src/HOL/Metis_Examples/Big_O.thy
 author haftmann Mon Jun 05 15:59:41 2017 +0200 (2017-06-05) changeset 66010 2f7d39285a1a parent 64267 b9a1486e79be child 66453 cc19f7ca2ed6 permissions -rw-r--r--
executable domain membership checks
```     1 (*  Title:      HOL/Metis_Examples/Big_O.thy
```
```     2     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
```
```     3     Author:     Jasmin Blanchette, TU Muenchen
```
```     4
```
```     5 Metis example featuring the Big O notation.
```
```     6 *)
```
```     7
```
```     8 section \<open>Metis Example Featuring the Big O Notation\<close>
```
```     9
```
```    10 theory Big_O
```
```    11 imports
```
```    12   "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
```
```    13   "~~/src/HOL/Library/Function_Algebras"
```
```    14   "~~/src/HOL/Library/Set_Algebras"
```
```    15 begin
```
```    16
```
```    17 subsection \<open>Definitions\<close>
```
```    18
```
```    19 definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
```
```    20   "O(f::('a => 'b)) == {h. \<exists>c. \<forall>x. \<bar>h x\<bar> <= c * \<bar>f x\<bar>}"
```
```    21
```
```    22 lemma bigo_pos_const:
```
```    23   "(\<exists>c::'a::linordered_idom.
```
```    24     \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
```
```    25     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
```
```    26   by (metis (no_types) abs_ge_zero
```
```    27       algebra_simps mult.comm_neutral
```
```    28       mult_nonpos_nonneg not_le_imp_less order_trans zero_less_one)
```
```    29
```
```    30 (*** Now various verions with an increasing shrink factor ***)
```
```    31
```
```    32 sledgehammer_params [isar_proofs, compress = 1]
```
```    33
```
```    34 lemma
```
```    35   "(\<exists>c::'a::linordered_idom.
```
```    36     \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
```
```    37     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
```
```    38   apply auto
```
```    39   apply (case_tac "c = 0", simp)
```
```    40   apply (rule_tac x = "1" in exI, simp)
```
```    41   apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
```
```    42 proof -
```
```    43   fix c :: 'a and x :: 'b
```
```    44   assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
```
```    45   have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 0 \<le> \<bar>x\<^sub>1\<bar>" by (metis abs_ge_zero)
```
```    46   have F2: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
```
```    47   have F3: "\<forall>x\<^sub>1 x\<^sub>3. x\<^sub>3 \<le> \<bar>h x\<^sub>1\<bar> \<longrightarrow> x\<^sub>3 \<le> c * \<bar>f x\<^sub>1\<bar>" by (metis A1 order_trans)
```
```    48   have F4: "\<forall>x\<^sub>2 x\<^sub>3::'a::linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
```
```    49     by (metis abs_mult)
```
```    50   have F5: "\<forall>x\<^sub>3 x\<^sub>1::'a::linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1"
```
```    51     by (metis abs_mult_pos)
```
```    52   hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = \<bar>1\<bar> * x\<^sub>1" by (metis F2)
```
```    53   hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F2 abs_one)
```
```    54   hence "\<forall>x\<^sub>3. 0 \<le> \<bar>h x\<^sub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F3)
```
```    55   hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F1)
```
```    56   hence "\<forall>x\<^sub>3. (0::'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F5)
```
```    57   hence "\<forall>x\<^sub>3. (0::'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F4)
```
```    58   hence "\<forall>x\<^sub>3. c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F1)
```
```    59   hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1)
```
```    60   thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F4)
```
```    61 qed
```
```    62
```
```    63 sledgehammer_params [isar_proofs, compress = 2]
```
```    64
```
```    65 lemma
```
```    66   "(\<exists>c::'a::linordered_idom.
```
```    67     \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
```
```    68     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
```
```    69   apply auto
```
```    70   apply (case_tac "c = 0", simp)
```
```    71   apply (rule_tac x = "1" in exI, simp)
```
```    72   apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
```
```    73 proof -
```
```    74   fix c :: 'a and x :: 'b
```
```    75   assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
```
```    76   have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
```
```    77   have F2: "\<forall>x\<^sub>2 x\<^sub>3::'a::linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
```
```    78     by (metis abs_mult)
```
```    79   have "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_mult_pos abs_one)
```
```    80   hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis A1 abs_ge_zero order_trans)
```
```    81   hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F2 abs_mult_pos)
```
```    82   hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero)
```
```    83   thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F2)
```
```    84 qed
```
```    85
```
```    86 sledgehammer_params [isar_proofs, compress = 3]
```
```    87
```
```    88 lemma
```
```    89   "(\<exists>c::'a::linordered_idom.
