(*  Title:      HOL/Real_Vector_Spaces.thy

     Author:     Brian Huffman

     Author:     Johannes Hölzl

 *)

 header {* Vector Spaces and Algebras over the Reals *}

 theory Real_Vector_Spaces

 imports Real Topological_Spaces

 begin

 subsection {* Locale for additive functions *}

 locale additive =

   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"

   assumes add: "f (x + y) = f x + f y"

 begin

 lemma zero: "f 0 = 0"

 proof -

   have "f 0 = f (0 + 0)" by simp

   also have "\<dots> = f 0 + f 0" by (rule add)

   finally show "f 0 = 0" by simp

 qed

 lemma minus: "f (- x) = - f x"

 proof -

   have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])

   also have "\<dots> = - f x + f x" by (simp add: zero)

   finally show "f (- x) = - f x" by (rule add_right_imp_eq)

 qed

 lemma diff: "f (x - y) = f x - f y"

   using add [of x "- y"] by (simp add: minus)

 lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"

 apply (cases "finite A")

 apply (induct set: finite)

 apply (simp add: zero)

 apply (simp add: add)

 apply (simp add: zero)

 done

 end

 subsection {* Vector spaces *}

 locale vector_space =

   fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"

   assumes scale_right_distrib [algebra_simps]:

     "scale a (x + y) = scale a x + scale a y"

   and scale_left_distrib [algebra_simps]:

     "scale (a + b) x = scale a x + scale b x"

   and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"

   and scale_one [simp]: "scale 1 x = x"

 begin

 lemma scale_left_commute:

   "scale a (scale b x) = scale b (scale a x)"

 by (simp add: mult_commute)

 lemma scale_zero_left [simp]: "scale 0 x = 0"

   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"

   and scale_left_diff_distrib [algebra_simps]:

         "scale (a - b) x = scale a x - scale b x"

   and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"

 proof -

   interpret s: additive "\<lambda>a. scale a x"

     proof qed (rule scale_left_distrib)

   show "scale 0 x = 0" by (rule s.zero)

   show "scale (- a) x = - (scale a x)" by (rule s.minus)

   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)

   show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)

 qed

 lemma scale_zero_right [simp]: "scale a 0 = 0"

   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"

   and scale_right_diff_distrib [algebra_simps]:

         "scale a (x - y) = scale a x - scale a y"

   and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"

 proof -

   interpret s: additive "\<lambda>x. scale a x"

     proof qed (rule scale_right_distrib)

   show "scale a 0 = 0" by (rule s.zero)

   show "scale a (- x) = - (scale a x)" by (rule s.minus)

   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)

   show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)

 qed

 lemma scale_eq_0_iff [simp]:

   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"

 proof cases

   assume "a = 0" thus ?thesis by simp

 next

   assume anz [simp]: "a \<noteq> 0"

   { assume "scale a x = 0"

     hence "scale (inverse a) (scale a x) = 0" by simp

     hence "x = 0" by simp }

   thus ?thesis by force

 qed

 lemma scale_left_imp_eq:

   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"

 proof -

   assume nonzero: "a \<noteq> 0"

   assume "scale a x = scale a y"

   hence "scale a (x - y) = 0"

      by (simp add: scale_right_diff_distrib)

   hence "x - y = 0" by (simp add: nonzero)

   thus "x = y" by (simp only: right_minus_eq)

 qed

 lemma scale_right_imp_eq:

   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"

 proof -

   assume nonzero: "x \<noteq> 0"

   assume "scale a x = scale b x"

   hence "scale (a - b) x = 0"

      by (simp add: scale_left_diff_distrib)

   hence "a - b = 0" by (simp add: nonzero)

   thus "a = b" by (simp only: right_minus_eq)

 qed

 lemma scale_cancel_left [simp]:

   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"

 by (auto intro: scale_left_imp_eq)

 lemma scale_cancel_right [simp]:

   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"

 by (auto intro: scale_right_imp_eq)

 end

 subsection {* Real vector spaces *}

 class scaleR =

   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)

 begin

 abbreviation

   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)

 where

   "x /\<^sub>R r == scaleR (inverse r) x"

 end

 class real_vector = scaleR + ab_group_add +

   assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"

   and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"

   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"

   and scaleR_one: "scaleR 1 x = x"

 interpretation real_vector:

   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"

 apply unfold_locales

 apply (rule scaleR_add_right)

 apply (rule scaleR_add_left)

 apply (rule scaleR_scaleR)

 apply (rule scaleR_one)

 done

 text {* Recover original theorem names *}

 lemmas scaleR_left_commute = real_vector.scale_left_commute

 lemmas scaleR_zero_left = real_vector.scale_zero_left

 lemmas scaleR_minus_left = real_vector.scale_minus_left

 lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib

 lemmas scaleR_setsum_left = real_vector.scale_setsum_left

 lemmas scaleR_zero_right = real_vector.scale_zero_right

 lemmas scaleR_minus_right = real_vector.scale_minus_right

 lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib

 lemmas scaleR_setsum_right = real_vector.scale_setsum_right

 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff

 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq

 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq

 lemmas scaleR_cancel_left = real_vector.scale_cancel_left

 lemmas scaleR_cancel_right = real_vector.scale_cancel_right

 text {* Legacy names *}

 lemmas scaleR_left_distrib = scaleR_add_left

 lemmas scaleR_right_distrib = scaleR_add_right

 lemmas scaleR_left_diff_distrib = scaleR_diff_left

 lemmas scaleR_right_diff_distrib = scaleR_diff_right

 lemma scaleR_minus1_left [simp]:

   fixes x :: "'a::real_vector"

   shows "scaleR (-1) x = - x"

   using scaleR_minus_left [of 1 x] by simp

 class real_algebra = real_vector + ring +

   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"

