Theory Real_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  inverse r *R x"

end

class real_vector = scaleR + ab_group_add +
  assumes scaleR_add_right: "a *R (x + y) = a *R x + a *R y"
  and scaleR_add_left: "(a + b) *R x = a *R x + b *R x"
  and scaleR_scaleR: "a *R b *R x = (a * b) *R x"
  and scaleR_one: "1 *R x = x"

class real_algebra = real_vector + ring +
  assumes mult_scaleR_left [simp]: "a *R x * y = a *R (x * y)"
    and mult_scaleR_right [simp]: "x * a *R y = a *R (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
proof unfold_locales
  show "Vector_Spaces.linear (*R) (*R) = linear" "Vector_Spaces.linear (*) (*R) = linear"
    by (force simp: linear_def real_scaleR_def[abs_def])+
qed (use scaleR_add_right scaleR_add_left scaleR_scaleR scaleR_one in auto)

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
proof unfold_locales
  show "Vector_Spaces.linear (*) (*R) = linear"
  unfolding linear_def real_scaleR_def by auto
qed (auto simp: linear_def)

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

lemma [field_simps]:
  "c  0  a = b /R c  c *R a = b"
  "c  0  b /R c = a  b = c *R a"
  "c  0  a + b /R c = (c *R a + b) /R c"
  "c  0  a /R c + b = (a + c *R b) /R c"
  "c  0  a - b /R c = (c *R a - b) /R c"
  "c  0  a /R c - b = (a - c *R b) /R c"
  "c  0  - (a /R c) + b = (- a + c *R b) /R c"
  "c  0  - (a /R c) - b = (- a - c *R b) /R c"
  for a b :: "'a :: real_vector"
  by (auto simp add: scaleR_add_right scaleR_add_left scaleR_diff_right scaleR_diff_left)


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

lemma linear_scale_real:
  fixes r::real shows "linear f  f (r * b) = r * f b"
  using linear_scale by fastforce

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"

lemma [vector_add_divide_simps]:
  "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"
  by (simp_all add: divide_inverse_commute scaleR_add_right scaleR_diff_right)


lemma eq_vector_fraction_iff [vector_add_divide_simps]:
  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)

lemma vector_fraction_eq_iff [vector_add_divide_simps]:
  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)

lemma of_real_sum[simp]: "of_real (sum f s) = (xs. of_real (f x))"
  by (induct s rule: infinite_finite_induct) auto

lemma of_real_prod[simp]: "of_real (prod f s) = (xs. 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_power_int [simp]:
  "of_real (power_int x n) = power_int (of_real x :: 'a :: {real_div_algebra,division_ring}) n"
  by (auto simp: power_int_def)

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

lemma numeral_power_int_eq_of_real_cancel_iff [simp]:
  "power_int (numeral x) n = (of_real y :: 'a :: {real_div_algebra, division_ring}) 
     power_int (numeral x) n = y"
proof -
  have "power_int (numeral x) n = (of_real (power_int (numeral x) n) :: 'a)"
    by simp
  also have " = of_real y  power_int (numeral x) n = y"
    by (subst of_real_eq_iff) auto
  finally show ?thesis .
qed

lemma of_real_eq_numeral_power_int_cancel_iff [simp]:
  "(of_real y :: 'a :: {real_div_algebra, division_ring}) = power_int (numeral x) n 
     y = power_int (numeral x) n"
  by (subst (1 2) eq_commute) simp

lemma of_real_eq_of_real_power_int_cancel_iff [simp]:
  "power_int (of_real b :: 'a :: {real_div_algebra, division_ring}) w = of_real x 
     power_int b w = x"
  by (metis of_real_power_int of_real_eq_iff)

lemma of_real_in_Ints_iff [simp]: "of_real x    x  "
proof safe
  fix x assume "(of_real x :: 'a)  "
  then obtain n where "(of_real x :: 'a) = of_int n"
    by (auto simp: Ints_def)
  also have "of_int n = of_real (real_of_int n)"
    by simp
  finally have "x = real_of_int n"
    by (subst (asm) of_real_eq_iff)
  thus "x  "
    by auto
qed (auto simp: Ints_def)

lemma Ints_of_real [intro]: "x    of_real x  "
  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_compose)
  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"  ("")
  where " = range of_real"

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

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

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

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

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

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

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

lemma Reals_minus_iff [simp]: "- a    a  "
  using Reals_minus by fastforce

lemma Reals_diff [simp]: "a    b    a - b  "
  by (metis Reals_add Reals_minus_iff add_uminus_conv_diff)

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

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

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

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

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

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

lemma Reals_power [simp]: "a    a ^ n  "
  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  "
  obtains (of_real) r where "q = of_real r"
  unfolding Reals_def
proof -
  from q   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  )  sum f s  "
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  )  prod f s  "
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. 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 order_trans scaleR_left_mono 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_divideR_eq [field_simps]:
  "a  b /R c  c *R a  b" (is "?P  ?Q") if "0 < c"
proof
  assume ?P
  with scaleR_left_mono that have "c *R a  c *R (b /R c)"
    by simp
  with that show ?Q
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
next
  assume ?Q
  with scaleR_left_mono that have "c *R a /R c  b /R c"
    by simp
  with that show ?P
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
qed

lemma pos_less_divideR_eq [field_simps]:
  "a < b /R c  c *R a < b" if "c > 0"
  using that pos_le_divideR_eq [of c a b]
  by (auto simp add: le_less scaleR_scaleR scaleR_one)

lemma pos_divideR_le_eq [field_simps]:
  "b /R c  a  b  c *R a" if "c > 0"
  using that pos_le_divideR_eq [of "inverse c" b a] by simp

lemma pos_divideR_less_eq [field_simps]:
  "b /R c < a  b < c *R a" if "c > 0"
  using that pos_less_divideR_eq [of "inverse c" b a] by simp

lemma pos_le_minus_divideR_eq [field_simps]:
  "a  - (b /R c)  c *R a  - b" if "c > 0"
  using that by (metis add_minus_cancel diff_0 left_minus minus_minus neg_le_iff_le
    scaleR_add_right uminus_add_conv_diff pos_le_divideR_eq)
  
lemma pos_less_minus_divideR_eq [field_simps]:
  "a < - (b /R c)  c *R a < - b" if "c > 0"
  using that by (metis le_less less_le_not_le pos_divideR_le_eq
    pos_divideR_less_eq pos_le_minus_divideR_eq)

lemma pos_minus_divideR_le_eq [field_simps]:
  "- (b /R c)  a  - b  c *R a" if "c > 0"
  using that by (metis pos_divideR_le_eq pos_le_minus_divideR_eq that
    inverse_positive_iff_positive le_imp_neg_le minus_minus)

lemma pos_minus_divideR_less_eq [field_simps]:
  "- (b /R c) < a  - b < c *R a" if "c > 0"
  using that by (simp add: less_le_not_le pos_le_minus_divideR_eq pos_minus_divideR_le_eq) 

lemma scaleR_image_atLeastAtMost: "c > 0  scaleR c ` {x..y} = {c *R x..c *R y}"
  apply (auto intro!: scaleR_left_mono simp: image_iff Bex_def)
  using pos_divideR_le_eq [of c] pos_le_divideR_eq [of c]
  apply (meson local.order_eq_iff) 
  done

end

lemma neg_le_divideR_eq [field_simps]:
  "a  b /R c  b  c *R a" (is "?P  ?Q") if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that pos_le_divideR_eq [of "- c" a "- b"] by simp

lemma neg_less_divideR_eq [field_simps]:
  "a < b /R c  b < c *R a" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that neg_le_divideR_eq [of c a b] by (auto simp add: le_less)

lemma neg_divideR_le_eq [field_simps]:
  "b /R c  a  c *R a  b" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that pos_divideR_le_eq [of "- c" "- b" a] by simp

lemma neg_divideR_less_eq [field_simps]:
  "b /R c < a  c *R a < b" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that neg_divideR_le_eq [of c b a] by (auto simp add: le_less)

lemma neg_le_minus_divideR_eq [field_simps]:
  "a  - (b /R c)  - b  c *R a" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that pos_le_minus_divideR_eq [of "- c" a "- b"] by (simp add: minus_le_iff)
  
lemma neg_less_minus_divideR_eq [field_simps]:
  "a < - (b /R c)  - b < c *R a" if "c < 0"
   for a b :: "'a :: ordered_real_vector"
proof -
  have *: "- b = c *R a  b = - (c *R a)"
    by (metis add.inverse_inverse)
  from that neg_le_minus_divideR_eq [of c a b]
  show ?thesis by (auto simp add: le_less *)
qed

lemma neg_minus_divideR_le_eq [field_simps]:
  "- (b /R c)  a  c *R a  - b" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that pos_minus_divideR_le_eq [of "- c" "- b" a] by (simp add: le_minus_iff) 

lemma neg_minus_divideR_less_eq [field_simps]:
  "- (b /R c) < a  c *R a < - b" if "c < 0"
    for a b :: "'a :: ordered_real_vector"
  using that by (simp add: less_le_not_le neg_le_minus_divideR_eq neg_minus_divideR_le_eq)

lemma [field_split_simps]:
  "a = b /R c  (if c = 0 then a = 0 else c *R a = b)"
  "b /R c = a  (if c = 0 then a = 0 else b = c *R a)"
  "a + b /R c = (if c = 0 then a else (c *R a + b) /R c)"
  "a /R c + b = (if c = 0 then b else (a + c *R b) /R c)"
  "a - b /R c = (if c = 0 then a else (c *R a - b) /R c)"
  "a /R c - b = (if c = 0 then - b else (a - c *R b) /R c)"
  "- (a /R c) + b = (if c = 0 then b else (- a + c *R b) /R c)"
  "- (a /R c) - b = (if c = 0 then - b else (- a - c *R b) /R c)"
    for a b :: "'a :: real_vector"
  by (auto simp add: field_simps)

lemma [field_split_simps]:
  "0 < c  a  b /R c  (if c > 0 then c *R a  b else if c < 0 then b  c *R a else a  0)"
  "0 < c  a < b /R c  (if c > 0 then c *R a < b else if c < 0 then b < c *R a else a < 0)"
  "0 < c  b /R c  a  (if c > 0 then b  c *R a else if c < 0 then c *R a  b else a  0)"
  "0 < c  b /R c < a  (if c > 0 then b < c *R a else if c < 0 then c *R a < b else a > 0)"
  "0 < c  a  - (b /R c)  (if c > 0 then c *R a  - b else if c < 0 then - b  c *R a else a  0)"
  "0 < c  a < - (b /R c)  (if c > 0 then c *R a < - b else if c < 0 then - b < c *R a else a < 0)"
  "0 < c  - (b /R c)  a  (if c > 0 then - b  c *R a else if c < 0 then c *R a  - b else a  0)"
  "0 < c  - (b /R c) < a  (if c > 0 then - b < c *R a else if c < 0 then c *R a < - b else a > 0)"
  for a b :: "'a :: ordered_real_vector"
  by (clarsimp intro!: field_simps)+

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

lemma bdd_below_norm_image: "bdd_below (norm ` A)"
  by (meson bdd_belowI2 norm_ge_zero)

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

context real_normed_vector begin

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

lemma zero_less_norm_iff [simp]: "norm x > 0  x  0"
  by (simp add: order_less_le)

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

lemma norm_le_zero_iff [simp]: "norm x  0  x = 0"
  by (simp add: order_le_less)

lemma norm_minus_cancel [simp]: "norm (- x) = norm x"
proof -
  have "- 1 *R x = - (1 *R x)"
    unfolding add_eq_0_iff2[symmetric] scaleR_add_left[symmetric]
    using norm_eq_zero
    by fastforce
  then have "norm (- x) = norm (scaleR (- 1) x)"
    by (simp only: 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)"
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"
  by (simp add: dist_norm)

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

lemma norm_uminus_minus: "norm (- x - y) = 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)"
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)"
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"
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: "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)"
proof -
  have "norm a - norm (- b)  norm (a - - b)"
    by (rule norm_triangle_ineq2)
  then show ?thesis by simp
qed

lemma norm_triangle_sub: "norm x  norm y + norm (x - y)"
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)

lemma norm_triangle_le: "norm x + norm y  e  norm (x + y)  e"
  by (rule norm_triangle_ineq [THEN order_trans])

lemma norm_triangle_lt: "norm x + norm y < e  norm (x + y) < e"
  by (rule norm_triangle_ineq [THEN le_less_trans])

lemma norm_add_leD: "norm (a + b)  c  norm b  norm a + c"
  by (metis ab_semigroup_add_class.add.commute 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)"
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: "norm (x - z)  e1 + e2"
  if "norm (x - y)  e1"  "norm (y - z)  e2"
proof -
  have "norm (x - (y + z - y))  norm (x - y) + norm (y - z)"
    using norm_diff_triangle_ineq that diff_diff_eq2 by presburger
  with that show ?thesis by simp
qed

lemma norm_diff_triangle_less: "norm (x - z) < e1 + e2"
  if "norm (x - y) < e1"  "norm (y - z) < e2"
proof -
  have "norm (x - z)  norm (x - y) + norm (y - z)"
    by (metis norm_diff_triangle_ineq add_diff_cancel_left' diff_diff_eq2)
  with that show ?thesis by auto
qed

lemma norm_triangle_mono:
  "norm a  r  norm b  s  norm (a + b)  r + s"
  by (metis (mono_tags) add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)

lemma norm_sum: "norm (sum f A)  (iA. norm (f i))"
  for f::"'b  'a"
  by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)

lemma sum_norm_le: "norm (sum f S)  sum g S"
  if "x. x  S  norm (f x)  g x"
  for f::"'b  'a"
  by (rule order_trans [OF norm_sum sum_mono]) (simp add: that)

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

lemma sum_norm_bound:
  "norm (sum f S)  of_nat (card S)*K"
  if "x. x  S  norm (f x)  K"
  for f :: "'b  'a"
  using sum_norm_le[OF that] sum_constant[symmetric]
  by simp

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

end

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_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)

lemma norm_of_real_addn [simp]:
  "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 norm_power_int: "norm (power_int x n) = power_int (norm x) n"
  for x :: "'a::real_normed_div_algebra"
  by (cases n rule: int_cases4) (auto simp: norm_power power_int_minus norm_inverse)

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 "x2 = 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)  (aA. 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)  (aA. 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 ((iI. z i) - (iI. w i))  (iI. 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 ((iinsert i I. z i) - (iinsert i I. w i)) =
    norm ((iI. z i) * (z i - w i) + ((iI. z i) - (iI. 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 (iI. z i) * norm (z i - w i)"
    by (rule norm_mult_ineq)
  also have "  (iI. norm (z i)) * norm(z i - w i)"
    by (rule mult_right_mono) (auto intro: norm_prod_le)
  also have "(iI. norm (z i))  (iI. 1)"
    by (intro prod_mono) (auto intro!: insert)
  also have "norm ?t2  norm ((iI. z i) - (iI. w i)) * norm (w i)"
    by (rule norm_mult_ineq)
  also have "norm (w i)  1"
    by (auto intro: insert)
  also have "norm ((iI. z i) - (iI. w i))  (iI. 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
  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 abs_dist_diff_le: "¦dist a b - dist b c¦  dist a c"
  using dist_triangle3[of b c a] dist_triangle2[of a b c] by simp

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  (xS. 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)  (xU. 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 "xS. 0 < f x  (y. dist y x < f x  y  S)"
      unfolding open_dist bchoice_iff ..
    then have *: "(xS. {x - f x <..}  {..< x + f x}) = S"