Theory BigO

theory BigO
imports Complex_Main Function_Algebras Set_Algebras
(*  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\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas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 => 'b::linordered_idom) => ('a => 'b) set"  ("(1O'(_'))")
  where "O(f:: 'a => 'b) = {h. ∃c. ∀x. abs (h x) ≤ c * abs (f x)}"

lemma bigo_pos_const:
  "(∃c::'a::linordered_idom. ∀x. abs (h x) ≤ c * abs (f x)) <->
    (∃c. 0 < c ∧ (∀x. abs (h x) ≤ 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) ≤ 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. ∃c. 0 < c ∧ (∀x. abs (h x) ≤ c * abs (f x))}"
  by (auto simp add: bigo_def bigo_pos_const)

lemma bigo_elt_subset [intro]: "f ∈ O(g) ==> O(f) ≤ 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) ≤ ca * (c * abs (g xa))")
  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 ∈ O(f)"
  apply(auto simp add: bigo_def)
  apply(rule_tac x = 1 in exI)
  apply simp
  done

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

lemma bigo_zero2: "O(λx. 0) = {λx. 0}"
  by (auto simp add: bigo_def)

lemma bigo_plus_self_subset [intro]: "O(f) + O(f) ⊆ 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) ⊆ 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 = "λn. if abs (g n) ≤ (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) ≤ (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) ≤ 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 = "λn. 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) ≤ (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) ≤ 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 ⊆ O(f) ==> B ⊆ O(f) ==> A + B ⊆ O(f)"
  apply (subgoal_tac "A + B ⊆ O(f) + O(f)")
  apply (erule order_trans)
  apply simp
  apply (auto del: subsetI simp del: bigo_plus_idemp)
  done

lemma bigo_plus_eq: "∀x. 0 ≤ f x ==> ∀x. 0 ≤ g x ==> 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 ≤ 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 ≤ f xa + g xa")
  apply (simp add: ring_distribs)
  apply (subgoal_tac "abs (a xa + b xa) ≤ abs (a xa) + abs (b xa)")
  apply (subgoal_tac "abs (a xa) + abs (b xa) ≤ max c ca * f xa + max c ca * g xa")
  apply force
  apply (rule add_mono)
  apply (subgoal_tac "c * f xa ≤ max c ca * f xa")
  apply force
  apply (rule mult_right_mono)
  apply (rule max.cobounded1)
  apply assumption
  apply (subgoal_tac "ca * g xa ≤ 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: "∀x. 0 ≤ f x ==> ∀x. f x ≤ c * g x ==> f ∈ 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: "∀x. 0 ≤ f x ==> ∀x. f x ≤ g x ==> f ∈ O(g)"
  apply (erule bigo_bounded_alt [of f 1 g])
  apply simp
  done

lemma bigo_bounded2: "∀x. lb x ≤ f x ==> ∀x. f x ≤ lb x + g x ==> f ∈ 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: "(λx. 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(λx. 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(λx. 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) ==> (λx. abs (f x)) =o (λx. 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 ∈ O(h)"
  have "(λx. abs (f x) - abs (g x)) =o O(λx. abs (abs (f x) - abs (g x)))"
    by (rule bigo_abs2)
  also have "… ⊆ O(λx. 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 "… ⊆ O(f - g)"
    apply (rule bigo_elt_subset)
    apply (subst fun_diff_def)
    apply (rule bigo_abs)
    done
  also from a have "… ⊆ O(h)"
    by (rule bigo_elt_subset)
  finally show "(λx. abs (f x) - abs (g x)) ∈ O(h)".
qed

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

lemma bigo_elt_subset2 [intro]: "f ∈ g +o O(h) ==> O(f) ⊆ O(g) + O(h)"
proof -
  assume "f ∈ g +o O(h)"
  also have "… ⊆ O(g) + O(h)"
    by (auto del: subsetI)
  also have "… = O(λx. abs (g x)) + O(λx. abs (h x))"
    apply (subst bigo_abs3 [symmetric])+
    apply (rule refl)
    done
  also have "… = O((λx. abs (g x)) + (λx. abs (h x)))"
    by (rule bigo_plus_eq [symmetric]) auto
  finally have "f ∈ …" .
  then have "O(f) ⊆ …"
    by (elim bigo_elt_subset)
  also have "… = O(λx. abs (g x)) + O(λx. 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) ⊆ 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: ac_simps abs_mult)
  done

lemma bigo_mult2 [intro]: "f *o O(g) ⊆ 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) ≤ abs (f x) * (c * abs (g x))")
  apply (force simp add: ac_simps)
  apply (rule mult_left_mono, assumption)
  apply (rule abs_ge_zero)
  done

lemma bigo_mult3: "f ∈ O(h) ==> g ∈ O(j) ==> f * g ∈ O(h * j)"
  apply (rule subsetD)
  apply (rule bigo_mult)
  apply (erule set_times_intro, assumption)
  done