   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"

 class real_algebra_1 = real_algebra + ring_1

 class real_div_algebra = real_algebra_1 + division_ring

 class real_field = real_div_algebra + field

 instantiation real :: real_field

 begin

 definition

   real_scaleR_def [simp]: "scaleR a x = a * x"

 instance proof

 qed (simp_all add: algebra_simps)

 end

 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"

 proof qed (rule scaleR_left_distrib)

 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"

 proof qed (rule scaleR_right_distrib)

 lemma nonzero_inverse_scaleR_distrib:

   fixes x :: "'a::real_div_algebra" shows

   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"

 by (rule inverse_unique, simp)

 lemma inverse_scaleR_distrib:

   fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"

   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"

 apply (case_tac "a = 0", simp)

 apply (case_tac "x = 0", simp)

 apply (erule (1) nonzero_inverse_scaleR_distrib)

 done

 subsection {* Embedding of the Reals into any @{text real_algebra_1}:

 @{term of_real} *}

 definition

   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where

   "of_real r = scaleR r 1"

 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"

 by (simp add: of_real_def)

 lemma of_real_0 [simp]: "of_real 0 = 0"

 by (simp add: of_real_def)

 lemma of_real_1 [simp]: "of_real 1 = 1"

 by (simp add: of_real_def)

 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"

 by (simp add: of_real_def scaleR_left_distrib)

 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"

 by (simp add: of_real_def)

 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"

 by (simp add: of_real_def scaleR_left_diff_distrib)

 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"

 by (simp add: of_real_def mult_commute)

 lemma nonzero_of_real_inverse:

   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =

    inverse (of_real x :: 'a::real_div_algebra)"

 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)

 lemma of_real_inverse [simp]:

   "of_real (inverse x) =

    inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"

 by (simp add: of_real_def inverse_scaleR_distrib)

 lemma nonzero_of_real_divide:

   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =

    (of_real x / of_real y :: 'a::real_field)"

 by (simp add: divide_inverse nonzero_of_real_inverse)

 lemma of_real_divide [simp]:

   "of_real (x / y) =

    (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"

 by (simp add: divide_inverse)

 lemma of_real_power [simp]:

   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"

 by (induct n) simp_all

 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"

 by (simp add: of_real_def)

 lemma inj_of_real:

   "inj of_real"

   by (auto intro: injI)

 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]

 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"

 proof

   fix r

   show "of_real r = id r"

     by (simp add: of_real_def)

 qed

 text{*Collapse nested embeddings*}

 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"

 by (induct n) auto

 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"

 by (cases z rule: int_diff_cases, simp)

 lemma of_real_numeral: "of_real (numeral w) = numeral w"

 using of_real_of_int_eq [of "numeral w"] by simp

 lemma of_real_neg_numeral: "of_real (- numeral w) = - numeral w"

 using of_real_of_int_eq [of "- numeral w"] by simp

 text{*Every real algebra has characteristic zero*}

 instance real_algebra_1 < ring_char_0

 proof

   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)

   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)

 qed

 instance real_field < field_char_0 ..

 subsection {* The Set of Real Numbers *}

 definition Reals :: "'a::real_algebra_1 set" where

   "Reals = range of_real"

 notation (xsymbols)

   Reals  ("\<real>")

 lemma Reals_of_real [simp]: "of_real r \<in> Reals"

 by (simp add: Reals_def)

 lemma Reals_of_int [simp]: "of_int z \<in> Reals"

 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)

 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"

 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)

 lemma Reals_numeral [simp]: "numeral w \<in> Reals"

 by (subst of_real_numeral [symmetric], rule Reals_of_real)

 lemma Reals_0 [simp]: "0 \<in> Reals"

 apply (unfold Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_0 [symmetric])

 done

 lemma Reals_1 [simp]: "1 \<in> Reals"

 apply (unfold Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_1 [symmetric])

 done

 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_add [symmetric])

 done

 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_minus [symmetric])

 done

 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_diff [symmetric])

 done

 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_mult [symmetric])

 done

 lemma nonzero_Reals_inverse:

   fixes a :: "'a::real_div_algebra"

   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (erule nonzero_of_real_inverse [symmetric])

 done

 lemma Reals_inverse:

   fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"

   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_inverse [symmetric])

 done

 lemma Reals_inverse_iff [simp]: 

   fixes x:: "'a :: {real_div_algebra, division_ring_inverse_zero}"

   shows "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"

 by (metis Reals_inverse inverse_inverse_eq)

 lemma nonzero_Reals_divide:

   fixes a b :: "'a::real_field"

   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (erule nonzero_of_real_divide [symmetric])

 done

 lemma Reals_divide [simp]:

   fixes a b :: "'a::{real_field, field_inverse_zero}"

   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_divide [symmetric])

 done

 lemma Reals_power [simp]:

   fixes a :: "'a::{real_algebra_1}"

   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"

 apply (auto simp add: Reals_def)

 apply (rule range_eqI)

 apply (rule of_real_power [symmetric])

 done

 lemma Reals_cases [cases set: Reals]:

   assumes "q \<in> \<real>"

   obtains (of_real) r where "q = of_real r"

   unfolding Reals_def

 proof -

   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .

   then obtain r where "q = of_real r" ..

   then show thesis ..

 qed

 lemma setsum_in_Reals: assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setsum f s \<in> \<real>"

 proof (cases "finite s")

   case True then show ?thesis using assms

     by (induct s rule: finite_induct) auto

 next

   case False then show ?thesis using assms

     by (metis Reals_0 setsum_infinite)

 qed

 lemma setprod_in_Reals: assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setprod f s \<in> \<real>"

 proof (cases "finite s")

   case True then show ?thesis using assms

     by (induct s rule: finite_induct) auto

 next

   case False then show ?thesis using assms

     by (metis Reals_1 setprod_infinite)

 qed

 lemma Reals_induct [case_names of_real, induct set: Reals]:

