# Theory FiniteProduct

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theory FiniteProduct
imports Group
begin

    (*  Title:      HOL/Algebra/FiniteProduct.thy
        ID:         $Id: FiniteProduct.thy,v 1.11 2005/07/01 12:02:58 berghofe Exp $
        Author:     Clemens Ballarin, started 19 November 2002

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

    header {* Product Operator for Commutative Monoids *}

    theory FiniteProduct imports Group begin

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

    consts
      foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"

    inductive "foldSetD D f e"
      intros
        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"

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

    lemma foldSetD_closed:
      "[| (A, z) ∈ foldSetD D f e ; e ∈ D; !!x y. [| x ∈ A; y ∈ D |] ==> f x y ∈ D
          |] ==> z ∈ D";
      by (erule foldSetD.elims) 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"
      apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
        apply auto
      done

    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 |] ==>
       EX x. (A, x) ∈ foldSetD D f e"
    proof (induct set: Finites)
      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

    subsection {* 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.elims) 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"
      apply (subgoal_tac "x ∈ B")
       prefer 2 apply fast
      apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
        apply auto
      done

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

    lemma (in LCD) finite_imp_foldSetD:
      "[| finite A; A ⊆ B; e ∈ D |] ==> EX x. (A, x) ∈ foldSetD D f e"
    proof (induct set: Finites)
      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:
      "e ∈ D ==> ∀A x. A ⊆ B & card A < n --> (A, x) ∈ foldSetD D f e -->
        (∀y. (A, y) ∈ foldSetD D f e --> y = x)"
      apply (induct n)
       apply (auto simp add: less_Suc_eq) (* slow *)
      apply (erule foldSetD.cases)
       apply blast
      apply (erule foldSetD.cases)
       apply blast
      apply clarify
      txt {* force simplification of @{text "card A < card (insert ...)"}. *}
      apply (erule rev_mp)
      apply (simp add: less_Suc_eq_le)
      apply (rule impI)
      apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
       apply (subgoal_tac "Aa = Ab")
        prefer 2 apply (blast elim!: equalityE)
       apply blast
      txt {* case @{prop "xa ∉ xb"}. *}
      apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb ∈ Aa & xa ∈ Ab")
       prefer 2 apply (blast elim!: equalityE)
      apply clarify
      apply (subgoal_tac "Aa = insert xb Ab - {xa}")
       prefer 2 apply blast
      apply (subgoal_tac "card Aa ≤ card Ab")
       prefer 2
       apply (rule Suc_le_mono [THEN subst])
       apply (simp add: card_Suc_Diff1)
      apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
         apply (blast intro: foldSetD_imp_finite finite_Diff)
        apply best
       apply assumption
      apply (frule (1) Diff1_foldSetD)
       apply best
      apply (subgoal_tac "ya = f xb x")
       prefer 2
       apply (subgoal_tac "Aa ⊆ B")
        prefer 2 apply best (* slow *)
       apply (blast del: equalityCE)
      apply (subgoal_tac "(Ab - {xa}, x) ∈ foldSetD D f e")
       prefer 2 apply simp
      apply (subgoal_tac "yb = f xa x")
       prefer 2
       apply (blast del: equalityCE dest: Diff1_foldSetD)
      apply (simp (no_asm_simp))
      apply (rule left_commute)
        apply assumption
       apply best (* slow *)
      apply best
      done

    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) =
        (EX y. (A, y) ∈ foldSetD D f e & v = f x y)"
      apply auto
      apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
         apply (fastsimp dest: foldSetD_imp_finite)
        apply assumption
       apply assumption
      apply (blast intro: foldSetD_determ)
      done

    lemma (in LCD) foldD_insert:
        "[| finite A; x ~: A; x ∈ B; e ∈ D; A ⊆ B |] ==>
         foldD D f e (insert x A) = f x (foldD D f e A)"
      apply (unfold foldD_def)
      apply (simp add: foldD_insert_aux)
      apply (rule the_equality)
       apply (auto intro: finite_imp_foldSetD
         cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
      done

    lemma (in LCD) foldD_closed [simp]:
      "[| finite A; e ∈ D; A ⊆ B |] ==> foldD D f e A ∈ D"
    proof (induct set: Finites)
      case empty then show ?case by (simp add: foldD_empty)
    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"
      apply (induct set: Finites)
       apply simp
      apply (auto simp add: left_commute foldD_insert)
      done

