(*  Title       : Lim.thy

     ID          : $Id$

     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::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"

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

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

      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s

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

 definition

   isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where

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

 definition

   isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where

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

 subsection {* Limits of Functions *}

 subsubsection {* Purely standard proofs *}

 lemma LIM_eq:

      "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 diff_minus)

 lemma LIM_I:

      "(!!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:

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

 apply (rule LIM_I)

 apply (drule_tac r="r" in LIM_D, safe)

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

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

 apply (simp add: compare_rls)

 done

 lemma LIM_offset_zero: "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: "(\<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_add:

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

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

   shows "(%x. f x + g(x)) -- a --> (L + M)"

 proof (rule LIM_I)

   fix r :: real

   assume r: "0 < r"

   from LIM_D [OF f half_gt_zero [OF r]]

   obtain fs

     where fs:    "0 < fs"

       and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"

   by blast

   from LIM_D [OF g half_gt_zero [OF r]]

   obtain gs

     where gs:    "0 < gs"

       and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"

   by blast

   show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"

   proof (intro exI conjI strip)

     show "0 < min fs gs"  by (simp add: fs gs)

     fix x :: 'a

     assume "x \<noteq> a \<and> norm (x-a) < min fs gs"

     hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp

     with fs_lt gs_lt

     have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+

     hence "norm (f x - L) + norm (g x - M) < r" by arith

     thus "norm (f x + g x - (L + M)) < r"

       by (blast intro: norm_diff_triangle_ineq order_le_less_trans)

   qed

 qed

 lemma LIM_add_zero:

   "\<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: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"

 by (simp only: LIM_eq minus_diff_minus norm_minus_cancel)

 lemma LIM_add_minus:

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

 by (intro LIM_add LIM_minus)

 lemma LIM_diff:

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

 by (simp only: diff_minus LIM_add LIM_minus)

 lemma LIM_zero: "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"

 by (simp add: LIM_def)

 lemma LIM_zero_cancel: "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"

 by (simp add: LIM_def)

 lemma LIM_zero_iff: "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"

 by (simp add: LIM_def)

 lemma LIM_imp_LIM:

   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 LIM_I, drule 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_norm: "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"

 by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)

 lemma LIM_norm_zero: "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"

 by (drule LIM_norm, simp)

 lemma LIM_norm_zero_cancel: "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"

 by (erule LIM_imp_LIM, simp)

 lemma LIM_norm_zero_iff: "(\<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_eq)

 apply (rule_tac x="norm (k - L)" in exI, simp, safe)

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

 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"

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

 apply (drule (1) LIM_diff)

 apply (auto dest!: LIM_const_eq)

 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 LIM_equal2:

   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, 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 Hyperreal/Transcendental.ML*}

 lemma LIM_trans:

      "[| (%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 LIM_I)

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

   obtain s where s: "0 < s"

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

     using LIM_D [OF g r] by fast

   obtain t where t: "0 < t"

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

     using LIM_D [OF f s] by fast

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

   proof (rule exI, safe)

     show "0 < t" using t .

   next

     fix x assume "x \<noteq> a" and "norm (x - a) < t"

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

     thus "norm (g (f x) - g l) < r"

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

   qed

 qed

 lemma LIM_compose2:

   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"

 proof (rule LIM_I)

   fix r :: real

   assume r: "0 < r"

   obtain s where s: "0 < s"

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

     using LIM_D [OF g r] by fast

   obtain t where t: "0 < t"

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

     using LIM_D [OF f s] by fast

   obtain d where d: "0 < d"

     and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"

     using inj by fast

   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (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 "norm (x - a) < min d t"

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

       using neq_b less_s by simp_all

     thus "norm (g (f x) - c) < r"

       by (rule less_r)

   qed

 qed

 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::real_normed_vector \<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) cont: "f -- a --> f a"

 proof (rule LIM_I)

   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. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"

   proof (rule exI, safe)

     from r K show "0 < r / K" by (rule divide_pos_pos)

   next

     fix x assume x: "norm (x - a) < r / K"

     have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)

     also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)

     also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)

     finally show "norm (f x - f a) < r" .

   qed

 qed

 lemma (in bounded_linear) LIM:

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

 by (rule LIM_compose [OF cont])

 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_prod_zero:

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

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

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

 proof (rule LIM_I)

   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)

   obtain s where s: "0 < s"

     and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"

     using LIM_D [OF f r] by auto

   obtain t where t: "0 < t"

     and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"

     using LIM_D [OF g K'] by auto

   show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"

   proof (rule exI, safe)

     from s t show "0 < min s t" by simp

   next

     fix x assume x: "x \<noteq> a"

     assume "norm (x - a) < min s t"

     hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all

     from x xs have 1: "norm (f x) < r" by (rule norm_f)

     from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)