   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"

   by (rule Reals_cases) auto

 subsection {* Ordered real vector spaces *}

 class ordered_real_vector = real_vector + ordered_ab_group_add +

   assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"

   assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"

 begin

 lemma scaleR_mono:

   "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"

 apply (erule scaleR_right_mono [THEN order_trans], assumption)

 apply (erule scaleR_left_mono, assumption)

 done

 lemma scaleR_mono':

   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d"

   by (rule scaleR_mono) (auto intro: order.trans)

 lemma pos_le_divideRI:

   assumes "0 < c"

   assumes "c *\<^sub>R a \<le> b"

   shows "a \<le> b /\<^sub>R c"

 proof -

   from scaleR_left_mono[OF assms(2)] assms(1)

   have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c"

     by simp

   with assms show ?thesis

     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)

 qed

 lemma pos_le_divideR_eq:

   assumes "0 < c"

   shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"

 proof rule

   assume "a \<le> b /\<^sub>R c"

   from scaleR_left_mono[OF this] assms

   have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"

     by simp

   with assms show "c *\<^sub>R a \<le> b"

     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)

 qed (rule pos_le_divideRI[OF assms])

 lemma scaleR_image_atLeastAtMost:

   "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"

   apply (auto intro!: scaleR_left_mono)

   apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)

   apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)

   done

 end

 lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"

   using scaleR_left_mono [of 0 x a]

   by simp

 lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"

   using scaleR_left_mono [of x 0 a] by simp

 lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"

   using scaleR_right_mono [of a 0 x] by simp

 lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>

   a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"

   by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)

 lemma le_add_iff1:

   fixes c d e::"'a::ordered_real_vector"

   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"

   by (simp add: algebra_simps)

 lemma le_add_iff2:

   fixes c d e::"'a::ordered_real_vector"

   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"

   by (simp add: algebra_simps)

 lemma scaleR_left_mono_neg:

   fixes a b::"'a::ordered_real_vector"

   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"

   apply (drule scaleR_left_mono [of _ _ "- c"])

   apply simp_all

   done

 lemma scaleR_right_mono_neg:

   fixes c::"'a::ordered_real_vector"

   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"

   apply (drule scaleR_right_mono [of _ _ "- c"])

   apply simp_all

   done

 lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"

 using scaleR_right_mono_neg [of a 0 b] by simp

 lemma split_scaleR_pos_le:

   fixes b::"'a::ordered_real_vector"

   shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"

   by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)

 lemma zero_le_scaleR_iff:

   fixes b::"'a::ordered_real_vector"

   shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0" (is "?lhs = ?rhs")

 proof cases

   assume "a \<noteq> 0"

   show ?thesis

   proof

     assume lhs: ?lhs

     {

       assume "0 < a"

       with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"

         by (intro scaleR_mono) auto

       hence ?rhs using `0 < a`

         by simp

     } moreover {

       assume "0 > a"

       with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"

         by (intro scaleR_mono) auto

       hence ?rhs using `0 > a`

         by simp

     } ultimately show ?rhs using `a \<noteq> 0` by arith

   qed (auto simp: not_le `a \<noteq> 0` intro!: split_scaleR_pos_le)

 qed simp

 lemma scaleR_le_0_iff:

   fixes b::"'a::ordered_real_vector"

   shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"

   by (insert zero_le_scaleR_iff [of "-a" b]) force

 lemma scaleR_le_cancel_left:

   fixes b::"'a::ordered_real_vector"

   shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

   by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg

     dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])

 lemma scaleR_le_cancel_left_pos:

   fixes b::"'a::ordered_real_vector"

   shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"

   by (auto simp: scaleR_le_cancel_left)

 lemma scaleR_le_cancel_left_neg:

   fixes b::"'a::ordered_real_vector"

   shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"

   by (auto simp: scaleR_le_cancel_left)

 lemma scaleR_left_le_one_le:

   fixes x::"'a::ordered_real_vector" and a::real

   shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"

   using scaleR_right_mono[of a 1 x] by simp

 subsection {* Real normed vector spaces *}

 class dist =

   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"

 class norm =

   fixes norm :: "'a \<Rightarrow> real"

 class sgn_div_norm = scaleR + norm + sgn +

   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"

 class dist_norm = dist + norm + minus +

   assumes dist_norm: "dist x y = norm (x - y)"

 class open_dist = "open" + dist +

   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +

   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"

   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"

   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"

 begin

 lemma norm_ge_zero [simp]: "0 \<le> norm x"

 proof -

   have "0 = norm (x + -1 *\<^sub>R x)" 

     using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)

   also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)

   finally show ?thesis by simp

 qed

 end

 class real_normed_algebra = real_algebra + real_normed_vector +

   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"

 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +

   assumes norm_one [simp]: "norm 1 = 1"

 class real_normed_div_algebra = real_div_algebra + real_normed_vector +

   assumes norm_mult: "norm (x * y) = norm x * norm y"

 class real_normed_field = real_field + real_normed_div_algebra

 instance real_normed_div_algebra < real_normed_algebra_1

 proof

   fix x y :: 'a

   show "norm (x * y) \<le> norm x * norm y"

     by (simp add: norm_mult)

 next

   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"

     by (rule norm_mult)

   thus "norm (1::'a) = 1" by simp

 qed

 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"

 by simp

 lemma zero_less_norm_iff [simp]:

   fixes x :: "'a::real_normed_vector"

   shows "(0 < norm x) = (x \<noteq> 0)"

 by (simp add: order_less_le)

 lemma norm_not_less_zero [simp]:

   fixes x :: "'a::real_normed_vector"

   shows "\<not> norm x < 0"

 by (simp add: linorder_not_less)

 lemma norm_le_zero_iff [simp]:

   fixes x :: "'a::real_normed_vector"

   shows "(norm x \<le> 0) = (x = 0)"

 by (simp add: order_le_less)

 lemma norm_minus_cancel [simp]:

   fixes x :: "'a::real_normed_vector"

   shows "norm (- x) = norm x"

 proof -

   have "norm (- x) = norm (scaleR (- 1) x)"

     by (simp only: scaleR_minus_left scaleR_one)

   also have "\<dots> = \<bar>- 1\<bar> * norm x"

     by (rule norm_scaleR)

   finally show ?thesis by simp

 qed

 lemma norm_minus_commute:

   fixes a b :: "'a::real_normed_vector"

