src/HOL/Algebra/FiniteProduct.thy
author wenzelm
Sun Mar 21 15:57:40 2010 +0100 (2010-03-21)
changeset 35847 19f1f7066917
parent 35416 d8d7d1b785af
child 35848 5443079512ea
permissions -rw-r--r--
eliminated old constdefs;
     1 (*  Title:      HOL/Algebra/FiniteProduct.thy
     2     Author:     Clemens Ballarin, started 19 November 2002
     3 
     4 This file is largely based on HOL/Finite_Set.thy.
     5 *)
     6 
     7 theory FiniteProduct imports Group begin
     8 
     9 
    10 subsection {* Product Operator for Commutative Monoids *}
    11 
    12 subsubsection {* Inductive Definition of a Relation for Products over Sets *}
    13 
    14 text {* Instantiation of locale @{text LC} of theory @{text Finite_Set} is not
    15   possible, because here we have explicit typing rules like 
    16   @{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain
    17   @{text D}. *}
    18 
    19 inductive_set
    20   foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
    21   for D :: "'a set" and f :: "'b => 'a => 'a" and e :: 'a
    22   where
    23     emptyI [intro]: "e \<in> D ==> ({}, e) \<in> foldSetD D f e"
    24   | insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>
    25                       (insert x A, f x y) \<in> foldSetD D f e"
    26 
    27 inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
    28 
    29 definition foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a" where
    30   "foldD D f e A == THE x. (A, x) \<in> foldSetD D f e"
    31 
    32 lemma foldSetD_closed:
    33   "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D 
    34       |] ==> z \<in> D";
    35   by (erule foldSetD.cases) auto
    36 
    37 lemma Diff1_foldSetD:
    38   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>
    39    (A, f x y) \<in> foldSetD D f e"
    40   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
    41     apply auto
    42   done
    43 
    44 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"
    45   by (induct set: foldSetD) auto
    46 
    47 lemma finite_imp_foldSetD:
    48   "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
    49    EX x. (A, x) \<in> foldSetD D f e"
    50 proof (induct set: finite)
    51   case empty then show ?case by auto
    52 next
    53   case (insert x F)
    54   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
    55   with insert have "y \<in> D" by (auto dest: foldSetD_closed)
    56   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
    57     by (intro foldSetD.intros) auto
    58   then show ?case ..
    59 qed
    60 
    61 
    62 text {* Left-Commutative Operations *}
    63 
    64 locale LCD =
    65   fixes B :: "'b set"
    66   and D :: "'a set"
    67   and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
    68   assumes left_commute:
    69     "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
    70   and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"
    71 
    72 lemma (in LCD) foldSetD_closed [dest]:
    73   "(A, z) \<in> foldSetD D f e ==> z \<in> D";
    74   by (erule foldSetD.cases) auto
    75 
    76 lemma (in LCD) Diff1_foldSetD:
    77   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
    78   (A, f x y) \<in> foldSetD D f e"
    79   apply (subgoal_tac "x \<in> B")
    80    prefer 2 apply fast
    81   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
    82     apply auto
    83   done
    84 
    85 lemma (in LCD) foldSetD_imp_finite [simp]:
    86   "(A, x) \<in> foldSetD D f e ==> finite A"
    87   by (induct set: foldSetD) auto
    88 
    89 lemma (in LCD) finite_imp_foldSetD:
    90   "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> foldSetD D f e"
    91 proof (induct set: finite)
    92   case empty then show ?case by auto
    93 next
    94   case (insert x F)
    95   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
    96   with insert have "y \<in> D" by auto
    97   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
    98     by (intro foldSetD.intros) auto
    99   then show ?case ..
