Theory FiniteProduct

(*  Title:      HOL/Algebra/FiniteProduct.thy
    Author:     Clemens Ballarin, started 19 November 2002

This file is largely based on HOL/Finite_Set.thy.
*)

theory FiniteProduct
imports Group
begin

subsection Product Operator for Commutative Monoids

subsubsection Inductive Definition of a Relation for Products over Sets

text Instantiation of locale LC› of theory Finite_Set› is not
  possible, because here we have explicit typing rules like
  x ∈ carrier G›.  We introduce an explicit argument for the domain
  D›.

inductive_set
  foldSetD :: "['a set, 'b  'a  'a, 'a]  ('b set * 'a) set"
  for D :: "'a set" and f :: "'b  'a  'a" and e :: 'a
  where
    emptyI [intro]: "e  D  ({}, e)  foldSetD D f e"
  | insertI [intro]: "x  A; f x y  D; (A, y)  foldSetD D f e 
                      (insert x A, f x y)  foldSetD D f e"

inductive_cases empty_foldSetDE [elim!]: "({}, x)  foldSetD D f e"

definition
  foldD :: "['a set, 'b  'a  'a, 'a, 'b set]  'a"
  where "foldD D f e A = (THE x. (A, x)  foldSetD D f e)"

lemma foldSetD_closed: "(A, z)  foldSetD D f e  z  D"
  by (erule foldSetD.cases) auto

lemma Diff1_foldSetD:
  "(A - {x}, y)  foldSetD D f e; x  A; f x y  D 
   (A, f x y)  foldSetD D f e"
  by (metis Diff_insert_absorb foldSetD.insertI mk_disjoint_insert)

lemma foldSetD_imp_finite [simp]: "(A, x)  foldSetD D f e  finite A"
  by (induct set: foldSetD) auto

lemma finite_imp_foldSetD:
  "finite A; e  D; x y. x  A; y  D  f x y  D
     x. (A, x)  foldSetD D f e"
proof (induct set: finite)
  case empty then show ?case by auto
next
  case (insert x F)
  then obtain y where y: "(F, y)  foldSetD D f e" by auto
  with insert have "y  D" by (auto dest: foldSetD_closed)
  with y and insert have "(insert x F, f x y)  foldSetD D f e"
    by (intro foldSetD.intros) auto
  then show ?case ..
qed

lemma foldSetD_backwards:
  assumes "A  {}" "(A, z)  foldSetD D f e"
  shows "x y. x  A  (A - { x }, y)  foldSetD D f e  z = f x y"
  using assms(2) by (cases) (simp add: assms(1), metis Diff_insert_absorb insertI1)

subsubsection Left-Commutative Operations

locale LCD =
  fixes B :: "'b set"
  and D :: "'a set"
  and f :: "'b  'a  'a"    (infixl "" 70)
  assumes left_commute:
    "x  B; y  B; z  D  x  (y  z) = y  (x  z)"
  and f_closed [simp, intro!]: "!!x y. x  B; y  D  f x y  D"

lemma (in LCD) foldSetD_closed [dest]: "(A, z)  foldSetD D f e  z  D"
  by (erule foldSetD.cases) auto

lemma (in LCD) Diff1_foldSetD:
  "(A - {x}, y)  foldSetD D f e; x  A; A  B 
  (A, f x y)  foldSetD D f e"
  by (meson Diff1_foldSetD f_closed local.foldSetD_closed subsetCE)

lemma (in LCD) finite_imp_foldSetD:
  "finite A; A  B; e  D  x. (A, x)  foldSetD D f e"
proof (induct set: finite)
  case empty then show ?case by auto
next
  case (insert x F)
  then obtain y where y: "(F, y)  foldSetD D f e" by auto
  with insert have "y  D" by auto
  with y and insert have "(insert x F, f x y)  foldSetD D f e"
    by (intro foldSetD.intros) auto
  then show ?case ..
qed


lemma (in LCD) foldSetD_determ_aux:
  assumes "e  D" and A: "card A < n" "A  B" "(A, x)  foldSetD D f e" "(A, y)  foldSetD D f e"
  shows "y = x"
  using A
proof (induction n arbitrary: A x y)
  case 0
  then show ?case
    by auto
next
  case (Suc n)
  then consider "card A = n" | "card A < n"
    by linarith
  then show ?case
  proof cases
    case 1
    show ?thesis
      using foldSetD.cases [OF (A,x)  foldSetD D (⋅) e]
    proof cases
      case 1
      then show ?thesis
        using (A,y)  foldSetD D (⋅) e by auto
    next
      case (2 x' A' y')
      note A' = this
      show ?thesis
        using foldSetD.cases [OF (A,y)  foldSetD D (⋅) e]
      proof cases
        case 1
        then show ?thesis
          using (A,x)  foldSetD D (⋅) e by auto
      next
        case (2 x'' A'' y'')
        note A'' = this
        show ?thesis
        proof (cases "x' = x''")
          case True
          show ?thesis
          proof (cases "y' = y''")
            case True
            then show ?thesis
              using A' A'' x' = x'' by (blast elim!: equalityE)
          next
            case False
            then show ?thesis
              using A' A'' x' = x'' 
              by (metis card A = n Suc.IH Suc.prems(2) card_insert_disjoint foldSetD_imp_finite insert_eq_iff insert_subset lessI)
          qed
        next
          case False
          then have *: "A' - {x''} = A'' - {x'}" "x''  A'" "x'  A''"
            using A' A'' by fastforce+
          then have "A' = insert x'' A'' - {x'}"
            using x'  A' by blast
          then have card: "card A'  card A''"
            using A' A'' * by (metis card_Suc_Diff1 eq_refl foldSetD_imp_finite)
          obtain u where u: "(A' - {x''}, u)  foldSetD D (⋅) e"
            using finite_imp_foldSetD [of "A' - {x''}"] A' Diff_insert A  B e  D by fastforce
          have "y' = f x'' u"
            using Diff1_foldSetD [OF u] x''  A' card A = n A' Suc.IH A  B by auto
          then have "(A'' - {x'}, u)  foldSetD D f e"
            using "*"(1) u by auto
          then have "y'' = f x' u"
            using A'' by (metis * card A = n A'(1) Diff1_foldSetD Suc.IH A  B
                card card_Suc_Diff1 card_insert_disjoint foldSetD_imp_finite insert_subset le_imp_less_Suc)
          then show ?thesis
            using A' A''
            by (metis A  B y' = x''  u insert_subset left_commute local.foldSetD_closed u)
        qed   
      qed
    qed
  next
    case 2 with Suc show ?thesis by blast
  qed
qed

lemma (in LCD) foldSetD_determ:
  "(A, x)  foldSetD D f e; (A, y)  foldSetD D f e; e  D; A  B
   y = x"
  by (blast intro: foldSetD_determ_aux [rule_format])

lemma (in LCD) foldD_equality:
  "(A, y)  foldSetD D f e; e  D; A  B  foldD D f e A = y"
  by (unfold foldD_def) (blast intro: foldSetD_determ)

lemma foldD_empty [simp]:
  "e  D  foldD D f e {} = e"
  by (unfold foldD_def) blast

lemma (in LCD) foldD_insert_aux:
  "x  A; x  B; e  D; A  B
     ((insert x A, v)  foldSetD D f e)  (y. (A, y)  foldSetD D f e  v = f x y)"
  apply auto
  by (metis Diff_insert_absorb f_closed finite_Diff foldSetD.insertI foldSetD_determ foldSetD_imp_finite insert_subset local.finite_imp_foldSetD local.foldSetD_closed)

lemma (in LCD) foldD_insert:
  assumes "finite A" "x  A" "x  B" "e  D" "A  B"
  shows "foldD D f e (insert x A) = f x (foldD D f e A)"
proof -
  have "(THE v. y. (A, y)  foldSetD D (⋅) e  v = x  y) = x  (THE y. (A, y)  foldSetD D (⋅) e)"
    by (rule the_equality) (use assms foldD_def foldD_equality foldD_def finite_imp_foldSetD in metis+)
  then show ?thesis
    unfolding foldD_def using assms by (simp add: foldD_insert_aux)
qed

lemma (in LCD) foldD_closed [simp]:
  "finite A; e  D; A  B  foldD D f e A  D"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case insert then show ?case by (simp add: foldD_insert)
qed

lemma (in LCD) foldD_commute:
  "finite A; x  B; e  D; A  B 
   f x (foldD D f e A) = foldD D f (f x e) A"
  by (induct set: finite) (auto simp add: left_commute foldD_insert)

