src/HOL/Real_Vector_Spaces.thy
 author paulson Fri, 13 Nov 2015 12:27:13 +0000 changeset 61649 268d88ec9087 parent 61609 77b453bd616f child 61762 d50b993b4fb9 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
```
(*  Title:      HOL/Real_Vector_Spaces.thy
Author:     Brian Huffman
Author:     Johannes Hölzl
*)

section \<open>Vector Spaces and Algebras over the Reals\<close>

theory Real_Vector_Spaces
imports Real Topological_Spaces
begin

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"

lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
apply (cases "finite A")
apply (induct set: finite)
done

end

subsection \<open>Vector spaces\<close>

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)"

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"
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"
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 \<open>Real vector spaces\<close>

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_scaleR)
apply (rule scaleR_one)
done

text \<open>Recover original theorem names\<close>

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 \<open>Legacy names\<close>

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

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}"
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

lemma setsum_constant_scaleR:
fixes y :: "'a::real_vector"
shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
apply (cases "finite A")
apply (induct set: finite)
done

lemma real_vector_affinity_eq:
fixes x :: "'a :: real_vector"
assumes m0: "m \<noteq> 0"
shows "m *\<^sub>R x + c = y \<longleftrightarrow> x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
proof
assume h: "m *\<^sub>R x + c = y"
hence "m *\<^sub>R x = y - c" by (simp add: field_simps)
hence "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp
then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
using m0
next
assume h: "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
show "m *\<^sub>R x + c = y" unfolding h
using m0  by (simp add: real_vector.scale_right_diff_distrib)
qed

lemma real_vector_eq_affinity:
fixes x :: "'a :: real_vector"
shows "m \<noteq> 0 ==> (y = m *\<^sub>R x + c \<longleftrightarrow> inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x)"
using real_vector_affinity_eq[where m=m and x=x and y=y and c=c]
by metis

subsection \<open>Embedding of the Reals into any @{text real_algebra_1}:
@{term of_real}\<close>

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"

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

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

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

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

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

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

lemma of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
by (induct s rule: infinite_finite_induct) auto

lemma of_real_setprod[simp]: "of_real (setprod f s) = (\<Prod>x\<in>s. of_real (f x))"
by (induct s rule: infinite_finite_induct) auto

lemma nonzero_of_real_inverse:
"x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
inverse (of_real x :: 'a::real_div_algebra)"

lemma of_real_inverse [simp]:
"of_real (inverse x) =
inverse (of_real x :: 'a::{real_div_algebra, division_ring})"

lemma nonzero_of_real_divide:
"y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
(of_real x / of_real y :: 'a::real_field)"

lemma of_real_divide [simp]:
"of_real (x / y) =
(of_real x / of_real y :: 'a::{real_field, field})"

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)"

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"
qed

text\<open>Collapse nested embeddings\<close>
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 [simp]: "of_real (numeral w) = numeral w"
using of_real_of_int_eq [of "numeral w"] by simp

lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w"
using of_real_of_int_eq [of "- numeral w"] by simp

text\<open>Every real algebra has characteristic zero\<close>

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 \<open>The Set of Real Numbers\<close>

definition Reals :: "'a::real_algebra_1 set"  ("\<real>")
where "\<real> = range of_real"

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

lemma Reals_of_int [simp]: "of_int z \<in> \<real>"
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)

lemma Reals_of_nat [simp]: "of_nat n \<in> \<real>"
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)

lemma Reals_numeral [simp]: "numeral w \<in> \<real>"
by (subst of_real_numeral [symmetric], rule Reals_of_real)

lemma Reals_0 [simp]: "0 \<in> \<real>"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_0 [symmetric])
done

lemma Reals_1 [simp]: "1 \<in> \<real>"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_1 [symmetric])
done

lemma Reals_add [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a + b \<in> \<real>"
apply (rule range_eqI)
done

lemma Reals_minus [simp]: "a \<in> \<real> \<Longrightarrow> - a \<in> \<real>"
apply (rule range_eqI)
apply (rule of_real_minus [symmetric])
done

lemma Reals_diff [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a - b \<in> \<real>"
