(* Title: HOL/Limits.thy
Author: Brian Huffman
Author: Jacques D. Fleuriot, University of Cambridge
Author: Lawrence C Paulson
Author: Jeremy Avigad
*)
section {* Limits on Real Vector Spaces *}
theory Limits
imports Real_Vector_Spaces
begin
subsection {* Filter going to infinity norm *}
definition at_infinity :: "'a::real_normed_vector filter" where
"at_infinity = (INF r. principal {x. r \<le> norm x})"
lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
unfolding at_infinity_def
by (subst eventually_INF_base)
(auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
lemma at_infinity_eq_at_top_bot:
"(at_infinity \<Colon> real filter) = sup at_top at_bot"
apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
eventually_at_top_linorder eventually_at_bot_linorder)
apply safe
apply (rule_tac x="b" in exI, simp)
apply (rule_tac x="- b" in exI, simp)
apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
done
lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
unfolding at_infinity_eq_at_top_bot by simp
lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
unfolding at_infinity_eq_at_top_bot by simp
lemma filterlim_at_top_imp_at_infinity:
fixes f :: "_ \<Rightarrow> real"
shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
lemma lim_infinity_imp_sequentially:
"(f ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> l) sequentially"
by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
subsubsection {* Boundedness *}
definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
"Bseq X \<equiv> Bfun X sequentially"
lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
lemma Bfun_def:
"Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
unfolding Bfun_metric_def norm_conv_dist
proof safe
fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
by (intro always_eventually) (metis dist_commute dist_triangle)
with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
by eventually_elim auto
with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
qed auto
lemma BfunI:
assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
unfolding Bfun_def
proof (intro exI conjI allI)
show "0 < max K 1" by simp
next
show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
using K by (rule eventually_elim1, simp)
qed
lemma BfunE:
assumes "Bfun f F"
obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
using assms unfolding Bfun_def by fast
lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
unfolding Cauchy_def Bfun_metric_def eventually_sequentially
apply (erule_tac x=1 in allE)
apply simp
apply safe
apply (rule_tac x="X M" in exI)
apply (rule_tac x=1 in exI)
apply (erule_tac x=M in allE)
apply simp
apply (rule_tac x=M in exI)
apply (auto simp: dist_commute)
done
subsubsection {* Bounded Sequences *}
lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
by (intro BfunI) (auto simp: eventually_sequentially)
lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
by (intro BfunI) (auto simp: eventually_sequentially)
lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
unfolding Bfun_def eventually_sequentially
proof safe
fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
(auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
qed auto
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 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 Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
proof (elim BseqE, intro bdd_aboveI2)
fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
by (auto elim!: allE[of _ n])
qed
lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
proof (elim BseqE, intro bdd_belowI2)
fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
by (auto elim!: allE[of _ n])
qed
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)
apply (drule order_trans [OF norm_triangle_ineq2])
apply simp
done
text{*alternative formulation for boundedness*}
lemma Bseq_iff3:
"Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
then obtain K
where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
by (auto intro: order_trans norm_triangle_ineq4)
then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
by simp
with `0 < K + norm (X 0)` show ?Q by blast
next
assume ?Q then show ?P by (auto simp add: Bseq_iff2)
qed
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_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
by (simp add: Bseq_def)
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
apply (simp add: subset_eq)
apply (rule BseqI'[where K="max (norm a) (norm b)"])
apply (erule_tac x=n in allE)
apply auto
done
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
subsection {* Bounded Monotonic Sequences *}
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)"
apply (rule_tac x="X m" in exI)
apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
unfolding eventually_sequentially
apply blast
done
subsection {* Convergence to Zero *}
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
lemma ZfunI:
"(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
unfolding Zfun_def by simp
lemma ZfunD:
"\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
unfolding Zfun_def by simp
lemma Zfun_ssubst:
"eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
unfolding Zfun_def by (auto elim!: eventually_rev_mp)
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
unfolding Zfun_def by simp
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
unfolding Zfun_def by simp
lemma Zfun_imp_Zfun:
assumes f: "Zfun f F"
assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
shows "Zfun (\<lambda>x. g x) F"
proof (cases)
assume K: "0 < K"
show ?thesis
proof (rule ZfunI)
fix r::real assume "0 < r"
hence "0 < r / K" using K by simp
then have "eventually (\<lambda>x. norm (f x) < r / K) F"
using ZfunD [OF f] by fast
with g show "eventually (\<lambda>x. norm (g x) < r) F"
proof eventually_elim
case (elim x)
hence "norm (f x) * K < r"
by (simp add: pos_less_divide_eq K)
thus ?case
by (simp add: order_le_less_trans [OF elim(1)])
qed
qed
next
assume "\<not> 0 < K"
hence K: "K \<le> 0" by (simp only: not_less)
show ?thesis
proof (rule ZfunI)
fix r :: real
assume "0 < r"
from g show "eventually (\<lambda>x. norm (g x) < r) F"
proof eventually_elim
case (elim x)
also have "norm (f x) * K \<le> norm (f x) * 0"
using K norm_ge_zero by (rule mult_left_mono)
finally show ?case
using `0 < r` by simp
qed
qed
qed
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
by (erule_tac K="1" in Zfun_imp_Zfun, simp)
lemma Zfun_add:
assumes f: "Zfun f F" and g: "Zfun g F"
shows "Zfun (\<lambda>x. f x + g x) F"
proof (rule ZfunI)
fix r::real assume "0 < r"
hence r: "0 < r / 2" by simp
have "eventually (\<lambda>x. norm (f x) < r/2) F"
using f r by (rule ZfunD)
moreover
have "eventually (\<lambda>x. norm (g x) < r/2) F"
using g r by (rule ZfunD)
ultimately
show "eventually (\<lambda>x. norm (f x + g x) < r) F"
proof eventually_elim
case (elim x)
have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
by (rule norm_triangle_ineq)
also have "\<dots> < r/2 + r/2"
using elim by (rule add_strict_mono)
finally show ?case
by simp
qed
qed
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
unfolding Zfun_def by simp
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
lemma (in bounded_linear) Zfun:
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f (g x)) F"
proof -
obtain K where "\<And>x. norm (f x) \<le> norm x * K"
using bounded by fast
then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
by simp
with g show ?thesis
by (rule Zfun_imp_Zfun)
qed
lemma (in bounded_bilinear) Zfun:
assumes f: "Zfun f F"
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
proof (rule ZfunI)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
using pos_bounded by fast
from K have K': "0 < inverse K"
by (rule positive_imp_inverse_positive)
have "eventually (\<lambda>x. norm (f x) < r) F"
using f r by (rule ZfunD)
moreover
have "eventually (\<lambda>x. norm (g x) < inverse K) F"
using g K' by (rule ZfunD)
ultimately
show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
proof eventually_elim
case (elim x)
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
by (rule norm_le)
also have "norm (f x) * norm (g x) * K < r * inverse K * K"
by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
also from K have "r * inverse K * K = r"
by simp
finally show ?case .
qed
qed
lemma (in bounded_bilinear) Zfun_left:
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
lemma (in bounded_bilinear) Zfun_right:
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
by (simp only: tendsto_iff Zfun_def dist_norm)
lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
\<Longrightarrow> (g ---> 0) F"
by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
subsubsection {* Distance and norms *}
lemma tendsto_dist [tendsto_intros]:
fixes l m :: "'a :: metric_space"
assumes f: "(f ---> l) F" and g: "(g ---> m) F"
shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
hence e2: "0 < e/2" by simp
from tendstoD [OF f e2] tendstoD [OF g e2]
show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
proof (eventually_elim)
case (elim x)
then show "dist (dist (f x) (g x)) (dist l m) < e"
unfolding dist_real_def
using dist_triangle2 [of "f x" "g x" "l"]
using dist_triangle2 [of "g x" "l" "m"]
using dist_triangle3 [of "l" "m" "f x"]
using dist_triangle [of "f x" "m" "g x"]
by arith
qed
qed
lemma continuous_dist[continuous_intros]:
fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
unfolding continuous_def by (rule tendsto_dist)
lemma continuous_on_dist[continuous_intros]:
fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
unfolding continuous_on_def by (auto intro: tendsto_dist)
lemma tendsto_norm [tendsto_intros]:
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
unfolding norm_conv_dist by (intro tendsto_intros)
lemma continuous_norm [continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
unfolding continuous_def by (rule tendsto_norm)
lemma continuous_on_norm [continuous_intros]:
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
unfolding continuous_on_def by (auto intro: tendsto_norm)
lemma tendsto_norm_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
by (drule tendsto_norm, simp)
lemma tendsto_norm_zero_cancel:
"((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma tendsto_norm_zero_iff:
"((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma tendsto_rabs [tendsto_intros]:
"(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
by (fold real_norm_def, rule tendsto_norm)
lemma continuous_rabs [continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
unfolding real_norm_def[symmetric] by (rule continuous_norm)
lemma continuous_on_rabs [continuous_intros]:
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
lemma tendsto_rabs_zero:
"(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero)
lemma tendsto_rabs_zero_cancel:
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero_cancel)
lemma tendsto_rabs_zero_iff:
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero_iff)
subsubsection {* Addition and subtraction *}
lemma tendsto_add [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
lemma continuous_add [continuous_intros]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
unfolding continuous_def by (rule tendsto_add)
lemma continuous_on_add [continuous_intros]:
fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
unfolding continuous_on_def by (auto intro: tendsto_add)
lemma tendsto_add_zero:
fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
by (drule (1) tendsto_add, simp)
lemma tendsto_minus [tendsto_intros]:
fixes a :: "'a::real_normed_vector"
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
lemma continuous_minus [continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
unfolding continuous_def by (rule tendsto_minus)
lemma continuous_on_minus [continuous_intros]:
fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
unfolding continuous_on_def by (auto intro: tendsto_minus)
lemma tendsto_minus_cancel:
fixes a :: "'a::real_normed_vector"
shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
by (drule tendsto_minus, simp)
lemma tendsto_minus_cancel_left:
"(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
using tendsto_minus_cancel[of f "- y" F] tendsto_minus[of f "- y" F]
by auto
lemma tendsto_diff [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
lemma continuous_diff [continuous_intros]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
unfolding continuous_def by (rule tendsto_diff)
lemma continuous_on_diff [continuous_intros]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
