(*  Title:      HOL/Algebra/FiniteProduct.thy

    ID:         $Id$

    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 \<in> 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 \<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"

constdefs

  foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"

  "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.elims) 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: Finites)

  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

subsection {* 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.elims) 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: Finites)

  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 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 \<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 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 \<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: 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 \<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: Finites)

   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: 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 \<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]

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 "\<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 D: "x \<in> D"

  have "x \<cdot> 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 \<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: 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 \<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]] 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>__\<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" == "finprod \<struct>\<index> (%i. b) A"

  -- {* Beware of argument permutation! *}

ML_setup {* 

  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;

*}

lemma (in comm_monoid) finprod_empty [simp]: 

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

  by (simp add: finprod_def)

ML_setup {* 

  simpset_ref() := simpset() setsubgoaler asm_simp_tac;

*}

declare funcsetI [intro]

  funcset_mem [dest]

lemma (in comm_monoid) 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 (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>) \<in> 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 \<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 (in comm_monoid) 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: Finites)

  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 Un_subset_iff) 

qed

lemma (in comm_monoid) 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 (in comm_monoid) 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: Finites)

  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 (in comm_monoid) 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" .

      next

	show "x ~: B" .

      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 (in comm_monoid) 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.

*}

declare funcsetI [rule del]

  funcset_mem [rule del]

lemma (in comm_monoid) finprod_0 [simp]:

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

by (simp add: Pi_def)

lemma (in comm_monoid) 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 (in comm_monoid) 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 (in comm_monoid) 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)

end