```
```    90     \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
```
```    91     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
```
```    92   apply auto
```
```    93   apply (case_tac "c = 0", simp)
```
```    94   apply (rule_tac x = "1" in exI, simp)
```
```    95   apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
```
```    96 proof -
```
```    97   fix c :: 'a and x :: 'b
```
```    98   assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
```
```    99   have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
```
```   100   have F2: "\<forall>x\<^sub>3 x\<^sub>1::'a::linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1" by (metis abs_mult_pos)
```
```   101   hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_one)
```
```   102   hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans)
```
```   103   thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis A1 abs_ge_zero)
```
```   104 qed
```
```   105
```
```   106 sledgehammer_params [isar_proofs, compress = 4]
```
```   107
```
```   108 lemma
```
```   109   "(\<exists>c::'a::linordered_idom.
```
```   110     \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
```
```   111     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
```
```   112   apply auto
```
```   113   apply (case_tac "c = 0", simp)
```
```   114   apply (rule_tac x = "1" in exI, simp)
```
```   115   apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
```
```   116 proof -
```
```   117   fix c :: 'a and x :: 'b
```
```   118   assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
```
```   119   have "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
```
```   120   hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>"
```
```   121     by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
```
```   122   hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
```
```   123   thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis abs_mult)
```
```   124 qed
```
```   125
```
```   126 sledgehammer_params [isar_proofs, compress = 1]
```
```   127
```
```   128 lemma bigo_alt_def: "O(f) = {h. \<exists>c. (0 < c \<and> (\<forall>x. \<bar>h x\<bar> <= c * \<bar>f x\<bar>))}"
```
```   129 by (auto simp add: bigo_def bigo_pos_const)
```
```   130
```
```   131 lemma bigo_elt_subset [intro]: "f : O(g) \<Longrightarrow> O(f) \<le> O(g)"
```
```   132 apply (auto simp add: bigo_alt_def)
```
```   133 apply (rule_tac x = "ca * c" in exI)
```
```   134 apply (metis algebra_simps mult_le_cancel_left_pos order_trans mult_pos_pos)
```
```   135 done
```
```   136
```
```   137 lemma bigo_refl [intro]: "f : O(f)"
```
```   138 unfolding bigo_def mem_Collect_eq
```
```   139 by (metis mult_1 order_refl)
```
```   140
```
```   141 lemma bigo_zero: "0 : O(g)"
```
```   142 apply (auto simp add: bigo_def func_zero)
```
```   143 by (metis mult_zero_left order_refl)
```
```   144
```
```   145 lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
```
```   146 by (auto simp add: bigo_def)
```
```   147
```
```   148 lemma bigo_plus_self_subset [intro]:
```
```   149   "O(f) + O(f) <= O(f)"
```
```   150 apply (auto simp add: bigo_alt_def set_plus_def)
```
```   151 apply (rule_tac x = "c + ca" in exI)
```
```   152 apply auto
```
```   153 apply (simp add: ring_distribs func_plus)
```
```   154 by (metis order_trans abs_triangle_ineq add_mono)
```
```   155
```
```   156 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
```
```   157 by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2)
```
```   158
```
```   159 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
```
```   160 apply (rule subsetI)
```
```   161 apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
```
```   162 apply (subst bigo_pos_const [symmetric])+
```
```   163 apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> <= \<bar>f n\<bar> then x n else 0" in exI)
```
```   164 apply (rule conjI)
```
```   165  apply (rule_tac x = "c + c" in exI)
```
```   166  apply clarsimp
```
```   167  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= (c + c) * \<bar>f xa\<bar>")
```
```   168   apply (metis mult_2 order_trans)
```
```   169  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
```
```   170   apply (erule order_trans)
```
```   171   apply (simp add: ring_distribs)
```
```   172  apply (rule mult_left_mono)
```
```   173   apply (simp add: abs_triangle_ineq)
```
```   174  apply (simp add: order_less_le)
```
```   175 apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
```
```   176 apply (rule conjI)
```
```   177  apply (rule_tac x = "c + c" in exI)
```
```   178  apply auto
```
```   179 apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= (c + c) * \<bar>g xa\<bar>")