    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)"
      apply (induct set: Finites)
       apply simp
      apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
        Int_mono2 Un_subset_iff)
      done

    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 @{text foldSetD} relation. *}

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

    subsection {* Commutative monoids *}

    text {*
      We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
      instead of @{text "'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 D: "x ∈ D"
      have "x · e = x" by (rule ident)
      with 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)"
      apply (induct set: Finites)
       apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
      apply (simp add: AC insert_absorb Int_insert_left
        LCD.foldD_insert [OF LCD.intro [of D]]
        LCD.foldD_closed [OF LCD.intro [of D]]
        Int_mono2 Un_subset_iff)
      done

    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]] Un_subset_iff)

    subsection {* Products over Finite Sets *}

    constdefs (structure G)
      finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
      "finprod G f A == if finite A
          then foldD (carrier G) (mult G o f) \<one> A
          else arbitrary"

    syntax
      "_finprod" :: "index => idt => 'a set => 'b => 'b"
          ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
    syntax (xsymbols)
      "_finprod" :: "index => idt => 'a set => 'b => 'b"
          ("(3\<Otimes>__∈_. _)" [1000, 0, 51, 10] 10)
    syntax (HTML output)
      "_finprod" :: "index => idt => 'a set => 'b => 'b"
          ("(3\<Otimes>__∈_. _)" [1000, 0, 51, 10] 10)
    translations
      "\<Otimes>\<index>i:A. b" == "finprod \<struct>\<index> (%i. b) A"
      -- {* Beware of argument permutation! *}

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

    declare funcsetI [intro]
      funcset_mem [dest]

    lemma (in comm_monoid) finprod_insert [simp]:
      "[| finite F; a ∉ F; f ∈ F -> carrier G; f a ∈ carrier G |] ==>
       finprod G f (insert a F) = f a ⊗ finprod G f F"
      apply (rule trans)
       apply (simp add: finprod_def)
      apply (rule trans)
       apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
             apply simp
             apply (rule m_lcomm)
               apply fast
              apply fast
             apply assumption
            apply (fastsimp intro: m_closed)
           apply simp+
       apply fast
      apply (auto simp add: finprod_def)
      done

    lemma (in comm_monoid) finprod_one [simp]:
      "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
    proof (induct set: Finites)
      case empty show ?case by simp
    next
      case (insert a A)
      have "(%i. \<one>) ∈ A -> carrier G" by auto
      with insert show ?case by simp
    qed

    lemma (in comm_monoid) finprod_closed [simp]:
      fixes A
      assumes fin: "finite A" and f: "f ∈ A -> carrier G"
      shows "finprod G f A ∈ carrier G"
    using fin f
    proof 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

    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 (in comm_monoid) 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: Finites)
      case empty then show ?case by (simp add: finprod_closed)
    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 finprod_closed
              Int_mono2 Un_subset_iff)
    qed

    lemma (in comm_monoid) 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"
      apply (subst finprod_Un_Int [symmetric])
          apply (auto simp add: finprod_closed)
      done

    lemma (in comm_monoid) finprod_multf:
      "[| finite A; f ∈ A -> carrier G; g ∈ A -> carrier G |] ==>
       finprod G (%x. f x ⊗ g x) A = (finprod G f A ⊗ finprod G g A)"
    proof (induct set: Finites)
      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

    lemma (in comm_monoid) 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" .
          next
            show "x ~: B" .
          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 fastsimp
          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 fastsimp
          qed
          also from insert have "... = g x ⊗ finprod G g B" by fastsimp
          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 add: finprod_def)
      qed
    qed

    lemma (in comm_monoid) finprod_cong:
      "[| A = B; f ∈ B -> carrier G = True;
          !!i. i ∈ B ==> 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') force+

