] = s(mark(X)) mark(take(X1,X2)) = [1 0 1 1] [1 0 1 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 1 1] [1 0 1 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] = a__take(mark(X1),mark(X2)) mark(tt()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = tt() mark(zeros()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = a__zeros() * Step 10: Failure MAYBE + Considered Problem: - Strict TRS: a__take(X1,X2) -> take(X1,X2) a__zeros() -> cons(0(),zeros()) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) a__zeros() -> zeros() mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(nil()) -> nil() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__take/2,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,take/2,tt/0 ,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__take,a__zeros ,mark} and constructors {0,and,cons,length,nil,s,take,tt,zeros} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE