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_ = WIDP} + Details: We add the following weak innermost dependency pairs: 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)) 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)) Weak DPs and mark the set of starting terms. * Step 2: 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(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)) 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)) - 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,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_13/1} - 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: 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) 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)) 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)) * Step 3: 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(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)) 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)) - Strict TRS: 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) - 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_13/1} - 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 = WgOnTrs} + 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(cond_link_t1_t2_2#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(Cons) = [1] x2 + [2] p(Elem) = [1] x1 + [0] p(False) = [0] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [0] p(S) = [1] x1 + [5] p(True) = [1] p(cond_insTree_t_xs_1) = [4] x3 + [2] x4 + [2] p(cond_link_t1_t2_2) = [2] x1 + [1] x2 + [4] x3 + [1] x4 + [1] x6 + [1] x7 + [4] p(leq#2) = [1] x1 + [0] p(lt#2) = [2] x1 + [1] p(main) = [1] x1 + [1] x2 + [1] p(cond_insTree_t_xs_1#) = [1] x1 + [1] x2 + [2] p(cond_link_t1_t2_2#) = [1] x1 + [0] p(leq#2#) = [1] p(lt#2#) = [3] p(main#) = [4] x1 + [4] x2 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [7] p(c_8) = [0] p(c_9) = [2] p(c_10) = [4] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [1] x1 + [1] Following rules are strictly oriented: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) = [1] x54 + [1] x8 + [2] > [1] x8 + [0] = 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)) cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) = [1] x5 + [1] x6 + [3] > [1] = c_2() cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) = [1] > [0] = c_4() leq#2#(S(x24),0()) = [1] > [0] = c_6() lt#2#(0(),0()) = [3] > [0] = c_8() lt#2#(0(),S(x20)) = [3] > [2] = c_9() main#(x4,Nil()) = [4] x4 + [2] > [0] = c_12() main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) = [4] x100 + [4] x132 + [4] x28 + [10] > [3] x100 + [1] x132 + [4] = c_13(cond_insTree_t_xs_1#(lt#2(x100,x48),Node(x100,x132,x164),Node(x48,x64,x80),x28)) leq#2(0(),x20) = [4] > [1] = True() leq#2(S(x24),0()) = [1] x24 + [5] > [0] = False() leq#2(S(x4),S(x2)) = [1] x4 + [5] > [1] x4 + [0] = leq#2(x4,x2) lt#2(0(),0()) = [9] > [0] = False() lt#2(0(),S(x20)) = [9] > [1] = True() lt#2(S(x20),0()) = [2] x20 + [11] > [0] = False() lt#2(S(x4),S(x2)) = [2] x4 + [11] > [2] x4 + [1] = lt#2(x4,x2) Following rules are (at-least) weakly oriented: cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) = [0] >= [0] = c_3() leq#2#(0(),x20) = [1] >= [1] = c_5() leq#2#(S(x4),S(x2)) = [1] >= [8] = c_7(leq#2#(x4,x2)) lt#2#(S(x20),0()) = [3] >= [4] = c_10() lt#2#(S(x4),S(x2)) = [3] >= [4] = c_11(lt#2#(x4,x2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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() leq#2#(0(),x20) -> c_5() leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) lt#2#(S(x20),0()) -> c_10() 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(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)) cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 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#(S(x24),0()) -> c_6() lt#2#(0(),0()) -> c_8() lt#2#(0(),S(x20)) -> c_9() 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)) - Weak TRS: 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) - 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_13/1} - 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 {2,4} by application of Pre({2,4}) = {3,5}. Here rules are labelled as follows: 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() 2: leq#2#(0(),x20) -> c_5() 3: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) 4: lt#2#(S(x20),0()) -> c_10() 5: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) 6: 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)) 7: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 8: 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() 9: leq#2#(S(x24),0()) -> c_6() 10: lt#2#(0(),0()) -> c_8() 11: lt#2#(0(),S(x20)) -> c_9() 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)) * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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() 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(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)) cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 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() 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)) - Weak TRS: 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) - 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_13/1} - 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_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() 2:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x24),0()) -> c_6():8 -->_1 leq#2#(0(),x20) -> c_5():7 -->_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():11 -->_1 lt#2#(0(),S(x20)) -> c_9():10 -->_1 lt#2#(0(),0()) -> c_8():9 -->_1 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3 4:W: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)) -->_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():6 -->_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():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#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4() 7:W:leq#2#(0(),x20) -> c_5() 8:W:leq#2#(S(x24),0()) -> c_6() 9:W:lt#2#(0(),0()) -> c_8() 10:W:lt#2#(0(),S(x20)) -> c_9() 11:W:lt#2#(S(x20),0()) -> c_10() 12:W:main#(x4,Nil()) -> c_12() 13:W: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)) -->_1 cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2():5 -->_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)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: main#(x4,Nil()) -> c_12() 5: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2() 6: 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() 9: lt#2#(0(),0()) -> c_8() 10: lt#2#(0(),S(x20)) -> c_9() 11: lt#2#(S(x20),0()) -> c_10() 7: leq#2#(0(),x20) -> c_5() 8: leq#2#(S(x24),0()) -> c_6() * Step 6: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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() 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(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)) 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)) - Weak TRS: 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) - 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_13/1} - 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: 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_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() - Weak 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#(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)) - Weak TRS: 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) - 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_13/1} - 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} Problem (S) - 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(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)) 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() 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)) - Weak TRS: 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) - 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_13/1} - 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} ** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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() - Weak 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#(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)) - Weak TRS: 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) - 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_13/1} - 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_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3() 2:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2 3:W: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:W: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)) -->_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():1 13:W: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)) -->_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)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) 2: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) ** Step 6.