(* Author: Florian Haftmann, TU Muenchen *)
header {* Comparing growth of functions on natural numbers by a preorder relation *}
theory Landau
imports Main Preorder
begin
text {*
We establish a preorder releation @{text "\<lesssim>"} on functions
from @{text "\<nat>"} to @{text "\<nat>"} such that @{text "f \<lesssim> g \<longleftrightarrow> f \<in> O(g)"}.
*}
subsection {* Auxiliary *}
lemma Ex_All_bounded:
fixes n :: nat
assumes "\<exists>n. \<forall>m\<ge>n. P m"
obtains m where "m \<ge> n" and "P m"
proof -
from assms obtain q where m_q: "\<forall>m\<ge>q. P m" ..
let ?m = "max q n"
have "?m \<ge> n" by auto
moreover from m_q have "P ?m" by auto
ultimately show thesis ..
qed
subsection {* The @{text "\<lesssim>"} relation *}
definition less_eq_fun :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" (infix "\<lesssim>" 50) where
"f \<lesssim> g \<longleftrightarrow> (\<exists>c n. \<forall>m\<ge>n. f m \<le> Suc c * g m)"
lemma less_eq_fun_intro:
assumes "\<exists>c n. \<forall>m\<ge>n. f m \<le> Suc c * g m"
shows "f \<lesssim> g"
unfolding less_eq_fun_def by (rule assms)
lemma less_eq_fun_not_intro:
assumes "\<And>c n. \<exists>m\<ge>n. Suc c * g m < f m"
shows "\<not> f \<lesssim> g"
using assms unfolding less_eq_fun_def linorder_not_le [symmetric]
by blast
lemma less_eq_fun_elim:
assumes "f \<lesssim> g"
obtains n c where "\<And>m. m \<ge> n \<Longrightarrow> f m \<le> Suc c * g m"
using assms unfolding less_eq_fun_def by blast
lemma less_eq_fun_not_elim:
assumes "\<not> f \<lesssim> g"
obtains m where "m \<ge> n" and "Suc c * g m < f m"
using assms unfolding less_eq_fun_def linorder_not_le [symmetric]
by blast
lemma less_eq_fun_refl:
"f \<lesssim> f"
proof (rule less_eq_fun_intro)
have "\<exists>n. \<forall>m\<ge>n. f m \<le> Suc 0 * f m" by auto
then show "\<exists>c n. \<forall>m\<ge>n. f m \<le> Suc c * f m" by blast
qed
lemma less_eq_fun_trans:
assumes f_g: "f \<lesssim> g" and g_h: "g \<lesssim> h"
shows f_h: "f \<lesssim> h"
proof -
from f_g obtain n\<^isub>1 c\<^isub>1
where P1: "\<And>m. m \<ge> n\<^isub>1 \<Longrightarrow> f m \<le> Suc c\<^isub>1 * g m"
by (erule less_eq_fun_elim)
moreover from g_h obtain n\<^isub>2 c\<^isub>2
where P2: "\<And>m. m \<ge> n\<^isub>2 \<Longrightarrow> g m \<le> Suc c\<^isub>2 * h m"
by (erule less_eq_fun_elim)
ultimately have "\<And>m. m \<ge> max n\<^isub>1 n\<^isub>2 \<Longrightarrow> f m \<le> Suc c\<^isub>1 * g m \<and> g m \<le> Suc c\<^isub>2 * h m"
by auto
moreover {
fix k l r :: nat
assume k_l: "k \<le> Suc c\<^isub>1 * l" and l_r: "l \<le> Suc c\<^isub>2 * r"
from l_r have "Suc c\<^isub>1 * l \<le> (Suc c\<^isub>1 * Suc c\<^isub>2) * r"
by (auto simp add: mult_le_cancel_left mult_assoc simp del: times_nat.simps mult_Suc_right)
with k_l have "k \<le> (Suc c\<^isub>1 * Suc c\<^isub>2) * r" by (rule preorder_class.order_trans)
}
ultimately have "\<And>m. m \<ge> max n\<^isub>1 n\<^isub>2 \<Longrightarrow> f m \<le> (Suc c\<^isub>1 * Suc c\<^isub>2) * h m" by auto
then have "\<And>m. m \<ge> max n\<^isub>1 n\<^isub>2 \<Longrightarrow> f m \<le> Suc ((Suc c\<^isub>1 * Suc c\<^isub>2) - 1) * h m" by auto
then show ?thesis unfolding less_eq_fun_def by blast
qed
subsection {* The @{text "\<approx>"} relation, the equivalence relation induced by @{text "\<lesssim>"} *}
definition equiv_fun :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" (infix "\<cong>" 50) where
"f \<cong> g \<longleftrightarrow> (\<exists>d c n. \<forall>m\<ge>n. g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m)"
lemma equiv_fun_intro:
assumes "\<exists>d c n. \<forall>m\<ge>n. g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m"
shows "f \<cong> g"
unfolding equiv_fun_def by (rule assms)
lemma equiv_fun_not_intro:
assumes "\<And>d c n. \<exists>m\<ge>n. Suc d * f m < g m \<or> Suc c * g m < f m"
shows "\<not> f \<cong> g"
unfolding equiv_fun_def
by (auto simp add: assms linorder_not_le
simp del: times_nat.simps mult_Suc_right)
lemma equiv_fun_elim:
assumes "f \<cong> g"
obtains n d c
where "\<And>m. m \<ge> n \<Longrightarrow> g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m"
using assms unfolding equiv_fun_def by blast
