(* Title: HOL/SEQ.thy
Author: Jacques D. Fleuriot, University of Cambridge
Author: Lawrence C Paulson
Author: Jeremy Avigad
Author: Brian Huffman
Convergence of sequences and series.
*)
header {* Sequences and Convergence *}
theory SEQ
imports Limits RComplete
begin
subsection {* Monotone sequences and subsequences *}
definition
monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
--{*Definition of monotonicity.
The use of disjunction here complicates proofs considerably.
One alternative is to add a Boolean argument to indicate the direction.
Another is to develop the notions of increasing and decreasing first.*}
"monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
definition
incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
--{*Increasing sequence*}
"incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
definition
decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
--{*Decreasing sequence*}
"decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
definition
subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
--{*Definition of subsequence*}
"subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
unfolding mono_def incseq_def by auto
lemma incseq_SucI:
"(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
using lift_Suc_mono_le[of X]
by (auto simp: incseq_def)
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
by (auto simp: incseq_def)
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
using incseqD[of A i "Suc i"] by auto
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
by (auto intro: incseq_SucI dest: incseq_SucD)
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
unfolding incseq_def by auto
lemma decseq_SucI:
"(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
using order.lift_Suc_mono_le[OF dual_order, of X]
by (auto simp: decseq_def)
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
by (auto simp: decseq_def)
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
using decseqD[of A i "Suc i"] by auto
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
by (auto intro: decseq_SucI dest: decseq_SucD)
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
unfolding decseq_def by auto
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
unfolding monoseq_def incseq_def decseq_def ..
lemma monoseq_Suc:
"monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
by (simp add: monoseq_def)
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
by (simp add: monoseq_def)
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
by (simp add: monoseq_Suc)
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
by (simp add: monoseq_Suc)
lemma monoseq_minus:
fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
assumes "monoseq a"
shows "monoseq (\<lambda> n. - a n)"
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
case True
hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
thus ?thesis by (rule monoI2)
next
case False
hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
thus ?thesis by (rule monoI1)
qed
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
apply (simp add: subseq_def)
apply (auto dest!: less_imp_Suc_add)
apply (induct_tac k)
apply (auto intro: less_trans)
done
text{* for any sequence, there is a monotonic subsequence *}
lemma seq_monosub:
fixes s :: "nat => 'a::linorder"
shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
proof cases
let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
assume *: "\<forall>n. \<exists>p. ?P p n"
def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
then have "subseq f" unfolding subseq_Suc_iff by auto
moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
proof (intro disjI2 allI)
fix n show "s (f (Suc n)) \<le> s (f n)"
proof (cases n)
case 0 with P_Suc[of 0] P_0 show ?thesis by auto
next
case (Suc m)
from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
with P_Suc Suc show ?thesis by simp
qed
qed
ultimately show ?thesis by auto
next
let "?P p m" = "m < p \<and> s m < s p"
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
have P_0: "?P (f 0) (Suc N)"
unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
{ fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
note P' = this
{ fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
by (induct i) (insert P_0 P', auto) }
then have "subseq f" "monoseq (\<lambda>x. s (f x))"
unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
then show ?thesis by auto
qed
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
proof(induct n)
case 0 thus ?case by simp
next
case (Suc n)
from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
have "n < f (Suc n)" by arith
thus ?case by arith
qed
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
unfolding filterlim_iff eventually_sequentially by (metis le_trans seq_suble)
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
unfolding subseq_def by simp
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
using assms by (auto simp: subseq_def)
lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X"
by (simp add: incseq_def monoseq_def)
lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X"
by (simp add: decseq_def monoseq_def)
lemma decseq_eq_incseq:
fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
by (simp add: decseq_def incseq_def)
lemma INT_decseq_offset:
assumes "decseq F"
shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
proof safe
fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
show "x \<in> F i"
proof cases
from x have "x \<in> F n" by auto
also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
unfolding decseq_def by simp
finally show ?thesis .
