Theory Fin_Map

(*  Title:      HOL/Probability/Fin_Map.thy
    Author:     Fabian Immler, TU München
*)

section ‹Finite Maps›

theory Fin_Map
  imports "HOL-Analysis.Finite_Product_Measure" "HOL-Library.Finite_Map"
begin

text ‹The \<^type>‹fmap› type can be instantiated to \<^class>‹polish_space›, needed for the proof of
  projective limit. \<^const>‹extensional› functions are used for the representation in order to
  stay close to the developments of (finite) products \<^const>‹Pi⇩E› and their sigma-algebra
  \<^const>‹Pi⇩M›.›

type_notation fmap ("(_ ⇒⇩F /_)" [22, 21] 21)

unbundle fmap.lifting


subsection ‹Domain and Application›

lift_definition domain::"('i ⇒⇩F 'a) ⇒ 'i set" is dom .

lemma finite_domain[simp, intro]: "finite (domain P)"
  by transfer simp

lift_definition proj :: "('i ⇒⇩F 'a) ⇒ 'i ⇒ 'a" ("'((_)')⇩F" [0] 1000) is
  "λf x. if x ∈ dom f then the (f x) else undefined" .

declare [[coercion proj]]

lemma extensional_proj[simp, intro]: "(P)⇩F ∈ extensional (domain P)"
  by transfer (auto simp: extensional_def)

lemma proj_undefined[simp, intro]: "i ∉ domain P ⟹ P i = undefined"
  using extensional_proj[of P] unfolding extensional_def by auto

lemma finmap_eq_iff: "P = Q ⟷ (domain P = domain Q ∧ (∀i∈domain P. P i = Q i))"
  apply transfer
  apply (safe intro!: ext)
  subgoal for P Q x
    by (cases "x ∈ dom P"; cases "P x") (auto dest!: bspec[where x=x])
  done


subsection ‹Constructor of Finite Maps›

lift_definition finmap_of::"'i set ⇒ ('i ⇒ 'a) ⇒ ('i ⇒⇩F 'a)" is
  "λI f x. if x ∈ I ∧ finite I then Some (f x) else None"
  by (simp add: dom_def)

lemma proj_finmap_of[simp]:
  assumes "finite inds"
  shows "(finmap_of inds f)⇩F = restrict f inds"
  using assms
  by transfer force

lemma domain_finmap_of[simp]:
  assumes "finite inds"
  shows "domain (finmap_of inds f) = inds"
  using assms
  by transfer (auto split: if_splits)

lemma finmap_of_eq_iff[simp]:
  assumes "finite i" "finite j"
  shows "finmap_of i m = finmap_of j n ⟷ i = j ∧ (∀k∈i. m k= n k)"
  using assms by (auto simp: finmap_eq_iff)

lemma finmap_of_inj_on_extensional_finite:
  assumes "finite K"
  assumes "S ⊆ extensional K"
  shows "inj_on (finmap_of K) S"
proof (rule inj_onI)
  fix x y::"'a ⇒ 'b"
  assume "finmap_of K x = finmap_of K y"
  hence "(finmap_of K x)⇩F = (finmap_of K y)⇩F" by simp
  moreover
  assume "x ∈ S" "y ∈ S" hence "x ∈ extensional K" "y ∈ extensional K" using assms by auto
  ultimately
  show "x = y" using assms by (simp add: extensional_restrict)
qed

subsection ‹Product set of Finite Maps›

text ‹This is \<^term>‹Pi› for Finite Maps, most of this is copied›

definition Pi' :: "'i set ⇒ ('i ⇒ 'a set) ⇒ ('i ⇒⇩F 'a) set" where
  "Pi' I A = { P. domain P = I ∧ (∀i. i ∈ I ⟶ (P)⇩F i ∈ A i) } "

syntax
  "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3Π'' _∈_./ _)"   10)
translations
  "Π' x∈A. B" == "CONST Pi' A (λx. B)"

subsubsection‹Basic Properties of \<^term>‹Pi'››

lemma Pi'_I[intro!]: "domain f = A ⟹ (⋀x. x ∈ A ⟹ f x ∈ B x) ⟹ f ∈ Pi' A B"
  by (simp add: Pi'_def)

lemma Pi'_I'[simp]: "domain f = A ⟹ (⋀x. x ∈ A ⟶ f x ∈ B x) ⟹ f ∈ Pi' A B"
  by (simp add:Pi'_def)

lemma Pi'_mem: "f∈ Pi' A B ⟹ x ∈ A ⟹ f x ∈ B x"
  by (simp add: Pi'_def)

lemma Pi'_iff: "f ∈ Pi' I X ⟷ domain f = I ∧ (∀i∈I. f i ∈ X i)"
  unfolding Pi'_def by auto

lemma Pi'E [elim]:
  "f ∈ Pi' A B ⟹ (f x ∈ B x ⟹ domain f = A ⟹ Q) ⟹ (x ∉ A ⟹ Q) ⟹ Q"
  by(auto simp: Pi'_def)

lemma in_Pi'_cong:
  "domain f = domain g ⟹ (⋀ w. w ∈ A ⟹ f w = g w) ⟹ f ∈ Pi' A B ⟷ g ∈ Pi' A B"
  by (auto simp: Pi'_def)

lemma Pi'_eq_empty[simp]:
  assumes "finite A" shows "(Pi' A B) = {} ⟷ (∃x∈A. B x = {})"
  using assms
  apply (simp add: Pi'_def, auto)
  apply (drule_tac x = "finmap_of A (λu. SOME y. y ∈ B u)" in spec, auto)
  apply (cut_tac P= "%y. y ∈ B i" in some_eq_ex, auto)
  done

lemma Pi'_mono: "(⋀x. x ∈ A ⟹ B x ⊆ C x) ⟹ Pi' A B ⊆ Pi' A C"
  by (auto simp: Pi'_def)

lemma Pi_Pi': "finite A ⟹ (Pi⇩E A B) = proj ` Pi' A B"
  apply (auto simp: Pi'_def Pi_def extensional_def)
  apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI)
  apply auto
  done

subsection ‹Topological Space of Finite Maps›

instantiation fmap :: (type, topological_space) topological_space
begin

definition open_fmap :: "('a ⇒⇩F 'b) set ⇒ bool" where
   [code del]: "open_fmap = generate_topology {Pi' a b|a b. ∀i∈a. open (b i)}"

lemma open_Pi'I: "(⋀i. i ∈ I ⟹ open (A i)) ⟹ open (Pi' I A)"
  by (auto intro: generate_topology.Basis simp: open_fmap_def)

instance using topological_space_generate_topology
  by intro_classes (auto simp: open_fmap_def class.topological_space_def)

end

lemma open_restricted_space:
  shows "open {m. P (domain m)}"
proof -
  have "{m. P (domain m)} = (⋃i ∈ Collect P. {m. domain m = i})" by auto
  also have "open …"
  proof (rule, safe, cases)
    fix i::"'a set"
    assume "finite i"
    hence "{m. domain m = i} = Pi' i (λ_. UNIV)" by (auto simp: Pi'_def)
    also have "open …" by (auto intro: open_Pi'I simp: ‹finite i›)
    finally show "open {m. domain m = i}" .
  next
    fix i::"'a set"
    assume "¬ finite i" hence "{m. domain m = i} = {}" by auto
    also have "open …" by simp
    finally show "open {m. domain m = i}" .
  qed
  finally show ?thesis .
qed

lemma closed_restricted_space:
  shows "closed {m. P (domain m)}"
  using open_restricted_space[of "λx. ¬ P x"]
  unfolding closed_def by (rule back_subst) auto

lemma tendsto_proj: "((λx. x) ⤏ a) F ⟹ ((λx. (x)⇩F i) ⤏ (a)⇩F i) F"
  unfolding tendsto_def
proof safe
  fix S::"'b set"
  let ?S = "Pi' (domain a) (λx. if x = i then S else UNIV)"
  assume "open S" hence "open ?S" by (auto intro!: open_Pi'I)
  moreover assume "∀S. open S ⟶ a ∈ S ⟶ eventually (λx. x ∈ S) F" "a i ∈ S"
  ultimately have "eventually (λx. x ∈ ?S) F" by auto
  thus "eventually (λx. (x)⇩F i ∈ S) F"
    by eventually_elim (insert ‹a i ∈ S›, force simp: Pi'_iff split: if_split_asm)
qed

lemma continuous_proj:
  shows "continuous_on s (λx. (x)⇩F i)"
  unfolding continuous_on_def by (safe intro!: tendsto_proj tendsto_ident_at)

instance fmap :: (type, first_countable_topology) first_countable_topology
proof
  fix x::"'a⇒⇩F'b"
  have "∀i. ∃A. countable A ∧ (∀a∈A. x i ∈ a) ∧ (∀a∈A. open a) ∧
    (∀S. open S ∧ x i ∈ S ⟶ (∃a∈A. a ⊆ S)) ∧ (∀a b. a ∈ A ⟶ b ∈ A ⟶ a ∩ b ∈ A)" (is "∀i. ?th i")
  proof
    fix i from first_countable_basis_Int_stableE[of "x i"]
    obtain A where
      "countable A"
      "⋀C. C ∈ A ⟹ (x)⇩F i ∈ C"
      "⋀C. C ∈ A ⟹ open C"
      "⋀S. open S ⟹ (x)⇩F i ∈ S ⟹ ∃A∈A. A ⊆ S"
      "⋀C D. C ∈ A ⟹ D ∈ A ⟹ C ∩ D ∈ A"
      by auto
    thus "?th i" by (intro exI[where x=A]) simp
  qed
  then obtain A
    where A: "∀i. countable (A i) ∧ Ball (A i) ((∈) ((x)⇩F i)) ∧ Ball (A i) open ∧
      (∀S. open S ∧ (x)⇩F i ∈ S ⟶ (∃a∈A i. a ⊆ S)) ∧ (∀a b. a ∈ A i ⟶ b ∈ A i ⟶ a ∩ b ∈ A i)"
    by (auto simp: choice_iff)
  hence open_sub: "⋀i S. i∈domain x ⟹ open (S i) ⟹ x i∈(S i) ⟹ (∃a∈A i. a⊆(S i))" by auto
  have A_notempty: "⋀i. i ∈ domain x ⟹ A i ≠ {}" using open_sub[of _ "λ_. UNIV"] by auto
  let ?A = "(λf. Pi' (domain x) f) ` (Pi⇩E (domain x) A)"
  show "∃A::nat ⇒ ('a⇒⇩F'b) set. (∀i. x ∈ (A i) ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))"
  proof (rule first_countableI[of "?A"], safe)
    show "countable ?A" using A by (simp add: countable_PiE)
  next
    fix S::"('a ⇒⇩F 'b) set" assume "open S" "x ∈ S"
    thus "∃a∈?A. a ⊆ S" unfolding open_fmap_def
    proof (induct rule: generate_topology.induct)
      case UNIV thus ?case by (auto simp add: ex_in_conv PiE_eq_empty_iff A_notempty)
    next
      case (Int a b)
      then obtain f g where
        "f ∈ Pi⇩E (domain x) A" "Pi' (domain x) f ⊆ a" "g ∈ Pi⇩E (domain x) A" "Pi' (domain x) g ⊆ b"
        by auto
      thus ?case using A
        by (auto simp: Pi'_iff PiE_iff extensional_def Int_stable_def
            intro!: bexI[where x="λi. f i ∩ g i"])
    next
      case (UN B)
      then obtain b where "x ∈ b" "b ∈ B" by auto
      hence "∃a∈?A. a ⊆ b" using UN by simp
      thus ?case using ‹b ∈ B› by (metis Sup_upper2)
    next
      case (Basis s)
      then obtain a b where xs: "x∈ Pi' a b" "s = Pi' a b" "⋀i. i∈a ⟹ open (b i)" by auto
      have "∀i. ∃a. (i ∈ domain x ∧ open (b i) ∧ (x)⇩F i ∈ b i) ⟶ (a∈A i ∧ a ⊆ b i)"
        using open_sub[of _ b] by auto
      then obtain b'
        where "⋀i. i ∈ domain x ⟹ open (b i) ⟹ (x)⇩F i ∈ b i ⟹ (b' i ∈A i ∧ b' i ⊆ b i)"
          unfolding choice_iff by auto
      with xs have "⋀i. i ∈ a ⟹ (b' i ∈A i ∧ b' i ⊆ b i)" "Pi' a b' ⊆ Pi' a b"
        by (auto simp: Pi'_iff intro!: Pi'_mono)
      thus ?case using xs
        by (intro bexI[where x="Pi' a b'"])
          (auto simp: Pi'_iff intro!: image_eqI[where x="restrict b' (domain x)"])
    qed
  qed (insert A,auto simp: PiE_iff intro!: open_Pi'I)
qed

subsection ‹Metric Space of Finite Maps›

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

instantiation fmap :: (type, metric_space) dist
begin

definition dist_fmap where
  "dist P Q = Max (range (λi. dist ((P)⇩F i) ((Q)⇩F i))) + (if domain P = domain Q then 0 else 1)"

instance ..
end

instantiation fmap :: (type, metric_space) uniformity_dist
begin

definition [code del]:
  "(uniformity :: (('a, 'b) fmap × ('a ⇒⇩F 'b)) filter) =
    (INF e∈{0 <..}. principal {(x, y). dist x y < e})"

instance
  by standard (rule uniformity_fmap_def)
end

declare uniformity_Abort[where 'a="('a ⇒⇩F 'b::metric_space)", code]

instantiation fmap :: (type, metric_space) metric_space
begin

lemma finite_proj_image': "x ∉ domain P ⟹ finite ((P)⇩F ` S)"
  by (rule finite_subset[of _ "proj P ` (domain P ∩ S ∪ {x})"]) auto

lemma finite_proj_image: "finite ((P)⇩F ` S)"
 by (cases "∃x. x ∉ domain P") (auto intro: finite_proj_image' finite_subset[where B="domain P"])

lemma finite_proj_diag: "finite ((λi. d ((P)⇩F i) ((Q)⇩F i)) ` S)"
proof -
  have "(λi. d ((P)⇩F i) ((Q)⇩F i)) ` S = (λ(i, j). d i j) ` ((λi. ((P)⇩F i, (Q)⇩F i)) ` S)" by auto
  moreover have "((λi. ((P)⇩F i, (Q)⇩F i)) ` S) ⊆ (λi. (P)⇩F i) ` S × (λi. (Q)⇩F i) ` S" by auto
  moreover have "finite …" using finite_proj_image[of P S] finite_proj_image[of Q S]
    by (intro finite_cartesian_product) simp_all
  ultimately show ?thesis by (simp add: finite_subset)
qed

lemma dist_le_1_imp_domain_eq:
  shows "dist P Q < 1 ⟹ domain P = domain Q"
  by (simp add: dist_fmap_def finite_proj_diag split: if_split_asm)

lemma dist_proj:
  shows "dist ((x)⇩F i) ((y)⇩F i) ≤ dist x y"
proof -
  have "dist (x i) (y i) ≤ Max (range (λi. dist (x i) (y i)))"
    by (simp add: Max_ge_iff finite_proj_diag)
  also have "… ≤ dist x y" by (simp add: dist_fmap_def)
  finally show ?thesis .
qed

lemma dist_finmap_lessI:
  assumes "domain P = domain Q"
  assumes "0 < e"
  assumes "⋀i. i ∈ domain P ⟹ dist (P i) (Q i) < e"
  shows "dist P Q < e"
proof -
  have "dist P Q = Max (range (λi. dist (P i) (Q i)))"
    using assms by (simp add: dist_fmap_def finite_proj_diag)
  also have "… < e"
  proof (subst Max_less_iff, safe)
    fix i
    show "dist ((P)⇩F i) ((Q)⇩F i) < e" using assms
      by (cases "i ∈ domain P") simp_all
  qed (simp add: finite_proj_diag)
  finally show ?thesis .
qed