     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)

     also from 1 2 K have "\<dots> < r * inverse K * K"

       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)

     also from K have "r * inverse K * K = r" by simp

     finally show "norm (f x ** g x - 0) < r" by simp

   qed

 qed

 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])

 lemma (in bounded_bilinear) LIM:

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

 apply (drule LIM_zero)

 apply (drule LIM_zero)

 apply (rule LIM_zero_cancel)

 apply (subst prod_diff_prod)

 apply (rule LIM_add_zero)

 apply (rule LIM_add_zero)

 apply (erule (1) LIM_prod_zero)

 apply (erule LIM_left_zero)

 apply (erule LIM_right_zero)

 done

 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::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"

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

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

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

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

 lemma LIM_inverse_lemma:

   fixes x :: "'a::real_normed_div_algebra"

   assumes r: "0 < r"

   assumes x: "norm (x - 1) < min (1/2) (r/2)"

   shows "norm (inverse x - 1) < r"

 proof -

   from r have r2: "0 < r/2" by simp

   from x have 0: "x \<noteq> 0" by clarsimp

   from x have x': "norm (1 - x) < min (1/2) (r/2)"

     by (simp only: norm_minus_commute)

   hence less1: "norm (1 - x) < r/2" by simp

   have "norm (1::'a) - norm x \<le> norm (1 - x)" by (rule norm_triangle_ineq2)

   also from x' have "norm (1 - x) < 1/2" by simp

   finally have "1/2 < norm x" by simp

   hence "inverse (norm x) < inverse (1/2)"

     by (rule less_imp_inverse_less, simp)

   hence less2: "norm (inverse x) < 2"

     by (simp add: nonzero_norm_inverse 0)

   from less1 less2 r2 norm_ge_zero

   have "norm (1 - x) * norm (inverse x) < (r/2) * 2"

     by (rule mult_strict_mono)

   thus "norm (inverse x - 1) < r"

     by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0)

 qed

 lemma LIM_inverse_fun:

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

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

 proof (rule LIM_equal2)

   from a show "0 < norm a" by simp

 next

   fix x assume "norm (x - a) < norm a"

   hence "x \<noteq> 0" by auto

   with a show "inverse x = inverse (inverse a * x) * inverse a"

     by (simp add: nonzero_inverse_mult_distrib

                   nonzero_imp_inverse_nonzero

                   nonzero_inverse_inverse_eq mult_assoc)

 next

   have 1: "inverse -- 1 --> inverse (1::'a)"

     apply (rule LIM_I)

     apply (rule_tac x="min (1/2) (r/2)" in exI)

     apply (simp add: LIM_inverse_lemma)

     done

   have "(\<lambda>x. inverse a * x) -- a --> inverse a * a"

     by (intro LIM_mult LIM_ident LIM_const)

   hence "(\<lambda>x. inverse a * x) -- a --> 1"

     by (simp add: a)

   with 1 have "(\<lambda>x. inverse (inverse a * x)) -- a --> inverse 1"

     by (rule LIM_compose)

   hence "(\<lambda>x. inverse (inverse a * x)) -- a --> 1"

     by simp

   hence "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a"

     by (intro LIM_mult LIM_const)

   thus "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> inverse a"

     by simp

 qed

 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"

 by (rule LIM_inverse_fun [THEN LIM_compose])

 subsection {* Continuity *}

 subsubsection {* Purely standard proofs *}

 lemma LIM_isCont_iff: "(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: "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: "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: "\<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: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"

   unfolding isCont_def by (rule LIM_minus)

 lemma isCont_diff: "\<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::real_normed_vector \<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::real_normed_vector \<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 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> 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)) a"

   unfolding isCont_def by (rule LIM_of_real)

 lemma isCont_power:

   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"

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

   unfolding isCont_def by (rule LIM_power)

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

 by (rule isCont_rabs [OF isCont_ident])

 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 CauchyI)

 apply (drule_tac x=e in spec, safe)

 apply (drule_tac e=s in CauchyD, safe)

 apply (rule_tac x=M in exI, simp)

 done

 lemma (in bounded_linear) isUCont: "isUCont f"

 unfolding isUCont_def

 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::real_normed_vector"

   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!: 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 LIM_D [OF X rgz] obtain s

     where sgz: "0 < s"

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

     by fast

   from LIMSEQ_D [OF S sgz]

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

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

   thus "\<exists>no. \<forall>n\<ge>no. norm (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 --> norm (X x - L) < r)" by (unfold LIM_def)

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

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

         by (rule F3)

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

     }

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

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

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