unfolding continuous_on_def by (auto intro: tendsto_diff)
lemma tendsto_setsum [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
proof (cases "finite S")
assume "finite S" thus ?thesis using assms
by (induct, simp, simp add: tendsto_add)
qed simp
lemma continuous_setsum [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
unfolding continuous_def by (rule tendsto_setsum)
lemma continuous_on_setsum [continuous_intros]:
fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
unfolding continuous_on_def by (auto intro: tendsto_setsum)
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
subsubsection {* Linear operators and multiplication *}
lemma (in bounded_linear) tendsto:
"(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
lemma (in bounded_linear) continuous:
"continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
using tendsto[of g _ F] by (auto simp: continuous_def)
lemma (in bounded_linear) continuous_on:
"continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
using tendsto[of g] by (auto simp: continuous_on_def)
lemma (in bounded_linear) tendsto_zero:
"(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
by (drule tendsto, simp only: zero)
lemma (in bounded_bilinear) tendsto:
"\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
by (simp only: tendsto_Zfun_iff prod_diff_prod
Zfun_add Zfun Zfun_left Zfun_right)
lemma (in bounded_bilinear) continuous:
"continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
using tendsto[of f _ F g] by (auto simp: continuous_def)
lemma (in bounded_bilinear) continuous_on:
"continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
lemma (in bounded_bilinear) tendsto_zero:
assumes f: "(f ---> 0) F"
assumes g: "(g ---> 0) F"
shows "((\<lambda>x. f x ** g x) ---> 0) F"
using tendsto [OF f g] by (simp add: zero_left)
lemma (in bounded_bilinear) tendsto_left_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
lemma (in bounded_bilinear) tendsto_right_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
lemmas tendsto_of_real [tendsto_intros] =
bounded_linear.tendsto [OF bounded_linear_of_real]
lemmas tendsto_scaleR [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
lemmas tendsto_mult [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_mult]
lemmas continuous_of_real [continuous_intros] =
bounded_linear.continuous [OF bounded_linear_of_real]
lemmas continuous_scaleR [continuous_intros] =
bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
lemmas continuous_mult [continuous_intros] =
bounded_bilinear.continuous [OF bounded_bilinear_mult]
lemmas continuous_on_of_real [continuous_intros] =
bounded_linear.continuous_on [OF bounded_linear_of_real]
lemmas continuous_on_scaleR [continuous_intros] =
bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
lemmas continuous_on_mult [continuous_intros] =
bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
lemmas tendsto_mult_zero =
bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
lemmas tendsto_mult_left_zero =
bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
lemmas tendsto_mult_right_zero =
bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
lemma tendsto_power [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
by (induct n) (simp_all add: tendsto_mult)
lemma continuous_power [continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
unfolding continuous_def by (rule tendsto_power)
lemma continuous_on_power [continuous_intros]:
fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
unfolding continuous_on_def by (auto intro: tendsto_power)
lemma tendsto_setprod [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
proof (cases "finite S")
assume "finite S" thus ?thesis using assms
by (induct, simp, simp add: tendsto_mult)
qed simp
lemma continuous_setprod [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
unfolding continuous_def by (rule tendsto_setprod)
lemma continuous_on_setprod [continuous_intros]:
fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
unfolding continuous_on_def by (auto intro: tendsto_setprod)
subsubsection {* Inverse and division *}
lemma (in bounded_bilinear) Zfun_prod_Bfun:
assumes f: "Zfun f F"
assumes g: "Bfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
proof -
obtain K where K: "0 \<le> K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
using nonneg_bounded by fast
obtain B where B: "0 < B"
and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
using g by (rule BfunE)
have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
using norm_g proof eventually_elim
case (elim x)
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
by (rule norm_le)
also have "\<dots> \<le> norm (f x) * B * K"
by (intro mult_mono' order_refl norm_g norm_ge_zero
mult_nonneg_nonneg K elim)
also have "\<dots> = norm (f x) * (B * K)"
by (rule mult.assoc)
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
qed
with f show ?thesis
by (rule Zfun_imp_Zfun)
qed
lemma (in bounded_bilinear) flip:
"bounded_bilinear (\<lambda>x y. y ** x)"
apply default
apply (rule add_right)
apply (rule add_left)
apply (rule scaleR_right)
apply (rule scaleR_left)
apply (subst mult.commute)
using bounded by fast
lemma (in bounded_bilinear) Bfun_prod_Zfun:
assumes f: "Bfun f F"
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
lemma Bfun_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 Bfun_inverse:
fixes a :: "'a::real_normed_div_algebra"
assumes f: "(f ---> a) F"
assumes a: "a \<noteq> 0"
shows "Bfun (\<lambda>x. inverse (f x)) F"
proof -
from a have "0 < norm a" by simp
hence "\<exists>r>0. r < norm a" by (rule dense)
then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
have "eventually (\<lambda>x. dist (f x) a < r) F"
using tendstoD [OF f r1] by fast
hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
proof eventually_elim
case (elim x)
hence 1: "norm (f x - a) < r"
by (simp add: dist_norm)
hence 2: "f x \<noteq> 0" using r2 by auto
hence "norm (inverse (f x)) = inverse (norm (f x))"
by (rule nonzero_norm_inverse)
also have "\<dots> \<le> inverse (norm a - r)"
proof (rule le_imp_inverse_le)
show "0 < norm a - r" using r2 by simp
next
have "norm a - norm (f x) \<le> norm (a - f x)"
by (rule norm_triangle_ineq2)
also have "\<dots> = norm (f x - a)"
by (rule norm_minus_commute)
also have "\<dots> < r" using 1 .
