WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {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,insert#3,leq#2,main ,sort#2} 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() insert#3#(x2,Nil()) -> c_3() insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#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#(x1) -> c_8(sort#2#(x1)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) sort#2#(Nil()) -> c_10() 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() insert#3#(x2,Nil()) -> c_3() insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#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#(x1) -> c_8(sort#2#(x1)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) sort#2#(Nil()) -> c_10() - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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 {2,3,5,6,10} by application of Pre({2,3,5,6,10}) = {1,4,7,8,9}. Here rules are labelled as follows: 1: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) 2: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 3: insert#3#(x2,Nil()) -> c_3() 4: insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#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#(x1) -> c_8(sort#2#(x1)) 9: sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) 10: sort#2#(Nil()) -> c_10() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(x1) -> c_8(sort#2#(x1)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) - Weak DPs: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() insert#3#(x2,Nil()) -> c_3() leq#2#(0(),x8) -> c_5() leq#2#(S(x12),0()) -> c_6() sort#2#(Nil()) -> c_10() - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2 -->_1 insert#3#(x2,Nil()) -> c_3():7 2:S:insert#3#(x6,Cons(x4,x2)) -> c_4(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_7(leq#2#(x4,x2)):3 -->_2 leq#2#(S(x12),0()) -> c_6():9 -->_2 leq#2#(0(),x8) -> c_5():8 -->_1 cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2():6 -->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1 3: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)):3 4:S:main#(x1) -> c_8(sort#2#(x1)) -->_1 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5 -->_1 sort#2#(Nil()) -> c_10():10 5:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) -->_2 sort#2#(Nil()) -> c_10():10 -->_1 insert#3#(x2,Nil()) -> c_3():7 -->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5 -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2 6:W:cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 7:W:insert#3#(x2,Nil()) -> c_3() 8:W:leq#2#(0(),x8) -> c_5() 9:W:leq#2#(S(x12),0()) -> c_6() 10:W:sort#2#(Nil()) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: sort#2#(Nil()) -> c_10() 7: insert#3#(x2,Nil()) -> c_3() 6: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 8: leq#2#(0(),x8) -> c_5() 9: leq#2#(S(x12),0()) -> c_6() * Step 4: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(x1) -> c_8(sort#2#(x1)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2 2:S:insert#3#(x6,Cons(x4,x2)) -> c_4(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_7(leq#2#(x4,x2)):3 -->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1 3: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)):3 4:S:main#(x1) -> c_8(sort#2#(x1)) -->_1 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5 5:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) -->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5 -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):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). [(4,main#(x1) -> c_8(sort#2#(x1)))] * Step 5: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) * Step 6: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) and a lower component cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)) -->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/1,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)) ** Step 6.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)) - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/1,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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) sort#2# :: ["A"(1)] -(15)-> "A"(0) c_9 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "c_9_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)) 2. Weak: ** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak DPs: sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) and a lower component leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> insert#3#(x3,x1) insert#3#(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert#3#(x6,Cons(x4,x2)) -> leq#2#(x6,x4) sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) *** Step 6.