Theory HOL-Algebra.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 A': (2 x' A' y')
      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 A'': (2 x'' A'' y'')
        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) 𝟭⇘G⇙ A
    else 𝟭⇘G⇙)"

syntax
  "_finprod" :: "index ⇒ idt ⇒ 'a set ⇒ 'b ⇒ 'b"
    (‹(‹indent=3 notation=‹binder ⨂››⨂__∈_. _)› [1000, 0, 51, 10] 10)
syntax_consts
  "_finprod" ⇌ finprod
translations
  "⨂⇘G⇙i∈A. 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]: "(⨂i∈A. 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 ⊗ (⨂x∈A. 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 "(⨂j∈A. 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 "(⨂j∈A. 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 "(⨂x∈carrier G. x) = (⨂x∈carrier G. a ⊗ x)"
    by (subst (2) finprod_reindex [symmetric],
      auto simp add: Pi_def inj_on_cmult surj_const_mult)
  also have "… = (⨂x∈carrier G. a) ⊗ (⨂x∈carrier G. x)"
    by (auto simp add: finprod_multf Pi_def)
  also have "(⨂x∈carrier 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 ∧ (∀x∈A. 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