finally show "norm a - r \<le> norm (f x)" by simp
qed
finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
qed
thus ?thesis by (rule BfunI)
qed
lemma tendsto_inverse [tendsto_intros]:
fixes a :: "'a::real_normed_div_algebra"
assumes f: "(f ---> a) F"
assumes a: "a \<noteq> 0"
shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
proof -
from a have "0 < norm a" by simp
with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
by (rule tendstoD)
then have "eventually (\<lambda>x. f x \<noteq> 0) F"
unfolding dist_norm by (auto elim!: eventually_elim1)
with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
- (inverse (f x) * (f x - a) * inverse a)) F"
by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
by (intro Zfun_minus Zfun_mult_left
bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
ultimately show ?thesis
unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
qed
lemma continuous_inverse:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. inverse (f x))"
using assms unfolding continuous_def by (rule tendsto_inverse)
lemma continuous_at_within_inverse[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
assumes "continuous (at a within s) f" and "f a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
using assms unfolding continuous_within by (rule tendsto_inverse)
lemma isCont_inverse[continuous_intros, simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
assumes "isCont f a" and "f a \<noteq> 0"
shows "isCont (\<lambda>x. inverse (f x)) a"
using assms unfolding continuous_at by (rule tendsto_inverse)
lemma continuous_on_inverse[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
shows "continuous_on s (\<lambda>x. inverse (f x))"
using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
lemma tendsto_divide [tendsto_intros]:
fixes a b :: "'a::real_normed_field"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
by (simp add: tendsto_mult tendsto_inverse divide_inverse)
lemma continuous_divide:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_def by (rule tendsto_divide)
lemma continuous_at_within_divide[continuous_intros]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_within by (rule tendsto_divide)
lemma isCont_divide[continuous_intros, simp]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
shows "isCont (\<lambda>x. (f x) / g x) a"
using assms unfolding continuous_at by (rule tendsto_divide)
lemma continuous_on_divide[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
shows "continuous_on s (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
lemma tendsto_sgn [tendsto_intros]:
fixes l :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
unfolding sgn_div_norm by (simp add: tendsto_intros)
lemma continuous_sgn:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. sgn (f x))"
using assms unfolding continuous_def by (rule tendsto_sgn)
lemma continuous_at_within_sgn[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
assumes "continuous (at a within s) f" and "f a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
using assms unfolding continuous_within by (rule tendsto_sgn)
lemma isCont_sgn[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
assumes "isCont f a" and "f a \<noteq> 0"
shows "isCont (\<lambda>x. sgn (f x)) a"
using assms unfolding continuous_at by (rule tendsto_sgn)
lemma continuous_on_sgn[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
shows "continuous_on s (\<lambda>x. sgn (f x))"
using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
lemma filterlim_at_infinity:
fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
assumes "0 \<le> c"
shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
unfolding filterlim_iff eventually_at_infinity
proof safe
fix P :: "'a \<Rightarrow> bool" and b
assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
have "max b (c + 1) > c" by auto
with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
by auto
then show "eventually (\<lambda>x. P (f x)) F"
proof eventually_elim
fix x assume "max b (c + 1) \<le> norm (f x)"
with P show "P (f x)" by auto
qed
qed force
subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
text {*
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
@{term "at_right x"} and also @{term "at_right 0"}.
