src/HOL/Algebra/FiniteProduct.thy
 author wenzelm Mon Feb 08 21:28:27 2010 +0100 (2010-02-08) changeset 35054 a5db9779b026 parent 32960 69916a850301 child 35416 d8d7d1b785af permissions -rw-r--r--
modernized some syntax translations;
```     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 constdefs
```
```    30   foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
```
```    31   "foldD D f e A == THE x. (A, x) \<in> foldSetD D f e"
```
```    32
```
```    33 lemma foldSetD_closed:
```
```    34   "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D
```
```    35       |] ==> z \<in> D";
```
```    36   by (erule foldSetD.cases) auto
```
```    37
```
```    38 lemma Diff1_foldSetD:
```
```    39   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>
```
```    40    (A, f x y) \<in> foldSetD D f e"
```
```    41   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
```
```    42     apply auto
```
```    43   done
```
```    44
```
```    45 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"
```
```    46   by (induct set: foldSetD) auto
```
```    47
```
```    48 lemma finite_imp_foldSetD:
```
```    49   "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
```
```    50    EX x. (A, x) \<in> foldSetD D f e"
```
```    51 proof (induct set: finite)
```
```    52   case empty then show ?case by auto
```
```    53 next
```
```    54   case (insert x F)
```
```    55   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
```
```    56   with insert have "y \<in> D" by (auto dest: foldSetD_closed)
```
```    57   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
```
```    58     by (intro foldSetD.intros) auto
```
```    59   then show ?case ..
```
```    60 qed
```
```    61
```
```    62
```
```    63 text {* Left-Commutative Operations *}
```
```    64
```
```    65 locale LCD =
```
```    66   fixes B :: "'b set"
```
```    67   and D :: "'a set"
```
```    68   and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
```
```    69   assumes left_commute:
```
```    70     "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```    71   and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"
```
```    72
```
```    73 lemma (in LCD) foldSetD_closed [dest]:
```
```    74   "(A, z) \<in> foldSetD D f e ==> z \<in> D";
```
```    75   by (erule foldSetD.cases) auto
```
```    76
```
```    77 lemma (in LCD) Diff1_foldSetD:
```
```    78   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
```
```    79   (A, f x y) \<in> foldSetD D f e"
```
```    80   apply (subgoal_tac "x \<in> B")
```
```    81    prefer 2 apply fast
```
```    82   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
```
```    83     apply auto
```
```    84   done
```
```    85
```
```    86 lemma (in LCD) foldSetD_imp_finite [simp]:
```
```    87   "(A, x) \<in> foldSetD D f e ==> finite A"
```
```    88   by (induct set: foldSetD) auto
```
```    89
```
```    90 lemma (in LCD) finite_imp_foldSetD:
```
```    91   "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> foldSetD D f e"
```
```    92 proof (induct set: finite)
```
```    93   case empty then show ?case by auto
```
```    94 next
```
```    95   case (insert x F)
```
```    96   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
```
```    97   with insert have "y \<in> D" by auto
```
```    98   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
```
```    99     by (intro foldSetD.intros) auto
```
```   100   then show ?case ..
```
```   101 qed
```
```   102
```
```   103 lemma (in LCD) foldSetD_determ_aux:
```
```   104   "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
```
```   105     (\<forall>y. (A, y) \<in> foldSetD D f e --> y = x)"
```
```   106   apply (induct n)
```
```   107    apply (auto simp add: less_Suc_eq) (* slow *)
```
```   108   apply (erule foldSetD.cases)
```
```   109    apply blast
```
```   110   apply (erule foldSetD.cases)
```
```   111    apply blast
```
```   112   apply clarify
```
```   113   txt {* force simplification of @{text "card A < card (insert ...)"}. *}
```
```   114   apply (erule rev_mp)
```
```   115   apply (simp add: less_Suc_eq_le)
```
```   116   apply (rule impI)
```
```   117   apply (rename_tac xa Aa ya xb Ab yb, case_tac "xa = xb")
```
```   118    apply (subgoal_tac "Aa = Ab")
```
```   119     prefer 2 apply (blast elim!: equalityE)
```
```   120    apply blast
```
```   121   txt {* case @{prop "xa \<notin> xb"}. *}
```
```   122   apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> Ab")
```
```   123    prefer 2 apply (blast elim!: equalityE)
```
```   124   apply clarify
```
```   125   apply (subgoal_tac "Aa = insert xb Ab - {xa}")
```
```   126    prefer 2 apply blast
```
```   127   apply (subgoal_tac "card Aa \<le> card Ab")
```
```   128    prefer 2
```
```   129    apply (rule Suc_le_mono [THEN subst])
```
```   130    apply (simp add: card_Suc_Diff1)
```
```   131   apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
```
```   132      apply (blast intro: foldSetD_imp_finite finite_Diff)
```
```   133     apply best
```
```   134    apply assumption
```
```   135   apply (frule (1) Diff1_foldSetD)
```
```   136    apply best
```
```   137   apply (subgoal_tac "ya = f xb x")
```
```   138    prefer 2
```
```   139    apply (subgoal_tac "Aa \<subseteq> B")
```
```   140     prefer 2 apply best (* slow *)
```
```   141    apply (blast del: equalityCE)
```
```   142   apply (subgoal_tac "(Ab - {xa}, x) \<in> foldSetD D f e")
```
```   143    prefer 2 apply simp
```
```   144   apply (subgoal_tac "yb = f xa x")
```
```   145    prefer 2
```
```   146    apply (blast del: equalityCE dest: Diff1_foldSetD)
```
```   147   apply (simp (no_asm_simp))
```
```   148   apply (rule left_commute)
```
```   149     apply assumption
```
```   150    apply best (* slow *)
```
```   151   apply best
```
```   152   done
```
```   153
```
```   154 lemma (in LCD) foldSetD_determ:
```
```   155   "[| (A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
```
```   156   ==> y = x"
```
```   157   by (blast intro: foldSetD_determ_aux [rule_format])
```
```   158
```