lemma bigo_mult4 [intro]: "f ∈ k +o O(h) ==> g * f ∈ (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 => 'b::linordered_field"
  assumes "∀x. f x ≠ 0"
  shows "O(f * g) ⊆ f *o O(g)"
proof
  fix h
  assume "h ∈ O(f * g)"
  then have "(λx. 1 / (f x)) * h ∈ (λx. 1 / f x) *o O(f * g)"
    by auto
  also have "… ⊆ O((λx. 1 / f x) * (f * g))"
    by (rule bigo_mult2)
  also have "(λx. 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 "(λx. (1::'b) / f x) * h ∈ O(g)" .
  then have "f * ((λx. (1::'b) / f x) * h) ∈ f *o O(g)"
    by auto
  also have "f * ((λx. (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 ∈ f *o O(g)" .
qed

lemma bigo_mult6:
  fixes f :: "'a => 'b::linordered_field"
  shows "∀x. f x ≠ 0 ==> 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 => 'b::linordered_field"
  shows "∀x. f x ≠ 0 ==> O(f * g) ⊆ 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 => 'b::linordered_field"
  shows "∀x. f x ≠ 0 ==> O(f * g) = O(f) * O(g)"
  apply (rule equalityI)
  apply (erule bigo_mult7)
  apply (rule bigo_mult)
  done

lemma bigo_minus [intro]: "f ∈ O(g) ==> - f ∈ O(g)"
  by (auto simp add: bigo_def fun_Compl_def)

lemma bigo_minus2: "f ∈ g +o O(h) ==> - f ∈ -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 ∈ O(g) ==> f +o O(g) ⊆ O(g)"
proof -
  assume a: "f ∈ O(g)"
  show "f +o O(g) ⊆ O(g)"
  proof -
    have "f ∈ O(f)" by auto
    then have "f +o O(g) ⊆ O(f) + O(g)"
      by (auto del: subsetI)
    also have "… ⊆ O(g) + O(g)"
    proof -
      from a have "O(f) ⊆ O(g)" by (auto del: subsetI)
      then show ?thesis by (auto del: subsetI)
    qed
    also have "… ⊆ O(g)" by simp
    finally show ?thesis .
  qed
qed

lemma bigo_plus_absorb_lemma2: "f ∈ O(g) ==> O(g) ⊆ f +o O(g)"
proof -
  assume a: "f ∈ O(g)"
  show "O(g) ⊆ f +o O(g)"
  proof -
    from a have "- f ∈ O(g)"
      by auto
    then have "- f +o O(g) ⊆ O(g)"
      by (elim bigo_plus_absorb_lemma1)
    then have "f +o (- f +o O(g)) ⊆ 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 ∈ O(g) ==> 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 ∈ O(g) ==> A ⊆ O(g) ==> f +o A ⊆ O(g)"
  apply (subgoal_tac "f +o A ⊆ f +o O(g)")
  apply force+
  done

lemma bigo_add_commute_imp: "f ∈ g +o O(h) ==> g ∈ 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 ∈ g +o O(h) <-> g ∈ f +o O(h)"
  apply (rule iffI)
  apply (erule bigo_add_commute_imp)+
  done

lemma bigo_const1: "(λx. c) ∈ O(λx. 1)"
  by (auto simp add: bigo_def ac_simps)

lemma bigo_const2 [intro]: "O(λx. c) ⊆ O(λx. 1)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const1)
  done

lemma bigo_const3:
  fixes c :: "'a::linordered_field"
  shows "c ≠ 0 ==> (λx. 1) ∈ O(λx. 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 ≠ 0 ==> O(λx. 1) ⊆ O(λx. c)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const3)
  apply assumption
  done

lemma bigo_const [simp]:
  fixes c :: "'a::linordered_field"
  shows "c ≠ 0 ==> O(λx. c) = O(λx. 1)"
  apply (rule equalityI)
  apply (rule bigo_const2)
  apply (rule bigo_const4)
  apply assumption
  done

lemma bigo_const_mult1: "(λx. c * f x) ∈ 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(λx. c * f x) ⊆ O(f)"
  apply (rule bigo_elt_subset)
  apply (rule bigo_const_mult1)
  done

lemma bigo_const_mult3:
  fixes c :: "'a::linordered_field"
  shows "c ≠ 0 ==> f ∈ O(λx. 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 ≠ 0 ==> O(f) ⊆ O(λx. 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 ≠ 0 ==> O(λx. 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 ≠ 0 ==> (λx. 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 = "λy. 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]: "(λx. c) *o O(f) ⊆ 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: ac_simps)
  done

lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (λx. c * f x) =o O(g)"
proof -
  assume "f =o O(g)"
  then have "(λx. c) * f =o (λx. c) *o O(g)"
    by auto
  also have "(λx. c) * f = (λx. c * f x)"
    by (simp add: func_times)
  also have "(λx. c) *o O(g) ⊆ O(g)"
    by (auto del: subsetI)
  finally show ?thesis .
qed