qed (insert x, simp)
qed auto
subsection {* Defintions of limits *}
abbreviation (in topological_space)
LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
("((_)/ ----> (_))" [60, 60] 60) where
"X ----> L \<equiv> (X ---> L) sequentially"
definition
lim :: "(nat \<Rightarrow> 'a::t2_space) \<Rightarrow> 'a" where
--{*Standard definition of limit using choice operator*}
"lim X = (THE L. X ----> L)"
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
"convergent X = (\<exists>L. X ----> L)"
definition
Bseq :: "(nat => 'a::real_normed_vector) => bool" where
--{*Standard definition for bounded sequence*}
"Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
"Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
subsection {* Bounded Sequences *}
lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
unfolding Bseq_def
proof (intro exI conjI allI)
show "0 < max K 1" by simp
next
fix n::nat
have "norm (X n) \<le> K" by (rule K)
thus "norm (X n) \<le> max K 1" by simp
qed
lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Bseq_def by auto
lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
proof (rule BseqI')
let ?A = "norm ` X ` {..N}"
have 1: "finite ?A" by simp
fix n::nat
show "norm (X n) \<le> max K (Max ?A)"
proof (cases rule: linorder_le_cases)
assume "n \<ge> N"
hence "norm (X n) \<le> K" using K by simp
thus "norm (X n) \<le> max K (Max ?A)" by simp
next
assume "n \<le> N"
hence "norm (X n) \<in> ?A" by simp
with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
thus "norm (X n) \<le> max K (Max ?A)" by simp
qed
qed
lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
unfolding Bseq_def by auto
lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
apply (erule BseqE)
apply (rule_tac N="k" and K="K" in BseqI2')
apply clarify
apply (drule_tac x="n - k" in spec, simp)
done
lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
unfolding Bfun_def eventually_sequentially
apply (rule iffI)
apply (simp add: Bseq_def)
apply (auto intro: BseqI2')
done
subsection {* Limits of Sequences *}
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
by simp
lemma LIMSEQ_def: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
unfolding tendsto_iff eventually_sequentially ..
lemma LIMSEQ_iff:
fixes L :: "'a::real_normed_vector"
shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
unfolding LIMSEQ_def dist_norm ..
lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
lemma metric_LIMSEQ_I:
"(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
by (simp add: LIMSEQ_def)
lemma metric_LIMSEQ_D:
"\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
by (simp add: LIMSEQ_def)
lemma LIMSEQ_I:
fixes L :: "'a::real_normed_vector"
shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
by (simp add: LIMSEQ_iff)
lemma LIMSEQ_D:
fixes L :: "'a::real_normed_vector"
shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
by (simp add: LIMSEQ_iff)
lemma LIMSEQ_const_iff:
fixes k l :: "'a::t2_space"
shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
using trivial_limit_sequentially by (rule tendsto_const_iff)
lemma LIMSEQ_ignore_initial_segment:
"f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_sequentially)
apply (erule exE, rename_tac N)
apply (rule_tac x=N in exI)
apply simp
done
lemma LIMSEQ_offset:
"(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_sequentially)
apply (erule exE, rename_tac N)
apply (rule_tac x="N + k" in exI)
apply clarify
apply (drule_tac x="n - k" in spec)
apply (simp add: le_diff_conv2)
done
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
unfolding tendsto_def eventually_sequentially
by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
lemma LIMSEQ_unique:
fixes a b :: "'a::t2_space"
shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
using trivial_limit_sequentially by (rule tendsto_unique)
lemma increasing_LIMSEQ:
fixes f :: "nat \<Rightarrow> real"
assumes inc: "\<And>n. f n \<le> f (Suc n)"
and bdd: "\<And>n. f n \<le> l"
and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
shows "f ----> l"
unfolding LIMSEQ_def
proof safe
fix r :: real assume "0 < r"
with bdd en[of "r / 2"] obtain n where n: "dist (f n) l \<le> r / 2"
by (auto simp add: field_simps dist_real_def)
{ fix N assume "n \<le> N"
then have "dist (f N) l \<le> dist (f n) l"
using incseq_SucI[of f] inc bdd by (auto dest!: incseqD simp: dist_real_def)
then have "dist (f N) l < r"
using `0 < r` n by simp }
with `0 < r` show "\<exists>no. \<forall>n\<ge>no. dist (f n) l < r"
by (auto simp add: LIMSEQ_def field_simps intro!: exI[of _ n])
qed
lemma Bseq_inverse_lemma:
fixes x :: "'a::real_normed_div_algebra"
shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
apply (subst nonzero_norm_inverse, clarsimp)
apply (erule (1) le_imp_inverse_le)
done
lemma Bseq_inverse:
fixes a :: "'a::real_normed_div_algebra"
shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
lemma LIMSEQ_diff_approach_zero:
fixes L :: "'a::real_normed_vector"
shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
by (drule (1) tendsto_add, simp)
lemma LIMSEQ_diff_approach_zero2:
fixes L :: "'a::real_normed_vector"
shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
by (drule (1) tendsto_diff, simp)
text{*An unbounded sequence's inverse tends to 0*}
lemma LIMSEQ_inverse_zero:
"\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
apply (rule filterlim_compose[OF tendsto_inverse_0])
apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
done
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
infinity is now easily proved*}
lemma LIMSEQ_inverse_real_of_nat_add:
"(%n. r + inverse(real(Suc n))) ----> r"
using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
lemma LIMSEQ_inverse_real_of_nat_add_minus:
"(%n. r + -inverse(real(Suc n))) ----> r"
using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
by auto
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