*}
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
lemma filtermap_homeomorph:
assumes f: "continuous (at a) f"
assumes g: "continuous (at (f a)) g"
assumes bij1: "\<forall>x. f (g x) = x" and bij2: "\<forall>x. g (f x) = x"
shows "filtermap f (nhds a) = nhds (f a)"
unfolding filter_eq_iff eventually_filtermap eventually_nhds
proof safe
fix P S assume S: "open S" "f a \<in> S" and P: "\<forall>x\<in>S. P x"
from continuous_within_topological[THEN iffD1, rule_format, OF f S] P
show "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P (f x))" by auto
next
fix P S assume S: "open S" "a \<in> S" and P: "\<forall>x\<in>S. P (f x)"
with continuous_within_topological[THEN iffD1, rule_format, OF g, of S] bij2
obtain A where "open A" "f a \<in> A" "(\<forall>y\<in>A. g y \<in> S)"
by (metis UNIV_I)
with P bij1 show "\<exists>S. open S \<and> f a \<in> S \<and> (\<forall>x\<in>S. P x)"
by (force intro!: exI[of _ A])
qed
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
by (rule filtermap_homeomorph[where g="\<lambda>x. x + d"]) (auto intro: continuous_intros)
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
by (rule filtermap_homeomorph[where g=uminus]) (auto intro: continuous_minus)
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
using filtermap_at_right_shift[of "-a" 0] by simp
lemma filterlim_at_right_to_0:
"filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
lemma eventually_at_right_to_0:
"eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
lemma filterlim_at_left_to_right:
"filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
lemma eventually_at_left_to_right:
"eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
by (metis le_minus_iff minus_minus)
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
unfolding filterlim_def at_top_mirror filtermap_filtermap ..
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
unfolding filterlim_at_top eventually_at_bot_dense
by (metis leI minus_less_iff order_less_asym)
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
unfolding filterlim_at_bot eventually_at_top_dense
by (metis leI less_minus_iff order_less_asym)
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
by auto
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
unfolding filterlim_uminus_at_top by simp
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
proof safe
fix Z :: real assume [arith]: "0 < Z"
then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
qed
lemma filterlim_inverse_at_top:
"(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
(simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
lemma filterlim_inverse_at_bot_neg:
"LIM x (at_left (0::real)). inverse x :> at_bot"
by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
lemma filterlim_inverse_at_bot:
"(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
lemma tendsto_inverse_0:
fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
shows "(inverse ---> (0::'a)) at_infinity"
unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
proof safe
fix r :: real assume "0 < r"
show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
proof (intro exI[of _ "inverse (r / 2)"] allI impI)
fix x :: 'a
from `0 < r` have "0 < inverse (r / 2)" by simp
also assume *: "inverse (r / 2) \<le> norm x"
finally show "norm (inverse x) < r"
using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
qed
qed
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
proof (rule antisym)
have "(inverse ---> (0::real)) at_top"
by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
then show "filtermap inverse at_top \<le> at_right (0::real)"
by (simp add: le_principal eventually_filtermap eventually_gt_at_top filterlim_def at_within_def)
next
have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
then show "at_right (0::real) \<le> filtermap inverse at_top"
by (simp add: filtermap_ident filtermap_filtermap)
qed
lemma eventually_at_right_to_top:
"eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
unfolding at_right_to_top eventually_filtermap ..
lemma filterlim_at_right_to_top:
"filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
unfolding filterlim_def at_right_to_top filtermap_filtermap ..
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
lemma eventually_at_top_to_right:
"eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
unfolding at_top_to_right eventually_filtermap ..
lemma filterlim_at_top_to_right:
"filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
unfolding filterlim_def at_top_to_right filtermap_filtermap ..
lemma filterlim_inverse_at_infinity:
fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring}"
shows "filterlim inverse at_infinity (at (0::'a))"
unfolding filterlim_at_infinity[OF order_refl]
proof safe
fix r :: real assume "0 < r"
then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
unfolding eventually_at norm_inverse
by (intro exI[of _ "inverse r"])
(auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
qed
lemma filterlim_inverse_at_iff:
fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring}"
shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
unfolding filterlim_def filtermap_filtermap[symmetric]
proof
assume "filtermap g F \<le> at_infinity"
then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
by (rule filtermap_mono)
also have "\<dots> \<le> at 0"
using tendsto_inverse_0[where 'a='b]
by (auto intro!: exI[of _ 1]
simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
finally show "filtermap inverse (filtermap g F) \<le> at 0" .
