(*  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 @{text LC} of theory @{text Finite_Set} is not

   possible, because here we have explicit typing rules like 

   @{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain

   @{text 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 \<in> D ==> ({}, e) \<in> foldSetD D f e"

   | insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>

                       (insert x A, f x y) \<in> foldSetD D f e"

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

 lemma foldSetD_closed:

   "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D 

       |] ==> z \<in> D";

   by (erule foldSetD.cases) auto

 lemma Diff1_foldSetD:

   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>

    (A, f x y) \<in> foldSetD D f e"

   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)

     apply auto

   done

 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"

   by (induct set: foldSetD) auto

 lemma finite_imp_foldSetD:

   "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>

    EX x. (A, x) \<in> 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) \<in> foldSetD D f e" by auto

   with insert have "y \<in> D" by (auto dest: foldSetD_closed)

   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"

     by (intro foldSetD.intros) auto

   then show ?case ..

 qed

 text {* Left-Commutative Operations *}

 locale LCD =

   fixes B :: "'b set"

   and D :: "'a set"

   and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)

   assumes left_commute:

     "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"

   and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"

 lemma (in LCD) foldSetD_closed [dest]:

   "(A, z) \<in> foldSetD D f e ==> z \<in> D";

   by (erule foldSetD.cases) auto

 lemma (in LCD) Diff1_foldSetD:

   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>

   (A, f x y) \<in> foldSetD D f e"

   apply (subgoal_tac "x \<in> 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) \<in> foldSetD D f e ==> finite A"

   by (induct set: foldSetD) auto

 lemma (in LCD) finite_imp_foldSetD:

   "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> 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) \<in> foldSetD D f e" by auto

   with insert have "y \<in> D" by auto

   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"

     by (intro foldSetD.intros) auto

   then show ?case ..

 qed

 lemma (in LCD) foldSetD_determ_aux:

   "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->

     (\<forall>y. (A, y) \<in> 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 xa Aa ya xb Ab yb, case_tac "xa = xb")

    apply (subgoal_tac "Aa = Ab")

     prefer 2 apply (blast elim!: equalityE)

    apply blast

   txt {* case @{prop "xa \<notin> xb"}. *}

   apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> 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 \<le> 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)

     apply best

    apply assumption

   apply (frule (1) Diff1_foldSetD)

    apply best

   apply (subgoal_tac "ya = f xb x")

    prefer 2

    apply (subgoal_tac "Aa \<subseteq> B")

     prefer 2 apply best (* slow *)

    apply (blast del: equalityCE)

   apply (subgoal_tac "(Ab - {xa}, x) \<in> 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) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]

   ==> y = x"

   by (blast intro: foldSetD_determ_aux [rule_format])

 lemma (in LCD) foldD_equality:

   "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"

   by (unfold foldD_def) (blast intro: foldSetD_determ)

 lemma foldD_empty [simp]:

   "e \<in> D ==> foldD D f e {} = e"

   by (unfold foldD_def) blast

 lemma (in LCD) foldD_insert_aux:

   "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>

     ((insert x A, v) \<in> foldSetD D f e) =

     (EX y. (A, y) \<in> 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 \<in> B; e \<in> D; A \<subseteq> 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 \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"

 proof (induct set: finite)

   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 \<in> B; e \<in> D; A \<subseteq> B |] ==>

    f x (foldD D f e A) = foldD D f (f x e) A"

   apply (induct set: finite)

    apply simp

   apply (auto simp add: left_commute foldD_insert)

   done

 lemma Int_mono2:

   "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"

   by blast

 lemma (in LCD) foldD_nest_Un_Int:

   "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> 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: finite)

    apply simp

   apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb

     Int_mono2)

   done

 lemma (in LCD) foldD_nest_Un_disjoint:

   "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> 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]

 text {* 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 "\<cdot>" 70)

     and e :: 'a

   assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"

     and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"

     and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"

     and e_closed [simp]: "e \<in> D"

     and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"

 lemma (in ACeD) left_commute:

   "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"

 proof -

   assume D: "x \<in> D" "y \<in> D" "z \<in> D"

   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)

   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)

   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)

   finally show ?thesis .

 qed

 lemmas (in ACeD) AC = assoc commute left_commute

 lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"

 proof -

   assume "x \<in> D"

   then have "x \<cdot> e = x" by (rule ident)

   with `x \<in> D` show ?thesis by (simp add: commute)

 qed

 lemma (in ACeD) foldD_Un_Int:

   "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>

     foldD D f e A \<cdot> foldD D f e B =

     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"

   apply (induct set: finite)

    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)

   done

 lemma (in ACeD) foldD_Un_disjoint:

   "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>

     foldD D f e (A Un B) = foldD D f e A \<cdot> 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 o f) \<one>\<^bsub>G\<^esub> A

     else undefined)"