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

 proof -

   have "norm (- (b - a)) = norm (b - a)"

     by (rule norm_minus_cancel)

   thus ?thesis by simp

 qed

 lemma norm_triangle_ineq2:

   fixes a b :: "'a::real_normed_vector"

   shows "norm a - norm b \<le> norm (a - b)"

 proof -

   have "norm (a - b + b) \<le> norm (a - b) + norm b"

     by (rule norm_triangle_ineq)

   thus ?thesis by simp

 qed

 lemma norm_triangle_ineq3:

   fixes a b :: "'a::real_normed_vector"

   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"

 apply (subst abs_le_iff)

 apply auto

 apply (rule norm_triangle_ineq2)

 apply (subst norm_minus_commute)

 apply (rule norm_triangle_ineq2)

 done

 lemma norm_triangle_ineq4:

   fixes a b :: "'a::real_normed_vector"

   shows "norm (a - b) \<le> norm a + norm b"

 proof -

   have "norm (a + - b) \<le> norm a + norm (- b)"

     by (rule norm_triangle_ineq)

   then show ?thesis by simp

 qed

 lemma norm_diff_ineq:

   fixes a b :: "'a::real_normed_vector"

   shows "norm a - norm b \<le> norm (a + b)"

 proof -

   have "norm a - norm (- b) \<le> norm (a - - b)"

     by (rule norm_triangle_ineq2)

   thus ?thesis by simp

 qed

 lemma norm_diff_triangle_ineq:

   fixes a b c d :: "'a::real_normed_vector"

   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"

 proof -

   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"

     by (simp add: algebra_simps)

   also have "\<dots> \<le> norm (a - c) + norm (b - d)"

     by (rule norm_triangle_ineq)

   finally show ?thesis .

 qed

 lemma norm_triangle_mono: 

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>norm a \<le> r; norm b \<le> s\<rbrakk> \<Longrightarrow> norm (a + b) \<le> r + s"

 by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)

 lemma abs_norm_cancel [simp]:

   fixes a :: "'a::real_normed_vector"

   shows "\<bar>norm a\<bar> = norm a"

 by (rule abs_of_nonneg [OF norm_ge_zero])

 lemma norm_add_less:

   fixes x y :: "'a::real_normed_vector"

   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"

 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])

 lemma norm_mult_less:

   fixes x y :: "'a::real_normed_algebra"

   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"

 apply (rule order_le_less_trans [OF norm_mult_ineq])

 apply (simp add: mult_strict_mono')

 done

 lemma norm_of_real [simp]:

   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"

 unfolding of_real_def by simp

 lemma norm_numeral [simp]:

   "norm (numeral w::'a::real_normed_algebra_1) = numeral w"

 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)

 lemma norm_neg_numeral [simp]:

   "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"

 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)

 lemma norm_of_int [simp]:

   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"

 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)

 lemma norm_of_nat [simp]:

   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"

 apply (subst of_real_of_nat_eq [symmetric])

 apply (subst norm_of_real, simp)

 done

 lemma nonzero_norm_inverse:

   fixes a :: "'a::real_normed_div_algebra"

   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"

 apply (rule inverse_unique [symmetric])

 apply (simp add: norm_mult [symmetric])

 done

 lemma norm_inverse:

   fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"

   shows "norm (inverse a) = inverse (norm a)"

 apply (case_tac "a = 0", simp)

 apply (erule nonzero_norm_inverse)

 done

 lemma nonzero_norm_divide:

   fixes a b :: "'a::real_normed_field"

   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"

 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)

 lemma norm_divide:

   fixes a b :: "'a::{real_normed_field, field_inverse_zero}"

   shows "norm (a / b) = norm a / norm b"

 by (simp add: divide_inverse norm_mult norm_inverse)

 lemma norm_power_ineq:

   fixes x :: "'a::{real_normed_algebra_1}"

   shows "norm (x ^ n) \<le> norm x ^ n"

 proof (induct n)

   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp

 next

   case (Suc n)

   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"

     by (rule norm_mult_ineq)

   also from Suc have "\<dots> \<le> norm x * norm x ^ n"

     using norm_ge_zero by (rule mult_left_mono)

   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"

     by simp

 qed

 lemma norm_power:

   fixes x :: "'a::{real_normed_div_algebra}"

   shows "norm (x ^ n) = norm x ^ n"

 by (induct n) (simp_all add: norm_mult)

 lemma setprod_norm:

   fixes f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"

   shows "setprod (\<lambda>x. norm(f x)) A = norm (setprod f A)"

 proof (cases "finite A")

   case True then show ?thesis 

     by (induct A rule: finite_induct) (auto simp: norm_mult)

 next

   case False then show ?thesis

     by (metis norm_one setprod.infinite) 

 qed

 subsection {* Metric spaces *}

 class metric_space = open_dist +

   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"

   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"

 begin

 lemma dist_self [simp]: "dist x x = 0"

 by simp

 lemma zero_le_dist [simp]: "0 \<le> dist x y"

 using dist_triangle2 [of x x y] by simp

 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"

 by (simp add: less_le)

 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"

 by (simp add: not_less)

 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"

 by (simp add: le_less)

 lemma dist_commute: "dist x y = dist y x"

 proof (rule order_antisym)

   show "dist x y \<le> dist y x"

     using dist_triangle2 [of x y x] by simp

   show "dist y x \<le> dist x y"

     using dist_triangle2 [of y x y] by simp

 qed

 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"

 using dist_triangle2 [of x z y] by (simp add: dist_commute)

