WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),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(x3) -> fold#3(x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) fold#3#(Nil()) -> c_4() insert_ord#2#(x2,Nil()) -> c_5() insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(x3)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) fold#3#(Nil()) -> c_4() insert_ord#2#(x2,Nil()) -> c_5() insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(x3)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),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(x3) -> fold#3(x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,5,7,8} by application of Pre({2,4,5,7,8}) = {1,3,6,9,10}. Here rules are labelled as follows: 1: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) 2: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 3: fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) 4: fold#3#(Nil()) -> c_4() 5: insert_ord#2#(x2,Nil()) -> c_5() 6: insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) 7: leq#2#(0(),x8) -> c_7() 8: leq#2#(S(x12),0()) -> c_8() 9: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) 10: main#(x3) -> c_10(fold#3#(x3)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(x3)) - Weak DPs: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(Nil()) -> c_4() insert_ord#2#(x2,Nil()) -> c_5() leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),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(x3) -> fold#3(x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_1 insert_ord#2#(x2,Nil()) -> c_5():8 2:S:fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_1 insert_ord#2#(x2,Nil()) -> c_5():8 -->_2 fold#3#(Nil()) -> c_4():7 -->_2 fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)):2 3:S:insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 -->_2 leq#2#(S(x12),0()) -> c_8():10 -->_2 leq#2#(0(),x8) -> c_7():9 -->_1 cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2():6 -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)):1 4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) -->_1 leq#2#(S(x12),0()) -> c_8():10 -->_1 leq#2#(0(),x8) -> c_7():9 -->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 5:S:main#(x3) -> c_10(fold#3#(x3)) -->_1 fold#3#(Nil()) -> c_4():7 -->_1 fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)):2 6:W:cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 7:W:fold#3#(Nil()) -> c_4() 8:W:insert_ord#2#(x2,Nil()) -> c_5() 9:W:leq#2#(0(),x8) -> c_7() 10:W:leq#2#(S(x12),0()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: fold#3#(Nil()) -> c_4() 8: insert_ord#2#(x2,Nil()) -> c_5() 6: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 9: leq#2#(0(),x8) -> c_7() 10: leq#2#(S(x12),0()) -> c_8() * Step 4: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(x3)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),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(x3) -> fold#3(x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 2:S:fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_2 fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)):2 3:S:insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)):1 4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 5:S:main#(x3) -> c_10(fold#3#(x3)) -->_1 fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)):2 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#(x3) -> c_10(fold#3#(x3)))] * Step 5: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),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(x3) -> fold#3(x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) * Step 6: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) and a lower component cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) -->_2 fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) ** Step 6.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(0) x "A"(1)] -(1)-> "A"(1) fold#3# :: ["A"(1)] -(15)-> "A"(0) c_3 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) 2. Weak: ** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak DPs: fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) and a lower component leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> insert_ord#2#(x0,x2) fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(x6,Cons(x4,x2)) -> leq#2#(x6,x4) *** Step 6.