src/HOL/Library/Groups_Big_Fun.thy
 author wenzelm Mon Dec 28 01:28:28 2015 +0100 (2015-12-28) changeset 61945 1135b8de26c3 parent 61776 57bb7da5c867 child 61955 e96292f32c3c permissions -rw-r--r--
more symbols;
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 section \<open>Big sum and product over function bodies\<close>
```
```     4
```
```     5 theory Groups_Big_Fun
```
```     6 imports
```
```     7   Main
```
```     8 begin
```
```     9
```
```    10 subsection \<open>Abstract product\<close>
```
```    11
```
```    12 no_notation times (infixl "*" 70)
```
```    13 no_notation Groups.one ("1")
```
```    14
```
```    15 locale comm_monoid_fun = comm_monoid
```
```    16 begin
```
```    17
```
```    18 definition G :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```    19 where
```
```    20   expand_set: "G g = comm_monoid_set.F f 1 g {a. g a \<noteq> 1}"
```
```    21
```
```    22 interpretation F: comm_monoid_set f 1
```
```    23   ..
```
```    24
```
```    25 lemma expand_superset:
```
```    26   assumes "finite A" and "{a. g a \<noteq> 1} \<subseteq> A"
```
```    27   shows "G g = F.F g A"
```
```    28   apply (simp add: expand_set)
```
```    29   apply (rule F.same_carrierI [of A])
```
```    30   apply (simp_all add: assms)
```
```    31   done
```
```    32
```
```    33 lemma conditionalize:
```
```    34   assumes "finite A"
```
```    35   shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else 1)"
```
```    36   using assms
```
```    37   apply (simp add: expand_set)
```
```    38   apply (rule F.same_carrierI [of A])
```
```    39   apply auto
```
```    40   done
```
```    41
```
```    42 lemma neutral [simp]:
```
```    43   "G (\<lambda>a. 1) = 1"
```
```    44   by (simp add: expand_set)
```
```    45
```
```    46 lemma update [simp]:
```
```    47   assumes "finite {a. g a \<noteq> 1}"
```
```    48   assumes "g a = 1"
```
```    49   shows "G (g(a := b)) = b * G g"
```
```    50 proof (cases "b = 1")
```
```    51   case True with \<open>g a = 1\<close> show ?thesis
```
```    52     by (simp add: expand_set) (rule F.cong, auto)
```
```    53 next
```
```    54   case False
```
```    55   moreover have "{a'. a' \<noteq> a \<longrightarrow> g a' \<noteq> 1} = insert a {a. g a \<noteq> 1}"
```
```    56     by auto
```
```    57   moreover from \<open>g a = 1\<close> have "a \<notin> {a. g a \<noteq> 1}"
```
```    58     by simp
```
```    59   moreover have "F.F (\<lambda>a'. if a' = a then b else g a') {a. g a \<noteq> 1} = F.F g {a. g a \<noteq> 1}"
```
```    60     by (rule F.cong) (auto simp add: \<open>g a = 1\<close>)
```
```    61   ultimately show ?thesis using \<open>finite {a. g a \<noteq> 1}\<close> by (simp add: expand_set)
```
```    62 qed
```
```    63
```
```    64 lemma infinite [simp]:
```
```    65   "\<not> finite {a. g a \<noteq> 1} \<Longrightarrow> G g = 1"
```
```    66   by (simp add: expand_set)
```
```    67
```
```    68 lemma cong:
```
```    69   assumes "\<And>a. g a = h a"
```
```    70   shows "G g = G h"
```
```    71   using assms by (simp add: expand_set)
```
```    72
```
```    73 lemma strong_cong [cong]:
```
```    74   assumes "\<And>a. g a = h a"
```
```    75   shows "G (\<lambda>a. g a) = G (\<lambda>a. h a)"
```
```    76   using assms by (fact cong)
```
```    77
```
```    78 lemma not_neutral_obtains_not_neutral:
```
```    79   assumes "G g \<noteq> 1"
```
```    80   obtains a where "g a \<noteq> 1"
```
```    81   using assms by (auto elim: F.not_neutral_contains_not_neutral simp add: expand_set)
```
```    82
```
```    83 lemma reindex_cong:
```
```    84   assumes "bij l"
```
```    85   assumes "g \<circ> l = h"
```
```    86   shows "G g = G h"
```
```    87 proof -
```
```    88   from assms have unfold: "h = g \<circ> l" by simp
```
```    89   from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
```
```    90   then have "inj_on l {a. h a \<noteq> 1}" by (rule subset_inj_on) simp
