# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1674126825 0
# Node ID 68f1fc53c8fda42e7e998078e30d46000b0a7a87
# Parent  d481dc154310782d2d86ab17c07329c9b723e97d
tidy up of this messy and obsolete theory

diff -r d481dc154310 -r 68f1fc53c8fd src/HOL/Library/BigO.thy
--- a/src/HOL/Library/BigO.thy	Mon Jan 16 22:41:00 2023 +0100
+++ b/src/HOL/Library/BigO.thy	Thu Jan 19 11:13:45 2023 +0000
@@ -77,11 +77,7 @@
   apply (rule conjI)
    apply (rule_tac x = "c + c" in exI)
    apply (clarsimp)
-   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
-    apply (metis mult_2 order_trans)
-   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
-    apply auto[1]
-  using abs_triangle_ineq mult_le_cancel_iff2 apply blast
+  apply (smt (verit, ccfv_threshold) mult.commute abs_triangle_ineq add_le_cancel_left dual_order.trans mult.left_commute mult_2 mult_le_cancel_iff2)
   apply (simp add: order_less_le)
   apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
   apply (rule conjI)
@@ -101,87 +97,40 @@
   apply (simp add: bigo_alt_def set_plus_def func_plus)
   apply clarify
   apply (rule_tac x = "max c ca" in exI)
-  apply (rule conjI)
-   apply (subgoal_tac "c \<le> max c ca")
-    apply linarith
-   apply (rule max.cobounded1)
-  apply clarify
-  apply (drule_tac x = "xa" in spec)+
-  apply (subgoal_tac "0 \<le> f xa + g xa")
-   apply (simp add: ring_distribs)
-   apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
-    apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
-     apply force
-    apply (metis add_mono le_max_iff_disj max_mult_distrib_right)
-  using abs_triangle_ineq apply blast
-  using add_nonneg_nonneg by blast
+  by (smt (verit, del_insts) add.commute abs_triangle_ineq add_mono_thms_linordered_field(3) distrib_left less_max_iff_disj linorder_not_less max.orderE max_mult_distrib_right order_le_less)
 
 lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
-  apply (auto simp add: bigo_def)
-  apply (rule_tac x = "\<bar>c\<bar>" in exI)
-  apply auto
-  apply (drule_tac x = x in spec)+
-  apply (simp flip: abs_mult)
-  done
+  by (simp add: bigo_def) (metis abs_mult abs_of_nonneg order_trans)
 
 lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)"
-  apply (erule bigo_bounded_alt [of f 1 g])
-  apply simp
-  done
+  by (metis bigo_bounded_alt mult_1)
 
 lemma bigo_bounded2: "\<forall>x. lb x \<le> f x \<Longrightarrow> \<forall>x. f x \<le> lb x + g x \<Longrightarrow> f \<in> lb +o O(g)"
-  apply (rule set_minus_imp_plus)
-  apply (rule bigo_bounded)
-   apply (auto simp add: fun_Compl_def func_plus)
-  apply (drule_tac x = x in spec)+
-  apply force
-  done
+  by (simp add: add.commute bigo_bounded diff_le_eq set_minus_imp_plus)
 
 lemma bigo_abs: "(\<lambda>x. \<bar>f x\<bar>) =o O(f)"
-  apply (unfold bigo_def)
-  apply auto
-  apply (rule_tac x = 1 in exI)
-  apply auto
-  done
+  by (smt (verit, del_insts) abs_abs bigo_def bigo_refl mem_Collect_eq)
 
 lemma bigo_abs2: "f =o O(\<lambda>x. \<bar>f x\<bar>)"
-  apply (unfold bigo_def)
-  apply auto
-  apply (rule_tac x = 1 in exI)
-  apply auto
-  done
+  by (smt (verit, del_insts) abs_abs bigo_def bigo_refl mem_Collect_eq)
 
 lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
-  apply (rule equalityI)
-   apply (rule bigo_elt_subset)
-   apply (rule bigo_abs2)
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_abs)
-  done
+  using bigo_abs bigo_abs2 bigo_elt_subset by blast
 