next
assume "filtermap inverse (filtermap g F) \<le> at 0"
then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
by (rule filtermap_mono)
with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
qed
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
lemma at_to_infinity:
fixes x :: "'a \<Colon> {real_normed_field,field}"
shows "(at (0::'a)) = filtermap inverse at_infinity"
proof (rule antisym)
have "(inverse ---> (0::'a)) at_infinity"
by (fact tendsto_inverse_0)
then show "filtermap inverse at_infinity \<le> at (0::'a)"
apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
apply (rule_tac x="1" in exI, auto)
done
next
have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
using filterlim_inverse_at_infinity unfolding filterlim_def
by (rule filtermap_mono)
then show "at (0::'a) \<le> filtermap inverse at_infinity"
by (simp add: filtermap_ident filtermap_filtermap)
qed
lemma lim_at_infinity_0:
fixes l :: "'a :: {real_normed_field,field}"
shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"
by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lemma lim_zero_infinity:
fixes l :: "'a :: {real_normed_field,field}"
shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"
by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
text {*
We only show rules for multiplication and addition when the functions are either against a real
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
*}
lemma filterlim_tendsto_pos_mult_at_top:
assumes f: "(f ---> c) F" and c: "0 < c"
assumes g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x * g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
fix Z :: real assume "0 < Z"
from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
simp: dist_real_def abs_real_def split: split_if_asm)
moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
proof eventually_elim
fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
by (intro mult_mono) (auto simp: zero_le_divide_iff)
with `0 < c` show "Z \<le> f x * g x"
by simp
qed
qed
lemma filterlim_at_top_mult_at_top:
assumes f: "LIM x F. f x :> at_top"
assumes g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x * g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
fix Z :: real assume "0 < Z"
from f have "eventually (\<lambda>x. 1 \<le> f x) F"
unfolding filterlim_at_top by auto
moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
proof eventually_elim
fix x assume "1 \<le> f x" "Z \<le> g x"
with `0 < Z` have "1 * Z \<le> f x * g x"
by (intro mult_mono) (auto simp: zero_le_divide_iff)
then show "Z \<le> f x * g x"
by simp
qed
qed
lemma filterlim_tendsto_pos_mult_at_bot:
assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
shows "LIM x F. f x * g x :> at_bot"
using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
unfolding filterlim_uminus_at_bot by simp
lemma filterlim_pow_at_top:
fixes f :: "real \<Rightarrow> real"
assumes "0 < n" and f: "LIM x F. f x :> at_top"
shows "LIM x F. (f x)^n :: real :> at_top"
using `0 < n` proof (induct n)
case (Suc n) with f show ?case
by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
qed simp
lemma filterlim_pow_at_bot_even:
fixes f :: "real \<Rightarrow> real"
shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
lemma filterlim_pow_at_bot_odd:
fixes f :: "real \<Rightarrow> real"
shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
lemma filterlim_tendsto_add_at_top:
assumes f: "(f ---> c) F"
assumes g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x + g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
fix Z :: real assume "0 < Z"
from f have "eventually (\<lambda>x. c - 1 < f x) F"
by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
by eventually_elim simp
qed
lemma LIM_at_top_divide:
fixes f g :: "'a \<Rightarrow> real"
assumes f: "(f ---> a) F" "0 < a"
assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
shows "LIM x F. f x / g x :> at_top"
unfolding divide_inverse
by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
lemma filterlim_at_top_add_at_top:
assumes f: "LIM x F. f x :> at_top"
assumes g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x + g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
fix Z :: real assume "0 < Z"
from f have "eventually (\<lambda>x. 0 \<le> f x) F"
unfolding filterlim_at_top by auto
moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
by eventually_elim simp
qed
lemma tendsto_divide_0:
fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring}"
assumes f: "(f ---> c) F"
assumes g: "LIM x F. g x :> at_infinity"
shows "((\<lambda>x. f x / g x) ---> 0) F"
using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
lemma linear_plus_1_le_power:
fixes x :: real
assumes x: "0 \<le> x"
shows "real n * x + 1 \<le> (x + 1) ^ n"
proof (induct n)
case (Suc n)
have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
by (simp add: field_simps real_of_nat_Suc x)
also have "\<dots> \<le> (x + 1)^Suc n"
using Suc x by (simp add: mult_left_mono)
finally show ?case .
qed simp
lemma filterlim_realpow_sequentially_gt1:
fixes x :: "'a :: real_normed_div_algebra"
assumes x[arith]: "1 < norm x"
shows "LIM n sequentially. x ^ n :> at_infinity"
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
fix y :: real assume "0 < y"
have "0 < norm x - 1" by simp
then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
also have "\<dots> = norm x ^ N" by simp
finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
by (metis order_less_le_trans power_increasing order_less_imp_le x)
then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
unfolding eventually_sequentially
by (auto simp: norm_power)
qed simp
subsection {* Limits of Sequences *}
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
by simp
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_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_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 mult.commute)
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))"
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
subsection {* Convergence on sequences *}
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
text {* A monotone sequence converges to its least upper bound. *}
lemma LIMSEQ_incseq_SUP:
fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
assumes u: "bdd_above (range X)"
assumes X: "incseq X"
shows "X ----> (SUP i. X i)"
by (rule order_tendstoI)
(auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
lemma LIMSEQ_decseq_INF:
fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
assumes u: "bdd_below (range X)"
assumes X: "decseq X"
shows "X ----> (INF i. X i)"
