# Theory Real_Vector_Spaces

theory Real_Vector_Spaces
imports Real Topological_Spaces Vector_Spaces
```(*  Title:      HOL/Real_Vector_Spaces.thy
Author:     Brian Huffman
Author:     Johannes Hölzl
*)

section ‹Vector Spaces and Algebras over the Reals›

theory Real_Vector_Spaces
imports Real Topological_Spaces Vector_Spaces
begin

subsection ‹Real vector spaces›

class scaleR =
fixes scaleR :: "real ⇒ 'a ⇒ 'a" (infixr "*⇩R" 75)
begin

abbreviation divideR :: "'a ⇒ real ⇒ 'a"  (infixl "'/⇩R" 70)
where "x /⇩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"

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
by standard (simp_all add: algebra_simps)

end

locale linear = Vector_Spaces.linear "scaleR::_⇒_⇒'a::real_vector" "scaleR::_⇒_⇒'b::real_vector"
begin
lemmas scaleR = scale
end

global_interpretation real_vector?: vector_space "scaleR :: real ⇒ 'a ⇒ 'a::real_vector"
rewrites "Vector_Spaces.linear ( *⇩R) ( *⇩R) = linear"
and "Vector_Spaces.linear ( *) ( *⇩R) = linear"
defines dependent_raw_def: dependent = real_vector.dependent
and representation_raw_def: representation = real_vector.representation
and subspace_raw_def: subspace = real_vector.subspace
and span_raw_def: span = real_vector.span
and extend_basis_raw_def: extend_basis = real_vector.extend_basis
and dim_raw_def: dim = real_vector.dim
apply unfold_locales
apply (rule scaleR_scaleR)
apply (rule scaleR_one)
apply (force simp: linear_def)
apply (force simp: linear_def real_scaleR_def[abs_def])
done

hide_const (open)― ‹locale constants›
real_vector.dependent
real_vector.independent
real_vector.representation
real_vector.subspace
real_vector.span
real_vector.extend_basis
real_vector.dim

abbreviation "independent x ≡ ¬ dependent x"

global_interpretation real_vector?: vector_space_pair "scaleR::_⇒_⇒'a::real_vector" "scaleR::_⇒_⇒'b::real_vector"
rewrites  "Vector_Spaces.linear ( *⇩R) ( *⇩R) = linear"
and "Vector_Spaces.linear ( *) ( *⇩R) = linear"
defines construct_raw_def: construct = real_vector.construct
apply unfold_locales
unfolding linear_def real_scaleR_def
by (rule refl)+

hide_const (open)― ‹locale constants›
real_vector.construct

lemma linear_compose: "linear f ⟹ linear g ⟹ linear (g ∘ f)"
unfolding linear_def by (rule Vector_Spaces.linear_compose)

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_sum_left = real_vector.scale_sum_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_sum_right = real_vector.scale_sum_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

lemmas linear_injective_0 = linear_inj_iff_eq_0
and linear_injective_on_subspace_0 = linear_inj_on_iff_eq_0
and linear_cmul = linear_scale
and linear_scaleR = linear_scale_self
and subspace_mul = subspace_scale
and span_linear_image = linear_span_image
and span_0 = span_zero
and span_mul = span_scale
and injective_scaleR = injective_scale

lemma scaleR_minus1_left [simp]: "scaleR (-1) x = - x"
for x :: "'a::real_vector"
using scaleR_minus_left [of 1 x] by simp

lemma scaleR_2:
fixes x :: "'a::real_vector"
shows "scaleR 2 x = x + x"
unfolding one_add_one [symmetric] scaleR_left_distrib by simp

lemma scaleR_half_double [simp]:
fixes a :: "'a::real_vector"
shows "(1 / 2) *⇩R (a + a) = a"
proof -
have "⋀r. r *⇩R (a + a) = (r * 2) *⇩R a"
by (metis scaleR_2 scaleR_scaleR)
then show ?thesis
by simp
qed

interpretation scaleR_left: additive "(λa. scaleR a x :: 'a::real_vector)"
by standard (rule scaleR_left_distrib)

interpretation scaleR_right: additive "(λx. scaleR a x :: 'a::real_vector)"
by standard (rule scaleR_right_distrib)

lemma nonzero_inverse_scaleR_distrib:
"a ≠ 0 ⟹ x ≠ 0 ⟹ inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
for x :: "'a::real_div_algebra"
by (rule inverse_unique) simp

lemma inverse_scaleR_distrib: "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
for x :: "'a::{real_div_algebra,division_ring}"
by (metis inverse_zero nonzero_inverse_scaleR_distrib scale_eq_0_iff)

lemmas sum_constant_scaleR = real_vector.sum_constant_scale― ‹legacy name›

named_theorems vector_add_divide_simps "to simplify sums of scaled vectors"

"v + (b / z) *⇩R w = (if z = 0 then v else (z *⇩R v + b *⇩R w) /⇩R z)"
"a *⇩R v + (b / z) *⇩R w = (if z = 0 then a *⇩R v else ((a * z) *⇩R v + b *⇩R w) /⇩R z)"
"(a / z) *⇩R v + w = (if z = 0 then w else (a *⇩R v + z *⇩R w) /⇩R z)"
"(a / z) *⇩R v + b *⇩R w = (if z = 0 then b *⇩R w else (a *⇩R v + (b * z) *⇩R w) /⇩R z)"
"v - (b / z) *⇩R w = (if z = 0 then v else (z *⇩R v - b *⇩R w) /⇩R z)"
"a *⇩R v - (b / z) *⇩R w = (if z = 0 then a *⇩R v else ((a * z) *⇩R v - b *⇩R w) /⇩R z)"
"(a / z) *⇩R v - w = (if z = 0 then -w else (a *⇩R v - z *⇩R w) /⇩R z)"
"(a / z) *⇩R v - b *⇩R w = (if z = 0 then -b *⇩R w else (a *⇩R v - (b * z) *⇩R w) /⇩R z)"
for v :: "'a :: real_vector"

fixes x :: "'a :: real_vector"
shows "(x = (u / v) *⇩R a) ⟷ (if v=0 then x = 0 else v *⇩R x = u *⇩R a)"
by auto (metis (no_types) divide_eq_1_iff divide_inverse_commute scaleR_one scaleR_scaleR)

fixes x :: "'a :: real_vector"
shows "((u / v) *⇩R a = x) ⟷ (if v=0 then x = 0 else u *⇩R a = v *⇩R x)"
by (metis eq_vector_fraction_iff)

lemma real_vector_affinity_eq:
fixes x :: "'a :: real_vector"
assumes m0: "m ≠ 0"
shows "m *⇩R x + c = y ⟷ x = inverse m *⇩R y - (inverse m *⇩R c)"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then have "m *⇩R x = y - c" by (simp add: field_simps)
then have "inverse m *⇩R (m *⇩R x) = inverse m *⇩R (y - c)" by simp
then show "x = inverse m *⇩R y - (inverse m *⇩R c)"
using m0
by (simp add: scaleR_diff_right)
next
assume ?rhs
with m0 show "m *⇩R x + c = y"
by (simp add: scaleR_diff_right)
qed

lemma real_vector_eq_affinity: "m ≠ 0 ⟹ y = m *⇩R x + c ⟷ inverse m *⇩R y - (inverse m *⇩R c) = x"
for x :: "'a::real_vector"
using real_vector_affinity_eq[where m=m and x=x and y=y and c=c]
by metis

lemma scaleR_eq_iff [simp]: "b + u *⇩R a = a + u *⇩R b ⟷ a = b ∨ u = 1"
for a :: "'a::real_vector"
proof (cases "u = 1")
case True
then show ?thesis by auto
next
case False
have "a = b" if "b + u *⇩R a = a + u *⇩R b"
proof -
from that have "(u - 1) *⇩R a = (u - 1) *⇩R b"
by (simp add: algebra_simps)
with False show ?thesis
by auto
qed
then show ?thesis by auto
qed

lemma scaleR_collapse [simp]: "(1 - u) *⇩R a + u *⇩R a = a"
for a :: "'a::real_vector"
by (simp add: algebra_simps)

subsection ‹Embedding of the Reals into any ‹real_algebra_1›: ‹of_real››

definition of_real :: "real ⇒ '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 of_real_sum[simp]: "of_real (sum f s) = (∑x∈s. of_real (f x))"
by (induct s rule: infinite_finite_induct) auto

lemma of_real_prod[simp]: "of_real (prod f s) = (∏x∈s. of_real (f x))"
by (induct s rule: infinite_finite_induct) auto

lemma nonzero_of_real_inverse:
"x ≠ 0 ⟹ 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})"
by (simp add: of_real_def inverse_scaleR_distrib)

lemma nonzero_of_real_divide:
"y ≠ 0 ⟹ 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_div_algebra)"
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]
lemmas of_real_eq_1_iff [simp] = of_real_eq_iff [of _ 1, simplified]

lemma minus_of_real_eq_of_real_iff [simp]: "-of_real x = of_real y ⟷ -x = y"
using of_real_eq_iff[of "-x" y] by (simp only: of_real_minus)

lemma of_real_eq_minus_of_real_iff [simp]: "of_real x = -of_real y ⟷ x = -y"
using of_real_eq_iff[of x "-y"] by (simp only: of_real_minus)

lemma of_real_eq_id [simp]: "of_real = (id :: real ⇒ real)"
by (rule ext) (simp add: of_real_def)

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 [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 ‹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 ∘ of_nat)"
by (rule inj_comp)
then show "inj (of_nat :: nat ⇒ 'a)"
by (simp add: comp_def)
qed

lemma fraction_scaleR_times [simp]:
fixes a :: "'a::real_algebra_1"
shows "(numeral u / numeral v) *⇩R (numeral w * a) = (numeral u * numeral w / numeral v) *⇩R a"
by (metis (no_types, lifting) of_real_numeral scaleR_conv_of_real scaleR_scaleR times_divide_eq_left)

lemma inverse_scaleR_times [simp]:
fixes a :: "'a::real_algebra_1"
shows "(1 / numeral v) *⇩R (numeral w * a) = (numeral w / numeral v) *⇩R a"
by (metis divide_inverse_commute inverse_eq_divide of_real_numeral scaleR_conv_of_real scaleR_scaleR)

lemma scaleR_times [simp]:
fixes a :: "'a::real_algebra_1"
shows "(numeral u) *⇩R (numeral w * a) = (numeral u * numeral w) *⇩R a"
by (simp add: scaleR_conv_of_real)

instance real_field < field_char_0 ..