lemma Int_mono2:
  "A  C; B  C  A Int B  C"
  by blast

lemma (in LCD) foldD_nest_Un_Int:
  "finite A; finite C; e  D; A  B; C  B 
   foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
proof (induction set: finite)
  case (insert x F)
  then show ?case 
    by (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb Int_mono2)
qed simp

lemma (in LCD) foldD_nest_Un_disjoint:
  "finite A; finite B; A Int B = {}; e  D; A  B; C  B
     foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
  by (simp add: foldD_nest_Un_Int)

― ‹Delete rules to do with foldSetD› relation.

declare foldSetD_imp_finite [simp del]
  empty_foldSetDE [rule del]
  foldSetD.intros [rule del]
declare (in LCD)
  foldSetD_closed [rule del]


text Commutative Monoids

text 
  We enter a more restrictive context, with f :: 'a ⇒ 'a ⇒ 'a›
  instead of 'b ⇒ 'a ⇒ 'a›.


locale ACeD =
  fixes D :: "'a set"
    and f :: "'a  'a  'a"    (infixl "" 70)
    and e :: 'a
  assumes ident [simp]: "x  D  x  e = x"
    and commute: "x  D; y  D  x  y = y  x"
    and assoc: "x  D; y  D; z  D  (x  y)  z = x  (y  z)"
    and e_closed [simp]: "e  D"
    and f_closed [simp]: "x  D; y  D  x  y  D"

lemma (in ACeD) left_commute:
  "x  D; y  D; z  D  x  (y  z) = y  (x  z)"
proof -
  assume D: "x  D" "y  D" "z  D"
  then have "x  (y  z) = (y  z)  x" by (simp add: commute)
  also from D have "... = y  (z  x)" by (simp add: assoc)
  also from D have "z  x = x  z" by (simp add: commute)
  finally show ?thesis .
qed

lemmas (in ACeD) AC = assoc commute left_commute

lemma (in ACeD) left_ident [simp]: "x  D  e  x = x"
proof -
  assume "x  D"
  then have "x  e = x" by (rule ident)
  with x  D show ?thesis by (simp add: commute)
qed

lemma (in ACeD) foldD_Un_Int:
  "finite A; finite B; A  D; B  D 
    foldD D f e A  foldD D f e B =
    foldD D f e (A Un B)  foldD D f e (A Int B)"
proof (induction set: finite)
  case empty
  then show ?case 
    by(simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
next
  case (insert x F)
  then show ?case
    by(simp add: AC insert_absorb Int_insert_left Int_mono2
                 LCD.foldD_insert [OF LCD.intro [of D]]
                 LCD.foldD_closed [OF LCD.intro [of D]])
qed

lemma (in ACeD) foldD_Un_disjoint:
  "finite A; finite B; A Int B = {}; A  D; B  D 
    foldD D f e (A Un B) = foldD D f e A  foldD D f e B"
  by (simp add: foldD_Un_Int
    left_commute LCD.foldD_closed [OF LCD.intro [of D]])


subsubsection Products over Finite Sets

definition
  finprod :: "[('b, 'm) monoid_scheme, 'a  'b, 'a set]  'b"
  where "finprod G f A =
   (if finite A
    then foldD (carrier G) (mult G  f) 𝟭GA
    else 𝟭G)"

syntax
  "_finprod" :: "index  idt  'a set  'b  'b"
      ("(3___. _)" [1000, 0, 51, 10] 10)
translations
  "GiA. b"  "CONST finprod G (%i. b) A"
  ― ‹Beware of argument permutation!