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

    declare funcsetI [rule del]
      funcset_mem [rule del]

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

    lemma (in comm_monoid) 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 (in comm_monoid) 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 (in comm_monoid) finprod_mult [simp]:
      "[| f ∈ {..n} -> carrier G; g ∈ {..n} -> carrier G |] ==>
         finprod G (%i. f i ⊗ g i) {..n::nat} =
         finprod G f {..n} ⊗ finprod G g {..n}"
      by (induct n) (simp_all add: m_ac Pi_def)

    end

lemmas empty_foldSetDE:

      [| ({}, x) ∈ foldSetD D f e; [| e ∈ D; x = e |] ==> P |] ==> P

lemma foldSetD_closed:

      [| (A, z) ∈ foldSetD D f e; e ∈ D; !!x y. [| x ∈ A; y ∈ D |] ==> f x y ∈ D |]
      ==> z ∈ D

lemma Diff1_foldSetD:

      [| (A - {x}, y) ∈ foldSetD D f e; x ∈ A; f x y ∈ D |]
      ==> (A, f x y) ∈ foldSetD D f e

lemma foldSetD_imp_finite:

      (A, x) ∈ foldSetD D f e ==> finite A

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

### Left-commutative operations

lemma foldSetD_closed:

      [| LCD B D f; (A, z) ∈ foldSetD D f e |] ==> z ∈ D

lemma Diff1_foldSetD:

      [| LCD B D f; (A - {x}, y) ∈ foldSetD D f e; x ∈ A; A ⊆ B |]
      ==> (A, f x y) ∈ foldSetD D f e

lemma foldSetD_imp_finite:

      [| LCD B D f; (A, x) ∈ foldSetD D f e |] ==> finite A

lemma finite_imp_foldSetD:

      [| LCD B D f; finite A; A ⊆ B; e ∈ D |] ==> ∃x. (A, x) ∈ foldSetD D f e

lemma foldSetD_determ_aux:

      [| LCD B D f; e ∈ D |]
      ==> ∀A x. A ⊆ B ∧ card A < n -->
                (A, x) ∈ foldSetD D f e --> (∀y. (A, y) ∈ foldSetD D f e --> y = x)

lemma foldSetD_determ:

      [| LCD B D f; (A, x) ∈ foldSetD D f e; (A, y) ∈ foldSetD D f e; e ∈ D; A ⊆ B |]
      ==> y = x

lemma foldD_equality:

      [| LCD B D f; (A, y) ∈ foldSetD D f e; e ∈ D; A ⊆ B |] ==> foldD D f e A = y

lemma foldD_empty:

      e ∈ D ==> foldD D f e {} = e

lemma foldD_insert_aux:

      [| LCD B D f; 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)

lemma foldD_insert:

      [| LCD B D f; finite A; x ∉ A; x ∈ B; e ∈ D; A ⊆ B |]
      ==> foldD D f e (insert x A) = f x (foldD D f e A)

lemma foldD_closed:

      [| LCD B D f; finite A; e ∈ D; A ⊆ B |] ==> foldD D f e A ∈ D

lemma foldD_commute:

      [| LCD B D f; finite A; x ∈ B; e ∈ D; A ⊆ B |]
      ==> f x (foldD D f e A) = foldD D f (f x e) A

lemma Int_mono2:

      [| A ⊆ C; B ⊆ C |] ==> A ∩ B ⊆ C

lemma foldD_nest_Un_Int:

      [| LCD B D f; 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 ∩ C)) (A ∪ C)

lemma foldD_nest_Un_disjoint:

      [| LCD B D f; finite A; finite B; A ∩ B = {}; e ∈ D; A ⊆ B; C ⊆ B |]
      ==> foldD D f e (A ∪ B) = foldD D f (foldD D f e B) A

### Commutative monoids

lemma left_commute:

      [| ACeD D f e; x ∈ D; y ∈ D; z ∈ D |] ==> f x (f y z) = f y (f x z)

lemmas AC:

