MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) foldl#3(x2,Nil()) -> x2 foldl#3(x6,Cons(x4,x2)) -> foldl#3(mrg#2(x6,x4),x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(MS(x4,x2)) -> foldl#3(Nil(),x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2,foldl#3,leq#2,main ,mrg#2} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x2,Nil()) -> c_3() foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x4,x2),Nil()) -> c_9() mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) mrg#2#(Nil(),x2) -> c_11() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x2,Nil()) -> c_3() foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x4,x2),Nil()) -> c_9() mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) mrg#2#(Nil(),x2) -> c_11() - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) foldl#3(x2,Nil()) -> x2 foldl#3(x6,Cons(x4,x2)) -> foldl#3(mrg#2(x6,x4),x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(MS(x4,x2)) -> foldl#3(Nil(),x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x2,Nil()) -> c_3() foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x4,x2),Nil()) -> c_9() mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) mrg#2#(Nil(),x2) -> c_11() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x2,Nil()) -> c_3() foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x4,x2),Nil()) -> c_9() mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) mrg#2#(Nil(),x2) -> c_11() - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,5,6,9,11} by application of Pre({3,5,6,9,11}) = {1,2,4,7,8,10}. Here rules are labelled as follows: 1: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) 2: cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) 3: foldl#3#(x2,Nil()) -> c_3() 4: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) 5: leq#2#(0(),x8) -> c_5() 6: leq#2#(S(x12),0()) -> c_6() 7: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) 8: main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) 9: mrg#2#(Cons(x4,x2),Nil()) -> c_9() 10: mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) 11: mrg#2#(Nil(),x2) -> c_11() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak DPs: foldl#3#(x2,Nil()) -> c_3() leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() mrg#2#(Cons(x4,x2),Nil()) -> c_9() mrg#2#(Nil(),x2) -> c_11() - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 -->_1 mrg#2#(Cons(x4,x2),Nil()) -> c_9():10 2:S:cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 -->_1 mrg#2#(Nil(),x2) -> c_11():11 3:S:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) -->_2 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 -->_2 mrg#2#(Nil(),x2) -> c_11():11 -->_2 mrg#2#(Cons(x4,x2),Nil()) -> c_9():10 -->_1 foldl#3#(x2,Nil()) -> c_3():7 -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):3 4:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x12),0()) -> c_6():9 -->_1 leq#2#(0(),x8) -> c_5():8 -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 5:S:main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) -->_1 foldl#3#(x2,Nil()) -> c_3():7 -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):3 6:S:mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) -->_2 leq#2#(S(x12),0()) -> c_6():9 -->_2 leq#2#(0(),x8) -> c_5():8 -->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 -->_1 cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))):2 -->_1 cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)):1 7:W:foldl#3#(x2,Nil()) -> c_3() 8:W:leq#2#(0(),x8) -> c_5() 9:W:leq#2#(S(x12),0()) -> c_6() 10:W:mrg#2#(Cons(x4,x2),Nil()) -> c_9() 11:W:mrg#2#(Nil(),x2) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: foldl#3#(x2,Nil()) -> c_3() 10: mrg#2#(Cons(x4,x2),Nil()) -> c_9() 11: mrg#2#(Nil(),x2) -> c_11() 8: leq#2#(0(),x8) -> c_5() 9: leq#2#(S(x12),0()) -> c_6() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 2:S:cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 3:S:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) -->_2 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):6 -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):3 4:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 5:S:main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)) -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):3 6:S:mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 -->_1 cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))):2 -->_1 cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,main#(MS(x4,x2)) -> c_8(foldl#3#(Nil(),x2)))] * Step 6: Decompose MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} Problem (S) - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} ** Step 6.