-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)"
-  apply (drule set_plus_imp_minus)
-  apply (rule set_minus_imp_plus)
-  apply (subst fun_diff_def)
+lemma bigo_abs4: assumes "f =o g +o O(h)" shows "(\<lambda>x. \<bar>f x\<bar>) =o (\<lambda>x. \<bar>g x\<bar>) +o O(h)"
 proof -
-  assume *: "f - g \<in> O(h)"
-  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>)"
-    by (rule bigo_abs2)
-  also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)"
-    apply (rule bigo_elt_subset)
-    apply (rule bigo_bounded)
-     apply force
-    apply (rule allI)
-    apply (rule abs_triangle_ineq3)
-    done
-  also have "\<dots> \<subseteq> O(f - g)"
-    apply (rule bigo_elt_subset)
-    apply (subst fun_diff_def)
-    apply (rule bigo_abs)
-    done
-  also from * have "\<dots> \<subseteq> O(h)"
-    by (rule bigo_elt_subset)
-  finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)".
+  { assume *: "f - g \<in> O(h)"
+    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>)"
+      by (rule bigo_abs2)
+    also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)"
+      by (simp add: abs_triangle_ineq3 bigo_bounded bigo_elt_subset)
+    also have "\<dots> \<subseteq> O(f - g)"
+      using bigo_abs3 by fastforce
+    also from * have "\<dots> \<subseteq> O(h)"
+      by (rule bigo_elt_subset)
+    finally have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)" . }
+  then show ?thesis
+    by (smt (verit) assms bigo_alt_def fun_diff_def mem_Collect_eq set_minus_imp_plus set_plus_imp_minus)
 qed
 
 lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
@@ -210,42 +159,18 @@
 lemma bigo_mult [intro]: "O(f)*O(g) \<subseteq> O(f * g)"
   apply (rule subsetI)
   apply (subst bigo_def)
-  apply (auto simp add: bigo_alt_def set_times_def func_times)
+  apply (clarsimp simp add: bigo_alt_def set_times_def func_times)
   apply (rule_tac x = "c * ca" in exI)
-  apply (rule allI)
-  apply (erule_tac x = x in allE)+
-  apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
-   apply (erule ssubst)
-   apply (subst abs_mult)
-   apply (rule mult_mono)
-      apply assumption+
-    apply auto
-  apply (simp add: ac_simps abs_mult)
-  done
+  by (smt (verit, ccfv_threshold) mult.commute mult.assoc abs_ge_zero abs_mult dual_order.trans mult_mono)
 
 lemma bigo_mult2 [intro]: "f *o O(g) \<subseteq> O(f * g)"
-  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
-  apply (rule_tac x = c in exI)
-  apply auto
-  apply (drule_tac x = x in spec)
-  apply (subgoal_tac "\<bar>f x\<bar> * \<bar>b x\<bar> \<le> \<bar>f x\<bar> * (c * \<bar>g x\<bar>)")
-   apply (force simp add: ac_simps)
-  apply (rule mult_left_mono, assumption)
-  apply (rule abs_ge_zero)
-  done
+  by (metis bigo_mult bigo_refl dual_order.trans mult.commute set_times_mono4)
 
 lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
-  apply (rule subsetD)
-   apply (rule bigo_mult)
-  apply (erule set_times_intro, assumption)
-  done
+  using bigo_mult mult.commute mult.commute set_times_intro subsetD by blast
 
 lemma bigo_mult4 [intro]: "f \<in> k +o O(h) \<Longrightarrow> g * f \<in> (g * k) +o O(g * h)"
-  apply (drule set_plus_imp_minus)
-  apply (rule set_minus_imp_plus)
-  apply (drule bigo_mult3 [where g = g and j = g])
-   apply (auto simp add: algebra_simps)
-  done
+  by (metis bigo_mult3 bigo_refl left_diff_distrib' mult.commute set_minus_imp_plus set_plus_imp_minus)
 
 lemma bigo_mult5:
   fixes f :: "'a \<Rightarrow> 'b::linordered_field"
@@ -259,52 +184,38 @@
   also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))"
     by (rule bigo_mult2)
   also have "(\<lambda>x. 1 / f x) * (f * g) = g"
-    apply (simp add: func_times)
-    apply (rule ext)
-    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
-    done
+    using assms by auto
   finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" .
   then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)"
     by auto
   also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
-    apply (simp add: func_times)
-    apply (rule ext)
-    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
-    done
+  by (simp add: assms times_fun_def)
   finally show "h \<in> f *o O(g)" .
 qed
 