 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"

 using dist_triangle2 [of x y a] by (simp add: dist_commute)

 lemma dist_triangle_alt:

   shows "dist y z <= dist x y + dist x z"

 by (rule dist_triangle3)

 lemma dist_pos_lt:

   shows "x \<noteq> y ==> 0 < dist x y"

 by (simp add: zero_less_dist_iff)

 lemma dist_nz:

   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"

 by (simp add: zero_less_dist_iff)

 lemma dist_triangle_le:

   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"

 by (rule order_trans [OF dist_triangle2])

 lemma dist_triangle_lt:

   shows "dist x z + dist y z < e ==> dist x y < e"

 by (rule le_less_trans [OF dist_triangle2])

 lemma dist_triangle_half_l:

   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"

 by (rule dist_triangle_lt [where z=y], simp)

 lemma dist_triangle_half_r:

   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"

 by (rule dist_triangle_half_l, simp_all add: dist_commute)

 subclass topological_space

 proof

   have "\<exists>e::real. 0 < e"

     by (fast intro: zero_less_one)

   then show "open UNIV"

     unfolding open_dist by simp

 next

   fix S T assume "open S" "open T"

   then show "open (S \<inter> T)"

     unfolding open_dist

     apply clarify

     apply (drule (1) bspec)+

     apply (clarify, rename_tac r s)

     apply (rule_tac x="min r s" in exI, simp)

     done

 next

   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"

     unfolding open_dist by fast

 qed

 lemma open_ball: "open {y. dist x y < d}"

 proof (unfold open_dist, intro ballI)

   fix y assume *: "y \<in> {y. dist x y < d}"

   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"

     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)

 qed

 subclass first_countable_topology

 proof

   fix x 

   show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

   proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])

     fix S assume "open S" "x \<in> S"

     then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"

       by (auto simp: open_dist subset_eq dist_commute)

     moreover

     from e obtain i where "inverse (Suc i) < e"

       by (auto dest!: reals_Archimedean)

     then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"

       by auto

     ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"

       by blast

   qed (auto intro: open_ball)

 qed

 end

 instance metric_space \<subseteq> t2_space

 proof

   fix x y :: "'a::metric_space"

   assume xy: "x \<noteq> y"

   let ?U = "{y'. dist x y' < dist x y / 2}"

   let ?V = "{x'. dist y x' < dist x y / 2}"

   have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y

                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith

   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"

     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]

     using open_ball[of _ "dist x y / 2"] by auto

   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"

     by blast

 qed

 text {* Every normed vector space is a metric space. *}

 instance real_normed_vector < metric_space

 proof

   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"

     unfolding dist_norm by simp

 next

   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"

     unfolding dist_norm

     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp

 qed

 subsection {* Class instances for real numbers *}

 instantiation real :: real_normed_field

 begin

 definition dist_real_def:

   "dist x y = \<bar>x - y\<bar>"

 definition open_real_def [code del]:

   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

 definition real_norm_def [simp]:

   "norm r = \<bar>r\<bar>"

 instance

 apply (intro_classes, unfold real_norm_def real_scaleR_def)

 apply (rule dist_real_def)

 apply (rule open_real_def)

 apply (simp add: sgn_real_def)

 apply (rule abs_eq_0)

 apply (rule abs_triangle_ineq)

 apply (rule abs_mult)

 apply (rule abs_mult)

 done

 end

 declare [[code abort: "open :: real set \<Rightarrow> bool"]]

 instance real :: linorder_topology

 proof

   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"

   proof (rule ext, safe)

     fix S :: "real set" assume "open S"

     then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"

       unfolding open_real_def bchoice_iff ..

     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"

       by (fastforce simp: dist_real_def)

     show "generate_topology (range lessThan \<union> range greaterThan) S"

       apply (subst *)

       apply (intro generate_topology_Union generate_topology.Int)

       apply (auto intro: generate_topology.Basis)

       done

   next

     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"

     moreover have "\<And>a::real. open {..<a}"

       unfolding open_real_def dist_real_def

     proof clarify

       fix x a :: real assume "x < a"

       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto

       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..

     qed

     moreover have "\<And>a::real. open {a <..}"

       unfolding open_real_def dist_real_def

     proof clarify

       fix x a :: real assume "a < x"

       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto

       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..

     qed

     ultimately show "open S"

       by induct auto

   qed

 qed

 instance real :: linear_continuum_topology ..

 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]

 lemmas open_real_lessThan = open_lessThan[where 'a=real]

 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]

 lemmas closed_real_atMost = closed_atMost[where 'a=real]

 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]

 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]

 subsection {* Extra type constraints *}

 text {* Only allow @{term "open"} in class @{text topological_space}. *}

 setup {* Sign.add_const_constraint

   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}

 text {* Only allow @{term dist} in class @{text metric_space}. *}

 setup {* Sign.add_const_constraint

   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}

 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}

 setup {* Sign.add_const_constraint

   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}

 subsection {* Sign function *}

 lemma norm_sgn:

   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"

 by (simp add: sgn_div_norm)

 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"

 by (simp add: sgn_div_norm)

 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"

 by (simp add: sgn_div_norm)

 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"

 by (simp add: sgn_div_norm)

 lemma sgn_scaleR:

   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"

 by (simp add: sgn_div_norm mult_ac)

 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"

 by (simp add: sgn_div_norm)

 lemma sgn_of_real:

   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"

 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)

 lemma sgn_mult:

   fixes x y :: "'a::real_normed_div_algebra"

   shows "sgn (x * y) = sgn x * sgn y"

 by (simp add: sgn_div_norm norm_mult mult_commute)

 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"

 by (simp add: sgn_div_norm divide_inverse)