      [| ACeD D f e; x ∈ D; y ∈ D; z ∈ D |] ==> f (f x y) z = f x (f y z)

      [| ACeD D f e; x ∈ D; y ∈ D |] ==> f x y = f y x

      [| ACeD D f e; x ∈ D; y ∈ D; z ∈ D |] ==> f x (f y z) = f y (f x z)

lemma left_ident:

      [| ACeD D f e; x ∈ D |] ==> f e x = x

lemma foldD_Un_Int:

      [| ACeD D f e; finite A; finite B; A ⊆ D; B ⊆ D |]
      ==> f (foldD D f e A) (foldD D f e B) =
          f (foldD D f e (A ∪ B)) (foldD D f e (A ∩ B))

lemma foldD_Un_disjoint:

      [| ACeD D f e; finite A; finite B; A ∩ B = {}; A ⊆ D; B ⊆ D |]
      ==> foldD D f e (A ∪ B) = f (foldD D f e A) (foldD D f e B)

### Products over Finite Sets

lemma finprod_empty:

      comm_monoid G ==> finprod G f {} = \<one>G

lemma finprod_insert:

      [| comm_monoid G; finite F; a ∉ F; f ∈ F -> carrier G; f a ∈ carrier G |]
      ==> finprod G f (insert a F) = f a ⊗G finprod G f F

lemma finprod_one:

      [| comm_monoid G; finite A |] ==> (\<Otimes>Gi∈A. \<one>G) = \<one>G

lemma finprod_closed:

      [| comm_monoid G; finite A; f ∈ A -> carrier G |] ==> finprod G f A ∈ carrier G

lemma funcset_Int_left:

      [| f ∈ A -> C; f ∈ B -> C |] ==> f ∈ A ∩ B -> C

lemma funcset_Un_left:

      (f ∈ A ∪ B -> C) = (f ∈ A -> C ∧ f ∈ B -> C)

lemma finprod_Un_Int:

      [| comm_monoid G; finite A; finite B; g ∈ A -> carrier G; g ∈ B -> carrier G |]
      ==> finprod G g (A ∪ B) ⊗G finprod G g (A ∩ B) = finprod G g A ⊗G finprod G g B

lemma finprod_Un_disjoint:

      [| comm_monoid G; finite A; finite B; A ∩ B = {}; g ∈ A -> carrier G;
         g ∈ B -> carrier G |]
      ==> finprod G g (A ∪ B) = finprod G g A ⊗G finprod G g B

lemma finprod_multf:

      [| comm_monoid G; finite A; f ∈ A -> carrier G; g ∈ A -> carrier G |]
      ==> (\<Otimes>Gx∈A. f x ⊗G g x) = finprod G f A ⊗G finprod G g A

lemma finprod_cong':

      [| comm_monoid G; A = B; g ∈ B -> carrier G; !!i. i ∈ B ==> f i = g i |]
      ==> finprod G f A = finprod G g B

lemma finprod_cong:

      [| comm_monoid G; A = B; (f ∈ B -> carrier G) = True;
         !!i. i ∈ B ==> f i = g i |]
      ==> finprod G f A = finprod G g B

lemma finprod_0:

      [| comm_monoid G; f ∈ {0} -> carrier G |] ==> finprod G f {..0} = f 0

lemma finprod_Suc:

      [| comm_monoid G; f ∈ {..Suc n} -> carrier G |]
      ==> finprod G f {..Suc n} = f (Suc n) ⊗G finprod G f {..n}

lemma finprod_Suc2:

      [| comm_monoid G; f ∈ {..Suc n} -> carrier G |]
      ==> finprod G f {..Suc n} = (\<Otimes>Gi∈{..n}. f (Suc i)) ⊗G f 0

lemma finprod_mult:

      [| comm_monoid G; f ∈ {..n} -> carrier G; g ∈ {..n} -> carrier G |]
      ==> (\<Otimes>Gi∈{..n}. f i ⊗G g i) = finprod G f {..n} ⊗G finprod G g {..n}