"(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
by auto
lemma LIMSEQ_le_const:
"\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
lemma LIMSEQ_le:
"\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
lemma LIMSEQ_le_const2:
"\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
subsection {* Convergence *}
lemma limI: "X ----> L ==> lim X = L"
apply (simp add: lim_def)
apply (blast intro: LIMSEQ_unique)
done
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
by (simp add: convergent_def)
lemma convergentI: "(X ----> L) ==> convergent X"
by (auto simp add: convergent_def)
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
lemma convergent_const: "convergent (\<lambda>n. c)"
by (rule convergentI, rule tendsto_const)
lemma convergent_add:
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
assumes "convergent (\<lambda>n. X n)"
assumes "convergent (\<lambda>n. Y n)"
shows "convergent (\<lambda>n. X n + Y n)"
using assms unfolding convergent_def by (fast intro: tendsto_add)
lemma convergent_setsum:
fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
proof (cases "finite A")
case True from this and assms show ?thesis
by (induct A set: finite) (simp_all add: convergent_const convergent_add)
qed (simp add: convergent_const)
lemma (in bounded_linear) convergent:
assumes "convergent (\<lambda>n. X n)"
shows "convergent (\<lambda>n. f (X n))"
using assms unfolding convergent_def by (fast intro: tendsto)
lemma (in bounded_bilinear) convergent:
assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
shows "convergent (\<lambda>n. X n ** Y n)"
using assms unfolding convergent_def by (fast intro: tendsto)
lemma convergent_minus_iff:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
apply (simp add: convergent_def)
apply (auto dest: tendsto_minus)
apply (drule tendsto_minus, auto)
done
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::real)) \<Longrightarrow> lim f \<le> x"
using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
lemma monoseq_le:
"monoseq a \<Longrightarrow> a ----> (x::real) \<Longrightarrow>
((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
lemma LIMSEQ_subseq_LIMSEQ:
"\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
lemma convergent_subseq_convergent:
"\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
subsection {* Bounded Monotonic Sequences *}
text{*Bounded Sequence*}
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
by (simp add: Bseq_def)
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
by (auto simp add: Bseq_def)
lemma lemma_NBseq_def:
"(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
proof safe
fix K :: real
from reals_Archimedean2 obtain n :: nat where "K < real n" ..
then have "K \<le> real (Suc n)" by auto
moreover assume "\<forall>m. norm (X m) \<le> K"
ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
by (blast intro: order_trans)
then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
qed (force simp add: real_of_nat_Suc)
text{* alternative definition for Bseq *}
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
apply (simp add: Bseq_def)
apply (simp (no_asm) add: lemma_NBseq_def)
done
lemma lemma_NBseq_def2:
"(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
apply (subst lemma_NBseq_def, auto)
apply (rule_tac x = "Suc N" in exI)
apply (rule_tac [2] x = N in exI)
apply (auto simp add: real_of_nat_Suc)
prefer 2 apply (blast intro: order_less_imp_le)
apply (drule_tac x = n in spec, simp)
done
(* yet another definition for Bseq *)
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
by (simp add: Bseq_def lemma_NBseq_def2)
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
text{*alternative formulation for boundedness*}
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
apply (unfold Bseq_def, safe)
apply (rule_tac [2] x = "k + norm x" in exI)
apply (rule_tac x = K in exI, simp)
apply (rule exI [where x = 0], auto)
apply (erule order_less_le_trans, simp)
apply (drule_tac x=n in spec, fold diff_minus)
apply (drule order_trans [OF norm_triangle_ineq2])
apply simp
done
text{*alternative formulation for boundedness*}
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
apply safe
apply (simp add: Bseq_def, safe)
apply (rule_tac x = "K + norm (X N)" in exI)
apply auto
apply (erule order_less_le_trans, simp)
apply (rule_tac x = N in exI, safe)
apply (drule_tac x = n in spec)
apply (rule order_trans [OF norm_triangle_ineq], simp)
apply (auto simp add: Bseq_iff2)
done
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
apply (simp add: Bseq_def)
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
apply (drule_tac x = n in spec, arith)
done
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
lemma Bseq_isUb:
"!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
text{* Use completeness of reals (supremum property)
to show that any bounded sequence has a least upper bound*}
lemma Bseq_isLub:
"!!(X::nat=>real). Bseq X ==>
\<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
by (blast intro: reals_complete Bseq_isUb)
subsubsection{*A Bounded and Monotonic Sequence Converges*}
(* TODO: delete *)
(* FIXME: one use in NSA/HSEQ.thy *)
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
unfolding tendsto_def eventually_sequentially
apply (rule_tac x = "X m" in exI, safe)
apply (rule_tac x = m in exI, safe)
apply (drule spec, erule impE, auto)
done
text {* A monotone sequence converges to its least upper bound. *}
lemma isLub_mono_imp_LIMSEQ:
fixes X :: "nat \<Rightarrow> real"
assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
shows "X ----> u"
proof (rule LIMSEQ_I)
have 1: "\<forall>n. X n \<le> u"
using isLubD2 [OF u] by auto
have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
hence 2: "\<forall>y<u. \<exists>n. y < X n"
by (metis not_le)
fix r :: real assume "0 < r"
hence "u - r < u" by simp
hence "\<exists>m. u - r < X m" using 2 by simp
then obtain m where "u - r < X m" ..
