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 <-> f ∈ O(g)"}.
*}
subsection {* Auxiliary *}
lemma Ex_All_bounded:
fixes n :: nat
assumes "∃n. ∀m≥n. P m"
obtains m where "m ≥ n" and "P m"
proof -
from assms obtain q where m_q: "∀m≥q. P m" ..
let ?m = "max q n"
have "?m ≥ 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 => nat) => (nat => nat) => bool" (infix "\<lesssim>" 50) where
"f \<lesssim> g <-> (∃c n. ∀m≥n. f m ≤ Suc c * g m)"
lemma less_eq_fun_intro:
assumes "∃c n. ∀m≥n. f m ≤ Suc c * g m"
shows "f \<lesssim> g"
unfolding less_eq_fun_def by (rule assms)
lemma less_eq_fun_not_intro:
assumes "!!c n. ∃m≥n. Suc c * g m < f m"
shows "¬ 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 "!!m. m ≥ n ==> f m ≤ Suc c * g m"
using assms unfolding less_eq_fun_def by blast
lemma less_eq_fun_not_elim:
assumes "¬ f \<lesssim> g"
obtains m where "m ≥ 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 "∃n. ∀m≥n. f m ≤ Suc 0 * f m" by auto
then show "∃c n. ∀m≥n. f m ≤ 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: "!!m. m ≥ n\<^isub>1 ==> f m ≤ 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: "!!m. m ≥ n\<^isub>2 ==> g m ≤ Suc c\<^isub>2 * h m"
by (erule less_eq_fun_elim)
ultimately have "!!m. m ≥ max n\<^isub>1 n\<^isub>2 ==> f m ≤ Suc c\<^isub>1 * g m ∧ g m ≤ Suc c\<^isub>2 * h m"
by auto
moreover {
fix k l r :: nat
assume k_l: "k ≤ Suc c\<^isub>1 * l" and l_r: "l ≤ Suc c\<^isub>2 * r"
from l_r have "Suc c\<^isub>1 * l ≤ (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 ≤ (Suc c\<^isub>1 * Suc c\<^isub>2) * r" by (rule preorder_class.order_trans)
}
ultimately have "!!m. m ≥ max n\<^isub>1 n\<^isub>2 ==> f m ≤ (Suc c\<^isub>1 * Suc c\<^isub>2) * h m" by auto
then have "!!m. m ≥ max n\<^isub>1 n\<^isub>2 ==> f m ≤ 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 "≈"} relation, the equivalence relation induced by @{text "\<lesssim>"} *}
definition equiv_fun :: "(nat => nat) => (nat => nat) => bool" (infix "≅" 50) where
"f ≅ g <-> (∃d c n. ∀m≥n. g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m)"
lemma equiv_fun_intro:
assumes "∃d c n. ∀m≥n. g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m"
shows "f ≅ g"
unfolding equiv_fun_def by (rule assms)
lemma equiv_fun_not_intro:
assumes "!!d c n. ∃m≥n. Suc d * f m < g m ∨ Suc c * g m < f m"
shows "¬ f ≅ 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 ≅ g"
obtains n d c
where "!!m. m ≥ n ==> g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m"
using assms unfolding equiv_fun_def by blast
lemma equiv_fun_not_elim:
fixes n d c
assumes "¬ f ≅ g"
obtains m where "m ≥ n"
and "Suc d * f m < g m ∨ 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 ≅ g <-> f \<lesssim> g ∧ g \<lesssim> f"
proof
assume x_y: "f ≅ g"
then obtain n d c
where interv: "!!m. m ≥ n ==> g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m"
by (erule equiv_fun_elim)
from interv have "∃c n. ∀m ≥ n. f m ≤ Suc c * g m" by auto
then have f_g: "f \<lesssim> g" by (rule less_eq_fun_intro)
from interv have "∃d n. ∀m ≥ n. g m ≤ 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 ∧ g \<lesssim> f" by auto
next
assume assm: "f \<lesssim> g ∧ g \<lesssim> f"
from assm less_eq_fun_elim obtain c n\<^isub>1 where
bound1: "!!m. m ≥ n\<^isub>1 ==> f m ≤ Suc c * g m"
by blast
from assm less_eq_fun_elim obtain d n\<^isub>2 where
bound2: "!!m. m ≥ n\<^isub>2 ==> g m ≤ Suc d * f m"
by blast
from bound2 have "!!m. m ≥ max n\<^isub>1 n\<^isub>2 ==> g m ≤ Suc d * f m"
by auto
with bound1
have "∀m ≥ max n\<^isub>1 n\<^isub>2. g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m"
by auto
then
have "∃d c n. ∀m≥n. g m ≤ Suc d * f m ∧ f m ≤ Suc c * g m"
by blast
then show "f ≅ g" by (rule equiv_fun_intro)
qed
subsection {* The @{text "\<prec>"} relation, the strict part of @{text "\<lesssim>"} *}
definition less_fun :: "(nat => nat) => (nat => nat) => bool" (infix "\<prec>" 50) where
"f \<prec> g <-> f \<lesssim> g ∧ ¬ g \<lesssim> f"
lemma less_fun_intro:
assumes "!!c. ∃n. ∀m≥n. Suc c * f m < g m"
shows "f \<prec> g"
proof (unfold less_fun_def, rule conjI)
from assms obtain n
where "∀m≥n. Suc 0 * f m < g m" ..
then have "∀m≥n. f m ≤ Suc 0 * g m" by auto
then have "∃c n. ∀m≥n. f m ≤ Suc c * g m" by blast
then show "f \<lesssim> g" by (rule less_eq_fun_intro)
next
show "¬ g \<lesssim> f"
proof (rule less_eq_fun_not_intro)
fix c n :: nat
from assms have "∃n. ∀m≥n. Suc c * f m < g m" by blast
then obtain m where "m ≥ n" and "Suc c * f m < g m"
by (rule Ex_All_bounded)
then show "∃m≥n. Suc c * f m < g m" by blast
qed
qed
text {*
We would like to show (or refute) that @{text "f \<prec> g <-> f ∈ o(g)"},
i.e.~@{prop "f \<prec> g <-> (∀c. ∃n. ∀m>n. f m < Suc c * g m)"} but did not
manage to do so.
*}
subsection {* Assert that @{text "\<lesssim>"} is indeed 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 "class.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 "(λ_. n) \<lesssim> (λn. n)"
proof (rule less_eq_fun_intro)
show "∃c q. ∀m≥q. n ≤ Suc c * m"
proof -
have "∀m≥n. n ≤ Suc 0 * m" by simp
then show ?thesis by blast
qed
qed
lemma "(λn. n) ≅ (λn. Suc k * n)"
proof (rule equiv_fun_intro)
show "∃d c n. ∀m≥n. Suc k * m ≤ Suc d * m ∧ m ≤ Suc c * (Suc k * m)"
proof -
have "∀m≥n. Suc k * m ≤ Suc k * m ∧ m ≤ Suc c * (Suc k * m)" by simp
then show ?thesis by blast
qed
qed
lemma "(λ_. n) \<prec> (λn. n)"
proof (rule less_fun_intro)
fix c
show "∃q. ∀m≥q. Suc c * n < m"
proof -
have "∀m≥Suc c * n + 1. Suc c * n < m" by simp
then show ?thesis by blast
qed
qed
end