src/HOL/Algebra/FiniteProduct.thy
 author paulson Sun Jun 17 22:00:43 2018 +0100 (11 months ago) changeset 68458 023b353911c5 parent 68447 0beb927eed89 child 68517 6b5f15387353 permissions -rw-r--r--
Algebra tidy-up
```     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
```
```     8 imports Group
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Product Operator for Commutative Monoids\<close>
```
```    12
```
```    13 subsubsection \<open>Inductive Definition of a Relation for Products over Sets\<close>
```
```    14
```
```    15 text \<open>Instantiation of locale \<open>LC\<close> of theory \<open>Finite_Set\<close> is not
```
```    16   possible, because here we have explicit typing rules like
```
```    17   \<open>x \<in> carrier G\<close>.  We introduce an explicit argument for the domain
```
```    18   \<open>D\<close>.\<close>
```
```    19
```
```    20 inductive_set
```
```    21   foldSetD :: "['a set, 'b \<Rightarrow> 'a \<Rightarrow> 'a, 'a] \<Rightarrow> ('b set * 'a) set"
```
```    22   for D :: "'a set" and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" and e :: 'a
```
```    23   where
```
```    24     emptyI [intro]: "e \<in> D \<Longrightarrow> ({}, e) \<in> foldSetD D f e"
```
```    25   | insertI [intro]: "\<lbrakk>x \<notin> A; f x y \<in> D; (A, y) \<in> foldSetD D f e\<rbrakk> \<Longrightarrow>
```
```    26                       (insert x A, f x y) \<in> foldSetD D f e"
```
```    27
```
```    28 inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
```
```    29
```
```    30 definition
```
```    31   foldD :: "['a set, 'b \<Rightarrow> 'a \<Rightarrow> 'a, 'a, 'b set] \<Rightarrow> 'a"
```
```    32   where "foldD D f e A = (THE x. (A, x) \<in> foldSetD D f e)"
```
```    33
```
```    34 lemma foldSetD_closed:
```
```    35   "\<lbrakk>(A, z) \<in> foldSetD D f e; e \<in> D; \<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D\<rbrakk>
```
```    36     \<Longrightarrow> z \<in> D"
```
```    37   by (erule foldSetD.cases) auto
```
```    38
```
```    39 lemma Diff1_foldSetD:
```
```    40   "\<lbrakk>(A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D\<rbrakk> \<Longrightarrow>
```
```    41    (A, f x y) \<in> foldSetD D f e"
```
```    42   by (metis Diff_insert_absorb foldSetD.insertI mk_disjoint_insert)
```
```    43
```
```    44 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e \<Longrightarrow> finite A"
```
```    45   by (induct set: foldSetD) auto
```
```    46
```
```    47 lemma finite_imp_foldSetD:
```
```    48   "\<lbrakk>finite A; e \<in> D; \<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D\<rbrakk>
```
```    49     \<Longrightarrow> \<exists>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 \<open>Left-Commutative Operations\<close>
```
```    63
```
```    64 locale LCD =
```
```    65   fixes B :: "'b set"
```
```    66   and D :: "'a set"
```
```    67   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<cdot>" 70)
```
```    68   assumes left_commute:
```
```    69     "\<lbrakk>x \<in> B; y \<in> B; z \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```    70   and f_closed [simp, intro!]: "!!x y. \<lbrakk>x \<in> B; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D"
```
```    71
```
```    72 lemma (in LCD) foldSetD_closed [dest]: "(A, z) \<in> foldSetD D f e \<Longrightarrow> z \<in> D"
```
```    73   by (erule foldSetD.cases) auto
```
```    74
```
```    75 lemma (in LCD) Diff1_foldSetD:
```
```    76   "\<lbrakk>(A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B\<rbrakk> \<Longrightarrow>
```
```    77   (A, f x y) \<in> foldSetD D f e"
```
```    78   by (meson Diff1_foldSetD f_closed local.foldSetD_closed subsetCE)
```
```    79
```
```    80 lemma (in LCD) finite_imp_foldSetD:
```
```    81   "\<lbrakk>finite A; A \<subseteq> B; e \<in> D\<rbrakk> \<Longrightarrow> \<exists>x. (A, x) \<in> foldSetD D f e"
```
```    82 proof (induct set: finite)
```
```    83   case empty then show ?case by auto
```
```    84 next
```
```    85   case (insert x F)
```
```    86   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
```
```    87   with insert have "y \<in> D" by auto
```
```    88   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
```
```    89     by (intro foldSetD.intros) auto
```
```    90   then show ?case ..
```
```    91 qed
```
```    92
```
```    93
```
```    94 lemma (in LCD) foldSetD_determ_aux:
```
```    95   assumes "e \<in> D" and A: "card A < n" "A \<subseteq> B" "(A, x) \<in> foldSetD D f e" "(A, y) \<in> foldSetD D f e"
```
```    96   shows "y = x"
```
```    97   using A
```
```    98 proof (induction n arbitrary: A x y)
```
```    99   case 0
```
```   100   then show ?case
```
```   101     by auto
```
```   102 next
```
```   103   case (Suc n)
```
```   104   then consider "card A = n" | "card A < n"
```
```   105     by linarith
```
```   106   then show ?case
```
```   107   proof cases
```
```   108     case 1
```
```   109     show ?thesis
```
```   110       using foldSetD.cases [OF \<open>(A,x) \<in> foldSetD D (\<cdot>) e\<close>]
```
```   111     proof cases
```
```   112       case 1
```
```   113       then show ?thesis
```
```   114         using \<open>(A,y) \<in> foldSetD D (\<cdot>) e\<close> by auto
```
```   115     next
```
```   116       case (2 x' A' y')
```
```   117       note A' = this
```
```   118       show ?thesis
```
```   119         using foldSetD.cases [OF \<open>(A,y) \<in> foldSetD D (\<cdot>) e\<close>]
```
```   120       proof cases
```
```   121         case 1
```
```   122         then show ?thesis
```
```   123           using \<open>(A,x) \<in> foldSetD D (\<cdot>) e\<close> by auto
```
```   124       next
```
```   125         case (2 x'' A'' y'')
```
```   126         note A'' = this
```
```   127         show ?thesis
```
```   128         proof (cases "x' = x''")
```
```   129           case True
```
```   130           show ?thesis
```
```   131           proof (cases "y' = y''")
```
```   132             case True
```
```   133             then show ?thesis
```
```   134               using A' A'' \<open>x' = x''\<close> by (blast elim!: equalityE)
```
```   135           next
```
```   136             case False
```
```   137             then show ?thesis
```
```   138               using A' A'' \<open>x' = x''\<close>
```
```   139               by (metis \<open>card A = n\<close> Suc.IH Suc.prems(2) card_insert_disjoint foldSetD_imp_finite insert_eq_iff insert_subset lessI)
```
```   140           qed
```
```   141         next
```
```   142           case False
```
```   143           then have *: "A' - {x''} = A'' - {x'}" "x'' \<in> A'" "x' \<in> A''"
```
```   144             using A' A'' by fastforce+
```
```   145           then have "A' = insert x'' A'' - {x'}"
```
```   146             using \<open>x' \<notin> A'\<close> by blast
```
```   147           then have card: "card A' \<le> card A''"
```
```   148             using A' A'' * by (metis card_Suc_Diff1 eq_refl foldSetD_imp_finite)
```
```   149           obtain u where u: "(A' - {x''}, u) \<in> foldSetD D (\<cdot>) e"
```
```   150             using finite_imp_foldSetD [of "A' - {x''}"] A' Diff_insert \<open>A \<subseteq> B\<close> \<open>e \<in> D\<close> by fastforce
```
```   151           have "y' = f x'' u"
```
```   152             using Diff1_foldSetD [OF u] \<open>x'' \<in> A'\<close> \<open>card A = n\<close> A' Suc.IH \<open>A \<subseteq> B\<close> by auto
```
```   153           then have "(A'' - {x'}, u) \<in> foldSetD D f e"
```
```   154             using "*"(1) u by auto
```
```   155           then have "y'' = f x' u"
```
```   156             using A'' by (metis * \<open>card A = n\<close> A'(1) Diff1_foldSetD Suc.IH \<open>A \<subseteq> B\<close>
```
```   157                 card card_Suc_Diff1 card_insert_disjoint foldSetD_imp_finite insert_subset le_imp_less_Suc)
```
```   158           then show ?thesis
```
```   159             using A' A''
```
```   160             by (metis \<open>A \<subseteq> B\<close> \<open>y' = x'' \<cdot> u\<close> insert_subset left_commute local.foldSetD_closed u)
