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 typefmap type can be instantiated to classpolish_space, needed for the proof of
  projective limit. constextensional functions are used for the representation in order to
  stay close to the developments of (finite) products constPiE and their sigma-algebra
  constPiM.›

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  (idomain 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  (ki. 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 termPi 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' xA. B" == "CONST Pi' A (λx. B)"

subsubsection‹Basic Properties of termPi'

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  (iI. 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) = {}  (xA. 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  (PiE 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. ia. 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::"'aF'b"
  have "i. A. countable A  (aA. x i  a)  (aA. open a) 
    (S. open S  x i  S  (aA. 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  AA. 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  (aA 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. idomain x  open (S i)  x i(S i)  (aA 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) ` (PiE (domain x) A)"
  show "A::nat  ('aF'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  PiE (domain x) A" "Pi' (domain x) f  a" "g  PiE (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. ia  open (b i)" by auto
      have "i. a. (i  domain x  open (b i)  (x)F i  b i)  (aA 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 = (xS. 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. ia  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 "xS. 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. xS. domain x = a  b = (λi. ball (x i) (e x))}"
    proof safe
      fix x assume "x  S"
      thus "x  {Pi' a b| a b. xS. 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' idomain 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 = (xS. 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  (aA. 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. idomain 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. iG. 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. nN  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. nN. 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. id. nni 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. nno. dist (p i n) (q i) < e" .
    qed
    then obtain ni where ni: "id. nni i. dist (p i n) (q i) < e" ..
    define N where "N = max Nd (Max (ni ` d))"
    show "N. nN. 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" "iI. 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' idomain f. from_nat_into basis_proj ((f)F i)) = Pi' I S 
    finite I  (iI. 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. idomain 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. idomain x  open (a i)"
      by force
    thus ?case by blast
  qed (auto simp: Pi'_def)
  have "B.
    (idomain 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: "idomain 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' jJ. space (M j))) {(Π' jJ. X j) |X J. J  I  X  (Π jJ. sets (M j))}"

abbreviation
  "PiF I M  PiF I M"

syntax
  "_PiF" :: "pttrn  'i set  'a measure  ('i => 'a) measure"  ("(3ΠF __./ _)"  10)
translations
  "ΠF xI. M" == "CONST PiF I (%x. M)"

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

lemma space_PiF: "space (PiF I M) = (J  I. (Π' jJ. 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' jJ. space (M j)))
    {(Π' jJ. X j) |X J. J  I  X  (Π jJ. 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' jI. space (M j))
    {(Π' jI. X j) |X. X  (Π jI. sets (M j))}"
  unfolding sets_PiF by simp

lemma in_sets_PiFI:
  assumes "X = (Pi' J S)" "J  I" "i. iJ  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. iJ  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) = (JI. 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  (idomain (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" "(iK. 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 jK. 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 
        (idomain (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 (iK. (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 (PiM 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)  (ΠE 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' iI. 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' iI. space (M i))
      {{fΠ' iI. space (M i). f i  A} | i A. i  I  A  sets (M i)}"
    (is "_ = sigma_sets  ?R")
  unfolding sets_PiF_singleton
proof (rule sigma_sets_eqI)
  interpret R: sigma_algebra  "sigma_sets  ?R" by (rule sigma_algebra_sigma_sets) auto
  fix A assume "A  {Pi' I X |X. X  (Π jI. sets (M j))}"
  then obtain X where X: "A = Pi' I X" "X  (Π jI. sets (M j))" by auto
  show "A  sigma_sets  ?R"
  proof -
    from I  {} X have "A = (jI. {fspace (PiF {I} M). f j  X j})"
      using sets.sets_into_space
      by (auto simp: space_PiF product_def) blast
    also have "  sigma_sets  ?R"
      using X I  {} assms by (intro R.finite_INT) (auto simp: space_PiF)
    finally show "A  sigma_sets  ?R" .
  qed
next
  fix A assume "A  ?R"
  then obtain i B where A: "A = {fΠ' iI. 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  {Pi' I X |X. X  (Π jI. 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  {Pi' I X |X. X  (Π jI. 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 "iI. 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. iI. F i  E i }"
  shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
proof
  let ?P = "sigma (space (PiF {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. iI  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 (PiF {I} M))"
    using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
  then have space_P: "space ?P = (Π' iI. space (M i))"
    by (simp add: space_PiF)
  have "sets (PiF {I} M) =
      sigma_sets (space ?P) {{f  Π' iI. 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 "AE 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
           (jI. (g)F j  (if i = j then A else S j (xs ! T j)))"
          if "domain g = I" and "jI. (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' jI. 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' jI. 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. jI. (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' jI. 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' jI. 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' iI. 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. (iI. 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]: "(JCollect finite. Π' jJ. 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 (PiF (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. (jJ. 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 (PiM J M) = (ΠE 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 (PiM J (λ_. M)) (PiF N (λ_. M))"
  unfolding fm_def
proof (rule measurable_comp, rule measurable_compose_inv)
  show "finmap_of (f ` J)  measurable (PiM (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 (PiM 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 (PiM 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 (PiM (f ` J) (λx. M)) (PiM 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 "