src/HOL/Library/BigO.thy

more formal comments;

(*  Title:      HOL/Library/BigO.thy
    Authors:    Jeremy Avigad and Kevin Donnelly
*)

section \<open>Big O notation\<close>

theory BigO
  imports
    Complex_Main
    Function_Algebras
    Set_Algebras
begin

text \<open>
  This library is designed to support asymptotic ``big O'' calculations,
  i.e.~reasoning with expressions of the form \<open>f = O(g)\<close> and \<open>f = g + O(h)\<close>.
  An earlier version of this library is described in detail in @{cite
  "Avigad-Donnelly"}.

  The main changes in this version are as follows:

    \<^item> We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
      to be inessential.)
    \<^item> We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
    \<^item> Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
      involving `\<open>sum\<close>.
    \<^item> The library has been expanded, with e.g.~support for expressions of
      the form \<open>f < g + O(h)\<close>.

  Note also since the Big O library includes rules that demonstrate set
  inclusion, to use the automated reasoners effectively with the library one
  should redeclare the theorem \<open>subsetI\<close> as an intro rule, rather than as an
  \<open>intro!\<close> rule, for example, using \<^theory_text>\<open>declare subsetI [del, intro]\<close>.
\<close>


subsection \<open>Definitions\<close>

definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
  where "O(f:: 'a \<Rightarrow> 'b) = {h. \<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"

lemma bigo_pos_const:
  "(\<exists>c::'a::linordered_idom. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) \<longleftrightarrow>
    (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
  apply auto
  apply (case_tac "c = 0")
   apply simp
   apply (rule_tac x = "1" in exI)
   apply simp
  apply (rule_tac x = "\<bar>c\<bar>" in exI)
  apply auto
  apply (subgoal_tac "c * \<bar>f x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>")
   apply (erule_tac x = x in allE)
   apply force
  apply (rule mult_right_mono)
   apply (rule abs_ge_self)
  apply (rule abs_ge_zero)
  done

lemma bigo_alt_def: "O(f) = {h. \<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)}"
  by (auto simp add: bigo_def bigo_pos_const)

lemma bigo_elt_subset [intro]: "f \<in> O(g) \<Longrightarrow> O(f) \<le> O(g)"
  apply (auto simp add: bigo_alt_def)
  apply (rule_tac x = "ca * c" in exI)
  apply (rule conjI)
   apply simp
  apply (rule allI)
  apply (drule_tac x = "xa" in spec)+
  apply (subgoal_tac "ca * \<bar>f xa\<bar> \<le> ca * (c * \<bar>g xa\<bar>)")
   apply (erule order_trans)
   apply (simp add: ac_simps)
  apply (rule mult_left_mono, assumption)
  apply (rule order_less_imp_le, assumption)
  done

lemma bigo_refl [intro]: "f \<in> O(f)"
  apply (auto simp add: bigo_def)
  apply (rule_tac x = 1 in exI)
  apply simp
  done

lemma bigo_zero: "0 \<in> O(g)"
  apply (auto simp add: bigo_def func_zero)
  apply (rule_tac x = 0 in exI)
  apply auto
  done

lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
  by (auto simp add: bigo_def)

lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \<subseteq> O(f)"
  apply (auto simp add: bigo_alt_def set_plus_def)
  apply (rule_tac x = "c + ca" in exI)
  apply auto
  apply (simp add: ring_distribs func_plus)
  apply (rule order_trans)
   apply (rule abs_triangle_ineq)
  apply (rule add_mono)
   apply force
  apply force
  done

lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
  apply (rule equalityI)
   apply (rule bigo_plus_self_subset)
  apply (rule set_zero_plus2)
  apply (rule bigo_zero)
  done

