src/HOL/Metis_Examples/Big_O.thy
 changeset 45575 3a865fc42bbf parent 45532 74b17a0881b3 child 45705 a25ff4283352
1.1 --- a/src/HOL/Metis_Examples/Big_O.thy	Fri Nov 18 11:47:12 2011 +0100
1.2 +++ b/src/HOL/Metis_Examples/Big_O.thy	Fri Nov 18 11:47:12 2011 +0100
1.3 @@ -10,36 +10,31 @@
1.4  theory Big_O
1.5  imports
1.6    "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
1.7 -  Main
1.8    "~~/src/HOL/Library/Function_Algebras"
1.9    "~~/src/HOL/Library/Set_Algebras"
1.10  begin
1.12 -declare [[metis_new_skolemizer]]
1.13 -
1.14  subsection {* Definitions *}
1.16 -definition bigo :: "('a => 'b::{linordered_idom,number_ring}) => ('a => 'b) set"    ("(1O'(_'))") where
1.17 -  "O(f::('a => 'b)) ==   {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
1.18 +definition bigo :: "('a => 'b\<Colon>{linordered_idom,number_ring}) => ('a => 'b) set" ("(1O'(_'))") where
1.19 +  "O(f\<Colon>('a => 'b)) == {h. \<exists>c. \<forall>x. abs (h x) <= c * abs (f x)}"
1.21 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_pos_const" ]]
1.22 -lemma bigo_pos_const: "(EX (c::'a::linordered_idom).
1.23 -    ALL x. (abs (h x)) <= (c * (abs (f x))))
1.24 -      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
1.25 -  apply auto
1.26 -  apply (case_tac "c = 0", simp)
1.27 -  apply (rule_tac x = "1" in exI, simp)
1.28 -  apply (rule_tac x = "abs c" in exI, auto)
1.29 -  apply (metis abs_ge_zero abs_of_nonneg Orderings.xt1(6) abs_mult)
1.30 -  done
1.31 +lemma bigo_pos_const:
1.32 +  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
1.33 +    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
1.34 +      = (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))"
1.35 +by (metis (hide_lams, no_types) abs_ge_zero
1.36 +      comm_semiring_1_class.normalizing_semiring_rules(7) mult.comm_neutral
1.37 +      mult_nonpos_nonneg not_leE order_trans zero_less_one)
1.39  (*** Now various verions with an increasing shrink factor ***)
1.41  sledgehammer_params [isar_proof, isar_shrink_factor = 1]
1.43 -lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
1.44 -    ALL x. (abs (h x)) <= (c * (abs (f x))))
1.45 -      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
1.46 +lemma
1.47 +  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
1.48 +    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
1.49 +      = (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))"
1.50    apply auto
1.51    apply (case_tac "c = 0", simp)
1.52    apply (rule_tac x = "1" in exI, simp)
1.53 @@ -67,9 +62,10 @@
1.55  sledgehammer_params [isar_proof, isar_shrink_factor = 2]
1.57 -lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
1.58 -    ALL x. (abs (h x)) <= (c * (abs (f x))))
1.59 -      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
1.60 +lemma
1.61 +  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
1.62 +    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
1.63 +      = (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))"
1.64    apply auto
1.65    apply (case_tac "c = 0", simp)
1.66    apply (rule_tac x = "1" in exI, simp)
1.67 @@ -89,9 +85,10 @@
1.69  sledgehammer_params [isar_proof, isar_shrink_factor = 3]
1.71 -lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
1.72 -    ALL x. (abs (h x)) <= (c * (abs (f x))))
1.73 -      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
1.74 +lemma
1.75 +  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
1.76 +    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
1.77 +      = (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))"
1.78    apply auto
1.79    apply (case_tac "c = 0", simp)
1.80    apply (rule_tac x = "1" in exI, simp)
1.81 @@ -108,9 +105,10 @@
1.83  sledgehammer_params [isar_proof, isar_shrink_factor = 4]
1.85 -lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
1.86 -    ALL x. (abs (h x)) <= (c * (abs (f x))))
1.87 -      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
1.88 +lemma
1.89 +  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
1.90 +    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
1.91 +      = (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))"
1.92    apply auto
1.93    apply (case_tac "c = 0", simp)
1.94    apply (rule_tac x = "1" in exI, simp)
1.95 @@ -127,142 +125,109 @@
1.97  sledgehammer_params [isar_proof, isar_shrink_factor = 1]
1.99 -lemma bigo_alt_def: "O(f) =
1.100 -    {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
1.101 +lemma bigo_alt_def: "O(f) = {h. \<exists>c. (0 < c & (\<forall>x. abs (h x) <= c * abs (f x)))}"
1.102  by (auto simp add: bigo_def bigo_pos_const)
1.104 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_elt_subset" ]]
1.105 -lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
1.106 -  apply (auto simp add: bigo_alt_def)
1.107 -  apply (rule_tac x = "ca * c" in exI)
1.108 -  apply (rule conjI)
1.109 -  apply (rule mult_pos_pos)
1.110 +lemma bigo_elt_subset [intro]: "f : O(g) \<Longrightarrow> O(f) <= O(g)"
1.111 +apply (auto simp add: bigo_alt_def)
1.112 +apply (rule_tac x = "ca * c" in exI)
1.113 +apply (rule conjI)
1.114 + apply (rule mult_pos_pos)
1.115    apply (assumption)+
1.116 -(*sledgehammer*)
1.117 -  apply (rule allI)
1.118 -  apply (drule_tac x = "xa" in spec)+
1.119 -  apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
1.120 -  apply (erule order_trans)
1.121 -  apply (simp add: mult_ac)
1.122 -  apply (rule mult_left_mono, assumption)
1.123 -  apply (rule order_less_imp_le, assumption)
1.124 -done
1.125 +(* sledgehammer *)
1.126 +apply (rule allI)
1.127 +apply (drule_tac x = "xa" in spec)+
1.128 +apply (subgoal_tac "ca * abs (f xa) <= ca * (c * abs (g xa))")
1.129 + apply (metis comm_semiring_1_class.normalizing_semiring_rules(19)
1.130 +          comm_semiring_1_class.normalizing_semiring_rules(7) order_trans)
1.131 +by (metis mult_le_cancel_left_pos)
1.134 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_refl" ]]
1.135  lemma bigo_refl [intro]: "f : O(f)"
1.136  apply (auto simp add: bigo_def)
1.137  by (metis mult_1 order_refl)
1.139 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_zero" ]]
1.140  lemma bigo_zero: "0 : O(g)"
1.141  apply (auto simp add: bigo_def func_zero)
1.142  by (metis mult_zero_left order_refl)
1.144 -lemma bigo_zero2: "O(%x.0) = {%x.0}"
1.145 -  by (auto simp add: bigo_def)
1.146 +lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
1.147 +by (auto simp add: bigo_def)
1.149  lemma bigo_plus_self_subset [intro]:
1.150    "O(f) \<oplus> O(f) <= O(f)"
1.151 -  apply (auto simp add: bigo_alt_def set_plus_def)
1.152 -  apply (rule_tac x = "c + ca" in exI)
1.153 -  apply auto
1.154 -  apply (simp add: ring_distribs func_plus)
1.155 -  apply (blast intro:order_trans abs_triangle_ineq add_mono elim:)
1.156 -done
1.157 +apply (auto simp add: bigo_alt_def set_plus_def)
1.158 +apply (rule_tac x = "c + ca" in exI)
1.159 +apply auto
1.160 +apply (simp add: ring_distribs func_plus)
1.161 +by (metis order_trans abs_triangle_ineq add_mono)
1.163  lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
1.164 -  apply (rule equalityI)
1.165 -  apply (rule bigo_plus_self_subset)
1.166 -  apply (rule set_zero_plus2)
1.167 -  apply (rule bigo_zero)
1.168 -done
1.169 +by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2)
1.171  lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
1.172 -  apply (rule subsetI)
1.173 -  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
1.174 -  apply (subst bigo_pos_const [symmetric])+
1.175 -  apply (rule_tac x =
1.176 -    "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
1.177 -  apply (rule conjI)
1.178 -  apply (rule_tac x = "c + c" in exI)
1.179 -  apply (clarsimp)
1.180 -  apply (auto)
1.181 +apply (rule subsetI)
1.182 +apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
1.183 +apply (subst bigo_pos_const [symmetric])+
1.184 +apply (rule_tac x = "\<lambda>n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
1.185 +apply (rule conjI)
1.186 + apply (rule_tac x = "c + c" in exI)
1.187 + apply clarsimp
1.188 + apply auto
1.189    apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
1.190 -  apply (erule_tac x = xa in allE)
1.191 -  apply (erule order_trans)
1.192 -  apply (simp)
1.193 +   apply (metis mult_2 order_trans)
1.194    apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
1.195 -  apply (erule order_trans)
1.196 -  apply (simp add: ring_distribs)
1.197 +   apply (erule order_trans)
1.198 +   apply (simp add: ring_distribs)
1.199    apply (rule mult_left_mono)
1.200 -  apply (simp add: abs_triangle_ineq)
1.201 +   apply (simp add: abs_triangle_ineq)
1.203 -  apply (rule mult_nonneg_nonneg)
1.204 + apply (rule mult_nonneg_nonneg)
1.205    apply auto
1.206 -  apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0"
1.207 -     in exI)
1.208 -  apply (rule conjI)
1.209 -  apply (rule_tac x = "c + c" in exI)
1.210 -  apply auto
1.211 -  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
1.212 -  apply (erule_tac x = xa in allE)
1.213 -  apply (erule order_trans)
1.214 -  apply (simp)
1.215 -  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
1.216 +apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI)
1.217 +apply (rule conjI)
1.218 + apply (rule_tac x = "c + c" in exI)
1.219 + apply auto
1.220 + apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
1.221 +  apply (metis order_trans semiring_mult_2)
1.222 + apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
1.223    apply (erule order_trans)
1.225 -  apply (rule mult_left_mono)
1.226 -  apply (rule abs_triangle_ineq)
1.227 -  apply (simp add: order_less_le)
1.228 -apply (metis abs_not_less_zero even_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
1.229 -done
1.230 + apply (metis abs_triangle_ineq mult_le_cancel_left_pos)
1.231 +by (metis abs_ge_zero abs_of_pos zero_le_mult_iff)
1.233 -lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
1.234 -  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
1.235 -  apply (erule order_trans)
1.236 -  apply simp
1.237 -  apply (auto del: subsetI simp del: bigo_plus_idemp)
1.238 -done
1.239 +lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> B <= O(f)"
1.240 +by (metis bigo_plus_idemp set_plus_mono2)
1.242 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq" ]]
1.243 -lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==>
1.244 -  O(f + g) = O(f) \<oplus> O(g)"
1.245 -  apply (rule equalityI)
1.246 -  apply (rule bigo_plus_subset)
1.247 -  apply (simp add: bigo_alt_def set_plus_def func_plus)
1.248 -  apply clarify
1.249 -(*sledgehammer*)
1.250 -  apply (rule_tac x = "max c ca" in exI)
1.251 -  apply (rule conjI)
1.252 -   apply (metis Orderings.less_max_iff_disj)
1.253 -  apply clarify
1.254 -  apply (drule_tac x = "xa" in spec)+
1.255 -  apply (subgoal_tac "0 <= f xa + g xa")
1.256 -  apply (simp add: ring_distribs)
1.257 -  apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
1.258 -  apply (subgoal_tac "abs(a xa) + abs(b xa) <=
1.259 -      max c ca * f xa + max c ca * g xa")
1.260 -  apply (blast intro: order_trans)
1.261 +lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> O(g)"
1.262 +apply (rule equalityI)
1.263 +apply (rule bigo_plus_subset)
1.264 +apply (simp add: bigo_alt_def set_plus_def func_plus)
1.265 +apply clarify
1.266 +(* sledgehammer *)
1.267 +apply (rule_tac x = "max c ca" in exI)
1.268 +apply (rule conjI)
1.269 + apply (metis less_max_iff_disj)
1.270 +apply clarify
1.271 +apply (drule_tac x = "xa" in spec)+
1.272 +apply (subgoal_tac "0 <= f xa + g xa")
1.273 + apply (simp add: ring_distribs)
1.274 + apply (subgoal_tac "abs (a xa + b xa) <= abs (a xa) + abs (b xa)")
1.275 +  apply (subgoal_tac "abs (a xa) + abs (b xa) <=
1.276 +           max c ca * f xa + max c ca * g xa")
1.277 +   apply (metis order_trans)
1.278    defer 1
1.279 -  apply (rule abs_triangle_ineq)
1.282 -using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]]
1.283 -  apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6))
1.284 -  apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
1.285 -done
1.286 +  apply (metis abs_triangle_ineq)
1.289 + apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6))
1.290 +by (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
1.292 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt" ]]
1.293 -lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
1.294 -    f : O(g)"
1.295 -  apply (auto simp add: bigo_def)
1.296 +lemma bigo_bounded_alt: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
1.297 +apply (auto simp add: bigo_def)
1.298  (* Version 1: one-line proof *)
1.299 -  apply (metis abs_le_D1 linorder_class.not_less  order_less_le  Orderings.xt1(12)  abs_mult)
1.300 -  done
1.301 +by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult)
1.303 -lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
1.304 -    f : O(g)"
1.305 +lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
1.306  apply (auto simp add: bigo_def)
1.307  (* Version 2: structured proof *)
1.308  proof -
1.309 @@ -270,32 +235,11 @@
1.310    thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
1.311  qed
1.313 -text{*So here is the easier (and more natural) problem using transitivity*}
1.314 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
1.315 -lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
1.316 -apply (auto simp add: bigo_def)
1.317 -(* Version 1: one-line proof *)
1.318 -by (metis abs_ge_self abs_mult order_trans)
1.319 +lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)"
1.320 +apply (erule bigo_bounded_alt [of f 1 g])
1.321 +by (metis mult_1)
1.323 -text{*So here is the easier (and more natural) problem using transitivity*}
1.324 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
1.325 -lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
1.326 -  apply (auto simp add: bigo_def)
1.327 -(* Version 2: structured proof *)
1.328 -proof -
1.329 -  assume "\<forall>x. f x \<le> c * g x"
1.330 -  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
1.331 -qed
1.333 -lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
1.334 -    f : O(g)"
1.335 -  apply (erule bigo_bounded_alt [of f 1 g])
1.336 -  apply simp
1.337 -done
1.339 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded2" ]]
1.340 -lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
1.341 -    f : lb +o O(g)"
1.342 +lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)"
1.343  apply (rule set_minus_imp_plus)
1.344  apply (rule bigo_bounded)
1.345   apply (auto simp add: diff_minus fun_Compl_def func_plus)
1.346 @@ -308,19 +252,17 @@
1.347    thus "(0\<Colon>'b) \<le> f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le)
1.348  qed
1.350 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]]
1.351 -lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
1.352 +lemma bigo_abs: "(\<lambda>x. abs(f x)) =o O(f)"
1.353  apply (unfold bigo_def)
1.354  apply auto
1.355  by (metis mult_1 order_refl)
1.357 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]]
1.358 -lemma bigo_abs2: "f =o O(%x. abs(f x))"
1.359 +lemma bigo_abs2: "f =o O(\<lambda>x. abs(f x))"
1.360  apply (unfold bigo_def)
1.361  apply auto
1.362  by (metis mult_1 order_refl)
1.364 -lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
1.365 +lemma bigo_abs3: "O(f) = O(\<lambda>x. abs(f x))"
1.366  proof -
1.367    have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
1.368    have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
1.369 @@ -328,16 +270,15 @@
1.370    thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
1.371  qed
1.373 -lemma bigo_abs4: "f =o g +o O(h) ==>
1.374 -    (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
1.375 +lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
1.376    apply (drule set_plus_imp_minus)
1.377    apply (rule set_minus_imp_plus)
1.378    apply (subst fun_diff_def)
1.379  proof -
1.380    assume a: "f - g : O(h)"
1.381 -  have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
1.382 +  have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs(abs (f x) - abs (g x)))"
1.383      by (rule bigo_abs2)
1.384 -  also have "... <= O(%x. abs (f x - g x))"
1.385 +  also have "... <= O(\<lambda>x. abs (f x - g x))"
1.386      apply (rule bigo_elt_subset)
1.387      apply (rule bigo_bounded)
1.388      apply force
1.389 @@ -351,45 +292,43 @@
1.390      done
1.391    also have "... <= O(h)"
1.392      using a by (rule bigo_elt_subset)
1.393 -  finally show "(%x. abs (f x) - abs (g x)) : O(h)".
1.394 +  finally show "(\<lambda>x. abs (f x) - abs (g x)) : O(h)".
1.395  qed
1.397 -lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)"
1.398 +lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)"
1.399  by (unfold bigo_def, auto)
1.401 -lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
1.402 +lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)"
1.403  proof -
1.404    assume "f : g +o O(h)"
1.405    also have "... <= O(g) \<oplus> O(h)"
1.406      by (auto del: subsetI)
1.407 -  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
1.408 +  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
1.409      apply (subst bigo_abs3 [symmetric])+
1.410      apply (rule refl)
1.411      done
1.412 -  also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
1.413 +  also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>x. abs(h x)))"
1.414      by (rule bigo_plus_eq [symmetric], auto)
1.415    finally have "f : ...".
1.416    then have "O(f) <= ..."
1.417      by (elim bigo_elt_subset)
1.418 -  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
1.419 +  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
1.420      by (rule bigo_plus_eq, auto)
1.421    finally show ?thesis
1.422      by (simp add: bigo_abs3 [symmetric])
1.423  qed
1.425 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]]
1.426  lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
1.427    apply (rule subsetI)
1.428    apply (subst bigo_def)
1.429    apply (auto simp del: abs_mult mult_ac
1.430                simp add: bigo_alt_def set_times_def func_times)
1.431 -(*sledgehammer*)
1.432 +(* sledgehammer *)
1.433    apply (rule_tac x = "c * ca" in exI)
1.434    apply(rule allI)
1.435    apply(erule_tac x = x in allE)+
1.436    apply(subgoal_tac "c * ca * abs(f x * g x) =
1.437        (c * abs(f x)) * (ca * abs(g x))")
1.438 -using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]]
1.439  prefer 2
1.440  apply (metis mult_assoc mult_left_commute
1.441    abs_of_pos mult_left_commute
1.442 @@ -400,14 +339,12 @@
1.443     abs_mult has just been done *)
1.444  by (metis abs_ge_zero mult_mono')
1.446 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult2" ]]
1.447  lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
1.448    apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
1.449 -(*sledgehammer*)
1.450 +(* sledgehammer *)
1.451    apply (rule_tac x = c in exI)
1.452    apply clarify
1.453    apply (drule_tac x = x in spec)
1.454 -using [[ sledgehammer_problem_prefix = "BigO__bigo_mult2_simpler" ]]
1.455  (*sledgehammer [no luck]*)
1.456    apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
1.458 @@ -415,36 +352,33 @@
1.459    apply (rule abs_ge_zero)
1.460  done
1.462 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]]
1.463 -lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
1.464 +lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)"
1.465  by (metis bigo_mult set_rev_mp set_times_intro)
1.467 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]]
1.468 -lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
1.469 +lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)"
1.470  by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
1.473 -lemma bigo_mult5: "ALL x. f x ~= 0 ==>
1.474 -    O(f * g) <= (f::'a => ('b::{linordered_field,number_ring})) *o O(g)"
1.475 +lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow>
1.476 +    O(f * g) <= (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)"
1.477  proof -
1.478 -  assume a: "ALL x. f x ~= 0"
1.479 +  assume a: "\<forall>x. f x ~= 0"
1.480    show "O(f * g) <= f *o O(g)"
1.481    proof
1.482      fix h
1.483      assume h: "h : O(f * g)"
1.484 -    then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
1.485 +    then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)"
1.486        by auto
1.487 -    also have "... <= O((%x. 1 / f x) * (f * g))"
1.488 +    also have "... <= O((\<lambda>x. 1 / f x) * (f * g))"
1.489        by (rule bigo_mult2)
1.490 -    also have "(%x. 1 / f x) * (f * g) = g"
1.491 +    also have "(\<lambda>x. 1 / f x) * (f * g) = g"
1.493        apply (rule ext)
1.494        apply (simp add: a h nonzero_divide_eq_eq mult_ac)
1.495        done
1.496 -    finally have "(%x. (1::'b) / f x) * h : O(g)".
1.497 -    then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
1.498 +    finally have "(\<lambda>x. (1\<Colon>'b) / f x) * h : O(g)".
1.499 +    then have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) : f *o O(g)"
1.500        by auto
1.501 -    also have "f * ((%x. (1::'b) / f x) * h) = h"
1.502 +    also have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) = h"
1.504        apply (rule ext)
1.505        apply (simp add: a h nonzero_divide_eq_eq mult_ac)
1.506 @@ -453,34 +387,32 @@
1.507    qed
1.508  qed
1.510 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]]
1.511 -lemma bigo_mult6: "ALL x. f x ~= 0 ==>
1.512 -    O(f * g) = (f::'a => ('b::{linordered_field,number_ring})) *o O(g)"
1.513 +lemma bigo_mult6: "\<forall>x. f x ~= 0 \<Longrightarrow>
1.514 +    O(f * g) = (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)"
1.515  by (metis bigo_mult2 bigo_mult5 order_antisym)
1.517  (*proof requires relaxing relevance: 2007-01-25*)
1.518 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult7" ]]
1.519    declare bigo_mult6 [simp]
1.520 -lemma bigo_mult7: "ALL x. f x ~= 0 ==>
1.521 -    O(f * g) <= O(f::'a => ('b::{linordered_field,number_ring})) \<otimes> O(g)"
1.522 -(*sledgehammer*)
1.523 +lemma bigo_mult7: "\<forall>x. f x ~= 0 \<Longrightarrow>
1.524 +    O(f * g) <= O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)"
1.525 +(* sledgehammer *)
1.526    apply (subst bigo_mult6)
1.527    apply assumption
1.528    apply (rule set_times_mono3)
1.529    apply (rule bigo_refl)
1.530  done
1.531 -  declare bigo_mult6 [simp del]
1.533 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult8" ]]
1.534 -  declare bigo_mult7[intro!]
1.535 -lemma bigo_mult8: "ALL x. f x ~= 0 ==>
1.536 -    O(f * g) = O(f::'a => ('b::{linordered_field,number_ring})) \<otimes> O(g)"
1.537 +declare bigo_mult6 [simp del]
1.538 +declare bigo_mult7 [intro!]
1.540 +lemma bigo_mult8: "\<forall>x. f x ~= 0 \<Longrightarrow>
1.541 +    O(f * g) = O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)"
1.542  by (metis bigo_mult bigo_mult7 order_antisym_conv)
1.544 -lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
1.545 +lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
1.546    by (auto simp add: bigo_def fun_Compl_def)
1.548 -lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
1.549 +lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)"
1.550    apply (rule set_minus_imp_plus)
1.551    apply (drule set_plus_imp_minus)
1.552    apply (drule bigo_minus)
1.553 @@ -490,7 +422,7 @@
1.554  lemma bigo_minus3: "O(-f) = O(f)"
1.555    by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel)
1.557 -lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
1.558 +lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) <= O(g)"
1.559  proof -
1.560    assume a: "f : O(g)"
1.561    show "f +o O(g) <= O(g)"
1.562 @@ -508,7 +440,7 @@
1.563    qed
1.564  qed
1.566 -lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
1.567 +lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) <= f +o O(g)"
1.568  proof -
1.569    assume a: "f : O(g)"
1.570    show "O(g) <= f +o O(g)"
1.571 @@ -522,23 +454,22 @@
1.572    qed
1.573  qed
1.575 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]]
1.576 -lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
1.577 +lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)"
1.578  by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
1.580 -lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
1.581 +lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A <= O(g)"
1.582    apply (subgoal_tac "f +o A <= f +o O(g)")
1.583    apply force+
1.584  done
1.586 -lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
1.587 +lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)"
1.588    apply (subst set_minus_plus [symmetric])
1.589    apply (subgoal_tac "g - f = - (f - g)")
1.590    apply (erule ssubst)
1.591    apply (rule bigo_minus)
1.592    apply (subst set_minus_plus)
1.593    apply assumption
1.596  done
1.598  lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
1.599 @@ -546,67 +477,60 @@
1.601  done
1.603 -lemma bigo_const1: "(%x. c) : O(%x. 1)"
1.604 +lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)"
1.605  by (auto simp add: bigo_def mult_ac)
1.607 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]]
1.608 -lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
1.609 +lemma (*bigo_const2 [intro]:*) "O(\<lambda>x. c) <= O(\<lambda>x. 1)"
1.610  by (metis bigo_const1 bigo_elt_subset)
1.612 -lemma bigo_const2 [intro]: "O(%x. c::'b::{linordered_idom,number_ring}) <= O(%x. 1)"
1.613 -(* "thus" had to be replaced by "show" with an explicit reference to "F1" *)
1.614 +lemma bigo_const2 [intro]: "O(\<lambda>x. c\<Colon>'b\<Colon>{linordered_idom,number_ring}) <= O(\<lambda>x. 1)"
1.615  proof -
1.616 -  have F1: "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
1.617 -  show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset)
1.618 +  have "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
1.619 +  thus "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis bigo_elt_subset)
1.620  qed
1.622 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]]
1.623 -lemma bigo_const3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> (%x. 1) : O(%x. c)"
1.624 +lemma bigo_const3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)"
1.626  by (metis abs_eq_0 left_inverse order_refl)
1.628 -lemma bigo_const4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> O(%x. 1) <= O(%x. c)"
1.629 +lemma bigo_const4: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)"
1.630  by (rule bigo_elt_subset, rule bigo_const3, assumption)
1.632 -lemma bigo_const [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
1.633 -    O(%x. c) = O(%x. 1)"
1.634 +lemma bigo_const [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
1.635 +    O(\<lambda>x. c) = O(\<lambda>x. 1)"
1.636  by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
1.638 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]]
1.639 -lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
1.640 +lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)"
1.641    apply (simp add: bigo_def abs_mult)
1.642  by (metis le_less)
1.644 -lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
1.645 +lemma bigo_const_mult2: "O(\<lambda>x. c * f x) <= O(f)"
1.646  by (rule bigo_elt_subset, rule bigo_const_mult1)
1.648 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]]
1.649 -lemma bigo_const_mult3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> f : O(%x. c * f x)"
1.650 -  apply (simp add: bigo_def)
1.651 -(*sledgehammer [no luck]*)
1.652 -  apply (rule_tac x = "abs(inverse c)" in exI)
1.653 -  apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
1.654 +lemma bigo_const_mult3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
1.656 +(* sledgehammer *)
1.657 +apply (rule_tac x = "abs(inverse c)" in exI)
1.658 +apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
1.659  apply (subst left_inverse)
1.660 -apply (auto )
1.661 -done
1.662 +by auto
1.664 -lemma bigo_const_mult4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
1.665 -    O(f) <= O(%x. c * f x)"
1.666 +lemma bigo_const_mult4: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
1.667 +    O(f) <= O(\<lambda>x. c * f x)"
1.668  by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
1.670 -lemma bigo_const_mult [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
1.671 -    O(%x. c * f x) = O(f)"
1.672 +lemma bigo_const_mult [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
1.673 +    O(\<lambda>x. c * f x) = O(f)"
1.674  by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
1.676 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]]
1.677 -lemma bigo_const_mult5 [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
1.678 -    (%x. c) *o O(f) = O(f)"
1.679 +lemma bigo_const_mult5 [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
1.680 +    (\<lambda>x. c) *o O(f) = O(f)"
1.681    apply (auto del: subsetI)
1.682    apply (rule order_trans)
1.683    apply (rule bigo_mult2)
1.685    apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
1.686 -  apply (rule_tac x = "%y. inverse c * x y" in exI)
1.687 +  apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
1.688    apply (rename_tac g d)
1.689    apply safe
1.690    apply (rule_tac [2] ext)
1.691 @@ -633,13 +557,11 @@
1.692      using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono)
1.693  qed
1.696 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]]
1.697 -lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
1.698 +lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)"
1.699    apply (auto intro!: subsetI
1.700      simp add: bigo_def elt_set_times_def func_times
1.701      simp del: abs_mult mult_ac)
1.702 -(*sledgehammer*)
1.703 +(* sledgehammer *)
1.704    apply (rule_tac x = "ca * (abs c)" in exI)
1.705    apply (rule allI)
1.706    apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
1.707 @@ -651,23 +573,23 @@
1.709  done
1.711 -lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
1.712 +lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
1.713  proof -
1.714    assume "f =o O(g)"
1.715 -  then have "(%x. c) * f =o (%x. c) *o O(g)"
1.716 +  then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
1.717      by auto
1.718 -  also have "(%x. c) * f = (%x. c * f x)"
1.719 +  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
1.721 -  also have "(%x. c) *o O(g) <= O(g)"
1.722 +  also have "(\<lambda>x. c) *o O(g) <= O(g)"
1.723      by (auto del: subsetI)
1.724    finally show ?thesis .
1.725  qed
1.727 -lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
1.728 +lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))"
1.729  by (unfold bigo_def, auto)
1.731 -lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o
1.732 -    O(%x. h(k x))"
1.733 +lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o
1.734 +    O(\<lambda>x. h(k x))"
1.735    apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
1.736        func_plus)
1.737    apply (erule bigo_compose1)
1.738 @@ -675,9 +597,9 @@
1.740  subsection {* Setsum *}
1.742 -lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
1.743 -    EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
1.744 -      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
1.745 +lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
1.746 +    \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) <= c * (h x y) \<Longrightarrow>
1.747 +      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
1.748    apply (auto simp add: bigo_def)
1.749    apply (rule_tac x = "abs c" in exI)
1.750    apply (subst abs_of_nonneg) back back
1.751 @@ -691,61 +613,50 @@
1.752  apply (blast intro: order_trans mult_right_mono abs_ge_self)
1.753  done
1.755 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum1" ]]
1.756 -lemma bigo_setsum1: "ALL x y. 0 <= h x y ==>
1.757 -    EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
1.758 -      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
1.759 -  apply (rule bigo_setsum_main)
1.760 -(*sledgehammer*)
1.761 -  apply force
1.762 -  apply clarsimp
1.763 -  apply (rule_tac x = c in exI)
1.764 -  apply force
1.765 -done
1.766 +lemma bigo_setsum1: "\<forall>x y. 0 <= h x y \<Longrightarrow>
1.767 +    \<exists>c. \<forall>x y. abs (f x y) <= c * (h x y) \<Longrightarrow>
1.768 +      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
1.769 +by (metis (no_types) bigo_setsum_main)
1.771 -lemma bigo_setsum2: "ALL y. 0 <= h y ==>
1.772 -    EX c. ALL y. abs(f y) <= c * (h y) ==>
1.773 -      (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
1.774 +lemma bigo_setsum2: "\<forall>y. 0 <= h y \<Longrightarrow>
1.775 +    \<exists>c. \<forall>y. abs(f y) <= c * (h y) \<Longrightarrow>
1.776 +      (\<lambda>x. SUM y : A x. f y) =o O(\<lambda>x. SUM y : A x. h y)"
1.777  by (rule bigo_setsum1, auto)
1.779 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum3" ]]
1.780 -lemma bigo_setsum3: "f =o O(h) ==>
1.781 -    (%x. SUM y : A x. (l x y) * f(k x y)) =o
1.782 -      O(%x. SUM y : A x. abs(l x y * h(k x y)))"
1.783 -  apply (rule bigo_setsum1)
1.784 -  apply (rule allI)+
1.785 -  apply (rule abs_ge_zero)
1.786 -  apply (unfold bigo_def)
1.787 -  apply (auto simp add: abs_mult)
1.788 -(*sledgehammer*)
1.789 -  apply (rule_tac x = c in exI)
1.790 -  apply (rule allI)+
1.791 -  apply (subst mult_left_commute)
1.792 -  apply (rule mult_left_mono)
1.793 -  apply (erule spec)
1.794 -  apply (rule abs_ge_zero)
1.795 -done
1.796 +lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
1.797 +    (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
1.798 +      O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))"
1.799 +apply (rule bigo_setsum1)
1.800 + apply (rule allI)+
1.801 + apply (rule abs_ge_zero)
1.802 +apply (unfold bigo_def)
1.803 +apply (auto simp add: abs_mult)
1.804 +(* sledgehammer *)
1.805 +apply (rule_tac x = c in exI)
1.806 +apply (rule allI)+
1.807 +apply (subst mult_left_commute)
1.808 +apply (rule mult_left_mono)
1.809 + apply (erule spec)
1.810 +by (rule abs_ge_zero)
1.812 -lemma bigo_setsum4: "f =o g +o O(h) ==>
1.813 -    (%x. SUM y : A x. l x y * f(k x y)) =o
1.814 -      (%x. SUM y : A x. l x y * g(k x y)) +o
1.815 -        O(%x. SUM y : A x. abs(l x y * h(k x y)))"
1.816 -  apply (rule set_minus_imp_plus)
1.817 -  apply (subst fun_diff_def)
1.818 -  apply (subst setsum_subtractf [symmetric])
1.819 -  apply (subst right_diff_distrib [symmetric])
1.820 -  apply (rule bigo_setsum3)
1.821 -  apply (subst fun_diff_def [symmetric])
1.822 -  apply (erule set_plus_imp_minus)
1.823 -done
1.824 +lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
1.825 +    (\<lambda>x. SUM y : A x. l x y * f(k x y)) =o
1.826 +      (\<lambda>x. SUM y : A x. l x y * g(k x y)) +o
1.827 +        O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))"
1.828 +apply (rule set_minus_imp_plus)
1.829 +apply (subst fun_diff_def)
1.830 +apply (subst setsum_subtractf [symmetric])
1.831 +apply (subst right_diff_distrib [symmetric])
1.832 +apply (rule bigo_setsum3)
1.833 +apply (subst fun_diff_def [symmetric])
1.834 +by (erule set_plus_imp_minus)
1.836 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum5" ]]
1.837 -lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==>
1.838 -    ALL x. 0 <= h x ==>
1.839 -      (%x. SUM y : A x. (l x y) * f(k x y)) =o
1.840 -        O(%x. SUM y : A x. (l x y) * h(k x y))"
1.841 -  apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) =
1.842 -      (%x. SUM y : A x. abs((l x y) * h(k x y)))")
1.843 +lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
1.844 +    \<forall>x. 0 <= h x \<Longrightarrow>
1.845 +      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
1.846 +        O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
1.847 +  apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) =