```
```    91   moreover from \<open>bij l\<close> have "{a. g a \<noteq> 1} = l ` {a. h a \<noteq> 1}"
```
```    92     by (auto simp add: image_Collect unfold elim: bij_pointE)
```
```    93   moreover have "\<And>x. x \<in> {a. h a \<noteq> 1} \<Longrightarrow> g (l x) = h x"
```
```    94     by (simp add: unfold)
```
```    95   ultimately have "F.F g {a. g a \<noteq> 1} = F.F h {a. h a \<noteq> 1}"
```
```    96     by (rule F.reindex_cong)
```
```    97   then show ?thesis by (simp add: expand_set)
```
```    98 qed
```
```    99
```
```   100 lemma distrib:
```
```   101   assumes "finite {a. g a \<noteq> 1}" and "finite {a. h a \<noteq> 1}"
```
```   102   shows "G (\<lambda>a. g a * h a) = G g * G h"
```
```   103 proof -
```
```   104   from assms have "finite ({a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1})" by simp
```
```   105   moreover have "{a. g a * h a \<noteq> 1} \<subseteq> {a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1}"
```
```   106     by auto (drule sym, simp)
```
```   107   ultimately show ?thesis
```
```   108     using assms
```
```   109     by (simp add: expand_superset [of "{a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1}"] F.distrib)
```
```   110 qed
```
```   111
```
```   112 lemma commute:
```
```   113   assumes "finite C"
```
```   114   assumes subset: "{a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
```
```   115   shows "G (\<lambda>a. G (g a)) = G (\<lambda>b. G (\<lambda>a. g a b))"
```
```   116 proof -
```
```   117   from \<open>finite C\<close> subset
```
```   118     have "finite ({a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1})"
```
```   119     by (rule rev_finite_subset)
```
```   120   then have fins:
```
```   121     "finite {b. \<exists>a. g a b \<noteq> 1}" "finite {a. \<exists>b. g a b \<noteq> 1}"
```
```   122     by (auto simp add: finite_cartesian_product_iff)
```
```   123   have subsets: "\<And>a. {b. g a b \<noteq> 1} \<subseteq> {b. \<exists>a. g a b \<noteq> 1}"
```
```   124     "\<And>b. {a. g a b \<noteq> 1} \<subseteq> {a. \<exists>b. g a b \<noteq> 1}"
```
```   125     "{a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1} \<noteq> 1} \<subseteq> {a. \<exists>b. g a b \<noteq> 1}"
```
```   126     "{a. F.F (\<lambda>aa. g aa a) {a. \<exists>b. g a b \<noteq> 1} \<noteq> 1} \<subseteq> {b. \<exists>a. g a b \<noteq> 1}"
```
```   127     by (auto elim: F.not_neutral_contains_not_neutral)
```
```   128   from F.commute have
```
```   129     "F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) {a. \<exists>b. g a b \<noteq> 1} =
```
```   130       F.F (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> 1}) {b. \<exists>a. g a b \<noteq> 1}" .
```
```   131   with subsets fins have "G (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) =
```
```   132     G (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> 1})"
```
```   133     by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> 1}"]
```
```   134       expand_superset [of "{a. \<exists>b. g a b \<noteq> 1}"])
```
```   135   with subsets fins show ?thesis
```
```   136     by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> 1}"]
```
```   137       expand_superset [of "{a. \<exists>b. g a b \<noteq> 1}"])
```
```   138 qed
```
```   139
```
```   140 lemma cartesian_product:
```
```   141   assumes "finite C"
```
```   142   assumes subset: "{a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
```
```   143   shows "G (\<lambda>a. G (g a)) = G (\<lambda>(a, b). g a b)"
```
```   144 proof -
```
```   145   from subset \<open>finite C\<close> have fin_prod: "finite (?A \<times> ?B)"
```
```   146     by (rule finite_subset)
```
```   147   from fin_prod have "finite ?A" and "finite ?B"
```
```   148     by (auto simp add: finite_cartesian_product_iff)
```
```   149   have *: "G (\<lambda>a. G (g a)) =
```
```   150     (F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) {a. \<exists>b. g a b \<noteq> 1})"