```   159 lemma (in LCD) foldD_equality:
```
```   160   "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"
```
```   161   by (unfold foldD_def) (blast intro: foldSetD_determ)
```
```   162
```
```   163 lemma foldD_empty [simp]:
```
```   164   "e \<in> D ==> foldD D f e {} = e"
```
```   165   by (unfold foldD_def) blast
```
```   166
```
```   167 lemma (in LCD) foldD_insert_aux:
```
```   168   "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   169     ((insert x A, v) \<in> foldSetD D f e) =
```
```   170     (EX y. (A, y) \<in> foldSetD D f e & v = f x y)"
```
```   171   apply auto
```
```   172   apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
```
```   173      apply (fastsimp dest: foldSetD_imp_finite)
```
```   174     apply assumption
```
```   175    apply assumption
```
```   176   apply (blast intro: foldSetD_determ)
```
```   177   done
```
```   178
```
```   179 lemma (in LCD) foldD_insert:
```
```   180     "[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   181      foldD D f e (insert x A) = f x (foldD D f e A)"
```
```   182   apply (unfold foldD_def)
```
```   183   apply (simp add: foldD_insert_aux)
```
```   184   apply (rule the_equality)
```
```   185    apply (auto intro: finite_imp_foldSetD
```
```   186      cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
```
```   187   done
```
```   188
```
```   189 lemma (in LCD) foldD_closed [simp]:
```
```   190   "[| finite A; e \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
```
```   191 proof (induct set: finite)
```
```   192   case empty then show ?case by (simp add: foldD_empty)
```
```   193 next
```
```   194   case insert then show ?case by (simp add: foldD_insert)
```
```   195 qed
```
```   196
```
```   197 lemma (in LCD) foldD_commute:
```
```   198   "[| finite A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
```
```   199    f x (foldD D f e A) = foldD D f (f x e) A"
```
```   200   apply (induct set: finite)
```
```   201    apply simp
```
```   202   apply (auto simp add: left_commute foldD_insert)
```
```   203   done
```
```   204
```
```   205 lemma Int_mono2:
```
```   206   "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
```
```   207   by blast
```
```   208
```
```   209 lemma (in LCD) foldD_nest_Un_Int:
```
```   210   "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B |] ==>
```
```   211    foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
```
```   212   apply (induct set: finite)
```
```   213    apply simp
```
```   214   apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
```
```   215     Int_mono2)
```
```   216   done
```
```   217
```
```   218 lemma (in LCD) foldD_nest_Un_disjoint:
```
```   219   "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B |]
```
```   220     ==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
```
```   221   by (simp add: foldD_nest_Un_Int)
```
```   222
```
```   223 -- {* Delete rules to do with @{text foldSetD} relation. *}
```
```   224
```
```   225 declare foldSetD_imp_finite [simp del]
```
```   226   empty_foldSetDE [rule del]
```
```   227   foldSetD.intros [rule del]
```
```   228 declare (in LCD)
```
```   229   foldSetD_closed [rule del]
```
```   230
```
```   231
```
```   232 text {* Commutative Monoids *}
```
```   233
```
```   234 text {*
```
```   235   We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
```
```   236   instead of @{text "'b => 'a => 'a"}.
```
```   237 *}
```
```   238
```
```   239 locale ACeD =
```
```   240   fixes D :: "'a set"
```
```   241     and f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   242     and e :: 'a
```
```   243   assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"
```
```   244     and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"
```
```   245     and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   246     and e_closed [simp]: "e \<in> D"
```
```   247     and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"
```
```   248
```
```   249 lemma (in ACeD) left_commute:
```
```   250   "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   251 proof -
```
```   252   assume D: "x \<in> D" "y \<in> D" "z \<in> D"
```
```   253   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
```
```   254   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
```
```   255   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
```
```   256   finally show ?thesis .
```
```   257 qed
```
```   258
```
```   259 lemmas (in ACeD) AC = assoc commute left_commute
```
```   260
```
```   261 lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"
```
```   262 proof -
```
```   263   assume "x \<in> D"
```
```   264   then have "x \<cdot> e = x" by (rule ident)
```
```   265   with `x \<in> D` show ?thesis by (simp add: commute)
```
```   266 qed
```
```   267
```
```   268 lemma (in ACeD) foldD_Un_Int:
```
```   269   "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
```
```   270     foldD D f e A \<cdot> foldD D f e B =
```
```   271     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
```
```   272   apply (induct set: finite)
```
```   273    apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
```
```   274   apply (simp add: AC insert_absorb Int_insert_left
```
```   275     LCD.foldD_insert [OF LCD.intro [of D]]
```
```   276     LCD.foldD_closed [OF LCD.intro [of D]]
```
```   277     Int_mono2)
```
```   278   done
```
```   279
```
```   280 lemma (in ACeD) foldD_Un_disjoint:
```
```   281   "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>
```
```   282     foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
```
```   283   by (simp add: foldD_Un_Int
```
```   284     left_commute LCD.foldD_closed [OF LCD.intro [of D]])
```
```   285
```
```   286
```
```   287 subsubsection {* Products over Finite Sets *}
```
```   288
```
```   289 constdefs (structure G)
```
```   290   finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
```
```   291   "finprod G f A == if finite A
```
```   292       then foldD (carrier G) (mult G o f) \<one> 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
```