by (rule order_tendstoI)
(auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
text{*Main monotonicity theorem*}
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_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 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 incseq_convergent:
fixes X :: "nat \<Rightarrow> real"
assumes "incseq X" and "\<forall>i. X i \<le> B"
obtains L where "X ----> L" "\<forall>i. X i \<le> L"
proof atomize_elim
from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
obtain L where "X ----> L"
by (auto simp: convergent_def monoseq_def incseq_def)
with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
by (auto intro!: exI[of _ L] incseq_le)
qed
lemma decseq_convergent:
fixes X :: "nat \<Rightarrow> real"
assumes "decseq X" and "\<forall>i. B \<le> X i"
obtains L where "X ----> L" "\<forall>i. L \<le> X i"
proof atomize_elim
from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
obtain L where "X ----> L"
by (auto simp: convergent_def monoseq_def decseq_def)
with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
by (auto intro!: exI[of _ L] decseq_le)
qed
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
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)
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
subsection {* Limits of Functions *}
lemma LIM_eq:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "f -- a --> L =
(\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
by (simp add: LIM_def dist_norm)
lemma LIM_I:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
==> f -- a --> L"
by (simp add: LIM_eq)
lemma LIM_D:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "[| f -- a --> L; 0<r |]
==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
by (simp add: LIM_eq)
lemma LIM_offset:
fixes a :: "'a::real_normed_vector"
shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
lemma LIM_offset_zero:
fixes a :: "'a::real_normed_vector"
shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
lemma LIM_offset_zero_cancel:
fixes a :: "'a::real_normed_vector"
shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
by (drule_tac k="- a" in LIM_offset, simp)
lemma LIM_offset_zero_iff:
fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
shows "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
lemma LIM_zero:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma LIM_zero_cancel:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
unfolding tendsto_iff dist_norm by simp
lemma LIM_zero_iff:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
unfolding tendsto_iff dist_norm by simp
lemma LIM_imp_LIM:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
assumes f: "f -- a --> l"
assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
shows "g -- a --> m"
by (rule metric_LIM_imp_LIM [OF f],
simp add: dist_norm le)
lemma LIM_equal2:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
assumes 1: "0 < R"
assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
shows "g -- a --> l \<Longrightarrow> f -- a --> l"
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
lemma LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f: "f -- a --> b"
assumes g: "g -- b --> c"
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
shows "(\<lambda>x. g (f x)) -- a --> c"
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
lemma real_LIM_sandwich_zero:
fixes f g :: "'a::topological_space \<Rightarrow> real"
assumes f: "f -- a --> 0"
assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
shows "g -- a --> 0"
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
fix x assume x: "x \<noteq> a"
have "norm (g x - 0) = g x" by (simp add: 1 x)
also have "g x \<le> f x" by (rule 2 [OF x])
also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
finally show "norm (g x - 0) \<le> norm (f x - 0)" .
qed
subsection {* Continuity *}
lemma LIM_isCont_iff:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
lemma isCont_iff:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
by (simp add: isCont_def LIM_isCont_iff)
lemma isCont_LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f [unfolded isCont_def]: "isCont f a"
assumes g: "g -- f a --> l"
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
shows "(\<lambda>x. g (f x)) -- a --> l"
by (rule LIM_compose2 [OF f g inj])
lemma isCont_norm [simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
by (fact continuous_norm)
lemma isCont_rabs [simp]:
fixes f :: "'a::t2_space \<Rightarrow> real"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
by (fact continuous_rabs)
lemma isCont_add [simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
by (fact continuous_add)
lemma isCont_minus [simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
by (fact continuous_minus)
lemma isCont_diff [simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
by (fact continuous_diff)
lemma isCont_mult [simp]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
by (fact continuous_mult)
lemma (in bounded_linear) isCont:
"isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
by (fact continuous)
lemma (in bounded_bilinear) isCont:
"\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
by (fact continuous)
lemmas isCont_scaleR [simp] =
bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
lemmas isCont_of_real [simp] =
bounded_linear.isCont [OF bounded_linear_of_real]
lemma isCont_power [simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
by (fact continuous_power)
lemma isCont_setsum [simp]:
fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
by (auto intro: continuous_setsum)
subsection {* Uniform Continuity *}
definition
isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
"isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
lemma isUCont_isCont: "isUCont f ==> isCont f x"
by (simp add: isUCont_def isCont_def LIM_def, force)
lemma isUCont_Cauchy:
"\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
unfolding isUCont_def
apply (rule metric_CauchyI)
apply (drule_tac x=e in spec, safe)
apply (drule_tac e=s in metric_CauchyD, safe)
apply (rule_tac x=M in exI, simp)
done
lemma (in bounded_linear) isUCont: "isUCont f"
unfolding isUCont_def dist_norm
proof (intro allI impI)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
using pos_bounded by fast
show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
proof (rule exI, safe)
from r K show "0 < r / K" by simp
next
fix x y :: 'a
assume xy: "norm (x - y) < r / K"
have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
finally show "norm (f x - f y) < r" .
