src/HOL/Library/BigO.thy

made mult_pos_pos a simp rule

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

header {* Big O notation *}

theory BigO
imports Complex_Main Function_Algebras Set_Algebras
begin

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

The main changes in this version are as follows:
\begin{itemize}
\item We have eliminated the @{text O} operator on sets. (Most uses of this seem
  to be inessential.)
\item We no longer use @{text "+"} as output syntax for @{text "+o"}
\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas
  involving `@{text "setsum"}.
\item The library has been expanded, with e.g.~support for expressions of
  the form @{text "f < g + O(h)"}.
\end{itemize}

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 @{text "subsetI"} as an intro rule,
rather than as an @{text "intro!"} rule, for example, using
\isa{\isakeyword{declare}}~@{text "subsetI [del, intro]"}.
*}

subsection {* Definitions *}

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

lemma bigo_pos_const:
  "(\<exists>c::'a::linordered_idom. \<forall>x. abs (h x) \<le> c * abs (f x)) \<longleftrightarrow>
    (\<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> c * abs (f x)))"
  apply auto
  apply (case_tac "c = 0")
  apply simp
  apply (rule_tac x = "1" in exI)
  apply simp
  apply (rule_tac x = "abs c" in exI)
  apply auto
  apply (subgoal_tac "c * abs (f x) \<le> abs c * abs (f x)")
  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. abs (h x) \<le> c * abs (f x))}"
  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 * abs (f xa) \<le> ca * (c * abs (g xa))")
  apply (erule order_trans)
  apply (simp add: mult_ac)
  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 abs (g n) \<le> (abs (f n)) then x n else 0" in exI)
  apply (rule conjI)
  apply (rule_tac x = "c + c" in exI)
  apply (clarsimp)
  apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (f xa)")
  apply (erule_tac x = xa in allE)
  apply (erule order_trans)
  apply (simp)
  apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
  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 (abs (f n)) < abs (g n) then x n else 0" in exI)
  apply (rule conjI)
  apply (rule_tac x = "c + c" in exI)
  apply auto
  apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (g xa)")
  apply (erule_tac x = xa in allE)
  apply (erule order_trans)
  apply simp
  apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
  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 "abs (a xa + b xa) \<le> abs (a xa) + abs (b xa)")
  apply (subgoal_tac "abs (a xa) + abs (b xa) \<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 = "abs c" 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
  apply (drule_tac x = x in spec)+
  apply force
  done

lemma bigo_abs: "(\<lambda>x. abs (f x)) =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. abs (f x))"
  apply (unfold bigo_def)
  apply auto
  apply (rule_tac x = 1 in exI)
  apply auto
  done

lemma bigo_abs3: "O(f) = O(\<lambda>x. abs (f x))"
  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. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
  apply (drule set_plus_imp_minus)
  apply (rule set_minus_imp_plus)
  apply (subst fun_diff_def)
proof -
  assume a: "f - g \<in> O(h)"
  have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs (abs (f x) - abs (g x)))"
    by (rule bigo_abs2)
  also have "\<dots> \<subseteq> O(\<lambda>x. abs (f x - g x))"
    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 a have "\<dots> \<subseteq> O(h)"
    by (rule bigo_elt_subset)
  finally show "(\<lambda>x. abs (f x) - abs (g x)) \<in> O(h)".
qed

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

lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
proof -
  assume "f \<in> g +o O(h)"
  also have "\<dots> \<subseteq> O(g) + O(h)"
    by (auto del: subsetI)
  also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
    apply (subst bigo_abs3 [symmetric])+
    apply (rule refl)
    done
  also have "\<dots> = O((\<lambda>x. abs (g x)) + (\<lambda>x. abs (h x)))"
    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. abs (g x)) + O(\<lambda>x. abs (h x))"
    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 * abs (f x * g x) = (c * abs (f x)) * (ca * abs (g x))")
  apply (erule ssubst)
  apply (subst abs_mult)
  apply (rule mult_mono)
  apply assumption+
  apply auto
  apply (simp add: mult_ac 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 "abs (f x) * abs (b x) \<le> abs (f x) * (c * abs (g x))")
  apply (force simp add: mult_ac)
  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 mult_ac)
    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 mult_ac)
    done
  finally show "h \<in> f *o O(g)" .
qed

lemma bigo_mult6:
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
  apply (rule equalityI)
  apply (erule bigo_mult5)
  apply (rule bigo_mult2)
  done

lemma bigo_mult7:
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
  apply (subst bigo_mult6)
  apply assumption
  apply (rule set_times_mono3)
  apply (rule bigo_refl)
  done

lemma bigo_mult8:
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
  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: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
proof -
  assume a: "f \<in> O(g)"
  show "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 a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
      thus ?thesis by (auto del: subsetI)
    qed
    also have "\<dots> \<subseteq> O(g)" by simp
    finally show ?thesis .
  qed
qed

lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
proof -
  assume a: "f \<in> O(g)"
  show "O(g) \<subseteq> f +o O(g)"
  proof -
    from a 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
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: add_ac)
  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 mult_ac)

