MAYBE * Step 1: DependencyPairs 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: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__and#(X1,X2) -> c_1() a__and#(tt(),X) -> c_2(mark#(X)) a__length#(X) -> c_3() a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) a__length#(nil()) -> c_5() a__zeros#() -> c_6() a__zeros#() -> c_7() mark#(0()) -> c_8() mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) mark#(nil()) -> c_12() mark#(s(X)) -> c_13(mark#(X)) mark#(tt()) -> c_14() mark#(zeros()) -> c_15(a__zeros#()) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__and#(X1,X2) -> c_1() a__and#(tt(),X) -> c_2(mark#(X)) a__length#(X) -> c_3() a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) a__length#(nil()) -> c_5() a__zeros#() -> c_6() a__zeros#() -> c_7() mark#(0()) -> c_8() mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) mark#(nil()) -> c_12() mark#(s(X)) -> c_13(mark#(X)) mark#(tt()) -> c_14() mark#(zeros()) -> c_15(a__zeros#()) - 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() -> 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,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,c_1/0,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/1,c_11/2,c_12/0 ,c_13/1,c_14/0,c_15/1} - 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: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,6,7,8,12,14} by application of Pre({1,3,5,6,7,8,12,14}) = {2,4,9,10,11,13,15}. Here rules are labelled as follows: 1: a__and#(X1,X2) -> c_1() 2: a__and#(tt(),X) -> c_2(mark#(X)) 3: a__length#(X) -> c_3() 4: a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) 5: a__length#(nil()) -> c_5() 6: a__zeros#() -> c_6() 7: a__zeros#() -> c_7() 8: mark#(0()) -> c_8() 9: mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) 10: mark#(cons(X1,X2)) -> c_10(mark#(X1)) 11: mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) 12: mark#(nil()) -> c_12() 13: mark#(s(X)) -> c_13(mark#(X)) 14: mark#(tt()) -> c_14() 15: mark#(zeros()) -> c_15(a__zeros#()) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__and#(tt(),X) -> c_2(mark#(X)) a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) mark#(zeros()) -> c_15(a__zeros#()) - Weak DPs: a__and#(X1,X2) -> c_1() a__length#(X) -> c_3() a__length#(nil()) -> c_5() a__zeros#() -> c_6() a__zeros#() -> c_7() mark#(0()) -> c_8() mark#(nil()) -> c_12() mark#(tt()) -> c_14() - 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() -> 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,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,c_1/0,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/1,c_11/2,c_12/0 ,c_13/1,c_14/0,c_15/1} - 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: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {7} by application of Pre({7}) = {1,2,3,4,5,6}. Here rules are labelled as follows: 1: a__and#(tt(),X) -> c_2(mark#(X)) 2: a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) 3: mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) 4: mark#(cons(X1,X2)) -> c_10(mark#(X1)) 5: mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) 6: mark#(s(X)) -> c_13(mark#(X)) 7: mark#(zeros()) -> c_15(a__zeros#()) 8: a__and#(X1,X2) -> c_1() 9: a__length#(X) -> c_3() 10: a__length#(nil()) -> c_5() 11: a__zeros#() -> c_6() 12: a__zeros#() -> c_7() 13: mark#(0()) -> c_8() 14: mark#(nil()) -> c_12() 15: mark#(tt()) -> c_14() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__and#(tt(),X) -> c_2(mark#(X)) a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) - Weak DPs: a__and#(X1,X2) -> c_1() a__length#(X) -> c_3() a__length#(nil()) -> c_5() a__zeros#() -> c_6() a__zeros#() -> c_7() mark#(0()) -> c_8() mark#(nil()) -> c_12() mark#(tt()) -> c_14() mark#(zeros()) -> c_15(a__zeros#()) - 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() -> 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,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,c_1/0,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/1,c_11/2,c_12/0 ,c_13/1,c_14/0,c_15/1} - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__and#(tt(),X) -> c_2(mark#(X)) -->_1 mark#(zeros()) -> c_15(a__zeros#()):15 -->_1 mark#(s(X)) -> c_13(mark#(X)):6 -->_1 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_1 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 -->_1 mark#(tt()) -> c_14():14 -->_1 mark#(nil()) -> c_12():13 -->_1 mark#(0()) -> c_8():12 2:S:a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) -->_2 mark#(zeros()) -> c_15(a__zeros#()):15 -->_2 mark#(s(X)) -> c_13(mark#(X)):6 -->_2 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_2 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 -->_2 mark#(tt()) -> c_14():14 -->_2 mark#(nil()) -> c_12():13 -->_2 mark#(0()) -> c_8():12 -->_1 a__length#(nil()) -> c_5():9 -->_1 a__length#(X) -> c_3():8 -->_1 a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)):2 3:S:mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) -->_2 mark#(zeros()) -> c_15(a__zeros#()):15 -->_2 mark#(s(X)) -> c_13(mark#(X)):6 -->_2 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_2 mark#(tt()) -> c_14():14 -->_2 mark#(nil()) -> c_12():13 -->_2 mark#(0()) -> c_8():12 -->_1 a__and#(X1,X2) -> c_1():7 -->_2 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 -->_1 a__and#(tt(),X) -> c_2(mark#(X)):1 4:S:mark#(cons(X1,X2)) -> c_10(mark#(X1)) -->_1 mark#(zeros()) -> c_15(a__zeros#()):15 -->_1 mark#(s(X)) -> c_13(mark#(X)):6 -->_1 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_1 mark#(tt()) -> c_14():14 -->_1 mark#(nil()) -> c_12():13 -->_1 mark#(0()) -> c_8():12 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_1 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 5:S:mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) -->_2 mark#(zeros()) -> c_15(a__zeros#()):15 -->_2 mark#(s(X)) -> c_13(mark#(X)):6 -->_2 mark#(tt()) -> c_14():14 -->_2 mark#(nil()) -> c_12():13 -->_2 mark#(0()) -> c_8():12 -->_1 a__length#(nil()) -> c_5():9 -->_1 a__length#(X) -> c_3():8 -->_2 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_2 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 -->_1 a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)):2 6:S:mark#(s(X)) -> c_13(mark#(X)) -->_1 mark#(zeros()) -> c_15(a__zeros#()):15 -->_1 mark#(tt()) -> c_14():14 -->_1 mark#(nil()) -> c_12():13 -->_1 mark#(0()) -> c_8():12 -->_1 mark#(s(X)) -> c_13(mark#(X)):6 -->_1 mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)):5 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):4 -->_1 mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)):3 7:W:a__and#(X1,X2) -> c_1() 8:W:a__length#(X) -> c_3() 9:W:a__length#(nil()) -> c_5() 10:W:a__zeros#() -> c_6() 11:W:a__zeros#() -> c_7() 12:W:mark#(0()) -> c_8() 13:W:mark#(nil()) -> c_12() 14:W:mark#(tt()) -> c_14() 15:W:mark#(zeros()) -> c_15(a__zeros#()) -->_1 a__zeros#() -> c_7():11 -->_1 a__zeros#() -> c_6():10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: a__and#(X1,X2) -> c_1() 8: a__length#(X) -> c_3() 9: a__length#(nil()) -> c_5() 12: mark#(0()) -> c_8() 13: mark#(nil()) -> c_12() 14: mark#(tt()) -> c_14() 15: mark#(zeros()) -> c_15(a__zeros#()) 10: a__zeros#() -> c_6() 11: a__zeros#() -> c_7() * Step 5: Failure MAYBE + Considered Problem: - Strict DPs: a__and#(tt(),X) -> c_2(mark#(X)) a__length#(cons(N,L)) -> c_4(a__length#(mark(L)),mark#(L)) mark#(and(X1,X2)) -> c_9(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(length(X)) -> c_11(a__length#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(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() -> 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,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,c_1/0,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/1,c_11/2,c_12/0 ,c_13/1,c_14/0,c_15/1} - 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