lemma (in comm_monoid) finprod_empty [simp]:
  "finprod G f {} = 𝟭"
  by (simp add: finprod_def)

lemma (in comm_monoid) finprod_infinite[simp]:
  "¬ finite A  finprod G f A = 𝟭"
  by (simp add: finprod_def)

declare funcsetI [intro]
  funcset_mem [dest]

context comm_monoid begin

lemma finprod_insert [simp]:
  assumes "finite F" "a  F" "f  F  carrier G" "f a  carrier G"
  shows "finprod G f (insert a F) = f a  finprod G f F"
proof -
  have "finprod G f (insert a F) = foldD (carrier G) ((⊗)  f) 𝟭 (insert a F)"
    by (simp add: finprod_def assms)
  also have "... = ((⊗)  f) a (foldD (carrier G) ((⊗)  f) 𝟭 F)"
    by (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
      (use assms in auto simp: m_lcomm Pi_iff)
  also have "... = f a  finprod G f F"
    using finite F by (auto simp add: finprod_def)
  finally show ?thesis .
qed

lemma finprod_one_eqI: "(x. x  A  f x = 𝟭)  finprod G f A = 𝟭"
proof (induct A rule: infinite_finite_induct)
  case empty show ?case by simp
next
  case (insert a A)
  have "(λi. 𝟭)  A  carrier G" by auto
  with insert show ?case by simp
qed simp

lemma finprod_one [simp]: "(iA. 𝟭) = 𝟭"
  by (simp add: finprod_one_eqI)

lemma finprod_closed [simp]:
  fixes A
  assumes f: "f  A  carrier G"
  shows "finprod G f A  carrier G"
using f
proof (induct A rule: infinite_finite_induct)
  case empty show ?case by simp
next
  case (insert a A)
  then have a: "f a  carrier G" by fast
  from insert have A: "f  A  carrier G" by fast
  from insert A a show ?case by simp
qed simp

lemma funcset_Int_left [simp, intro]:
  "f  A  C; f  B  C  f  A Int B  C"
  by fast

lemma funcset_Un_left [iff]:
  "(f  A Un B  C) = (f  A  C  f  B  C)"
  by fast

lemma finprod_Un_Int:
  "finite A; finite B; g  A  carrier G; g  B  carrier G 
     finprod G g (A Un B)  finprod G g (A Int B) =
     finprod G g A  finprod G g B"
― ‹The reversed orientation looks more natural, but LOOPS as a simprule!
proof (induct set: finite)
  case empty then show ?case by simp
next
  case (insert a A)
  then have a: "g a  carrier G" by fast
  from insert have A: "g  A  carrier G" by fast
  from insert A a show ?case
    by (simp add: m_ac Int_insert_left insert_absorb Int_mono2)
qed

lemma finprod_Un_disjoint:
  "finite A; finite B; A Int B = {};
      g  A  carrier G; g  B  carrier G
    finprod G g (A Un B) = finprod G g A  finprod G g B"
  by (metis Pi_split_domain finprod_Un_Int finprod_closed finprod_empty r_one)

lemma finprod_multf [simp]:
  "f  A  carrier G; g  A  carrier G 
   finprod G (λx. f x  g x) A = (finprod G f A  finprod G g A)"
proof (induct A rule: infinite_finite_induct)
  case empty show ?case by simp
next
  case (insert a A) then
  have fA: "f  A  carrier G" by fast
  from insert have fa: "f a  carrier G" by fast
  from insert have gA: "g  A  carrier G" by fast
  from insert have ga: "g a  carrier G" by fast
  from insert have fgA: "(%x. f x  g x)  A  carrier G"
    by (simp add: Pi_def)
  show ?case
    by (simp add: insert fA fa gA ga fgA m_ac)
qed simp

lemma finprod_cong':
  "A = B; g  B  carrier G;
      !!i. i  B  f i = g i  finprod G f A = finprod G g B"
proof -
  assume prems: "A = B" "g  B  carrier G"
    "!!i. i  B  f i = g i"
  show ?thesis
  proof (cases "finite B")
    case True
    then have "!!A. A = B; g  B  carrier G;
      !!i. i  B  f i = g i  finprod G f A = finprod G g B"
    proof induct
      case empty thus ?case by simp
    next
      case (insert x B)
      then have "finprod G f A = finprod G f (insert x B)" by simp
      also from insert have "... = f x  finprod G f B"
      proof (intro finprod_insert)
        show "finite B" by fact
      next
        show "x  B" by fact
      next
        assume "x  B" "!!i. i  insert x B  f i = g i"
          "g  insert x B  carrier G"
        thus "f  B  carrier G" by fastforce
      next
        assume "x  B" "!!i. i  insert x B  f i = g i"
          "g  insert x B  carrier G"
        thus "f x  carrier G" by fastforce
      qed
      also from insert have "... = g x  finprod G g B" by fastforce
      also from insert have "... = finprod G g (insert x B)"
      by (intro finprod_insert [THEN sym]) auto
      finally show ?case .
    qed
    with prems show ?thesis by simp
  next
    case False with prems show ?thesis by simp
  qed
qed