instance
proof
  fix S::"('a ⇒⇩F 'b) set"
  have *: "open S = (∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S)" (is "_ = ?od")
  proof
    assume "open S"
    thus ?od
      unfolding open_fmap_def
    proof (induct rule: generate_topology.induct)
      case UNIV thus ?case by (auto intro: zero_less_one)
    next
      case (Int a b)
      show ?case
      proof safe
        fix x assume x: "x ∈ a" "x ∈ b"
        with Int x obtain e1 e2 where
          "e1>0" "∀y. dist y x < e1 ⟶ y ∈ a" "e2>0" "∀y. dist y x < e2 ⟶ y ∈ b" by force
        thus "∃e>0. ∀y. dist y x < e ⟶ y ∈ a ∩ b"
          by (auto intro!: exI[where x="min e1 e2"])
      qed
    next
      case (UN K)
      show ?case
      proof safe
        fix x X assume "x ∈ X" and X: "X ∈ K"
        with UN obtain e where "e>0" "⋀y. dist y x < e ⟶ y ∈ X" by force
        with X show "∃e>0. ∀y. dist y x < e ⟶ y ∈ ⋃K" by auto
      qed
    next
      case (Basis s) then obtain a b where s: "s = Pi' a b" and b: "⋀i. i∈a ⟹ open (b i)" by auto
      show ?case
      proof safe
        fix x assume "x ∈ s"
        hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff)
        obtain es where es: "∀i ∈ a. es i > 0 ∧ (∀y. dist y (proj x i) < es i ⟶ y ∈ b i)"
          using b ‹x ∈ s› by atomize_elim (intro bchoice, auto simp: open_dist s)
        hence in_b: "⋀i y. i ∈ a ⟹ dist y (proj x i) < es i ⟹ y ∈ b i" by auto
        show "∃e>0. ∀y. dist y x < e ⟶ y ∈ s"
        proof (cases, rule, safe)
          assume "a ≠ {}"
          show "0 < min 1 (Min (es ` a))" using es by (auto simp: ‹a ≠ {}›)
          fix y assume d: "dist y x < min 1 (Min (es ` a))"
          show "y ∈ s" unfolding s
          proof
            show "domain y = a" using d s ‹a ≠ {}› by (auto simp: dist_le_1_imp_domain_eq a_dom)
            fix i assume i: "i ∈ a"
            hence "dist ((y)⇩F i) ((x)⇩F i) < es i" using d
              by (auto simp: dist_fmap_def ‹a ≠ {}› intro!: le_less_trans[OF dist_proj])
            with i show "y i ∈ b i" by (rule in_b)
          qed
        next
          assume "¬a ≠ {}"
          thus "∃e>0. ∀y. dist y x < e ⟶ y ∈ s"
            using s ‹x ∈ s› by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1])
        qed
      qed
    qed
  next
    assume "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S"
    then obtain e where e_pos: "⋀x. x ∈ S ⟹ e x > 0" and
      e_in:  "⋀x y . x ∈ S ⟹ dist y x < e x ⟹ y ∈ S"
      unfolding bchoice_iff
      by auto
    have S_eq: "S = ⋃{Pi' a b| a b. ∃x∈S. domain x = a ∧ b = (λi. ball (x i) (e x))}"
    proof safe
      fix x assume "x ∈ S"
      thus "x ∈ ⋃{Pi' a b| a b. ∃x∈S. domain x = a ∧ b = (λi. ball (x i) (e x))}"
        using e_pos by (auto intro!: exI[where x="Pi' (domain x) (λi. ball (x i) (e x))"])
    next
      fix x y
      assume "y ∈ S"
      moreover
      assume "x ∈ (Π' i∈domain y. ball (y i) (e y))"
      hence "dist x y < e y" using e_pos ‹y ∈ S›
        by (auto simp: dist_fmap_def Pi'_iff finite_proj_diag dist_commute)
      ultimately show "x ∈ S" by (rule e_in)
    qed
    also have "open …"
      unfolding open_fmap_def
      by (intro generate_topology.UN) (auto intro: generate_topology.Basis)
    finally show "open S" .
  qed
  show "open S = (∀x∈S. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ S)"
    unfolding * eventually_uniformity_metric
    by (simp del: split_paired_All add: dist_fmap_def dist_commute eq_commute)
next
  fix P Q::"'a ⇒⇩F 'b"
  have Max_eq_iff: "⋀A m. finite A ⟹ A ≠ {} ⟹ (Max A = m) = (m ∈ A ∧ (∀a∈A. a ≤ m))"
    by (auto intro: Max_in Max_eqI)
  show "dist P Q = 0 ⟷ P = Q"
    by (auto simp: finmap_eq_iff dist_fmap_def Max_ge_iff finite_proj_diag Max_eq_iff
        add_nonneg_eq_0_iff
      intro!: Max_eqI image_eqI[where x=undefined])
next
  fix P Q R::"'a ⇒⇩F 'b"
  let ?dists = "λP Q i. dist ((P)⇩F i) ((Q)⇩F i)"
  let ?dpq = "?dists P Q" and ?dpr = "?dists P R" and ?dqr = "?dists Q R"
  let ?dom = "λP Q. (if domain P = domain Q then 0 else 1::real)"
  have "dist P Q = Max (range ?dpq) + ?dom P Q"
    by (simp add: dist_fmap_def)
  also obtain t where "t ∈ range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag)
  then obtain i where "Max (range ?dpq) = ?dpq i" by auto
  also have "?dpq i ≤ ?dpr i + ?dqr i" by (rule dist_triangle2)
  also have "?dpr i ≤ Max (range ?dpr)" by (simp add: finite_proj_diag)
  also have "?dqr i ≤ Max (range ?dqr)" by (simp add: finite_proj_diag)
  also have "?dom P Q ≤ ?dom P R + ?dom Q R" by simp
  finally show "dist P Q ≤ dist P R + dist Q R" by (simp add: dist_fmap_def ac_simps)
qed

end

subsection ‹Complete Space of Finite Maps›

lemma tendsto_finmap:
  fixes f::"nat ⇒ ('i ⇒⇩F ('a::metric_space))"
  assumes ind_f:  "⋀n. domain (f n) = domain g"
  assumes proj_g:  "⋀i. i ∈ domain g ⟹ (λn. (f n) i) ⇢ g i"
  shows "f ⇢ g"
  unfolding tendsto_iff
proof safe
  fix e::real assume "0 < e"
  let ?dists = "λx i. dist ((f x)⇩F i) ((g)⇩F i)"
  have "eventually (λx. ∀i∈domain g. ?dists x i < e) sequentially"
    using finite_domain[of g] proj_g
  proof induct
    case (insert i G)
    with ‹0 < e› have "eventually (λx. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff)
    moreover
    from insert have "eventually (λx. ∀i∈G. dist ((f x)⇩F i) ((g)⇩F i) < e) sequentially" by simp
    ultimately show ?case by eventually_elim auto
  qed simp
  thus "eventually (λx. dist (f x) g < e) sequentially"
    by eventually_elim (auto simp add: dist_fmap_def finite_proj_diag ind_f ‹0 < e›)
qed

instance fmap :: (type, complete_space) complete_space
proof
  fix P::"nat ⇒ 'a ⇒⇩F 'b"
  assume "Cauchy P"
  then obtain Nd where Nd: "⋀n. n ≥ Nd ⟹ dist (P n) (P Nd) < 1"
    by (force simp: Cauchy_altdef2)
  define d where "d = domain (P Nd)"
  with Nd have dim: "⋀n. n ≥ Nd ⟹ domain (P n) = d" using dist_le_1_imp_domain_eq by auto
  have [simp]: "finite d" unfolding d_def by simp
  define p where "p i n = P n i" for i n
  define q where "q i = lim (p i)" for i
  define Q where "Q = finmap_of d q"
  have q: "⋀i. i ∈ d ⟹ q i = Q i" by (auto simp add: Q_def Abs_fmap_inverse)
  {
    fix i assume "i ∈ d"
    have "Cauchy (p i)" unfolding Cauchy_altdef2 p_def
    proof safe
      fix e::real assume "0 < e"
      with ‹Cauchy P› obtain N where N: "⋀n. n≥N ⟹ dist (P n) (P N) < min e 1"
        by (force simp: Cauchy_altdef2 min_def)
      hence "⋀n. n ≥ N ⟹ domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
      with dim have dim: "⋀n. n ≥ N ⟹ domain (P n) = d" by (metis nat_le_linear)
      show "∃N. ∀n≥N. dist ((P n) i) ((P N) i) < e"
      proof (safe intro!: exI[where x="N"])
        fix n assume "N ≤ n" have "N ≤ N" by simp
        have "dist ((P n) i) ((P N) i) ≤ dist (P n) (P N)"
          using dim[OF ‹N ≤ n›]  dim[OF ‹N ≤ N›] ‹i ∈ d›
          by (auto intro!: dist_proj)
        also have "… < e" using N[OF ‹N ≤ n›] by simp
        finally show "dist ((P n) i) ((P N) i) < e" .
      qed
    qed
    hence "convergent (p i)" by (metis Cauchy_convergent_iff)
    hence "p i ⇢ q i" unfolding q_def convergent_def by (metis limI)
  } note p = this
  have "P ⇢ Q"
  proof (rule metric_LIMSEQ_I)
    fix e::real assume "0 < e"
    have "∃ni. ∀i∈d. ∀n≥ni i. dist (p i n) (q i) < e"
    proof (safe intro!: bchoice)
      fix i assume "i ∈ d"
      from p[OF ‹i ∈ d›, THEN metric_LIMSEQ_D, OF ‹0 < e›]
      show "∃no. ∀n≥no. dist (p i n) (q i) < e" .
    qed
    then obtain ni where ni: "∀i∈d. ∀n≥ni i. dist (p i n) (q i) < e" ..
    define N where "N = max Nd (Max (ni ` d))"
    show "∃N. ∀n≥N. dist (P n) Q < e"
    proof (safe intro!: exI[where x="N"])
      fix n assume "N ≤ n"
      hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
        using dim by (simp_all add: N_def Q_def dim_def Abs_fmap_inverse)
      show "dist (P n) Q < e"
      proof (rule dist_finmap_lessI[OF dom(3) ‹0 < e›])
        fix i
        assume "i ∈ domain (P n)"
        hence "ni i ≤ Max (ni ` d)" using dom by simp
        also have "… ≤ N" by (simp add: N_def)
        finally show "dist ((P n)⇩F i) ((Q)⇩F i) < e" using ni ‹i ∈ domain (P n)› ‹N ≤ n› dom
          by (auto simp: p_def q N_def less_imp_le)
      qed
    qed
  qed
  thus "convergent P" by (auto simp: convergent_def)
qed