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:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
  apply (simp add: bigo_def)
  apply (rule_tac x = "abs (inverse c)" in exI)
  apply (simp add: abs_mult [symmetric])
  done

lemma bigo_const4:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const3)
  apply assumption
  done

lemma bigo_const [simp]:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
  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 = "abs c" 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:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
  apply (simp add: bigo_def)
  apply (rule_tac x = "abs (inverse c)" in exI)
  apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
  done

lemma bigo_const_mult4:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const_mult3)
  apply assumption
  done

lemma bigo_const_mult [simp]:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
  apply (rule equalityI)
  apply (rule bigo_const_mult2)
  apply (erule bigo_const_mult4)
  done

lemma bigo_const_mult5 [simp]:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
  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 = "abs (inverse c) * ca" in exI)
  apply (rule allI)
  apply (subst mult_assoc)
  apply (rule mult_left_mono)
  apply (erule spec)
  apply force
  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 * abs c" in exI)
  apply (rule allI)
  apply (subgoal_tac "ca * abs c * abs (f x) = abs c * (ca * abs (f x))")
  apply (erule ssubst)
  apply (subst abs_mult)
  apply (rule mult_left_mono)
  apply (erule spec)
  apply simp
  apply(simp add: mult_ac)
  done

lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
proof -
  assume "f =o O(g)"
  then 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))"
  unfolding bigo_def by auto

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 {* Setsum *}

lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
    \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) \<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 = "abs c" in exI)
  apply (subst abs_of_nonneg) back back
  apply (rule setsum_nonneg)
  apply force
  apply (subst setsum_right_distrib)
  apply (rule allI)
  apply (rule order_trans)
  apply (rule setsum_abs)
  apply (rule setsum_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_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
    \<exists>c. \<forall>x y. abs (f x y) \<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_setsum_main)
  apply force
  apply clarsimp
  apply (rule_tac x = c in exI)
  apply force
  done

lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
    \<exists>c. \<forall>y. abs (f y) \<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_setsum1) auto

lemma bigo_setsum3: "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. abs (l x y * h (k x y)))"
  apply (rule bigo_setsum1)
  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_setsum4: "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. abs (l x y * h (k x y)))"
  apply (rule set_minus_imp_plus)
  apply (subst fun_diff_def)
  apply (subst setsum_subtractf [symmetric])
  apply (subst right_diff_distrib [symmetric])
  apply (rule bigo_setsum3)
  apply (subst fun_diff_def [symmetric])
  apply (erule set_plus_imp_minus)
  done

lemma bigo_setsum5: "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. abs (l x y * h (k x y)))")
  apply (erule ssubst)
  apply (erule bigo_setsum3)
  apply (rule ext)
  apply (rule setsum_cong2)
  apply (subst abs_of_nonneg)
  apply auto
  done

lemma bigo_setsum6: "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 setsum_subtractf [symmetric])
  apply (subst right_diff_distrib [symmetric])
  apply (rule bigo_setsum5)
  apply (subst fun_diff_def [symmetric])
  apply (drule set_plus_imp_minus)
  apply auto
  done


subsection {* Misc useful stuff *}

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:
  fixes c :: "'a::linordered_field"
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
  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 {* Less than or equal to *}

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. abs (g x) \<le> abs (f x) \<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. abs (g x) \<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> abs (f x) \<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:
  fixes k :: "'a \<Rightarrow> 'b::linordered_field"
  shows "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
  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 * abs (h x)"
  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 "abs (max (f x - g x) 0) = 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 ----> 0 \<Longrightarrow> f ----> (0::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 * abs (g n) < 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 ----> 0 \<Longrightarrow> f ----> a \<Longrightarrow> g ----> (a::real)"
  apply (drule set_plus_imp_minus)
  apply (drule bigo_LIMSEQ1)
  apply assumption
  apply (simp only: fun_diff_def)
  apply (erule LIMSEQ_diff_approach_zero2)
  apply assumption
  done

end