1.848 +      (\<lambda>x. SUM y : A x. abs((l x y) * h(k x y)))")
1.849    apply (erule ssubst)
1.850    apply (erule bigo_setsum3)
1.851    apply (rule ext)
1.852 @@ -754,11 +665,11 @@
1.853  apply (metis abs_of_nonneg zero_le_mult_iff)
1.854  done
1.856 -lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
1.857 -    ALL x. 0 <= h x ==>
1.858 -      (%x. SUM y : A x. (l x y) * f(k x y)) =o
1.859 -        (%x. SUM y : A x. (l x y) * g(k x y)) +o
1.860 -          O(%x. SUM y : A x. (l x y) * h(k x y))"
1.861 +lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
1.862 +    \<forall>x. 0 <= h x \<Longrightarrow>
1.863 +      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
1.864 +        (\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o
1.865 +          O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
1.866    apply (rule set_minus_imp_plus)
1.867    apply (subst fun_diff_def)
1.868    apply (subst setsum_subtractf [symmetric])
1.869 @@ -771,50 +682,39 @@
1.871  subsection {* Misc useful stuff *}
1.873 -lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
1.874 +lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
1.875    A \<oplus> B <= O(f)"
1.876    apply (subst bigo_plus_idemp [symmetric])
1.877    apply (rule set_plus_mono2)
1.878    apply assumption+
1.879  done
1.881 -lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
1.882 +lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
1.883    apply (subst bigo_plus_idemp [symmetric])
1.884    apply (rule set_plus_intro)
1.885    apply assumption+
1.886  done
1.888 -lemma bigo_useful_const_mult: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
1.889 -    (%x. c) * f =o O(h) ==> f =o O(h)"
1.890 +lemma bigo_useful_const_mult: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
1.891 +    (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
1.892    apply (rule subsetD)
1.893 -  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
1.894 +  apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)")
1.895    apply assumption
1.896    apply (rule bigo_const_mult6)
1.897 -  apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
1.898 +  apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
1.899    apply (erule ssubst)
1.900    apply (erule set_times_intro2)
1.902  done
1.904 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]]
1.905 -lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
1.906 +lemma bigo_fix: "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow>
1.907      f =o O(h)"
1.908 -  apply (simp add: bigo_alt_def)
1.909 -(*sledgehammer*)
1.910 -  apply clarify
1.911 -  apply (rule_tac x = c in exI)
1.912 -  apply safe
1.913 -  apply (case_tac "x = 0")
1.914 -apply (metis abs_ge_zero  abs_zero  order_less_le  split_mult_pos_le)
1.915 -  apply (subgoal_tac "x = Suc (x - 1)")
1.916 -  apply metis
1.917 -  apply simp
1.918 -  done
1.921 +by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc)
1.923  lemma bigo_fix2:
1.924 -    "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==>
1.925 -       f 0 = g 0 ==> f =o g +o O(h)"
1.926 +    "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
1.927 +       f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
1.928    apply (rule set_minus_imp_plus)
1.929    apply (rule bigo_fix)
1.930    apply (subst fun_diff_def)
1.931 @@ -826,23 +726,23 @@
1.933  subsection {* Less than or equal to *}
1.935 -definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
1.936 -  "f <o g == (%x. max (f x - g x) 0)"
1.937 +definition lesso :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
1.938 +  "f <o g == (\<lambda>x. max (f x - g x) 0)"
1.940 -lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
1.941 +lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= abs (f x) \<Longrightarrow>
1.942      g =o O(h)"
1.943    apply (unfold bigo_def)
1.944    apply clarsimp
1.945  apply (blast intro: order_trans)
1.946  done
1.948 -lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
1.949 +lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= f x \<Longrightarrow>
1.950        g =o O(h)"
1.951    apply (erule bigo_lesseq1)
1.952  apply (blast intro: abs_ge_self order_trans)
1.953  done
1.955 -lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
1.956 +lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow>
1.957        g =o O(h)"
1.958    apply (erule bigo_lesseq2)
1.959    apply (rule allI)
1.960 @@ -850,8 +750,8 @@
1.961    apply (erule spec)+
1.962  done
1.964 -lemma bigo_lesseq4: "f =o O(h) ==>
1.965 -    ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
1.966 +lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
1.967 +    \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= abs (f x) \<Longrightarrow>
1.968        g =o O(h)"
1.969    apply (erule bigo_lesseq1)
1.970    apply (rule allI)
1.971 @@ -859,23 +759,15 @@
1.972    apply (erule spec)+
1.973  done
1.975 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]]
1.976 -lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
1.977 +lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)"
1.978  apply (unfold lesso_def)
1.979 -apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
1.980 -proof -
1.981 -  assume "(\<lambda>x. max (f x - g x) 0) = 0"
