src/HOL/Metis_Examples/Big_O.thy
author blanchet
Fri Feb 24 11:23:36 2012 +0100 (2012-02-24)
changeset 46644 bd03e0890699
parent 46369 9ac0c64ad8e7
child 47108 2a1953f0d20d
permissions -rw-r--r--
rephrase some slow "metis" calls
     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 header {* Metis Example Featuring the Big O Notation *}
     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 {* Definitions *}
    18 
    19 definition bigo :: "('a => 'b\<Colon>{linordered_idom,number_ring}) => ('a => 'b) set" ("(1O'(_'))") where
    20   "O(f\<Colon>('a => 'b)) == {h. \<exists>c. \<forall>x. abs (h x) <= c * abs (f x)}"
    21 
    22 lemma bigo_pos_const:
    23   "(\<exists>c\<Colon>'a\<Colon>linordered_idom.
    24     \<forall>x. abs (h x) \<le> c * abs (f x))
    25     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))"
    26 by (metis (no_types) abs_ge_zero
    27       comm_semiring_1_class.normalizing_semiring_rules(7) mult.comm_neutral
    28       mult_nonpos_nonneg not_leE order_trans zero_less_one)
    29 
    30 (*** Now various verions with an increasing shrink factor ***)
    31 
    32 sledgehammer_params [isar_proof, isar_shrink_factor = 1]
    33 
    34 lemma
    35   "(\<exists>c\<Colon>'a\<Colon>linordered_idom.
    36     \<forall>x. abs (h x) \<le> c * abs (f x))
    37     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))"
    38   apply auto
    39   apply (case_tac "c = 0", simp)
    40   apply (rule_tac x = "1" in exI, simp)
    41   apply (rule_tac x = "abs c" 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\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_ge_zero)
    46   have F2: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
    47   have F3: "\<forall>x\<^isub>1 x\<^isub>3. x\<^isub>3 \<le> \<bar>h x\<^isub>1\<bar> \<longrightarrow> x\<^isub>3 \<le> c * \<bar>f x\<^isub>1\<bar>" by (metis A1 order_trans)
    48   have F4: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
    49     by (metis abs_mult)
    50   have F5: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1"
    51     by (metis abs_mult_pos)
    52   hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = \<bar>1\<bar> * x\<^isub>1" by (metis F2)
    53   hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F2 abs_one)
    54   hence "\<forall>x\<^isub>3. 0 \<le> \<bar>h x\<^isub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F3)
    55   hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F1)
    56   hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F5)
    57   hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F4)
    58   hence "\<forall>x\<^isub>3. c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>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_proof, isar_shrink_factor = 2]
    64 
    65 lemma
    66   "(\<exists>c\<Colon>'a\<Colon>linordered_idom.
    67     \<forall>x. abs (h x) \<le> c * abs (f x))
    68     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))"
    69   apply auto
    70   apply (case_tac "c = 0", simp)
    71   apply (rule_tac x = "1" in exI, simp)
    72   apply (rule_tac x = "abs c" 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\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
    77   have F2: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
    78     by (metis abs_mult)
    79   have "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_mult_pos abs_one)
    80   hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis A1 abs_ge_zero order_trans)
    81   hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>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_proof, isar_shrink_factor = 3]
    87 
    88 lemma
    89   "(\<exists>c\<Colon>'a\<Colon>linordered_idom.
    90     \<forall>x. abs (h x) \<le> c * abs (f x))
    91     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))"
    92   apply auto
    93   apply (case_tac "c = 0", simp)
    94   apply (rule_tac x = "1" in exI, simp)
    95   apply (rule_tac x = "abs c" 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\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
   100   have F2: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1" by (metis abs_mult_pos)
   101   hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_one)
   102   hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>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_proof, isar_shrink_factor = 4]
   107 
   108 lemma
   109   "(\<exists>c\<Colon>'a\<Colon>linordered_idom.
