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\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas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. EX c. ALL x. abs (h x) <= c * abs (f x)}"


lemma bigo_pos_const: "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL 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. EX c. (0 < c & (ALL 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 (rule mult_pos_pos)
apply (assumption)+
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: mult_ac)
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(%x.0) = {%x.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 =
"%n. 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 (auto)
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 mult_nonneg_nonneg)
apply auto
apply (rule_tac x = "%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) <= (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)
apply (rule mult_nonneg_nonneg)
apply (erule order_less_imp_le)
apply simp
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: "ALL x. 0 <= f x ==> ALL 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 le_maxI1)
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 le_maxI1)
apply assumption
apply (subgoal_tac "ca * g xa <= max c ca * g xa")
apply (force)
apply (rule mult_right_mono)
apply (rule le_maxI2)
apply assumption
apply (rule abs_triangle_ineq)
apply (rule add_nonneg_nonneg)
apply assumption+
done

lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL 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: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
f : O(g)"

apply (erule bigo_bounded_alt [of f 1 g])
apply simp
done

lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL 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: diff_minus 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: "(%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(%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(%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) ==>
(%x. abs (f x)) =o (%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 : O(h)"
have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
by (rule bigo_abs2)
also have "... <= O(%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 "... <= 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 "(%x. abs (f x) - abs (g x)) : O(h)".
qed

lemma bigo_abs5: "f =o O(g) ==> (%x. 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(%x. abs(g x)) + O(%x. abs(h x))"
apply (subst bigo_abs3 [symmetric])+
apply (rule refl)
done
also have "... = O((%x. abs(g x)) + (%x. 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(%x. abs(g x)) + O(%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) <= 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 (rule mult_nonneg_nonneg)
apply auto
apply (simp add: mult_ac 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: mult_ac)
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:
assumes "ALL x. f x ~= 0"
shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
proof
fix h
assume "h : O(f * g)"
then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
by auto
also have "... <= O((%x. 1 / f x) * (f * g))"
by (rule bigo_mult2)
also have "(%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 "(%x. (1::'b) / f x) * h : O(g)" .
then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
by auto
also have "f * ((%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 : f *o O(g)" .
qed

lemma bigo_mult6: "ALL x. f x ~= 0 ==>
O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"

apply (rule equalityI)
apply (erule bigo_mult5)
apply (rule bigo_mult2)
done

lemma bigo_mult7: "ALL x. f x ~= 0 ==>
O(f * g) <= O(f::'a => ('b::linordered_field)) * O(g)"

apply (subst bigo_mult6)
apply assumption
apply (rule set_times_mono3)
apply (rule bigo_refl)
done

lemma bigo_mult8: "ALL x. f x ~= 0 ==>
O(f * g) = O(f::'a => ('b::linordered_field)) * 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 add: diff_minus)
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)
thus ?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: diff_minus add_ac)
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: "(%x. c) : O(%x. 1)"
by (auto simp add: bigo_def mult_ac)

lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
apply (rule bigo_elt_subset)
apply (rule bigo_const1)
done

lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%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: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
by (rule bigo_elt_subset, rule bigo_const3, assumption)

lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c) = O(%x. 1)"

by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)

lemma bigo_const_mult1: "(%x. 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(%x. c * f x) <= O(f)"
by (rule bigo_elt_subset, rule bigo_const_mult1)

lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%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: "(c::'a::linordered_field) ~= 0 ==>
O(f) <= O(%x. c * f x)"

by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)

lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c * f x) = O(f)"

by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)

lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==>
(%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 = "%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]: "(%x. 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: mult_ac)
done

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

lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
by (unfold bigo_def, auto)

lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o
O(%x. h(k x))"

apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
func_plus)
apply (erule bigo_compose1)
done


subsection {* Setsum *}

lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
(%x. SUM y : A x. f x y) =o O(%x. SUM 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: "ALL x y. 0 <= h x y ==>
EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
(%x. SUM y : A x. f x y) =o O(%x. SUM 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: "ALL y. 0 <= h y ==>
EX c. ALL y. abs(f y) <= c * (h y) ==>
(%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"

by (rule bigo_setsum1, auto)

lemma bigo_setsum3: "f =o O(h) ==>
(%x. SUM y : A x. (l x y) * f(k x y)) =o
O(%x. SUM 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) ==>
(%x. SUM y : A x. l x y * f(k x y)) =o
(%x. SUM y : A x. l x y * g(k x y)) +o
O(%x. SUM 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) ==> ALL x y. 0 <= l x y ==>
ALL x. 0 <= h x ==>
(%x. SUM y : A x. (l x y) * f(k x y)) =o
O(%x. SUM y : A x. (l x y) * h(k x y))"

apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) =
(%x. SUM y : 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 (rule mult_nonneg_nonneg)
apply auto
done

lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
ALL x. 0 <= h x ==>
(%x. SUM y : A x. (l x y) * f(k x y)) =o
(%x. SUM y : A x. (l x y) * g(k x y)) +o
O(%x. SUM 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: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) * f =o O(h) ==> f =o O(h)"

apply (rule subsetD)
apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
apply assumption
apply (rule bigo_const_mult6)
apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
apply (erule ssubst)
apply (erule set_times_intro2)
apply (simp add: func_times)
done

lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. 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 (rule mult_nonneg_nonneg)
apply force
apply force
apply (subgoal_tac "x = Suc (x - 1)")
apply (erule ssubst) back
apply (erule spec)
apply simp
done

lemma bigo_fix2:
"(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. 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 = (%x. max (f x - g x) 0)"

lemma bigo_lesseq1: "f =o O(h) ==> ALL 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) ==> ALL 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) ==> ALL x. 0 <= g x ==> ALL 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) ==>
ALL x. 0 <= g x ==> ALL 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: "ALL x. f x <= g x ==> f <o g =o O(h)"
apply (unfold lesso_def)
apply (subgoal_tac "(%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) ==>
ALL x. 0 <= k x ==> ALL 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 le_maxI2)
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_minus)+
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) ==>
ALL x. 0 <= k x ==> ALL 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 le_maxI2)
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_minus)+
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::'a=>'b::linordered_field) ==>
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) ==>
EX C. ALL 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 le_maxI2)
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 (erule divide_pos_pos)
apply assumption
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