subsection ‹Second Countable Space of Finite Maps›

instantiation fmap :: (countable, second_countable_topology) second_countable_topology
begin

definition basis_proj::"'b set set"
  where "basis_proj = (SOME B. countable B ∧ topological_basis B)"

lemma countable_basis_proj: "countable basis_proj" and basis_proj: "topological_basis basis_proj"
  unfolding basis_proj_def by (intro is_basis countable_basis)+

definition basis_finmap::"('a ⇒⇩F 'b) set set"
  where "basis_finmap = {Pi' I S|I S. finite I ∧ (∀i ∈ I. S i ∈ basis_proj)}"

lemma in_basis_finmapI:
  assumes "finite I" assumes "⋀i. i ∈ I ⟹ S i ∈ basis_proj"
  shows "Pi' I S ∈ basis_finmap"
  using assms unfolding basis_finmap_def by auto

lemma basis_finmap_eq:
  assumes "basis_proj ≠ {}"
  shows "basis_finmap = (λf. Pi' (domain f) (λi. from_nat_into basis_proj ((f)⇩F i))) `
    (UNIV::('a ⇒⇩F nat) set)" (is "_ = ?f ` _")
  unfolding basis_finmap_def
proof safe
  fix I::"'a set" and S::"'a ⇒ 'b set"
  assume "finite I" "∀i∈I. S i ∈ basis_proj"
  hence "Pi' I S = ?f (finmap_of I (λx. to_nat_on basis_proj (S x)))"
    by (force simp: Pi'_def countable_basis_proj)
  thus "Pi' I S ∈ range ?f" by simp
next
  fix x and f::"'a ⇒⇩F nat"
  show "∃I S. (Π' i∈domain f. from_nat_into basis_proj ((f)⇩F i)) = Pi' I S ∧
    finite I ∧ (∀i∈I. S i ∈ basis_proj)"
    using assms by (auto intro: from_nat_into)
qed

lemma basis_finmap_eq_empty: "basis_proj = {} ⟹ basis_finmap = {Pi' {} undefined}"
  by (auto simp: Pi'_iff basis_finmap_def)

lemma countable_basis_finmap: "countable basis_finmap"
  by (cases "basis_proj = {}") (auto simp: basis_finmap_eq basis_finmap_eq_empty)

lemma finmap_topological_basis:
  "topological_basis basis_finmap"
proof (subst topological_basis_iff, safe)
  fix B' assume "B' ∈ basis_finmap"
  thus "open B'"
    by (auto intro!: open_Pi'I topological_basis_open[OF basis_proj]
      simp: topological_basis_def basis_finmap_def Let_def)
next
  fix O'::"('a ⇒⇩F 'b) set" and x
  assume O': "open O'" "x ∈ O'"
  then obtain a where a:
    "x ∈ Pi' (domain x) a" "Pi' (domain x) a ⊆ O'" "⋀i. i∈domain x ⟹ open (a i)"
    unfolding open_fmap_def
  proof (atomize_elim, induct rule: generate_topology.induct)
    case (Int a b)
    let ?p="λa f. x ∈ Pi' (domain x) f ∧ Pi' (domain x) f ⊆ a ∧ (∀i. i ∈ domain x ⟶ open (f i))"
    from Int obtain f g where "?p a f" "?p b g" by auto
    thus ?case by (force intro!: exI[where x="λi. f i ∩ g i"] simp: Pi'_def)
  next
    case (UN k)
    then obtain kk a where "x ∈ kk" "kk ∈ k" "x ∈ Pi' (domain x) a" "Pi' (domain x) a ⊆ kk"
      "⋀i. i∈domain x ⟹ open (a i)"
      by force
    thus ?case by blast
  qed (auto simp: Pi'_def)
  have "∃B.
    (∀i∈domain x. x i ∈ B i ∧ B i ⊆ a i ∧ B i ∈ basis_proj)"
  proof (rule bchoice, safe)
    fix i assume "i ∈ domain x"
    hence "open (a i)" "x i ∈ a i" using a by auto
    from topological_basisE[OF basis_proj this] obtain b'
      where "b' ∈ basis_proj" "(x)⇩F i ∈ b'" "b' ⊆ a i"
      by blast
    thus "∃y. x i ∈ y ∧ y ⊆ a i ∧ y ∈ basis_proj" by auto
  qed
  then obtain B where B: "∀i∈domain x. (x)⇩F i ∈ B i ∧ B i ⊆ a i ∧ B i ∈ basis_proj"
    by auto
  define B' where "B' = Pi' (domain x) (λi. (B i)::'b set)"
  have "B' ⊆ Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def)
  also note ‹… ⊆ O'›
  finally show "∃B'∈basis_finmap. x ∈ B' ∧ B' ⊆ O'" using B
    by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def)
qed

lemma range_enum_basis_finmap_imp_open:
  assumes "x ∈ basis_finmap"
  shows "open x"
  using finmap_topological_basis assms by (auto simp: topological_basis_def)

instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap topological_basis_imp_subbasis)

end

subsection ‹Polish Space of Finite Maps›

instance fmap :: (countable, polish_space) polish_space proof qed


subsection ‹Product Measurable Space of Finite Maps›

definition "PiF I M ≡
  sigma (⋃J ∈ I. (Π' j∈J. space (M j))) {(Π' j∈J. X j) |X J. J ∈ I ∧ X ∈ (Π j∈J. sets (M j))}"

abbreviation
  "Pi⇩F I M ≡ PiF I M"

syntax
  "_PiF" :: "pttrn ⇒ 'i set ⇒ 'a measure ⇒ ('i => 'a) measure"  ("(3Π⇩F _∈_./ _)"  10)
translations
  "Π⇩F x∈I. M" == "CONST PiF I (%x. M)"

lemma PiF_gen_subset: "{(Π' j∈J. X j) |X J. J ∈ I ∧ X ∈ (Π j∈J. sets (M j))} ⊆
    Pow (⋃J ∈ I. (Π' j∈J. space (M j)))"
  by (auto simp: Pi'_def) (blast dest: sets.sets_into_space)

lemma space_PiF: "space (PiF I M) = (⋃J ∈ I. (Π' j∈J. space (M j)))"
  unfolding PiF_def using PiF_gen_subset by (rule space_measure_of)

lemma sets_PiF:
  "sets (PiF I M) = sigma_sets (⋃J ∈ I. (Π' j∈J. space (M j)))
    {(Π' j∈J. X j) |X J. J ∈ I ∧ X ∈ (Π j∈J. sets (M j))}"
  unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of)

lemma sets_PiF_singleton:
  "sets (PiF {I} M) = sigma_sets (Π' j∈I. space (M j))
    {(Π' j∈I. X j) |X. X ∈ (Π j∈I. sets (M j))}"
  unfolding sets_PiF by simp

lemma in_sets_PiFI:
  assumes "X = (Pi' J S)" "J ∈ I" "⋀i. i∈J ⟹ S i ∈ sets (M i)"
  shows "X ∈ sets (PiF I M)"
  unfolding sets_PiF
  using assms by blast

lemma product_in_sets_PiFI:
  assumes "J ∈ I" "⋀i. i∈J ⟹ S i ∈ sets (M i)"
  shows "(Pi' J S) ∈ sets (PiF I M)"
  unfolding sets_PiF
  using assms by blast

lemma singleton_space_subset_in_sets:
  fixes J
  assumes "J ∈ I"
  assumes "finite J"
  shows "space (PiF {J} M) ∈ sets (PiF I M)"
  using assms
  by (intro in_sets_PiFI[where J=J and S="λi. space (M i)"])
      (auto simp: product_def space_PiF)

