WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> Cons(cond_link_t1_t2_2(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,Nil()) cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1)) cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1)) cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3)) leq#2(0(),x20) -> True() leq#2(S(x24),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x4,Nil()) -> Cons(x4,Nil()) main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3 ,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1,cond_link_t1_t2_2,leq#2,lt#2 ,main} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() leq#2#(0(),x20) -> c_5() leq#2#(S(x24),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(0(),0()) -> c_8() lt#2#(0(),S(x20)) -> c_9() lt#2#(S(x20),0()) -> c_10() lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(x4,Nil()) -> c_12() main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() leq#2#(0(),x20) -> c_5() leq#2#(S(x24),0()) -> c_6() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(0(),0()) -> c_8() lt#2#(0(),S(x20)) -> c_9() lt#2#(S(x20),0()) -> c_10() lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(x4,Nil()) -> c_12() main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> Cons(cond_link_t1_t2_2(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,Nil()) cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1)) cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1)) cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3)) leq#2(0(),x20) -> True() leq#2(S(x24),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x4,Nil()) -> Cons(x4,Nil()) main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,6,8,9,10,12} by application of Pre({2,3,4,5,6,8,9,10,12}) = {1,7,11,13}. Here rules are labelled as follows: 1: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) 2: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 3: cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() 4: cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() 5: leq#2#(0(),x20) -> c_5() 6: leq#2#(S(x24),0()) -> c_6() 7: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) 8: lt#2#(0(),0()) -> c_8() 9: lt#2#(0(),S(x20)) -> c_9() 10: lt#2#(S(x20),0()) -> c_10() 11: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) 12: main#(x4,Nil()) -> c_12() 13: main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak DPs: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() leq#2#(0(),x20) -> c_5() leq#2#(S(x24),0()) -> c_6() lt#2#(0(),0()) -> c_8() lt#2#(0(),S(x20)) -> c_9() lt#2#(S(x20),0()) -> c_10() main#(x4,Nil()) -> c_12() - Weak TRS: cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> Cons(cond_link_t1_t2_2(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,Nil()) cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1)) cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1)) cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3)) leq#2(0(),x20) -> True() leq#2(S(x24),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x4,Nil()) -> Cons(x4,Nil()) main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2 -->_2 leq#2#(S(x24),0()) -> c_6():9 -->_2 leq#2#(0(),x20) -> c_5():8 -->_1 cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4():7 -->_1 cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3():6 2:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x24),0()) -> c_6():9 -->_1 leq#2#(0(),x20) -> c_5():8 -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2 3:S:lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) -->_1 lt#2#(S(x20),0()) -> c_10():12 -->_1 lt#2#(0(),S(x20)) -> c_9():11 -->_1 lt#2#(0(),0()) -> c_8():10 -->_1 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3 4:S:main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) -->_2 lt#2#(S(x20),0()) -> c_10():12 -->_2 lt#2#(0(),S(x20)) -> c_9():11 -->_2 lt#2#(0(),0()) -> c_8():10 -->_1 cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2():5 -->_2 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3 -->_1 cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)):1 5:W:cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 6:W:cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() 7:W:cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() 8:W:leq#2#(0(),x20) -> c_5() 9:W:leq#2#(S(x24),0()) -> c_6() 10:W:lt#2#(0(),0()) -> c_8() 11:W:lt#2#(0(),S(x20)) -> c_9() 12:W:lt#2#(S(x20),0()) -> c_10() 13:W:main#(x4,Nil()) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: main#(x4,Nil()) -> c_12() 5: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 10: lt#2#(0(),0()) -> c_8() 11: lt#2#(0(),S(x20)) -> c_9() 12: lt#2#(S(x20),0()) -> c_10() 6: cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() 7: cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() 8: leq#2#(0(),x20) -> c_5() 9: leq#2#(S(x24),0()) -> c_6() * Step 4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> Cons(cond_link_t1_t2_2(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,Nil()) cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1)) cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1)) cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3)) leq#2(0(),x20) -> True() leq#2(S(x24),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x4,Nil()) -> Cons(x4,Nil()) main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2 