with X have "\<forall>n\<ge>m. u - r < X n"
by (fast intro: less_le_trans)
hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
using 1 by (simp add: diff_less_eq add_commute)
qed
text{*A standard proof of the theorem for monotone increasing sequence*}
lemma Bseq_mono_convergent:
"Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
by (simp add: Bseq_def)
text{*Main monotonicity theorem*}
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
Bseq_mono_convergent)
subsubsection{*Increasing and Decreasing Series*}
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::real)"
by (metis incseq_def LIMSEQ_le_const)
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::real) \<le> X n"
by (metis decseq_def LIMSEQ_le_const2)
subsection {* Cauchy Sequences *}
lemma metric_CauchyI:
"(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
by (simp add: Cauchy_def)
lemma metric_CauchyD:
"Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
by (simp add: Cauchy_def)
lemma Cauchy_iff:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
unfolding Cauchy_def dist_norm ..
lemma Cauchy_iff2:
"Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
apply (simp add: Cauchy_iff, auto)
apply (drule reals_Archimedean, safe)
apply (drule_tac x = n in spec, auto)
apply (rule_tac x = M in exI, auto)
apply (drule_tac x = m in spec, simp)
apply (drule_tac x = na in spec, auto)
done
lemma CauchyI:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
by (simp add: Cauchy_iff)
lemma CauchyD:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
by (simp add: Cauchy_iff)
lemma Cauchy_subseq_Cauchy:
"\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
apply (auto simp add: Cauchy_def)
apply (drule_tac x=e in spec, clarify)
apply (rule_tac x=M in exI, clarify)
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
done
subsubsection {* Cauchy Sequences are Bounded *}
text{*A Cauchy sequence is bounded -- this is the standard
proof mechanization rather than the nonstandard proof*}
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
==> \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
apply (clarify, drule spec, drule (1) mp)
apply (simp only: norm_minus_commute)
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
apply simp
done
lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
apply (simp add: Cauchy_iff)
apply (drule spec, drule mp, rule zero_less_one, safe)
apply (drule_tac x="M" in spec, simp)
apply (drule lemmaCauchy)
apply (rule_tac k="M" in Bseq_offset)
apply (simp add: Bseq_def)
apply (rule_tac x="1 + norm (X M)" in exI)
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
apply (simp add: order_less_imp_le)
done
subsubsection {* Cauchy Sequences are Convergent *}
class complete_space = metric_space +
assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
class banach = real_normed_vector + complete_space
theorem LIMSEQ_imp_Cauchy:
assumes X: "X ----> a" shows "Cauchy X"
proof (rule metric_CauchyI)
fix e::real assume "0 < e"
hence "0 < e/2" by simp
with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
proof (intro exI allI impI)
fix m assume "N \<le> m"
hence m: "dist (X m) a < e/2" using N by fast
fix n assume "N \<le> n"
hence n: "dist (X n) a < e/2" using N by fast
have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
by (rule dist_triangle2)
also from m n have "\<dots> < e" by simp
finally show "dist (X m) (X n) < e" .
