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__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(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,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__and) = {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(a__and) = [1] x1 + [1] x2 + [0] p(a__length) = [1] x1 + [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 + [8] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [9] p(zeros) = [0] Following rules are strictly oriented: a__and(tt(),X) = [1] X + [9] > [1] X + [8] = mark(X) mark(0()) = [8] > [0] = 0() mark(nil()) = [8] > [0] = nil() mark(tt()) = [17] > [9] = tt() mark(zeros()) = [8] > [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__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [0] >= [1] L + [8] = s(a__length(mark(L))) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(and(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = cons(mark(X1),X2) mark(length(X)) = [1] X + [8] >= [1] X + [8] = a__length(mark(X)) mark(s(X)) = [1] X + [8] >= [1] X + [8] = s(mark(X)) 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__length(X) -> length(X) a__length(cons(N,L)) -> s(a__length(mark(L))) a__length(nil()) -> 0() 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(tt(),X) -> mark(X) mark(0()) -> 0() mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,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__and) = {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) = [2] p(a__and) = [1] x1 + [1] p(a__length) = [1] x1 + [9] p(a__zeros) = [2] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(length) = [1] x1 + [0] p(mark) = [2] p(nil) = [2] p(s) = [1] x1 + [0] p(tt) = [1] p(zeros) = [1] Following rules are strictly oriented: a__and(X1,X2) = [1] X1 + [1] > [1] X1 + [0] = and(X1,X2) a__length(X) = [1] X + [9] > [1] X + [0] = length(X) a__length(nil()) = [11] > [2] = 0() a__zeros() = [2] > [1] = zeros() Following rules are (at-least) weakly oriented: a__and(tt(),X) = [2] >= [2] = mark(X) a__length(cons(N,L)) = [1] N + [9] >= [11] = s(a__length(mark(L))) a__zeros() = [2] >= [2] = cons(0(),zeros()) mark(0()) = [2] >= [2] = 0() mark(and(X1,X2)) = [2] >= [3] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [2] = cons(mark(X1),X2) mark(length(X)) = [2] >= [11] = a__length(mark(X)) mark(nil()) = [2] >= [2] = nil() mark(s(X)) = [2] >= [2] = s(mark(X)) mark(tt()) = [2] >= [1] = tt() mark(zeros()) = [2] >= [2] = 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__length(cons(N,L)) -> s(a__length(mark(L))) 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(nil()) -> 0() a__zeros() -> zeros() mark(0()) -> 0() mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,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__and) = {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(a__and) = [1] x1 + [9] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(and) = [1] p(cons) = [1] x1 + [9] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [8] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__length(cons(N,L)) = [1] N + [9] > [8] = s(a__length(mark(L))) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [9] >= [1] = and(X1,X2) a__and(tt(),X) = [9] >= [0] = mark(X) a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [0] >= [9] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [0] >= [9] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [9] = cons(mark(X1),X2) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [8] = 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 4: NaturalMI MAYBE + Considered Problem: - Strict TRS: 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__zeros() -> zeros() mark(0()) -> 0() mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 2, 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: The following argument positions are considered usable: uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__and,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__and) = [1 0] x1 + [0 1] x2 + [3] [0 1] [0 1] [4] p(a__length) = [1 0] x1 + [0] [0 1] [0] p(a__zeros) = [0] [0] p(and) = [1 0] x1 + [0 1] x2 + [0] [0 1] [0 1] [4] p(cons) = [1 0] x1 + [0 1] x2 + [0] [0 1] [0 1] [0] p(length) = [1 0] x1 + [0] [0 1] [0] p(mark) = [0 1] x1 + [0] [0 1] [0] p(nil) = [0] [1] p(s) = [1 0] x1 + [0] [0 1] [0] p(tt) = [3] [6] p(zeros) = [0] [0] Following rules are strictly oriented: mark(and(X1,X2)) = [0 1] X1 + [0 1] X2 + [4] [0 1] [0 1] [4] > [0 1] X1 + [0 1] X2 + [3] [0 1] [0 1] [4] = a__and(mark(X1),X2) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1 0] X1 + [0 1] X2 + [3] [0 1] [0 1] [4] >= [1 0] X1 + [0 1] X2 + [0] [0 1] [0 1] [4] = and(X1,X2) a__and(tt(),X) = [0 1] X + [6] [0 1] [10] >= [0 1] X + [0] [0 1] [0] = mark(X) a__length(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = length(X) a__length(cons(N,L)) = [0 1] L + [1 0] N + [0] [0 1] [0 1] [0] >= [0 1] L + [0] [0 1] [0] = s(a__length(mark(L))) a__length(nil()) = [0] [1] >= [0] [0] = 