 lemma bigo_mult6: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
   for f :: "'a \<Rightarrow> 'b::linordered_field"
-  apply (rule equalityI)
-   apply (erule bigo_mult5)
-  apply (rule bigo_mult2)
-  done
+  by (simp add: bigo_mult2 bigo_mult5 subset_antisym)
 
 lemma bigo_mult7: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
   for f :: "'a \<Rightarrow> 'b::linordered_field"
-  apply (subst bigo_mult6)
-   apply assumption
-  apply (rule set_times_mono3)
-  apply (rule bigo_refl)
-  done
+  by (metis bigo_mult6 bigo_refl mult.commute set_times_mono4)
 
 lemma bigo_mult8: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
   for f :: "'a \<Rightarrow> 'b::linordered_field"
-  apply (rule equalityI)
-   apply (erule bigo_mult7)
-  apply (rule bigo_mult)
-  done
+  by (simp add: bigo_mult bigo_mult7 subset_antisym)
 
 lemma bigo_minus [intro]: "f \<in> O(g) \<Longrightarrow> - f \<in> O(g)"
   by (auto simp add: bigo_def fun_Compl_def)
 
-lemma bigo_minus2: "f \<in> g +o O(h) \<Longrightarrow> - f \<in> -g +o O(h)"
-  apply (rule set_minus_imp_plus)
-  apply (drule set_plus_imp_minus)
-  apply (drule bigo_minus)
-  apply simp
-  done
+lemma bigo_minus2:
+  assumes "f \<in> g +o O(h)" shows "- f \<in> -g +o O(h)"
+proof -
+   have "- f + g \<in> O(h)"
+    by (metis assms bigo_minus minus_diff_eq set_plus_imp_minus uminus_add_conv_diff)
+  then show ?thesis
+    by (simp add: set_minus_imp_plus)
+qed
 
 lemma bigo_minus3: "O(- f) = O(f)"
   by (auto simp add: bigo_def fun_Compl_def)
@@ -312,20 +223,7 @@
 lemma bigo_plus_absorb_lemma1:
   assumes *: "f \<in> O(g)"
   shows "f +o O(g) \<subseteq> O(g)"
-proof -
-  have "f \<in> O(f)" by auto
-  then have "f +o O(g) \<subseteq> O(f) + O(g)"
-    by (auto del: subsetI)
-  also have "\<dots> \<subseteq> O(g) + O(g)"
-  proof -
-    from * have "O(f) \<subseteq> O(g)"
-      by (auto del: subsetI)
-    then show ?thesis
-      by (auto del: subsetI)
-  qed
-  also have "\<dots> \<subseteq> O(g)" by simp
-  finally show ?thesis .
-qed
+  using assms bigo_plus_idemp set_plus_mono4 by blast
 
 lemma bigo_plus_absorb_lemma2:
   assumes *: "f \<in> O(g)"
@@ -343,118 +241,75 @@
 qed
 
 lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
-  apply (rule equalityI)
-   apply (erule bigo_plus_absorb_lemma1)
-  apply (erule bigo_plus_absorb_lemma2)
-  done
+  by (simp add: bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 subset_antisym)
 
 lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
-  apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
-   apply force+
-  done
+  using bigo_plus_absorb set_plus_mono by blast
 
 lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
-  apply (subst set_minus_plus [symmetric])
-  apply (subgoal_tac "g - f = - (f - g)")
-   apply (erule ssubst)
-   apply (rule bigo_minus)
-   apply (subst set_minus_plus)
-   apply assumption
-  apply (simp add: ac_simps)
-  done
+  by (metis bigo_minus minus_diff_eq set_minus_imp_plus set_plus_imp_minus)
 
 lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
-  apply (rule iffI)
-   apply (erule bigo_add_commute_imp)+
-  done
+  using bigo_add_commute_imp by blast
 
 lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
   by (auto simp add: bigo_def ac_simps)
 
 lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)"
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_const1)
-  done
+  by (metis bigo_elt_subset bigo_const1)
 
 lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
   for c :: "'a::linordered_field"
-  apply (simp add: bigo_def)
-  apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
-  apply (simp flip: abs_mult)
-  done
+  by (metis bigo_bounded_alt le_numeral_extra(4) nonzero_divide_eq_eq zero_less_one_class.zero_le_one)
 
 lemma bigo_const4: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
   for c :: "'a::linordered_field"
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_const3)
-  apply assumption
-  done
+  by (metis bigo_elt_subset bigo_const3)
 
 lemma bigo_const [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
   for c :: "'a::linordered_field"
-  apply (rule equalityI)
-   apply (rule bigo_const2)
-  apply (rule bigo_const4)
-  apply assumption
-  done
+  by (metis equalityI bigo_const2 bigo_const4)
 
 lemma bigo_const_mult1: "(\<lambda>x. c * f x) \<in> O(f)"
-  apply (simp add: bigo_def)
-  apply (rule_tac x = "\<bar>c\<bar>" in exI)
-  apply (auto simp flip: abs_mult)
-  done
+  by (smt (z3) abs_mult bigo_def bigo_refl mem_Collect_eq mult.left_commute mult_commute_abs)
 
 lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_const_mult1)
-  done
+  by (metis bigo_elt_subset bigo_const_mult1)
 
 lemma bigo_const_mult3: "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
   for c :: "'a::linordered_field"
-  apply (simp add: bigo_def)
-  apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
-  apply (simp add: abs_mult mult.assoc [symmetric])
-  done
+  by (simp add: bigo_def) (metis abs_mult field_class.field_divide_inverse mult.commute nonzero_divide_eq_eq order_refl)
 
 lemma bigo_const_mult4: "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
   for c :: "'a::linordered_field"
-  apply (rule bigo_elt_subset)
-  apply (rule bigo_const_mult3)
-  apply assumption
-  done
+  by (simp add: bigo_const_mult3 bigo_elt_subset)
 
 lemma bigo_const_mult [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
   for c :: "'a::linordered_field"
-  apply (rule equalityI)
-   apply (rule bigo_const_mult2)
-  apply (erule bigo_const_mult4)
-  done
+  by (simp add: bigo_const_mult2 bigo_const_mult4 subset_antisym)
 
-lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
+lemma bigo_const_mult5 [simp]: "(\<lambda>x. c) *o O(f) = O(f)" if "c \<noteq> 0"
   for c :: "'a::linordered_field"
-  apply (auto del: subsetI)
-   apply (rule order_trans)
-    apply (rule bigo_mult2)
-   apply (simp add: func_times)
-  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
-  apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
-  apply (simp add: mult.assoc [symmetric] abs_mult)
-  apply (rule_tac x = "\<bar>inverse c\<bar> * ca" in exI)
-  apply auto
-  done
+proof 
+  show "O(f) \<subseteq> (\<lambda>x. c) *o O(f)"
+    using that
+    apply (clarsimp simp add: bigo_def elt_set_times_def func_times)
+    apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
+    apply (simp add: mult.assoc [symmetric] abs_mult)
+    apply (rule_tac x = "\<bar>inverse c\<bar> * ca" in exI)
+    apply auto
+    done
+  have "O(\<lambda>x. c * f x) \<subseteq> O(f)"
+    by (simp add: bigo_const_mult2)
+  then show "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
+    using order_trans[OF bigo_mult2] by (force simp add: times_fun_def)
+qed
+
 
 lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
-  apply (rule allI)
-  apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
-   apply (erule ssubst)
-   apply (subst abs_mult)
-   apply (rule mult_left_mono)
-    apply (erule spec)
-   apply simp
-  apply (simp add: ac_simps)
-  done
+  by (smt (verit, ccfv_SIG) ab_semigroup_mult_class.mult_ac(1) abs_abs abs_le_self_iff abs_mult le_cases3 mult.commute mult_left_mono)
 
 lemma bigo_const_mult7 [intro]:
   assumes *: "f =o O(g)"
@@ -473,47 +328,25 @@
   by (auto simp: bigo_def)
 
