(*  Title       : Lim.thy

     Author      : Jacques D. Fleuriot

     Copyright   : 1998  University of Cambridge

     Conversion to Isar and new proofs by Lawrence C Paulson, 2004

 *)

 header{* Limits and Continuity *}

 theory Lim

 imports SEQ

 begin

 text{*Standard Definitions*}

 definition

   LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"

         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where

   [code del]: "f -- a --> L =

      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s

         --> dist (f x) L < r)"

 definition

   isCont :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a] \<Rightarrow> bool" where

   "isCont f a = (f -- a --> (f a))"

 definition

   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where

   [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"

 subsection {* Limits of Functions *}

 lemma LIM_conv_tendsto: "(f -- a --> L) \<longleftrightarrow> (f ---> L) (at a)"

 unfolding LIM_def tendsto_iff eventually_at ..

 lemma metric_LIM_I:

   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)

     \<Longrightarrow> f -- a --> L"

 by (simp add: LIM_def)

 lemma metric_LIM_D:

   "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>

     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"

 by (simp add: LIM_def)

 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" and L :: "'b::metric_space"

   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"

 unfolding LIM_def dist_norm

 apply clarify

 apply (drule_tac x="r" in spec, safe)

 apply (rule_tac x="s" in exI, safe)

 apply (drule_tac x="x + k" in spec)

 apply (simp add: algebra_simps)

 done

 lemma LIM_offset_zero:

   fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"

   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" and L :: "'b::metric_space"

   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"

 by (drule_tac k="- a" in LIM_offset, simp)

 lemma LIM_const [simp]: "(%x. k) -- x --> k"

 by (simp add: LIM_def)

 lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp

 lemma LIM_add:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   assumes f: "f -- a --> L" and g: "g -- a --> M"

   shows "(\<lambda>x. f x + g x) -- a --> (L + M)"

 using assms unfolding LIM_conv_tendsto by (rule tendsto_add)

 lemma LIM_add_zero:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"

 by (drule (1) LIM_add, simp)

 lemma minus_diff_minus:

   fixes a b :: "'a::ab_group_add"

   shows "(- a) - (- b) = - (a - b)"

 by simp

 lemma LIM_minus:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"

 unfolding LIM_conv_tendsto by (rule tendsto_minus)

 (* TODO: delete *)

 lemma LIM_add_minus:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"

 by (intro LIM_add LIM_minus)

 lemma LIM_diff:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"

 unfolding LIM_conv_tendsto by (rule tendsto_diff)

 lemma LIM_zero:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"

 by (simp add: LIM_def dist_norm)

 lemma LIM_zero_cancel:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"

 by (simp add: LIM_def dist_norm)

 lemma LIM_zero_iff:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"

 by (simp add: LIM_def dist_norm)

 lemma metric_LIM_imp_LIM:

   assumes f: "f -- a --> l"

   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"

   shows "g -- a --> m"

 apply (rule metric_LIM_I, drule metric_LIM_D [OF f], safe)

 apply (rule_tac x="s" in exI, safe)

 apply (drule_tac x="x" in spec, safe)

 apply (erule (1) order_le_less_trans [OF le])

 done

 lemma LIM_imp_LIM:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   fixes g :: "'a::metric_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"

 apply (rule metric_LIM_imp_LIM [OF f])

 apply (simp add: dist_norm le)

 done

 lemma LIM_norm:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"

 unfolding LIM_conv_tendsto by (rule tendsto_norm)

 lemma LIM_norm_zero:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"

 by (drule LIM_norm, simp)

 lemma LIM_norm_zero_cancel:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"

 by (erule LIM_imp_LIM, simp)

 lemma LIM_norm_zero_iff:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"

 by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])

 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"

 by (fold real_norm_def, rule LIM_norm)

 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"

 by (fold real_norm_def, rule LIM_norm_zero)

 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"

 by (fold real_norm_def, rule LIM_norm_zero_cancel)

 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"

 by (fold real_norm_def, rule LIM_norm_zero_iff)

 lemma LIM_const_not_eq:

   fixes a :: "'a::real_normed_algebra_1"

   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"

 apply (simp add: LIM_def)

 apply (rule_tac x="dist k L" in exI, simp add: zero_less_dist_iff, safe)

 apply (rule_tac x="a + of_real (s/2)" in exI, simp add: dist_norm)

 done

 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]

 lemma LIM_const_eq:

   fixes a :: "'a::real_normed_algebra_1"