lemma equiv_fun_not_elim:
fixes n d c
assumes "\<not> f \<cong> g"
obtains m where "m \<ge> n"
and "Suc d * f m < g m \<or> Suc c * g m < f m"
using assms unfolding equiv_fun_def
by (auto simp add: linorder_not_le, blast)
lemma equiv_fun_less_eq_fun:
"f \<cong> g \<longleftrightarrow> f \<lesssim> g \<and> g \<lesssim> f"
proof
assume x_y: "f \<cong> g"
then obtain n d c
where interv: "\<And>m. m \<ge> n \<Longrightarrow> g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m"
by (erule equiv_fun_elim)
from interv have "\<exists>c n. \<forall>m \<ge> n. f m \<le> Suc c * g m" by auto
then have f_g: "f \<lesssim> g" by (rule less_eq_fun_intro)
from interv have "\<exists>d n. \<forall>m \<ge> n. g m \<le> Suc d * f m" by auto
then have g_f: "g \<lesssim> f" by (rule less_eq_fun_intro)
from f_g g_f show "f \<lesssim> g \<and> g \<lesssim> f" by auto
next
assume assm: "f \<lesssim> g \<and> g \<lesssim> f"
from assm less_eq_fun_elim obtain c n\<^isub>1 where
bound1: "\<And>m. m \<ge> n\<^isub>1 \<Longrightarrow> f m \<le> Suc c * g m"
by blast
from assm less_eq_fun_elim obtain d n\<^isub>2 where
bound2: "\<And>m. m \<ge> n\<^isub>2 \<Longrightarrow> g m \<le> Suc d * f m"
by blast
from bound2 have "\<And>m. m \<ge> max n\<^isub>1 n\<^isub>2 \<Longrightarrow> g m \<le> Suc d * f m"
by auto
with bound1
have "\<forall>m \<ge> max n\<^isub>1 n\<^isub>2. g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m"
by auto
then
have "\<exists>d c n. \<forall>m\<ge>n. g m \<le> Suc d * f m \<and> f m \<le> Suc c * g m"
by blast
then show "f \<cong> g" by (rule equiv_fun_intro)
qed
subsection {* The @{text "\<prec>"} relation, the strict part of @{text "\<lesssim>"} *}
definition less_fun :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" (infix "\<prec>" 50) where
"f \<prec> g \<longleftrightarrow> f \<lesssim> g \<and> \<not> g \<lesssim> f"
lemma less_fun_intro:
assumes "\<And>c. \<exists>n. \<forall>m\<ge>n. Suc c * f m < g m"
shows "f \<prec> g"
proof (unfold less_fun_def, rule conjI)
from assms obtain n
where "\<forall>m\<ge>n. Suc 0 * f m < g m" ..
then have "\<forall>m\<ge>n. f m \<le> Suc 0 * g m" by auto
then have "\<exists>c n. \<forall>m\<ge>n. f m \<le> Suc c * g m" by blast
then show "f \<lesssim> g" by (rule less_eq_fun_intro)
next
show "\<not> g \<lesssim> f"
proof (rule less_eq_fun_not_intro)
fix c n :: nat
from assms have "\<exists>n. \<forall>m\<ge>n. Suc c * f m < g m" by blast
then obtain m where "m \<ge> n" and "Suc c * f m < g m"
by (rule Ex_All_bounded)
then show "\<exists>m\<ge>n. Suc c * f m < g m" by blast
qed
qed
text {*
We would like to show (or refute) that @{text "f \<prec> g \<longleftrightarrow> f \<in> o(g)"},
i.e.~@{prop "f \<prec> g \<longleftrightarrow> (\<forall>c. \<exists>n. \<forall>m>n. f m < Suc c * g m)"} but did not manage to
do so.
*}
subsection {* Assert that @{text "\<lesssim>"} is ineed a preorder *}
interpretation fun_order: preorder_equiv less_eq_fun less_fun
where "preorder_equiv.equiv less_eq_fun = equiv_fun"
proof -
interpret preorder_equiv less_eq_fun less_fun proof
qed (simp_all add: less_fun_def less_eq_fun_refl, auto intro: less_eq_fun_trans)
show "preorder_equiv less_eq_fun less_fun" using preorder_equiv_axioms .
show "preorder_equiv.equiv less_eq_fun = equiv_fun"
by (simp add: expand_fun_eq equiv_def equiv_fun_less_eq_fun)
qed
subsection {* Simple examples *}
lemma "(\<lambda>_. n) \<lesssim> (\<lambda>n. n)"
proof (rule less_eq_fun_intro)
show "\<exists>c q. \<forall>m\<ge>q. n \<le> Suc c * m"
proof -
have "\<forall>m\<ge>n. n \<le> Suc 0 * m" by simp
then show ?thesis by blast
qed
qed
lemma "(\<lambda>n. n) \<cong> (\<lambda>n. Suc k * n)"
proof (rule equiv_fun_intro)
show "\<exists>d c n. \<forall>m\<ge>n. Suc k * m \<le> Suc d * m \<and> m \<le> Suc c * (Suc k * m)"
proof -
have "\<forall>m\<ge>n. Suc k * m \<le> Suc k * m \<and> m \<le> Suc c * (Suc k * m)" by simp
then show ?thesis by blast
qed
qed
lemma "(\<lambda>_. n) \<prec> (\<lambda>n. n)"
proof (rule less_fun_intro)
fix c
show "\<exists>q. \<forall>m\<ge>q. Suc c * n < m"
proof -
have "\<forall>m\<ge>Suc c * n + 1. Suc c * n < m" by simp
then show ?thesis by blast
qed
qed
end