qed
qed
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
by (rule isUCont [THEN isUCont_Cauchy])
lemma LIM_less_bound:
fixes f :: "real \<Rightarrow> real"
assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
shows "0 \<le> f x"
proof (rule tendsto_le_const)
show "(f ---> f x) (at_left x)"
using `isCont f x` by (simp add: filterlim_at_split isCont_def)
show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
qed simp
subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
lemma nested_sequence_unique:
assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) ----> 0"
shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
proof -
have "incseq f" unfolding incseq_Suc_iff by fact
have "decseq g" unfolding decseq_Suc_iff by fact
{ fix n
from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
using incseq_convergent[OF `incseq f`] by auto
moreover
{ fix n
from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
using decseq_convergent[OF `decseq g`] by auto
moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
ultimately show ?thesis by auto
qed
lemma Bolzano[consumes 1, case_names trans local]:
fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
assumes [arith]: "a \<le> b"
assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
shows "P a b"
proof -
def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
by (simp_all add: l_def u_def bisect_def split: prod.split)
{ fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
proof (safe intro!: nested_sequence_unique)
fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
next
{ fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
qed fact
then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
show "P a b"
proof (rule ccontr)
assume "\<not> P a b"
{ fix n have "\<not> P (l n) (u n)"
proof (induct n)
case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
qed (simp add: `\<not> P a b`) }
moreover
{ have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
proof eventually_elim
fix n assume "x - d / 2 < l n" "u n < x + d / 2"
from add_strict_mono[OF this] have "u n - l n < d" by simp
with x show "P (l n) (u n)" by (rule d)
qed }
ultimately show False by simp
qed
qed
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
proof (cases "a \<le> b", rule compactI)
fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
def T == "{a .. b}"
from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
proof (induct rule: Bolzano)
case (trans a b c)
then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
by (auto simp: *)
with trans show ?case
unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
next
case (local x)
then have "x \<in> \<Union>C" using C by auto
with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
with `c \<in> C` show ?case
by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
qed
qed simp
lemma continuous_image_closed_interval:
fixes a b and f :: "real \<Rightarrow> real"
defines "S \<equiv> {a..b}"
assumes "a \<le> b" and f: "continuous_on S f"
shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
proof -
have S: "compact S" "S \<noteq> {}"
using `a \<le> b` by (auto simp: S_def)
obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
using continuous_attains_sup[OF S f] by auto
moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
using continuous_attains_inf[OF S f] by auto
moreover have "connected (f`S)"
using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
by (auto simp: connected_iff_interval)
then show ?thesis
by auto
qed
subsection {* Boundedness of continuous functions *}
text{*By bisection, function continuous on closed interval is bounded above*}
lemma isCont_eq_Ub:
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
using continuous_attains_sup[of "{a .. b}" f]
by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
lemma isCont_eq_Lb:
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
using continuous_attains_inf[of "{a .. b}" f]
by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
lemma isCont_bounded:
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
using isCont_eq_Ub[of a b f] by auto
lemma isCont_has_Ub:
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
using isCont_eq_Ub[of a b f] by auto
(*HOL style here: object-level formulations*)
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
by (blast intro: IVT)
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
by (blast intro: IVT2)
lemma isCont_Lb_Ub:
fixes f :: "real \<Rightarrow> real"
assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
(\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
proof -
obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
using isCont_eq_Ub[OF assms] by auto
obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
using isCont_eq_Lb[OF assms] by auto
show ?thesis
using IVT[of f L _ M] IVT2[of f L _ M] M L assms
apply (rule_tac x="f L" in exI)
apply (rule_tac x="f M" in exI)
apply (cases "L \<le> M")
apply (simp, metis order_trans)
apply (simp, metis order_trans)
done
qed
text{*Continuity of inverse function*}
lemma isCont_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes d: "0 < d"
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
shows "isCont g (f x)"
proof -
let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
have f: "continuous_on ?D f"
using cont by (intro continuous_at_imp_continuous_on ballI) auto
then have g: "continuous_on (f`?D) g"
using inj by (intro continuous_on_inv) auto
from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
by (rule continuous_on_subset)
moreover
have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
by auto
ultimately
show ?thesis
by (simp add: continuous_on_eq_continuous_at)
qed
lemma isCont_inverse_function2:
fixes f g :: "real \<Rightarrow> real" shows
"\<lbrakk>a < x; x < b;
\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
\<Longrightarrow> isCont g (f x)"
apply (rule isCont_inverse_function
[where f=f and d="min (x - a) (b - x)"])
apply (simp_all add: abs_le_iff)
done
(* need to rename second isCont_inverse *)
lemma isCont_inv_fun:
fixes f g :: "real \<Rightarrow> real"
shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
\<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
==> isCont g (f x)"
by (rule isCont_inverse_function)
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
lemma LIM_fun_gt_zero:
fixes f :: "real \<Rightarrow> real"
shows "f -- c --> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
apply (drule (1) LIM_D, clarify)
apply (rule_tac x = s in exI)
apply (simp add: abs_less_iff)
done
lemma LIM_fun_less_zero:
fixes f :: "real \<Rightarrow> real"
shows "f -- c --> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
apply (drule LIM_D [where r="-l"], simp, clarify)
apply (rule_tac x = s in exI)
apply (simp add: abs_less_iff)
done
lemma LIM_fun_not_zero:
fixes f :: "real \<Rightarrow> real"
shows "f -- c --> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
end