```
```   180  apply (metis order_trans mult_2)
```
```   181 apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
```
```   182  apply (erule order_trans)
```
```   183  apply (simp add: ring_distribs)
```
```   184 by (metis abs_triangle_ineq mult_le_cancel_left_pos)
```
```   185
```
```   186 lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A + B <= O(f)"
```
```   187 by (metis bigo_plus_idemp set_plus_mono2)
```
```   188
```
```   189 lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
```
```   190 apply (rule equalityI)
```
```   191 apply (rule bigo_plus_subset)
```
```   192 apply (simp add: bigo_alt_def set_plus_def func_plus)
```
```   193 apply clarify
```
```   194 (* sledgehammer *)
```
```   195 apply (rule_tac x = "max c ca" in exI)
```
```   196
```
```   197 apply (rule conjI)
```
```   198  apply (metis less_max_iff_disj)
```
```   199 apply clarify
```
```   200 apply (drule_tac x = "xa" in spec)+
```
```   201 apply (subgoal_tac "0 <= f xa + g xa")
```
```   202  apply (simp add: ring_distribs)
```
```   203  apply (subgoal_tac "\<bar>a xa + b xa\<bar> <= \<bar>a xa\<bar> + \<bar>b xa\<bar>")
```
```   204   apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> <= max c ca * f xa + max c ca * g xa")
```
```   205    apply (metis order_trans)
```
```   206   defer 1
```
```   207   apply (metis abs_triangle_ineq)
```
```   208  apply (metis add_nonneg_nonneg)
```
```   209 apply (rule add_mono)
```
```   210  apply (metis max.cobounded2 linorder_linear max.absorb1 mult_right_mono xt1(6))
```
```   211 by (metis max.cobounded2 linorder_not_le mult_le_cancel_right order_trans)
```
```   212
```
```   213 lemma bigo_bounded_alt: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
```
```   214 apply (auto simp add: bigo_def)
```
```   215 (* Version 1: one-line proof *)
```
```   216 by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult)
```
```   217
```
```   218 lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
```
```   219 apply (auto simp add: bigo_def)
```
```   220 (* Version 2: structured proof *)
```
```   221 proof -
```
```   222   assume "\<forall>x. f x \<le> c * g x"
```
```   223   thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
```
```   224 qed
```
```   225
```
```   226 lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)"
```
```   227 apply (erule bigo_bounded_alt [of f 1 g])
```
```   228 by (metis mult_1)
```
```   229
```
```   230 lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)"
```
```   231 apply (rule set_minus_imp_plus)
```
```   232 apply (rule bigo_bounded)
```
```   233  apply (metis add_le_cancel_left diff_add_cancel diff_self minus_apply
```
```   234               algebra_simps)
```
```   235 by (metis add_le_cancel_left diff_add_cancel func_plus le_fun_def
```
```   236           algebra_simps)
```
```   237
```
```   238 lemma bigo_abs: "(\<lambda>x. \<bar>f x\<bar>) =o O(f)"
```
```   239 apply (unfold bigo_def)
```
```   240 apply auto
```
```   241 by (metis mult_1 order_refl)
```
```   242
```
```   243 lemma bigo_abs2: "f =o O(\<lambda>x. \<bar>f x\<bar>)"
```
```   244 apply (unfold bigo_def)
```
```   245 apply auto
```
```   246 by (metis mult_1 order_refl)
```
```   247
```
```   248 lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
```
```   249 proof -
```
```   250   have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
```
```   251   have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
```
```   252   have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2)
```
```   253   thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
```
```   254 qed
```
```   255
```
```   256 lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o (\<lambda>x. \<bar>g x\<bar>) +o O(h)"
```
```   257   apply (drule set_plus_imp_minus)
```
```   258   apply (rule set_minus_imp_plus)
```
```   259   apply (subst fun_diff_def)
```
```   260 proof -
```
```   261   assume a: "f - g : O(h)"
```
```   262   have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)"
```
```   263     by (rule bigo_abs2)
```
```   264   also have "... <= O(\<lambda>x. \<bar>f x - g x\<bar>)"
```
```   265     apply (rule bigo_elt_subset)
```
```   266     apply (rule bigo_bounded)
```
```   267      apply (metis abs_ge_zero)
```
```   268     by (metis abs_triangle_ineq3)
```
```   269   also have "... <= O(f - g)"
```
```   270     apply (rule bigo_elt_subset)
```
```   271     apply (subst fun_diff_def)
```
```   272     apply (rule bigo_abs)
```
```   273     done
```
```   274   also have "... <= O(h)"
```
```   275     using a by (rule bigo_elt_subset)
```
```   276   finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) : O(h)" .