   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"

 apply (rule ccontr)

 apply (blast dest: LIM_const_not_eq)

 done

 lemma LIM_unique:

   fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}

   shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"

 apply (rule ccontr)

 apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)

 apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)

 apply (clarify, rename_tac r s)

 apply (subgoal_tac "min r s \<noteq> 0")

 apply (subgoal_tac "dist L M < dist L M / 2 + dist L M / 2", simp)

 apply (subgoal_tac "dist L M \<le> dist (f (a + of_real (min r s / 2))) L +

                                dist (f (a + of_real (min r s / 2))) M")

 apply (erule le_less_trans, rule add_strict_mono)

 apply (drule spec, erule mp, simp add: dist_norm)

 apply (drule spec, erule mp, simp add: dist_norm)

 apply (subst dist_commute, rule dist_triangle)

 apply simp

 done

 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"

 by (auto simp add: LIM_def)

 text{*Limits are equal for functions equal except at limit point*}

 lemma LIM_equal:

      "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"

 by (simp add: LIM_def)

 lemma LIM_cong:

   "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>

    \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"

 by (simp add: LIM_def)

 lemma metric_LIM_equal2:

   assumes 1: "0 < R"

   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"

   shows "g -- a --> l \<Longrightarrow> f -- a --> l"

 apply (unfold LIM_def, safe)

 apply (drule_tac x="r" in spec, safe)

 apply (rule_tac x="min s R" in exI, safe)

 apply (simp add: 1)

 apply (simp add: 2)

 done

 lemma LIM_equal2:

   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::metric_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"

 apply (unfold LIM_def dist_norm, safe)

 apply (drule_tac x="r" in spec, safe)

 apply (rule_tac x="min s R" in exI, safe)

 apply (simp add: 1)

 apply (simp add: 2)

 done

 text{*Two uses in Transcendental.ML*}

 lemma LIM_trans:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"

 apply (drule LIM_add, assumption)

 apply (auto simp add: add_assoc)

 done

 lemma LIM_compose:

   assumes g: "g -- l --> g l"

   assumes f: "f -- a --> l"

   shows "(\<lambda>x. g (f x)) -- a --> g l"

 proof (rule metric_LIM_I)

   fix r::real assume r: "0 < r"

   obtain s where s: "0 < s"

     and less_r: "\<And>y. \<lbrakk>y \<noteq> l; dist y l < s\<rbrakk> \<Longrightarrow> dist (g y) (g l) < r"

     using metric_LIM_D [OF g r] by fast

   obtain t where t: "0 < t"

     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) l < s"

     using metric_LIM_D [OF f s] by fast

   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) (g l) < r"

   proof (rule exI, safe)

     show "0 < t" using t .

   next

     fix x assume "x \<noteq> a" and "dist x a < t"

     hence "dist (f x) l < s" by (rule less_s)

     thus "dist (g (f x)) (g l) < r"

       using r less_r by (case_tac "f x = l", simp_all)

   qed

 qed

 lemma metric_LIM_compose2:

   assumes f: "f -- a --> b"

   assumes g: "g -- b --> c"

   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"

   shows "(\<lambda>x. g (f x)) -- a --> c"

 proof (rule metric_LIM_I)

   fix r :: real

   assume r: "0 < r"

   obtain s where s: "0 < s"

     and less_r: "\<And>y. \<lbrakk>y \<noteq> b; dist y b < s\<rbrakk> \<Longrightarrow> dist (g y) c < r"

     using metric_LIM_D [OF g r] by fast

   obtain t where t: "0 < t"

     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) b < s"

     using metric_LIM_D [OF f s] by fast

   obtain d where d: "0 < d"

     and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"

     using inj by fast

   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) c < r"

   proof (safe intro!: exI)

     show "0 < min d t" using d t by simp

   next

     fix x

     assume "x \<noteq> a" and "dist x a < min d t"

     hence "f x \<noteq> b" and "dist (f x) b < s"

       using neq_b less_s by simp_all

     thus "dist (g (f x)) c < r"

       by (rule less_r)

   qed

 qed

 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 LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"

 unfolding o_def by (rule LIM_compose)

 lemma real_LIM_sandwich_zero:

   fixes f g :: "'a::metric_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])