   110     \<forall>x. abs (h x) \<le> c * abs (f x))
   111     \<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))"
   112   apply auto
   113   apply (case_tac "c = 0", simp)
   114   apply (rule_tac x = "1" in exI, simp)
   115   apply (rule_tac x = "abs c" 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\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
   120   hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>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_proof, isar_shrink_factor = 1]
   127 
   128 lemma bigo_alt_def: "O(f) = {h. \<exists>c. (0 < c \<and> (\<forall>x. abs (h x) <= c * abs (f x)))}"
   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 by (metis comm_semiring_1_class.normalizing_semiring_rules(7,19)
   135           mult_le_cancel_left_pos order_trans mult_pos_pos)
   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) \<oplus> 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) \<oplus> 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) \<oplus> 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 abs (g n) <= (abs (f n)) 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 auto
   168   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
   169    apply (metis mult_2 order_trans)
   170   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   171    apply (erule order_trans)
   172    apply (simp add: ring_distribs)
   173   apply (rule mult_left_mono)
   174    apply (simp add: abs_triangle_ineq)
   175   apply (simp add: order_less_le)
   176  apply (rule mult_nonneg_nonneg)
   177   apply auto
   178 apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI)
   179 apply (rule conjI)
   180  apply (rule_tac x = "c + c" in exI)
   181  apply auto
   182  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   183   apply (metis order_trans semiring_mult_2)
   184  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   185   apply (erule order_trans)
   186   apply (simp add: ring_distribs)
   187  apply (metis abs_triangle_ineq mult_le_cancel_left_pos)
   188 by (metis abs_ge_zero abs_of_pos zero_le_mult_iff)
   189 
   190 lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> B <= O(f)"
   191 by (metis bigo_plus_idemp set_plus_mono2)
   192 
   193 lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> O(g)"
   194 apply (rule equalityI)
   195 apply (rule bigo_plus_subset)
   196 apply (simp add: bigo_alt_def set_plus_def func_plus)
   197 apply clarify
   198 (* sledgehammer *)
   199 apply (rule_tac x = "max c ca" in exI)
   200 
   201 apply (rule conjI)
   202  apply (metis less_max_iff_disj)
   203 apply clarify
   204 apply (drule_tac x = "xa" in spec)+
   205 apply (subgoal_tac "0 <= f xa + g xa")
   206  apply (simp add: ring_distribs)
   207  apply (subgoal_tac "abs (a xa + b xa) <= abs (a xa) + abs (b xa)")
   208   apply (subgoal_tac "abs (a xa) + abs (b xa) <=
   209            max c ca * f xa + max c ca * g xa")
   210    apply (metis order_trans)
   211   defer 1
   212   apply (metis abs_triangle_ineq)
   213  apply (metis add_nonneg_nonneg)
   214 apply (rule add_mono)
   215  apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6))
   216 by (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
   217 
   218 lemma bigo_bounded_alt: "\<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 1: one-line proof *)
   221 by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult)
   222 
   223 lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
   224 apply (auto simp add: bigo_def)
   225 (* Version 2: structured proof *)
   226 proof -
   227   assume "\<forall>x. f x \<le> c * g x"
   228   thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
   229 qed
   230 
   231 lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)"
   232 apply (erule bigo_bounded_alt [of f 1 g])
   233 by (metis mult_1)
   234 
   235 lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)"
   236 apply (rule set_minus_imp_plus)
   237 apply (rule bigo_bounded)
   238  apply (metis add_le_cancel_left diff_add_cancel diff_self minus_apply
   239               comm_semiring_1_class.normalizing_semiring_rules(24))
   240 by (metis add_le_cancel_left diff_add_cancel func_plus le_fun_def
   241           comm_semiring_1_class.normalizing_semiring_rules(24))
   242 
   243 lemma bigo_abs: "(\<lambda>x. abs(f x)) =o O(f)"
   244 apply (unfold bigo_def)
   245 apply auto
   246 by (metis mult_1 order_refl)
   247 
   248 lemma bigo_abs2: "f =o O(\<lambda>x. abs(f x))"
   249 apply (unfold bigo_def)
   250 apply auto
   251 by (metis mult_1 order_refl)
   252 
   253 lemma bigo_abs3: "O(f) = O(\<lambda>x. abs(f x))"
   254 proof -
   255   have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
   256   have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
   257   have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2)
   258   thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
   259 qed
   260 
   261 lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
   262   apply (drule set_plus_imp_minus)
   263   apply (rule set_minus_imp_plus)
   264   apply (subst fun_diff_def)
   265 proof -
   266   assume a: "f - g : O(h)"
   267   have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs(abs (f x) - abs (g x)))"
   268     by (rule bigo_abs2)
   269   also have "... <= O(\<lambda>x. abs (f x - g x))"
   270     apply (rule bigo_elt_subset)
   271     apply (rule bigo_bounded)
   272      apply (metis abs_ge_zero)
   273     by (metis abs_triangle_ineq3)
   274   also have "... <= O(f - g)"
   275     apply (rule bigo_elt_subset)
   276     apply (subst fun_diff_def)
   277     apply (rule bigo_abs)
   278     done
   279   also have "... <= O(h)"
   280     using a by (rule bigo_elt_subset)
   281   finally show "(\<lambda>x. abs (f x) - abs (g x)) : O(h)".