```
```   277 qed
```
```   278
```
```   279 lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
```
```   280 by (unfold bigo_def, auto)
```
```   281
```
```   282 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) + O(h)"
```
```   283 proof -
```
```   284   assume "f : g +o O(h)"
```
```   285   also have "... <= O(g) + O(h)"
```
```   286     by (auto del: subsetI)
```
```   287   also have "... = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
```
```   288     by (metis bigo_abs3)
```
```   289   also have "... = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
```
```   290     by (rule bigo_plus_eq [symmetric], auto)
```
```   291   finally have "f : ...".
```
```   292   then have "O(f) <= ..."
```
```   293     by (elim bigo_elt_subset)
```
```   294   also have "... = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
```
```   295     by (rule bigo_plus_eq, auto)
```
```   296   finally show ?thesis
```
```   297     by (simp add: bigo_abs3 [symmetric])
```
```   298 qed
```
```   299
```
```   300 lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)"
```
```   301 apply (rule subsetI)
```
```   302 apply (subst bigo_def)
```
```   303 apply (auto simp del: abs_mult ac_simps
```
```   304             simp add: bigo_alt_def set_times_def func_times)
```
```   305 (* sledgehammer *)
```
```   306 apply (rule_tac x = "c * ca" in exI)
```
```   307 apply (rule allI)
```
```   308 apply (erule_tac x = x in allE)+
```
```   309 apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
```
```   310  apply (metis (no_types) abs_ge_zero abs_mult mult_mono')
```
```   311 by (metis mult.assoc mult.left_commute abs_of_pos mult.left_commute abs_mult)
```
```   312
```
```   313 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
```
```   314 by (metis bigo_mult bigo_refl set_times_mono3 subset_trans)
```
```   315
```
```   316 lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)"
```
```   317 by (metis bigo_mult set_rev_mp set_times_intro)
```
```   318
```
```   319 lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)"
```
```   320 by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
```
```   321
```
```   322 lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow>
```
```   323     O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
```
```   324 proof -
```
```   325   assume a: "\<forall>x. f x ~= 0"
```
```   326   show "O(f * g) <= f *o O(g)"
```
```   327   proof
```
```   328     fix h
```
```   329     assume h: "h : O(f * g)"
```
```   330     then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)"
```
```   331       by auto
```
```   332     also have "... <= O((\<lambda>x. 1 / f x) * (f * g))"
```
```   333       by (rule bigo_mult2)
```
```   334     also have "(\<lambda>x. 1 / f x) * (f * g) = g"
```
```   335       by (simp add: fun_eq_iff a)
```
```   336     finally have "(\<lambda>x. (1::'b) / f x) * h : O(g)".
```
```   337     then have "f * ((\<lambda>x. (1::'b) / f x) * h) : f *o O(g)"
```
```   338       by auto
```
```   339     also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
```
```   340       by (simp add: func_times fun_eq_iff a)
```
```   341     finally show "h : f *o O(g)".
```
```   342   qed
```
```   343 qed
```
```   344
```
```   345 lemma bigo_mult6:
```
```   346 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = (f::'a \<Rightarrow> ('b::linordered_field)) *o O(g)"
```
```   347 by (metis bigo_mult2 bigo_mult5 order_antisym)
```
```   348
```
```   349 (*proof requires relaxing relevance: 2007-01-25*)
```
```   350 declare bigo_mult6 [simp]
```
```   351
```
```   352 lemma bigo_mult7:
```
```   353 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f::'a \<Rightarrow> ('b::linordered_field)) * O(g)"
```
```   354 by (metis bigo_refl bigo_mult6 set_times_mono3)
```
```   355
```
```   356 declare bigo_mult6 [simp del]
```
```   357 declare bigo_mult7 [intro!]