   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

 text {* Bounded Linear Operators *}

 lemma (in bounded_linear) LIM:

   "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"

 unfolding LIM_conv_tendsto by (rule tendsto)

 lemma (in bounded_linear) cont: "f -- a --> f a"

 by (rule LIM [OF LIM_ident])

 lemma (in bounded_linear) LIM_zero:

   "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"

 by (drule LIM, simp only: zero)

 text {* Bounded Bilinear Operators *}

 lemma (in bounded_bilinear) LIM:

   "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"

 unfolding LIM_conv_tendsto by (rule tendsto)

 lemma (in bounded_bilinear) LIM_prod_zero:

   fixes a :: "'d::metric_space"

   assumes f: "f -- a --> 0"

   assumes g: "g -- a --> 0"

   shows "(\<lambda>x. f x ** g x) -- a --> 0"

 using LIM [OF f g] by (simp add: zero_left)

 lemma (in bounded_bilinear) LIM_left_zero:

   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"

 by (rule bounded_linear.LIM_zero [OF bounded_linear_left])

 lemma (in bounded_bilinear) LIM_right_zero:

   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"

 by (rule bounded_linear.LIM_zero [OF bounded_linear_right])

 lemmas LIM_mult = mult.LIM

 lemmas LIM_mult_zero = mult.LIM_prod_zero

 lemmas LIM_mult_left_zero = mult.LIM_left_zero

 lemmas LIM_mult_right_zero = mult.LIM_right_zero

 lemmas LIM_scaleR = scaleR.LIM

 lemmas LIM_of_real = of_real.LIM

 lemma LIM_power:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"

   assumes f: "f -- a --> l"

   shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"

 by (induct n, simp, simp add: LIM_mult f)

 subsubsection {* Derived theorems about @{term LIM} *}

 lemma LIM_inverse:

   fixes L :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"

 unfolding LIM_conv_tendsto

 by (rule tendsto_inverse)

 lemma LIM_inverse_fun:

   assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"

   shows "inverse -- a --> inverse a"

 by (rule LIM_inverse [OF LIM_ident a])

 lemma LIM_sgn:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"

 unfolding sgn_div_norm

 by (simp add: LIM_scaleR LIM_inverse LIM_norm)

 subsection {* Continuity *}

 lemma LIM_isCont_iff:

   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::metric_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::metric_space"

   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"

 by (simp add: isCont_def LIM_isCont_iff)

 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"

   unfolding isCont_def by (rule LIM_ident)

 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"

   unfolding isCont_def by (rule LIM_const)

 lemma isCont_norm:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"

   unfolding isCont_def by (rule LIM_norm)

 lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"

   unfolding isCont_def by (rule LIM_rabs)

 lemma isCont_add:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"

   unfolding isCont_def by (rule LIM_add)

 lemma isCont_minus:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"

   unfolding isCont_def by (rule LIM_minus)

 lemma isCont_diff:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"

   unfolding isCont_def by (rule LIM_diff)

 lemma isCont_mult:

   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_algebra"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"

   unfolding isCont_def by (rule LIM_mult)

 lemma isCont_inverse:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_div_algebra"

   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"

   unfolding isCont_def by (rule LIM_inverse)

 lemma isCont_LIM_compose:

   "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"

   unfolding isCont_def by (rule LIM_compose)

 lemma metric_isCont_LIM_compose2:

   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> dist x a < d \<longrightarrow> f x \<noteq> f a"

   shows "(\<lambda>x. g (f x)) -- a --> l"

 by (rule metric_LIM_compose2 [OF f g inj])

 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_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"

   unfolding isCont_def by (rule LIM_compose)

 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"

   unfolding o_def by (rule isCont_o2)

 lemma (in bounded_linear) isCont: "isCont f a"

   unfolding isCont_def by (rule cont)

 lemma (in bounded_bilinear) isCont:

   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"

   unfolding isCont_def by (rule LIM)

 lemmas isCont_scaleR = scaleR.isCont

 lemma isCont_of_real:

   "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"

   unfolding isCont_def by (rule LIM_of_real)

 lemma isCont_power:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"

   unfolding isCont_def by (rule LIM_power)

 lemma isCont_sgn:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"

   unfolding isCont_def by (rule LIM_sgn)

 lemma isCont_abs [simp]: "isCont abs (a::real)"

 by (rule isCont_rabs [OF isCont_ident])

 lemma isCont_setsum:

   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"

   fixes A :: "'a set" assumes "finite A"

   shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x"

   using `finite A`

 proof induct

   case (insert a F) show "isCont (\<lambda> x. \<Sum> i \<in> (insert a F). f i x) x" 

     unfolding setsum_insert[OF `finite F` `a \<notin> F`] by (rule isCont_add, auto simp add: insert)

 qed auto

 lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"

   and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"

   shows "0 \<le> f x"

 proof (rule ccontr)

   assume "\<not> 0 \<le> f x" hence "f x < 0" by auto

   hence "0 < - f x / 2" by auto

   from isCont[unfolded isCont_def, THEN LIM_D, OF this]

   obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x' - x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x' - f x \<bar> < - f x / 2" by auto

   let ?x = "x - min (s / 2) ((x - b) / 2)"

   have "?x < x" and "\<bar> ?x - x \<bar> < s"

     using `b < x` and `0 < s` by auto

   have "b < ?x"

   proof (cases "s < x - b")

     case True thus ?thesis using `0 < s` by auto

   next

     case False hence "s / 2 \<ge> (x - b) / 2" by auto

     hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)

     thus ?thesis using `b < x` by auto

   qed

   hence "0 \<le> f ?x" using all_le `?x < x` by auto

   moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"

     using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto

   hence "f ?x - f x < - f x / 2" by auto

   hence "f ?x < f x / 2" by auto

   hence "f ?x < 0" using `f x < 0` by auto

   thus False using `0 \<le> f ?x` by auto

 qed

 subsection {* Uniform Continuity *}

 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 (rule divide_pos_pos)

   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])

 subsection {* Relation of LIM and LIMSEQ *}

 lemma LIMSEQ_SEQ_conv1:

   fixes a :: "'a::metric_space"

   assumes X: "X -- a --> L"

   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

 proof (safe intro!: metric_LIMSEQ_I)

   fix S :: "nat \<Rightarrow> 'a"

   fix r :: real

   assume rgz: "0 < r"

   assume as: "\<forall>n. S n \<noteq> a"

   assume S: "S ----> a"

   from metric_LIM_D [OF X rgz] obtain s

     where sgz: "0 < s"

     and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"

     by fast

   from metric_LIMSEQ_D [OF S sgz]

   obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast

   hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)

   thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..

 qed

 lemma LIMSEQ_SEQ_conv2:

   fixes a :: real

   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

   shows "X -- a --> L"

 proof (rule ccontr)

   assume "\<not> (X -- a --> L)"

   hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"

     unfolding LIM_def dist_norm .

   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> dist (X x) L < r)" by simp

   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less)

   then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto

   let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"

   have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"

     using rdef by simp

   hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"

     by (rule someI_ex)

   hence F1: "\<And>n. ?F n \<noteq> a"

     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"

     and F3: "\<And>n. dist (X (?F n)) L \<ge> r"

     by fast+

   have "?F ----> a"

   proof (rule LIMSEQ_I, unfold real_norm_def)

       fix e::real

       assume "0 < e"

         (* choose no such that inverse (real (Suc n)) < e *)

       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)

       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto

       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"

       proof (intro exI allI impI)

         fix n

         assume mlen: "m \<le> n"

         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"

           by (rule F2)

         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"

           using mlen by auto

         also from nodef have

           "inverse (real (Suc m)) < e" .

         finally show "\<bar>?F n - a\<bar> < e" .

       qed

   qed

   moreover have "\<forall>n. ?F n \<noteq> a"

     by (rule allI) (rule F1)

   moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp

   ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp

   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"

   proof -

     {

       fix no::nat

       obtain n where "n = no + 1" by simp

       then have nolen: "no \<le> n" by simp

         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)

       have "dist (X (?F n)) L \<ge> r"

         by (rule F3)

       with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast

     }

     then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp

     with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto

     thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)

   qed

   ultimately show False by simp

 qed

 lemma LIMSEQ_SEQ_conv:

   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =

    (X -- a --> L)"

 proof

   assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

   thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)

 next

   assume "(X -- a --> L)"

   thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)

 qed

 end