```
```   161         qed
```
```   162       qed
```
```   163     qed
```
```   164   next
```
```   165     case 2 with Suc show ?thesis by blast
```
```   166   qed
```
```   167 qed
```
```   168
```
```   169 lemma (in LCD) foldSetD_determ:
```
```   170   "\<lbrakk>(A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B\<rbrakk>
```
```   171   \<Longrightarrow> y = x"
```
```   172   by (blast intro: foldSetD_determ_aux [rule_format])
```
```   173
```
```   174 lemma (in LCD) foldD_equality:
```
```   175   "\<lbrakk>(A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow> foldD D f e A = y"
```
```   176   by (unfold foldD_def) (blast intro: foldSetD_determ)
```
```   177
```
```   178 lemma foldD_empty [simp]:
```
```   179   "e \<in> D \<Longrightarrow> foldD D f e {} = e"
```
```   180   by (unfold foldD_def) blast
```
```   181
```
```   182 lemma (in LCD) foldD_insert_aux:
```
```   183   "\<lbrakk>x \<notin> A; x \<in> B; e \<in> D; A \<subseteq> B\<rbrakk>
```
```   184     \<Longrightarrow> ((insert x A, v) \<in> foldSetD D f e) \<longleftrightarrow> (\<exists>y. (A, y) \<in> foldSetD D f e \<and> v = f x y)"
```
```   185   apply auto
```
```   186   by (metis Diff_insert_absorb f_closed finite_Diff foldSetD.insertI foldSetD_determ foldSetD_imp_finite insert_subset local.finite_imp_foldSetD local.foldSetD_closed)
```
```   187
```
```   188 lemma (in LCD) foldD_insert:
```
```   189   assumes "finite A" "x \<notin> A" "x \<in> B" "e \<in> D" "A \<subseteq> B"
```
```   190   shows "foldD D f e (insert x A) = f x (foldD D f e A)"
```
```   191 proof -
```
```   192   have "(THE v. \<exists>y. (A, y) \<in> foldSetD D (\<cdot>) e \<and> v = x \<cdot> y) = x \<cdot> (THE y. (A, y) \<in> foldSetD D (\<cdot>) e)"
```
```   193     by (rule the_equality) (use assms foldD_def foldD_equality foldD_def finite_imp_foldSetD in \<open>metis+\<close>)
```
```   194   then show ?thesis
```
```   195     unfolding foldD_def using assms by (simp add: foldD_insert_aux)
```
```   196 qed
```
```   197
```
```   198 lemma (in LCD) foldD_closed [simp]:
```
```   199   "\<lbrakk>finite A; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow> foldD D f e A \<in> D"
```
```   200 proof (induct set: finite)
```
```   201   case empty then show ?case by simp
```
```   202 next
```
```   203   case insert then show ?case by (simp add: foldD_insert)
```
```   204 qed
```
```   205
```
```   206 lemma (in LCD) foldD_commute:
```
```   207   "\<lbrakk>finite A; x \<in> B; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow>
```
```   208    f x (foldD D f e A) = foldD D f (f x e) A"
```
```   209   by (induct set: finite) (auto simp add: left_commute foldD_insert)
```
```   210
```
```   211 lemma Int_mono2:
```
```   212   "\<lbrakk>A \<subseteq> C; B \<subseteq> C\<rbrakk> \<Longrightarrow> A Int B \<subseteq> C"
```
```   213   by blast
```
```   214
```
```   215 lemma (in LCD) foldD_nest_Un_Int:
```
```   216   "\<lbrakk>finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B\<rbrakk> \<Longrightarrow>
```
```   217    foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
```
```   218 proof (induction set: finite)
```
```   219   case (insert x F)
```
```   220   then show ?case
```
```   221     by (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb Int_mono2)
```
```   222 qed simp
```
```   223
```
```   224 lemma (in LCD) foldD_nest_Un_disjoint:
```
```   225   "\<lbrakk>finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B\<rbrakk>
```
```   226     \<Longrightarrow> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
```
```   227   by (simp add: foldD_nest_Un_Int)
```
```   228
```
```   229 \<comment> \<open>Delete rules to do with \<open>foldSetD\<close> relation.\<close>
```
```   230
```
```   231 declare foldSetD_imp_finite [simp del]
```
```   232   empty_foldSetDE [rule del]
```
```   233   foldSetD.intros [rule del]
```
```   234 declare (in LCD)
```
```   235   foldSetD_closed [rule del]
```
```   236
```
```   237
```
```   238 text \<open>Commutative Monoids\<close>
```
```   239
```
```   240 text \<open>
```
```   241   We enter a more restrictive context, with \<open>f :: 'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>
```
```   242   instead of \<open>'b \<Rightarrow> 'a \<Rightarrow> 'a\<close>.
```
```   243 \<close>
```
```   244
```
```   245 locale ACeD =
```
```   246   fixes D :: "'a set"
```
```   247     and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<cdot>" 70)
```
```   248     and e :: 'a
```
```   249   assumes ident [simp]: "x \<in> D \<Longrightarrow> x \<cdot> e = x"
```
```   250     and commute: "\<lbrakk>x \<in> D; y \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> y = y \<cdot> x"
```
```   251     and assoc: "\<lbrakk>x \<in> D; y \<in> D; z \<in> D\<rbrakk> \<Longrightarrow> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   252     and e_closed [simp]: "e \<in> D"
```
```   253     and f_closed [simp]: "\<lbrakk>x \<in> D; y \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> y \<in> D"
```
```   254
```
```   255 lemma (in ACeD) left_commute:
```
```   256   "\<lbrakk>x \<in> D; y \<in> D; z \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   257 proof -
```
```   258   assume D: "x \<in> D" "y \<in> D" "z \<in> D"
```
```   259   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
```
```   260   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
```
```   261   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
```
```   262   finally show ?thesis .
```
```   263 qed
```
```   264
```
```   265 lemmas (in ACeD) AC = assoc commute left_commute
```
```   266
```
```   267 lemma (in ACeD) left_ident [simp]: "x \<in> D \<Longrightarrow> e \<cdot> x = x"
```
```   268 proof -
```
```   269   assume "x \<in> D"
```
```   270   then have "x \<cdot> e = x" by (rule ident)
```
```   271   with \<open>x \<in> D\<close> show ?thesis by (simp add: commute)
```
```   272 qed
```
```   273
```
```   274 lemma (in ACeD) foldD_Un_Int:
```
```   275   "\<lbrakk>finite A; finite B; A \<subseteq> D; B \<subseteq> D\<rbrakk> \<Longrightarrow>
```
```   276     foldD D f e A \<cdot> foldD D f e B =
```
```   277     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
```
```   278 proof (induction set: finite)
```
```   279   case empty
```
```   280   then show ?case
```
```   281     by(simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
```
```   282 next
```
```   283   case (insert x F)
```
```   284   then show ?case
```
```   285     by(simp add: AC insert_absorb Int_insert_left Int_mono2
```
```   286                  LCD.foldD_insert [OF LCD.intro [of D]]
```
```   287                  LCD.foldD_closed [OF LCD.intro [of D]])
```
```   288 qed
```
```   289
```
```   290 lemma (in ACeD) foldD_Un_disjoint:
```
```   291   "\<lbrakk>finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D\<rbrakk> \<Longrightarrow>
```
```   292     foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
```
```   293   by (simp add: foldD_Un_Int
```
```   294     left_commute LCD.foldD_closed [OF LCD.intro [of D]])
```
```   295
```
```   296
```
```   297 subsubsection \<open>Products over Finite Sets\<close>
```
```   298
```
```   299 definition
```
```   300   finprod :: "[('b, 'm) monoid_scheme, 'a \<Rightarrow> 'b, 'a set] \<Rightarrow> 'b"
```
```   301   where "finprod G f A =
```
```   302    (if finite A
```
```   303     then foldD (carrier G) (mult G \<circ> f) \<one>\<^bsub>G\<^esub> A
```
```   304     else \<one>\<^bsub>G\<^esub>)"
```
```   305
```
```   306 syntax
```
```   307   "_finprod" :: "index \<Rightarrow> idt \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   308       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```   309 translations
```
```   310   "\<Otimes>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finprod G (%i. b) A"
```
```   311   \<comment> \<open>Beware of argument permutation!\<close>
```
```   312
```
```   313 lemma (in comm_monoid) finprod_empty [simp]:
```
```   314   "finprod G f {} = \<one>"
```
```   315   by (simp add: finprod_def)
```
```   316
```
```   317 lemma (in comm_monoid) finprod_infinite[simp]:
```
```   318   "\<not> finite A \<Longrightarrow> finprod G f A = \<one>"
```
```   319   by (simp add: finprod_def)
```
```   320
```
```   321 declare funcsetI [intro]
```
```   322   funcset_mem [dest]
```
```   323
```
```   324 context comm_monoid begin
```
```   325
```
```   326 lemma finprod_insert [simp]:
```
```   327   assumes "finite F" "a \<notin> F" "f \<in> F \<rightarrow> carrier G" "f a \<in> carrier G"
```
```   328   shows "finprod G f (insert a F) = f a \<otimes> finprod G f F"
```
```   329 proof -
```
```   330   have "finprod G f (insert a F) = foldD (carrier G) ((\<otimes>) \<circ> f) \<one> (insert a F)"
```
```   331     by (simp add: finprod_def assms)
```
```   332   also have "... = ((\<otimes>) \<circ> f) a (foldD (carrier G) ((\<otimes>) \<circ> f) \<one> F)"
```
```   333     by (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
```
```   334       (use assms in \<open>auto simp: m_lcomm Pi_iff\<close>)
```
```   335   also have "... = f a \<otimes> finprod G f F"
```
```   336     using \<open>finite F\<close> by (auto simp add: finprod_def)
```
```   337   finally show ?thesis .
```
```   338 qed
```
```   339
```
```   340 lemma finprod_one_eqI: "(\<And>x. x \<in> A \<Longrightarrow> f x = \<one>) \<Longrightarrow> finprod G f A = \<one>"
```
```   341 proof (induct A rule: infinite_finite_induct)
```
```   342   case empty show ?case by simp
```
```   343 next
```
```   344   case (insert a A)
```
```   345   have "(\<lambda>i. \<one>) \<in> A \<rightarrow> carrier G" by auto
```
```   346   with insert show ?case by simp
```
```   347 qed simp
```
```   348
```
```   349 lemma finprod_one [simp]: "(\<Otimes>i\<in>A. \<one>) = \<one>"
```
```   350   by (simp add: finprod_one_eqI)
```
```   351
```
```   352 lemma finprod_closed [simp]:
```
```   353   fixes A
```
```   354   assumes f: "f \<in> A \<rightarrow> carrier G"
```
```   355   shows "finprod G f A \<in> carrier G"
```
```   356 using f
```
```   357 proof (induct A rule: infinite_finite_induct)
```
```   358   case empty show ?case by simp
```
```   359 next
```
```   360   case (insert a A)
```
```   361   then have a: "f a \<in> carrier G" by fast
```
```   362   from insert have A: "f \<in> A \<rightarrow> carrier G" by fast
```
```   363   from insert A a show ?case by simp
```
```   364 qed simp
```
```   365
```
```   366 lemma funcset_Int_left [simp, intro]:
```
```   367   "\<lbrakk>f \<in> A \<rightarrow> C; f \<in> B \<rightarrow> C\<rbrakk> \<Longrightarrow> f \<in> A Int B \<rightarrow> C"
```
```   368   by fast
```
```   369
```
```   370 lemma funcset_Un_left [iff]:
```
```   371   "(f \<in> A Un B \<rightarrow> C) = (f \<in> A \<rightarrow> C \<and> f \<in> B \<rightarrow> C)"
```
```   372   by fast
```
```   373
```
```   374 lemma finprod_Un_Int:
```
```   375   "\<lbrakk>finite A; finite B; g \<in> A \<rightarrow> carrier G; g \<in> B \<rightarrow> carrier G\<rbrakk> \<Longrightarrow>
```
```   376      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
```
```   377      finprod G g A \<otimes> finprod G g B"