```
```   358
```
```   359 lemma bigo_mult8:
```
```   360 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f::'a \<Rightarrow> ('b::linordered_field)) * O(g)"
```
```   361 by (metis bigo_mult bigo_mult7 order_antisym_conv)
```
```   362
```
```   363 lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
```
```   364 by (auto simp add: bigo_def fun_Compl_def)
```
```   365
```
```   366 lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)"
```
```   367 by (metis (no_types, lifting) bigo_minus diff_minus_eq_add minus_add_distrib
```
```   368     minus_minus set_minus_imp_plus set_plus_imp_minus)
```
```   369
```
```   370 lemma bigo_minus3: "O(-f) = O(f)"
```
```   371 by (metis bigo_elt_subset bigo_minus bigo_refl equalityI minus_minus)
```
```   372
```
```   373 lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) \<le> O(g)"
```
```   374 by (metis bigo_plus_idemp set_plus_mono3)
```
```   375
```
```   376 lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) \<le> f +o O(g)"
```
```   377 by (metis (no_types) bigo_minus bigo_plus_absorb_lemma1 right_minus
```
```   378           set_plus_mono set_plus_rearrange2 set_zero_plus subsetD subset_refl
```
```   379           subset_trans)
```
```   380
```
```   381 lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)"
```
```   382 by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
```
```   383
```
```   384 lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A \<le> O(g)"
```
```   385 by (metis bigo_plus_absorb set_plus_mono)
```
```   386
```
```   387 lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)"
```
```   388 by (metis bigo_minus minus_diff_eq set_plus_imp_minus set_minus_plus)
```
```   389
```
```   390 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
```
```   391 by (metis bigo_add_commute_imp)
```
```   392
```
```   393 lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)"
```
```   394 by (auto simp add: bigo_def ac_simps)
```
```   395
```
```   396 lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<le> O(\<lambda>x. 1)"
```
```   397 by (metis bigo_const1 bigo_elt_subset)
```
```   398
```
```   399 lemma bigo_const3: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)"
```
```   400 apply (simp add: bigo_def)
```
```   401 by (metis abs_eq_0 left_inverse order_refl)
```
```   402
```
```   403 lemma bigo_const4: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)"
```
```   404 by (metis bigo_elt_subset bigo_const3)
```
```   405
```
```   406 lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
```
```   407     O(\<lambda>x. c) = O(\<lambda>x. 1)"
```
```   408 by (metis bigo_const2 bigo_const4 equalityI)
```
```   409
```
```   410 lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)"
```
```   411 apply (simp add: bigo_def abs_mult)
```
```   412 by (metis le_less)
```
```   413
```
```   414 lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<le> O(f)"
```
```   415 by (rule bigo_elt_subset, rule bigo_const_mult1)
```
```   416
```
```   417 lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
```
```   418 apply (simp add: bigo_def)
```
```   419 by (metis (no_types) abs_mult mult.assoc mult_1 order_refl left_inverse)
```
```   420
```
```   421 lemma bigo_const_mult4:
```
```   422 "(c::'a::linordered_field) \<noteq> 0 \<Longrightarrow> O(f) \<le> O(\<lambda>x. c * f x)"
```
```   423 by (metis bigo_elt_subset bigo_const_mult3)
```
```   424
```
```   425 lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
```
```   426     O(\<lambda>x. c * f x) = O(f)"
```
```   427 by (metis equalityI bigo_const_mult2 bigo_const_mult4)
```
```   428
```
```   429 lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
```
```   430     (\<lambda>x. c) *o O(f) = O(f)"
```
```   431   apply (auto del: subsetI)
```
```   432   apply (rule order_trans)
```
```   433   apply (rule bigo_mult2)
```
```   434   apply (simp add: func_times)
```
```   435   apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
```
```   436   apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
```
```   437   apply (rename_tac g d)
```
```   438   apply safe
```
```   439   apply (rule_tac [2] ext)
```
```   440    prefer 2
```
```   441    apply simp
```
```   442   apply (simp add: mult.assoc [symmetric] abs_mult)
```
```   443   (* couldn't get this proof without the step above *)
```
```   444 proof -
```