lemma singleton_subspace_set_in_sets:
  assumes A: "A ∈ sets (PiF {J} M)"
  assumes "finite J"
  assumes "J ∈ I"
  shows "A ∈ sets (PiF I M)"
  using A[unfolded sets_PiF]
  apply (induct A)
  unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
  using assms
  by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets)

lemma finite_measurable_singletonI:
  assumes "finite I"
  assumes "⋀J. J ∈ I ⟹ finite J"
  assumes MN: "⋀J. J ∈ I ⟹ A ∈ measurable (PiF {J} M) N"
  shows "A ∈ measurable (PiF I M) N"
  unfolding measurable_def
proof safe
  fix y assume "y ∈ sets N"
  have "A -` y ∩ space (PiF I M) = (⋃J∈I. A -` y ∩ space (PiF {J} M))"
    by (auto simp: space_PiF)
  also have "… ∈ sets (PiF I M)"
  proof (rule sets.finite_UN)
    show "finite I" by fact
    fix J assume "J ∈ I"
    with assms have "finite J" by simp
    show "A -` y ∩ space (PiF {J} M) ∈ sets (PiF I M)"
      by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+
  qed
  finally show "A -` y ∩ space (PiF I M) ∈ sets (PiF I M)" .
next
  fix x assume "x ∈ space (PiF I M)" thus "A x ∈ space N"
    using MN[of "domain x"]
    by (auto simp: space_PiF measurable_space Pi'_def)
qed

lemma countable_finite_comprehension:
  fixes f :: "'a::countable set ⇒ _"
  assumes "⋀s. P s ⟹ finite s"
  assumes "⋀s. P s ⟹ f s ∈ sets M"
  shows "⋃{f s|s. P s} ∈ sets M"
proof -
  have "⋃{f s|s. P s} = (⋃n::nat. let s = set (from_nat n) in if P s then f s else {})"
  proof safe
    fix x X s assume *: "x ∈ f s" "P s"
    with assms obtain l where "s = set l" using finite_list by blast
    with * show "x ∈ (⋃n. let s = set (from_nat n) in if P s then f s else {})" using ‹P s›
      by (auto intro!: exI[where x="to_nat l"])
  next
    fix x n assume "x ∈ (let s = set (from_nat n) in if P s then f s else {})"
    thus "x ∈ ⋃{f s|s. P s}" using assms by (auto simp: Let_def split: if_split_asm)
  qed
  hence "⋃{f s|s. P s} = (⋃n. let s = set (from_nat n) in if P s then f s else {})" by simp
  also have "… ∈ sets M" using assms by (auto simp: Let_def)
  finally show ?thesis .
qed

lemma space_subset_in_sets:
  fixes J::"'a::countable set set"
  assumes "J ⊆ I"
  assumes "⋀j. j ∈ J ⟹ finite j"
  shows "space (PiF J M) ∈ sets (PiF I M)"
proof -
  have "space (PiF J M) = ⋃{space (PiF {j} M)|j. j ∈ J}"
    unfolding space_PiF by blast
  also have "… ∈ sets (PiF I M)" using assms
    by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets)
  finally show ?thesis .
qed

lemma subspace_set_in_sets:
  fixes J::"'a::countable set set"
  assumes A: "A ∈ sets (PiF J M)"
  assumes "J ⊆ I"
  assumes "⋀j. j ∈ J ⟹ finite j"
  shows "A ∈ sets (PiF I M)"
  using A[unfolded sets_PiF]
  apply (induct A)
  unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
  using assms
  by (auto intro: in_sets_PiFI intro!: space_subset_in_sets)

lemma countable_measurable_PiFI:
  fixes I::"'a::countable set set"
  assumes MN: "⋀J. J ∈ I ⟹ finite J ⟹ A ∈ measurable (PiF {J} M) N"
  shows "A ∈ measurable (PiF I M) N"
  unfolding measurable_def
proof safe
  fix y assume "y ∈ sets N"
  have "A -` y = (⋃{A -` y ∩ {x. domain x = J}|J. finite J})" by auto
  { fix x::"'a ⇒⇩F 'b"
    from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto
    hence "∃n. domain x = set (from_nat n)"
      by (intro exI[where x="to_nat xs"]) auto }
  hence "A -` y ∩ space (PiF I M) = (⋃n. A -` y ∩ space (PiF ({set (from_nat n)}∩I) M))"
    by (auto simp: space_PiF Pi'_def)
  also have "… ∈ sets (PiF I M)"
    apply (intro sets.Int sets.countable_nat_UN subsetI, safe)
    apply (case_tac "set (from_nat i) ∈ I")
    apply simp_all
    apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
    using assms ‹y ∈ sets N›
    apply (auto simp: space_PiF)
    done
  finally show "A -` y ∩ space (PiF I M) ∈ sets (PiF I M)" .
next
  fix x assume "x ∈ space (PiF I M)" thus "A x ∈ space N"
    using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def)
qed

lemma measurable_PiF:
  assumes f: "⋀x. x ∈ space N ⟹ domain (f x) ∈ I ∧ (∀i∈domain (f x). (f x) i ∈ space (M i))"
  assumes S: "⋀J S. J ∈ I ⟹ (⋀i. i ∈ J ⟹ S i ∈ sets (M i)) ⟹
    f -` (Pi' J S) ∩ space N ∈ sets N"
  shows "f ∈ measurable N (PiF I M)"
  unfolding PiF_def
  using PiF_gen_subset
  apply (rule measurable_measure_of)
  using f apply force
  apply (insert S, auto)
  done

lemma restrict_sets_measurable:
  assumes A: "A ∈ sets (PiF I M)" and "J ⊆ I"
  shows "A ∩ {m. domain m ∈ J} ∈ sets (PiF J M)"
  using A[unfolded sets_PiF]
proof (induct A)
  case (Basic a)
  then obtain K S where S: "a = Pi' K S" "K ∈ I" "(∀i∈K. S i ∈ sets (M i))"
    by auto
  show ?case
  proof cases
    assume "K ∈ J"
    hence "a ∩ {m. domain m ∈ J} ∈ {Pi' K X |X K. K ∈ J ∧ X ∈ (Π j∈K. sets (M j))}" using S
      by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def)
    also have "… ⊆ sets (PiF J M)" unfolding sets_PiF by auto
    finally show ?thesis .
  next
    assume "K ∉ J"
    hence "a ∩ {m. domain m ∈ J} = {}" using S by (auto simp: Pi'_def)
    also have "… ∈ sets (PiF J M)" by simp
    finally show ?thesis .
  qed
next
  case (Union a)
  have "⋃(a ` UNIV) ∩ {m. domain m ∈ J} = (⋃i. (a i ∩ {m. domain m ∈ J}))"
    by simp
  also have "… ∈ sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto
  finally show ?case .
next
  case (Compl a)
  have "(space (PiF I M) - a) ∩ {m. domain m ∈ J} = (space (PiF J M) - (a ∩ {m. domain m ∈ J}))"
    using ‹J ⊆ I› by (auto simp: space_PiF Pi'_def)
  also have "… ∈ sets (PiF J M)" using Compl by auto
  finally show ?case by (simp add: space_PiF)
qed simp

lemma measurable_finmap_of:
  assumes f: "⋀i. (∃x ∈ space N. i ∈ J x) ⟹ (λx. f x i) ∈ measurable N (M i)"
  assumes J: "⋀x. x ∈ space N ⟹ J x ∈ I" "⋀x. x ∈ space N ⟹ finite (J x)"
  assumes JN: "⋀S. {x. J x = S} ∩ space N ∈ sets N"
  shows "(λx. finmap_of (J x) (f x)) ∈ measurable N (PiF I M)"
proof (rule measurable_PiF)
  fix x assume "x ∈ space N"
  with J[of x] measurable_space[OF f]
  show "domain (finmap_of (J x) (f x)) ∈ I ∧
        (∀i∈domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i ∈ space (M i))"
    by auto
next
  fix K S assume "K ∈ I" and *: "⋀i. i ∈ K ⟹ S i ∈ sets (M i)"
  with J have eq: "(λx. finmap_of (J x) (f x)) -` Pi' K S ∩ space N =
    (if ∃x ∈ space N. K = J x ∧ finite K then if K = {} then {x ∈ space N. J x = K}
      else (⋂i∈K. (λx. f x i) -` S i ∩ {x ∈ space N. J x = K}) else {})"
    by (auto simp: Pi'_def)
  have r: "{x ∈ space N. J x = K} = space N ∩ ({x. J x = K} ∩ space N)" by auto
  show "(λx. finmap_of (J x) (f x)) -` Pi' K S ∩ space N ∈ sets N"
    unfolding eq r
    apply (simp del: INT_simps add: )
    apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top])
    apply simp apply assumption
    apply (subst Int_assoc[symmetric])
    apply (rule sets.Int)
    apply (intro measurable_sets[OF f] *) apply force apply assumption
    apply (intro JN)
    done
qed