2: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)):2 3:S:lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3 4:S:main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) -->_2 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3 -->_1 cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(cond_link_t1_t2_2#(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,leq#2#(x8,x4)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) * Step 5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> Cons(cond_link_t1_t2_2(leq#2(x8,x4) ,Node(x54,Elem(x8),x38) ,Node(x30,Elem(x4),x14) ,x54 ,Elem(x8) ,x38 ,Elem(x4) ,x14) ,Nil()) cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1)) cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1)) cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3)) leq#2(0(),x20) -> True() leq#2(S(x24),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x4,Nil()) -> Cons(x4,Nil()) main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cond_insTree_t_xs_1#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [0] p(Elem) = [1] x1 + [2] p(False) = [3] p(Nil) = [1] p(Node) = [1] x1 + [0] p(S) = [1] x1 + [0] p(True) = [2] p(cond_insTree_t_xs_1) = [0] p(cond_link_t1_t2_2) = [0] p(leq#2) = [0] p(lt#2) = [1] x1 + [5] p(main) = [0] p(cond_insTree_t_xs_1#) = [1] x1 + [3] x2 + [3] x3 + [4] x4 + [0] p(cond_link_t1_t2_2#) = [2] x6 + [2] x7 + [1] x8 + [0] p(leq#2#) = [2] p(lt#2#) = [2] x1 + [1] x2 + [1] p(main#) = [6] x1 + [4] x2 + [7] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) = [3] x30 + [3] x54 + [7] > [2] = c_1(leq#2#(x8,x4)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) = [6] x100 + [4] x28 + [4] x48 + [7] > [6] x100 + [4] x28 + [4] x48 + [6] = c_13(cond_insTree_t_xs_1#(lt#2(x100,x48),Node(x100,x132,x164),Node(x48,x64,x80),x28),lt#2#(x100,x48)) Following rules are (at-least) weakly oriented: leq#2#(S(x4),S(x2)) = [2] >= [2] = c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) = [1] x2 + [2] x4 + [1] >= [1] x2 + [2] x4 + [1] = c_11(lt#2#(x4,x2)) lt#2(0(),0()) = [6] >= [3] = False() lt#2(0(),S(x20)) = [6] >= [2] = True() lt#2(S(x20),0()) = [1] x20 + [5] >= [3] = False() lt#2(S(x4),S(x2)) = [1] x4 + [5] >= [1] x4 + [5] = lt#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cond_insTree_t_xs_1#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(Cons) = [1] x2 + [2] p(Elem) = [1] x1 + [1] p(False) = [1] p(Nil) = [7] p(Node) = [1] x2 + [0] p(S) = [1] x1 + [2] p(True) = [1] p(cond_insTree_t_xs_1) = [1] x1 + [1] x3 + [0] p(cond_link_t1_t2_2) = [1] x2 + [2] x3 + [1] x4 + [2] p(leq#2) = [4] x1 + [2] p(lt#2) = [1] p(main) = [1] x2 + [0] p(cond_insTree_t_xs_1#) = [1] x1 + [4] x2 + [1] x4 + [3] p(cond_link_t1_t2_2#) = [1] x1 + [1] x4 + [2] x5 + [0] p(leq#2#) = [1] x1 + [0] p(lt#2#) = [3] p(main#) = [4] x1 + [4] x2 + [5] p(c_1) = [1] x1 + [2] p(c_2) = [1] p(c_3) = [4] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [1] p(c_8) = [2] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [4] p(c_12) = [1] p(c_13) = [1] x1 + [1] x2 + [4] Following rules are strictly oriented: leq#2#(S(x4),S(x2)) = [1] x4 + [2] > [1] x4 + [1] = c_7(leq#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) = [4] x8 + [15] >= [1] x8 + [2] = c_1(leq#2#(x8,x4)) lt#2#(S(x4),S(x2)) = [3] >= [7] = c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) = [4] x132 + [4] x28 + [13] >= [4] x132 + [1] x28 + [11] = c_13(cond_insTree_t_xs_1#(lt#2(x100,x48),Node(x100,x132,x164),Node(x48,x64,x80),x28),lt#2#(x100,x48)) lt#2(0(),0()) = [1] >= [1] = False() lt#2(0(),S(x20)) = [1] >= [1] = True() lt#2(S(x20),0()) = [1] >= [1] = False() lt#2(S(x4),S(x2)) = [1] >= [1] = lt#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cond_insTree_t_xs_1#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [0] p(Elem) = [0] p(False) = [0] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [0] p(S) = [1] x1 + [3] p(True) = [0] p(cond_insTree_t_xs_1) = [0] p(cond_link_t1_t2_2) = [0] p(leq#2) = [0] p(lt#2) = [5] x2 + [0] p(main) = [1] x1 + [0] p(cond_insTree_t_xs_1#) = [1] x1 + [4] x2 + [0] p(cond_link_t1_t2_2#) = [2] x8 + [0] p(leq#2#) = [0] p(lt#2#) = [1] x2 + [0] p(main#) = [4] x1 + [6] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: lt#2#(S(x4),S(x2)) = [1] x2 + [3] > [1] x2 + [0] = c_11(lt#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) = [4] x54 + [0] >= [0] = c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) = [0] >= [0] = c_7(leq#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) = [4] x100 + [4] x132 + [6] x48 + [6] x64 + [0] >= [4] x100 + [4] x132 + [6] x48 + [0] = c_13(cond_insTree_t_xs_1#(lt#2(x100,x48),Node(x100,x132,x164),Node(x48,x64,x80),x28),lt#2#(x100,x48)) lt#2(0(),0()) = [0] >= [0] = False() lt#2(0(),S(x20)) = [5] x20 + [15] >= [0] = True() lt#2(S(x20),0()) = [0] >= [0] = False() lt#2(S(x4),S(x2)) = [5] x2 + [15] >= [5] x2 + [0] = lt#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4)) leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48) ,Node(x100,x132,x164) ,Node(x48,x64,x80) ,x28) ,lt#2#(x100,x48)) - Weak TRS: lt#2(0(),0()) -> False() lt#2(0(),S(x20)) -> True() lt#2(S(x20),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) - Signature: {cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8 ,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2# ,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))