subsection ‹The Set of Real Numbers›

definition Reals :: "'a::real_algebra_1 set"  ("ℝ")
where "ℝ = range of_real"

lemma Reals_of_real [simp]: "of_real r ∈ ℝ"
by (simp add: Reals_def)

lemma Reals_of_int [simp]: "of_int z ∈ ℝ"
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)

lemma Reals_of_nat [simp]: "of_nat n ∈ ℝ"
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)

lemma Reals_numeral [simp]: "numeral w ∈ ℝ"
by (subst of_real_numeral [symmetric], rule Reals_of_real)

lemma Reals_0 [simp]: "0 ∈ ℝ" and Reals_1 [simp]: "1 ∈ ℝ"
by (simp_all add: Reals_def)

lemma Reals_add [simp]: "a ∈ ℝ ⟹ b ∈ ℝ ⟹ a + b ∈ ℝ"
by (metis (no_types, hide_lams) Reals_def Reals_of_real imageE of_real_add)

lemma Reals_minus [simp]: "a ∈ ℝ ⟹ - a ∈ ℝ"
by (auto simp: Reals_def)

lemma Reals_minus_iff [simp]: "- a ∈ ℝ ⟷ a ∈ ℝ"
apply (auto simp: Reals_def)
by (metis add.inverse_inverse of_real_minus rangeI)

lemma Reals_diff [simp]: "a ∈ ℝ ⟹ b ∈ ℝ ⟹ a - b ∈ ℝ"

lemma Reals_mult [simp]: "a ∈ ℝ ⟹ b ∈ ℝ ⟹ a * b ∈ ℝ"
by (metis (no_types, lifting) Reals_def Reals_of_real imageE of_real_mult)

lemma nonzero_Reals_inverse: "a ∈ ℝ ⟹ a ≠ 0 ⟹ inverse a ∈ ℝ"
for a :: "'a::real_div_algebra"
by (metis Reals_def Reals_of_real imageE of_real_inverse)

lemma Reals_inverse: "a ∈ ℝ ⟹ inverse a ∈ ℝ"
for a :: "'a::{real_div_algebra,division_ring}"
using nonzero_Reals_inverse by fastforce

lemma Reals_inverse_iff [simp]: "inverse x ∈ ℝ ⟷ x ∈ ℝ"
for x :: "'a::{real_div_algebra,division_ring}"
by (metis Reals_inverse inverse_inverse_eq)

lemma nonzero_Reals_divide: "a ∈ ℝ ⟹ b ∈ ℝ ⟹ b ≠ 0 ⟹ a / b ∈ ℝ"
for a b :: "'a::real_field"
by (simp add: divide_inverse)

lemma Reals_divide [simp]: "a ∈ ℝ ⟹ b ∈ ℝ ⟹ a / b ∈ ℝ"
for a b :: "'a::{real_field,field}"
using nonzero_Reals_divide by fastforce

lemma Reals_power [simp]: "a ∈ ℝ ⟹ a ^ n ∈ ℝ"
for a :: "'a::real_algebra_1"
by (metis Reals_def Reals_of_real imageE of_real_power)

lemma Reals_cases [cases set: Reals]:
assumes "q ∈ ℝ"
obtains (of_real) r where "q = of_real r"
unfolding Reals_def
proof -
from ‹q ∈ ℝ› have "q ∈ range of_real" unfolding Reals_def .
then obtain r where "q = of_real r" ..
then show thesis ..
qed

lemma sum_in_Reals [intro,simp]: "(⋀i. i ∈ s ⟹ f i ∈ ℝ) ⟹ sum f s ∈ ℝ"
proof (induct s rule: infinite_finite_induct)
case infinite
then show ?case by (metis Reals_0 sum.infinite)
qed simp_all

lemma prod_in_Reals [intro,simp]: "(⋀i. i ∈ s ⟹ f i ∈ ℝ) ⟹ prod f s ∈ ℝ"
proof (induct s rule: infinite_finite_induct)
case infinite
then show ?case by (metis Reals_1 prod.infinite)
qed simp_all

lemma Reals_induct [case_names of_real, induct set: Reals]:
"q ∈ ℝ ⟹ (⋀r. P (of_real r)) ⟹ 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 ≤ y ⟹ 0 ≤ a ⟹ a *⇩R x ≤ a *⇩R y"
and scaleR_right_mono: "a ≤ b ⟹ 0 ≤ x ⟹ a *⇩R x ≤ b *⇩R x"
begin

lemma scaleR_mono: "a ≤ b ⟹ x ≤ y ⟹ 0 ≤ b ⟹ 0 ≤ x ⟹ a *⇩R x ≤ b *⇩R y"
by (meson local.dual_order.trans local.scaleR_left_mono local.scaleR_right_mono)

lemma scaleR_mono': "a ≤ b ⟹ c ≤ d ⟹ 0 ≤ a ⟹ 0 ≤ c ⟹ a *⇩R c ≤ b *⇩R d"
by (rule scaleR_mono) (auto intro: order.trans)

lemma pos_le_divideRI:
assumes "0 < c"
and "c *⇩R a ≤ b"
shows "a ≤ b /⇩R c"
proof -
from scaleR_left_mono[OF assms(2)] assms(1)
have "c *⇩R a /⇩R c ≤ b /⇩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 ≤ b /⇩R c ⟷ c *⇩R a ≤ b"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
from scaleR_left_mono[OF this] assms have "c *⇩R a ≤ c *⇩R (b /⇩R c)"
by simp
with assms show ?rhs
by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
next
assume ?rhs
with assms show ?lhs by (rule pos_le_divideRI)
qed

lemma scaleR_image_atLeastAtMost: "c > 0 ⟹ scaleR c ` {x..y} = {c *⇩R x..c *⇩R y}"
apply (auto intro!: scaleR_left_mono)
apply (rule_tac x = "inverse c *⇩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 ≤ b /⇩R c ⟷ b ≤ c *⇩R a"
using pos_le_divideR_eq [of "-c" a "-b"] assms by simp

lemma scaleR_nonneg_nonneg: "0 ≤ a ⟹ 0 ≤ x ⟹ 0 ≤ a *⇩R x"
for x :: "'a::ordered_real_vector"
using scaleR_left_mono [of 0 x a] by simp

lemma scaleR_nonneg_nonpos: "0 ≤ a ⟹ x ≤ 0 ⟹ a *⇩R x ≤ 0"
for x :: "'a::ordered_real_vector"
using scaleR_left_mono [of x 0 a] by simp

lemma scaleR_nonpos_nonneg: "a ≤ 0 ⟹ 0 ≤ x ⟹ a *⇩R x ≤ 0"
for x :: "'a::ordered_real_vector"
using scaleR_right_mono [of a 0 x] by simp

lemma split_scaleR_neg_le: "(0 ≤ a ∧ x ≤ 0) ∨ (a ≤ 0 ∧ 0 ≤ x) ⟹ a *⇩R x ≤ 0"
for x :: "'a::ordered_real_vector"
by (auto simp: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)

lemma le_add_iff1: "a *⇩R e + c ≤ b *⇩R e + d ⟷ (a - b) *⇩R e + c ≤ d"
for c d e :: "'a::ordered_real_vector"
by (simp add: algebra_simps)

lemma le_add_iff2: "a *⇩R e + c ≤ b *⇩R e + d ⟷ c ≤ (b - a) *⇩R e + d"
for c d e :: "'a::ordered_real_vector"
by (simp add: algebra_simps)

lemma scaleR_left_mono_neg: "b ≤ a ⟹ c ≤ 0 ⟹ c *⇩R a ≤ c *⇩R b"
for a b :: "'a::ordered_real_vector"
by (drule scaleR_left_mono [of _ _ "- c"], simp_all)

lemma scaleR_right_mono_neg: "b ≤ a ⟹ c ≤ 0 ⟹ a *⇩R c ≤ b *⇩R c"
for c :: "'a::ordered_real_vector"
by (drule scaleR_right_mono [of _ _ "- c"], simp_all)

lemma scaleR_nonpos_nonpos: "a ≤ 0 ⟹ b ≤ 0 ⟹ 0 ≤ a *⇩R b"
for b :: "'a::ordered_real_vector"
using scaleR_right_mono_neg [of a 0 b] by simp

lemma split_scaleR_pos_le: "(0 ≤ a ∧ 0 ≤ b) ∨ (a ≤ 0 ∧ b ≤ 0) ⟹ 0 ≤ a *⇩R b"
for b :: "'a::ordered_real_vector"
by (auto simp: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)

lemma zero_le_scaleR_iff:
fixes b :: "'a::ordered_real_vector"
shows "0 ≤ a *⇩R b ⟷ 0 < a ∧ 0 ≤ b ∨ a < 0 ∧ b ≤ 0 ∨ a = 0"
(is "?lhs = ?rhs")
proof (cases "a = 0")
case True
then show ?thesis by simp
next
case False
show ?thesis
proof
assume ?lhs
from ‹a ≠ 0› consider "a > 0" | "a < 0" by arith
then show ?rhs
proof cases
case 1
with ‹?lhs› have "inverse a *⇩R 0 ≤ inverse a *⇩R (a *⇩R b)"
by (intro scaleR_mono) auto
with 1 show ?thesis
by simp
next
case 2
with ‹?lhs› have "- inverse a *⇩R 0 ≤ - inverse a *⇩R (a *⇩R b)"
by (intro scaleR_mono) auto
with 2 show ?thesis
by simp
qed
next
assume ?rhs
then show ?lhs
by (auto simp: not_le ‹a ≠ 0› intro!: split_scaleR_pos_le)
qed
qed

lemma scaleR_le_0_iff: "a *⇩R b ≤ 0 ⟷ 0 < a ∧ b ≤ 0 ∨ a < 0 ∧ 0 ≤ b ∨ a = 0"
for b::"'a::ordered_real_vector"
by (insert zero_le_scaleR_iff [of "-a" b]) force

lemma scaleR_le_cancel_left: "c *⇩R a ≤ c *⇩R b ⟷ (0 < c ⟶ a ≤ b) ∧ (c < 0 ⟶ b ≤ a)"
for b :: "'a::ordered_real_vector"
by (auto simp: 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: "0 < c ⟹ c *⇩R a ≤ c *⇩R b ⟷ a ≤ b"
for b :: "'a::ordered_real_vector"
by (auto simp: scaleR_le_cancel_left)

lemma scaleR_le_cancel_left_neg: "c < 0 ⟹ c *⇩R a ≤ c *⇩R b ⟷ b ≤ a"
for b :: "'a::ordered_real_vector"
by (auto simp: scaleR_le_cancel_left)

lemma scaleR_left_le_one_le: "0 ≤ x ⟹ a ≤ 1 ⟹ a *⇩R x ≤ x"
for x :: "'a::ordered_real_vector" and a :: real
using scaleR_right_mono[of a 1 x] by simp

subsection ‹Real normed vector spaces›

class dist =
fixes dist :: "'a ⇒ 'a ⇒ real"

class norm =
fixes norm :: "'a ⇒ real"

class sgn_div_norm = scaleR + norm + sgn +
assumes sgn_div_norm: "sgn x = x /⇩R norm x"

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

class uniformity_dist = dist + uniformity +
assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
begin

lemma eventually_uniformity_metric:
"eventually P uniformity ⟷ (∃e>0. ∀x y. dist x y < e ⟶ P (x, y))"
unfolding uniformity_dist
by (subst eventually_INF_base)
(auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"])

end

class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
assumes norm_eq_zero [simp]: "norm x = 0 ⟷ x = 0"
and norm_triangle_ineq: "norm (x + y) ≤ norm x + norm y"
and norm_scaleR [simp]: "norm (scaleR a x) = ¦a¦ * norm x"
begin

lemma norm_ge_zero [simp]: "0 ≤ norm x"
proof -
have "0 = norm (x + -1 *⇩R x)"
using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