```
```   378 \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
```
```   379 proof (induct set: finite)
```
```   380   case empty then show ?case by simp
```
```   381 next
```
```   382   case (insert a A)
```
```   383   then have a: "g a \<in> carrier G" by fast
```
```   384   from insert have A: "g \<in> A \<rightarrow> carrier G" by fast
```
```   385   from insert A a show ?case
```
```   386     by (simp add: m_ac Int_insert_left insert_absorb Int_mono2)
```
```   387 qed
```
```   388
```
```   389 lemma finprod_Un_disjoint:
```
```   390   "\<lbrakk>finite A; finite B; A Int B = {};
```
```   391       g \<in> A \<rightarrow> carrier G; g \<in> B \<rightarrow> carrier G\<rbrakk>
```
```   392    \<Longrightarrow> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
```
```   393   by (metis Pi_split_domain finprod_Un_Int finprod_closed finprod_empty r_one)
```
```   394
```
```   395 lemma finprod_multf:
```
```   396   "\<lbrakk>f \<in> A \<rightarrow> carrier G; g \<in> A \<rightarrow> carrier G\<rbrakk> \<Longrightarrow>
```
```   397    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
```
```   398 proof (induct A rule: infinite_finite_induct)
```
```   399   case empty show ?case by simp
```
```   400 next
```
```   401   case (insert a A) then
```
```   402   have fA: "f \<in> A \<rightarrow> carrier G" by fast
```
```   403   from insert have fa: "f a \<in> carrier G" by fast
```
```   404   from insert have gA: "g \<in> A \<rightarrow> carrier G" by fast
```
```   405   from insert have ga: "g a \<in> carrier G" by fast
```
```   406   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A \<rightarrow> carrier G"
```
```   407     by (simp add: Pi_def)
```
```   408   show ?case
```
```   409     by (simp add: insert fA fa gA ga fgA m_ac)
```
```   410 qed simp
```
```   411
```
```   412 lemma finprod_cong':
```
```   413   "\<lbrakk>A = B; g \<in> B \<rightarrow> carrier G;
```
```   414       !!i. i \<in> B \<Longrightarrow> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
```
```   415 proof -
```
```   416   assume prems: "A = B" "g \<in> B \<rightarrow> carrier G"
```
```   417     "!!i. i \<in> B \<Longrightarrow> f i = g i"
```
```   418   show ?thesis
```
```   419   proof (cases "finite B")
```
```   420     case True
```
```   421     then have "!!A. \<lbrakk>A = B; g \<in> B \<rightarrow> carrier G;
```
```   422       !!i. i \<in> B \<Longrightarrow> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
```
```   423     proof induct
```
```   424       case empty thus ?case by simp
```
```   425     next
```
```   426       case (insert x B)
```
```   427       then have "finprod G f A = finprod G f (insert x B)" by simp
```
```   428       also from insert have "... = f x \<otimes> finprod G f B"
```
```   429       proof (intro finprod_insert)
```
```   430         show "finite B" by fact
```
```   431       next
```
```   432         show "x \<notin> B" by fact
```
```   433       next
```
```   434         assume "x \<notin> B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   435           "g \<in> insert x B \<rightarrow> carrier G"
```
```   436         thus "f \<in> B \<rightarrow> carrier G" by fastforce
```
```   437       next
```
```   438         assume "x \<notin> B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   439           "g \<in> insert x B \<rightarrow> carrier G"
```
```   440         thus "f x \<in> carrier G" by fastforce
```
```   441       qed
```
```   442       also from insert have "... = g x \<otimes> finprod G g B" by fastforce
```
```   443       also from insert have "... = finprod G g (insert x B)"
```
```   444       by (intro finprod_insert [THEN sym]) auto
```
```   445       finally show ?case .
```
```   446     qed
```
```   447     with prems show ?thesis by simp
```
```   448   next
```
```   449     case False with prems show ?thesis by simp
```
```   450   qed
```
```   451 qed
```
```   452
```
```   453 lemma finprod_cong:
```
```   454   "\<lbrakk>A = B; f \<in> B \<rightarrow> carrier G = True;
```
```   455       \<And>i. i \<in> B =simp=> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
```
```   456   (* This order of prems is slightly faster (3%) than the last two swapped. *)
```
```   457   by (rule finprod_cong') (auto simp add: simp_implies_def)
```
```   458
```
```   459 text \<open>Usually, if this rule causes a failed congruence proof error,
```
```   460   the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
```
```   461   Adding @{thm [source] Pi_def} to the simpset is often useful.
```
```   462   For this reason, @{thm [source] finprod_cong}
```
```   463   is not added to the simpset by default.
```
```   464 \<close>
```
```   465
```
```   466 end
```
```   467
```
```   468 declare funcsetI [rule del]
```
```   469   funcset_mem [rule del]
```
```   470
```
```   471 context comm_monoid begin
```
```   472
```
```   473 lemma finprod_0 [simp]:
```
```   474   "f \<in> {0::nat} \<rightarrow> carrier G \<Longrightarrow> finprod G f {..0} = f 0"
```
```   475 by (simp add: Pi_def)
```
```   476
```
```   477 lemma finprod_Suc [simp]:
```
```   478   "f \<in> {..Suc n} \<rightarrow> carrier G \<Longrightarrow>
```
```   479    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
```
```   480 by (simp add: Pi_def atMost_Suc)
```
```   481
```
```   482 lemma finprod_Suc2:
```
```   483   "f \<in> {..Suc n} \<rightarrow> carrier G \<Longrightarrow>
```
```   484    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
```
```   485 proof (induct n)
```
```   486   case 0 thus ?case by (simp add: Pi_def)
```
```   487 next
```
```   488   case Suc thus ?case by (simp add: m_assoc Pi_def)
```
```   489 qed
```
```   490
```
```   491 lemma finprod_mult [simp]:
```
```   492   "\<lbrakk>f \<in> {..n} \<rightarrow> carrier G; g \<in> {..n} \<rightarrow> carrier G\<rbrakk> \<Longrightarrow>
```
```   493      finprod G (%i. f i \<otimes> g i) {..n::nat} =
```
```   494      finprod G f {..n} \<otimes> finprod G g {..n}"
```
```   495   by (induct n) (simp_all add: m_ac Pi_def)
```
```   496
```
```   497 (* The following two were contributed by Jeremy Avigad. *)
```
```   498
```
```   499 lemma finprod_reindex:
```
```   500   "f \<in> (h ` A) \<rightarrow> carrier G \<Longrightarrow>
```
```   501         inj_on h A \<Longrightarrow> finprod G f (h ` A) = finprod G (\<lambda>x. f (h x)) A"
```
```   502 proof (induct A rule: infinite_finite_induct)
```
```   503   case (infinite A)
```
```   504   hence "\<not> finite (h ` A)"
```
```   505     using finite_imageD by blast
```
```   506   with \<open>\<not> finite A\<close> show ?case by simp
```
```   507 qed (auto simp add: Pi_def)
```
```   508
```
```   509 lemma finprod_const:
```
```   510   assumes a [simp]: "a \<in> carrier G"
```
```   511     shows "finprod G (\<lambda>x. a) A = a [^] card A"
```
```   512 proof (induct A rule: infinite_finite_induct)
```
```   513   case (insert b A)
```
```   514   show ?case
```
```   515   proof (subst finprod_insert[OF insert(1-2)])
```
```   516     show "a \<otimes> (\<Otimes>x\<in>A. a) = a [^] card (insert b A)"
```
```   517       by (insert insert, auto, subst m_comm, auto)
```
```   518   qed auto
```
```   519 qed auto
```
```   520
```
```   521 (* The following lemma was contributed by Jesus Aransay. *)
```
```   522
```
```   523 lemma finprod_singleton:
```
```   524   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
```
```   525   shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
```
```   526   using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]
```
```   527     fin_A f_Pi finprod_one [of "A - {i}"]
```
```   528     finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"]
```
```   529   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
```
```   530
```
```   531 end
```
```   532
```
```   533 (* Jeremy Avigad. This should be generalized to arbitrary groups, not just commutative
```
```   534    ones, using Lagrange's theorem. *)
```
```   535
```
```   536 lemma (in comm_group) power_order_eq_one:
```
```   537   assumes fin [simp]: "finite (carrier G)"
```
```   538     and a [simp]: "a \<in> carrier G"
```
```   539   shows "a [^] card(carrier G) = one G"
```
```   540 proof -
```
```   541   have "(\<Otimes>x\<in>carrier G. x) = (\<Otimes>x\<in>carrier G. a \<otimes> x)"
```
```   542     by (subst (2) finprod_reindex [symmetric],
```
```   543       auto simp add: Pi_def inj_on_cmult surj_const_mult)
```
```   544   also have "\<dots> = (\<Otimes>x\<in>carrier G. a) \<otimes> (\<Otimes>x\<in>carrier G. x)"
```
```   545     by (auto simp add: finprod_multf Pi_def)
```
```   546   also have "(\<Otimes>x\<in>carrier G. a) = a [^] card(carrier G)"
```
```   547     by (auto simp add: finprod_const)
```
```   548   finally show ?thesis
```
```   549 (* uses the preceeding lemma *)
```
```   550     by auto
```
```   551 qed
```
```   552
```
```   553 lemma (in comm_monoid) finprod_UN_disjoint:
```
```   554   assumes
```
```   555     "finite I" "\<And>i. i \<in> I \<Longrightarrow> finite (A i)" "pairwise (\<lambda>i j. disjnt (A i) (A j)) I"
```
```   556     "\<And>i x. i \<in> I \<Longrightarrow> x \<in> A i \<Longrightarrow> g x \<in> carrier G"
```
```   557 shows "finprod G g (UNION I A) = finprod G (\<lambda>i. finprod G g (A i)) I"
```
```   558   using assms
```
```   559 proof (induction set: finite)
```
```   560   case empty
```
```   561   then show ?case
```
```   562     by force
```
```   563 next
```
```   564   case (insert i I)
```
```   565   then show ?case
```
```   566     unfolding pairwise_def disjnt_def
```
```   567     apply clarsimp
```
```   568     apply (subst finprod_Un_disjoint)
```
```   569          apply (fastforce intro!: funcsetI finprod_closed)+
```
```   570     done
```
```   571 qed
```
```   572
```
```   573 lemma (in comm_monoid) finprod_Union_disjoint:
```
```   574   "\<lbrakk>finite C; \<And>A. A \<in> C \<Longrightarrow> finite A \<and> (\<forall>x\<in>A. f x \<in> carrier G); pairwise disjnt C\<rbrakk> \<Longrightarrow>
```
```   575     finprod G f (\<Union>C) = finprod G (finprod G f) C"
```
```   576   apply (frule finprod_UN_disjoint [of C id f])
```
```   577   apply auto
```
```   578   done
```
```   579
```
```   580 end
```