lemma measurable_PiM_finmap_of:
  assumes "finite J"
  shows "finmap_of J ∈ measurable (Pi⇩M J M) (PiF {J} M)"
  apply (rule measurable_finmap_of)
  apply (rule measurable_component_singleton)
  apply simp
  apply rule
  apply (rule ‹finite J›)
  apply simp
  done

lemma proj_measurable_singleton:
  assumes "A ∈ sets (M i)"
  shows "(λx. (x)⇩F i) -` A ∩ space (PiF {I} M) ∈ sets (PiF {I} M)"
proof cases
  assume "i ∈ I"
  hence "(λx. (x)⇩F i) -` A ∩ space (PiF {I} M) =
    Pi' I (λx. if x = i then A else space (M x))"
    using sets.sets_into_space[OF ] ‹A ∈ sets (M i)› assms
    by (auto simp: space_PiF Pi'_def)
  thus ?thesis  using assms ‹A ∈ sets (M i)›
    by (intro in_sets_PiFI) auto
next
  assume "i ∉ I"
  hence "(λx. (x)⇩F i) -` A ∩ space (PiF {I} M) =
    (if undefined ∈ A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def)
  thus ?thesis by simp
qed

lemma measurable_proj_singleton:
  assumes "i ∈ I"
  shows "(λx. (x)⇩F i) ∈ measurable (PiF {I} M) (M i)"
  by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
     (insert ‹i ∈ I›, auto simp: space_PiF)

lemma measurable_proj_countable:
  fixes I::"'a::countable set set"
  assumes "y ∈ space (M i)"
  shows "(λx. if i ∈ domain x then (x)⇩F i else y) ∈ measurable (PiF I M) (M i)"
proof (rule countable_measurable_PiFI)
  fix J assume "J ∈ I" "finite J"
  show "(λx. if i ∈ domain x then x i else y) ∈ measurable (PiF {J} M) (M i)"
    unfolding measurable_def
  proof safe
    fix z assume "z ∈ sets (M i)"
    have "(λx. if i ∈ domain x then x i else y) -` z ∩ space (PiF {J} M) =
      (λx. if i ∈ J then (x)⇩F i else y) -` z ∩ space (PiF {J} M)"
      by (auto simp: space_PiF Pi'_def)
    also have "… ∈ sets (PiF {J} M)" using ‹z ∈ sets (M i)› ‹finite J›
      by (cases "i ∈ J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
    finally show "(λx. if i ∈ domain x then x i else y) -` z ∩ space (PiF {J} M) ∈
      sets (PiF {J} M)" .
  qed (insert ‹y ∈ space (M i)›, auto simp: space_PiF Pi'_def)
qed

lemma measurable_restrict_proj:
  assumes "J ∈ II" "finite J"
  shows "finmap_of J ∈ measurable (PiM J M) (PiF II M)"
  using assms
  by (intro measurable_finmap_of measurable_component_singleton) auto

lemma measurable_proj_PiM:
  fixes J K ::"'a::countable set" and I::"'a set set"
  assumes "finite J" "J ∈ I"
  assumes "x ∈ space (PiM J M)"
  shows "proj ∈ measurable (PiF {J} M) (PiM J M)"
proof (rule measurable_PiM_single)
  show "proj ∈ space (PiF {J} M) → (Π⇩E i ∈ J. space (M i))"
    using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def)
next
  fix A i assume A: "i ∈ J" "A ∈ sets (M i)"
  show "{ω ∈ space (PiF {J} M). (ω)⇩F i ∈ A} ∈ sets (PiF {J} M)"
  proof
    have "{ω ∈ space (PiF {J} M). (ω)⇩F i ∈ A} =
      (λω. (ω)⇩F i) -` A ∩ space (PiF {J} M)" by auto
    also have "… ∈ sets (PiF {J} M)"
      using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM)
    finally show ?thesis .
  qed simp
qed

lemma space_PiF_singleton_eq_product:
  assumes "finite I"
  shows "space (PiF {I} M) = (Π' i∈I. space (M i))"
  by (auto simp: product_def space_PiF assms)

text ‹adapted from @{thm sets_PiM_single}›

lemma sets_PiF_single:
  assumes "finite I" "I ≠ {}"
  shows "sets (PiF {I} M) =
    sigma_sets (Π' i∈I. space (M i))
      {{f∈Π' i∈I. space (M i). f i ∈ A} | i A. i ∈ I ∧ A ∈ sets (M i)}"
    (is "_ = sigma_sets ?Ω ?R")
  unfolding sets_PiF_singleton
proof (rule sigma_sets_eqI)
  interpret R: sigma_algebra ?Ω "sigma_sets ?Ω ?R" by (rule sigma_algebra_sigma_sets) auto
  fix A assume "A ∈ {Pi' I X |X. X ∈ (Π j∈I. sets (M j))}"
  then obtain X where X: "A = Pi' I X" "X ∈ (Π j∈I. sets (M j))" by auto
  show "A ∈ sigma_sets ?Ω ?R"
  proof -
    from ‹I ≠ {}› X have "A = (⋂j∈I. {f∈space (PiF {I} M). f j ∈ X j})"
      using sets.sets_into_space
      by (auto simp: space_PiF product_def) blast
    also have "… ∈ sigma_sets ?Ω ?R"
      using X ‹I ≠ {}› assms by (intro R.finite_INT) (auto simp: space_PiF)
    finally show "A ∈ sigma_sets ?Ω ?R" .
  qed
next
  fix A assume "A ∈ ?R"
  then obtain i B where A: "A = {f∈Π' i∈I. space (M i). f i ∈ B}" "i ∈ I" "B ∈ sets (M i)"
    by auto
  then have "A = (Π' j ∈ I. if j = i then B else space (M j))"
    using sets.sets_into_space[OF A(3)]
    apply (auto simp: Pi'_iff split: if_split_asm)
    apply blast
    done
  also have "… ∈ sigma_sets ?Ω {Pi' I X |X. X ∈ (Π j∈I. sets (M j))}"
    using A
    by (intro sigma_sets.Basic )
       (auto intro: exI[where x="λj. if j = i then B else space (M j)"])
  finally show "A ∈ sigma_sets ?Ω {Pi' I X |X. X ∈ (Π j∈I. sets (M j))}" .
qed

text ‹adapted from @{thm PiE_cong}›

lemma Pi'_cong:
  assumes "finite I"
  assumes "⋀i. i ∈ I ⟹ f i = g i"
  shows "Pi' I f = Pi' I g"
using assms by (auto simp: Pi'_def)

text ‹adapted from @{thm Pi_UN}›

lemma Pi'_UN:
  fixes A :: "nat ⇒ 'i ⇒ 'a set"
  assumes "finite I"
  assumes mono: "⋀i n m. i ∈ I ⟹ n ≤ m ⟹ A n i ⊆ A m i"
  shows "(⋃n. Pi' I (A n)) = Pi' I (λi. ⋃n. A n i)"
proof (intro set_eqI iffI)
  fix f assume "f ∈ Pi' I (λi. ⋃n. A n i)"
  then have "∀i∈I. ∃n. f i ∈ A n i" "domain f = I" by (auto simp: ‹finite I› Pi'_def)
  from bchoice[OF this(1)] obtain n where n: "⋀i. i ∈ I ⟹ f i ∈ (A (n i) i)" by auto
  obtain k where k: "⋀i. i ∈ I ⟹ n i ≤ k"
    using ‹finite I› finite_nat_set_iff_bounded_le[of "n`I"] by auto
  have "f ∈ Pi' I (λi. A k i)"
  proof
    fix i assume "i ∈ I"
    from mono[OF this, of "n i" k] k[OF this] n[OF this] ‹domain f = I› ‹i ∈ I›
    show "f i ∈ A k i " by (auto simp: ‹finite I›)
  qed (simp add: ‹domain f = I› ‹finite I›)
  then show "f ∈ (⋃n. Pi' I (A n))" by auto
qed (auto simp: Pi'_def ‹finite I›)