   282 qed
   283 
   284 lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)"
   285 by (unfold bigo_def, auto)
   286 
   287 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)"
   288 proof -
   289   assume "f : g +o O(h)"
   290   also have "... <= O(g) \<oplus> O(h)"
   291     by (auto del: subsetI)
   292   also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
   293     by (metis bigo_abs3)
   294   also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>x. abs(h x)))"
   295     by (rule bigo_plus_eq [symmetric], auto)
   296   finally have "f : ...".
   297   then have "O(f) <= ..."
   298     by (elim bigo_elt_subset)
   299   also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
   300     by (rule bigo_plus_eq, auto)
   301   finally show ?thesis
   302     by (simp add: bigo_abs3 [symmetric])
   303 qed
   304 
   305 lemma bigo_mult [intro]: "O(f) \<otimes> O(g) <= O(f * g)"
   306 apply (rule subsetI)
   307 apply (subst bigo_def)
   308 apply (auto simp del: abs_mult mult_ac
   309             simp add: bigo_alt_def set_times_def func_times)
   310 (* sledgehammer *)
   311 apply (rule_tac x = "c * ca" in exI)
   312 apply (rule allI)
   313 apply (erule_tac x = x in allE)+
   314 apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs(f x)) * (ca * abs (g x))")
   315  apply (metis (no_types) abs_ge_zero abs_mult mult_mono')
   316 by (metis mult_assoc mult_left_commute abs_of_pos mult_left_commute abs_mult)
   317 
   318 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
   319 by (metis bigo_mult bigo_refl set_times_mono3 subset_trans)
   320 
   321 lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)"
   322 by (metis bigo_mult set_rev_mp set_times_intro)
   323 
   324 lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)"
   325 by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
   326 
   327 lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow>
   328     O(f * g) <= (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)"
   329 proof -
   330   assume a: "\<forall>x. f x ~= 0"
   331   show "O(f * g) <= f *o O(g)"
   332   proof
   333     fix h
   334     assume h: "h : O(f * g)"
   335     then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)"
   336       by auto
   337     also have "... <= O((\<lambda>x. 1 / f x) * (f * g))"
   338       by (rule bigo_mult2)
   339     also have "(\<lambda>x. 1 / f x) * (f * g) = g"
   340       apply (simp add: func_times)
   341       by (metis (lifting, no_types) a ext mult_ac(2) nonzero_divide_eq_eq)
   342     finally have "(\<lambda>x. (1\<Colon>'b) / f x) * h : O(g)".
   343     then have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) : f *o O(g)"
   344       by auto
   345     also have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) = h"
   346       apply (simp add: func_times)
   347       by (metis (lifting, no_types) a eq_divide_imp ext
   348                 comm_semiring_1_class.normalizing_semiring_rules(7))
   349     finally show "h : f *o O(g)".
   350   qed
   351 qed
   352 
   353 lemma bigo_mult6:
   354 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = (f\<Colon>'a \<Rightarrow> ('b\<Colon>{linordered_field,number_ring})) *o O(g)"
   355 by (metis bigo_mult2 bigo_mult5 order_antisym)
   356 
   357 (*proof requires relaxing relevance: 2007-01-25*)
   358 declare bigo_mult6 [simp]
   359 
   360 lemma bigo_mult7:
   361 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)"
   362 by (metis bigo_refl bigo_mult6 set_times_mono3)
   363 
   364 declare bigo_mult6 [simp del]
   365 declare bigo_mult7 [intro!]