apply (rule range_eqI)
apply (rule of_real_diff [symmetric])
done

lemma Reals_mult [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a * b \<in> \<real>"
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> \<real>; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> \<real>"
apply (rule range_eqI)
apply (erule nonzero_of_real_inverse [symmetric])
done

lemma Reals_inverse:
fixes a :: "'a::{real_div_algebra, division_ring}"
shows "a \<in> \<real> \<Longrightarrow> inverse a \<in> \<real>"
apply (rule range_eqI)
apply (rule of_real_inverse [symmetric])
done

lemma Reals_inverse_iff [simp]:
fixes x:: "'a :: {real_div_algebra, division_ring}"
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> \<real>; b \<in> \<real>; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
apply (rule range_eqI)
apply (erule nonzero_of_real_divide [symmetric])
done

lemma Reals_divide [simp]:
fixes a b :: "'a::{real_field, field}"
shows "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
apply (rule range_eqI)
apply (rule of_real_divide [symmetric])
done

lemma Reals_power [simp]:
fixes a :: "'a::{real_algebra_1}"
shows "a \<in> \<real> \<Longrightarrow> a ^ n \<in> \<real>"
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 \<open>q \<in> \<real>\<close> 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 [intro,simp]:
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 [intro,simp]:
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 \<open>Ordered real vector spaces\<close>

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 neg_le_divideR_eq:
fixes a :: "'a :: ordered_real_vector"
assumes "c < 0"
shows "a \<le> b /\<^sub>R c \<longleftrightarrow> b \<le> c *\<^sub>R a"
using pos_le_divideR_eq [of "-c" a "-b"] assms
by simp

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)

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"

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"

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 \<open>0 < a\<close>
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 \<open>0 > a\<close>
by simp
} ultimately show ?rhs using \<open>a \<noteq> 0\<close> by arith
qed (auto simp: not_le \<open>a \<noteq> 0\<close> 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 \<open>Real normed vector spaces\<close>

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"
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)"

lemma norm_not_less_zero [simp]:
fixes x :: "'a::real_normed_vector"
shows "\<not> norm x < 0"

lemma norm_le_zero_iff [simp]:
fixes x :: "'a::real_normed_vector"
shows "(norm x \<le> 0) = (x = 0)"

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_uminus_minus: "norm (-x - y :: 'a :: real_normed_vector) = norm (x + y)"
by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp

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))"
also have "\<dots> \<le> norm (a - c) + norm (b - d)"
by (rule norm_triangle_ineq)
finally show ?thesis .
qed

lemma norm_diff_triangle_le:
fixes x y z :: "'a::real_normed_vector"
assumes "norm (x - y) \<le> e1"  "norm (y - z) \<le> e2"
shows "norm (x - z) \<le> e1 + e2"
using norm_diff_triangle_ineq [of x y y z] assms by simp

lemma norm_diff_triangle_less:
fixes x y z :: "'a::real_normed_vector"
assumes "norm (x - y) < e1"  "norm (y - z) < e2"
shows "norm (x - z) < e1 + e2"
using norm_diff_triangle_ineq [of x y y z] assms by simp

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"

lemma norm_setsum:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)

lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)

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])

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])
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])
done

lemma norm_inverse:
fixes a :: "'a::{real_normed_div_algebra, division_ring}"
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}"
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 norm_mult_numeral1 [simp]:
fixes a b :: "'a::{real_normed_field, field}"
shows "norm (numeral w * a) = numeral w * norm a"

lemma norm_mult_numeral2 [simp]:
fixes a b :: "'a::{real_normed_field, field}"
shows "norm (a * numeral w) = norm a * numeral w"

lemma norm_divide_numeral [simp]:
fixes a b :: "'a::{real_normed_field, field}"
shows "norm (a / numeral w) = norm a / numeral w"

lemma norm_of_real_diff [simp]:
"norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \<le> \<bar>b - a\<bar>"
by (metis norm_of_real of_real_diff order_refl)

text\<open>Despite a superficial resemblance, @{text norm_eq_1} is not relevant.\<close>
lemma square_norm_one:
fixes x :: "'a::real_normed_div_algebra"
assumes "x^2 = 1" shows "norm x = 1"
by (metis assms norm_minus_cancel norm_one power2_eq_1_iff)

lemma norm_less_p1:
fixes x :: "'a::real_normed_algebra_1"
shows "norm x < norm (of_real (norm x) + 1 :: 'a)"
proof -
have "norm x < norm (of_real (norm x + 1) :: 'a)"
then show ?thesis
by simp
qed

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)"
by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)

lemma norm_setprod_le:
"norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))"
proof (induction A rule: infinite_finite_induct)
case (insert a A)
then have "norm (setprod f (insert a A)) \<le> norm (f a) * norm (setprod f A)"
also have "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a))"
by (rule insert)
finally show ?case
qed simp_all

lemma norm_setprod_diff:
fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}"
shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow>
norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
proof (induction I rule: infinite_finite_induct)
case (insert i I)
note insert.hyps[simp]

have "norm ((\<Prod>i\<in>insert i I. z i) - (\<Prod>i\<in>insert i I. w i)) =
norm ((\<Prod>i\<in>I. z i) * (z i - w i) + ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * w i)"
(is "_ = norm (?t1 + ?t2)")
also have "... \<le> norm ?t1 + norm ?t2"
by (rule norm_triangle_ineq)
also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)"
by (rule norm_mult_ineq)
also have "\<dots> \<le> (\<Prod>i\<in>I. norm (z i)) * norm(z i - w i)"
by (rule mult_right_mono) (auto intro: norm_setprod_le)
also have "(\<Prod>i\<in>I. norm (z i)) \<le> (\<Prod>i\<in>I. 1)"
by (intro setprod_mono) (auto intro!: insert)
also have "norm ?t2 \<le> norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * norm (w i)"
by (rule norm_mult_ineq)
also have "norm (w i) \<le> 1"
by (auto intro: insert)
also have "norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
using insert by auto
finally show ?case
by (auto simp add: ac_simps mult_right_mono mult_left_mono)
qed simp_all

lemma norm_power_diff:
fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
assumes "norm z \<le> 1" "norm w \<le> 1"
shows "norm (z^m - w^m) \<le> m * norm (z - w)"
proof -
have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))"
also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
by (intro norm_setprod_diff) (auto simp add: assms)
also have "\<dots> = m * norm (z - w)"
by simp
finally show ?thesis .
qed

subsection \<open>Metric spaces\<close>

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"

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

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

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"

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

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 (blast 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 (meson UnionE UnionI)
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 \<open>Every normed vector space is a metric space.\<close>

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 \<open>Class instances for real numbers\<close>

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 (rule abs_eq_0)
apply (rule abs_triangle_ineq)
apply (rule abs_mult)
apply (rule abs_mult)
done

end

lemma dist_of_real [simp]:
fixes a :: "'a::real_normed_div_algebra"
shows "dist (of_real x :: 'a) (of_real y) = dist x y"
by (metis dist_norm norm_of_real of_real_diff real_norm_def)

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 \<open>Extra type constraints\<close>

text \<open>Only allow @{term "open"} in class @{text topological_space}.\<close>

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

text \<open>Only allow @{term dist} in class @{text metric_space}.\<close>

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

text \<open>Only allow @{term norm} in class @{text real_normed_vector}.\<close>

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

subsection \<open>Sign function\<close>

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

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

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

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

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

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

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>"

lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> (x::real)"
by (cases "0::real" x rule: linorder_cases) simp_all

lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> (x::real) \<le> 0"
by (cases "0::real" x rule: linorder_cases) simp_all

lemma norm_conv_dist: "norm x = dist x 0"
unfolding dist_norm by simp

lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"

lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \<bar>m - n\<bar>"
proof -
have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))"
by simp
also have "\<dots> = of_int \<bar>m - n\<bar>" by (subst dist_diff, subst norm_of_int) simp
finally show ?thesis .
qed

lemma dist_of_nat:
"dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \<bar>int m - int n\<bar>"
by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int)

subsection \<open>Bounded Linear and Bilinear Operators\<close>

locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
assumes scaleR: "f (scaleR r x) = scaleR r (f x)"

lemma linear_imp_scaleR:
assumes "linear D" obtains d where "D = (\<lambda>x. x *\<^sub>R d)"
by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def)

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 standard (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 blast
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

lemma linear: "linear f" ..

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 standard (blast 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 zero_left: "prod 0 b = 0"

lemma zero_right: "prod a 0 = 0"

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

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

lemma diff_left:
"prod (a - a') b = prod a b - prod a' b"

lemma diff_right:
"prod a (b - b') = prod a b - prod a b'"

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 scaleR_left)
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 scaleR_right)
done

lemma prod_diff_prod:
"(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"

end

lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
by standard (auto intro!: exI[of _ 1])

lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
by standard (auto intro!: exI[of _ 1])

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"
by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
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)
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)"
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 blast
from g.pos_bounded
obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
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)
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

lemma bij_linear_imp_inv_linear:
assumes "linear f" "bij f" shows "linear (inv f)"
using assms unfolding linear_def linear_axioms_def additive_def
by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!:  Hilbert_Choice.inv_f_eq)

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 \<open>Filters and Limits on Metric Space\<close>

lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})"
unfolding nhds_def
proof (safe intro!: INF_eq)
fix S assume "open S" "x \<in> S"
then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e"
by (auto simp: open_dist subset_eq)
then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S"
by auto
qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)

lemma (in metric_space) tendsto_iff:
"(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
unfolding nhds_metric filterlim_INF filterlim_principal by auto

lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f ---> l) F"
by (auto simp: tendsto_iff)

lemma (in metric_space) tendstoD: "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
by (auto simp: tendsto_iff)

lemma (in metric_space) eventually_nhds_metric:
"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
unfolding nhds_metric
by (subst eventually_INF_base)
(auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b])

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 eventually_at_left_real: "a > (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {b<..<a}) (at_left a)"
by (subst eventually_at, rule exI[of _ "a - b"]) (force simp: dist_real_def)

lemma eventually_at_right_real: "a < (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {a<..<b}) (at_right a)"
by (subst eventually_at, rule exI[of _ "b - a"]) (force simp: dist_real_def)

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="nat(ceiling (Z + 1))" in eventually_sequentiallyI)
by linarith

subsubsection \<open>Limits of Sequences\<close>

lemma lim_sequentially: "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 ..

lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*)

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 lim_sequentially 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)"

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"

subsubsection \<open>Limits of Functions\<close>

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)"

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"

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 (rule_tac x="min d R" in exI, safe)
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 \<open>Complete metric spaces\<close>

subsection \<open>Cauchy sequences\<close>

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)"

lemma Cauchy_altdef:
"Cauchy f = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
proof
assume A: "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
show "Cauchy f" unfolding Cauchy_def
proof (intro allI impI)
fix e :: real assume e: "e > 0"
with A obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n > m \<Longrightarrow> dist (f m) (f n) < e" by blast
have "dist (f m) (f n) < e" if "m \<ge> M" "n \<ge> M" for m n
using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute)
thus "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m) (f n) < e" by blast
qed
next
assume "Cauchy f"
show "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
proof (intro allI impI)
fix e :: real assume e: "e > 0"
with `Cauchy f` obtain M where "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> dist (f m) (f n) < e"
unfolding Cauchy_def by blast
thus "\<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e" by (intro exI[of _ M]) force
qed
qed

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"

lemma CauchyI': "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
unfolding Cauchy_altdef by blast

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"

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 (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 (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 blast
fix n assume "N \<le> n"
hence n: "dist (X n) a < e/2" using N by blast
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 \<open>Cauchy Sequences are Convergent\<close>

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 (blast intro: Cauchy_convergent convergent_Cauchy)

subsection \<open>The set of real numbers is a complete metric space\<close>

text \<open>
Proof that Cauchy sequences converge based on the one from
@{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
\<close>

text \<open>
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}"}
\<close>

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 \<open>0 < e\<close>] obtain n where "l - e \<le> f n"
by (auto simp: field_simps)
with \<open>e < l - x\<close> \<open>0 < e\<close> 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"
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"
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"
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 blast
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 blast
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
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 ..

lemma tendsto_at_topI_sequentially:
fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) ----> y"
shows "(f ---> y) at_top"
proof -
from nhds_countable[of y] guess A . note A = this

have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m"
proof (rule ccontr)
assume "\<not> (\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m)"
then obtain m where "\<And>k. \<exists>x\<ge>k. f x \<notin> A m"
by auto
then have "\<exists>X. \<forall>n. (f (X n) \<notin> A m) \<and> max n (X n) + 1 \<le> X (Suc n)"
by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)"
by auto
{ fix n have "1 \<le> n \<longrightarrow> real n \<le> X n"
using X[of "n - 1"] by auto }
then have "filterlim X at_top sequentially"
by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially]
simp: eventually_sequentially)
from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False
by auto
qed
then obtain k where "\<And>m x. k m \<le> x \<Longrightarrow> f x \<in> A m"
by metis
then show ?thesis
unfolding at_top_def A
by (intro filterlim_base[where i=k]) auto
qed

lemma tendsto_at_topI_sequentially_real:
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: lim_sequentially 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])
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

```