```
```   151     apply (subst expand_superset [of "?B"])
```
```   152     apply (rule \<open>finite ?B\<close>)
```
```   153     apply auto
```
```   154     apply (subst expand_superset [of "?A"])
```
```   155     apply (rule \<open>finite ?A\<close>)
```
```   156     apply auto
```
```   157     apply (erule F.not_neutral_contains_not_neutral)
```
```   158     apply auto
```
```   159     done
```
```   160   have "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> 1} \<subseteq> ?A \<times> ?B"
```
```   161     by auto
```
```   162   with subset have **: "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> 1} \<subseteq> C"
```
```   163     by blast
```
```   164   show ?thesis
```
```   165     apply (simp add: *)
```
```   166     apply (simp add: F.cartesian_product)
```
```   167     apply (subst expand_superset [of C])
```
```   168     apply (rule \<open>finite C\<close>)
```
```   169     apply (simp_all add: **)
```
```   170     apply (rule F.same_carrierI [of C])
```
```   171     apply (rule \<open>finite C\<close>)
```
```   172     apply (simp_all add: subset)
```
```   173     apply auto
```
```   174     done
```
```   175 qed
```
```   176
```
```   177 lemma cartesian_product2:
```
```   178   assumes fin: "finite D"
```
```   179   assumes subset: "{(a, b). \<exists>c. g a b c \<noteq> 1} \<times> {c. \<exists>a b. g a b c \<noteq> 1} \<subseteq> D" (is "?AB \<times> ?C \<subseteq> D")
```
```   180   shows "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>(a, b, c). g a b c)"
```
```   181 proof -
```
```   182   have bij: "bij (\<lambda>(a, b, c). ((a, b), c))"
```
```   183     by (auto intro!: bijI injI simp add: image_def)
```
```   184   have "{p. \<exists>c. g (fst p) (snd p) c \<noteq> 1} \<times> {c. \<exists>p. g (fst p) (snd p) c \<noteq> 1} \<subseteq> D"
```
```   185     by auto (insert subset, blast)
```
```   186   with fin have "G (\<lambda>p. G (g (fst p) (snd p))) = G (\<lambda>(p, c). g (fst p) (snd p) c)"
```
```   187     by (rule cartesian_product)
```
```   188   then have "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>((a, b), c). g a b c)"
```
```   189     by (auto simp add: split_def)
```
```   190   also have "G (\<lambda>((a, b), c). g a b c) = G (\<lambda>(a, b, c). g a b c)"
```
```   191     using bij by (rule reindex_cong [of "\<lambda>(a, b, c). ((a, b), c)"]) (simp add: fun_eq_iff)
```
```   192   finally show ?thesis .
```
```   193 qed
```
```   194
```
```   195 lemma delta [simp]:
```
```   196   "G (\<lambda>b. if b = a then g b else 1) = g a"
```
```   197 proof -
```
```   198   have "{b. (if b = a then g b else 1) \<noteq> 1} \<subseteq> {a}" by auto
```
```   199   then show ?thesis by (simp add: expand_superset [of "{a}"])
```
```   200 qed
```
```   201
```
```   202 lemma delta' [simp]:
```
```   203   "G (\<lambda>b. if a = b then g b else 1) = g a"
```
```   204 proof -
```
```   205   have "(\<lambda>b. if a = b then g b else 1) = (\<lambda>b. if b = a then g b else 1)"
```
```   206     by (simp add: fun_eq_iff)
```
```   207   then have "G (\<lambda>b. if a = b then g b else 1) = G (\<lambda>b. if b = a then g b else 1)"
```
```   208     by (simp cong del: strong_cong)
```
```   209   then show ?thesis by simp
```
```   210 qed
```
```   211
```
```   212 end
```
```   213
```
```   214 notation times (infixl "*" 70)
```
```   215 notation Groups.one ("1")
```
```   216
```
```   217
```
```   218 subsection \<open>Concrete sum\<close>
```
```   219
```
```   220 context comm_monoid_add
```
```   221 begin
```
```   222
```
```   223 sublocale Sum_any: comm_monoid_fun plus 0
```
```   224 defines
```
```   225   Sum_any = Sum_any.G
```
```   226 rewrites
```
```   227   "comm_monoid_set.F plus 0 = setsum"
```
```   228 proof -
```
```   229   show "comm_monoid_fun plus 0" ..
```
```   230   then interpret Sum_any: comm_monoid_fun plus 0 .