```   445   fix g :: "'b \<Rightarrow> 'a" and d :: 'a
```
```   446   assume A1: "c \<noteq> (0::'a)"
```
```   447   assume A2: "\<forall>x::'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
```
```   448   have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse)
```
```   449   have F2: "(0::'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff)
```
```   450   have "(0::'a) < \<bar>c\<bar> \<longrightarrow> (0::'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive)
```
```   451   hence "(0::'a) < \<bar>inverse c\<bar>" using F2 by metis
```
```   452   hence F3: "(0::'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less)
```
```   453   have "\<exists>(u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar>"
```
```   454     using A2 by metis
```
```   455   hence F4: "\<exists>(u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0::'a) \<le> \<bar>inverse c\<bar>"
```
```   456     using F3 by metis
```
```   457   hence "\<exists>(v::'a) (u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^sub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^sub>7 (u * v))\<bar>)"
```
```   458     by (metis mult_left_mono)
```
```   459   then show "\<exists>ca::'a. \<forall>x::'b. inverse \<bar>c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>"
```
```   460     using A2 F4 by (metis F1 \<open>0 < \<bar>inverse c\<bar>\<close> linordered_field_class.sign_simps(23) mult_le_cancel_left_pos)
```
```   461 qed
```
```   462
```
```   463 lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)"
```
```   464   apply (auto intro!: subsetI
```
```   465     simp add: bigo_def elt_set_times_def func_times
```
```   466     simp del: abs_mult ac_simps)
```
```   467 (* sledgehammer *)
```
```   468   apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
```
```   469   apply (rule allI)
```
```   470   apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
```
```   471   apply (erule ssubst)
```
```   472   apply (subst abs_mult)
```
```   473   apply (rule mult_left_mono)
```
```   474   apply (erule spec)
```
```   475   apply simp
```
```   476   apply (simp add: ac_simps)
```
```   477 done
```
```   478
```
```   479 lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
```
```   480 by (metis bigo_const_mult1 bigo_elt_subset order_less_le psubsetD)
```
```   481
```
```   482 lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))"
```
```   483 by (unfold bigo_def, auto)
```
```   484
```
```   485 lemma bigo_compose2:
```
```   486 "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o O(\<lambda>x. h(k x))"
```
```   487 apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
```
```   488 apply (drule bigo_compose1 [of "f - g" h k])
```
```   489 apply (simp add: fun_diff_def)
```
```   490 done
```
```   491
```
```   492 subsection \<open>Sum\<close>
```
```   493
```
```   494 lemma bigo_sum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
```
```   495     \<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
```
```   496       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
```
```   497 apply (auto simp add: bigo_def)
```
```   498 apply (rule_tac x = "\<bar>c\<bar>" in exI)
```
```   499 apply (subst abs_of_nonneg) back back
```
```   500  apply (rule sum_nonneg)
```
```   501  apply force
```
```   502 apply (subst sum_distrib_left)
```
```   503 apply (rule allI)
```
```   504 apply (rule order_trans)
```
```   505  apply (rule sum_abs)
```
```   506 apply (rule sum_mono)
```
```   507 by (metis abs_ge_self abs_mult_pos order_trans)
```
```   508
```
```   509 lemma bigo_sum1: "\<forall>x y. 0 <= h x y \<Longrightarrow>
```
```   510     \<exists>c. \<forall>x y. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
```
```   511       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
```
```   512 by (metis (no_types) bigo_sum_main)
```
```   513
```
```   514 lemma bigo_sum2: "\<forall>y. 0 <= h y \<Longrightarrow>
```
```   515     \<exists>c. \<forall>y. \<bar>f y\<bar> <= c * (h y) \<Longrightarrow>
```
```   516       (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
```
```   517 apply (rule bigo_sum1)
```
```   518 by metis+
```
```   519
```
```   520 lemma bigo_sum3: "f =o O(h) \<Longrightarrow>
```
```   521     (\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
```
```   522       O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h(k x y)\<bar>)"