   100 qed
   101 
   102 lemma (in LCD) foldSetD_determ_aux:
   103   "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
   104     (\<forall>y. (A, y) \<in> foldSetD D f e --> y = x)"
   105   apply (induct n)
   106    apply (auto simp add: less_Suc_eq) (* slow *)
   107   apply (erule foldSetD.cases)
   108    apply blast
   109   apply (erule foldSetD.cases)
   110    apply blast
   111   apply clarify
   112   txt {* force simplification of @{text "card A < card (insert ...)"}. *}
   113   apply (erule rev_mp)
   114   apply (simp add: less_Suc_eq_le)
   115   apply (rule impI)
   116   apply (rename_tac xa Aa ya xb Ab yb, case_tac "xa = xb")
   117    apply (subgoal_tac "Aa = Ab")
   118     prefer 2 apply (blast elim!: equalityE)
   119    apply blast
   120   txt {* case @{prop "xa \<notin> xb"}. *}
   121   apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> Ab")
   122    prefer 2 apply (blast elim!: equalityE)
   123   apply clarify
   124   apply (subgoal_tac "Aa = insert xb Ab - {xa}")
   125    prefer 2 apply blast
   126   apply (subgoal_tac "card Aa \<le> card Ab")
   127    prefer 2
   128    apply (rule Suc_le_mono [THEN subst])
   129    apply (simp add: card_Suc_Diff1)
   130   apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
   131      apply (blast intro: foldSetD_imp_finite finite_Diff)
   132     apply best
   133    apply assumption
   134   apply (frule (1) Diff1_foldSetD)
   135    apply best
   136   apply (subgoal_tac "ya = f xb x")
   137    prefer 2
   138    apply (subgoal_tac "Aa \<subseteq> B")
   139     prefer 2 apply best (* slow *)
   140    apply (blast del: equalityCE)
   141   apply (subgoal_tac "(Ab - {xa}, x) \<in> foldSetD D f e")
   142    prefer 2 apply simp
   143   apply (subgoal_tac "yb = f xa x")
   144    prefer 2 
   145    apply (blast del: equalityCE dest: Diff1_foldSetD)
   146   apply (simp (no_asm_simp))
   147   apply (rule left_commute)
   148     apply assumption
   149    apply best (* slow *)
   150   apply best
   151   done
   152 
   153 lemma (in LCD) foldSetD_determ:
   154   "[| (A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
   155   ==> y = x"
   156   by (blast intro: foldSetD_determ_aux [rule_format])
   157 
   158 lemma (in LCD) foldD_equality:
   159   "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"
   160   by (unfold foldD_def) (blast intro: foldSetD_determ)
   161 
   162 lemma foldD_empty [simp]:
   163   "e \<in> D ==> foldD D f e {} = e"
   164   by (unfold foldD_def) blast
   165 
   166 lemma (in LCD) foldD_insert_aux:
   167   "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
   168     ((insert x A, v) \<in> foldSetD D f e) =
   169     (EX y. (A, y) \<in> foldSetD D f e & v = f x y)"
   170   apply auto
   171   apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
   172      apply (fastsimp dest: foldSetD_imp_finite)
   173     apply assumption
   174    apply assumption
   175   apply (blast intro: foldSetD_determ)
   176   done
   177 
   178 lemma (in LCD) foldD_insert:
   179     "[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
   180      foldD D f e (insert x A) = f x (foldD D f e A)"
   181   apply (unfold foldD_def)
   182   apply (simp add: foldD_insert_aux)
   183   apply (rule the_equality)
   184    apply (auto intro: finite_imp_foldSetD
   185      cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
   186   done
   187 
   188 lemma (in LCD) foldD_closed [simp]:
   189   "[| finite A; e \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
   190 proof (induct set: finite)
   191   case empty then show ?case by (simp add: foldD_empty)
   192 next
   193   case insert then show ?case by (simp add: foldD_insert)
   194 qed
   195 
   196 lemma (in LCD) foldD_commute:
   197   "[| finite A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
   198    f x (foldD D f e A) = foldD D f (f x e) A"
   199   apply (induct set: finite)
   200    apply simp
   201   apply (auto simp add: left_commute foldD_insert)
   202   done
   203 
   204 lemma Int_mono2:
   205   "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
   206   by blast
   207 
   208 lemma (in LCD) foldD_nest_Un_Int:
   209   "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B |] ==>
   210    foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
   211   apply (induct set: finite)
   212    apply simp
   213   apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
   214     Int_mono2)
   215   done
   216 
   217 lemma (in LCD) foldD_nest_Un_disjoint:
   218   "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B |]
   219     ==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
   220   by (simp add: foldD_nest_Un_Int)
   221 
   222 -- {* Delete rules to do with @{text foldSetD} relation. *}
   223 
   224 declare foldSetD_imp_finite [simp del]
   225   empty_foldSetDE [rule del]
   226   foldSetD.intros [rule del]
   227 declare (in LCD)
   228   foldSetD_closed [rule del]
   229 
   230 
   231 text {* Commutative Monoids *}
   232 
   233 text {*
   234   We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
   235   instead of @{text "'b => 'a => 'a"}.
   236 *}
   237 
   238 locale ACeD =
   239   fixes D :: "'a set"
   240     and f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
   241     and e :: 'a
   242   assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"
   243     and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"
   244     and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
   245     and e_closed [simp]: "e \<in> D"
   246     and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"
   247 
   248 lemma (in ACeD) left_commute:
   249   "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
   250 proof -
   251   assume D: "x \<in> D" "y \<in> D" "z \<in> D"
   252   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
   253   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
   254   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
   255   finally show ?thesis .