 lemma bigo_compose2: "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))"
-  apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
-  apply (drule bigo_compose1)
-  apply (simp add: fun_diff_def)
-  done
+  by (smt (verit, best) set_minus_plus bigo_def fun_diff_def mem_Collect_eq)
 
 
 subsection \<open>Sum\<close>
 
-lemma bigo_sum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
-    \<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
-      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
-  apply (auto simp add: bigo_def)
-  apply (rule_tac x = "\<bar>c\<bar>" in exI)
-  apply (subst abs_of_nonneg) back back
-   apply (rule sum_nonneg)
-   apply force
-  apply (subst sum_distrib_left)
-  apply (rule allI)
-  apply (rule order_trans)
-   apply (rule sum_abs)
-  apply (rule sum_mono)
-  apply (rule order_trans)
-   apply (drule spec)+
-   apply (drule bspec)+
-     apply assumption+
-   apply (drule bspec)
-    apply assumption+
-  apply (rule mult_right_mono)
-   apply (rule abs_ge_self)
-  apply force
-  done
+lemma bigo_sum_main:
+  assumes "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y" and "\<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> \<le> c * h x y"
+  shows "(\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
+proof -
+  have "(\<Sum>i\<in>A x. \<bar>f x i\<bar>) \<le> \<bar>c\<bar> * sum (h x) (A x)" for x
+    by (smt (verit, ccfv_threshold) assms abs_mult_pos abs_of_nonneg abs_of_nonpos dual_order.trans le_cases3 neg_0_le_iff_le sum_distrib_left sum_mono)
+  then show ?thesis
+    using assms by (fastforce simp add: bigo_def sum_nonneg intro: order_trans [OF sum_abs])
+qed
 
 lemma bigo_sum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
     \<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
-  apply (rule bigo_sum_main)
-   apply force
-  apply clarsimp
-  apply (rule_tac x = c in exI)
-  apply force
-  done
+  by (metis (no_types) bigo_sum_main)
 
 lemma bigo_sum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
     \<exists>c. \<forall>y. \<bar>f y\<bar> \<le> c * (h y) \<Longrightarrow>
@@ -523,112 +356,52 @@
 lemma bigo_sum3: "f =o O(h) \<Longrightarrow>
     (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
   apply (rule bigo_sum1)
-   apply (rule allI)+
-   apply (rule abs_ge_zero)
-  apply (unfold bigo_def)
-  apply auto
-  apply (rule_tac x = c in exI)
-  apply (rule allI)+
-  apply (subst abs_mult)+
-  apply (subst mult.left_commute)
-  apply (rule mult_left_mono)
-   apply (erule spec)
-  apply (rule abs_ge_zero)
-  done
+  using abs_ge_zero apply blast
+  apply (clarsimp simp: bigo_def)
+  by (smt (verit, ccfv_threshold) abs_mult abs_not_less_zero mult.left_commute mult_le_cancel_left)
 
 lemma bigo_sum4: "f =o g +o O(h) \<Longrightarrow>
     (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
       (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
         O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
-  apply (rule set_minus_imp_plus)
-  apply (subst fun_diff_def)
-  apply (subst sum_subtractf [symmetric])
-  apply (subst right_diff_distrib [symmetric])
-  apply (rule bigo_sum3)
-  apply (subst fun_diff_def [symmetric])
-  apply (erule set_plus_imp_minus)
-  done
+  using bigo_sum3 [of "f-g" h l k A]
+  apply (simp add: algebra_simps sum_subtractf)
+  by (smt (verit) bigo_alt_def minus_apply set_minus_imp_plus set_plus_imp_minus mem_Collect_eq)
 
 lemma bigo_sum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
     \<forall>x. 0 \<le> h x \<Longrightarrow>
       (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
         O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
-  apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
-      (\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)")
-   apply (erule ssubst)
-   apply (erule bigo_sum3)
-  apply (rule ext)
-  apply (rule sum.cong)
-   apply (rule refl)
-  apply (subst abs_of_nonneg)
-   apply auto
-  done
+  using bigo_sum3 [of f h l k A] by simp
 