 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"

 unfolding real_sgn_eq by simp

 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"

 unfolding real_sgn_eq by simp

 lemma norm_conv_dist: "norm x = dist x 0"

   unfolding dist_norm by simp

 subsection {* Bounded Linear and Bilinear Operators *}

 locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +

   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"

 lemma linearI:

   assumes "\<And>x y. f (x + y) = f x + f y"

   assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"

   shows "linear f"

   by default (rule assms)+

 locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +

   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"

 begin

 lemma pos_bounded:

   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"

 proof -

   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"

     using bounded by fast

   show ?thesis

   proof (intro exI impI conjI allI)

     show "0 < max 1 K"

       by (rule order_less_le_trans [OF zero_less_one max.cobounded1])

   next

     fix x

     have "norm (f x) \<le> norm x * K" using K .

     also have "\<dots> \<le> norm x * max 1 K"

       by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])

     finally show "norm (f x) \<le> norm x * max 1 K" .

   qed

 qed

 lemma nonneg_bounded:

   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"

 proof -

   from pos_bounded

   show ?thesis by (auto intro: order_less_imp_le)

 qed

 end

 lemma bounded_linear_intro:

   assumes "\<And>x y. f (x + y) = f x + f y"

   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"

   assumes "\<And>x. norm (f x) \<le> norm x * K"

   shows "bounded_linear f"

   by default (fast intro: assms)+

 locale bounded_bilinear =

   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]

                  \<Rightarrow> 'c::real_normed_vector"

     (infixl "**" 70)

   assumes add_left: "prod (a + a') b = prod a b + prod a' b"

   assumes add_right: "prod a (b + b') = prod a b + prod a b'"

   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"

   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"

   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"

 begin

 lemma pos_bounded:

   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"

 apply (cut_tac bounded, erule exE)

 apply (rule_tac x="max 1 K" in exI, safe)

 apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])

 apply (drule spec, drule spec, erule order_trans)

 apply (rule mult_left_mono [OF max.cobounded2])

 apply (intro mult_nonneg_nonneg norm_ge_zero)

 done

 lemma nonneg_bounded:

   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"

 proof -

   from pos_bounded

   show ?thesis by (auto intro: order_less_imp_le)

 qed

 lemma additive_right: "additive (\<lambda>b. prod a b)"

 by (rule additive.intro, rule add_right)

 lemma additive_left: "additive (\<lambda>a. prod a b)"

 by (rule additive.intro, rule add_left)

 lemma zero_left: "prod 0 b = 0"

 by (rule additive.zero [OF additive_left])

 lemma zero_right: "prod a 0 = 0"

 by (rule additive.zero [OF additive_right])

 lemma minus_left: "prod (- a) b = - prod a b"

 by (rule additive.minus [OF additive_left])

 lemma minus_right: "prod a (- b) = - prod a b"

 by (rule additive.minus [OF additive_right])

 lemma diff_left:

   "prod (a - a') b = prod a b - prod a' b"

 by (rule additive.diff [OF additive_left])

 lemma diff_right:

   "prod a (b - b') = prod a b - prod a b'"

 by (rule additive.diff [OF additive_right])

 lemma bounded_linear_left:

   "bounded_linear (\<lambda>a. a ** b)"

 apply (cut_tac bounded, safe)

 apply (rule_tac K="norm b * K" in bounded_linear_intro)

 apply (rule add_left)

 apply (rule scaleR_left)

 apply (simp add: mult_ac)

 done

 lemma bounded_linear_right:

   "bounded_linear (\<lambda>b. a ** b)"

 apply (cut_tac bounded, safe)

 apply (rule_tac K="norm a * K" in bounded_linear_intro)

 apply (rule add_right)

 apply (rule scaleR_right)

 apply (simp add: mult_ac)

 done

 lemma prod_diff_prod:

   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"

 by (simp add: diff_left diff_right)

 end

 lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"

   by default (auto intro!: exI[of _ 1])

 lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"

   by default (auto intro!: exI[of _ 1])

 lemma bounded_linear_add:

   assumes "bounded_linear f"

   assumes "bounded_linear g"

   shows "bounded_linear (\<lambda>x. f x + g x)"

 proof -

   interpret f: bounded_linear f by fact

   interpret g: bounded_linear g by fact

   show ?thesis

   proof

     from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast

     from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast

     show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"

       using add_mono[OF Kf Kg]

       by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)

   qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)

 qed

 lemma bounded_linear_minus:

   assumes "bounded_linear f"

   shows "bounded_linear (\<lambda>x. - f x)"

 proof -

   interpret f: bounded_linear f by fact

   show ?thesis apply (unfold_locales)

     apply (simp add: f.add)

     apply (simp add: f.scaleR)

     apply (simp add: f.bounded)

     done

 qed

 lemma bounded_linear_compose:

   assumes "bounded_linear f"

   assumes "bounded_linear g"

   shows "bounded_linear (\<lambda>x. f (g x))"

 proof -

   interpret f: bounded_linear f by fact

   interpret g: bounded_linear g by fact

   show ?thesis proof (unfold_locales)

     fix x y show "f (g (x + y)) = f (g x) + f (g y)"

       by (simp only: f.add g.add)

   next

     fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"

       by (simp only: f.scaleR g.scaleR)

   next

     from f.pos_bounded

     obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast

     from g.pos_bounded

     obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast

     show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"

     proof (intro exI allI)

       fix x

       have "norm (f (g x)) \<le> norm (g x) * Kf"

         using f .

       also have "\<dots> \<le> (norm x * Kg) * Kf"

         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)

       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"

         by (rule mult_assoc)

       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .

     qed

   qed

 qed

 lemma bounded_bilinear_mult:

   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"

 apply (rule bounded_bilinear.intro)

 apply (rule distrib_right)

 apply (rule distrib_left)

 apply (rule mult_scaleR_left)

 apply (rule mult_scaleR_right)

 apply (rule_tac x="1" in exI)

 apply (simp add: norm_mult_ineq)

 done

 lemma bounded_linear_mult_left:

   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"

   using bounded_bilinear_mult

   by (rule bounded_bilinear.bounded_linear_left)

 lemma bounded_linear_mult_right:

   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"

   using bounded_bilinear_mult

   by (rule bounded_bilinear.bounded_linear_right)

 lemmas bounded_linear_mult_const =

   bounded_linear_mult_left [THEN bounded_linear_compose]

 lemmas bounded_linear_const_mult =

   bounded_linear_mult_right [THEN bounded_linear_compose]

 lemma bounded_linear_divide:

   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"

   unfolding divide_inverse by (rule bounded_linear_mult_left)

 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"

 apply (rule bounded_bilinear.intro)

 apply (rule scaleR_left_distrib)

 apply (rule scaleR_right_distrib)

 apply simp

 apply (rule scaleR_left_commute)

 apply (rule_tac x="1" in exI, simp)

 done

 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"

   using bounded_bilinear_scaleR

   by (rule bounded_bilinear.bounded_linear_left)

 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"

   using bounded_bilinear_scaleR

   by (rule bounded_bilinear.bounded_linear_right)

 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"

   unfolding of_real_def by (rule bounded_linear_scaleR_left)

 lemma real_bounded_linear:

   fixes f :: "real \<Rightarrow> real"

   shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"

 proof -

   { fix x assume "bounded_linear f"

     then interpret bounded_linear f .

     from scaleR[of x 1] have "f x = x * f 1"

       by simp }

   then show ?thesis

     by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)

 qed

 instance real_normed_algebra_1 \<subseteq> perfect_space

 proof

   fix x::'a

   show "\<not> open {x}"

     unfolding open_dist dist_norm

     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)

 qed

 subsection {* Filters and Limits on Metric Space *}

 lemma eventually_nhds_metric:

   fixes a :: "'a :: metric_space"

   shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"

 unfolding eventually_nhds open_dist

 apply safe

 apply fast

 apply (rule_tac x="{x. dist x a < d}" in exI, simp)

 apply clarsimp

 apply (rule_tac x="d - dist x a" in exI, clarsimp)

 apply (simp only: less_diff_eq)

 apply (erule le_less_trans [OF dist_triangle])

 done

 lemma eventually_at:

   fixes a :: "'a :: metric_space"

   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"

   unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)

 lemma eventually_at_le:

   fixes a :: "'a::metric_space"

   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"

   unfolding eventually_at_filter eventually_nhds_metric

   apply auto

   apply (rule_tac x="d / 2" in exI)

   apply auto

   done

 lemma tendstoI:

   fixes l :: "'a :: metric_space"

   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"

   shows "(f ---> l) F"

   apply (rule topological_tendstoI)

   apply (simp add: open_dist)

   apply (drule (1) bspec, clarify)

   apply (drule assms)

   apply (erule eventually_elim1, simp)

   done

 lemma tendstoD:

   fixes l :: "'a :: metric_space"

   shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"

   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)

   apply (clarsimp simp add: open_dist)

   apply (rule_tac x="e - dist x l" in exI, clarsimp)

   apply (simp only: less_diff_eq)

   apply (erule le_less_trans [OF dist_triangle])

   apply simp

   apply simp

   done

 lemma tendsto_iff:

   fixes l :: "'a :: metric_space"

   shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"

   using tendstoI tendstoD by fast

 lemma metric_tendsto_imp_tendsto:

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

   assumes f: "(f ---> a) F"

   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"

   shows "(g ---> b) F"

 proof (rule tendstoI)

   fix e :: real assume "0 < e"

   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)

   with le show "eventually (\<lambda>x. dist (g x) b < e) F"

     using le_less_trans by (rule eventually_elim2)

 qed

 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"

   unfolding filterlim_at_top

   apply (intro allI)

   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)

   apply (auto simp: natceiling_le_eq)

   done

 subsubsection {* Limits of Sequences *}

 lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"

   unfolding tendsto_iff eventually_sequentially ..

 lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"

   unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)

 lemma metric_LIMSEQ_I:

   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"

 by (simp add: LIMSEQ_def)

 lemma metric_LIMSEQ_D:

   "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"

 by (simp add: LIMSEQ_def)

 subsubsection {* Limits of Functions *}

 lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =

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

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

   unfolding tendsto_iff eventually_at by simp

 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::'a::metric_space) --> (L::'b::metric_space)"

 by (simp add: LIM_def)

 lemma metric_LIM_D:

   "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 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 metric_LIM_imp_LIM:

   assumes f: "f -- a --> (l::'a::metric_space)"

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

   shows "g -- a --> (m::'b::metric_space)"

   by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)

 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::'a::metric_space) --> l"

 apply (rule topological_tendstoI)

 apply (drule (2) topological_tendstoD)

 apply (simp add: eventually_at, safe)

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

 apply (simp add: 1)

 apply (simp add: 2)

 done

 lemma metric_LIM_compose2:

   assumes f: "f -- (a::'a::metric_space) --> 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"

   using inj

   by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)

 lemma metric_isCont_LIM_compose2:

   fixes f :: "'a :: metric_space \<Rightarrow> _"

   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])

 subsection {* Complete metric spaces *}

 subsection {* Cauchy sequences *}

 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where

   "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"

 subsection {* Cauchy Sequences *}

 lemma metric_CauchyI:

   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"

   by (simp add: Cauchy_def)

 lemma metric_CauchyD:

   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"

   by (simp add: Cauchy_def)

 lemma metric_Cauchy_iff2:

   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"

 apply (simp add: Cauchy_def, auto)

 apply (drule reals_Archimedean, safe)

 apply (drule_tac x = n in spec, auto)

 apply (rule_tac x = M in exI, auto)

 apply (drule_tac x = m in spec, simp)

 apply (drule_tac x = na in spec, auto)

 done

 lemma Cauchy_iff2:

   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"

   unfolding metric_Cauchy_iff2 dist_real_def ..

 lemma Cauchy_subseq_Cauchy:

   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"

 apply (auto simp add: Cauchy_def)

 apply (drule_tac x=e in spec, clarify)

 apply (rule_tac x=M in exI, clarify)

 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)

 done

 theorem LIMSEQ_imp_Cauchy:

   assumes X: "X ----> a" shows "Cauchy X"

 proof (rule metric_CauchyI)

   fix e::real assume "0 < e"

   hence "0 < e/2" by simp

   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)

   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..

