src/HOL/Groups_List.thy
 author haftmann Fri Oct 10 19:55:32 2014 +0200 (2014-10-10) changeset 58646 cd63a4b12a33 parent 58368 fe083c681ed8 child 58889 5b7a9633cfa8 permissions -rw-r--r--
specialized specification: avoid trivial instances
```     1
```
```     2 (* Author: Tobias Nipkow, TU Muenchen *)
```
```     3
```
```     4 header {* Sum and product over lists *}
```
```     5
```
```     6 theory Groups_List
```
```     7 imports List
```
```     8 begin
```
```     9
```
```    10 no_notation times (infixl "*" 70)
```
```    11 no_notation Groups.one ("1")
```
```    12
```
```    13 locale monoid_list = monoid
```
```    14 begin
```
```    15
```
```    16 definition F :: "'a list \<Rightarrow> 'a"
```
```    17 where
```
```    18   eq_foldr [code]: "F xs = foldr f xs 1"
```
```    19
```
```    20 lemma Nil [simp]:
```
```    21   "F [] = 1"
```
```    22   by (simp add: eq_foldr)
```
```    23
```
```    24 lemma Cons [simp]:
```
```    25   "F (x # xs) = x * F xs"
```
```    26   by (simp add: eq_foldr)
```
```    27
```
```    28 lemma append [simp]:
```
```    29   "F (xs @ ys) = F xs * F ys"
```
```    30   by (induct xs) (simp_all add: assoc)
```
```    31
```
```    32 end
```
```    33
```
```    34 locale comm_monoid_list = comm_monoid + monoid_list
```
```    35 begin
```
```    36
```
```    37 lemma rev [simp]:
```
```    38   "F (rev xs) = F xs"
```
```    39   by (simp add: eq_foldr foldr_fold  fold_rev fun_eq_iff assoc left_commute)
```
```    40
```
```    41 end
```
```    42
```
```    43 locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set
```
```    44 begin
```
```    45
```
```    46 lemma distinct_set_conv_list:
```
```    47   "distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)"
```
```    48   by (induct xs) simp_all
```
```    49
```
```    50 lemma set_conv_list [code]:
```
```    51   "set.F g (set xs) = list.F (map g (remdups xs))"
```
```    52   by (simp add: distinct_set_conv_list [symmetric])
```
```    53
```
```    54 end
```
```    55
```
```    56 notation times (infixl "*" 70)
```
```    57 notation Groups.one ("1")
```
```    58
```
```    59
```
```    60 subsection {* List summation *}
```
```    61
```
```    62 context monoid_add
```
```    63 begin
```
```    64
```
```    65 definition listsum :: "'a list \<Rightarrow> 'a"
```
```    66 where
```
```    67   "listsum  = monoid_list.F plus 0"
```
```    68
```
```    69 sublocale listsum!: monoid_list plus 0
```
```    70 where
```
```    71  "monoid_list.F plus 0 = listsum"
```
```    72 proof -
```
```    73   show "monoid_list plus 0" ..
```
```    74   then interpret listsum!: monoid_list plus 0 .
```
```    75   from listsum_def show "monoid_list.F plus 0 = listsum" by rule
```
```    76 qed
```
```    77
```
```    78 end
```
```    79
```
```    80 context comm_monoid_add
```
```    81 begin
```
```    82
```
```    83 sublocale listsum!: comm_monoid_list plus 0
```
```    84 where
```
```    85   "monoid_list.F plus 0 = listsum"
```
```    86 proof -
```
```    87   show "comm_monoid_list plus 0" ..
```
```    88   then interpret listsum!: comm_monoid_list plus 0 .
```
```    89   from listsum_def show "monoid_list.F plus 0 = listsum" by rule
```
```    90 qed
```
```    91
```
```    92 sublocale setsum!: comm_monoid_list_set plus 0
```
```    93 where
```
```    94   "monoid_list.F plus 0 = listsum"
```
```    95   and "comm_monoid_set.F plus 0 = setsum"
```
```    96 proof -
```
```    97   show "comm_monoid_list_set plus 0" ..
```
```    98   then interpret setsum!: comm_monoid_list_set plus 0 .
