src/HOL/Algebra/FiniteProduct.thy
 author haftmann Mon Mar 01 13:40:23 2010 +0100 (2010-03-01) changeset 35416 d8d7d1b785af parent 35054 a5db9779b026 child 35847 19f1f7066917 permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
```     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 constdefs (structure G)
```
```   289   finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
```
```   290   "finprod G f A == if finite A
```
```   291       then foldD (carrier G) (mult G o f) \<one> A
```
```   292       else undefined"
```
```   293
```
```   294 syntax
```
```   295   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   296       ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
```
```   297 syntax (xsymbols)
```
```   298   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   299       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```   300 syntax (HTML output)
```
```   301   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   302       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```   303 translations
```
```   304   "\<Otimes>\<index>i:A. b" == "CONST finprod \<struct>\<index> (%i. b) A"
```
```   305   -- {* Beware of argument permutation! *}
```
```   306
```
```   307 lemma (in comm_monoid) finprod_empty [simp]:
```
```   308   "finprod G f {} = \<one>"
```
```   309   by (simp add: finprod_def)
```
```   310
```
```   311 declare funcsetI [intro]
```
```   312   funcset_mem [dest]
```
```   313
```
```   314 context comm_monoid begin
```
```   315
```
```   316 lemma finprod_insert [simp]:
```
```   317   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
```
```   318    finprod G f (insert a F) = f a \<otimes> finprod G f F"
```
```   319   apply (rule trans)
```
```   320    apply (simp add: finprod_def)
```
```   321   apply (rule trans)
```
```   322    apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
```
```   323          apply simp
```
```   324          apply (rule m_lcomm)
```
```   325            apply fast
```
```   326           apply fast
```
```   327          apply assumption
```
```   328         apply (fastsimp intro: m_closed)
```
```   329        apply simp+
```
```   330    apply fast
```
```   331   apply (auto simp add: finprod_def)
```
```   332   done
```
```   333
```
```   334 lemma finprod_one [simp]:
```
```   335   "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
```
```   336 proof (induct set: finite)
```
```   337   case empty show ?case by simp
```
```   338 next
```
```   339   case (insert a A)
```
```   340   have "(%i. \<one>) \<in> A -> carrier G" by auto
```
```   341   with insert show ?case by simp
```
```   342 qed
```
```   343
```
```   344 lemma finprod_closed [simp]:
```
```   345   fixes A
```
```   346   assumes fin: "finite A" and f: "f \<in> A -> carrier G"
```
```   347   shows "finprod G f A \<in> carrier G"
```
```   348 using fin f
```
```   349 proof induct
```
```   350   case empty show ?case by simp
```
```   351 next
```
```   352   case (insert a A)
```
```   353   then have a: "f a \<in> carrier G" by fast
```
```   354   from insert have A: "f \<in> A -> carrier G" by fast
```
```   355   from insert A a show ?case by simp
```
```   356 qed
```
```   357
```
```   358 lemma funcset_Int_left [simp, intro]:
```
```   359   "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
```
```   360   by fast
```
```   361
```
```   362 lemma funcset_Un_left [iff]:
```
```   363   "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
```
```   364   by fast
```
```   365
```
```   366 lemma finprod_Un_Int:
```
```   367   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
```
```   368      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
```
```   369      finprod G g A \<otimes> finprod G g B"
```
```   370 -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   371 proof (induct set: finite)
```
```   372   case empty then show ?case by (simp add: finprod_closed)
```
```   373 next
```
```   374   case (insert a A)
```
```   375   then have a: "g a \<in> carrier G" by fast
```
```   376   from insert have A: "g \<in> A -> carrier G" by fast
```
```   377   from insert A a show ?case
```
```   378     by (simp add: m_ac Int_insert_left insert_absorb finprod_closed
```
```   379           Int_mono2)
```
```   380 qed
```
```   381
```
```   382 lemma finprod_Un_disjoint:
```
```   383   "[| finite A; finite B; A Int B = {};
```
```   384       g \<in> A -> carrier G; g \<in> B -> carrier G |]
```
```   385    ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
```
```   386   apply (subst finprod_Un_Int [symmetric])
```
```   387       apply (auto simp add: finprod_closed)
```
```   388   done
```
```   389
```
```   390 lemma finprod_multf:
```
```   391   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
```
```   392    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
```
```   393 proof (induct set: finite)
```
```   394   case empty show ?case by simp
```
```   395 next
```
```   396   case (insert a A) then
```
```   397   have fA: "f \<in> A -> carrier G" by fast
```
```   398   from insert have fa: "f a \<in> carrier G" by fast
```
```   399   from insert have gA: "g \<in> A -> carrier G" by fast
```
```   400   from insert have ga: "g a \<in> carrier G" by fast
```
```   401   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"
```
```   402     by (simp add: Pi_def)
```
```   403   show ?case
```
```   404     by (simp add: insert fA fa gA ga fgA m_ac)
```
```   405 qed
```
```   406
```
```   407 lemma finprod_cong':
```
```   408   "[| A = B; g \<in> B -> carrier G;
```
```   409       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   410 proof -
```
```   411   assume prems: "A = B" "g \<in> B -> carrier G"
```
```   412     "!!i. i \<in> B ==> f i = g i"
```
```   413   show ?thesis
```
```   414   proof (cases "finite B")
```
```   415     case True
```
```   416     then have "!!A. [| A = B; g \<in> B -> carrier G;
```
```   417       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   418     proof induct
```
```   419       case empty thus ?case by simp
```
```   420     next
```
```   421       case (insert x B)
```
```   422       then have "finprod G f A = finprod G f (insert x B)" by simp
```
```   423       also from insert have "... = f x \<otimes> finprod G f B"
```
```   424       proof (intro finprod_insert)
```
```   425         show "finite B" by fact
```
```   426       next
```
```   427         show "x ~: B" by fact
```
```   428       next
```
```   429         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   430           "g \<in> insert x B \<rightarrow> carrier G"
```
```   431         thus "f \<in> B -> carrier G" by fastsimp
```
```   432       next
```
```   433         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   434           "g \<in> insert x B \<rightarrow> carrier G"
```
```   435         thus "f x \<in> carrier G" by fastsimp
```
```   436       qed
```
```   437       also from insert have "... = g x \<otimes> finprod G g B" by fastsimp
```
```   438       also from insert have "... = finprod G g (insert x B)"
```
```   439       by (intro finprod_insert [THEN sym]) auto
```
```   440       finally show ?case .
```
```   441     qed
```
```   442     with prems show ?thesis by simp
```
```   443   next
```
```   444     case False with prems show ?thesis by (simp add: finprod_def)
```
```   445   qed
```
```   446 qed
```
```   447
```
```   448 lemma finprod_cong:
```
```   449   "[| A = B; f \<in> B -> carrier G = True;
```
```   450       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   451   (* This order of prems is slightly faster (3%) than the last two swapped. *)
```
```   452   by (rule finprod_cong') force+
```
```   453
```
```   454 text {*Usually, if this rule causes a failed congruence proof error,
```
```   455   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
```
```   456   Adding @{thm [source] Pi_def} to the simpset is often useful.
```
```   457   For this reason, @{thm [source] comm_monoid.finprod_cong}
```
```   458   is not added to the simpset by default.
```
```   459 *}
```
```   460
```
```   461 end
```
```   462
```
```   463 declare funcsetI [rule del]
```
```   464   funcset_mem [rule del]
```
```   465
```
```   466 context comm_monoid begin
```
```   467
```
```   468 lemma finprod_0 [simp]:
```
```   469   "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"
```
```   470 by (simp add: Pi_def)
```
```   471
```
```   472 lemma finprod_Suc [simp]:
```
```   473   "f \<in> {..Suc n} -> carrier G ==>
```
```   474    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
```
```   475 by (simp add: Pi_def atMost_Suc)
```
```   476
```
```   477 lemma finprod_Suc2:
```
```   478   "f \<in> {..Suc n} -> carrier G ==>
```
```   479    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
```
```   480 proof (induct n)
```
```   481   case 0 thus ?case by (simp add: Pi_def)
```
```   482 next
```
```   483   case Suc thus ?case by (simp add: m_assoc Pi_def)
```
```   484 qed
```
```   485
```
```   486 lemma finprod_mult [simp]:
```
```   487   "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
```
```   488      finprod G (%i. f i \<otimes> g i) {..n::nat} =
```
```   489      finprod G f {..n} \<otimes> finprod G g {..n}"
```
```   490   by (induct n) (simp_all add: m_ac Pi_def)
```
```   491
```
```   492 (* The following two were contributed by Jeremy Avigad. *)
```
```   493
```
```   494 lemma finprod_reindex:
```
```   495   assumes fin: "finite A"
```
```   496     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
```
```   497         inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"
```
```   498   using fin apply induct
```
```   499   apply (auto simp add: finprod_insert Pi_def)
```
```   500 done
```
```   501
```
```   502 lemma finprod_const:
```
```   503   assumes fin [simp]: "finite A"
```
```   504       and a [simp]: "a : carrier G"
```
```   505     shows "finprod G (%x. a) A = a (^) card A"
```
```   506   using fin apply induct
```
```   507   apply force
```
```   508   apply (subst finprod_insert)
```
```   509   apply auto
```
```   510   apply (subst m_comm)
```
```   511   apply auto
```
```   512 done
```
```   513
```
```   514 (* The following lemma was contributed by Jesus Aransay. *)
```
```   515
```
```   516 lemma finprod_singleton:
```
```   517   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
```
```   518   shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
```
```   519   using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]
```
```   520     fin_A f_Pi finprod_one [of "A - {i}"]
```
```   521     finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"]
```
```   522   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
```
```   523
```
```   524 end
```
```   525
```
```   526 end
```