   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"

   proof (intro exI allI impI)

     fix m assume "N \<le> m"

     hence m: "dist (X m) a < e/2" using N by fast

     fix n assume "N \<le> n"

     hence n: "dist (X n) a < e/2" using N by fast

     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"

       by (rule dist_triangle2)

     also from m n have "\<dots> < e" by simp

     finally show "dist (X m) (X n) < e" .

   qed

 qed

 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"

 unfolding convergent_def

 by (erule exE, erule LIMSEQ_imp_Cauchy)

 subsubsection {* Cauchy Sequences are Convergent *}

 class complete_space = metric_space +

   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"

 lemma Cauchy_convergent_iff:

   fixes X :: "nat \<Rightarrow> 'a::complete_space"

   shows "Cauchy X = convergent X"

 by (fast intro: Cauchy_convergent convergent_Cauchy)

 subsection {* The set of real numbers is a complete metric space *}

 text {*

 Proof that Cauchy sequences converge based on the one from

 @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}

 *}

 text {*

   If sequence @{term "X"} is Cauchy, then its limit is the lub of

   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}

 *}

 lemma increasing_LIMSEQ:

   fixes f :: "nat \<Rightarrow> real"

   assumes inc: "\<And>n. f n \<le> f (Suc n)"

       and bdd: "\<And>n. f n \<le> l"

       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"

   shows "f ----> l"

 proof (rule increasing_tendsto)

   fix x assume "x < l"

   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"

     by auto

   from en[OF `0 < e`] obtain n where "l - e \<le> f n"

     by (auto simp: field_simps)

   with `e < l - x` `0 < e` have "x < f n" by simp

   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"

     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)

 qed (insert bdd, auto)

 lemma real_Cauchy_convergent:

   fixes X :: "nat \<Rightarrow> real"

   assumes X: "Cauchy X"

   shows "convergent X"

 proof -

   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"

   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto

   { fix N x assume N: "\<forall>n\<ge>N. X n < x"

   fix y::real assume "y \<in> S"

   hence "\<exists>M. \<forall>n\<ge>M. y < X n"

     by (simp add: S_def)

   then obtain M where "\<forall>n\<ge>M. y < X n" ..

   hence "y < X (max M N)" by simp

   also have "\<dots> < x" using N by simp

   finally have "y \<le> x"

     by (rule order_less_imp_le) }

   note bound_isUb = this 

   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"

     using X[THEN metric_CauchyD, OF zero_less_one] by auto

   hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp

   have [simp]: "S \<noteq> {}"

   proof (intro exI ex_in_conv[THEN iffD1])

     from N have "\<forall>n\<ge>N. X N - 1 < X n"

       by (simp add: abs_diff_less_iff dist_real_def)

     thus "X N - 1 \<in> S" by (rule mem_S)

   qed

   have [simp]: "bdd_above S"

   proof

     from N have "\<forall>n\<ge>N. X n < X N + 1"

       by (simp add: abs_diff_less_iff dist_real_def)

     thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"

       by (rule bound_isUb)

   qed

   have "X ----> Sup S"

   proof (rule metric_LIMSEQ_I)

   fix r::real assume "0 < r"

   hence r: "0 < r/2" by simp

   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"

     using metric_CauchyD [OF X r] by auto

   hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp

   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"

     by (simp only: dist_real_def abs_diff_less_iff)

   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast

   hence "X N - r/2 \<in> S" by (rule mem_S)

   hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)

   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast

   from bound_isUb[OF this]

   have 2: "Sup S \<le> X N + r/2"

     by (intro cSup_least) simp_all

   show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"

   proof (intro exI allI impI)

     fix n assume n: "N \<le> n"

     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+

     thus "dist (X n) (Sup S) < r" using 1 2

       by (simp add: abs_diff_less_iff dist_real_def)

   qed

   qed

   then show ?thesis unfolding convergent_def by auto

 qed

 instance real :: complete_space

   by intro_classes (rule real_Cauchy_convergent)

 class banach = real_normed_vector + complete_space

 instance real :: banach by default

 lemma tendsto_at_topI_sequentially:

   fixes f :: "real \<Rightarrow> real"

   assumes mono: "mono f"

   assumes limseq: "(\<lambda>n. f (real n)) ----> y"

   shows "(f ---> y) at_top"

 proof (rule tendstoI)

   fix e :: real assume "0 < e"

   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"

     by (auto simp: LIMSEQ_def dist_real_def)

   { fix x :: real

     obtain n where "x \<le> real_of_nat n"

       using ex_le_of_nat[of x] ..

     note monoD[OF mono this]

     also have "f (real_of_nat n) \<le> y"

       by (rule LIMSEQ_le_const[OF limseq])

          (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])

     finally have "f x \<le> y" . }

   note le = this

   have "eventually (\<lambda>x. real N \<le> x) at_top"

     by (rule eventually_ge_at_top)

   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"

   proof eventually_elim

     fix x assume N': "real N \<le> x"

     with N[of N] le have "y - f (real N) < e" by auto

     moreover note monoD[OF mono N']

     ultimately show "dist (f x) y < e"

       using le[of x] by (auto simp: dist_real_def field_simps)

   qed

 qed

 end