also have "… ≤ norm x + norm (-1 *⇩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) ≤ norm x * norm y"

class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
assumes norm_one [simp]: "norm 1 = 1"

lemma (in real_normed_algebra_1) scaleR_power [simp]: "(scaleR x y) ^ n = scaleR (x^n) (y^n)"
by (induct n) (simp_all add: scaleR_one scaleR_scaleR mult_ac)

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
show "norm (x * y) ≤ norm x * norm y" for x y :: 'a
by (simp add: norm_mult)
next
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
by (rule norm_mult)
then show "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]: "norm x > 0 ⟷ x ≠ 0"
for x :: "'a::real_normed_vector"
by (simp add: order_less_le)

lemma norm_not_less_zero [simp]: "¬ norm x < 0"
for x :: "'a::real_normed_vector"
by (simp add: linorder_not_less)

lemma norm_le_zero_iff [simp]: "norm x ≤ 0 ⟷ x = 0"
for x :: "'a::real_normed_vector"
by (simp add: order_le_less)

lemma norm_minus_cancel [simp]: "norm (- x) = norm x"
for x :: "'a::real_normed_vector"
proof -
have "norm (- x) = norm (scaleR (- 1) x)"
by (simp only: scaleR_minus_left scaleR_one)
also have "… = ¦- 1¦ * norm x"
by (rule norm_scaleR)
finally show ?thesis by simp
qed

lemma norm_minus_commute: "norm (a - b) = norm (b - a)"
for a b :: "'a::real_normed_vector"
proof -
have "norm (- (b - a)) = norm (b - a)"
by (rule norm_minus_cancel)
then show ?thesis by simp
qed

lemma dist_add_cancel [simp]: "dist (a + b) (a + c) = dist b c"
for a :: "'a::real_normed_vector"
by (simp add: dist_norm)

lemma dist_add_cancel2 [simp]: "dist (b + a) (c + a) = dist b c"
for a :: "'a::real_normed_vector"
by (simp add: dist_norm)

lemma dist_scaleR [simp]: "dist (x *⇩R a) (y *⇩R a) = ¦x - y¦ * norm a"
for a :: "'a::real_normed_vector"
by (metis dist_norm norm_scaleR scaleR_left.diff)

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: "norm a - norm b ≤ norm (a - b)"
for a b :: "'a::real_normed_vector"
proof -
have "norm (a - b + b) ≤ norm (a - b) + norm b"
by (rule norm_triangle_ineq)
then show ?thesis by simp
qed

lemma norm_triangle_ineq3: "¦norm a - norm b¦ ≤ norm (a - b)"
for a b :: "'a::real_normed_vector"
proof -
have "norm a - norm b ≤ norm (a - b)"
by (simp add: norm_triangle_ineq2)
moreover have "norm b - norm a ≤ norm (a - b)"
by (metis norm_minus_commute norm_triangle_ineq2)
ultimately show ?thesis
by (simp add: abs_le_iff)
qed

lemma norm_triangle_ineq4: "norm (a - b) ≤ norm a + norm b"
for a b :: "'a::real_normed_vector"
proof -
have "norm (a + - b) ≤ norm a + norm (- b)"
by (rule norm_triangle_ineq)
then show ?thesis by simp
qed

lemma norm_triangle_le_diff:
fixes x y :: "'a::real_normed_vector"
shows "norm x + norm y ≤ e ⟹ norm (x - y) ≤ e"
by (meson norm_triangle_ineq4 order_trans)

lemma norm_diff_ineq: "norm a - norm b ≤ norm (a + b)"
for a b :: "'a::real_normed_vector"
proof -
have "norm a - norm (- b) ≤ norm (a - - b)"
by (rule norm_triangle_ineq2)
then show ?thesis by simp
qed

lemma norm_add_leD: "norm (a + b) ≤ c ⟹ norm b ≤ norm a + c"
for a b :: "'a::real_normed_vector"
by (metis add.commute diff_le_eq norm_diff_ineq order.trans)

lemma norm_diff_triangle_ineq: "norm ((a + b) - (c + d)) ≤ norm (a - c) + norm (b - d)"
for a b c d :: "'a::real_normed_vector"
proof -
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
by (simp add: algebra_simps)
also have "… ≤ 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) ≤ 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_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 "norm a ≤ r ⟹ norm b ≤ s ⟹ norm (a + b) ≤ r + s"
by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)

lemma norm_sum:
fixes f :: "'a ⇒ 'b::real_normed_vector"
shows "norm (sum f A) ≤ (∑i∈A. norm (f i))"
by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)

lemma sum_norm_le:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes fg: "⋀x. x ∈ S ⟹ norm (f x) ≤ g x"
shows "norm (sum f S) ≤ sum g S"
by (rule order_trans [OF norm_sum sum_mono]) (simp add: fg)

lemma abs_norm_cancel [simp]: "¦norm a¦ = norm a"
for a :: "'a::real_normed_vector"
by (rule abs_of_nonneg [OF norm_ge_zero])

lemma norm_add_less: "norm x < r ⟹ norm y < s ⟹ norm (x + y) < r + s"
for x y :: "'a::real_normed_vector"
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])

lemma norm_mult_less: "norm x < r ⟹ norm y < s ⟹ norm (x * y) < r * s"
for x y :: "'a::real_normed_algebra"
by (rule order_le_less_trans [OF norm_mult_ineq]) (simp add: mult_strict_mono')

lemma norm_of_real [simp]: "norm (of_real r :: 'a::real_normed_algebra_1) = ¦r¦"
by (simp add: of_real_def)

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_real_add1 [simp]: "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = ¦x + 1¦"
by (metis norm_of_real of_real_1 of_real_add)

"norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = ¦x + numeral b¦"
by (metis norm_of_real of_real_add of_real_numeral)

lemma norm_of_int [simp]: "norm (of_int z::'a::real_normed_algebra_1) = ¦of_int z¦"
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"
by (metis abs_of_nat norm_of_real of_real_of_nat_eq)

lemma nonzero_norm_inverse: "a ≠ 0 ⟹ norm (inverse a) = inverse (norm a)"
for a :: "'a::real_normed_div_algebra"
by (metis inverse_unique norm_mult norm_one right_inverse)

lemma norm_inverse: "norm (inverse a) = inverse (norm a)"
for a :: "'a::{real_normed_div_algebra,division_ring}"
by (metis inverse_zero nonzero_norm_inverse norm_zero)

lemma nonzero_norm_divide: "b ≠ 0 ⟹ norm (a / b) = norm a / norm b"
for a b :: "'a::real_normed_field"
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)

lemma norm_divide: "norm (a / b) = norm a / norm b"
for a b :: "'a::{real_normed_field,field}"
by (simp add: divide_inverse norm_mult norm_inverse)

lemma norm_inverse_le_norm:
fixes x :: "'a::real_normed_div_algebra"
shows "r ≤ norm x ⟹ 0 < r ⟹ norm (inverse x) ≤ inverse r"
by (simp add: le_imp_inverse_le norm_inverse)

lemma norm_power_ineq: "norm (x ^ n) ≤ norm x ^ n"
for x :: "'a::real_normed_algebra_1"
proof (induct n)
case 0
show "norm (x ^ 0) ≤ norm x ^ 0" by simp
next
case (Suc n)
have "norm (x * x ^ n) ≤ norm x * norm (x ^ n)"
by (rule norm_mult_ineq)
also from Suc have "… ≤ norm x * norm x ^ n"
using norm_ge_zero by (rule mult_left_mono)
finally show "norm (x ^ Suc n) ≤ norm x ^ Suc n"
by simp
qed

lemma norm_power: "norm (x ^ n) = norm x ^ n"
for x :: "'a::real_normed_div_algebra"
by (induct n) (simp_all add: norm_mult)

lemma power_eq_imp_eq_norm:
fixes w :: "'a::real_normed_div_algebra"
assumes eq: "w ^ n = z ^ n" and "n > 0"
shows "norm w = norm z"
proof -
have "norm w ^ n = norm z ^ n"
by (metis (no_types) eq norm_power)
then show ?thesis
using assms by (force intro: power_eq_imp_eq_base)
qed

lemma power_eq_1_iff:
fixes w :: "'a::real_normed_div_algebra"
shows "w ^ n = 1 ⟹ norm w = 1 ∨ n = 0"
by (metis norm_one power_0_left power_eq_0_iff power_eq_imp_eq_norm power_one)

lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a"
for a b :: "'a::{real_normed_field,field}"
by (simp add: norm_mult)

lemma norm_mult_numeral2 [simp]: "norm (a * numeral w) = norm a * numeral w"
for a b :: "'a::{real_normed_field,field}"
by (simp add: norm_mult)

lemma norm_divide_numeral [simp]: "norm (a / numeral w) = norm a / numeral w"
for a b :: "'a::{real_normed_field,field}"
by (simp add: norm_divide)

lemma norm_of_real_diff [simp]:
"norm (of_real b - of_real a :: 'a::real_normed_algebra_1) ≤ ¦b - a¦"
by (metis norm_of_real of_real_diff order_refl)

text ‹Despite a superficial resemblance, ‹norm_eq_1› is not relevant.›
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: "norm x < norm (of_real (norm x) + 1 :: 'a)"
for x :: "'a::real_normed_algebra_1"
proof -
have "norm x < norm (of_real (norm x + 1) :: 'a)"
by (simp add: of_real_def)