   256 qed
   257 
   258 lemmas (in ACeD) AC = assoc commute left_commute
   259 
   260 lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"
   261 proof -
   262   assume "x \<in> D"
   263   then have "x \<cdot> e = x" by (rule ident)
   264   with `x \<in> D` show ?thesis by (simp add: commute)
   265 qed
   266 
   267 lemma (in ACeD) foldD_Un_Int:
   268   "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
   269     foldD D f e A \<cdot> foldD D f e B =
   270     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
   271   apply (induct set: finite)
   272    apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
   273   apply (simp add: AC insert_absorb Int_insert_left
   274     LCD.foldD_insert [OF LCD.intro [of D]]
   275     LCD.foldD_closed [OF LCD.intro [of D]]
   276     Int_mono2)
   277   done
   278 
   279 lemma (in ACeD) foldD_Un_disjoint:
   280   "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>
   281     foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
   282   by (simp add: foldD_Un_Int
   283     left_commute LCD.foldD_closed [OF LCD.intro [of D]])
   284 
   285 
   286 subsubsection {* Products over Finite Sets *}
   287 
   288 definition
   289   finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b" where
   290   "finprod G f A ==
   291     if finite A
   292     then foldD (carrier G) (mult G o f) \<one>\<^bsub>G\<^esub> A
   293     else undefined"
   294 
   295 syntax
   296   "_finprod" :: "index => idt => 'a set => 'b => 'b"
   297       ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
   298 syntax (xsymbols)
   299   "_finprod" :: "index => idt => 'a set => 'b => 'b"
   300       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
   301 syntax (HTML output)
   302   "_finprod" :: "index => idt => 'a set => 'b => 'b"
   303       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
   304 translations
   305   "\<Otimes>\<index>i:A. b" == "CONST finprod \<struct>\<index> (%i. b) A"
   306   -- {* Beware of argument permutation! *}
   307 
   308 lemma (in comm_monoid) finprod_empty [simp]: 
   309   "finprod G f {} = \<one>"
   310   by (simp add: finprod_def)
   311 
   312 declare funcsetI [intro]
   313   funcset_mem [dest]
   314 
   315 context comm_monoid begin
   316 
   317 lemma finprod_insert [simp]:
   318   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
   319    finprod G f (insert a F) = f a \<otimes> finprod G f F"
   320   apply (rule trans)
   321    apply (simp add: finprod_def)
   322   apply (rule trans)
   323    apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
   324          apply simp
   325          apply (rule m_lcomm)
   326            apply fast
   327           apply fast
   328          apply assumption
   329         apply (fastsimp intro: m_closed)
   330        apply simp+
   331    apply fast
   332   apply (auto simp add: finprod_def)
   333   done
   334 
   335 lemma finprod_one [simp]:
   336   "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
   337 proof (induct set: finite)
   338   case empty show ?case by simp
   339 next
   340   case (insert a A)
   341   have "(%i. \<one>) \<in> A -> carrier G" by auto
   342   with insert show ?case by simp
   343 qed
   344 
   345 lemma finprod_closed [simp]:
   346   fixes A
   347   assumes fin: "finite A" and f: "f \<in> A -> carrier G" 
   348   shows "finprod G f A \<in> carrier G"
   349 using fin f
   350 proof induct
   351   case empty show ?case by simp
   352 next
   353   case (insert a A)
   354   then have a: "f a \<in> carrier G" by fast
   355   from insert have A: "f \<in> A -> carrier G" by fast
   356   from insert A a show ?case by simp
   357 qed
   358 
   359 lemma funcset_Int_left [simp, intro]:
   360   "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
   361   by fast
   362 
   363 lemma funcset_Un_left [iff]:
   364   "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
   365   by fast
   366 
   367 lemma finprod_Un_Int:
   368   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
   369      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
   370      finprod G g A \<otimes> finprod G g B"
   371 -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
   372 proof (induct set: finite)
   373   case empty then show ?case by (simp add: finprod_closed)
   374 next
   375   case (insert a A)
   376   then have a: "g a \<in> carrier G" by fast
   377   from insert have A: "g \<in> A -> carrier G" by fast
   378   from insert A a show ?case
   379     by (simp add: m_ac Int_insert_left insert_absorb finprod_closed
   380           Int_mono2) 
   381 qed
   382 
   383 lemma finprod_Un_disjoint:
   384   "[| finite A; finite B; A Int B = {};
   385       g \<in> A -> carrier G; g \<in> B -> carrier G |]
   386    ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
   387   apply (subst finprod_Un_Int [symmetric])
   388       apply (auto simp add: finprod_closed)
   389   done
   390 
   391 lemma finprod_multf:
   392   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
   393    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