   366 
   367 lemma bigo_mult8:
   368 "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)"
   369 by (metis bigo_mult bigo_mult7 order_antisym_conv)
   370 
   371 lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
   372 by (auto simp add: bigo_def fun_Compl_def)
   373 
   374 lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)"
   375 by (metis (no_types) bigo_elt_subset bigo_minus bigo_mult4 bigo_refl
   376           comm_semiring_1_class.normalizing_semiring_rules(11) minus_mult_left
   377           set_plus_mono_b)
   378 
   379 lemma bigo_minus3: "O(-f) = O(f)"
   380 by (metis bigo_elt_subset bigo_minus bigo_refl equalityI minus_minus)
   381 
   382 lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) \<le> O(g)"
   383 by (metis bigo_plus_idemp set_plus_mono3)
   384 
   385 lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) \<le> f +o O(g)"
   386 by (metis (no_types) bigo_minus bigo_plus_absorb_lemma1 right_minus
   387           set_plus_mono set_plus_rearrange2 set_zero_plus subsetD subset_refl
   388           subset_trans)
   389 
   390 lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)"
   391 by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
   392 
   393 lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A \<le> O(g)"
   394 by (metis bigo_plus_absorb set_plus_mono)
   395 
   396 lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)"
   397 by (metis bigo_minus minus_diff_eq set_plus_imp_minus set_minus_plus)
   398 
   399 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
   400 by (metis bigo_add_commute_imp)
   401 
   402 lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)"
   403 by (auto simp add: bigo_def mult_ac)
   404 
   405 lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<le> O(\<lambda>x. 1)"
   406 by (metis bigo_const1 bigo_elt_subset)
   407 
   408 lemma bigo_const3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)"
   409 apply (simp add: bigo_def)
   410 by (metis abs_eq_0 left_inverse order_refl)
   411 
   412 lemma bigo_const4: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)"
   413 by (metis bigo_elt_subset bigo_const3)
   414 
   415 lemma bigo_const [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
   416     O(\<lambda>x. c) = O(\<lambda>x. 1)"
   417 by (metis bigo_const2 bigo_const4 equalityI)
   418 
   419 lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)"
   420 apply (simp add: bigo_def abs_mult)
   421 by (metis le_less)
   422 
   423 lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<le> O(f)"
   424 by (rule bigo_elt_subset, rule bigo_const_mult1)
   425 
   426 lemma bigo_const_mult3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
   427 apply (simp add: bigo_def)
   428 by (metis (no_types) abs_mult mult_assoc mult_1 order_refl left_inverse)
   429 
   430 lemma bigo_const_mult4:
   431 "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) \<noteq> 0 \<Longrightarrow> O(f) \<le> O(\<lambda>x. c * f x)"
   432 by (metis bigo_elt_subset bigo_const_mult3)
   433 
   434 lemma bigo_const_mult [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
   435     O(\<lambda>x. c * f x) = O(f)"
   436 by (metis equalityI bigo_const_mult2 bigo_const_mult4)
   437 
   438 lemma bigo_const_mult5 [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
   439     (\<lambda>x. c) *o O(f) = O(f)"
   440   apply (auto del: subsetI)
   441   apply (rule order_trans)
   442   apply (rule bigo_mult2)
   443   apply (simp add: func_times)
   444   apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
   445   apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
   446   apply (rename_tac g d)
   447   apply safe
   448   apply (rule_tac [2] ext)
   449    prefer 2
   450    apply simp
   451   apply (simp add: mult_assoc [symmetric] abs_mult)
   452   (* couldn't get this proof without the step above *)
   453 proof -
   454   fix g :: "'b \<Rightarrow> 'a" and d :: 'a
   455   assume A1: "c \<noteq> (0\<Colon>'a)"
   456   assume A2: "\<forall>x\<Colon>'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
   457   have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse)
   458   have F2: "(0\<Colon>'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff)
   459   have "(0\<Colon>'a) < \<bar>c\<bar> \<longrightarrow> (0\<Colon>'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive)
   460   hence "(0\<Colon>'a) < \<bar>inverse c\<bar>" using F2 by metis
   461   hence F3: "(0\<Colon>'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less)
   462   have "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar>"
   463     using A2 by metis
   464   hence F4: "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0\<Colon>'a) \<le> \<bar>inverse c\<bar>"
   465     using F3 by metis
   466   hence "\<exists>(v\<Colon>'a) (u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^isub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^isub>7 (u * v))\<bar>)"
   467     by (metis comm_mult_left_mono)