1.982 -  thus "(\<lambda>x. max (f x - g x) 0) \<in> O(h)" by (metis bigo_zero)
1.983 -next
1.984 -  show "\<forall>x\<Colon>'a. f x \<le> g x \<Longrightarrow> (\<lambda>x\<Colon>'a. max (f x - g x) (0\<Colon>'b)) = (0\<Colon>'a \<Rightarrow> 'b)"
1.985 -  apply (unfold func_zero)
1.986 -  apply (rule ext)
1.987 -  by (simp split: split_max)
1.988 -qed
1.989 +apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
1.990 + apply (metis bigo_zero)
1.991 +by (metis (lam_lifting, no_types) func_zero le_fun_def le_iff_diff_le_0
1.992 +      min_max.sup_absorb2 order_eq_iff)
1.994 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso2" ]]
1.995 -lemma bigo_lesso2: "f =o g +o O(h) ==>
1.996 -    ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
1.997 +lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
1.998 +    \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow>
1.999        k <o g =o O(h)"
1.1000    apply (unfold lesso_def)
1.1001    apply (rule bigo_lesseq4)
1.1002 @@ -885,33 +777,15 @@
1.1003    apply (rule allI)
1.1004    apply (subst fun_diff_def)
1.1005  apply (erule thin_rl)
1.1006 -(*sledgehammer*)
1.1007 -  apply (case_tac "0 <= k x - g x")
1.1008 -(* apply (metis abs_le_iff add_le_imp_le_right diff_minus le_less
1.1009 -                le_max_iff_disj min_max.le_supE min_max.sup_absorb2
1.1010 -                min_max.sup_commute) *)
1.1011 -proof -
1.1012 -  fix x :: 'a
1.1013 -  assume "\<forall>x\<Colon>'a. k x \<le> f x"
1.1014 -  hence F1: "\<forall>x\<^isub>1\<Colon>'a. max (k x\<^isub>1) (f x\<^isub>1) = f x\<^isub>1" by (metis min_max.sup_absorb2)
1.1015 -  assume "(0\<Colon>'b) \<le> k x - g x"
1.1016 -  hence F2: "max (0\<Colon>'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2)
1.1017 -  have F3: "\<forall>x\<^isub>1\<Colon>'b. x\<^isub>1 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_le_iff le_less)
1.1018 -  have "\<forall>(x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>2 \<or> max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>1" by (metis le_less le_max_iff_disj)
1.1019 -  hence "\<forall>(x\<^isub>3\<Colon>'b) (x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. x\<^isub>1 - x\<^isub>2 \<le> x\<^isub>3 - x\<^isub>2 \<or> x\<^isub>3 \<le> x\<^isub>1" by (metis add_le_imp_le_right diff_minus min_max.le_supE)
1.1020 -  hence "k x - g x \<le> f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute)
1.1021 -  hence "k x - g x \<le> \<bar>f x - g x\<bar>" by (metis F3 le_max_iff_disj min_max.sup_absorb2)
1.1022 -  thus "max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" by (metis F2 min_max.sup_commute)
1.1023 -next
1.1024 -  show "\<And>x\<Colon>'a.
1.1025 -       \<lbrakk>\<forall>x\<Colon>'a. (0\<Colon>'b) \<le> k x; \<forall>x\<Colon>'a. k x \<le> f x; \<not> (0\<Colon>'b) \<le> k x - g x\<rbrakk>
1.1026 -       \<Longrightarrow> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
1.1027 -    by (metis abs_ge_zero le_cases min_max.sup_absorb2)
1.1028 -qed
1.1029 +(* sledgehammer *)
1.1030 +apply (case_tac "0 <= k x - g x")
1.1031 + apply (metis (hide_lams, no_types) abs_le_iff add_le_imp_le_right diff_minus le_less
1.1032 +          le_max_iff_disj min_max.le_supE min_max.sup_absorb2
1.1033 +          min_max.sup_commute)
1.1034 +by (metis abs_ge_zero le_cases min_max.sup_absorb2)
1.1036 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3" ]]
1.1037 -lemma bigo_lesso3: "f =o g +o O(h) ==>
1.1038 -    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
1.1039 +lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
1.1040 +    \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow>
1.1041        f <o k =o O(h)"
1.1042    apply (unfold lesso_def)
1.1043    apply (rule bigo_lesseq4)
1.1044 @@ -920,20 +794,19 @@
1.1045    apply (rule le_maxI2)
1.1046    apply (rule allI)
1.1047    apply (subst fun_diff_def)
1.1048 -apply (erule thin_rl)
1.1049 -(*sledgehammer*)
1.1050 +  apply (erule thin_rl)
1.1051 +  (* sledgehammer *)
1.1052    apply (case_tac "0 <= f x - k x")
1.1053 -  apply (simp)
1.1054 +  apply simp
1.1055    apply (subst abs_of_nonneg)
1.1056    apply (drule_tac x = x in spec) back
1.1057 -using [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3_simpler" ]]
1.1058 -apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
1.1060 +  apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
1.1062  apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
1.1063  done
1.1065 -lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::{linordered_field,number_ring}) ==>
1.1066 -    g =o h +o O(k) ==> f <o h =o O(k)"
1.1067 +lemma bigo_lesso4: "f <o g =o O(k\<Colon>'a=>'b\<Colon>{linordered_field,number_ring}) \<Longrightarrow>
1.1068 +    g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
1.1069    apply (unfold lesso_def)
1.1070    apply (drule set_plus_imp_minus)
1.1071    apply (drule bigo_abs5) back
1.1072 @@ -946,9 +819,7 @@
1.1073      split: split_max abs_split)
1.1074  done
1.1076 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso5" ]]
1.1077 -lemma bigo_lesso5: "f <o g =o O(h) ==>
1.1078 -    EX C. ALL x. f x <= g x + C * abs(h x)"
1.1079 +lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs(h x)"
1.1080    apply (simp only: lesso_def bigo_alt_def)
1.1081    apply clarsimp
1.1082    apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)