```
```    99   from listsum_def show "monoid_list.F plus 0 = listsum" by rule
```
```   100   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
```
```   101 qed
```
```   102
```
```   103 end
```
```   104
```
```   105 text {* Some syntactic sugar for summing a function over a list: *}
```
```   106
```
```   107 syntax
```
```   108   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
```
```   109 syntax (xsymbols)
```
```   110   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```   111 syntax (HTML output)
```
```   112   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```   113
```
```   114 translations -- {* Beware of argument permutation! *}
```
```   115   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
```
```   116   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
```
```   117
```
```   118 text {* TODO duplicates *}
```
```   119 lemmas listsum_simps = listsum.Nil listsum.Cons
```
```   120 lemmas listsum_append = listsum.append
```
```   121 lemmas listsum_rev = listsum.rev
```
```   122
```
```   123 lemma (in monoid_add) fold_plus_listsum_rev:
```
```   124   "fold plus xs = plus (listsum (rev xs))"
```
```   125 proof
```
```   126   fix x
```
```   127   have "fold plus xs x = listsum (rev xs @ [x])"
```
```   128     by (simp add: foldr_conv_fold listsum.eq_foldr)
```
```   129   also have "\<dots> = listsum (rev xs) + x"
```
```   130     by simp
```
```   131   finally show "fold plus xs x = listsum (rev xs) + x"
```
```   132     .
```
```   133 qed
```
```   134
```
```   135 lemma (in comm_monoid_add) listsum_map_remove1:
```
```   136   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
```
```   137   by (induct xs) (auto simp add: ac_simps)
```
```   138
```
```   139 lemma (in monoid_add) size_list_conv_listsum:
```
```   140   "size_list f xs = listsum (map f xs) + size xs"
```
```   141   by (induct xs) auto
```
```   142
```
```   143 lemma (in monoid_add) length_concat:
```
```   144   "length (concat xss) = listsum (map length xss)"
```
```   145   by (induct xss) simp_all
```
```   146
```
```   147 lemma (in monoid_add) length_product_lists:
```
```   148   "length (product_lists xss) = foldr op * (map length xss) 1"
```
```   149 proof (induct xss)
```
```   150   case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
```
```   151 qed simp
```
```   152
```
```   153 lemma (in monoid_add) listsum_map_filter:
```
```   154   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
```
```   155   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
```
```   156   using assms by (induct xs) auto
```
```   157
```
```   158 lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
```
```   159   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
```
```   160   by (induct xs) simp_all
```
```   161
```
```   162 lemma listsum_eq_0_nat_iff_nat [simp]:
```
```   163   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
```
```   164   by (induct ns) simp_all
```
```   165
```
```   166 lemma member_le_listsum_nat:
```
```   167   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
```
```   168   by (induct ns) auto
```
```   169
```
```   170 lemma elem_le_listsum_nat:
```
```   171   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
```
```   172   by (rule member_le_listsum_nat) simp
```
```   173
```
```   174 lemma listsum_update_nat:
```
```   175   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
```
```   176 apply(induct ns arbitrary:k)
```
```   177  apply (auto split:nat.split)
```
```   178 apply(drule elem_le_listsum_nat)
```
```   179 apply arith
```
```   180 done
```
```   181
```
```   182 lemma (in monoid_add) listsum_triv:
```
```   183   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
```
```   184   by (induct xs) (simp_all add: distrib_right)
```
```   185
```
```   186 lemma (in monoid_add) listsum_0 [simp]:
```
```   187   "(\<Sum>x\<leftarrow>xs. 0) = 0"
```
```   188   by (induct xs) (simp_all add: distrib_right)
```
```   189
```
```   190 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
```
```   191 lemma (in ab_group_add) uminus_listsum_map:
```
```   192   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
```
```   193   by (induct xs) simp_all
```
```   194
```
```   195 lemma (in comm_monoid_add) listsum_addf:
```
```   196   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
```
```   197   by (induct xs) (simp_all add: algebra_simps)
```
```   198
```
```   199 lemma (in ab_group_add) listsum_subtractf:
```
```   200   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
```
```   201   by (induct xs) (simp_all add: algebra_simps)
```
```   202
```
```   203 lemma (in semiring_0) listsum_const_mult:
```
```   204   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
```
```   205   by (induct xs) (simp_all add: algebra_simps)
```
```   206
```
```   207 lemma (in semiring_0) listsum_mult_const:
```
```   208   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
```
```   209   by (induct xs) (simp_all add: algebra_simps)
```
```   210
```
```   211 lemma (in ordered_ab_group_add_abs) listsum_abs:
```
```   212   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
```
```   213   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
```
```   214
```
```   215 lemma listsum_mono:
```
```   216   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
```
```   217   shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
```
```   218   by (induct xs) (simp, simp add: add_mono)
```
```   219
```
```   220 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
```
```   221   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
```
```   222   by (induct xs) simp_all
```
```   223
```
```   224 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
```
```   225   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
```
```   226   by (simp add: listsum_distinct_conv_setsum_set)
```
```   227
```
```   228 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
```
```   229   "listsum (map f [k..l]) = setsum f (set [k..l])"
```
```   230   by (simp add: listsum_distinct_conv_setsum_set)
```
```   231
```
```   232 text {* General equivalence between @{const listsum} and @{const setsum} *}
```
```   233 lemma (in monoid_add) listsum_setsum_nth:
```
```   234   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
```
```   235   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
```
```   236
```
```   237
```
```   238 subsection {* Further facts about @{const List.n_lists} *}
```
```   239
```
```   240 lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
```
```   241   by (induct n) (auto simp add: comp_def length_concat listsum_triv)
```
```   242
```
```   243 lemma distinct_n_lists:
```
```   244   assumes "distinct xs"
```
```   245   shows "distinct (List.n_lists n xs)"
```
```   246 proof (rule card_distinct)
```
```   247   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   248   have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
```
```   249   proof (induct n)
```
```   250     case 0 then show ?case by simp
```
```   251   next
```
```   252     case (Suc n)
```
```   253     moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   254       = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   255       by (rule card_UN_disjoint) auto
```
```   256     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   257       by (rule card_image) (simp add: inj_on_def)
```
```   258     ultimately show ?case by auto
```
```   259   qed
```
```   260   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   261   finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
```
```   262     by (simp add: length_n_lists)
```
```   263 qed
```
```   264
```
```   265
```
```   266 subsection {* Tools setup *}
```
```   267
```
```   268 lemmas setsum_code = setsum.set_conv_list
```
```   269
```
```   270 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
```
```   271   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
```
```   272   by (simp add: interv_listsum_conv_setsum_set_int)
```
```   273
```
```   274 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
```
```   275   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
```
```   276   by (simp add: interv_listsum_conv_setsum_set_nat)
```
```   277
```
```   278 context
```
```   279 begin
```
```   280
```
```   281 interpretation lifting_syntax .
```
```   282
```
```   283 lemma listsum_transfer[transfer_rule]:
```
```   284   assumes [transfer_rule]: "A 0 0"
```
```   285   assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
```
```   286   shows "(list_all2 A ===> A) listsum listsum"
```
```   287   unfolding listsum.eq_foldr [abs_def]
```
```   288   by transfer_prover
```
```   289
```
```   290 end
```
```   291
```
```   292
```
```   293 subsection {* List product *}
```
```   294
```
```   295 context monoid_mult
```
```   296 begin
```
```   297
```
```   298 definition listprod :: "'a list \<Rightarrow> 'a"
```
```   299 where
```
```   300   "listprod  = monoid_list.F times 1"
```
```   301
```
```   302 sublocale listprod!: monoid_list times 1
```
```   303 where
```
```   304   "monoid_list.F times 1 = listprod"
```
```   305 proof -
```
```   306   show "monoid_list times 1" ..
```
```   307   then interpret listprod!: monoid_list times 1 .
```
```   308   from listprod_def show "monoid_list.F times 1 = listprod" by rule
```
```   309 qed
```
```   310
```
```   311 end
```
```   312
```
```   313 context comm_monoid_mult
```
```   314 begin
```
```   315
```
```   316 sublocale listprod!: comm_monoid_list times 1
```
```   317 where
```
```   318   "monoid_list.F times 1 = listprod"
```
```   319 proof -
```
```   320   show "comm_monoid_list times 1" ..
```
```   321   then interpret listprod!: comm_monoid_list times 1 .
```
```   322   from listprod_def show "monoid_list.F times 1 = listprod" by rule
```
```   323 qed
```
```   324
```
```   325 sublocale setprod!: comm_monoid_list_set times 1
```
```   326 where
```
```   327   "monoid_list.F times 1 = listprod"
```
```   328   and "comm_monoid_set.F times 1 = setprod"
```
```   329 proof -
```
```   330   show "comm_monoid_list_set times 1" ..
```
```   331   then interpret setprod!: comm_monoid_list_set times 1 .
```
```   332   from listprod_def show "monoid_list.F times 1 = listprod" by rule
```
```   333   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
```
```   334 qed
```
```   335
```
```   336 end
```
```   337
```
```   338 text {* Some syntactic sugar: *}
```
```   339
```
```   340 syntax
```
```   341   "_listprod" :: "pttrn => 'a list => 'b => 'b"    ("(3PROD _<-_. _)" [0, 51, 10] 10)
```
```   342 syntax (xsymbols)
```
```   343   "_listprod" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```   344 syntax (HTML output)
```
```   345   "_listprod" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```   346
```
```   347 translations -- {* Beware of argument permutation! *}
```
```   348   "PROD x<-xs. b" == "CONST listprod (CONST map (%x. b) xs)"
```
```   349   "\<Prod>x\<leftarrow>xs. b" == "CONST listprod (CONST map (%x. b) xs)"
```
```   350
```
```   351 end
```