   468   thus "\<exists>ca\<Colon>'a. \<forall>x\<Colon>'b. \<bar>inverse c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>"
   469     using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono)
   470 qed
   471 
   472 lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)"
   473   apply (auto intro!: subsetI
   474     simp add: bigo_def elt_set_times_def func_times
   475     simp del: abs_mult mult_ac)
   476 (* sledgehammer *)
   477   apply (rule_tac x = "ca * (abs c)" in exI)
   478   apply (rule allI)
   479   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
   480   apply (erule ssubst)
   481   apply (subst abs_mult)
   482   apply (rule mult_left_mono)
   483   apply (erule spec)
   484   apply simp
   485   apply (simp add: mult_ac)
   486 done
   487 
   488 lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
   489 by (metis bigo_const_mult1 bigo_elt_subset order_less_le psubsetD)
   490 
   491 lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))"
   492 by (unfold bigo_def, auto)
   493 
   494 lemma bigo_compose2:
   495 "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))"
   496 apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def func_plus)
   497 by (erule bigo_compose1)
   498 
   499 subsection {* Setsum *}
   500 
   501 lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
   502     \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) <= c * (h x y) \<Longrightarrow>
   503       (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
   504 apply (auto simp add: bigo_def)
   505 apply (rule_tac x = "abs c" in exI)
   506 apply (subst abs_of_nonneg) back back
   507  apply (rule setsum_nonneg)
   508  apply force
   509 apply (subst setsum_right_distrib)
   510 apply (rule allI)
   511 apply (rule order_trans)
   512  apply (rule setsum_abs)
   513 apply (rule setsum_mono)
   514 by (metis abs_ge_self abs_mult_pos order_trans)
   515 
   516 lemma bigo_setsum1: "\<forall>x y. 0 <= h x y \<Longrightarrow>
   517     \<exists>c. \<forall>x y. abs (f x y) <= c * (h x y) \<Longrightarrow>
   518       (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
   519 by (metis (no_types) bigo_setsum_main)
   520 
   521 lemma bigo_setsum2: "\<forall>y. 0 <= h y \<Longrightarrow>
   522     \<exists>c. \<forall>y. abs (f y) <= c * (h y) \<Longrightarrow>
   523       (\<lambda>x. SUM y : A x. f y) =o O(\<lambda>x. SUM y : A x. h y)"
   524 apply (rule bigo_setsum1)
   525 by metis+
   526 
   527 lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
   528     (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
   529       O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))"
   530 apply (rule bigo_setsum1)
   531  apply (rule allI)+
   532  apply (rule abs_ge_zero)
   533 apply (unfold bigo_def)
   534 apply (auto simp add: abs_mult)
   535 by (metis abs_ge_zero mult_left_commute mult_left_mono)
   536 
   537 lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
   538     (\<lambda>x. SUM y : A x. l x y * f(k x y)) =o
   539       (\<lambda>x. SUM y : A x. l x y * g(k x y)) +o
   540         O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))"
   541 apply (rule set_minus_imp_plus)
   542 apply (subst fun_diff_def)
   543 apply (subst setsum_subtractf [symmetric])
   544 apply (subst right_diff_distrib [symmetric])
   545 apply (rule bigo_setsum3)
   546 by (metis (lifting, no_types) fun_diff_def set_plus_imp_minus ext)
   547 
   548 lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
   549     \<forall>x. 0 <= h x \<Longrightarrow>
   550       (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
   551         O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
   552 apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) =
   553       (\<lambda>x. SUM y : A x. abs((l x y) * h(k x y)))")
   554  apply (erule ssubst)
   555  apply (erule bigo_setsum3)
   556 apply (rule ext)
   557 apply (rule setsum_cong2)
   558 by (metis abs_of_nonneg zero_le_mult_iff)
   559 
   560 lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
   561     \<forall>x. 0 <= h x \<Longrightarrow>
   562       (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
   563         (\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o
   564           O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
   565   apply (rule set_minus_imp_plus)
   566   apply (subst fun_diff_def)
   567   apply (subst setsum_subtractf [symmetric])
   568   apply (subst right_diff_distrib [symmetric])
   569   apply (rule bigo_setsum5)
   570   apply (subst fun_diff_def [symmetric])
   571   apply (drule set_plus_imp_minus)
   572   apply auto
   573 done
   574 
   575 subsection {* Misc useful stuff *}
   576 
   577 lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
   578   A \<oplus> B <= O(f)"
   579   apply (subst bigo_plus_idemp [symmetric])
   580   apply (rule set_plus_mono2)
   581   apply assumption+
   582 done
   583 
   584 lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
   585   apply (subst bigo_plus_idemp [symmetric])