lemma bigo_plus_subset [intro]: "O(f + g) \<subseteq> O(f) + O(g)"
  apply (rule subsetI)
  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
  apply (subst bigo_pos_const [symmetric])+
  apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> \<le> \<bar>f n\<bar> then x n else 0" in exI)
  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 (erule_tac x = xa in allE)
    apply (erule order_trans)
    apply (simp)
   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
    apply (erule order_trans)
    apply (simp add: ring_distribs)
   apply (rule mult_left_mono)
    apply (simp add: abs_triangle_ineq)
   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)
   apply (rule_tac x = "c + c" in exI)
   apply auto
  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>g xa\<bar>")
   apply (erule_tac x = xa in allE)
   apply (erule order_trans)
   apply simp
  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
   apply (erule order_trans)
   apply (simp add: ring_distribs)
  apply (rule mult_left_mono)
   apply (rule abs_triangle_ineq)
  apply (simp add: order_less_le)
  done

lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
  apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
   apply (erule order_trans)
   apply simp
  apply (auto del: subsetI simp del: bigo_plus_idemp)
  done

lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
  apply (rule equalityI)
   apply (rule bigo_plus_subset)
  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 (erule order_less_le_trans)
    apply assumption
   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 (rule add_mono)
     apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
      apply force
     apply (rule mult_right_mono)
      apply (rule max.cobounded1)
     apply assumption
    apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
     apply force
    apply (rule mult_right_mono)
     apply (rule max.cobounded2)
    apply assumption
   apply (rule abs_triangle_ineq)
  apply (rule add_nonneg_nonneg)
   apply assumption+
  done

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 add: abs_mult [symmetric])
  done

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

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

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

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

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

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)
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)".
qed

lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
  by (auto simp: bigo_def)

lemma bigo_elt_subset2 [intro]:
  assumes *: "f \<in> g +o O(h)"
  shows "O(f) \<subseteq> O(g) + O(h)"
proof -
  note *
  also have "g +o O(h) \<subseteq> O(g) + O(h)"
    by (auto del: subsetI)
  also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
    by (subst bigo_abs3 [symmetric])+ (rule refl)
  also have "\<dots> = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
    by (rule bigo_plus_eq [symmetric]) auto
  finally have "f \<in> \<dots>" .
  then have "O(f) \<subseteq> \<dots>"
    by (elim bigo_elt_subset)
  also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
    by (rule bigo_plus_eq, auto)
  finally show ?thesis
    by (simp add: bigo_abs3 [symmetric])
qed

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 (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

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

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

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

lemma bigo_mult5:
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
  assumes "\<forall>x. f x \<noteq> 0"
  shows "O(f * g) \<subseteq> f *o O(g)"
proof
  fix h
  assume "h \<in> O(f * g)"
  then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)"
    by auto
  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
  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
  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

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

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

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_minus3: "O(- f) = O(f)"
  by (auto simp add: bigo_def fun_Compl_def)

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

lemma bigo_plus_absorb_lemma2:
  assumes *: "f \<in> O(g)"
  shows "O(g) \<subseteq> f +o O(g)"
proof -
  from * have "- f \<in> O(g)"
    by auto
  then have "- f +o O(g) \<subseteq> O(g)"
    by (elim bigo_plus_absorb_lemma1)
  then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
    by auto
  also have "f +o (- f +o O(g)) = O(g)"
    by (simp add: set_plus_rearranges)
  finally show ?thesis .
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

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

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

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

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

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 add: abs_mult [symmetric])
  done

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

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

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 add: abs_mult [symmetric])
  done

lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const_mult1)
  done

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

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

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

lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
  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

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

lemma bigo_const_mult7 [intro]:
  assumes *: "f =o O(g)"
  shows "(\<lambda>x. c * f x) =o O(g)"
proof -
  from * have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
    by auto
  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
    by (simp add: func_times)
  also have "(\<lambda>x. c) *o O(g) \<subseteq> O(g)"
    by (auto del: subsetI)
  finally show ?thesis .
qed

lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
  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


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_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

lemma bigo_sum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
    \<exists>c. \<forall>y. \<bar>f y\<bar> \<le> c * (h y) \<Longrightarrow>
      (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
  by (rule bigo_sum1) auto

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

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

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

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


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

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

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

lemma bigo_fix2:
    "(\<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)
  done


subsection \<open>Less than or equal to\<close>

definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"  (infixl "<o" 70)
  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

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

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

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_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

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)
    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

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)
    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

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_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 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

lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<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

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

end