   394 proof (induct set: finite)
   395   case empty show ?case by simp
   396 next
   397   case (insert a A) then
   398   have fA: "f \<in> A -> carrier G" by fast
   399   from insert have fa: "f a \<in> carrier G" by fast
   400   from insert have gA: "g \<in> A -> carrier G" by fast
   401   from insert have ga: "g a \<in> carrier G" by fast
   402   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"
   403     by (simp add: Pi_def)
   404   show ?case
   405     by (simp add: insert fA fa gA ga fgA m_ac)
   406 qed
   407 
   408 lemma finprod_cong':
   409   "[| A = B; g \<in> B -> carrier G;
   410       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
   411 proof -
   412   assume prems: "A = B" "g \<in> B -> carrier G"
   413     "!!i. i \<in> B ==> f i = g i"
   414   show ?thesis
   415   proof (cases "finite B")
   416     case True
   417     then have "!!A. [| A = B; g \<in> B -> carrier G;
   418       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
   419     proof induct
   420       case empty thus ?case by simp
   421     next
   422       case (insert x B)
   423       then have "finprod G f A = finprod G f (insert x B)" by simp
   424       also from insert have "... = f x \<otimes> finprod G f B"
   425       proof (intro finprod_insert)
   426         show "finite B" by fact
   427       next
   428         show "x ~: B" by fact
   429       next
   430         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
   431           "g \<in> insert x B \<rightarrow> carrier G"
   432         thus "f \<in> B -> carrier G" by fastsimp
   433       next
   434         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
   435           "g \<in> insert x B \<rightarrow> carrier G"
   436         thus "f x \<in> carrier G" by fastsimp
   437       qed
   438       also from insert have "... = g x \<otimes> finprod G g B" by fastsimp
   439       also from insert have "... = finprod G g (insert x B)"
   440       by (intro finprod_insert [THEN sym]) auto
   441       finally show ?case .
   442     qed
   443     with prems show ?thesis by simp
   444   next
   445     case False with prems show ?thesis by (simp add: finprod_def)
   446   qed
   447 qed
   448 
   449 lemma finprod_cong:
   450   "[| A = B; f \<in> B -> carrier G = True;
   451       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
   452   (* This order of prems is slightly faster (3%) than the last two swapped. *)
   453   by (rule finprod_cong') force+
   454 
   455 text {*Usually, if this rule causes a failed congruence proof error,
   456   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
   457   Adding @{thm [source] Pi_def} to the simpset is often useful.
   458   For this reason, @{thm [source] comm_monoid.finprod_cong}
   459   is not added to the simpset by default.
   460 *}
   461 
   462 end
   463 
   464 declare funcsetI [rule del]
   465   funcset_mem [rule del]
   466 
   467 context comm_monoid begin
   468 
   469 lemma finprod_0 [simp]:
   470   "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"
   471 by (simp add: Pi_def)
   472 
   473 lemma finprod_Suc [simp]:
   474   "f \<in> {..Suc n} -> carrier G ==>
   475    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
   476 by (simp add: Pi_def atMost_Suc)
   477 
   478 lemma finprod_Suc2:
   479   "f \<in> {..Suc n} -> carrier G ==>
   480    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
   481 proof (induct n)
   482   case 0 thus ?case by (simp add: Pi_def)
   483 next
   484   case Suc thus ?case by (simp add: m_assoc Pi_def)
   485 qed
   486 
   487 lemma finprod_mult [simp]:
   488   "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
   489      finprod G (%i. f i \<otimes> g i) {..n::nat} =
   490      finprod G f {..n} \<otimes> finprod G g {..n}"
   491   by (induct n) (simp_all add: m_ac Pi_def)
   492 
   493 (* The following two were contributed by Jeremy Avigad. *)
   494 
   495 lemma finprod_reindex:
   496   assumes fin: "finite A"
   497     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow> 
   498         inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"
   499   using fin apply induct
   500   apply (auto simp add: finprod_insert Pi_def)
   501 done
   502 
   503 lemma finprod_const:
   504   assumes fin [simp]: "finite A"
   505       and a [simp]: "a : carrier G"
   506     shows "finprod G (%x. a) A = a (^) card A"
   507   using fin apply induct
   508   apply force
   509   apply (subst finprod_insert)
   510   apply auto
   511   apply (subst m_comm)
   512   apply auto
   513 done
   514 
   515 (* The following lemma was contributed by Jesus Aransay. *)
   516 
   517 lemma finprod_singleton:
   518   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
   519   shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
   520   using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]
   521     fin_A f_Pi finprod_one [of "A - {i}"]
   522     finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"] 
   523   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
   524 
   525 end
   526 
   527 end