 lemma bigo_sum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
     \<forall>x. 0 \<le> h x \<Longrightarrow>
       (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
         (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
           O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
-  apply (rule set_minus_imp_plus)
-  apply (subst fun_diff_def)
-  apply (subst sum_subtractf [symmetric])
-  apply (subst right_diff_distrib [symmetric])
-  apply (rule bigo_sum5)
-    apply (subst fun_diff_def [symmetric])
-    apply (drule set_plus_imp_minus)
-    apply auto
-  done
+  using bigo_sum5 [of "f-g" h l k A]
+  apply (simp add: algebra_simps sum_subtractf)
+  by (smt (verit, del_insts) bigo_alt_def set_minus_imp_plus minus_apply set_plus_imp_minus mem_Collect_eq)
 
 
 subsection \<open>Misc useful stuff\<close>
 
-lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
-  apply (subst bigo_plus_idemp [symmetric])
-  apply (rule set_plus_mono2)
-   apply assumption+
-  done
-
 lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
-  apply (subst bigo_plus_idemp [symmetric])
-  apply (rule set_plus_intro)
-   apply assumption+
-  done
+  using bigo_plus_idemp set_plus_intro by blast
 
 lemma bigo_useful_const_mult: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
   for c :: "'a::linordered_field"
-  apply (rule subsetD)
-   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
-    apply assumption
-   apply (rule bigo_const_mult6)
-  apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
-   apply (erule ssubst)
-   apply (erule set_times_intro2)
-  apply (simp add: func_times)
-  done
+  using bigo_elt_subset bigo_mult6 by fastforce
 
 lemma bigo_fix: "(\<lambda>x::nat. f (x + 1)) =o O(\<lambda>x. h (x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> f =o O(h)"
-  apply (simp add: bigo_alt_def)
-  apply auto
-  apply (rule_tac x = c in exI)
-  apply auto
-  apply (case_tac "x = 0")
-   apply simp
-  apply (subgoal_tac "x = Suc (x - 1)")
-   apply (erule ssubst) back
-   apply (erule spec)
-  apply simp
-  done
+  by (simp add: bigo_alt_def) (metis abs_eq_0_iff abs_ge_zero abs_mult abs_of_pos not0_implies_Suc)
 
 lemma bigo_fix2:
-    "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
+  "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
        f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
-  apply (rule set_minus_imp_plus)
-  apply (rule bigo_fix)
-   apply (subst fun_diff_def)
-   apply (subst fun_diff_def [symmetric])
-   apply (rule set_plus_imp_minus)
-   apply simp
-  apply (simp add: fun_diff_def)
+  apply (rule set_minus_imp_plus [OF bigo_fix])
+   apply (smt (verit, del_insts) bigo_alt_def fun_diff_def set_plus_imp_minus mem_Collect_eq)
+  apply simp
   done
 
 
@@ -638,159 +411,93 @@
   where "f <o g = (\<lambda>x. max (f x - g x) 0)"
 
 lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)"
-  apply (unfold bigo_def)
-  apply clarsimp
-  apply (rule_tac x = c in exI)
-  apply (rule allI)
-  apply (rule order_trans)
-   apply (erule spec)+
-  done
+  by (smt (verit, del_insts) bigo_def mem_Collect_eq order_trans)
 
 lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> f x \<Longrightarrow> g =o O(h)"
-  apply (erule bigo_lesseq1)
-  apply (rule allI)
-  apply (drule_tac x = x in spec)
-  apply (rule order_trans)
-   apply assumption
-  apply (rule abs_ge_self)
-  done
+  by (metis (mono_tags, lifting) abs_ge_zero abs_of_nonneg bigo_lesseq1 dual_order.trans)
 
 lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> f x \<Longrightarrow> g =o O(h)"
-  apply (erule bigo_lesseq2)
-  apply (rule allI)
-  apply (subst abs_of_nonneg)
-   apply (erule spec)+
-  done
+  by (meson bigo_bounded bigo_elt_subset subsetD)
 
-lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
-    \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)"
-  apply (erule bigo_lesseq1)
-  apply (rule allI)
-  apply (subst abs_of_nonneg)
-   apply (erule spec)+
-  done
+lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)"
+  by (metis abs_of_nonneg bigo_lesseq1)
 
 lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
-  apply (unfold lesso_def)
-  apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
-   apply (erule ssubst)
-   apply (rule bigo_zero)
-  apply (unfold func_zero)
-  apply (rule ext)
-  apply (simp split: split_max)
-  done
+  by (smt (verit, del_insts) abs_ge_zero add_0 bigo_abs3 bigo_bounded diff_le_eq lesso_def max_def order_refl)
 
 lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
-  apply (unfold lesso_def)
-  apply (rule bigo_lesseq4)
+  unfolding lesso_def
+  apply (rule bigo_lesseq4 [of "f-g"])
     apply (erule set_plus_imp_minus)
-   apply (rule allI)
-   apply (rule max.cobounded2)
-  apply (rule allI)
-  apply (subst fun_diff_def)
-  apply (case_tac "0 \<le> k x - g x")
-   apply simp
-   apply (subst abs_of_nonneg)
-    apply (drule_tac x = x in spec) back
-    apply (simp add: algebra_simps)
-   apply (subst diff_conv_add_uminus)+
-   apply (rule add_right_mono)
-   apply (erule spec)
-  apply (rule order_trans)
-   prefer 2
-   apply (rule abs_ge_zero)
-  apply (simp add: algebra_simps)
-  done
+  using max.cobounded2 apply blast
+  by (smt (verit) abs_ge_zero abs_of_nonneg diff_ge_0_iff_ge diff_mono diff_self fun_diff_def order_refl max.coboundedI2 max_def)
 
 lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
-  apply (unfold lesso_def)
-  apply (rule bigo_lesseq4)
+  unfolding lesso_def
+  apply (rule bigo_lesseq4 [of "f-g"])
     apply (erule set_plus_imp_minus)
-   apply (rule allI)
-   apply (rule max.cobounded2)
-  apply (rule allI)
-  apply (subst fun_diff_def)
-  apply (case_tac "0 \<le> f x - k x")
-   apply simp
-   apply (subst abs_of_nonneg)
-    apply (drule_tac x = x in spec) back
-    apply (simp add: algebra_simps)
-   apply (subst diff_conv_add_uminus)+
-   apply (rule add_left_mono)
-   apply (rule le_imp_neg_le)
-   apply (erule spec)
-  apply (rule order_trans)
-   prefer 2
-   apply (rule abs_ge_zero)
-  apply (simp add: algebra_simps)
-  done
+  using max.cobounded2 apply blast
+  by (smt (verit) abs_eq_iff abs_ge_zero abs_if abs_minus_le_zero diff_left_mono fun_diff_def le_max_iff_disj order.trans order_eq_refl)
 
-lemma bigo_lesso4: "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
-  for k :: "'a \<Rightarrow> 'b::linordered_field"
-  apply (unfold lesso_def)
-  apply (drule set_plus_imp_minus)
-  apply (drule bigo_abs5) back
-  apply (simp add: fun_diff_def)
-  apply (drule bigo_useful_add)
-   apply assumption
-  apply (erule bigo_lesseq2) back
-  apply (rule allI)
-  apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
-  done
+lemma bigo_lesso4:
+  fixes k :: "'a \<Rightarrow> 'b::linordered_field"
+  assumes f: "f <o g =o O(k)" and g: "g =o h +o O(k)"
+  shows "f <o h =o O(k)"
+proof -
+  have "g - h \<in> O(k)"
+    by (simp add: g set_plus_imp_minus)
+  then have "(\<lambda>x. \<bar>g x - h x\<bar>) \<in> O(k)"
+    using bigo_abs5 by force
+  then have \<section>: "(\<lambda>x. max (f x - g x) 0) + (\<lambda>x. \<bar>g x - h x\<bar>) \<in> O(k)"
+    by (metis (mono_tags, lifting) bigo_lesseq1 bigo_useful_add dual_order.eq_iff f lesso_def)
+  have "\<bar>max (f x - h x) 0\<bar> \<le> ((\<lambda>x. max (f x - g x) 0) + (\<lambda>x. \<bar>g x - h x\<bar>)) x" for x
+    by (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
+  then show ?thesis
+    by (smt (verit, ccfv_SIG) \<section> bigo_lesseq2 lesso_def)
+qed
 
-lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * \<bar>h x\<bar>"
-  apply (simp only: lesso_def bigo_alt_def)
-  apply clarsimp
-  apply (rule_tac x = c in exI)
-  apply (rule allI)
-  apply (drule_tac x = x in spec)
-  apply (subgoal_tac "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0")
-   apply (clarsimp simp add: algebra_simps)
-  apply (rule abs_of_nonneg)
-  apply (rule max.cobounded2)
-  done
+
+lemma bigo_lesso5: 
+  assumes "f <o g =o O(h)" shows "\<exists>C. \<forall>x. f x \<le> g x + C * \<bar>h x\<bar>"
+proof -
+  obtain c where "0 < c" and c: "\<And>x. f x - g x \<le> c * \<bar>h x\<bar>"
+    using assms by (auto simp: lesso_def bigo_alt_def)
+  have "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0" for x
+    by (auto simp add: algebra_simps)
+  then show ?thesis
+    by (metis c add.commute diff_le_eq)
+qed
 
 lemma lesso_add: "f <o g =o O(h) \<Longrightarrow> k <o l =o O(h) \<Longrightarrow> (f + k) <o (g + l) =o O(h)"
-  apply (unfold lesso_def)
-  apply (rule bigo_lesseq3)
-    apply (erule bigo_useful_add)
-    apply assumption
-   apply (force split: split_max)
-  apply (auto split: split_max simp add: func_plus)
-  done
+  unfolding lesso_def
+  using bigo_useful_add by (fastforce split: split_max intro: bigo_lesseq3)
 
-lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> 0"
+lemma bigo_LIMSEQ1: "f \<longlonglongrightarrow> 0" if f: "f =o O(g)" and g: "g \<longlonglongrightarrow> 0"
   for f g :: "nat \<Rightarrow> real"
-  apply (simp add: LIMSEQ_iff bigo_alt_def)
-  apply clarify
-  apply (drule_tac x = "r / c" in spec)
-  apply (drule mp)
-   apply simp
-  apply clarify
-  apply (rule_tac x = no in exI)
-  apply (rule allI)
-  apply (drule_tac x = n in spec)+
-  apply (rule impI)
-  apply (drule mp)
-   apply assumption
-  apply (rule order_le_less_trans)
-   apply assumption
-  apply (rule order_less_le_trans)
-   apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
-    apply assumption
-   apply (erule mult_strict_left_mono)
-   apply assumption
-  apply simp
-  done
+proof -
+  { fix r::real
+    assume "0 < r"
+    obtain c::real where "0 < c"  and rc: "\<And>x. \<bar>f x\<bar> \<le> c * \<bar>g x\<bar>"
+      using f by (auto simp: LIMSEQ_iff bigo_alt_def)
+    with g \<open>0 < r\<close> obtain no where "\<forall>n\<ge>no. \<bar>g n\<bar> < r/c"
+      by (fastforce simp: LIMSEQ_iff)
+    then have "\<exists>no. \<forall>n\<ge>no. \<bar>f n\<bar> < r"
+      by (metis \<open>0 < c\<close> mult.commute order_le_less_trans pos_less_divide_eq rc) }
+  then show ?thesis
+    by (auto simp: LIMSEQ_iff)
+qed
 
-lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> a"
-  for f g h :: "nat \<Rightarrow> real"
-  apply (drule set_plus_imp_minus)
-  apply (drule bigo_LIMSEQ1)
-   apply assumption
-  apply (simp only: fun_diff_def)
-  apply (erule Lim_transform2)
-  apply assumption
-  done
+lemma bigo_LIMSEQ2: 
+  fixes f g :: "nat \<Rightarrow> real"
+  assumes "f =o g +o O(h)" "h \<longlonglongrightarrow> 0" and f: "f \<longlonglongrightarrow> a"
+  shows  "g \<longlonglongrightarrow> a"
+proof -
+  have "f - g \<longlonglongrightarrow> 0"
+    using assms bigo_LIMSEQ1 set_plus_imp_minus by blast
+  then have "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
+    by (simp add: fun_diff_def)
+  then show ?thesis
+    using Lim_transform_eq f by blast
+qed
 
 end