```
```   231   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by (auto intro: sym)
```
```   232 qed
```
```   233
```
```   234 end
```
```   235
```
```   236 syntax
```
```   237   "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"    ("(3SUM _. _)" [0, 10] 10)
```
```   238 syntax (xsymbols)
```
```   239   "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"    ("(3\<Sum>_. _)" [0, 10] 10)
```
```   240 translations
```
```   241   "\<Sum>a. b" == "CONST Sum_any (\<lambda>a. b)"
```
```   242
```
```   243 lemma Sum_any_left_distrib:
```
```   244   fixes r :: "'a :: semiring_0"
```
```   245   assumes "finite {a. g a \<noteq> 0}"
```
```   246   shows "Sum_any g * r = (\<Sum>n. g n * r)"
```
```   247 proof -
```
```   248   note assms
```
```   249   moreover have "{a. g a * r \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
```
```   250   ultimately show ?thesis
```
```   251     by (simp add: setsum_left_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
```
```   252 qed
```
```   253
```
```   254 lemma Sum_any_right_distrib:
```
```   255   fixes r :: "'a :: semiring_0"
```
```   256   assumes "finite {a. g a \<noteq> 0}"
```
```   257   shows "r * Sum_any g = (\<Sum>n. r * g n)"
```
```   258 proof -
```
```   259   note assms
```
```   260   moreover have "{a. r * g a \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
```
```   261   ultimately show ?thesis
```
```   262     by (simp add: setsum_right_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
```
```   263 qed
```
```   264
```
```   265 lemma Sum_any_product:
```
```   266   fixes f g :: "'b \<Rightarrow> 'a::semiring_0"
```
```   267   assumes "finite {a. f a \<noteq> 0}" and "finite {b. g b \<noteq> 0}"
```
```   268   shows "Sum_any f * Sum_any g = (\<Sum>a. \<Sum>b. f a * g b)"
```
```   269 proof -
```
```   270   have subset_f: "{a. (\<Sum>b. f a * g b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}"
```
```   271     by rule (simp, rule, auto)
```
```   272   moreover have subset_g: "\<And>a. {b. f a * g b \<noteq> 0} \<subseteq> {b. g b \<noteq> 0}"
```
```   273     by rule (simp, rule, auto)
```
```   274   ultimately show ?thesis using assms
```
```   275     by (auto simp add: Sum_any.expand_set [of f] Sum_any.expand_set [of g]
```
```   276       Sum_any.expand_superset [of "{a. f a \<noteq> 0}"] Sum_any.expand_superset [of "{b. g b \<noteq> 0}"]
```
```   277       setsum_product)
```
```   278 qed
```
```   279
```
```   280 lemma Sum_any_eq_zero_iff [simp]:
```
```   281   fixes f :: "'a \<Rightarrow> nat"
```
```   282   assumes "finite {a. f a \<noteq> 0}"
```
```   283   shows "Sum_any f = 0 \<longleftrightarrow> f = (\<lambda>_. 0)"
```
```   284   using assms by (simp add: Sum_any.expand_set fun_eq_iff)
```
```   285
```
```   286
```
```   287 subsection \<open>Concrete product\<close>
```
```   288
```
```   289 context comm_monoid_mult
```
```   290 begin
```
```   291
```
```   292 sublocale Prod_any: comm_monoid_fun times 1
```
```   293 defines
```
```   294   Prod_any = Prod_any.G
```
```   295 rewrites
```
```   296   "comm_monoid_set.F times 1 = setprod"
```
```   297 proof -
```
```   298   show "comm_monoid_fun times 1" ..
```
```   299   then interpret Prod_any: comm_monoid_fun times 1 .
```
```   300   from setprod_def show "comm_monoid_set.F times 1 = setprod" by (auto intro: sym)
```
```   301 qed
```
```   302
```
```   303 end
```
```   304
```
```   305 syntax
```
```   306   "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"    ("(3PROD _. _)" [0, 10] 10)
```
```   307 syntax (xsymbols)
```
```   308   "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"    ("(3\<Prod>_. _)" [0, 10] 10)
```
```   309 translations
```
```   310   "\<Prod>a. b" == "CONST Prod_any (\<lambda>a. b)"
```
```   311
```
```   312 lemma Prod_any_zero:
```
```   313   fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
```
```   314   assumes "finite {a. f a \<noteq> 1}"
```
```   315   assumes "f a = 0"
```
```   316   shows "(\<Prod>a. f a) = 0"
```
```   317 proof -
```
```   318   from \<open>f a = 0\<close> have "f a \<noteq> 1" by simp
```
```   319   with \<open>f a = 0\<close> have "\<exists>a. f a \<noteq> 1 \<and> f a = 0" by blast
```
```   320   with \<open>finite {a. f a \<noteq> 1}\<close> show ?thesis
```
```   321     by (simp add: Prod_any.expand_set setprod_zero)
```
```   322 qed
```
```   323
```
```   324 lemma Prod_any_not_zero:
```
```   325   fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
```
```   326   assumes "finite {a. f a \<noteq> 1}"
```
```   327   assumes "(\<Prod>a. f a) \<noteq> 0"
```
```   328   shows "f a \<noteq> 0"
```
```   329   using assms Prod_any_zero [of f] by blast
```
```   330
```
```   331 lemma power_Sum_any:
```
```   332   assumes "finite {a. f a \<noteq> 0}"
```
```   333   shows "c ^ (\<Sum>a. f a) = (\<Prod>a. c ^ f a)"
```
```   334 proof -
```
```   335   have "{a. c ^ f a \<noteq> 1} \<subseteq> {a. f a \<noteq> 0}"
```
```   336     by (auto intro: ccontr)
```
```   337   with assms show ?thesis
```
```   338     by (simp add: Sum_any.expand_set Prod_any.expand_superset power_setsum)
```
```   339 qed
```
```   340
```
```   341 end
```
```   342
```