 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>__\<in>_. _)" [1000, 0, 51, 10] 10)

 syntax (HTML output)

   "_finprod" :: "index => idt => 'a set => 'b => 'b"

       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)

 translations

   "\<Otimes>\<index>i:A. b" == "CONST 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]

 context comm_monoid begin

 lemma finprod_insert [simp]:

   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>

    finprod G f (insert a F) = f a \<otimes> 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 finprod_one [simp]:

   "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"

 proof (induct set: finite)

   case empty show ?case by simp

 next

   case (insert a A)

   have "(%i. \<one>) \<in> A -> carrier G" by auto

   with insert show ?case by simp

 qed

 lemma finprod_closed [simp]:

   fixes A

   assumes fin: "finite A" and f: "f \<in> A -> carrier G" 

   shows "finprod G f A \<in> carrier G"

 using fin f

 proof induct

   case empty show ?case by simp

 next

   case (insert a A)

   then have a: "f a \<in> carrier G" by fast

   from insert have A: "f \<in> A -> carrier G" by fast

   from insert A a show ?case by simp

 qed

 lemma funcset_Int_left [simp, intro]:

   "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"

   by fast

 lemma funcset_Un_left [iff]:

   "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"

   by fast

 lemma finprod_Un_Int:

   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>

      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =

      finprod G g A \<otimes> 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 add: finprod_closed)

 next

   case (insert a A)

   then have a: "g a \<in> carrier G" by fast

   from insert have A: "g \<in> 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) 

 qed

 lemma finprod_Un_disjoint:

   "[| finite A; finite B; A Int B = {};

       g \<in> A -> carrier G; g \<in> B -> carrier G |]

    ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"

   apply (subst finprod_Un_Int [symmetric])

       apply (auto simp add: finprod_closed)

   done

 lemma finprod_multf:

   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>

    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"

 proof (induct set: finite)

   case empty show ?case by simp

 next

   case (insert a A) then

   have fA: "f \<in> A -> carrier G" by fast

   from insert have fa: "f a \<in> carrier G" by fast

   from insert have gA: "g \<in> A -> carrier G" by fast

   from insert have ga: "g a \<in> carrier G" by fast

   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"

     by (simp add: Pi_def)

   show ?case

     by (simp add: insert fA fa gA ga fgA m_ac)

 qed

 lemma finprod_cong':

   "[| A = B; g \<in> B -> carrier G;

       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"

 proof -

   assume prems: "A = B" "g \<in> B -> carrier G"

     "!!i. i \<in> B ==> f i = g i"

   show ?thesis

   proof (cases "finite B")

     case True

     then have "!!A. [| A = B; g \<in> B -> carrier G;

       !!i. i \<in> 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 \<otimes> 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 \<in> insert x B \<Longrightarrow> f i = g i"

           "g \<in> insert x B \<rightarrow> carrier G"

         thus "f \<in> B -> carrier G" by fastsimp

       next

         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"

           "g \<in> insert x B \<rightarrow> carrier G"

         thus "f x \<in> carrier G" by fastsimp

       qed

       also from insert have "... = g x \<otimes> 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 finprod_cong:

   "[| A = B; f \<in> B -> carrier G = True;

       !!i. i \<in> 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 \<in> 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.

 *}

 end

 declare funcsetI [rule del]

   funcset_mem [rule del]

 context comm_monoid begin

 lemma finprod_0 [simp]:

   "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"

 by (simp add: Pi_def)

 lemma finprod_Suc [simp]:

   "f \<in> {..Suc n} -> carrier G ==>

    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"

 by (simp add: Pi_def atMost_Suc)

 lemma finprod_Suc2:

   "f \<in> {..Suc n} -> carrier G ==>

    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> 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_mult [simp]:

   "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>

      finprod G (%i. f i \<otimes> g i) {..n::nat} =

      finprod G f {..n} \<otimes> finprod G g {..n}"

   by (induct n) (simp_all add: m_ac Pi_def)

 (* The following two were contributed by Jeremy Avigad. *)

 lemma finprod_reindex:

   assumes fin: "finite A"

     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow> 

         inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"

   using fin apply induct

   apply (auto simp add: finprod_insert Pi_def)

 done

 lemma finprod_const:

   assumes fin [simp]: "finite A"

       and a [simp]: "a : carrier G"

     shows "finprod G (%x. a) A = a (^) card A"

   using fin apply induct

   apply force

   apply (subst finprod_insert)

   apply auto

   apply (subst m_comm)

   apply auto

 done

 (* The following lemma was contributed by Jesus Aransay. *)

 lemma finprod_singleton:

   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"

   shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"

   using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]

     fin_A f_Pi finprod_one [of "A - {i}"]

     finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"] 

   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)

 end

 end