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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() - Weak 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)) 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)) - Weak TRS: 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) - 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_13/1} - 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: Trivial + Details: Consider the dependency graph 1:S: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:W: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)) -->_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():1 13:W: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)) -->_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)):4 The dependency graph contains no loops, we remove all dependency pairs. ** Step 6.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 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) - 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_13/1} - 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). ** Step 6.b:1: RemoveWeakSuffixes 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(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)) 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() 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)) - Weak TRS: 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) - 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_13/1} - 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: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)):1 2: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)):2 3:W: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)) -->_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():4 4: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() 5:W: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)) -->_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)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: 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)) 3: 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)) 4: 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() ** Step 6.b:2: Decompose 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 TRS: 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) - 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_13/1} - 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: 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: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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} Problem (S) - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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} *** Step 6.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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: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)):1 2:W: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)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) *** Step 6.b:2.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) *** Step 6.b:2.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#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_13/1} - 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: 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: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) The strictly oriented rules are moved into the weak component. **** Step 6.b:2.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#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_13/1} - 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: 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_7) = {1} Following symbols are considered usable: {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#,main#} TcT has computed the following interpretation: p(0) = [2] p(Cons) = [1] x1 + [1] x2 + [1] p(Elem) = [1] p(False) = [0] p(Nil) = [4] p(Node) = [1] x1 + [1] x3 + [4] p(S) = [1] x1 + [2] p(True) = [8] p(cond_insTree_t_xs_1) = [1] x1 + [1] x2 + [1] p(cond_link_t1_t2_2) = [1] x1 + [2] x2 + [1] x3 + [2] x7 + [1] x8 + [0] p(leq#2) = [8] x1 + [1] x2 + [0] p(lt#2) = [1] p(main) = [8] x2 + [0] p(cond_insTree_t_xs_1#) = [8] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(cond_link_t1_t2_2#) = [2] x2 + [1] x4 + [1] x5 + [1] x8 + [1] p(leq#2#) = [8] x1 + [0] p(lt#2#) = [1] x1 + [1] x2 + [1] p(main#) = [1] x1 + [1] p(c_1) = [2] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [1] x1 + [12] p(c_8) = [1] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [2] p(c_12) = [0] p(c_13) = [2] x1 + [0] Following rules are strictly oriented: leq#2#(S(x4),S(x2)) = [8] x4 + [16] > [8] x4 + [12] = c_7(leq#2#(x4,x2)) Following rules are (at-least) weakly oriented: **** Step 6.b:2.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#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_13/1} - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:2.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#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_13/1} - 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:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) **** Step 6.b:2.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_13/1} - 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). *** Step 6.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak DPs: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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: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)):1 2:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)) *** Step 6.b:2.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) - Weak TRS: 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) - 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_13/1} - 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#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) *** Step 6.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(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_13/1} - 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: 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: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) The strictly oriented rules are moved into the weak component. **** Step 6.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#2#(S(x4),S(x2)) -> c_11(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_13/1} - 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: 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_11) = {1} Following symbols are considered usable: {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#,main#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(Elem) = [1] x1 + [0] p(False) = [0] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [1] x3 + [0] p(S) = [1] x1 + [8] p(True) = [2] p(cond_insTree_t_xs_1) = [1] x3 + [1] p(cond_link_t1_t2_2) = [1] x1 + [1] x4 + [8] x5 + [2] x6 + [1] x7 + [2] p(leq#2) = [1] x1 + [1] x2 + [0] p(lt#2) = [2] x1 + [1] p(main) = [8] x2 + [2] p(cond_insTree_t_xs_1#) = [4] x1 + [2] x2 + [4] x3 + [1] x4 + [4] p(cond_link_t1_t2_2#) = [1] x2 + [8] x7 + [1] p(leq#2#) = [1] x1 + [0] p(lt#2#) = [2] x1 + [9] p(main#) = [2] x2 + [0] p(c_1) = [2] x1 + [0] p(c_2) = [1] p(c_3) = [4] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] x1 + [8] p(c_12) = [0] p(c_13) = [1] Following rules are strictly oriented: lt#2#(S(x4),S(x2)) = [2] x4 + [25] > [2] x4 + [17] = c_11(lt#2#(x4,x2)) Following rules are (at-least) weakly oriented: **** Step 6.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: lt#2#(S(x4),S(x2)) -> c_11(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_13/1} - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: lt#2#(S(x4),S(x2)) -> c_11(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_13/1} - 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:W: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)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)) **** Step 6.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_13/1} - 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))