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) - Weak DPs: sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2 2:S:insert#3#(x6,Cons(x4,x2)) -> c_4(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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1 3:W:sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) -->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2 4:W:sort#2#(Cons(x4,x2)) -> sort#2#(x2) -->_1 sort#2#(Cons(x4,x2)) -> sort#2#(x2):4 -->_1 sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert#3#(x6,Cons(x4,x2)) -> c_4(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(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak DPs: sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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"(7) x "A"(7)] -(7)-> "A"(7) Cons :: ["A"(10) x "A"(10)] -(10)-> "A"(10) Cons :: ["A"(13) x "A"(13)] -(13)-> "A"(13) False :: [] -(0)-> "A"(0) False :: [] -(0)-> "A"(1) False :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(7) Nil :: [] -(0)-> "A"(10) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(13) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(1) True :: [] -(0)-> "A"(15) cond_insert_ord_x_ys_1 :: ["A"(1) x "A"(10) x "A"(7) x "A"(7)] -(14)-> "A"(7) insert#3 :: ["A"(10) x "A"(7)] -(7)-> "A"(7) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(13) sort#2 :: ["A"(10)] -(5)-> "A"(7) cond_insert_ord_x_ys_1# :: ["A"(0) x "A"(10) x "A"(1) x "A"(7)] -(0)-> "A"(1) insert#3# :: ["A"(10) x "A"(7)] -(0)-> "A"(1) sort#2# :: ["A"(13)] -(13)-> "A"(1) c_1 :: ["A"(0)] -(0)-> "A"(3) c_4 :: ["A"(0)] -(2)-> "A"(2) 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_4_A" :: ["A"(0)] -(1)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) 2. Weak: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) *** Step 6.b:1.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) - Weak DPs: insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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"(5)] -(5)-> "A"(5) Cons :: ["A"(0) x "A"(9)] -(9)-> "A"(9) Cons :: ["A"(0) x "A"(8)] -(8)-> "A"(8) False :: [] -(0)-> "A"(1) False :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(5) Nil :: [] -(0)-> "A"(9) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(11) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(1) True :: [] -(0)-> "A"(15) cond_insert_ord_x_ys_1 :: ["A"(1) x "A"(0) x "A"(0) x "A"(5)] -(13)-> "A"(5) insert#3 :: ["A"(0) x "A"(5)] -(8)-> "A"(5) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(15) sort#2 :: ["A"(9)] -(5)-> "A"(5) cond_insert_ord_x_ys_1# :: ["A"(1) x "A"(0) x "A"(0) x "A"(5)] -(6)-> "A"(3) insert#3# :: ["A"(0) x "A"(5)] -(2)-> "A"(11) sort#2# :: ["A"(9)] -(1)-> "A"(3) c_1 :: ["A"(0)] -(0)-> "A"(15) c_4 :: ["A"(0)] -(0)-> "A"(15) 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"(1)] -(0)-> "A"(1) "True_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_4_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)) 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_7(leq#2#(x4,x2)) - Weak DPs: cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> insert#3#(x3,x1) insert#3#(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert#3#(x6,Cons(x4,x2)) -> leq#2#(x6,x4) sort#2#(Cons(x4,x2)) -> insert#3#(x4,sort#2(x2)) sort#2#(Cons(x4,x2)) -> sort#2#(x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(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) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2 ,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1 ,c_9/2,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main# ,sort#2#} 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"(1) x "A"(1)] -(1)-> "A"(1) Cons :: ["A"(13) x "A"(13)] -(13)-> "A"(13) Cons :: ["A"(15) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(3) x "A"(3)] -(3)-> "A"(3) False :: [] -(0)-> "A"(1) False :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(1) Nil :: [] -(0)-> "A"(13) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(7) S :: ["A"(6)] -(6)-> "A"(6) S :: ["A"(1)] -(1)-> "A"(1) S :: ["A"(0)] -(0)-> "A"(0) True :: [] -(0)-> "A"(1) True :: [] -(0)-> "A"(15) cond_insert_ord_x_ys_1 :: ["A"(1) x "A"(4) x "A"(1) x "A"(1)] -(6)-> "A"(1) insert#3 :: ["A"(4) x "A"(1)] -(5)-> "A"(1) leq#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(15) sort#2 :: ["A"(13)] -(1)-> "A"(1) cond_insert_ord_x_ys_1# :: ["A"(1) x "A"(10) x "A"(0) x "A"(1)] -(5)-> "A"(5) insert#3# :: ["A"(10) x "A"(1)] -(5)-> "A"(5) leq#2# :: ["A"(6) x "A"(1)] -(0)-> "A"(5) sort#2# :: ["A"(15)] -(9)-> "A"(1) c_7 :: ["A"(0)] -(0)-> "A"(14) 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_7_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) 2. Weak: WORST_CASE(?,O(n^3))