src/HOL/Lim.thy

+(*  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 "~~/src/HOL/Hyperreal/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