lemma finprod_cong:
  "A = B; f  B  carrier G = True;
      i. i  B =simp=> f i = g i  finprod G f A = finprod G g B"
  (* This order of prems is slightly faster (3%) than the last two swapped. *)
  by (rule finprod_cong') (auto simp add: simp_implies_def)

text Usually, if this rule causes a failed congruence proof error,
  the reason is that the premise g ∈ B → carrier G› cannot be shown.
  Adding @{thm [source] Pi_def} to the simpset is often useful.
  For this reason, @{thm [source] finprod_cong}
  is not added to the simpset by default.


end

declare funcsetI [rule del]
  funcset_mem [rule del]

context comm_monoid begin

lemma finprod_0 [simp]:
  "f  {0::nat}  carrier G  finprod G f {..0} = f 0"
  by (simp add: Pi_def)

lemma finprod_0':
  "f  {..n}  carrier G  (f 0)  finprod G f {Suc 0..n} = finprod G f {..n}"
proof -
  assume A: "f  {.. n}  carrier G"
  hence "(f 0)  finprod G f {Suc 0..n} = finprod G f {..0}  finprod G f {Suc 0..n}"
    using finprod_0[of f] by (simp add: funcset_mem)
  also have " ... = finprod G f ({..0}  {Suc 0..n})"
    using finprod_Un_disjoint[of "{..0}" "{Suc 0..n}" f] A by (simp add: funcset_mem)
  also have " ... = finprod G f {..n}"
    by (simp add: atLeastAtMost_insertL atMost_atLeast0)
  finally show ?thesis .
qed

lemma finprod_Suc [simp]:
  "f  {..Suc n}  carrier G 
   finprod G f {..Suc n} = (f (Suc n)  finprod G f {..n})"
by (simp add: Pi_def atMost_Suc)

lemma finprod_Suc2:
  "f  {..Suc n}  carrier G 
   finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n}  f 0)"
proof (induct n)
  case 0 thus ?case by (simp add: Pi_def)
next
  case Suc thus ?case by (simp add: m_assoc Pi_def)
qed

lemma finprod_Suc3:
  assumes "f  {..n :: nat}  carrier G"
  shows "finprod G f {.. n} = (f n)  finprod G f {..< n}"
proof (cases "n = 0")
  case True thus ?thesis
   using assms atMost_Suc by simp
next
  case False
  then obtain k where "n = Suc k"
    using not0_implies_Suc by blast
  thus ?thesis
    using finprod_Suc[of f k] assms atMost_Suc lessThan_Suc_atMost by simp
qed

lemma finprod_reindex: contributor Jeremy Avigad
  "f  (h ` A)  carrier G 
        inj_on h A  finprod G f (h ` A) = finprod G (λx. f (h x)) A"
proof (induct A rule: infinite_finite_induct)
  case (infinite A)
  hence "¬ finite (h ` A)"
    using finite_imageD by blast
  with ¬ finite A show ?case by simp
qed (auto simp add: Pi_def)

lemma finprod_const: contributor Jeremy Avigad
  assumes a [simp]: "a  carrier G"
    shows "finprod G (λx. a) A = a [^] card A"
proof (induct A rule: infinite_finite_induct)
  case (insert b A)
  show ?case
  proof (subst finprod_insert[OF insert(1-2)])
    show "a  (xA. a) = a [^] card (insert b A)"
      by (insert insert, auto, subst m_comm, auto)
  qed auto
qed auto

lemma finprod_singleton: contributor Jesus Aransay
  assumes i_in_A: "i  A" and fin_A: "finite A" and f_Pi: "f  A  carrier G"
  shows "(jA. if i = j then f j else 𝟭) = f i"
  using i_in_A finprod_insert [of "A - {i}" i "(λj. if i = j then f j else 𝟭)"]
    fin_A f_Pi finprod_one [of "A - {i}"]
    finprod_cong [of "A - {i}" "A - {i}" "(λj. if i = j then f j else 𝟭)" "(λi. 𝟭)"]
  unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)