```
```   523 apply (rule bigo_sum1)
```
```   524  apply (rule allI)+
```
```   525  apply (rule abs_ge_zero)
```
```   526 apply (unfold bigo_def)
```
```   527 apply (auto simp add: abs_mult)
```
```   528 by (metis abs_ge_zero mult.left_commute mult_left_mono)
```
```   529
```
```   530 lemma bigo_sum4: "f =o g +o O(h) \<Longrightarrow>
```
```   531     (\<lambda>x. \<Sum>y \<in> A x. l x y * f(k x y)) =o
```
```   532       (\<lambda>x. \<Sum>y \<in> A x. l x y * g(k x y)) +o
```
```   533         O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h(k x y)\<bar>)"
```
```   534 apply (rule set_minus_imp_plus)
```
```   535 apply (subst fun_diff_def)
```
```   536 apply (subst sum_subtractf [symmetric])
```
```   537 apply (subst right_diff_distrib [symmetric])
```
```   538 apply (rule bigo_sum3)
```
```   539 by (metis (lifting, no_types) fun_diff_def set_plus_imp_minus ext)
```
```   540
```
```   541 lemma bigo_sum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
```
```   542     \<forall>x. 0 <= h x \<Longrightarrow>
```
```   543       (\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
```
```   544         O(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y))"
```
```   545 apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y)) =
```
```   546       (\<lambda>x. \<Sum>y \<in> A x. \<bar>(l x y) * h(k x y)\<bar>)")
```
```   547  apply (erule ssubst)
```
```   548  apply (erule bigo_sum3)
```
```   549 apply (rule ext)
```
```   550 apply (rule sum.cong)
```
```   551 apply (rule refl)
```
```   552 by (metis abs_of_nonneg zero_le_mult_iff)
```
```   553
```
```   554 lemma bigo_sum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
```
```   555     \<forall>x. 0 <= h x \<Longrightarrow>
```
```   556       (\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
```
```   557         (\<lambda>x. \<Sum>y \<in> A x. (l x y) * g(k x y)) +o
```
```   558           O(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y))"
```
```   559   apply (rule set_minus_imp_plus)
```
```   560   apply (subst fun_diff_def)
```
```   561   apply (subst sum_subtractf [symmetric])
```
```   562   apply (subst right_diff_distrib [symmetric])
```
```   563   apply (rule bigo_sum5)
```
```   564   apply (subst fun_diff_def [symmetric])
```
```   565   apply (drule set_plus_imp_minus)
```
```   566   apply auto
```
```   567 done
```
```   568
```
```   569 subsection \<open>Misc useful stuff\<close>
```
```   570
```
```   571 lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
```
```   572   A + B <= O(f)"
```
```   573   apply (subst bigo_plus_idemp [symmetric])
```
```   574   apply (rule set_plus_mono2)
```
```   575   apply assumption+
```
```   576 done
```
```   577
```
```   578 lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
```
```   579   apply (subst bigo_plus_idemp [symmetric])
```
```   580   apply (rule set_plus_intro)
```
```   581   apply assumption+
```
```   582 done
```
```   583
```
```   584 lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
```
```   585     (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
```
```   586   apply (rule subsetD)
```
```   587   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)")
```
```   588   apply assumption
```
```   589   apply (rule bigo_const_mult6)
```
```   590   apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
```
```   591   apply (erule ssubst)
```
```   592   apply (erule set_times_intro2)
```
```   593   apply (simp add: func_times)
```
```   594 done
```
```   595
```
```   596 lemma bigo_fix: "(\<lambda>x. f ((x::nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow>
```
```   597     f =o O(h)"
```
```   598 apply (simp add: bigo_alt_def)
```
```   599 by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc)
```
```   600
```
```   601 lemma bigo_fix2:
```
```   602     "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
```
```   603        f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
```
```   604   apply (rule set_minus_imp_plus)
```
```   605   apply (rule bigo_fix)
```
```   606   apply (subst fun_diff_def)
```
```   607   apply (subst fun_diff_def [symmetric])
```
```   608   apply (rule set_plus_imp_minus)
```
```   609   apply simp
```
```   610   apply (simp add: fun_diff_def)
```
```   611 done
```
```   612
```
```   613 subsection \<open>Less than or equal to\<close>
```
```   614
```
```   615 definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
```
```   616   "f <o g == (\<lambda>x. max (f x - g x) 0)"
```
```   617
```
```   618 lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> <= \<bar>f x\<bar> \<Longrightarrow>
```
```   619     g =o O(h)"
```
```   620   apply (unfold bigo_def)
```
```   621   apply clarsimp
```
```   622 apply (blast intro: order_trans)
```
```   623 done
```
```   624
```
```   625 lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> <= f x \<Longrightarrow>
```
```   626       g =o O(h)"
```
```   627   apply (erule bigo_lesseq1)
```
```   628 apply (blast intro: abs_ge_self order_trans)
```
```   629 done
```
```   630
```
```   631 lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow>
```
```   632       g =o O(h)"
```
```   633   apply (erule bigo_lesseq2)
```
```   634   apply (rule allI)
```
```   635   apply (subst abs_of_nonneg)
```
```   636   apply (erule spec)+
```
```   637 done
```
```   638
```
```   639 lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
```
```   640     \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= \<bar>f x\<bar> \<Longrightarrow>
```
```   641       g =o O(h)"
```
```   642   apply (erule bigo_lesseq1)
```
```   643   apply (rule allI)
```
```   644   apply (subst abs_of_nonneg)
```
```   645   apply (erule spec)+
```
```   646 done
```
```   647
```
```   648 lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)"
```
```   649 apply (unfold lesso_def)
```
```   650 apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
```
```   651  apply (metis bigo_zero)
```
```   652 by (metis (lifting, no_types) func_zero le_fun_def le_iff_diff_le_0
```
```   653       max.absorb2 order_eq_iff)
```
```   654
```
```   655 lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
```
```   656     \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow>
```
```   657       k <o g =o O(h)"
```
```   658   apply (unfold lesso_def)
```
```   659   apply (rule bigo_lesseq4)
```
```   660   apply (erule set_plus_imp_minus)
```
```   661   apply (rule allI)
```
```   662   apply (rule max.cobounded2)
```
```   663   apply (rule allI)
```
```   664   apply (subst fun_diff_def)
```
```   665 apply (erule thin_rl)
```
```   666 (* sledgehammer *)
```
```   667 apply (case_tac "0 <= k x - g x")
```
```   668  apply (metis (lifting) abs_le_D1 linorder_linear min_diff_distrib_left
```
```   669           min.absorb1 min.absorb2 max.absorb1)
```
```   670 by (metis abs_ge_zero le_cases max.absorb2)
```
```   671
```
```   672 lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
```
```   673     \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow>
```
```   674       f <o k =o O(h)"
```
```   675 apply (unfold lesso_def)
```
```   676 apply (rule bigo_lesseq4)
```
```   677   apply (erule set_plus_imp_minus)
```
```   678  apply (rule allI)
```
```   679  apply (rule max.cobounded2)
```
```   680 apply (rule allI)
```
```   681 apply (subst fun_diff_def)
```
```   682 apply (erule thin_rl)
```
```   683 (* sledgehammer *)
```
```   684 apply (case_tac "0 <= f x - k x")
```
```   685  apply simp
```
```   686  apply (subst abs_of_nonneg)
```
```   687   apply (drule_tac x = x in spec) back
```
```   688   apply (metis diff_less_0_iff_less linorder_not_le not_le_imp_less xt1(12) xt1(6))
```
```   689  apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
```
```   690 by (metis abs_ge_zero linorder_linear max.absorb1 max.commute)
```
```   691
```
```   692 lemma bigo_lesso4:
```
```   693   "f <o g =o O(k::'a=>'b::{linordered_field}) \<Longrightarrow>
```
```   694    g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
```
```   695 apply (unfold lesso_def)
```
```   696 apply (drule set_plus_imp_minus)
```
```   697 apply (drule bigo_abs5) back
```
```   698 apply (simp add: fun_diff_def)
```
```   699 apply (drule bigo_useful_add, assumption)
```
```   700 apply (erule bigo_lesseq2) back
```
```   701 apply (rule allI)
```
```   702 by (auto simp add: func_plus fun_diff_def algebra_simps
```
```   703     split: split_max abs_split)
```
```   704
```
```   705 lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * \<bar>h x\<bar>"
```
```   706 apply (simp only: lesso_def bigo_alt_def)
```
```   707 apply clarsimp
```
```   708 by (metis add.commute diff_le_eq)
```
```   709
```
```   710 end
```