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) and a lower component cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) Further, following extension rules are added to the lower component. foldl#3#(x6,Cons(x4,x2)) -> foldl#3#(mrg#2(x6,x4),x2) foldl#3#(x6,Cons(x4,x2)) -> mrg#2#(x6,x4) *** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1} Following symbols are considered usable: {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main#,mrg#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [8] p(False) = [0] p(MS) = [1] x1 + [1] x2 + [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(cond_mrg_xs_ys_2) = [2] x2 + [0] p(foldl#3) = [0] p(leq#2) = [0] p(main) = [0] p(mrg#2) = [0] p(cond_mrg_xs_ys_2#) = [0] p(foldl#3#) = [2] x2 + [0] p(leq#2#) = [0] p(main#) = [0] p(mrg#2#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [8] p(c_5) = [0] p(c_6) = [0] p(c_7) = [4] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [1] x2 + [0] p(c_11) = [0] Following rules are strictly oriented: foldl#3#(x6,Cons(x4,x2)) = [2] x2 + [2] x4 + [16] > [2] x2 + [8] = c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) Following rules are (at-least) weakly oriented: **** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) **** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:1.b:1: Failure MAYBE + Considered Problem: - Strict DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> foldl#3#(mrg#2(x6,x4),x2) foldl#3#(x6,Cons(x4,x2)) -> mrg#2#(x6,x4) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak DPs: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) ,leq#2#(x8,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) -->_2 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):5 -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):1 2:W:cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):5 3:W:cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) -->_1 mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)):5 4:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 5:W:mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4 -->_1 cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))):3 -->_1 cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: mrg#2#(Cons(x8,x6),Cons(x4,x2)) -> c_10(cond_mrg_xs_ys_2#(leq#2(x8,x4) ,Cons(x8,x6) ,Cons(x4,x2) ,x8 ,x6 ,x4 ,x2) ,leq#2#(x8,x4)) 3: cond_mrg_xs_ys_2#(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_2(mrg#2#(x3,Cons(x5,x6))) 2: cond_mrg_xs_ys_2#(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> c_1(mrg#2#(Cons(x7,x8),x1)) 4: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)) -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2),mrg#2#(x6,x4)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) ** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) The strictly oriented rules are moved into the weak component. *** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1} Following symbols are considered usable: {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main#,mrg#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [2] p(False) = [0] p(MS) = [0] p(Nil) = [0] p(S) = [0] p(True) = [0] p(cond_mrg_xs_ys_2) = [0] p(foldl#3) = [0] p(leq#2) = [0] p(main) = [0] p(mrg#2) = [3] p(cond_mrg_xs_ys_2#) = [0] p(foldl#3#) = [4] x2 + [0] p(leq#2#) = [0] p(main#) = [0] p(mrg#2#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [4] x2 + [0] p(c_11) = [0] Following rules are strictly oriented: foldl#3#(x6,Cons(x4,x2)) = [4] x2 + [8] > [4] x2 + [0] = c_4(foldl#3#(mrg#2(x6,x4),x2)) Following rules are (at-least) weakly oriented: *** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) -->_1 foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: foldl#3#(x6,Cons(x4,x2)) -> c_4(foldl#3#(mrg#2(x6,x4),x2)) *** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: cond_mrg_xs_ys_2(False(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x2,mrg#2(Cons(x7,x8),x1)) cond_mrg_xs_ys_2(True(),Cons(x7,x8),Cons(x5,x6),x4,x3,x2,x1) -> Cons(x4,mrg#2(x3,Cons(x5,x6))) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) mrg#2(Cons(x4,x2),Nil()) -> Cons(x4,x2) mrg#2(Cons(x8,x6),Cons(x4,x2)) -> cond_mrg_xs_ys_2(leq#2(x8,x4),Cons(x8,x6),Cons(x4,x2),x8,x6,x4,x2) mrg#2(Nil(),x2) -> x2 - Signature: {cond_mrg_xs_ys_2/7,foldl#3/2,leq#2/2,main/1,mrg#2/2,cond_mrg_xs_ys_2#/7,foldl#3#/2,leq#2#/2,main#/1 ,mrg#2#/2} / {0/0,Cons/2,False/0,MS/2,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0 ,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_mrg_xs_ys_2#,foldl#3#,leq#2#,main# ,mrg#2#} and constructors {0,Cons,False,MS,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE