WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + 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: UsableRules 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)) 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: 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)) 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)) * Step 3: PredecessorEstimation 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)) 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) - 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 4: RemoveWeakSuffixes 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)) 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) - 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 5: RemoveHeads 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)) 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) - 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 6: Decompose 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)) 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: 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_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)) -> 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} Problem (S) - Strict DPs: fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) - Weak 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 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} ** Step 6.a:1: PredecessorEstimationCP 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)) -> 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: 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)) 4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Consider the set of all dependency pairs 1: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) 2: fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) 3: 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)) 4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {3,4} These cover all (indirect) predecessors of dependency pairs {1,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 6.a:1.a:1: NaturalPI 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)) -> 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: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1,2}, uargs(c_6) = {1,2}, uargs(c_9) = {1} Following symbols are considered usable: {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#} TcT has computed the following interpretation: p(0) = 0 p(Cons) = 2 + x1 + x2 p(False) = 1 p(Nil) = 0 p(S) = 2 + x1 p(True) = 2 p(cond_insert_ord_x_ys_1) = 4 + x2 + x3 + x4 p(fold#3) = x1 p(insert_ord#2) = 2 + x1 + x2 p(leq#2) = 0 p(main) = x1 + x1^2 p(cond_insert_ord_x_ys_1#) = 2 + 6*x2 + 6*x2*x3 + 6*x2*x4 + 3*x3 + 4*x4 p(fold#3#) = 1 + 3*x1^2 p(insert_ord#2#) = 2 + 6*x1*x2 + 4*x2 p(leq#2#) = 4 + 4*x1 p(main#) = 0 p(c_1) = x1 p(c_2) = 4 p(c_3) = 7 + x1 + x2 p(c_4) = 0 p(c_5) = 0 p(c_6) = x1 + x2 p(c_7) = 0 p(c_8) = 0 p(c_9) = 1 + x1 p(c_10) = 0 Following rules are strictly oriented: insert_ord#2#(x6,Cons(x4,x2)) = 10 + 4*x2 + 6*x2*x6 + 4*x4 + 6*x4*x6 + 12*x6 > 6 + 4*x2 + 6*x2*x6 + 3*x4 + 6*x4*x6 + 10*x6 = 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)) = 12 + 4*x4 > 5 + 4*x4 = c_9(leq#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = 2 + 6*x0 + 6*x0*x2 + 6*x0*x5 + 4*x2 + 3*x5 >= 2 + 6*x0*x2 + 4*x2 = c_1(insert_ord#2#(x0,x2)) fold#3#(Cons(x2,x1)) = 13 + 12*x1 + 6*x1*x2 + 3*x1^2 + 12*x2 + 3*x2^2 >= 10 + 4*x1 + 6*x1*x2 + 3*x1^2 = c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) cond_insert_ord_x_ys_1(False(),x0,x5,x2) = 4 + x0 + x2 + x5 >= 4 + x0 + x2 + x5 = Cons(x5,insert_ord#2(x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = 4 + x1 + x2 + x3 >= 4 + x1 + x2 + x3 = Cons(x3,Cons(x2,x1)) fold#3(Cons(x2,x1)) = 2 + x1 + x2 >= 2 + x1 + x2 = insert_ord#2(x2,fold#3(x1)) fold#3(Nil()) = 0 >= 0 = Nil() insert_ord#2(x2,Nil()) = 2 + x2 >= 2 + x2 = Cons(x2,Nil()) insert_ord#2(x6,Cons(x4,x2)) = 4 + x2 + x4 + x6 >= 4 + x2 + x4 + x6 = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) *** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(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)) -> 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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:W: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:W: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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: fold#3#(Cons(x2,x1)) -> c_3(insert_ord#2#(x2,fold#3(x1)),fold#3#(x1)) 1: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) 3: 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)) 4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) *** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes 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 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 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1: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)):1 2:W: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 3:W: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)):2 4:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: 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: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(x0,x2)) 4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) ** Step 6.b:2: 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.b:3: 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.b:4: PredecessorEstimationCP 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: 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: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) The strictly oriented rules are moved into the weak component. *** Step 6.b:4.a:1: NaturalMI 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: 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_3) = {1} Following symbols are considered usable: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [3] p(False) = [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [0] p(fold#3) = [0] p(insert_ord#2) = [0] p(leq#2) = [0] p(main) = [0] p(cond_insert_ord_x_ys_1#) = [0] p(fold#3#) = [8] x1 + [5] p(insert_ord#2#) = [0] p(leq#2#) = [0] p(main#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [2] x2 + [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [1] Following rules are strictly oriented: fold#3#(Cons(x2,x1)) = [8] x1 + [8] x2 + [29] > [8] x1 + [5] = c_3(fold#3#(x1)) Following rules are (at-least) weakly oriented: *** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) -->_1 fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fold#3#(Cons(x2,x1)) -> c_3(fold#3#(x1)) *** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))