   586   apply (rule set_plus_intro)
   587   apply assumption+
   588 done
   589 
   590 lemma bigo_useful_const_mult: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
   591     (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
   592   apply (rule subsetD)
   593   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)")
   594   apply assumption
   595   apply (rule bigo_const_mult6)
   596   apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
   597   apply (erule ssubst)
   598   apply (erule set_times_intro2)
   599   apply (simp add: func_times)
   600 done
   601 
   602 lemma bigo_fix: "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow>
   603     f =o O(h)"
   604 apply (simp add: bigo_alt_def)
   605 by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc)
   606 
   607 lemma bigo_fix2:
   608     "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
   609        f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
   610   apply (rule set_minus_imp_plus)
   611   apply (rule bigo_fix)
   612   apply (subst fun_diff_def)
   613   apply (subst fun_diff_def [symmetric])
   614   apply (rule set_plus_imp_minus)
   615   apply simp
   616   apply (simp add: fun_diff_def)
   617 done
   618 
   619 subsection {* Less than or equal to *}
   620 
   621 definition lesso :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
   622   "f <o g == (\<lambda>x. max (f x - g x) 0)"
   623 
   624 lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= abs (f x) \<Longrightarrow>
   625     g =o O(h)"
   626   apply (unfold bigo_def)
   627   apply clarsimp
   628 apply (blast intro: order_trans)
   629 done
   630 
   631 lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= f x \<Longrightarrow>
   632       g =o O(h)"
   633   apply (erule bigo_lesseq1)
   634 apply (blast intro: abs_ge_self order_trans)
   635 done
   636 
   637 lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow>
   638       g =o O(h)"
   639   apply (erule bigo_lesseq2)
   640   apply (rule allI)
   641   apply (subst abs_of_nonneg)
   642   apply (erule spec)+
   643 done
   644 
   645 lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
   646     \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= abs (f x) \<Longrightarrow>
   647       g =o O(h)"
   648   apply (erule bigo_lesseq1)
   649   apply (rule allI)
   650   apply (subst abs_of_nonneg)
   651   apply (erule spec)+
   652 done
   653 
   654 lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)"
   655 apply (unfold lesso_def)
   656 apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
   657  apply (metis bigo_zero)
   658 by (metis (lifting, no_types) func_zero le_fun_def le_iff_diff_le_0
   659       min_max.sup_absorb2 order_eq_iff)
   660 
   661 lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
   662     \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow>
   663       k <o g =o O(h)"
   664   apply (unfold lesso_def)
   665   apply (rule bigo_lesseq4)
   666   apply (erule set_plus_imp_minus)
   667   apply (rule allI)
   668   apply (rule le_maxI2)
   669   apply (rule allI)
   670   apply (subst fun_diff_def)
   671 apply (erule thin_rl)
   672 (* sledgehammer *)
   673 apply (case_tac "0 <= k x - g x")
   674  apply (metis (lifting) abs_le_D1 linorder_linear min_diff_distrib_left
   675           min_max.inf_absorb1 min_max.inf_absorb2 min_max.sup_absorb1)
   676 by (metis abs_ge_zero le_cases min_max.sup_absorb2)
   677 
   678 lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
   679     \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow>
   680       f <o k =o O(h)"
   681 apply (unfold lesso_def)
   682 apply (rule bigo_lesseq4)
   683   apply (erule set_plus_imp_minus)
   684  apply (rule allI)
   685  apply (rule le_maxI2)
   686 apply (rule allI)
   687 apply (subst fun_diff_def)
   688 apply (erule thin_rl)
   689 (* sledgehammer *)
   690 apply (case_tac "0 <= f x - k x")
   691  apply simp
   692  apply (subst abs_of_nonneg)
   693   apply (drule_tac x = x in spec) back
   694   apply (metis diff_less_0_iff_less linorder_not_le not_leE xt1(12) xt1(6))
   695  apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
   696 by (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
   697 
   698 lemma bigo_lesso4:
   699   "f <o g =o O(k\<Colon>'a=>'b\<Colon>{linordered_field,number_ring}) \<Longrightarrow>
   700    g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
   701 apply (unfold lesso_def)
   702 apply (drule set_plus_imp_minus)
   703 apply (drule bigo_abs5) back
   704 apply (simp add: fun_diff_def)
   705 apply (drule bigo_useful_add, assumption)
   706 apply (erule bigo_lesseq2) back
   707 apply (rule allI)
   708 by (auto simp add: func_plus fun_diff_def algebra_simps
   709     split: split_max abs_split)
   710 
   711 lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs (h x)"
   712 apply (simp only: lesso_def bigo_alt_def)
   713 apply clarsimp
   714 by (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)
   715 
   716 end