# Theory FiniteProduct

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theory FiniteProduct
imports Group
`(*  Title:      HOL/Algebra/FiniteProduct.thy    Author:     Clemens Ballarin, started 19 November 2002This file is largely based on HOL/Finite_Set.thy.*)theory FiniteProductimports Groupbeginsubsection {* 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 ∈ 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 ∈ 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 ; e ∈ D; !!x y. [| x ∈ A; y ∈ D |] ==> f x y ∈ D       |] ==> z ∈ D";  by (erule foldSetD.cases) autolemma 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  donelemma foldSetD_imp_finite [simp]: "(A, x) ∈ foldSetD D f e ==> finite A"  by (induct set: foldSetD) autolemma 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: finite)  case empty then show ?case by autonext  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 ..qedtext {* 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) autolemma (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  donelemma (in LCD) foldSetD_imp_finite [simp]:  "(A, x) ∈ foldSetD D f e ==> finite A"  by (induct set: foldSetD) autolemma (in LCD) finite_imp_foldSetD:  "[| finite A; A ⊆ B; e ∈ D |] ==> EX x. (A, x) ∈ foldSetD D f e"proof (induct set: finite)  case empty then show ?case by autonext  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 ..qedlemma (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 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 ∉ 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)    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  donelemma (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) blastlemma (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 (fastforce dest: foldSetD_imp_finite)    apply assumption   apply assumption  apply (blast intro: foldSetD_determ)  donelemma (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)  donelemma (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 simpnext  case insert then show ?case by (simp add: foldD_insert)qedlemma (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: finite)   apply simp  apply (auto simp add: left_commute foldD_insert)  donelemma Int_mono2:  "[| A ⊆ C; B ⊆ C |] ==> A Int B ⊆ C"  by blastlemma (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: finite)   apply simp  apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb    Int_mono2)  donelemma (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]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 "·" 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 .qedlemmas (in ACeD) AC = assoc commute left_commutelemma (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)qedlemma (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: 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)  donelemma (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 o f) \<one>⇘G⇙ 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>__∈_. _)" [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" == "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 beginlemma 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 fastforce       apply simp+   apply fast  apply (auto simp add: finprod_def)  donelemma finprod_one [simp]:  "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"proof (induct set: finite)  case empty show ?case by simpnext  case (insert a A)  have "(%i. \<one>) ∈ A -> carrier G" by auto  with insert show ?case by simpqedlemma finprod_closed [simp]:  fixes A  assumes fin: "finite A" and f: "f ∈ A -> carrier G"   shows "finprod G f A ∈ carrier G"using fin fproof induct  case empty show ?case by simpnext  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 simpqedlemma funcset_Int_left [simp, intro]:  "[| f ∈ A -> C; f ∈ B -> C |] ==> f ∈ A Int B -> C"  by fastlemma funcset_Un_left [iff]:  "(f ∈ A Un B -> C) = (f ∈ A -> C & f ∈ B -> C)"  by fastlemma 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 simpnext  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) qedlemma 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  donelemma 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: finite)  case empty show ?case by simpnext  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)qedlemma 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 add: finprod_def)  qedqedlemma 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 @{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.*}enddeclare funcsetI [rule del]  funcset_mem [rule del]context comm_monoid beginlemma finprod_0 [simp]:  "f ∈ {0::nat} -> carrier G ==> finprod G f {..0} = f 0"by (simp add: Pi_def)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)qedlemma 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)(* The following two were contributed by Jeremy Avigad. *)lemma finprod_reindex:  assumes fin: "finite A"    shows "f : (h ` A) -> carrier G ==>         inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"  using fin  by induct (auto simp add: Pi_def)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 ∈ A" and fin_A: "finite A" and f_Pi: "f ∈ A -> carrier G"  shows "(\<Otimes>j∈A. if i = j then f j else \<one>) = f i"  using i_in_A finprod_insert [of "A - {i}" i "(λj. if i = j then f j else \<one>)"]    fin_A f_Pi finprod_one [of "A - {i}"]    finprod_cong [of "A - {i}" "A - {i}" "(λj. if i = j then f j else \<one>)" "(λi. \<one>)"]   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)endend`