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) - Weak DPs: fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):2 2:S:insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)):1 3:W:fold#3#(Cons(x2,x1)) -> fold#3#(x1) -->_1 fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)):4 -->_1 fold#3#(Cons(x2,x1)) -> fold#3#(x1):3 4:W:fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) -->_1 insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) *** Step 6.b:1.a:2: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak DPs: fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) Cons :: ["A"(4) x "A"(4)] -(4)-> "A"(4) Cons :: ["A"(13) x "A"(13)] -(13)-> "A"(13) Cons :: ["A"(6) x "A"(6)] -(6)-> "A"(6) False :: [] -(0)-> "A"(0) False :: [] -(0)-> "A"(1) False :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(13) Nil :: [] -(0)-> "A"(6) Nil :: [] -(0)-> "A"(15) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(1) True :: [] -(0)-> "A"(15) cond_insert_ord_x_ys_1 :: ["A"(1) x "A"(12) x "A"(6) x "A"(6)] -(14)-> "A"(6) fold#3 :: ["A"(13)] -(1)-> "A"(6) insert_ord#2 :: ["A"(12) x "A"(6)] -(8)-> "A"(6) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(13) cond_insert_ord_x_ys_1# :: ["A"(0) x "A"(1) x "A"(1) x "A"(4)] -(14)-> "A"(9) fold#3# :: ["A"(13)] -(15)-> "A"(3) insert_ord#2# :: ["A"(1) x "A"(4)] -(13)-> "A"(9) c_1 :: ["A"(9)] -(0)-> "A"(9) c_6 :: ["A"(9)] -(0)-> "A"(9) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "Cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "False_A" :: [] -(0)-> "A"(1) "Nil_A" :: [] -(0)-> "A"(1) "S_A" :: ["A"(1)] -(0)-> "A"(1) "True_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) 2. Weak: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) *** Step 6.b:1.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) - Weak DPs: fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) Cons :: ["A"(0) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(0) x "A"(7)] -(7)-> "A"(7) Cons :: ["A"(0) x "A"(8)] -(8)-> "A"(8) False :: [] -(0)-> "A"(0) False :: [] -(0)-> "A"(14) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(7) Nil :: [] -(0)-> "A"(14) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(0) True :: [] -(0)-> "A"(14) cond_insert_ord_x_ys_1 :: ["A"(0) x "A"(0) x "A"(0) x "A"(7)] -(15)-> "A"(7) fold#3 :: ["A"(15)] -(0)-> "A"(7) insert_ord#2 :: ["A"(0) x "A"(7)] -(8)-> "A"(7) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(14) cond_insert_ord_x_ys_1# :: ["A"(0) x "A"(0) x "A"(0) x "A"(7)] -(4)-> "A"(2) fold#3# :: ["A"(15)] -(14)-> "A"(0) insert_ord#2# :: ["A"(0) x "A"(7)] -(3)-> "A"(0) c_1 :: ["A"(0)] -(0)-> "A"(2) c_6 :: ["A"(1)] -(0)-> "A"(1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "False_A" :: [] -(0)-> "A"(1) "Nil_A" :: [] -(0)-> "A"(1) "S_A" :: ["A"(0)] -(0)-> "A"(1) "True_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) 2. Weak: *** Step 6.b:1.b:1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> insert_ord#2#(x0,x2) fold#3#(Cons(x2,x1)) -> fold#3#(x1) fold#3#(Cons(x2,x1)) -> insert_ord#2#(x2,fold#3(x1)) insert_ord#2#(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(x6,Cons(x4,x2)) -> leq#2#(x6,x4) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) -> insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) -> Nil() insert_ord#2(x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/1,insert_ord#2/2,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/1 ,insert_ord#2#/2,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0 ,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) Cons :: ["A"(15) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(7) x "A"(7)] -(7)-> "A"(7) Cons :: ["A"(8) x "A"(8)] -(8)-> "A"(8) False :: [] -(0)-> "A"(1) False :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(7) Nil :: [] -(0)-> "A"(13) S :: ["A"(11)] -(11)-> "A"(11) S :: ["A"(7)] -(7)-> "A"(7) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(1) True :: [] -(0)-> "A"(15) cond_insert_ord_x_ys_1 :: ["A"(1) x "A"(13) x "A"(7) x "A"(7)] -(15)-> "A"(7) fold#3 :: ["A"(15)] -(4)-> "A"(7) insert_ord#2 :: ["A"(13) x "A"(7)] -(8)-> "A"(7) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(15) cond_insert_ord_x_ys_1# :: ["A"(1) x "A"(13) x "A"(2) x "A"(7)] -(9)-> "A"(1) fold#3# :: ["A"(15)] -(15)-> "A"(1) insert_ord#2# :: ["A"(13) x "A"(7)] -(9)-> "A"(1) leq#2# :: ["A"(11) x "A"(7)] -(0)-> "A"(1) c_9 :: ["A"(0)] -(0)-> "A"(11) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "Cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "False_A" :: [] -(0)-> "A"(1) "Nil_A" :: [] -(0)-> "A"(1) "S_A" :: ["A"(1)] -(1)-> "A"(1) "True_A" :: [] -(0)-> "A"(1) "c_9_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) 2. Weak: WORST_CASE(?,O(n^3))