text ‹adapted from @{thm sets_PiM_sigma}›

lemma sigma_fprod_algebra_sigma_eq:
  fixes E :: "'i ⇒ 'a set set" and S :: "'i ⇒ nat ⇒ 'a set"
  assumes [simp]: "finite I" "I ≠ {}"
    and S_union: "⋀i. i ∈ I ⟹ (⋃j. S i j) = space (M i)"
    and S_in_E: "⋀i. i ∈ I ⟹ range (S i) ⊆ E i"
  assumes E_closed: "⋀i. i ∈ I ⟹ E i ⊆ Pow (space (M i))"
    and E_generates: "⋀i. i ∈ I ⟹ sets (M i) = sigma_sets (space (M i)) (E i)"
  defines "P == { Pi' I F | F. ∀i∈I. F i ∈ E i }"
  shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
proof
  let ?P = "sigma (space (Pi⇩F {I} M)) P"
  from ‹finite I›[THEN ex_bij_betw_finite_nat] obtain T where "bij_betw T I {0..<card I}" ..
  then have T: "⋀i. i ∈ I ⟹ T i < card I" "⋀i. i∈I ⟹ the_inv_into I T (T i) = i"
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: ‹finite I›)
  have P_closed: "P ⊆ Pow (space (Pi⇩F {I} M))"
    using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
  then have space_P: "space ?P = (Π' i∈I. space (M i))"
    by (simp add: space_PiF)
  have "sets (PiF {I} M) =
      sigma_sets (space ?P) {{f ∈ Π' i∈I. space (M i). f i ∈ A} |i A. i ∈ I ∧ A ∈ sets (M i)}"
    using sets_PiF_single[of I M] by (simp add: space_P)
  also have "… ⊆ sets (sigma (space (PiF {I} M)) P)"
  proof (safe intro!: sets.sigma_sets_subset)
    fix i A assume "i ∈ I" and A: "A ∈ sets (M i)"
    have "(λx. (x)⇩F i) ∈ measurable ?P (sigma (space (M i)) (E i))"
    proof (subst measurable_iff_measure_of)
      show "E i ⊆ Pow (space (M i))" using ‹i ∈ I› by fact
      from space_P ‹i ∈ I› show "(λx. (x)⇩F i) ∈ space ?P → space (M i)"
        by auto
      show "∀A∈E i. (λx. (x)⇩F i) -` A ∩ space ?P ∈ sets ?P"
      proof
        fix A assume A: "A ∈ E i"
        from T have *: "∃xs. length xs = card I
          ∧ (∀j∈I. (g)⇩F j ∈ (if i = j then A else S j (xs ! T j)))"
          if "domain g = I" and "∀j∈I. (g)⇩F j ∈ (if i = j then A else S j (f j))" for g f
          using that by (auto intro!: exI [of _ "map (λn. f (the_inv_into I T n)) [0..<card I]"])
        from A have "(λx. (x)⇩F i) -` A ∩ space ?P = (Π' j∈I. if i = j then A else space (M j))"
          using E_closed ‹i ∈ I› by (auto simp: space_P Pi_iff subset_eq split: if_split_asm)
        also have "… = (Π' j∈I. ⋃n. if i = j then A else S j n)"
          by (intro Pi'_cong) (simp_all add: S_union)
        also have "… = {g. domain g = I ∧ (∃f. ∀j∈I. (g)⇩F j ∈ (if i = j then A else S j (f j)))}"
          by (rule set_eqI) (simp del: if_image_distrib add: Pi'_iff bchoice_iff)
        also have "… = (⋃xs∈{xs. length xs = card I}. Π' j∈I. if i = j then A else S j (xs ! T j))"
          using * by (auto simp add: Pi'_iff split del: if_split)
        also have "… ∈ sets ?P"
        proof (safe intro!: sets.countable_UN)
          fix xs show "(Π' j∈I. if i = j then A else S j (xs ! T j)) ∈ sets ?P"
            using A S_in_E
            by (simp add: P_closed)
               (auto simp: P_def subset_eq intro!: exI[of _ "λj. if i = j then A else S j (xs ! T j)"])
        qed
        finally show "(λx. (x)⇩F i) -` A ∩ space ?P ∈ sets ?P"
          using P_closed by simp
      qed
    qed
    from measurable_sets[OF this, of A] A ‹i ∈ I› E_closed
    have "(λx. (x)⇩F i) -` A ∩ space ?P ∈ sets ?P"
      by (simp add: E_generates)
    also have "(λx. (x)⇩F i) -` A ∩ space ?P = {f ∈ Π' i∈I. space (M i). f i ∈ A}"
      using P_closed by (auto simp: space_PiF)
    finally show "… ∈ sets ?P" .
  qed
  finally show "sets (PiF {I} M) ⊆ sigma_sets (space (PiF {I} M)) P"
    by (simp add: P_closed)
  show "sigma_sets (space (PiF {I} M)) P ⊆ sets (PiF {I} M)"
    using ‹finite I› ‹I ≠ {}›
    by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
qed

lemma product_open_generates_sets_PiF_single:
  assumes "I ≠ {}"
  assumes [simp]: "finite I"
  shows "sets (PiF {I} (λ_. borel::'b::second_countable_topology measure)) =
    sigma_sets (space (PiF {I} (λ_. borel))) {Pi' I F |F. (∀i∈I. F i ∈ Collect open)}"
proof -
  from open_countable_basisE[OF open_UNIV] obtain S::"'b set set"
    where S: "S ⊆ (SOME B. countable B ∧ topological_basis B)" "UNIV = ⋃ S"
    by auto
  show ?thesis
  proof (rule sigma_fprod_algebra_sigma_eq)
    show "finite I" by simp
    show "I ≠ {}" by fact
    define S' where "S' = from_nat_into S"
    show "(⋃j. S' j) = space borel"
      using S
      apply (auto simp add: from_nat_into countable_basis_proj S'_def basis_proj_def)
      apply (metis (lifting, mono_tags) UNIV_I UnionE basis_proj_def countable_basis_proj countable_subset from_nat_into_surj)
      done
    show "range S' ⊆ Collect open"
      using S
      apply (auto simp add: from_nat_into countable_basis_proj S'_def)
      apply (metis UNIV_not_empty Union_empty from_nat_into subsetD topological_basis_open[OF basis_proj] basis_proj_def)
      done
    show "Collect open ⊆ Pow (space borel)" by simp
    show "sets borel = sigma_sets (space borel) (Collect open)"
      by (simp add: borel_def)
  qed
qed

lemma finmap_UNIV[simp]: "(⋃J∈Collect finite. Π' j∈J. UNIV) = UNIV" by auto

lemma borel_eq_PiF_borel:
  shows "(borel :: ('i::countable ⇒⇩F 'a::polish_space) measure) =
    PiF (Collect finite) (λ_. borel :: 'a measure)"
  unfolding borel_def PiF_def
proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI)
  fix a::"('i ⇒⇩F 'a) set" assume "a ∈ Collect open" hence "open a" by simp
  then obtain B' where B': "B'⊆basis_finmap" "a = ⋃B'"
    using finmap_topological_basis by (force simp add: topological_basis_def)
  have "a ∈ sigma UNIV {Pi' J X |X J. finite J ∧ X ∈ J → sigma_sets UNIV (Collect open)}"
    unfolding ‹a = ⋃B'›
  proof (rule sets.countable_Union)
    from B' countable_basis_finmap show "countable B'" by (metis countable_subset)
  next
    show "B' ⊆ sets (sigma UNIV
      {Pi' J X |X J. finite J ∧ X ∈ J → sigma_sets UNIV (Collect open)})" (is "_ ⊆ sets ?s")
    proof
      fix x assume "x ∈ B'" with B' have "x ∈ basis_finmap" by auto
      then obtain J X where "x = Pi' J X" "finite J" "X ∈ J → sigma_sets UNIV (Collect open)"
        by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj])
      thus "x ∈ sets ?s" by auto
    qed
  qed
  thus "a ∈ sigma_sets UNIV {Pi' J X |X J. finite J ∧ X ∈ J → sigma_sets UNIV (Collect open)}"
    by simp
next
  fix b::"('i ⇒⇩F 'a) set"
  assume "b ∈ {Pi' J X |X J. finite J ∧ X ∈ J → sigma_sets UNIV (Collect open)}"
  hence b': "b ∈ sets (Pi⇩F (Collect finite) (λ_. borel))" by (auto simp: sets_PiF borel_def)
  let ?b = "λJ. b ∩ {x. domain x = J}"
  have "b = ⋃((λJ. ?b J) ` Collect finite)" by auto
  also have "… ∈ sets borel"
  proof (rule sets.countable_Union, safe)
    fix J::"'i set" assume "finite J"
    { assume ef: "J = {}"
      have "?b J ∈ sets borel"
      proof cases
        assume "?b J ≠ {}"
        then obtain f where "f ∈ b" "domain f = {}" using ef by auto
        hence "?b J = {f}" using ‹J = {}›
          by (auto simp: finmap_eq_iff)
        also have "{f} ∈ sets borel" by simp
        finally show ?thesis .
      qed simp
    } moreover {
      assume "J ≠ ({}::'i set)"
      have "(?b J) = b ∩ {m. domain m ∈ {J}}" by auto
      also have "… ∈ sets (PiF {J} (λ_. borel))"
        using b' by (rule restrict_sets_measurable) (auto simp: ‹finite J›)
      also have "… = sigma_sets (space (PiF {J} (λ_. borel)))
        {Pi' (J) F |F. (∀j∈J. F j ∈ Collect open)}"
        (is "_ = sigma_sets _ ?P")
       by (rule product_open_generates_sets_PiF_single[OF ‹J ≠ {}› ‹finite J›])
      also have "… ⊆ sigma_sets UNIV (Collect open)"
        by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF)
      finally have "(?b J) ∈ sets borel" by (simp add: borel_def)
    } ultimately show "(?b J) ∈ sets borel" by blast
  qed (simp add: countable_Collect_finite)
  finally show "b ∈ sigma_sets UNIV (Collect open)" by (simp add: borel_def)
qed (simp add: emeasure_sigma borel_def PiF_def)

subsection ‹Isomorphism between Functions and Finite Maps›

lemma measurable_finmap_compose:
  shows "(λm. compose J m f) ∈ measurable (PiM (f ` J) (λ_. M)) (PiM J (λ_. M))"
  unfolding compose_def by measurable

lemma measurable_compose_inv:
  assumes inj: "⋀j. j ∈ J ⟹ f' (f j) = j"
  shows "(λm. compose (f ` J) m f') ∈ measurable (PiM J (λ_. M)) (PiM (f ` J) (λ_. M))"
  unfolding compose_def by (rule measurable_restrict) (auto simp: inj)

locale function_to_finmap =
  fixes J::"'a set" and f :: "'a ⇒ 'b::countable" and f'
  assumes [simp]: "finite J"
  assumes inv: "i ∈ J ⟹ f' (f i) = i"
begin

text ‹to measure finmaps›

definition "fm = (finmap_of (f ` J)) o (λg. compose (f ` J) g f')"

lemma domain_fm[simp]: "domain (fm x) = f ` J"
  unfolding fm_def by simp

lemma fm_restrict[simp]: "fm (restrict y J) = fm y"
  unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext)

lemma fm_product:
  assumes "⋀i. space (M i) = UNIV"
  shows "fm -` Pi' (f ` J) S ∩ space (Pi⇩M J M) = (Π⇩E j ∈ J. S (f j))"
  using assms
  by (auto simp: inv fm_def compose_def space_PiM Pi'_def)

lemma fm_measurable:
  assumes "f ` J ∈ N"
  shows "fm ∈ measurable (Pi⇩M J (λ_. M)) (Pi⇩F N (λ_. M))"
  unfolding fm_def
proof (rule measurable_comp, rule measurable_compose_inv)
  show "finmap_of (f ` J) ∈ measurable (Pi⇩M (f ` J) (λ_. M)) (PiF N (λ_. M)) "
    using assms by (intro measurable_finmap_of measurable_component_singleton) auto
qed (simp_all add: inv)

lemma proj_fm:
  assumes "x ∈ J"
  shows "fm m (f x) = m x"
  using assms by (auto simp: fm_def compose_def o_def inv)

lemma inj_on_compose_f': "inj_on (λg. compose (f ` J) g f') (extensional J)"
proof (rule inj_on_inverseI)
  fix x::"'a ⇒ 'c" assume "x ∈ extensional J"
  thus "(λx. compose J x f) (compose (f ` J) x f') = x"
    by (auto simp: compose_def inv extensional_def)
qed

lemma inj_on_fm:
  assumes "⋀i. space (M i) = UNIV"
  shows "inj_on fm (space (Pi⇩M J M))"
  using assms
  apply (auto simp: fm_def space_PiM PiE_def)
  apply (rule comp_inj_on)
  apply (rule inj_on_compose_f')
  apply (rule finmap_of_inj_on_extensional_finite)
  apply simp
  apply (auto)
  done

text ‹to measure functions›

definition "mf = (λg. compose J g f) o proj"

lemma mf_fm:
  assumes "x ∈ space (Pi⇩M J (λ_. M))"
  shows "mf (fm x) = x"
proof -
  have "mf (fm x) ∈ extensional J"
    by (auto simp: mf_def extensional_def compose_def)
  moreover
  have "x ∈ extensional J" using assms sets.sets_into_space
    by (force simp: space_PiM PiE_def)
  moreover
  { fix i assume "i ∈ J"
    hence "mf (fm x) i = x i"
      by (auto simp: inv mf_def compose_def fm_def)
  }
  ultimately
  show ?thesis by (rule extensionalityI)
qed

lemma mf_measurable:
  assumes "space M = UNIV"
  shows "mf ∈ measurable (PiF {f ` J} (λ_. M)) (PiM J (λ_. M))"
  unfolding mf_def
proof (rule measurable_comp, rule measurable_proj_PiM)
  show "(λg. compose J g f) ∈ measurable (Pi⇩M (f ` J) (λx. M)) (Pi⇩M J (λ_. M))"
    by (rule measurable_finmap_compose)
qed (auto simp add: space_PiM extensional_def assms)

lemma fm_image_measurable:
  assumes "space M = UNIV"
  assumes "X ∈ sets (Pi⇩M J (λ_. M))"
  shows "fm ` X ∈ sets (PiF {f ` J} (λ_. M))"
proof -
  have "fm ` X = (mf) -` X ∩ space (PiF {f ` J} (λ_. M))"
  proof safe
    fix x assume "x ∈ X"
    with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x ∈ mf -` X" by auto
    show "fm x ∈ space (PiF {f ` J} (λ_. M))" by (simp add: space_PiF assms)
  next
    fix y x
    assume x: "mf y ∈ X"
    assume y: "y ∈ space (PiF {f ` J} (λ_. M))"
    thus "y ∈ fm ` X"
      by (intro image_eqI[OF _ x], unfold finmap_eq_iff)
         (auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def)
  qed
  also have "… ∈ sets (PiF {f ` J} (λ_. M))"
    using assms
    by (intro measurable_sets[OF mf_measurable]) auto
  finally show ?thesis .
qed

lemma fm_image_measurable_finite:
  assumes "space M = UNIV"
  assumes "X ∈ sets (Pi⇩M J (λ_. M::'c measure))"
  shows "fm ` X ∈ sets (PiF (Collect finite) (λ_. M::'c measure))"
  using fm_image_measurable[OF assms]
  by (rule subspace_set_in_sets) (auto simp: finite_subset)

text ‹measure on finmaps›

definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"

lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)"
  unfolding mapmeasure_def by simp

lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
  unfolding mapmeasure_def by simp

lemma mapmeasure_PiF:
  assumes s1: "space M = space (Pi⇩M J (λ_. N))"
  assumes s2: "sets M = sets (Pi⇩M J (λ_. N))"
  assumes "space N = UNIV"
  assumes "X ∈ sets (PiF (Collect finite) (λ_. N))"
  shows "emeasure (mapmeasure M (λ_. N)) X = emeasure M ((fm -` X ∩ extensional J))"
  using assms
  by (auto simp: measurable_cong_sets[OF s2 refl] mapmeasure_def emeasure_distr
    fm_measurable space_PiM PiE_def)

lemma mapmeasure_PiM:
  fixes N::"'c measure"
  assumes s1: "space M = space (Pi⇩M J (λ_. N))"
  assumes s2: "sets M = (Pi⇩M J (λ_. N))"
  assumes N: "space N = UNIV"
  assumes X: "X ∈ sets M"
  shows "emeasure M X = emeasure (mapmeasure M (λ_. N)) (fm ` X)"
  unfolding mapmeasure_def
proof (subst emeasure_distr, subst measurable_cong_sets[OF s2 refl], rule fm_measurable)
  have "X ⊆ space (Pi⇩M J (λ_. N))" using assms by (simp add: sets.sets_into_space)
  from assms inj_on_fm[of "λ_. N"] subsetD[OF this] have "fm -` fm ` X ∩ space (Pi⇩M J (λ_. N)) = X"
    by (auto simp: vimage_image_eq inj_on_def)
  thus "emeasure M X = emeasure M (fm -` fm ` X ∩ space M)" using s1
    by simp
  show "fm ` X ∈ sets (PiF (Collect finite) (λ_. N))"
    by (rule fm_image_measurable_finite[OF N X[simplified s2]])
qed simp

end

end