0() a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(cons(X1,X2)) = [0 1] X1 + [0 1] X2 + [0] [0 1] [0 1] [0] >= [0 1] X1 + [0 1] X2 + [0] [0 1] [0 1] [0] = cons(mark(X1),X2) mark(length(X)) = [0 1] X + [0] [0 1] [0] >= [0 1] X + [0] [0 1] [0] = a__length(mark(X)) mark(nil()) = [1] [1] >= [0] [1] = nil() mark(s(X)) = [0 1] X + [0] [0 1] [0] >= [0 1] X + [0] [0 1] [0] = s(mark(X)) mark(tt()) = [6] [6] >= [3] [6] = tt() mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() * Step 5: NaturalMI MAYBE + Considered Problem: - Strict TRS: 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__zeros() -> zeros() mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 2, 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: The following argument positions are considered usable: uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__and,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__and) = [1 0] x1 + [0 4] x2 + [2] [0 1] [0 1] [2] p(a__length) = [1 0] x1 + [0] [0 1] [0] p(a__zeros) = [0] [2] p(and) = [0 0] x1 + [0 4] x2 + [1] [0 1] [0 1] [2] p(cons) = [1 0] x1 + [0 4] x2 + [0] [0 1] [0 1] [2] p(length) = [0 0] x1 + [0] [0 1] [0] p(mark) = [0 4] x1 + [0] [0 1] [2] p(nil) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [0] p(tt) = [0] [2] p(zeros) = [0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [0 4] X1 + [0 4] X2 + [8] [0 1] [0 1] [4] > [0 4] X1 + [0 4] X2 + [0] [0 1] [0 1] [4] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1 0] X1 + [0 4] X2 + [2] [0 1] [0 1] [2] >= [0 0] X1 + [0 4] X2 + [1] [0 1] [0 1] [2] = and(X1,X2) a__and(tt(),X) = [0 4] X + [2] [0 1] [4] >= [0 4] X + [0] [0 1] [2] = mark(X) a__length(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = length(X) a__length(cons(N,L)) = [0 4] L + [1 0] N + [0] [0 1] [0 1] [2] >= [0 4] L + [0] [0 1] [2] = s(a__length(mark(L))) a__length(nil()) = [0] [0] >= [0] [0] = 0() a__zeros() = [0] [2] >= [0] [2] = cons(0(),zeros()) a__zeros() = [0] [2] >= [0] [0] = zeros() mark(0()) = [0] [2] >= [0] [0] = 0() mark(and(X1,X2)) = [0 4] X1 + [0 4] X2 + [8] [0 1] [0 1] [4] >= [0 4] X1 + [0 4] X2 + [2] [0 1] [0 1] [4] = a__and(mark(X1),X2) mark(length(X)) = [0 4] X + [0] [0 1] [2] >= [0 4] X + [0] [0 1] [2] = a__length(mark(X)) mark(nil()) = [0] [2] >= [0] [0] = nil() mark(s(X)) = [0 4] X + [0] [0 1] [2] >= [0 4] X + [0] [0 1] [2] = s(mark(X)) mark(tt()) = [8] [4] >= [0] [2] = tt() mark(zeros()) = [0] [2] >= [0] [2] = a__zeros() * Step 6: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__zeros() -> cons(0(),zeros()) 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__zeros() -> zeros() mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 2, 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: The following argument positions are considered usable: uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__and,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__and) = [1 0] x1 + [1 4] x2 + [2] [0 1] [0 1] [0] p(a__length) = [1 4] x1 + [6] [0 1] [2] p(a__zeros) = [0] [0] p(and) = [1 0] x1 + [1 4] x2 + [2] [0 1] [0 1] [0] p(cons) = [1 3] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] p(length) = [1 4] x1 + [0] [0 1] [2] p(mark) = [1 4] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [0] p(tt) = [0] [1] p(zeros) = [0] [0] Following rules are strictly oriented: mark(length(X)) = [1 8] X + [8] [0 1] [2] > [1 8] X + [6] [0 1] [2] = a__length(mark(X)) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1 0] X1 + [1 4] X2 + [2] [0 1] [0 1] [0] >= [1 0] X1 + [1 4] X2 + [2] [0 1] [0 1] [0] = and(X1,X2) a__and(tt(),X) = [1 4] X + [2] [0 1] [1] >= [1 4] X + [0] [0 1] [0] = mark(X) a__length(X) = [1 4] X + [6] [0 1] [2] >= [1 4] X + [0] [0 1] [2] = length(X) a__length(cons(N,L)) = [1 8] L + [1 7] N + [6] [0 1] [0 1] [2] >= [1 8] L + [6] [0 1] [2] = s(a__length(mark(L))) a__length(nil()) = [6] [2] >= [0] [0] = 0() a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(and(X1,X2)) = [1 4] X1 + [1 8] X2 + [2] [0 1] [0 1] [0] >= [1 4] X1 + [1 4] X2 + [2] [0 1] [0 1] [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1 7] X1 + [1 8] X2 + [0] [0 1] [0 1] [0] >= [1 7] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = cons(mark(X1),X2) mark(nil()) = [0] [0] >= [0] [0] = nil() mark(s(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = s(mark(X)) mark(tt()) = [4] [1] >= [0] [1] = tt() mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() * Step 7: Failure MAYBE + Considered Problem: - Strict TRS: a__zeros() -> cons(0(),zeros()) 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__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(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__and/2,a__length/1,a__zeros/0,mark/1} / {0/0,and/2,cons/2,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__length,a__zeros,mark} and constructors {0,and ,cons,length,nil,s,tt,zeros} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE