src/HOL/Algebra/FiniteProduct.thy
 author haftmann Fri Apr 20 11:21:42 2007 +0200 (2007-04-20) changeset 22744 5cbe966d67a2 parent 22265 3c5c6bdf61de child 23350 50c5b0912a0c permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
```     1 (*  Title:      HOL/Algebra/FiniteProduct.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Clemens Ballarin, started 19 November 2002
```
```     4
```
```     5 This file is largely based on HOL/Finite_Set.thy.
```
```     6 *)
```
```     7
```
```     8 theory FiniteProduct imports Group begin
```
```     9
```
```    10
```
```    11 section {* Product Operator for Commutative Monoids *}
```
```    12
```
```    13
```
```    14 subsection {* Inductive Definition of a Relation for Products over Sets *}
```
```    15
```
```    16 text {* Instantiation of locale @{text LC} of theory @{text Finite_Set} is not
```
```    17   possible, because here we have explicit typing rules like
```
```    18   @{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain
```
```    19   @{text D}. *}
```
```    20
```
```    21 consts
```
```    22   foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
```
```    23
```
```    24 inductive "foldSetD D f e"
```
```    25   intros
```
```    26     emptyI [intro]: "e \<in> D ==> ({}, e) \<in> foldSetD D f e"
```
```    27     insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>
```
```    28                       (insert x A, f x y) \<in> foldSetD D f e"
```
```    29
```
```    30 inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
```
```    31
```
```    32 constdefs
```
```    33   foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
```
```    34   "foldD D f e A == THE x. (A, x) \<in> foldSetD D f e"
```
```    35
```
```    36 lemma foldSetD_closed:
```
```    37   "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D
```
```    38       |] ==> z \<in> D";
```
```    39   by (erule foldSetD.elims) auto
```
```    40
```
```    41 lemma Diff1_foldSetD:
```
```    42   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>
```
```    43    (A, f x y) \<in> foldSetD D f e"
```
```    44   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
```
```    45     apply auto
```
```    46   done
```
```    47
```
```    48 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"
```
```    49   by (induct set: foldSetD) auto
```
```    50
```
```    51 lemma finite_imp_foldSetD:
```
```    52   "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
```
```    53    EX x. (A, x) \<in> foldSetD D f e"
```
```    54 proof (induct set: finite)
```
```    55   case empty then show ?case by auto
```
```    56 next
```
```    57   case (insert x F)
```
```    58   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
```
```    59   with insert have "y \<in> D" by (auto dest: foldSetD_closed)
```
```    60   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
```
```    61     by (intro foldSetD.intros) auto
```
```    62   then show ?case ..
```
```    63 qed
```
```    64
```
```    65
```
```    66 subsection {* Left-Commutative Operations *}
```
```    67
```
```    68 locale LCD =
```
```    69   fixes B :: "'b set"
```
```    70   and D :: "'a set"
```
```    71   and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
```
```    72   assumes left_commute:
```
```    73     "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```    74   and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"
```
```    75
```
```    76 lemma (in LCD) foldSetD_closed [dest]:
```
```    77   "(A, z) \<in> foldSetD D f e ==> z \<in> D";
```
```    78   by (erule foldSetD.elims) auto
```
```    79
```
```    80 lemma (in LCD) Diff1_foldSetD:
```
```    81   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
```
```    82   (A, f x y) \<in> foldSetD D f e"
```
```    83   apply (subgoal_tac "x \<in> B")
```
```    84    prefer 2 apply fast
```
```    85   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
```
```    86     apply auto
```
```    87   done
```
```    88
```
```    89 lemma (in LCD) foldSetD_imp_finite [simp]:
```
```    90   "(A, x) \<in> foldSetD D f e ==> finite A"
```
```    91   by (induct set: foldSetD) auto
```
```    92
```
```    93 lemma (in LCD) finite_imp_foldSetD:
```
```    94   "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> foldSetD D f e"
```
```    95 proof (induct set: finite)
```
```    96   case empty then show ?case by auto
```
```    97 next
```
```    98   case (insert x F)
```
```    99   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
```
```   100   with insert have "y \<in> D" by auto
```
```   101   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
```
```   102     by (intro foldSetD.intros) auto
```
```   103   then show ?case ..
```
```   104 qed
```
```   105
```
```   106 lemma (in LCD) foldSetD_determ_aux:
```
```   107   "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
```
```   108     (\<forall>y. (A, y) \<in> foldSetD D f e --> y = x)"
```
```   109   apply (induct n)
```
```   110    apply (auto simp add: less_Suc_eq) (* slow *)
```
```   111   apply (erule foldSetD.cases)
```
```   112    apply blast
```
```   113   apply (erule foldSetD.cases)
```
```   114    apply blast
```
```   115   apply clarify
```
```   116   txt {* force simplification of @{text "card A < card (insert ...)"}. *}
```
```   117   apply (erule rev_mp)
```
```   118   apply (simp add: less_Suc_eq_le)
```
```   119   apply (rule impI)
```
```   120   apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
```
```   121    apply (subgoal_tac "Aa = Ab")
```
```   122     prefer 2 apply (blast elim!: equalityE)
```
```   123    apply blast
```
```   124   txt {* case @{prop "xa \<notin> xb"}. *}
```
```   125   apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> Ab")
```
```   126    prefer 2 apply (blast elim!: equalityE)
```
```   127   apply clarify
```
```   128   apply (subgoal_tac "Aa = insert xb Ab - {xa}")
```
```   129    prefer 2 apply blast
```
```   130   apply (subgoal_tac "card Aa \<le> card Ab")
```
```   131    prefer 2
```
```   132    apply (rule Suc_le_mono [THEN subst])
```
```   133    apply (simp add: card_Suc_Diff1)
```
```   134   apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
```
```   135      apply (blast intro: foldSetD_imp_finite finite_Diff)
```
```   136     apply best
```
```   137    apply assumption
```
```   138   apply (frule (1) Diff1_foldSetD)
```
```   139    apply best
```
```   140   apply (subgoal_tac "ya = f xb x")
```
```   141    prefer 2
```
```   142    apply (subgoal_tac "Aa \<subseteq> B")
```
```   143     prefer 2 apply best (* slow *)
```
```   144    apply (blast del: equalityCE)
```
```   145   apply (subgoal_tac "(Ab - {xa}, x) \<in> foldSetD D f e")
```
```   146    prefer 2 apply simp
```
```   147   apply (subgoal_tac "yb = f xa x")
```
```   148    prefer 2
```
```   149    apply (blast del: equalityCE dest: Diff1_foldSetD)
```
```   150   apply (simp (no_asm_simp))
```
```   151   apply (rule left_commute)
```
```   152     apply assumption
```
```   153    apply best (* slow *)
```
```   154   apply best
```
```   155   done
```
```   156
```
```   157 lemma (in LCD) foldSetD_determ:
```
```   158   "[| (A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
```
```   159   ==> y = x"
```
```   160   by (blast intro: foldSetD_determ_aux [rule_format])
```
```   161
```
```   162 lemma (in LCD) foldD_equality:
```
```   163   "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"
```
```   164   by (unfold foldD_def) (blast intro: foldSetD_determ)
```
```   165
```
```   166 lemma foldD_empty [simp]:
```
```   167   "e \<in> D ==> foldD D f e {} = e"
```
```   168   by (unfold foldD_def) blast
```
```   169
```
```   170 lemma (in LCD) foldD_insert_aux:
```
```   171   "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   172     ((insert x A, v) \<in> foldSetD D f e) =
```
```   173     (EX y. (A, y) \<in> foldSetD D f e & v = f x y)"
```
```   174   apply auto
```
```   175   apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
```
```   176      apply (fastsimp dest: foldSetD_imp_finite)
```
```   177     apply assumption
```
```   178    apply assumption
```
```   179   apply (blast intro: foldSetD_determ)
```
```   180   done
```
```   181
```
```   182 lemma (in LCD) foldD_insert:
```
```   183     "[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   184      foldD D f e (insert x A) = f x (foldD D f e A)"
```
```   185   apply (unfold foldD_def)
```
```   186   apply (simp add: foldD_insert_aux)
```
```   187   apply (rule the_equality)
```
```   188    apply (auto intro: finite_imp_foldSetD
```
```   189      cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
```
```   190   done
```
```   191
```
```   192 lemma (in LCD) foldD_closed [simp]:
```
```   193   "[| finite A; e \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
```
```   194 proof (induct set: finite)
```
```   195   case empty then show ?case by (simp add: foldD_empty)
```
```   196 next
```
```   197   case insert then show ?case by (simp add: foldD_insert)
```
```   198 qed
```
```   199
```
```   200 lemma (in LCD) foldD_commute:
```
```   201   "[| finite A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   202    f x (foldD D f e A) = foldD D f (f x e) A"
```
```   203   apply (induct set: finite)
```
```   204    apply simp
```
```   205   apply (auto simp add: left_commute foldD_insert)
```
```   206   done
```
```   207
```
```   208 lemma Int_mono2:
```
```   209   "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
```
```   210   by blast
```
```   211
```
```   212 lemma (in LCD) foldD_nest_Un_Int:
```
```   213   "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B |] ==>
```
```   214    foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
```
```   215   apply (induct set: finite)
```
```   216    apply simp
```
```   217   apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
```
```   218     Int_mono2 Un_subset_iff)
```
```   219   done
```
```   220
```
```   221 lemma (in LCD) foldD_nest_Un_disjoint:
```
```   222   "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B |]
```
```   223     ==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
```
```   224   by (simp add: foldD_nest_Un_Int)
```
```   225
```
```   226 -- {* Delete rules to do with @{text foldSetD} relation. *}
```
```   227
```
```   228 declare foldSetD_imp_finite [simp del]
```
```   229   empty_foldSetDE [rule del]
```
```   230   foldSetD.intros [rule del]
```
```   231 declare (in LCD)
```
```   232   foldSetD_closed [rule del]
```
```   233
```
```   234
```
```   235 subsection {* Commutative Monoids *}
```
```   236
```
```   237 text {*
```
```   238   We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
```
```   239   instead of @{text "'b => 'a => 'a"}.
```
```   240 *}
```
```   241
```
```   242 locale ACeD =
```
```   243   fixes D :: "'a set"
```
```   244     and f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   245     and e :: 'a
```
```   246   assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"
```
```   247     and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"
```
```   248     and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   249     and e_closed [simp]: "e \<in> D"
```
```   250     and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"
```
```   251
```
```   252 lemma (in ACeD) left_commute:
```
```   253   "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   254 proof -
```
```   255   assume D: "x \<in> D" "y \<in> D" "z \<in> D"
```
```   256   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
```
```   257   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
```
```   258   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
```
```   259   finally show ?thesis .
```
```   260 qed
```
```   261
```
```   262 lemmas (in ACeD) AC = assoc commute left_commute
```
```   263
```
```   264 lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"
```
```   265 proof -
```
```   266   assume D: "x \<in> D"
```
```   267   have "x \<cdot> e = x" by (rule ident)
```
```   268   with D show ?thesis by (simp add: commute)
```
```   269 qed
```
```   270
```
```   271 lemma (in ACeD) foldD_Un_Int:
```
```   272   "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
```
```   273     foldD D f e A \<cdot> foldD D f e B =
```
```   274     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
```
```   275   apply (induct set: finite)
```
```   276    apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
```
```   277   apply (simp add: AC insert_absorb Int_insert_left
```
```   278     LCD.foldD_insert [OF LCD.intro [of D]]
```
```   279     LCD.foldD_closed [OF LCD.intro [of D]]
```
```   280     Int_mono2 Un_subset_iff)
```
```   281   done
```
```   282
```
```   283 lemma (in ACeD) foldD_Un_disjoint:
```
```   284   "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>
```
```   285     foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
```
```   286   by (simp add: foldD_Un_Int
```
```   287     left_commute LCD.foldD_closed [OF LCD.intro [of D]] Un_subset_iff)
```
```   288
```
```   289
```
```   290 subsection {* Products over Finite Sets *}
```
```   291
```
```   292 constdefs (structure G)
```
```   293   finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
```
```   294   "finprod G f A == if finite A
```
```   295       then foldD (carrier G) (mult G o f) \<one> A
```
```   296       else arbitrary"
```
```   297
```
```   298 syntax
```
```   299   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   300       ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
```
```   301 syntax (xsymbols)
```
```   302   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   303       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```   304 syntax (HTML output)
```
```   305   "_finprod" :: "index => idt => 'a set => 'b => 'b"
```
```   306       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```   307 translations
```
```   308   "\<Otimes>\<index>i:A. b" == "finprod \<struct>\<index> (%i. b) A"
```
```   309   -- {* Beware of argument permutation! *}
```
```   310
```
```   311 lemma (in comm_monoid) finprod_empty [simp]:
```
```   312   "finprod G f {} = \<one>"
```
```   313   by (simp add: finprod_def)
```
```   314
```
```   315 declare funcsetI [intro]
```
```   316   funcset_mem [dest]
```
```   317
```
```   318 lemma (in comm_monoid) finprod_insert [simp]:
```
```   319   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
```
```   320    finprod G f (insert a F) = f a \<otimes> finprod G f F"
```
```   321   apply (rule trans)
```
```   322    apply (simp add: finprod_def)
```
```   323   apply (rule trans)
```
```   324    apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
```
```   325          apply simp
```
```   326          apply (rule m_lcomm)
```
```   327            apply fast
```
```   328           apply fast
```
```   329          apply assumption
```
```   330         apply (fastsimp intro: m_closed)
```
```   331        apply simp+
```
```   332    apply fast
```
```   333   apply (auto simp add: finprod_def)
```
```   334   done
```
```   335
```
```   336 lemma (in comm_monoid) finprod_one [simp]:
```
```   337   "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
```
```   338 proof (induct set: finite)
```
```   339   case empty show ?case by simp
```
```   340 next
```
```   341   case (insert a A)
```
```   342   have "(%i. \<one>) \<in> A -> carrier G" by auto
```
```   343   with insert show ?case by simp
```
```   344 qed
```
```   345
```
```   346 lemma (in comm_monoid) finprod_closed [simp]:
```
```   347   fixes A
```
```   348   assumes fin: "finite A" and f: "f \<in> A -> carrier G"
```
```   349   shows "finprod G f A \<in> carrier G"
```
```   350 using fin f
```
```   351 proof induct
```
```   352   case empty show ?case by simp
```
```   353 next
```
```   354   case (insert a A)
```
```   355   then have a: "f a \<in> carrier G" by fast
```
```   356   from insert have A: "f \<in> A -> carrier G" by fast
```
```   357   from insert A a show ?case by simp
```
```   358 qed
```
```   359
```
```   360 lemma funcset_Int_left [simp, intro]:
```
```   361   "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
```
```   362   by fast
```
```   363
```
```   364 lemma funcset_Un_left [iff]:
```
```   365   "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
```
```   366   by fast
```
```   367
```
```   368 lemma (in comm_monoid) finprod_Un_Int:
```
```   369   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
```
```   370      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
```
```   371      finprod G g A \<otimes> finprod G g B"
```
```   372 -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   373 proof (induct set: finite)
```
```   374   case empty then show ?case by (simp add: finprod_closed)
```
```   375 next
```
```   376   case (insert a A)
```
```   377   then have a: "g a \<in> carrier G" by fast
```
```   378   from insert have A: "g \<in> A -> carrier G" by fast
```
```   379   from insert A a show ?case
```
```   380     by (simp add: m_ac Int_insert_left insert_absorb finprod_closed
```
```   381           Int_mono2 Un_subset_iff)
```
```   382 qed
```
```   383
```
```   384 lemma (in comm_monoid) finprod_Un_disjoint:
```
```   385   "[| finite A; finite B; A Int B = {};
```
```   386       g \<in> A -> carrier G; g \<in> B -> carrier G |]
```
```   387    ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
```
```   388   apply (subst finprod_Un_Int [symmetric])
```
```   389       apply (auto simp add: finprod_closed)
```
```   390   done
```
```   391
```
```   392 lemma (in comm_monoid) finprod_multf:
```
```   393   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
```
```   394    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
```
```   395 proof (induct set: finite)
```
```   396   case empty show ?case by simp
```
```   397 next
```
```   398   case (insert a A) then
```
```   399   have fA: "f \<in> A -> carrier G" by fast
```
```   400   from insert have fa: "f a \<in> carrier G" by fast
```
```   401   from insert have gA: "g \<in> A -> carrier G" by fast
```
```   402   from insert have ga: "g a \<in> carrier G" by fast
```
```   403   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"
```
```   404     by (simp add: Pi_def)
```
```   405   show ?case
```
```   406     by (simp add: insert fA fa gA ga fgA m_ac)
```
```   407 qed
```
```   408
```
```   409 lemma (in comm_monoid) finprod_cong':
```
```   410   "[| A = B; g \<in> B -> carrier G;
```
```   411       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   412 proof -
```
```   413   assume prems: "A = B" "g \<in> B -> carrier G"
```
```   414     "!!i. i \<in> B ==> f i = g i"
```
```   415   show ?thesis
```
```   416   proof (cases "finite B")
```
```   417     case True
```
```   418     then have "!!A. [| A = B; g \<in> B -> carrier G;
```
```   419       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   420     proof induct
```
```   421       case empty thus ?case by simp
```
```   422     next
```
```   423       case (insert x B)
```
```   424       then have "finprod G f A = finprod G f (insert x B)" by simp
```
```   425       also from insert have "... = f x \<otimes> finprod G f B"
```
```   426       proof (intro finprod_insert)
```
```   427 	show "finite B" .