lemma finprod_singleton_swap:
  assumes i_in_A: "i  A" and fin_A: "finite A" and f_Pi: "f  A  carrier G"
  shows "(jA. if j = i then f j else 𝟭) = f i"
  using finprod_singleton [OF assms] by (simp add: eq_commute)

lemma finprod_mono_neutral_cong_left:
  assumes "finite B"
    and "A  B"
    and 1: "i. i  B - A  h i = 𝟭"
    and gh: "x. x  A  g x = h x"
    and h: "h  B  carrier G"
  shows "finprod G g A = finprod G h B"
proof-
  have eq: "A  (B - A) = B" using A  B by blast
  have d: "A  (B - A) = {}" using A  B by blast
  from finite B A  B have f: "finite A" "finite (B - A)"
    by (auto intro: finite_subset)
  have "h  A  carrier G" "h  B - A  carrier G"
    using assms by (auto simp: image_subset_iff_funcset)
  moreover have "finprod G g A = finprod G h A  finprod G h (B - A)"
  proof -
    have "finprod G h (B - A) = 𝟭"
      using "1" finprod_one_eqI by blast
    moreover have "finprod G g A = finprod G h A"
      using h  A  carrier G finprod_cong' gh by blast
    ultimately show ?thesis
      by (simp add: h  A  carrier G)
  qed
  ultimately show ?thesis
    by (simp add: finprod_Un_disjoint [OF f d, unfolded eq])
qed

lemma finprod_mono_neutral_cong_right:
  assumes "finite B"
    and "A  B" "i. i  B - A  g i = 𝟭" "x. x  A  g x = h x" "g  B  carrier G"
  shows "finprod G g B = finprod G h A"
  using assms  by (auto intro!: finprod_mono_neutral_cong_left [symmetric])

lemma finprod_mono_neutral_cong:
  assumes [simp]: "finite B" "finite A"
    and *: "i. i  B - A  h i = 𝟭" "i. i  A - B  g i = 𝟭"
    and gh: "x. x  A  B  g x = h x"
    and g: "g  A  carrier G"
    and h: "h  B  carrier G"
 shows "finprod G g A = finprod G h B"
proof-
  have "finprod G g A = finprod G g (A  B)"
    by (rule finprod_mono_neutral_cong_right) (use assms in auto)
  also have " = finprod G h (A  B)"
    by (rule finprod_cong) (use assms in auto)
  also have " = finprod G h B"
    by (rule finprod_mono_neutral_cong_left) (use assms in auto)
  finally show ?thesis .
qed

end

(* Jeremy Avigad. This should be generalized to arbitrary groups, not just commutative
   ones, using Lagrange's theorem. *)

lemma (in comm_group) power_order_eq_one:
  assumes fin [simp]: "finite (carrier G)"
    and a [simp]: "a  carrier G"
  shows "a [^] card(carrier G) = one G"
proof -
  have "(xcarrier G. x) = (xcarrier G. a  x)"
    by (subst (2) finprod_reindex [symmetric],
      auto simp add: Pi_def inj_on_cmult surj_const_mult)
  also have " = (xcarrier G. a)  (xcarrier G. x)"
    by (auto simp add: finprod_multf Pi_def)
  also have "(xcarrier G. a) = a [^] card(carrier G)"
    by (auto simp add: finprod_const)
  finally show ?thesis
    by auto
qed

lemma (in comm_monoid) finprod_UN_disjoint:
  assumes
    "finite I" "i. i  I  finite (A i)" "pairwise (λi j. disjnt (A i) (A j)) I"
    "i x. i  I  x  A i  g x  carrier G"
shows "finprod G g ((A ` I)) = finprod G (λi. finprod G g (A i)) I"
  using assms
proof (induction set: finite)
  case empty
  then show ?case
    by force
next
  case (insert i I)
  then show ?case
    unfolding pairwise_def disjnt_def
    apply clarsimp
    apply (subst finprod_Un_disjoint)
         apply (fastforce intro!: funcsetI finprod_closed)+
    done
qed

lemma (in comm_monoid) finprod_Union_disjoint:
  "finite C; A. A  C  finite A  (xA. f x  carrier G); pairwise disjnt C 
    finprod G f (C) = finprod G (finprod G f) C"
  by (frule finprod_UN_disjoint [of C id f]) auto

end