Quelle  Product_Vector.thy

(*  Title:      HOL/Analysis/Product_Vector.thy
  Author: Brian Huffman
  Dominique Unruh, University of Tartu
*)

section ‹Cartesian Products as Vector Spaces›

theory Product_Vector
  imports
    Complex_Main
    "HOL-Library.Product_Plus"
begin

lemma Times_eq_image_sum:
  fixes S :: "'a :: comm_monoid_add set" and T :: "'b :: comm_monoid_add set"
  shows "S × T = {u + v |u v. u ∈ (λx. (x, 0)) ` S ∧ v ∈ Pair 0 ` T}"
  by force


subsection ‹Product is a Module›

locale module_prod = module_pair begin

definition scale :: "'a ==> 'b × 'c ==> 'b × 'c"
  where "scale a v = (s1 a (fst v), s2 a (snd v))"

lemma🍋‹tag important› scale_prod: "scale x (a, b) = (s1 x a, s2 x b)"
  by (auto simp: scale_def)

sublocale🍋‹tag important› p: module scale
proof qed (simp_all add: scale_def
  m1.scale_left_distrib m1.scale_right_distrib m2.scale_left_distrib m2.scale_right_distrib)

lemma subspace_Times: "m1.subspace A ==> m2.subspace B ==> p.subspace (A × B)"
  unfolding m1.subspace_def m2.subspace_def p.subspace_def
  by (auto simp: zero_prod_def scale_def)

lemma module_hom_fst: "module_hom scale s1 fst"
  by unfold_locales (auto simp: scale_def)

lemma module_hom_snd: "module_hom scale s2 snd"
  by unfold_locales (auto simp: scale_def)

end

locale vector_space_prod = vector_space_pair begin

sublocale module_prod s1 s2
  rewrites "module_hom = Vector_Spaces.linear"
  by unfold_locales (fact module_hom_eq_linear)

sublocale p: vector_space scale by unfold_locales (auto simp: algebra_simps)

lemmas linear_fst = module_hom_fst
  and linear_snd = module_hom_snd

end


subsection ‹Product is a Real Vector Space›

instantiation prod :: (real_vector, real_vector) real_vector
begin

definition scaleR_prod_def:
  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"

lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
  unfolding scaleR_prod_def by simp

lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
  unfolding scaleR_prod_def by simp

proposition scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
  unfolding scaleR_prod_def by simp

instance
proof
  fix a b :: real and x y :: "'a × 'b"
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
    by (simp add: prod_eq_iff scaleR_right_distrib)
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
    by (simp add: prod_eq_iff scaleR_left_distrib)
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
    by (simp add: prod_eq_iff)
  show "scaleR 1 x = x"
    by (simp add: prod_eq_iff)
qed

end

lemma module_prod_scale_eq_scaleR: "module_prod.scale (*🪙R) (*🪙R) = scaleR"
  using module_pair_axioms module_prod.scale_def module_prod_def by fastforce

interpretation real_vector?: vector_space_prod "scaleR::_==>_==>'a::real_vector" "scaleR::_==>_==>'b::real_vector"
  rewrites "scale = ((*🪙R)::_==>_==>('a × 'b))"
    and "module.dependent (*🪙R) = dependent"
    and "module.representation (*🪙R) = representation"
    and "module.subspace (*🪙R) = subspace"
    and "module.span (*🪙R) = span"
    and "vector_space.extend_basis (*🪙R) = extend_basis"
    and "vector_space.dim (*🪙R) = dim"
    and "Vector_Spaces.linear (*🪙R) (*🪙R) = linear"
  subgoal by unfold_locales
  subgoal by (fact module_prod_scale_eq_scaleR)
  unfolding dependent_raw_def representation_raw_def subspace_raw_def span_raw_def
    extend_basis_raw_def dim_raw_def linear_def
  by (rule refl)+

subsection ‹Product is a Metric Space›

(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)

instantiation🍋‹tag unimportant› prod :: (metric_space, metric_space) dist
begin

definition dist_prod_def[code del]:
  "dist x y = sqrt ((dist (fst x) (fst y))🪙2 + (dist (snd x) (snd y))🪙2)"

instance ..
end

instantiation🍋‹tag unimportant› prod :: (uniformity, uniformity) uniformity begin

definition [code del]: ‹(uniformity :: (('a × 'b) × ('a × 'b)) filter) =
  filtermap (λ((x1,x2),(y1,y2)). ((x1,y1),(x2,y2))) (uniformity ×🪙F uniformity)›

instance..
end

subsubsection ‹Uniform spaces›

instantiation🍋‹tag unimportant› prod :: (uniform_space, uniform_space) uniform_space 
begin
instance 
proof standard
  fix U :: ‹('a × 'b) set›
  show ‹open U ⟷ (∀x∈U. ∀🪙F (x', y) in uniformity. x' = x ⟶ y ∈ U)›
  proof (intro iffI ballI)
    fix x assume ‹open U› and ‹x ∈ U›
    then obtain A B where ‹open A› ‹open B› ‹x ∈ A×B› ‹A×B ⊆ U›
      by (metis open_prod_elim)
    define UA where ‹UA = (λ(x'::'a,y). x' = fst x ⟶ y ∈ A)›
    from ‹open A› ‹x ∈ A×B›
    have ‹eventually UA uniformity›
      unfolding open_uniformity UA_def by auto
    define UB where ‹UB = (λ(x'::'b,y). x' = snd x ⟶ y ∈ B)›
    from ‹open A› ‹open B› ‹x ∈ A×B›
    have ‹eventually UA uniformity› ‹eventually UB uniformity›
      unfolding open_uniformity UA_def UB_def by auto
    then have ‹∀🪙F ((x'1, y1), (x'2, y2)) in uniformity ×🪙F uniformity. (x'1,x'2) = x ⟶ (y1,y2) ∈ U›
      apply (auto intro!: exI[of _ UA] exI[of _ UB] simp add: eventually_prod_filter)
      using ‹A×B ⊆ U› by (auto simp: UA_def UB_def)
    then show ‹∀🪙F (x', y) in uniformity. x' = x ⟶ y ∈ U›
      by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold)
  next
    assume asm: ‹∀x∈U. ∀🪙F (x', y) in uniformity. x' = x ⟶ y ∈ U›
    show ‹open U›
    proof (unfold open_prod_def, intro ballI)
      fix x assume ‹x ∈ U›
      with asm have ‹∀🪙F (x', y) in uniformity. x' = x ⟶ y ∈ U›
        by auto
      then have ‹∀🪙F ((x'1, y1), (x'2, y2)) in uniformity ×🪙F uniformity. (x'1,x'2) = x ⟶ (y1,y2) ∈ U›
        by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold)
      then obtain UA UB where ‹eventually UA uniformity› and ‹eventually UB uniformity›
           and UA_UB_U: ‹UA (a1, a2) ==> UB (b1, b2) ==> (a1, b1) = x ==> (a2, b2) ∈ U› for a1 a2 b1 b2
         by (force simp: case_prod_beta eventually_prod_filter)
      have ‹eventually (λa. UA (fst x, a)) (nhds (fst x))›
        using ‹eventually UA uniformity› eventually_mono eventually_nhds_uniformity by fastforce
      then obtain A where ‹open A› and A_UA: ‹A ⊆ {a. UA (fst x, a)}› and ‹fst x ∈ A›
        by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI)
      have ‹eventually (λb. UB (snd x, b)) (nhds (snd x))›
        using ‹eventually UB uniformity› eventually_mono eventually_nhds_uniformity by fastforce
      then obtain B where ‹open B› and B_UB: ‹B ⊆ {b. UB (snd x, b)}› and ‹snd x ∈ B›
        by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI)
      have ‹x ∈ A × B›
        by (simp add: ‹fst x ∈ A› ‹snd x ∈ B› mem_Times_iff)
      have ‹A × B ⊆ U›
        using A_UA B_UB UA_UB_U by fastforce
      show ‹∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ U›
        using ‹A × B ⊆ U› ‹open A› ‹open B› ‹x ∈ A × B› by auto
    qed
  qed
next
  show ‹eventually E uniformity ==> E (x, x)› for E and x :: ‹'a × 'b›
    apply (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter)
    by (metis surj_pair uniformity_refl)
next
  show ‹eventually E uniformity ==> ∀🪙F (x::'a×'b, y) in uniformity. E (y, x)› for E
    unfolding uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter
    apply clarify
    subgoal for Pf Pg
      apply (rule_tac x=‹λ(x,y). Pf (y,x)› in exI)
      apply (rule_tac x=‹λ(x,y). Pg (y,x)› in exI)
      by (auto simp add: uniformity_sym)
    done
next
  show ‹∃D. eventually D uniformity ∧ (∀x y z. D (x::'a×'b, y) ⟶ D (y, z) ⟶ E (x, z))›
    if ‹eventually E uniformity› for E
  proof -
    from that
    obtain EA EB where ‹eventually EA uniformity› and ‹eventually EB uniformity›
               and EA_EB_E: ‹EA (a1, a2) ==> EB (b1, b2) ==> E ((a1, b1), (a2, b2))› for a1 a2 b1 b2
      by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter)
    obtain DA where ‹eventually DA uniformity› and DA_EA: ‹DA (x,y) ==> DA (y,z) ==> EA (x,z)› for x y z
      using ‹eventually EA uniformity› uniformity_transE by blast
    obtain DB where ‹eventually DB uniformity› and DB_EB: ‹DB (x,y) ==> DB (y,z) ==> EB (x,z)› for x y z
      using ‹eventually EB uniformity› uniformity_transE by blast
    define D where ‹D = (λ((a1,b1),(a2,b2)). DA (a1,a2) ∧ DB (b1,b2))›
    have ‹eventually D uniformity›
      using ‹eventually DA uniformity› ‹eventually DB uniformity›
      by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter D_def)
    moreover have ‹D ((a1, b1), (a2, b2)) ==> D ((a2, b2), (a3, b3)) ==> E ((a1, b1), (a3, b3))› for a1 b1 a2 b2 a3 b3
      using DA_EA DB_EB D_def EA_EB_E by blast
    ultimately show ?thesis
      by auto
  qed
qed
end


lemma (in uniform_space) nhds_eq_comap_uniformity: "nhds x = filtercomap (λy. (x, y)) uniformity"
proof -
  have *: "eventually P (filtercomap (λy. (x, y)) F) ⟷
           eventually (λz. fst z = x ⟶ P (snd z)) F" for P :: "'a ==> bool" and F
    unfolding eventually_filtercomap
    by (smt (verit, best) eventually_mono split_pairs2)  
  thus ?thesis
    unfolding filter_eq_iff *
    by (auto simp: eventually_nhds_uniformity case_prod_unfold)
qed

lemma uniformity_of_uniform_continuous_invariant:
  fixes f :: "'a :: uniform_space ==> 'a ==> 'a"
  assumes "filterlim (λ((a,b),(c,d)). (f a c, f b d)) uniformity (uniformity ×🪙F uniformity)"
  assumes "eventually P uniformity"
  obtains Q where "eventually Q uniformity" "∧a b c. Q (a, b) ==> P (f a c, f b c)"
  using eventually_compose_filterlim[OF assms(2,1)] uniformity_refl
    by (fastforce simp: case_prod_unfold eventually_filtercomap eventually_prod_same)

class uniform_topological_monoid_add = topological_monoid_add + uniform_space +
  assumes uniformly_continuous_add':
    "filterlim (λ((a,b), (c,d)). (a + c, b + d)) uniformity (uniformity ×🪙F uniformity)"

lemma uniformly_continuous_add:
  "uniformly_continuous_on UNIV (λ(x :: 'a :: uniform_topological_monoid_add,y). x + y)"
  using uniformly_continuous_add'[where ?'a = 'a]
  by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)

lemma filterlim_fst: "filterlim fst F (F ×🪙F G)"
  by (simp add: filterlim_def filtermap_fst_prod_filter)

lemma filterlim_snd: "filterlim snd G (F ×🪙F G)"
  by (simp add: filterlim_def filtermap_snd_prod_filter)

class uniform_topological_group_add = topological_group_add + uniform_topological_monoid_add +
  assumes uniformly_continuous_uminus': "filterlim (λ(a, b). (-a, -b)) uniformity uniformity"
begin

lemma uniformly_continuous_minus':
  "filterlim (λ((a,b), (c,d)). (a - c, b - d)) uniformity (uniformity ×🪙F uniformity)"
proof -
  have "filterlim ((λ((a,b), (c,d)). (a + c, b + d)) ∘ (λ((a,b), (c,d)). ((a, b), (-c, -d))))
          uniformity (uniformity ×🪙F uniformity)"
    unfolding o_def using uniformly_continuous_uminus'
    by (intro filterlim_compose[OF uniformly_continuous_add'])
       (auto simp: case_prod_unfold intro!: filterlim_Pair
          filterlim_fst filterlim_compose[OF _ filterlim_snd])
  thus ?thesis
    by (simp add: o_def case_prod_unfold)
qed

end

lemma uniformly_continuous_uminus:
  "uniformly_continuous_on UNIV (λx :: 'a :: uniform_topological_group_add. -x)"
  using uniformly_continuous_uminus'[where ?'a = 'a]
  by (simp add: uniformly_continuous_on_uniformity)

lemma uniformly_continuous_minus:
  "uniformly_continuous_on UNIV (λ(x :: 'a :: uniform_topological_group_add,y). x - y)"
  using uniformly_continuous_minus'[where ?'a = 'a]
  by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)



lemma real_normed_vector_is_uniform_topological_group_add [Pure.intro]:
  "OFCLASS('a :: real_normed_vector, uniform_topological_group_add_class)"
proof
  show "filterlim (λ((a::'a,b), (c,d)). (a + c, b + d)) uniformity (uniformity ×🪙F uniformity)"
    unfolding filterlim_def le_filter_def eventually_filtermap case_prod_unfold
  proof safe
    fix P :: "'a × 'a ==> bool"
    assume "eventually P uniformity"
    then obtain ε where ε: "ε > 0" "∧x y. dist x y < ε ==> P (x, y)"
      by (auto simp: eventually_uniformity_metric)
    define Q where "Q = (λ(x::'a,y). dist x y < ε / 2)"
    have Q: "eventually Q uniformity"
      unfolding eventually_uniformity_metric Q_def using ‹ε > 0›
      by (meson case_prodI divide_pos_pos zero_less_numeral)
    have "P (a + c, b + d)" if "Q (a, b)" "Q (c, d)" for a b c d
    proof -
      have "dist (a + c) (b + d) ≤ dist a b + dist c d"
        by (simp add: dist_norm norm_diff_triangle_ineq)
      also have "… < ε"
        using that by (auto simp: Q_def)
      finally show ?thesis
        by (intro ε)
    qed
    thus "∀🪙F x in uniformity ×🪙F uniformity. P (fst (fst x) + fst (snd x), snd (fst x) + snd (snd x))"
      unfolding eventually_prod_filter by (intro exI[of _ Q] conjI Q) auto
  qed
next
  show "filterlim (λ((a::'a), b). (-a, -b)) uniformity uniformity"
    unfolding filterlim_def le_filter_def eventually_filtermap
  proof safe
    fix P :: "'a × 'a ==> bool"
    assume "eventually P uniformity"
    then obtain ε where ε: "ε > 0" "∧x y. dist x y < ε ==> P (x, y)"
      by (auto simp: eventually_uniformity_metric)
    show "∀🪙F x in uniformity. P (case x of (a, b) ==> (- a, - b))"
      unfolding eventually_uniformity_metric
      by (intro exI[of _ ε]) (auto intro!: ε simp: dist_norm norm_minus_commute)
  qed
qed

instance real :: uniform_topological_group_add ..
instance complex :: uniform_topological_group_add ..

lemma cauchy_seq_finset_iff_vanishing:
  "uniformity = filtercomap (λ(x,y). y - x :: 'a :: uniform_topological_group_add) (nhds 0)"
proof -
  have "filtercomap (λx. (0, case x of (x, y) ==> y - (x :: 'a))) uniformity ≤ uniformity"
    apply (simp add: le_filter_def eventually_filtercomap)
    using uniformity_of_uniform_continuous_invariant[OF uniformly_continuous_add']
    by (metis diff_self eq_diff_eq)
  moreover
  have "uniformity ≤ filtercomap (λx. (0, case x of (x, y) ==> y - (x :: 'a))) uniformity"
    apply (simp add: le_filter_def eventually_filtercomap)
    using uniformity_of_uniform_continuous_invariant[OF uniformly_continuous_minus']
    by (metis (mono_tags) diff_self eventually_mono surjective_pairing)
  ultimately show ?thesis
    by (simp add: nhds_eq_comap_uniformity filtercomap_filtercomap)
qed

subsubsection ‹Metric spaces›

instantiation🍋‹tag unimportant› prod :: (metric_space, metric_space) uniformity_dist begin
instance
proof
  show ‹uniformity = (INF e∈{0 🚫}. principal {(x::'a×'b, y). dist x y 🚫})›
  proof (subst filter_eq_iff, intro allI iffI)
    fix P :: ‹('a × 'b) × ('a × 'b) ==> bool›

    have 1: ‹∃e∈{0🚫}.
  {(x,y). dist x y 🚫} ⊆ {(x,y). dist x y 🚫} ∧
  {(x,y). dist x y 🚫} ⊆ {(x,y). dist x y 🚫}›
      using that by (auto intro: bexI[of _ ‹min a b›])
    have 2: ‹mono (λP. eventually (λx. P (Q x)) F)› for F :: ‹'z filter› and Q :: ‹'z ==> 'y›
      unfolding mono_def using eventually_mono le_funD by fastforce
    have ‹∀🪙F ((x1::'a,y1),(x2::'b,y2)) in uniformity ×🪙F uniformity. dist x1 y1 🚫/2 ∧ dist x2 y2 🚫/2› if ‹e>0› for e
      by (auto intro!: eventually_prodI exI[of _ ‹e/2›] simp: case_prod_unfold eventually_uniformity_metric that)
    then have 3: ‹∀🪙F ((x1::'a,y1),(x2::'b,y2)) in uniformity ×🪙F uniformity. dist (x1,x2) (y1,y2) 🚫› if ‹e>0› for e
      apply (rule eventually_rev_mp)
      by (auto intro!: that eventuallyI simp: case_prod_unfold dist_prod_def sqrt_sum_squares_half_less)
    show ‹eventually P (INF e∈{0🚫}. principal {(x, y). dist x y 🚫}) ==> eventually P uniformity›
      apply (subst (asm) eventually_INF_base)
      using 1 3 apply (auto simp: uniformity_prod_def case_prod_unfold eventually_filtermap 2 eventually_principal)
      by (smt (verit, best) eventually_mono)
  next
    fix P :: ‹('a × 'b) × ('a × 'b) ==> bool›
    assume ‹eventually P uniformity›
    then obtain P1 P2 where ‹eventually P1 uniformity› ‹eventually P2 uniformity›
      and P1P2P: ‹P1 (x1, y1) ==> P2 (x2, y2) ==> P ((x1, x2), (y1, y2))› for x1 y1 x2 y2
      by (auto simp: eventually_filtermap case_prod_beta eventually_prod_filter uniformity_prod_def)
    from ‹eventually P1 uniformity› obtain e1 where ‹e1>0› and e1P1: ‹dist x y 🚫 ==> P1 (x,y)› for x y
      using eventually_uniformity_metric by blast
    from ‹eventually P2 uniformity› obtain e2 where ‹e2>0› and e2P2: ‹dist x y 🚫 ==> P2 (x,y)› for x y
      using eventually_uniformity_metric by blast
    define e where ‹e = min e1 e2›
    have ‹e > 0›
      using ‹0 🚫› ‹0 🚫› e_def by auto
    have ‹dist (x1,x2) (y1,y2) 🚫 ==> dist x1 y1 🚫› for x1 y1 :: 'a and x2 y2 :: 'b
      unfolding dist_prod_def e_def
      by (metis real_sqrt_sum_squares_ge1 fst_conv min_less_iff_conj order_le_less_trans snd_conv)
    moreover have ‹dist (x1,x2) (y1,y2) 🚫 ==> dist x2 y2 🚫› for x1 y1 :: 'a and x2 y2 :: 'b
      unfolding dist_prod_def e_def
      using real_sqrt_sum_squares_ge1 [of "dist x1 y1" "dist x2 y2"]
      by (metis min_less_iff_conj order_le_less_trans real_sqrt_sum_squares_ge2 snd_conv)
    ultimately have *: ‹dist (x1,x2) (y1,y2) 🚫 ==> P ((x1, x2), (y1, y2))› for x1 y1 x2 y2
      using e1P1 e2P2 P1P2P by auto

    show ‹eventually P (INF e∈{0🚫}. principal {(x, y). dist x y 🚫})›
      using ‹e > 0› * 
      by (auto simp: eventually_principal intro: eventually_INF1)
  qed
qed
end

declare uniformity_Abort[where 'a="'a :: metric_space × 'b :: metric_space", code]

instantiation prod :: (metric_space, metric_space) metric_space
begin

proposition dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)🪙2 + (dist b d)🪙2)"
  unfolding dist_prod_def by simp

lemma dist_fst_le: "dist (fst x) (fst y) ≤ dist x y"
  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)

lemma dist_snd_le: "dist (snd x) (snd y) ≤ dist x y"
  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)

instance
proof
  fix x y :: "'a × 'b"
  show "dist x y = 0 ⟷ x = y"
    unfolding dist_prod_def prod_eq_iff by simp
next
  fix x y z :: "'a × 'b"
  show "dist x y ≤ dist x z + dist y z"
    unfolding dist_prod_def
    by (meson add_mono dist_triangle2 order_trans power_mono real_sqrt_le_iff
        real_sqrt_sum_squares_triangle_ineq zero_le_dist)
next
  fix S :: "('a × 'b) set"
  have *: "open S ⟷ (∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S)"
  proof
    assume "open S" show "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S"
    proof
      fix x assume "x ∈ S"
      obtain A B where "open A" "open B" "x ∈ A × B" "A × B ⊆ S"
        using ‹open S› and ‹x ∈ S› by (rule open_prod_elim)
      obtain r where r: "0 < r" "∀y. dist y (fst x) < r ⟶ y ∈ A"
        using ‹open A› and ‹x ∈ A × B› unfolding open_dist by auto
      obtain s where s: "0 < s" "∀y. dist y (snd x) < s ⟶ y ∈ B"
        using ‹open B› and ‹x ∈ A × B› unfolding open_dist by auto
      let ?e = "min r s"
      have "0 < ?e ∧ (∀y. dist y x < ?e ⟶ y ∈ S)"
      proof (intro allI impI conjI)
        show "0 < min r s" by (simp add: r(1) s(1))
      next
        fix y assume "dist y x < min r s"
        hence "dist y x < r" and "dist y x < s"
          by simp_all
        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
        hence "fst y ∈ A" and "snd y ∈ B"
          by (simp_all add: r(2) s(2))
        hence "y ∈ A × B" by (induct y, simp)
        with ‹A × B ⊆ S› show "y ∈ S" ..
      qed
      thus "∃e>0. ∀y. dist y x < e ⟶ y ∈ S" ..
    qed
  next
    assume *: "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S" show "open S"
    proof (rule open_prod_intro)
      fix x assume "x ∈ S"
      then obtain e where "0 < e" and S: "∀y. dist y x < e ⟶ y ∈ S"
        using * by fast
      define r where "r = e / sqrt 2"
      define s where "s = e / sqrt 2"
      from ‹0 🚫› have "0 < r" and "0 < s"
        unfolding r_def s_def by simp_all
      from ‹0 🚫› have "e = sqrt (r🪙2 + s🪙2)"
        unfolding r_def s_def by (simp add: power_divide)
      define A where "A = {y. dist (fst x) y < r}"
      define B where "B = {y. dist (snd x) y < s}"
      have "open A" and "open B"
        unfolding A_def B_def by (simp_all add: open_ball)
      moreover have "x ∈ A × B"
        unfolding A_def B_def mem_Times_iff
        using ‹0 🚫› and ‹0 🚫› by simp
      moreover have "A × B ⊆ S"
      proof (clarify)
        fix a b assume "a ∈ A" and "b ∈ B"
        hence "dist a (fst x) < r" and "dist b (snd x) < s"
          unfolding A_def B_def by (simp_all add: dist_commute)
        hence "dist (a, b) x < e"
          unfolding dist_prod_def ‹e = sqrt (r🪙2 + s🪙2)›
          by (simp add: add_strict_mono power_strict_mono)
        thus "(a, b) ∈ S"
          by (simp add: S)
      qed
      ultimately show "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S" by fast
    qed
  qed
qed

end

declare [[code abort: "dist::('a::metric_space*'b::metric_space)==>('a*'b) ==> real"]]

lemma Cauchy_fst: "Cauchy X ==> Cauchy (λn. fst (X n :: 'a::metric_space × 'b::metric_space))"
  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])

lemma Cauchy_snd: "Cauchy X ==> Cauchy (λn. snd (X n :: 'a::metric_space × 'b::metric_space))"
  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])

lemma Cauchy_Pair:
  assumes "Cauchy X" and "Cauchy Y"
  shows "Cauchy (λn. (X n :: 'a::metric_space, Y n :: 'a::metric_space))"
proof (rule metric_CauchyI)
  fix r :: real assume "0 < r"
  hence "0 < r / sqrt 2" (is "0 < ?s") by simp
  obtain M where M: "∀m≥M. ∀n≥M. dist (X m) (X n) < ?s"
    using metric_CauchyD [OF ‹Cauchy X› ‹0 🚫s›] ..
  obtain N where N: "∀m≥N. ∀n≥N. dist (Y m) (Y n) < ?s"
    using metric_CauchyD [OF ‹Cauchy Y› ‹0 🚫s›] ..
  have "∀m≥max M N. ∀n≥max M N. dist (X m, Y m) (X n, Y n) < r"
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
  then show "∃n0. ∀m≥n0. ∀n≥n0. dist (X m, Y m) (X n, Y n) < r" ..
qed

text ‹Analogue to @{thm [source] uniformly_continuous_on_def} for two-argument functions.›
lemma uniformly_continuous_on_prod_metric:
  fixes f :: ‹('a::metric_space × 'b::metric_space) ==> 'c::metric_space›
  shows ‹uniformly_continuous_on (S×T) f ⟷ (∀e>0. ∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y 🚫 ⟶ dist x' y' 🚫 ⟶ dist (f (x, x')) (f (y, y')) 🚫)›
proof (unfold uniformly_continuous_on_def, intro iffI impI allI)
  fix e :: real 
  assume ‹e > 0› and ‹∀e>0. ∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y 🚫 ⟶ dist x' y' 🚫 ⟶ dist (f (x, x')) (f (y, y')) 🚫›
  then obtain d where ‹d > 0›
    and d: ‹∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y 🚫 ⟶ dist x' y' 🚫 ⟶ dist (f (x, x')) (f (y, y')) 🚫›
    by auto
  show ‹∃d>0. ∀x∈S×T. ∀y∈S×T. dist y x 🚫 ⟶ dist (f y) (f x) 🚫›
    apply (rule exI[of _ d])
    by (metis SigmaE ‹0 🚫› d dist_fst_le dist_snd_le fst_eqD order_le_less_trans
        snd_conv)
next
  fix e :: real 
  assume ‹e > 0› and ‹∀e>0. ∃d>0. ∀x∈S×T. ∀x'∈S×T. dist x' x 🚫 ⟶ dist (f x') (f x) 🚫›
  then obtain d where ‹d > 0› and d: ‹∀x∈S×T. ∀x'∈S×T. dist x' x 🚫 ⟶ dist (f x') (f x) 🚫›
    by auto
  show ‹∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y 🚫 ⟶ dist x' y' 🚫 ⟶ dist (f (x, x')) (f (y, y')) 🚫›
  proof (intro exI conjI impI ballI)
    from ‹d > 0› show ‹d / 2 > 0› by auto
    fix x y x' y'
    assume [simp]: ‹x ∈ S› ‹y ∈ S› ‹x' ∈ T› ‹y' ∈ T›
    assume ‹dist x y 🚫 / 2› and ‹dist x' y' 🚫 / 2›
    then have ‹dist (x, x') (y, y') 🚫›
      by (simp add: dist_Pair_Pair sqrt_sum_squares_half_less)
    with d show ‹dist (f (x, x')) (f (y, y')) 🚫›
      by auto
  qed
qed

text ‹Analogue to @{thm [source] isUCont_def} for two-argument functions.›
lemma isUCont_prod_metric:
  fixes f :: ‹('a::metric_space × 'b::metric_space) ==> 'c::metric_space›
  shows ‹isUCont f ⟷ (∀e>0. ∃d>0. ∀x. ∀y. ∀x'. ∀y'. dist x y 🚫 ⟶ dist x' y' 🚫 ⟶ dist (f (x, x')) (f (y, y')) 🚫)›
  using uniformly_continuous_on_prod_metric[of UNIV UNIV]
  by auto

text ‹This logically belong with the real vector spaces but we only have the necessary lemmas now.›
lemma isUCont_plus[simp]:
  shows ‹isUCont (λ(x::'a::real_normed_vector,y). x+y)›
proof (rule isUCont_prod_metric[THEN iffD2], intro allI impI, simp)
  fix e :: real assume ‹0 🚫›
  show ‹∃d>0. ∀x y :: 'a. dist x y 🚫 ⟶ (∀x' y'. dist x' y' 🚫 ⟶ dist (x + x') (y + y') 🚫)›
    apply (rule exI[of _ ‹e/2›])
    using ‹0 🚫›
    by (smt (verit) dist_add_cancel dist_add_cancel2 dist_commute 
        dist_triangle_lt field_sum_of_halves)
qed

subsection ‹Product is a Complete Metric Space›

instance🍋‹tag important› prod :: (complete_space, complete_space) complete_space
proof
  fix X :: "nat ==> 'a × 'b" assume "Cauchy X"
  have 1: "(λn. fst (X n)) <---- lim (λn. fst (X n))"
    using Cauchy_fst [OF ‹Cauchy X›]
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
  have 2: "(λn. snd (X n)) <---- lim (λn. snd (X n))"
    using Cauchy_snd [OF ‹Cauchy X›]
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
  have "X <---- (lim (λn. fst (X n)), lim (λn. snd (X n)))"
    using tendsto_Pair [OF 1 2] by simp
  then show "convergent X"
    by (rule convergentI)
qed

subsection ‹Product is a Normed Vector Space›

instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
begin

definition norm_prod_def[code del]:
  "norm x = sqrt ((norm (fst x))🪙2 + (norm (snd x))🪙2)"

definition sgn_prod_def:
  "sgn (x::'a × 'b) = scaleR (inverse (norm x)) x"

proposition norm_Pair: "norm (a, b) = sqrt ((norm a)🪙2 + (norm b)🪙2)"
  unfolding norm_prod_def by simp

instance
proof
  fix r :: real and x y :: "'a × 'b"
  show "norm x = 0 ⟷ x = 0"
    unfolding norm_prod_def
    by (simp add: prod_eq_iff)
  show "norm (x + y) ≤ norm x + norm y"
    unfolding norm_prod_def
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
    apply (simp add: add_mono power_mono norm_triangle_ineq)
    done
  show "norm (scaleR r x) = ∣r∣ * norm x"
    unfolding norm_prod_def
    by (simp add: power_mult_distrib real_sqrt_mult flip: distrib_left)
  show "sgn x = scaleR (inverse (norm x)) x"
    by (rule sgn_prod_def)
  show "dist x y = norm (x - y)"
    unfolding dist_prod_def norm_prod_def
    by (simp add: dist_norm)
qed

end

declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) ==> real"]]

instance🍋‹tag important› prod :: (banach, banach) banach ..

subsubsection🍋‹tag unimportant› ‹Pair operations are linear›

lemma bounded_linear_fst: "bounded_linear fst"
  using fst_add fst_scaleR
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)

lemma bounded_linear_snd: "bounded_linear snd"
  using snd_add snd_scaleR
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)

lemmas bounded_linear_fst_comp = bounded_linear_fst[THEN bounded_linear_compose]

lemmas bounded_linear_snd_comp = bounded_linear_snd[THEN bounded_linear_compose]

lemma bounded_linear_Pair:
  assumes f: "bounded_linear f"
  assumes g: "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
  fix x y and r :: real
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
    by (simp add: f.add g.add)
  show "(f (r *🪙R x), g (r *🪙R x)) = r *🪙R (f x, g x)"
    by (simp add: f.scale g.scale)
  obtain Kf where "0 < Kf" and norm_f: "∧x. norm (f x) ≤ norm x * Kf"
    using f.pos_bounded by fast
  obtain Kg where "0 < Kg" and norm_g: "∧x. norm (g x) ≤ norm x * Kg"
    using g.pos_bounded by fast
  have "∧x. sqrt ((norm (f x))🪙2) + sqrt ((norm (g x))🪙2) ≤ norm x * (Kf + Kg)"
    by (simp add: add_mono distrib_left norm_f norm_g)
  then have "∀x. norm (f x, g x) ≤ norm x * (Kf + Kg)"
    by (smt (verit) norm_ge_zero norm_prod_def prod.sel sqrt_add_le_add_sqrt zero_le_power)
  then show "∃K. ∀x. norm (f x, g x) ≤ norm x * K" ..
qed

subsubsection🍋‹tag unimportant› ‹Frechet derivatives involving pairs›

text🍋‹tag important› ‹%whitespace›
proposition has_derivative_Pair [derivative_intros]:
  assumes f: "(f has_derivative f') (at x within s)"
    and g: "(g has_derivative g') (at x within s)"
  shows "((λx. (f x, g x)) has_derivative (λh. (f' h, g' h))) (at x within s)"
proof (rule has_derivativeI_sandwich[of 1])
  show "bounded_linear (λh. (f' h, g' h))"
    using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
  let ?Rf = "λy. f y - f x - f' (y - x)"
  let ?Rg = "λy. g y - g x - g' (y - x)"
  let ?R = "λy. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"

  show "((λy. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) ---> 0) (at x within s)"
    using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)

  fix y :: 'a assume "y ≠ x"
  show "norm (?R y) / norm (y - x) ≤ norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)"
    unfolding add_divide_distrib [symmetric]
    by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt])
qed simp

lemma differentiable_Pair [simp, derivative_intros]:
  "f differentiable at x within s ==> g differentiable at x within s ==>
    (λx. (f x, g x)) differentiable at x within s"
  unfolding differentiable_def by (blast intro: has_derivative_Pair)

lemmas has_derivative_fst [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_fst]
lemmas has_derivative_snd [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_snd]

lemma has_derivative_split [derivative_intros]:
  "((λp. f (fst p) (snd p)) has_derivative f') F ==> ((λ(a, b). f a b) has_derivative f') F"
  unfolding split_beta' .


subsubsection🍋‹tag unimportant› ‹Vector derivatives involving pairs›

lemma has_vector_derivative_Pair[derivative_intros]:
  assumes "(f has_vector_derivative f') (at x within s)"
    "(g has_vector_derivative g') (at x within s)"
  shows "((λx. (f x, g x)) has_vector_derivative (f', g')) (at x within s)"
  using assms
  by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros)

lemma
  fixes x :: "'a::real_normed_vector"
  shows norm_Pair1 [simp]: "norm (0,x) = norm x"
    and norm_Pair2 [simp]: "norm (x,0) = norm x"
by (auto simp: norm_Pair)

lemma norm_commute: "norm (x,y) = norm (y,x)"
  by (simp add: norm_Pair)

lemma norm_fst_le: "norm x ≤ norm (x,y)"
  by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)

lemma norm_snd_le: "norm y ≤ norm (x,y)"
  by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)

lemma norm_Pair_le:
  shows "norm (x, y) ≤ norm x + norm y"
  unfolding norm_Pair
  by (metis norm_ge_zero sqrt_sum_squares_le_sum)

lemma (in vector_space_prod) span_Times_sing1: "p.span ({0} × B) = {0} × vs2.span B"
proof (rule p.span_unique)
  show "{0} × B ⊆ {0} × vs2.span B"
    by (auto intro!: vs1.span_base vs2.span_base)
  show "p.subspace ({0} × vs2.span B)"
    using vs1.subspace_single_0 vs2.subspace_span by (rule subspace_Times)
  fix T :: "('b × 'c) set"
  assume subset_T: "{0} × B ⊆ T" and subspace: "p.subspace T" 
  show "{0} × vs2.span B ⊆ T"
  proof clarify
    fix b
    assume b_span: "b ∈ vs2.span B"
    then obtain t r where b: "b = (∑a∈t. r a *b a)" and t: "finite t" "t ⊆ B"
      by (auto simp: vs2.span_explicit)
    have "(0, b) = (∑b∈t. scale (r b) (0, b))"
      unfolding b scale_prod sum_prod by simp
    also have "… ∈ T"
      using ‹t ⊆ B› subset_T
      by (auto intro!: p.subspace_sum p.subspace_scale subspace)
    finally show "(0, b) ∈ T" .
  qed
qed

lemma (in vector_space_prod) span_Times_sing2: "p.span (A × {0}) = vs1.span A × {0}"
proof (rule p.span_unique)
  show "A × {0} ⊆ vs1.span A × {0}"
    by (auto intro!: vs1.span_base vs2.span_base)
  show "p.subspace (vs1.span A × {0})"
    using vs1.subspace_span vs2.subspace_single_0 by (rule subspace_Times)
  fix T :: "('b × 'c) set"
  assume subset_T: "A × {0} ⊆ T" and subspace: "p.subspace T"
  show "vs1.span A × {0} ⊆ T"
  proof clarify
    fix a
    assume a_span: "a ∈ vs1.span A"
    then obtain t r where a: "a = (∑a∈t. r a *a a)" and t: "finite t" "t ⊆ A"
      by (auto simp: vs1.span_explicit)
    have "(a, 0) = (∑a∈t. scale (r a) (a, 0))"
      unfolding a scale_prod sum_prod by simp
    also have "… ∈ T"
      using ‹t ⊆ A› subset_T
      by (auto intro!: p.subspace_sum p.subspace_scale subspace)
    finally show "(a, 0) ∈ T" .
  qed
qed

subsection ‹Product is Finite Dimensional›

lemma (in finite_dimensional_vector_space) zero_not_in_Basis[simp]: "0 ∉ Basis"
  using dependent_zero local.independent_Basis by blast

locale finite_dimensional_vector_space_prod = vector_space_prod + finite_dimensional_vector_space_pair begin

definition "Basis_pair = B1 × {0} ∪ {0} × B2"

sublocale p: finite_dimensional_vector_space scale Basis_pair
proof unfold_locales
  show "finite Basis_pair"
    by (auto intro!: finite_cartesian_product vs1.finite_Basis vs2.finite_Basis simp: Basis_pair_def)
  show "p.independent Basis_pair"
    unfolding p.dependent_def Basis_pair_def
  proof safe
    fix a
    assume a: "a ∈ B1"
    assume "(a, 0) ∈ p.span (B1 × {0} ∪ {0} × B2 - {(a, 0)})"
    also have "B1 × {0} ∪ {0} × B2 - {(a, 0)} = (B1 - {a}) × {0} ∪ {0} × B2"
      by auto
    finally show False
      using a vs1.dependent_def vs1.independent_Basis
      by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
  next
    fix b
    assume b: "b ∈ B2"
    assume "(0, b) ∈ p.span (B1 × {0} ∪ {0} × B2 - {(0, b)})"
    also have "(B1 × {0} ∪ {0} × B2 - {(0, b)}) = B1 × {0} ∪ {0} × (B2 - {b})"
      by auto
    finally show False
      using b vs2.dependent_def vs2.independent_Basis
      by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
  qed
  show "p.span Basis_pair = UNIV"
    by (auto simp: p.span_Un span_Times_sing2 span_Times_sing1 vs1.span_Basis vs2.span_Basis
        Basis_pair_def)
qed

proposition dim_Times:
  assumes "vs1.subspace S" "vs2.subspace T"
  shows "p.dim(S × T) = vs1.dim S + vs2.dim T"
proof -
  interpret p1: Vector_Spaces.linear s1 scale "(λx. (x, 0))"
    by unfold_locales (auto simp: scale_def)
  interpret pair1: finite_dimensional_vector_space_pair "(*a)" B1 scale Basis_pair
    by unfold_locales
  interpret p2: Vector_Spaces.linear s2 scale "(λx. (0, x))"
    by unfold_locales (auto simp: scale_def)
  interpret pair2: finite_dimensional_vector_space_pair "(*b)" B2 scale Basis_pair
  by unfold_locales
  have ss: "p.subspace ((λx. (x, 0)) ` S)" "p.subspace (Pair 0 ` T)"
  by (rule p1.subspace_image p2.subspace_image assms)+
  have "p.dim(S × T) = p.dim({u + v |u v. u ∈ (λx. (x, 0)) ` S ∧ v ∈ Pair 0 ` T})"
  by (simp add: Times_eq_image_sum)
  moreover have "p.dim ((λx. (x, 0::'c)) ` S) = vs1.dim S" "p.dim (Pair (0::'b) ` T) = vs2.dim T"
  by (simp_all add: inj_on_def p1.linear_axioms pair1.dim_image_eq p2.linear_axioms pair2.dim_image_eq)
  moreover have "p.dim ((λx. (x, 0)) ` S ∩ Pair 0 ` T) = 0"
  by (subst p.dim_eq_0) auto
  ultimately show ?thesis
  using p.dim_sums_Int [OF ss] by linarith
 qed
 
 lemma dimension_pair: "p.dimension = vs1.dimension + vs2.dimension"
  using dim_Times[OF vs1.subspace_UNIV vs2.subspace_UNIV]
  by (auto simp: p.dimension_def vs1.dimension_def vs2.dimension_def)
 
 end
 
 end