lemma bigo_compose1: "f =o O(g) ==> (λx. f (k x)) =o O(λx. g (k x))"
  unfolding bigo_def by auto

lemma bigo_compose2: "f =o g +o O(h) ==>
    (λx. f (k x)) =o (λx. g (k x)) +o O(λx. 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: "∀x. ∀y ∈ A x. 0 ≤ h x y ==>
    ∃c. ∀x. ∀y ∈ A x. abs (f x y) ≤ c * (h x y) ==>
      (λx. ∑y ∈ A x. f x y) =o O(λx. ∑y ∈ 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: "∀x y. 0 ≤ h x y ==>
    ∃c. ∀x y. abs (f x y) ≤ c * h x y ==>
      (λx. ∑y ∈ A x. f x y) =o O(λx. ∑y ∈ 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: "∀y. 0 ≤ h y ==>
    ∃c. ∀y. abs (f y) ≤ c * (h y) ==>
      (λx. ∑y ∈ A x. f y) =o O(λx. ∑y ∈ A x. h y)"
  by (rule bigo_setsum1) auto

lemma bigo_setsum3: "f =o O(h) ==>
    (λx. ∑y ∈ A x. l x y * f (k x y)) =o O(λx. ∑y ∈ 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) ==>
    (λx. ∑y ∈ A x. l x y * f (k x y)) =o
      (λx. ∑y ∈ A x. l x y * g (k x y)) +o
        O(λx. ∑y ∈ 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) ==> ∀x y. 0 ≤ l x y ==>
    ∀x. 0 ≤ h x ==>
      (λx. ∑y ∈ A x. l x y * f (k x y)) =o
        O(λx. ∑y ∈ A x. l x y * h (k x y))"
  apply (subgoal_tac "(λx. ∑y ∈ A x. l x y * h (k x y)) =
      (λx. ∑y ∈ A x. abs (l x y * h (k x y)))")
  apply (erule ssubst)
  apply (erule bigo_setsum3)
  apply (rule ext)
  apply (rule setsum.cong)
  apply (rule refl)
  apply (subst abs_of_nonneg)
  apply auto
  done

lemma bigo_setsum6: "f =o g +o O(h) ==> ∀x y. 0 ≤ l x y ==>
    ∀x. 0 ≤ h x ==>
      (λx. ∑y ∈ A x. l x y * f (k x y)) =o
        (λx. ∑y ∈ A x. l x y * g (k x y)) +o
          O(λx. ∑y ∈ 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 ⊆ O(f) ==> B ⊆ O(f) ==> A + B ⊆ O(f)"
  apply (subst bigo_plus_idemp [symmetric])
  apply (rule set_plus_mono2)
  apply assumption+
  done

lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> 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 ≠ 0 ==> (λx. c) * f =o O(h) ==> f =o O(h)"
  apply (rule subsetD)
  apply (subgoal_tac "(λx. 1 / c) *o O(h) ⊆ O(h)")
  apply assumption
  apply (rule bigo_const_mult6)
  apply (subgoal_tac "f = (λx. 1 / c) * ((λx. c) * f)")
  apply (erule ssubst)
  apply (erule set_times_intro2)
  apply (simp add: func_times)
  done

lemma bigo_fix: "(λx::nat. f (x + 1)) =o O(λx. h (x + 1)) ==> f 0 = 0 ==> 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:
    "(λx. f ((x::nat) + 1)) =o (λx. g(x + 1)) +o O(λx. h(x + 1)) ==>
       f 0 = g 0 ==> 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 => 'b::linordered_idom) => ('a => 'b) => 'a => 'b"  (infixl "<o" 70)
  where "f <o g = (λx. max (f x - g x) 0)"

lemma bigo_lesseq1: "f =o O(h) ==> ∀x. abs (g x) ≤ abs (f x) ==> 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) ==> ∀x. abs (g x) ≤ f x ==> 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) ==> ∀x. 0 ≤ g x ==> ∀x. g x ≤ f x ==> 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) ==>
    ∀x. 0 ≤ g x ==> ∀x. g x ≤ abs (f x) ==> g =o O(h)"
  apply (erule bigo_lesseq1)
  apply (rule allI)
  apply (subst abs_of_nonneg)
  apply (erule spec)+
  done

lemma bigo_lesso1: "∀x. f x ≤ g x ==> f <o g =o O(h)"
  apply (unfold lesso_def)
  apply (subgoal_tac "(λx. 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) ==>
    ∀x. 0 ≤ k x ==> ∀x. k x ≤ f x ==> 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 ≤ 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) ==>
    ∀x. 0 ≤ k x ==> ∀x. g x ≤ k x ==> 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 ≤ 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 => 'b::linordered_field"
  shows "f <o g =o O(k) ==> g =o h +o O(k) ==> 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) ==> ∃C. ∀x. f x ≤ 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) ==> k <o l =o O(h) ==> (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) ==> g ----> 0 ==> 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) ==> h ----> 0 ==> f ----> a ==> 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