then show ?thesis
by simp
qed

lemma prod_norm: "prod (λx. norm (f x)) A = norm (prod f A)"
for f :: "'a ⇒ 'b::{comm_semiring_1,real_normed_div_algebra}"
by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)

lemma norm_prod_le:
"norm (prod f A) ≤ (∏a∈A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))"
proof (induct A rule: infinite_finite_induct)
case empty
then show ?case by simp
next
case (insert a A)
then have "norm (prod f (insert a A)) ≤ norm (f a) * norm (prod f A)"
by (simp add: norm_mult_ineq)
also have "norm (prod f A) ≤ (∏a∈A. norm (f a))"
by (rule insert)
finally show ?case
by (simp add: insert mult_left_mono)
next
case infinite
then show ?case by simp
qed

lemma norm_prod_diff:
fixes z w :: "'i ⇒ 'a::{real_normed_algebra_1, comm_monoid_mult}"
shows "(⋀i. i ∈ I ⟹ norm (z i) ≤ 1) ⟹ (⋀i. i ∈ I ⟹ norm (w i) ≤ 1) ⟹
norm ((∏i∈I. z i) - (∏i∈I. w i)) ≤ (∑i∈I. norm (z i - w i))"
proof (induction I rule: infinite_finite_induct)
case empty
then show ?case by simp
next
case (insert i I)
note insert.hyps[simp]

have "norm ((∏i∈insert i I. z i) - (∏i∈insert i I. w i)) =
norm ((∏i∈I. z i) * (z i - w i) + ((∏i∈I. z i) - (∏i∈I. w i)) * w i)"
(is "_ = norm (?t1 + ?t2)")
by (auto simp: field_simps)
also have "… ≤ norm ?t1 + norm ?t2"
by (rule norm_triangle_ineq)
also have "norm ?t1 ≤ norm (∏i∈I. z i) * norm (z i - w i)"
by (rule norm_mult_ineq)
also have "… ≤ (∏i∈I. norm (z i)) * norm(z i - w i)"
by (rule mult_right_mono) (auto intro: norm_prod_le)
also have "(∏i∈I. norm (z i)) ≤ (∏i∈I. 1)"
by (intro prod_mono) (auto intro!: insert)
also have "norm ?t2 ≤ norm ((∏i∈I. z i) - (∏i∈I. w i)) * norm (w i)"
by (rule norm_mult_ineq)
also have "norm (w i) ≤ 1"
by (auto intro: insert)
also have "norm ((∏i∈I. z i) - (∏i∈I. w i)) ≤ (∑i∈I. norm (z i - w i))"
using insert by auto
finally show ?case
by (auto simp: ac_simps mult_right_mono mult_left_mono)
next
case infinite
then show ?case by simp
qed

lemma norm_power_diff:
fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
assumes "norm z ≤ 1" "norm w ≤ 1"
shows "norm (z^m - w^m) ≤ m * norm (z - w)"
proof -
have "norm (z^m - w^m) = norm ((∏ i < m. z) - (∏ i < m. w))"
by (simp add: prod_constant)
also have "… ≤ (∑i<m. norm (z - w))"
by (intro norm_prod_diff) (auto simp: assms)
also have "… = m * norm (z - w)"
by simp
finally show ?thesis .
qed

subsection ‹Metric spaces›

class metric_space = uniformity_dist + open_uniformity +
assumes dist_eq_0_iff [simp]: "dist x y = 0 ⟷ x = y"
and dist_triangle2: "dist x y ≤ dist x z + dist y z"
begin

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

lemma zero_le_dist [simp]: "0 ≤ dist x y"
using dist_triangle2 [of x x y] by simp

lemma zero_less_dist_iff: "0 < dist x y ⟷ x ≠ y"
by (simp add: less_le)

lemma dist_not_less_zero [simp]: "¬ dist x y < 0"
by (simp add: not_less)

lemma dist_le_zero_iff [simp]: "dist x y ≤ 0 ⟷ x = y"
by (simp add: le_less)

lemma dist_commute: "dist x y = dist y x"
proof (rule order_antisym)
show "dist x y ≤ dist y x"
using dist_triangle2 [of x y x] by simp
show "dist y x ≤ dist x y"
using dist_triangle2 [of y x y] by simp
qed

lemma dist_commute_lessI: "dist y x < e ⟹ dist x y < e"
by (simp add: dist_commute)

lemma dist_triangle: "dist x z ≤ dist x y + dist y z"
using dist_triangle2 [of x z y] by (simp add: dist_commute)

lemma dist_triangle3: "dist x y ≤ dist a x + dist a y"
using dist_triangle2 [of x y a] by (simp add: dist_commute)

lemma dist_pos_lt: "x ≠ y ⟹ 0 < dist x y"
by (simp add: zero_less_dist_iff)

lemma dist_nz: "x ≠ y ⟷ 0 < dist x y"
by (simp add: zero_less_dist_iff)

declare dist_nz [symmetric, simp]

lemma dist_triangle_le: "dist x z + dist y z ≤ e ⟹ dist x y ≤ e"
by (rule order_trans [OF dist_triangle2])

lemma dist_triangle_lt: "dist x z + dist y z < e ⟹ dist x y < e"
by (rule le_less_trans [OF dist_triangle2])

lemma dist_triangle_less_add: "dist x1 y < e1 ⟹ dist x2 y < e2 ⟹ dist x1 x2 < e1 + e2"
by (rule dist_triangle_lt [where z=y]) simp

lemma dist_triangle_half_l: "dist x1 y < e / 2 ⟹ dist x2 y < e / 2 ⟹ dist x1 x2 < e"
by (rule dist_triangle_lt [where z=y]) simp

lemma dist_triangle_half_r: "dist y x1 < e / 2 ⟹ dist y x2 < e / 2 ⟹ dist x1 x2 < e"
by (rule dist_triangle_half_l) (simp_all add: dist_commute)

lemma dist_triangle_third:
assumes "dist x1 x2 < e/3" "dist x2 x3 < e/3" "dist x3 x4 < e/3"
shows "dist x1 x4 < e"
proof -
have "dist x1 x3 < e/3 + e/3"
by (metis assms(1) assms(2) dist_commute dist_triangle_less_add)
then have "dist x1 x4 < (e/3 + e/3) + e/3"
by (metis assms(3) dist_commute dist_triangle_less_add)
then show ?thesis
by simp
qed

subclass uniform_space
proof
fix E x
assume "eventually E uniformity"
then obtain e where E: "0 < e" "⋀x y. dist x y < e ⟹ E (x, y)"
by (auto simp: eventually_uniformity_metric)
then show "E (x, x)" "∀⇩F (x, y) in uniformity. E (y, x)"
by (auto simp: eventually_uniformity_metric dist_commute)
show "∃D. eventually D uniformity ∧ (∀x y z. D (x, y) ⟶ D (y, z) ⟶ E (x, z))"
using E dist_triangle_half_l[where e=e]
unfolding eventually_uniformity_metric
by (intro exI[of _ "λ(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI)
(auto simp: dist_commute)
qed

lemma open_dist: "open S ⟷ (∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S)"
by (simp add: dist_commute open_uniformity eventually_uniformity_metric)

lemma open_ball: "open {y. dist x y < d}"
unfolding open_dist
proof (intro ballI)
fix y
assume *: "y ∈ {y. dist x y < d}"
then show "∃e>0. ∀z. dist z y < e ⟶ z ∈ {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 "∃A::nat ⇒ 'a set. (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))"
proof (safe intro!: exI[of _ "λn. {y. dist x y < inverse (Suc n)}"])
fix S
assume "open S" "x ∈ S"
then obtain e where e: "0 < e" and "{y. dist x y < e} ⊆ 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)} ⊆ {y. dist x y < e}"
by auto
ultimately show "∃i. {y. dist x y < inverse (Suc i)} ⊆ S"
by blast
qed (auto intro: open_ball)
qed

end

instance metric_space ⊆ t2_space
proof
fix x y :: "'a::metric_space"
assume xy: "x ≠ y"
let ?U = "{y'. dist x y' < dist x y / 2}"
let ?V = "{x'. dist y x' < dist x y / 2}"
have *: "d x z ≤ d x y + d y z ⟹ d y z = d z y ⟹ ¬ (d x y * 2 < d x z ∧ d z y * 2 < d x z)"
for d :: "'a ⇒ 'a ⇒ real" and x y z :: 'a
by arith
have "open ?U ∧ open ?V ∧ x ∈ ?U ∧ y ∈ ?V ∧ ?U ∩ ?V = {}"
using dist_pos_lt[OF xy] *[of dist, OF dist_triangle dist_commute]
using open_ball[of _ "dist x y / 2"] by auto
then show "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}"
by blast
qed

text ‹Every normed vector space is a metric space.›
instance real_normed_vector < metric_space
proof
fix x y z :: 'a
show "dist x y = 0 ⟷ x = y"
by (simp add: dist_norm)
show "dist x y ≤ dist x z + dist y z"
using norm_triangle_ineq4 [of "x - z" "y - z"] by (simp add: dist_norm)
qed

subsection ‹Class instances for real numbers›

instantiation real :: real_normed_field
begin

definition dist_real_def: "dist x y = ¦x - y¦"

definition uniformity_real_def [code del]:
"(uniformity :: (real × real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

definition open_real_def [code del]:
"open (U :: real set) ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)"

definition real_norm_def [simp]: "norm r = ¦r¦"

instance
by intro_classes (auto simp: abs_mult open_real_def dist_real_def sgn_real_def uniformity_real_def)

end

declare uniformity_Abort[where 'a=real, code]

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

declare [[code abort: "open :: real set ⇒ bool"]]

instance real :: linorder_topology
proof
show "(open :: real set ⇒ bool) = generate_topology (range lessThan ∪ range greaterThan)"
proof (rule ext, safe)
fix S :: "real set"
assume "open S"
then obtain f where "∀x∈S. 0 < f x ∧ (∀y. dist y x < f x ⟶ y ∈ S)"
unfolding open_dist bchoice_iff ..
then have *: "S = (⋃x∈S. {x - f x <..} ∩ {..< x + f x})"
by (fastforce simp: dist_real_def)
show "generate_topology (range lessThan ∪ 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 ∪ range greaterThan) S"
moreover have "⋀a::real. open {..<a}"
unfolding open_dist dist_real_def
proof clarify
fix x a :: real
assume "x < a"
then have "0 < a - x ∧ (∀y. ¦y - x¦ < a - x ⟶ y ∈ {..<a})" by auto
then show "∃e>0. ∀y. ¦y - x¦ < e ⟶ y ∈ {..<a}" ..
qed
moreover have "⋀a::real. open {a <..}"
unfolding open_dist dist_real_def
proof clarify
fix x a :: real
assume "a < x"
then have "0 < x - a ∧ (∀y. ¦y - x¦ < x - a ⟶ y ∈ {a<..})" by auto
then show "∃e>0. ∀y. ¦y - x¦ < e ⟶ y ∈ {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 ‹topological_space›.›
(@{const_name "open"}, SOME @{typ "'a::topological_space set ⇒ bool"})›

text ‹Only allow @{term "uniformity"} in class ‹uniform_space›.›
(@{const_name "uniformity"}, SOME @{typ "('a::uniformity × 'a) filter"})›

text ‹Only allow @{term dist} in class ‹metric_space›.›
(@{const_name dist}, SOME @{typ "'a::metric_space ⇒ 'a ⇒ real"})›

text ‹Only allow @{term norm} in class ‹real_normed_vector›.›
(@{const_name norm}, SOME @{typ "'a::real_normed_vector ⇒ real"})›

subsection ‹Sign function›

lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)"
for x :: "'a::real_normed_vector"
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 = 0 ⟷ x = 0"
for x :: "'a::real_normed_vector"
by (simp add: sgn_div_norm)

lemma sgn_minus: "sgn (- x) = - sgn x"
for x :: "'a::real_normed_vector"
by (simp add: sgn_div_norm)

lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)"
for x :: "'a::real_normed_vector"
by (simp add: sgn_div_norm ac_simps)

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: "sgn (x * y) = sgn x * sgn y"
for x y :: "'a::real_normed_div_algebra"
by (simp add: sgn_div_norm norm_mult mult.commute)

hide_fact (open) sgn_mult

lemma real_sgn_eq: "sgn x = x / ¦x¦"
for x :: real
by (simp add: sgn_div_norm divide_inverse)

lemma zero_le_sgn_iff [simp]: "0 ≤ sgn x ⟷ 0 ≤ x"
for x :: real
by (cases "0::real" x rule: linorder_cases) simp_all

lemma sgn_le_0_iff [simp]: "sgn x ≤ 0 ⟷ x ≤ 0"
for x :: real
by (cases "0::real" x rule: linorder_cases) simp_all

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

declare norm_conv_dist [symmetric, simp]

lemma dist_0_norm [simp]: "dist 0 x = norm x"
for x :: "'a::real_normed_vector"
by (simp add: dist_norm)

lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"
by (simp_all add: dist_norm)

lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int ¦m - n¦"
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 "… = of_int ¦m - n¦" 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 ¦int m - int n¦"
by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int)

subsection ‹Bounded Linear and Bilinear Operators›

lemma linearI: "linear f"
if "⋀b1 b2. f (b1 + b2) = f b1 + f b2"
"⋀r b. f (r *⇩R b) = r *⇩R f b"
using that
by unfold_locales (auto simp: algebra_simps)

lemma linear_iff:
"linear f ⟷ (∀x y. f (x + y) = f x + f y) ∧ (∀c x. f (c *⇩R x) = c *⇩R f x)"
(is "linear f ⟷ ?rhs")
proof
assume "linear f"
then interpret f: linear f .
show "?rhs" by (simp add: f.add f.scale)
next
assume "?rhs"
then show "linear f" by (intro linearI) auto
qed

lemmas linear_scaleR_left = linear_scale_left
lemmas linear_imp_scaleR = linear_imp_scale

corollary real_linearD:
fixes f :: "real ⇒ real"
assumes "linear f" obtains c where "f = ( *) c"
by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real)

lemma linear_times_of_real: "linear (λx. a * of_real x)"
by (auto intro!: linearI simp: distrib_left)
(metis mult_scaleR_right scaleR_conv_of_real)

locale bounded_linear = linear f for f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" +
assumes bounded: "∃K. ∀x. norm (f x) ≤ norm x * K"
begin

lemma pos_bounded: "∃K>0. ∀x. norm (f x) ≤ norm x * K"
proof -
obtain K where K: "⋀x. norm (f x) ≤ 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) ≤ norm x * K" using K .
also have "… ≤ norm x * max 1 K"
by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])
finally show "norm (f x) ≤ norm x * max 1 K" .
qed
qed

lemma nonneg_bounded: "∃K≥0. ∀x. norm (f x) ≤ norm x * K"
using pos_bounded by (auto intro: order_less_imp_le)

lemma linear: "linear f"
by (fact local.linear_axioms)

end

lemma bounded_linear_intro:
assumes "⋀x y. f (x + y) = f x + f y"
and "⋀r x. f (scaleR r x) = scaleR r (f x)"
and "⋀x. norm (f x) ≤ 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 ⇒ 'c::real_normed_vector"
(infixl "**" 70)
assumes add_left: "prod (a + a') b = prod a b + prod a' b"
and add_right: "prod a (b + b') = prod a b + prod a b'"
and scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
and scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
and bounded: "∃K. ∀a b. norm (prod a b) ≤ norm a * norm b * K"
begin

lemma pos_bounded: "∃K>0. ∀a b. norm (a ** b) ≤ norm a * norm b * K"
proof -
obtain K where "⋀a b. norm (a ** b) ≤ norm a * norm b * K"
using bounded by blast
then have "norm (a ** b) ≤ norm a * norm b * (max 1 K)" for a b
by (rule order.trans) (simp add: mult_left_mono)
then show ?thesis
by force
qed

lemma nonneg_bounded: "∃K≥0. ∀a b. norm (a ** b) ≤ norm a * norm b * K"
using pos_bounded by (auto intro: order_less_imp_le)

lemma additive_right: "additive (λb. prod a b)"

lemma additive_left: "additive (λa. prod a b)"

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 sum_left: "prod (sum g S) x = sum ((λi. prod (g i) x)) S"

lemma sum_right: "prod x (sum g S) = sum ((λi. (prod x (g i)))) S"

lemma bounded_linear_left: "bounded_linear (λa. a ** b)"
proof -
obtain K where "⋀a b. norm (a ** b) ≤ norm a * norm b * K"
using pos_bounded by blast
then show ?thesis
by (rule_tac K="norm b * K" in bounded_linear_intro) (auto simp: algebra_simps scaleR_left add_left)
qed

lemma bounded_linear_right: "bounded_linear (λb. a ** b)"
proof -
obtain K where "⋀a b. norm (a ** b) ≤ norm a * norm b * K"
using pos_bounded by blast
then show ?thesis
by (rule_tac K="norm a * K" in bounded_linear_intro) (auto simp: algebra_simps scaleR_right add_right)
qed

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)

lemma flip: "bounded_bilinear (λx y. y ** x)"
apply standard
by (metis bounded mult.commute)

lemma comp1:
assumes "bounded_linear g"
shows "bounded_bilinear (λx. ( **) (g x))"
proof unfold_locales
interpret g: bounded_linear g by fact
show "⋀a a' b. g (a + a') ** b = g a ** b + g a' ** b"
"⋀a b b'. g a ** (b + b') = g a ** b + g a ** b'"
"⋀r a b. g (r *⇩R a) ** b = r *⇩R (g a ** b)"
"⋀a r b. g a ** (r *⇩R b) = r *⇩R (g a ** b)"
from g.nonneg_bounded nonneg_bounded obtain K L
where nn: "0 ≤ K" "0 ≤ L"
and K: "⋀x. norm (g x) ≤ norm x * K"
and L: "⋀a b. norm (a ** b) ≤ norm a * norm b * L"
by auto
have "norm (g a ** b) ≤ norm a * K * norm b * L" for a b
by (auto intro!:  order_trans[OF K] order_trans[OF L] mult_mono simp: nn)
then show "∃K. ∀a b. norm (g a ** b) ≤ norm a * norm b * K"
by (auto intro!: exI[where x="K * L"] simp: ac_simps)
qed

lemma comp: "bounded_linear f ⟹ bounded_linear g ⟹ bounded_bilinear (λx y. f x ** g y)"
by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]])

end

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

lemma bounded_linear_zero[simp]: "bounded_linear (λx. 0)"
by standard (auto intro!: exI[of _ 1])

assumes "bounded_linear f"
and "bounded_linear g"
shows "bounded_linear (λx. 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: "norm (f x) ≤ norm x * Kf" for x
by blast
from g.bounded obtain Kg where Kg: "norm (g x) ≤ norm x * Kg" for x
by blast
show "∃K. ∀x. norm (f x + g x) ≤ 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

lemma bounded_linear_minus:
assumes "bounded_linear f"
shows "bounded_linear (λx. - f x)"
proof -
interpret f: bounded_linear f by fact
show ?thesis
by unfold_locales (simp_all add: f.add f.scaleR f.bounded)
qed

lemma bounded_linear_sub: "bounded_linear f ⟹ bounded_linear g ⟹ bounded_linear (λx. f x - g x)"
using bounded_linear_add[of f "λx. - g x"] bounded_linear_minus[of g]
by (auto simp: algebra_simps)

lemma bounded_linear_sum:
fixes f :: "'i ⇒ 'a::real_normed_vector ⇒ 'b::real_normed_vector"
shows "(⋀i. i ∈ I ⟹ bounded_linear (f i)) ⟹ bounded_linear (λx. ∑i∈I. f i x)"
by (induct I rule: infinite_finite_induct) (auto intro!: bounded_linear_add)

lemma bounded_linear_compose:
assumes "bounded_linear f"
and "bounded_linear g"
shows "bounded_linear (λx. f (g x))"
proof -
interpret f: bounded_linear f by fact
interpret g: bounded_linear g by fact
show ?thesis
proof unfold_locales
show "f (g (x + y)) = f (g x) + f (g y)" for x y
show "f (g (scaleR r x)) = scaleR r (f (g x))" for r x
by (simp only: f.scaleR g.scaleR)
from f.pos_bounded obtain Kf where f: "⋀x. norm (f x) ≤ norm x * Kf" and Kf: "0 < Kf"
by blast
from g.pos_bounded obtain Kg where g: "⋀x. norm (g x) ≤ norm x * Kg"
by blast
show "∃K. ∀x. norm (f (g x)) ≤ norm x * K"
proof (intro exI allI)
fix x
have "norm (f (g x)) ≤ norm (g x) * Kf"
using f .
also have "… ≤ (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)) ≤ norm x * (Kg * Kf)" .
qed
qed
qed

lemma bounded_bilinear_mult: "bounded_bilinear (( *) :: 'a ⇒ 'a ⇒ 'a::real_normed_algebra)"
apply (rule bounded_bilinear.intro)
apply (auto simp: algebra_simps)
apply (rule_tac x=1 in exI)
apply (simp add: norm_mult_ineq)
done

lemma bounded_linear_mult_left: "bounded_linear (λx::'a::real_normed_algebra. x * y)"
using bounded_bilinear_mult
by (rule bounded_bilinear.bounded_linear_left)

lemma bounded_linear_mult_right: "bounded_linear (λy::'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 (λx. x / y)"
for y :: "'a::real_normed_field"
unfolding divide_inverse by (rule bounded_linear_mult_left)

lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
apply (rule bounded_bilinear.intro)
apply (auto simp: algebra_simps)
apply (rule_tac x=1 in exI, simp)
done

lemma bounded_linear_scaleR_left: "bounded_linear (λr. scaleR r x)"
using bounded_bilinear_scaleR
by (rule bounded_bilinear.bounded_linear_left)

lemma bounded_linear_scaleR_right: "bounded_linear (λx. scaleR r x)"
using bounded_bilinear_scaleR
by (rule bounded_bilinear.bounded_linear_right)

lemmas bounded_linear_scaleR_const =
bounded_linear_scaleR_left[THEN bounded_linear_compose]

lemmas bounded_linear_const_scaleR =
bounded_linear_scaleR_right[THEN bounded_linear_compose]

lemma bounded_linear_of_real: "bounded_linear (λr. of_real r)"
unfolding of_real_def by (rule bounded_linear_scaleR_left)

lemma real_bounded_linear: "bounded_linear f ⟷ (∃c::real. f = (λx. x * c))"
for f :: "real ⇒ real"
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 ⊆ perfect_space
proof
show "¬ open {x}" for x :: 'a
apply (clarsimp simp: open_dist dist_norm)
apply (rule_tac x = "x + of_real (e/2)" in exI)
apply simp
done
qed

subsection ‹Filters and Limits on Metric Space›

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 ∈ S"
then obtain e where "{y. dist y x < e} ⊆ S" "0 < e"
by (auto simp: open_dist subset_eq)
then show "∃e∈{0<..}. principal {y. dist y x < e} ≤ 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 ⟷ (∀e>0. eventually (λx. dist (f x) l < e) F)"
unfolding nhds_metric filterlim_INF filterlim_principal by auto

lemma tendsto_dist_iff:
"((f ⤏ l) F) ⟷ (((λx. dist (f x) l) ⤏ 0) F)"
unfolding tendsto_iff by simp

lemma (in metric_space) tendstoI [intro?]:
"(⋀e. 0 < e ⟹ eventually (λx. dist (f x) l < e) F) ⟹ (f ⤏ l) F"
by (auto simp: tendsto_iff)

lemma (in metric_space) tendstoD: "(f ⤏ l) F ⟹ 0 < e ⟹ eventually (λx. dist (f x) l < e) F"
by (auto simp: tendsto_iff)

lemma (in metric_space) eventually_nhds_metric:
"eventually P (nhds a) ⟷ (∃d>0. ∀x. dist x a < d ⟶ 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: "eventually P (at a within S) ⟷ (∃d>0. ∀x∈S. x ≠ a ∧ dist x a < d ⟶ P x)"
for a :: "'a :: metric_space"
by (auto simp: eventually_at_filter eventually_nhds_metric)

lemma frequently_at: "frequently P (at a within S) ⟷ (∀d>0. ∃x∈S. x ≠ a ∧ dist x a < d ∧ P x)"
for a :: "'a :: metric_space"
unfolding frequently_def eventually_at by auto

lemma eventually_at_le: "eventually P (at a within S) ⟷ (∃d>0. ∀x∈S. x ≠ a ∧ dist x a ≤ d ⟶ P x)"
for a :: "'a::metric_space"
unfolding eventually_at_filter eventually_nhds_metric
apply safe
apply (rule_tac x="d / 2" in exI, auto)
done

lemma eventually_at_left_real: "a > (b :: real) ⟹ eventually (λx. x ∈ {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) ⟹ eventually (λx. x ∈ {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"
and le: "eventually (λx. dist (g x) b ≤ dist (f x) a) F"
shows "(g ⤏ b) F"
proof (rule tendstoI)
fix e :: real
assume "0 < e"
with f have "eventually (λx. dist (f x) a < e) F" by (rule tendstoD)
with le show "eventually (λx. 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"
apply (clarsimp simp: filterlim_at_top)
apply (rule_tac c="nat ⌈Z + 1⌉" in eventually_sequentiallyI, linarith)
done

lemma filterlim_nat_sequentially: "filterlim nat sequentially at_top"
proof -
have "∀⇩F x in at_top. Z ≤ nat x" for Z
by (auto intro!: eventually_at_top_linorderI[where c="int Z"])
then show ?thesis
unfolding filterlim_at_top ..
qed

lemma filterlim_floor_sequentially: "filterlim floor at_top at_top"
proof -
have "∀⇩F x in at_top. Z ≤ ⌊x⌋" for Z
by (auto simp: le_floor_iff intro!: eventually_at_top_linorderI[where c="of_int Z"])
then show ?thesis
unfolding filterlim_at_top ..
qed

lemma filterlim_sequentially_iff_filterlim_real:
"filterlim f sequentially F ⟷ filterlim (λx. real (f x)) at_top F"
apply (rule iffI)
subgoal using filterlim_compose filterlim_real_sequentially by blast
subgoal premises prems
proof -
have "filterlim (λx. nat (floor (real (f x)))) sequentially F"
by (intro filterlim_compose[OF filterlim_nat_sequentially]
filterlim_compose[OF filterlim_floor_sequentially] prems)
then show ?thesis by simp
qed
done

subsubsection ‹Limits of Sequences›

lemma lim_sequentially: "X ⇢ L ⟷ (∀r>0. ∃no. ∀n≥no. dist (X n) L < r)"
for L :: "'a::metric_space"
unfolding tendsto_iff eventually_sequentially ..

lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*)

lemma LIMSEQ_iff_nz: "X ⇢ L ⟷ (∀r>0. ∃no>0. ∀n≥no. dist (X n) L < r)"
for L :: "'a::metric_space"
unfolding lim_sequentially by (metis Suc_leD zero_less_Suc)

lemma metric_LIMSEQ_I: "(⋀r. 0 < r ⟹ ∃no. ∀n≥no. dist (X n) L < r) ⟹ X ⇢ L"
for L :: "'a::metric_space"
by (simp add: lim_sequentially)

lemma metric_LIMSEQ_D: "X ⇢ L ⟹ 0 < r ⟹ ∃no. ∀n≥no. dist (X n) L < r"
for L :: "'a::metric_space"
by (simp add: lim_sequentially)

lemma LIMSEQ_norm_0:
assumes  "⋀n::nat. norm (f n) < 1 / real (Suc n)"
shows "f ⇢ 0"
proof (rule metric_LIMSEQ_I)
fix ε :: "real"
assume "ε > 0"
then obtain N::nat where "ε > inverse N" "N > 0"
by (metis neq0_conv real_arch_inverse)
then have "norm (f n) < ε" if "n ≥ N" for n
proof -
have "1 / (Suc n) ≤ 1 / N"
using ‹0 < N› inverse_of_nat_le le_SucI that by blast
also have "… < ε"
by (metis (no_types) ‹inverse (real N) < ε› inverse_eq_divide)
finally show ?thesis
by (meson assms less_eq_real_def not_le order_trans)
qed
then show "∃no. ∀n≥no. dist (f n) 0 < ε"
by auto
qed

subsubsection ‹Limits of Functions›

lemma LIM_def: "f ─a→ L ⟷ (∀r > 0. ∃s > 0. ∀x. x ≠ a ∧ dist x a < s ⟶ dist (f x) L < r)"
for a :: "'a::metric_space" and L :: "'b::metric_space"
unfolding tendsto_iff eventually_at by simp

lemma metric_LIM_I:
"(⋀r. 0 < r ⟹ ∃s>0. ∀x. x ≠ a ∧ dist x a < s ⟶ dist (f x) L < r) ⟹ f ─a→ L"
for a :: "'a::metric_space" and L :: "'b::metric_space"
by (simp add: LIM_def)

lemma metric_LIM_D: "f ─a→ L ⟹ 0 < r ⟹ ∃s>0. ∀x. x ≠ a ∧ dist x a < s ⟶ dist (f x) L < r"
for a :: "'a::metric_space" and L :: "'b::metric_space"
by (simp add: LIM_def)

lemma metric_LIM_imp_LIM:
fixes l :: "'a::metric_space"
and m :: "'b::metric_space"
assumes f: "f ─a→ l"
and le: "⋀x. x ≠ a ⟹ dist (g x) m ≤ dist (f x) l"
shows "g ─a→ m"
by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp: eventually_at_topological le)

lemma metric_LIM_equal2:
fixes a :: "'a::metric_space"
assumes "g ─a→ l" "0 < R"
and "⋀x. x ≠ a ⟹ dist x a < R ⟹ f x = g x"
shows "f ─a→ l"
proof -
have "⋀S. ⟦open S; l ∈ S; ∀⇩F x in at a. g x ∈ S⟧ ⟹ ∀⇩F x in at a. f x ∈ S"
apply (clarsimp simp add: eventually_at)
apply (rule_tac x="min d R" in exI)
apply (auto simp: assms)
done
then show ?thesis
using assms by (simp add: tendsto_def)
qed

lemma metric_LIM_compose2:
fixes a :: "'a::metric_space"
assumes f: "f ─a→ b"
and g: "g ─b→ c"
and inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d ⟶ f x ≠ b"
shows "(λx. 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 ⇒ _"
assumes f [unfolded isCont_def]: "isCont f a"
and g: "g ─f a→ l"
and inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d ⟶ f x ≠ f a"
shows "(λx. g (f x)) ─a→ l"
by (rule metric_LIM_compose2 [OF f g inj])

subsection ‹Complete metric spaces›

subsection ‹Cauchy sequences›

lemma (in metric_space) Cauchy_def: "Cauchy X = (∀e>0. ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e)"
proof -
have *: "eventually P (INF M. principal {(X m, X n) | n m. m ≥ M ∧ n ≥ M}) ⟷
(∃M. ∀m≥M. ∀n≥M. P (X m, X n))" for P
apply (subst eventually_INF_base)
subgoal by simp
subgoal for a b
by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq)
subgoal by (auto simp: eventually_principal, blast)
done
have "Cauchy X ⟷ (INF M. principal {(X m, X n) | n m. m ≥ M ∧ n ≥ M}) ≤ uniformity"
unfolding Cauchy_uniform_iff le_filter_def * ..
also have "… = (∀e>0. ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e)"
unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal)
finally show ?thesis .
qed

lemma (in metric_space) Cauchy_altdef: "Cauchy f ⟷ (∀e>0. ∃M. ∀m≥M. ∀n>m. dist (f m) (f n) < e)"
(is "?lhs ⟷ ?rhs")
proof
assume ?rhs
show ?lhs
unfolding Cauchy_def
proof (intro allI impI)
fix e :: real assume e: "e > 0"
with ‹?rhs› obtain M where M: "m ≥ M ⟹ n > m ⟹ dist (f m) (f n) < e" for m n
by blast
have "dist (f m) (f n) < e" if "m ≥ M" "n ≥ 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)
then show "∃M. ∀m≥M. ∀n≥M. dist (f m) (f n) < e"
by blast
qed
next
assume ?lhs
show ?rhs
proof (intro allI impI)
fix e :: real
assume e: "e > 0"
with ‹Cauchy f› obtain M where "⋀m n. m ≥ M ⟹ n ≥ M ⟹ dist (f m) (f n) < e"
unfolding Cauchy_def by blast
then show "∃M. ∀m≥M. ∀n>m. dist (f m) (f n) < e"
by (intro exI[of _ M]) force
qed
qed

lemma (in metric_space) Cauchy_altdef2: "Cauchy s ⟷ (∀e>0. ∃N::nat. ∀n≥N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
proof
assume "Cauchy s"
then show ?rhs by (force simp: Cauchy_def)
next
assume ?rhs
{
fix e::real
assume "e>0"
with ‹?rhs› obtain N where N: "∀n≥N. dist (s n) (s N) < e/2"
by (erule_tac x="e/2" in allE) auto
{
fix n m
assume nm: "N ≤ m ∧ N ≤ n"
then have "dist (s m) (s n) < e" using N
using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
by blast
}
then have "∃N. ∀m n. N ≤ m ∧ N ≤ n ⟶ dist (s m) (s n) < e"
by blast
}
then have ?lhs
unfolding Cauchy_def by blast
then show ?lhs
by blast
qed

lemma (in metric_space) metric_CauchyI:
"(⋀e. 0 < e ⟹ ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e) ⟹ Cauchy X"
by (simp add: Cauchy_def)

lemma (in metric_space) CauchyI':
"(⋀e. 0 < e ⟹ ∃M. ∀m≥M. ∀n>m. dist (X m) (X n) < e) ⟹ Cauchy X"
unfolding Cauchy_altdef by blast

lemma (in metric_space) metric_CauchyD:
"Cauchy X ⟹ 0 < e ⟹ ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e"
by (simp add: Cauchy_def)

lemma (in metric_space) metric_Cauchy_iff2:
"Cauchy X = (∀j. (∃M. ∀m ≥ M. ∀n ≥ M. dist (X m) (X n) < inverse(real (Suc j))))"
apply (auto simp add: Cauchy_def)
by (metis less_trans of_nat_Suc reals_Archimedean)

lemma Cauchy_iff2: "Cauchy X ⟷ (∀j. (∃M. ∀m ≥ M. ∀n ≥ M. ¦X m - X n¦ < inverse (real (Suc j))))"
by (simp only: metric_Cauchy_iff2 dist_real_def)

lemma lim_1_over_n: "((λn. 1 / of_nat n) ⤏ (0::'a::real_normed_field)) sequentially"
proof (subst lim_sequentially, intro allI impI exI)
fix e :: real
assume e: "e > 0"
fix n :: nat
assume n: "n ≥ nat ⌈inverse e + 1⌉"
have "inverse e < of_nat (nat ⌈inverse e + 1⌉)" by linarith
also note n
finally show "dist (1 / of_nat n :: 'a) 0 < e"
using e by (simp add: divide_simps mult.commute norm_divide)
qed

lemma (in metric_space) complete_def:
shows "complete S = (∀f. (∀n. f n ∈ S) ∧ Cauchy f ⟶ (∃l∈S. f ⇢ l))"
unfolding complete_uniform
proof safe
fix f :: "nat ⇒ 'a"
assume f: "∀n. f n ∈ S" "Cauchy f"
and *: "∀F≤principal S. F ≠ bot ⟶ cauchy_filter F ⟶ (∃x∈S. F ≤ nhds x)"
then show "∃l∈S. f ⇢ l"
unfolding filterlim_def using f
by (intro *[rule_format])
(auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform)
next
fix F :: "'a filter"
assume "F ≤ principal S" "F ≠ bot" "cauchy_filter F"
assume seq: "∀f. (∀n. f n ∈ S) ∧ Cauchy f ⟶ (∃l∈S. f ⇢ l)"

from ‹F ≤ principal S› ‹cauchy_filter F›
have FF_le: "F ×⇩F F ≤ uniformity_on S"
by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono)

let ?P = "λP e. eventually P F ∧ (∀x. P x ⟶ x ∈ S) ∧ (∀x y. P x ⟶ P y ⟶ dist x y < e)"
have P: "∃P. ?P P ε" if "0 < ε" for ε :: real
proof -
from that have "eventually (λ(x, y). x ∈ S ∧ y ∈ S ∧ dist x y < ε) (uniformity_on S)"
by (auto simp: eventually_inf_principal eventually_uniformity_metric)
from filter_leD[OF FF_le this] show ?thesis
by (auto simp: eventually_prod_same)
qed

have "∃P. ∀n. ?P (P n) (1 / Suc n) ∧ P (Suc n) ≤ P n"
proof (rule dependent_nat_choice)
show "∃P. ?P P (1 / Suc 0)"
using P[of 1] by auto
next
fix P n assume "?P P (1/Suc n)"
moreover obtain Q where "?P Q (1 / Suc (Suc n))"
using P[of "1/Suc (Suc n)"] by auto
ultimately show "∃Q. ?P Q (1 / Suc (Suc n)) ∧ Q ≤ P"
by (intro exI[of _ "λx. P x ∧ Q x"]) (auto simp: eventually_conj_iff)
qed
then obtain P where P: "eventually (P n) F" "P n x ⟹ x ∈ S"
"P n x ⟹ P n y ⟹ dist x y < 1 / Suc n" "P (Suc n) ≤ P n"
for n x y
by metis
have "antimono P"
using P(4) unfolding decseq_Suc_iff le_fun_def by blast

obtain X where X: "P n (X n)" for n
using P(1)[THEN eventually_happens'[OF ‹F ≠ bot›]] by metis
have "Cauchy X"
unfolding metric_Cauchy_iff2 inverse_eq_divide
proof (intro exI allI impI)
fix j m n :: nat
assume "j ≤ m" "j ≤ n"
with ‹antimono P› X have "P j (X m)" "P j (X n)"
by (auto simp: antimono_def)
then show "dist (X m) (X n) < 1 / Suc j"
by (rule P)
qed
moreover have "∀n. X n ∈ S"
using P(2) X by auto
ultimately obtain x where "X ⇢ x" "x ∈ S"
using seq by blast

show "∃x∈S. F ≤ nhds x"
proof (rule bexI)
have "eventually (λy. dist y x < e) F" if "0 < e" for e :: real
proof -
from that have "(λn. 1 / Suc n :: real) ⇢ 0 ∧ 0 < e / 2"
by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n)
then have "∀⇩F n in sequentially. dist (X n) x < e / 2 ∧ 1 / Suc n < e / 2"
using ‹X ⇢ x›
unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff
by blast
then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2"
by (auto simp: eventually_sequentially dist_commute)
show ?thesis
using ‹eventually (P n) F›
proof eventually_elim
case (elim y)
then have "dist y (X n) < 1 / Suc n"
by (intro X P)
also have "… < e / 2" by fact
finally show "dist y x < e"
by (rule dist_triangle_half_l) fact
qed
qed
then show "F ≤ nhds x"
unfolding nhds_metric le_INF_iff le_principal by auto
qed fact
qed

text‹apparently unused›
lemma (in metric_space) totally_bounded_metric:
"totally_bounded S ⟷ (∀e>0. ∃k. finite k ∧ S ⊆ (⋃x∈k. {y. dist x y < e}))"
unfolding totally_bounded_def eventually_uniformity_metric imp_ex
apply (subst all_comm)
apply (intro arg_cong[where f=All] ext, safe)
subgoal for e
apply (erule allE[of _ "λ(x, y). dist x y < e"])
apply auto
done
subgoal for e P k
apply (intro exI[of _ k])
apply (force simp: subset_eq)
done
done

subsubsection ‹Cauchy Sequences are Convergent›

(* TODO: update to uniform_space *)
class complete_space = metric_space +
assumes Cauchy_convergent: "Cauchy X ⟹ convergent X"

lemma Cauchy_convergent_iff: "Cauchy X ⟷ convergent X"
for X :: "nat ⇒ 'a::complete_space"
by (blast intro: Cauchy_convergent convergent_Cauchy)

text ‹To prove that a Cauchy sequence converges, it suffices to show that a subsequence converges.›

lemma Cauchy_converges_subseq:
fixes u::"nat ⇒ 'a::metric_space"
assumes "Cauchy u"
"strict_mono r"
"(u ∘ r) ⇢ l"
shows "u ⇢ l"
proof -
have *: "eventually (λn. dist (u n) l < e) sequentially" if "e > 0" for e
proof -
have "e/2 > 0" using that by auto
then obtain N1 where N1: "⋀m n. m ≥ N1 ⟹ n ≥ N1 ⟹ dist (u m) (u n) < e/2"
using ‹Cauchy u› unfolding Cauchy_def by blast
obtain N2 where N2: "⋀n. n ≥ N2 ⟹ dist ((u ∘ r) n) l < e / 2"
using order_tendstoD(2)[OF iffD1[OF tendsto_dist_iff ‹(u ∘ r) ⇢ l›] ‹e/2 > 0›]
unfolding eventually_sequentially by auto
have "dist (u n) l < e" if "n ≥ max N1 N2" for n
proof -
have "dist (u n) l ≤ dist (u n) ((u ∘ r) n) + dist ((u ∘ r) n) l"
by (rule dist_triangle)
also have "… < e/2 + e/2"
using N1[of n "r n"] N2[of n] that unfolding comp_def
by (auto simp: less_imp_le) (meson assms(2) less_imp_le order.trans seq_suble)
finally show ?thesis by simp
qed
then show ?thesis unfolding eventually_sequentially by blast
qed
have "(λn. dist (u n) l) ⇢ 0"
apply (rule order_tendstoI)
using * by auto (meson eventually_sequentiallyI less_le_trans zero_le_dist)
then show ?thesis using tendsto_dist_iff by auto
qed

subsection ‹The set of real numbers is a complete metric space›

text ‹
Proof that Cauchy sequences converge based on the one from
🌐‹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. ∃N. ∀n≥N. r < X n}"}
›
lemma increasing_LIMSEQ:
fixes f :: "nat ⇒ real"
assumes inc: "⋀n. f n ≤ f (Suc n)"
and bdd: "⋀n. f n ≤ l"
and en: "⋀e. 0 < e ⟹ ∃n. l ≤ 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 ≤ 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 (λn. x < f n) sequentially"
by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
qed (use bdd in auto)

lemma real_Cauchy_convergent:
fixes X :: "nat ⇒ real"
assumes X: "Cauchy X"
shows "convergent X"
proof -
define S :: "real set" where "S = {x. ∃N. ∀n≥N. x < X n}"
then have mem_S: "⋀N x. ∀n≥N. x < X n ⟹ x ∈ S"
by auto

have bound_isUb: "y ≤ x" if N: "∀n≥N. X n < x" and "y ∈ S" for N and x y :: real
proof -
from that have "∃M. ∀n≥M. y < X n"
by (simp add: S_def)
then obtain M where "∀n≥M. y < X n" ..
then have "y < X (max M N)" by simp
also have "… < x" using N by simp
finally show ?thesis by (rule order_less_imp_le)
qed

obtain N where "∀m≥N. ∀n≥N. dist (X m) (X n) < 1"
using X[THEN metric_CauchyD, OF zero_less_one] by auto
then have N: "∀n≥N. dist (X n) (X N) < 1" by simp
have [simp]: "S ≠ {}"
proof (intro exI ex_in_conv[THEN iffD1])
from N have "∀n≥N. X N - 1 < X n"
by (simp add: abs_diff_less_iff dist_real_def)
then show "X N - 1 ∈ S" by (rule mem_S)
qed
have [simp]: "bdd_above S"
proof
from N have "∀n≥N. X n < X N + 1"
by (simp add: abs_diff_less_iff dist_real_def)
then show "⋀s. s ∈ S ⟹  s ≤ X N + 1"
by (rule bound_isUb)
qed
have "X ⇢ Sup S"
proof (rule metric_LIMSEQ_I)
fix r :: real
assume "0 < r"
then have r: "0 < r/2" by simp
obtain N where "∀n≥N. ∀m≥N. dist (X n) (X m) < r/2"
using metric_CauchyD [OF X r] by auto
then have "∀n≥N. dist (X n) (X N) < r/2" by simp
then have N: "∀n≥N. X N - r/2 < X n ∧ X n < X N + r/2"
by (simp only: dist_real_def abs_diff_less_iff)

from N have "∀n≥N. X N - r/2 < X n" by blast
then have "X N - r/2 ∈ S" by (rule mem_S)
then have 1: "X N - r/2 ≤ Sup S" by (simp add: cSup_upper)

from N have "∀n≥N. X n < X N + r/2" by blast
from bound_isUb[OF this]
have 2: "Sup S ≤ X N + r/2"
by (intro cSup_least) simp_all

show "∃N. ∀n≥N. dist (X n) (Sup S) < r"
proof (intro exI allI impI)
fix n
assume n: "N ≤ n"
from N n have "X n < X N + r/2" and "X N - r/2 < X n"
by simp_all
then show "dist (X n) (Sup S) < r" using 1 2
by (simp add: abs_diff_less_iff dist_real_def)
qed
qed
then show ?thesis by (auto simp: convergent_def)
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 ⇒ 'b::first_countable_topology"
assumes *: "⋀X. filterlim X at_top sequentially ⟹ (λn. f (X n)) ⇢ y"
shows "(f ⤏ y) at_top"
proof -
obtain A where A: "decseq A" "open (A n)" "y ∈ A n" "nhds y = (INF n. principal (A n))" for n
by (rule nhds_countable[of y]) (rule that)

have "∀m. ∃k. ∀x≥k. f x ∈ A m"
proof (rule ccontr)
assume "¬ (∀m. ∃k. ∀x≥k. f x ∈ A m)"
then obtain m where "⋀k. ∃x≥k. f x ∉ A m"
by auto
then have "∃X. ∀n. (f (X n) ∉ A m) ∧ max n (X n) + 1 ≤ X (Suc n)"
by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
then obtain X where X: "⋀n. f (X n) ∉ A m" "⋀n. max n (X n) + 1 ≤ X (Suc n)"
by auto
have "1 ≤ n ⟹ real n ≤ X n" for 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 "k m ≤ x ⟹ f x ∈ A m" for m x
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 ⇒ real"
assumes mono: "mono f"
and limseq: "(λn. 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: "N ≤ n ⟹ ¦f (real n) - y¦ < e" for n
by (auto simp: lim_sequentially dist_real_def)
have le: "f x ≤ y" for x :: real
proof -
obtain n where "x ≤ real_of_nat n"
using real_arch_simple[of x] ..
note monoD[OF mono this]
also have "f (real_of_nat n) ≤ y"
by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono])
finally show ?thesis .
qed
have "eventually (λx. real N ≤ x) at_top"
by (rule eventually_ge_at_top)
then show "eventually (λx. dist (f x) y < e) at_top"
proof eventually_elim
case (elim x)
with N[of N] le have "y - f (real N) < e" by auto
moreover note monoD[OF mono elim]
ultimately show "dist (f x) y < e"
using le[of x] by (auto simp: dist_real_def field_simps)
qed
qed

end
```