qed
qed
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
unfolding convergent_def
by (erule exE, erule LIMSEQ_imp_Cauchy)
lemma Cauchy_convergent_iff:
fixes X :: "nat \<Rightarrow> 'a::complete_space"
shows "Cauchy X = convergent X"
by (fast intro: Cauchy_convergent convergent_Cauchy)
text {*
Proof that Cauchy sequences converge based on the one from
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
*}
text {*
If sequence @{term "X"} is Cauchy, then its limit is the lub of
@{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
*}
lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
by (simp add: isUbI setleI)
lemma real_Cauchy_convergent:
fixes X :: "nat \<Rightarrow> real"
assumes X: "Cauchy X"
shows "convergent X"
proof -
def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
{ fix N x assume N: "\<forall>n\<ge>N. X n < x"
have "isUb UNIV S x"
proof (rule isUb_UNIV_I)
fix y::real assume "y \<in> S"
hence "\<exists>M. \<forall>n\<ge>M. y < X n"
by (simp add: S_def)
then obtain M where "\<forall>n\<ge>M. y < X n" ..
hence "y < X (max M N)" by simp
also have "\<dots> < x" using N by simp
finally show "y \<le> x"
by (rule order_less_imp_le)
qed }
note bound_isUb = this
have "\<exists>u. isLub UNIV S u"
proof (rule reals_complete)
obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
using CauchyD [OF X zero_less_one] by auto
hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
show "\<exists>x. x \<in> S"
proof
from N have "\<forall>n\<ge>N. X N - 1 < X n"
by (simp add: abs_diff_less_iff)
thus "X N - 1 \<in> S" by (rule mem_S)
qed
show "\<exists>u. isUb UNIV S u"
proof
from N have "\<forall>n\<ge>N. X n < X N + 1"
by (simp add: abs_diff_less_iff)
thus "isUb UNIV S (X N + 1)"
by (rule bound_isUb)
qed
qed
then obtain x where x: "isLub UNIV S x" ..
have "X ----> x"
proof (rule LIMSEQ_I)
fix r::real assume "0 < r"
hence r: "0 < r/2" by simp
obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
using CauchyD [OF X r] by auto
hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
by (simp only: real_norm_def abs_diff_less_iff)
from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
hence "X N - r/2 \<in> S" by (rule mem_S)
hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
proof (intro exI allI impI)
fix n assume n: "N \<le> n"
from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
thus "norm (X n - x) < r" using 1 2
by (simp add: abs_diff_less_iff)
qed
qed
then show ?thesis unfolding convergent_def by auto
qed
instance real :: banach
by intro_classes (rule real_Cauchy_convergent)
subsection {* Power Sequences *}
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
"x<1"}. Proof will use (NS) Cauchy equivalence for convergence and
also fact that bounded and monotonic sequence converges.*}
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
apply (simp add: Bseq_def)
apply (rule_tac x = 1 in exI)
apply (simp add: power_abs)
apply (auto dest: power_mono)
done
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
apply (clarify intro!: mono_SucI2)
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
done
lemma convergent_realpow:
"[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
lemma LIMSEQ_realpow_zero:
"\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
proof cases
assume "0 \<le> x" and "x \<noteq> 0"
hence x0: "0 < x" by simp
assume x1: "x < 1"
from x0 x1 have "1 < inverse x"
by (rule one_less_inverse)
hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
by (rule LIMSEQ_inverse_realpow_zero)
thus ?thesis by (simp add: power_inverse)
qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
lemma LIMSEQ_power_zero:
fixes x :: "'a::{real_normed_algebra_1}"
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
apply (simp add: power_abs norm_power_ineq)
done
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
by (rule LIMSEQ_power_zero) simp
lemma tendsto_at_topI_sequentially:
fixes f :: "real \<Rightarrow> real"
assumes mono: "mono f"
assumes limseq: "(\<lambda>n. f (real n)) ----> y"
shows "(f ---> y) at_top"
proof (rule tendstoI)
fix e :: real assume "0 < e"
with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
by (auto simp: LIMSEQ_def dist_real_def)
{ fix x :: real
from ex_le_of_nat[of x] guess n ..
note monoD[OF mono this]
also have "f (real_of_nat n) \<le> y"
by (rule LIMSEQ_le_const[OF limseq])
(auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
finally have "f x \<le> y" . }
note le = this
have "eventually (\<lambda>x. real N \<le> x) at_top"
by (rule eventually_ge_at_top)
then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
proof eventually_elim
fix x assume N': "real N \<le> x"
with N[of N] le have "y - f (real N) < e" by auto
moreover note monoD[OF mono N']
ultimately show "dist (f x) y < e"
using le[of x] by (auto simp: dist_real_def field_simps)
qed
qed
end