```
```   428       next
```
```   429 	show "x ~: B" .
```
```   430       next
```
```   431 	assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   432 	  "g \<in> insert x B \<rightarrow> carrier G"
```
```   433 	thus "f \<in> B -> carrier G" by fastsimp
```
```   434       next
```
```   435 	assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
```
```   436 	  "g \<in> insert x B \<rightarrow> carrier G"
```
```   437 	thus "f x \<in> carrier G" by fastsimp
```
```   438       qed
```
```   439       also from insert have "... = g x \<otimes> finprod G g B" by fastsimp
```
```   440       also from insert have "... = finprod G g (insert x B)"
```
```   441       by (intro finprod_insert [THEN sym]) auto
```
```   442       finally show ?case .
```
```   443     qed
```
```   444     with prems show ?thesis by simp
```
```   445   next
```
```   446     case False with prems show ?thesis by (simp add: finprod_def)
```
```   447   qed
```
```   448 qed
```
```   449
```
```   450 lemma (in comm_monoid) finprod_cong:
```
```   451   "[| A = B; f \<in> B -> carrier G = True;
```
```   452       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
```
```   453   (* This order of prems is slightly faster (3%) than the last two swapped. *)
```
```   454   by (rule finprod_cong') force+
```
```   455
```
```   456 text {*Usually, if this rule causes a failed congruence proof error,
```
```   457   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
```
```   458   Adding @{thm [source] Pi_def} to the simpset is often useful.
```
```   459   For this reason, @{thm [source] comm_monoid.finprod_cong}
```
```   460   is not added to the simpset by default.
```
```   461 *}
```
```   462
```
```   463 declare funcsetI [rule del]
```
```   464   funcset_mem [rule del]
```
```   465
```
```   466 lemma (in comm_monoid) finprod_0 [simp]:
```
```   467   "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"
```
```   468 by (simp add: Pi_def)
```
```   469
```
```   470 lemma (in comm_monoid) finprod_Suc [simp]:
```
```   471   "f \<in> {..Suc n} -> carrier G ==>
```
```   472    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
```
```   473 by (simp add: Pi_def atMost_Suc)
```
```   474
```
```   475 lemma (in comm_monoid) finprod_Suc2:
```
```   476   "f \<in> {..Suc n} -> carrier G ==>
```
```   477    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
```
```   478 proof (induct n)
```
```   479   case 0 thus ?case by (simp add: Pi_def)
```
```   480 next
```
```   481   case Suc thus ?case by (simp add: m_assoc Pi_def)
```
```   482 qed
```
```   483
```
```   484 lemma (in comm_monoid) finprod_mult [simp]:
```
```   485   "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
```
```   486      finprod G (%i. f i \<otimes> g i) {..n::nat} =
```
```   487      finprod G f {..n} \<otimes> finprod G g {..n}"
```
```   488   by (induct n) (simp_all add: m_ac Pi_def)
```
```   489
```
```   490 end
```
```   491
```