src/HOL/Groups_List.thy
 author blanchet Wed Sep 03 00:06:24 2014 +0200 (2014-09-03) changeset 58152 6fe60a9a5bad parent 58101 e7ebe5554281 child 58320 351810c45a48 permissions -rw-r--r--
use 'datatype_new' in 'Main'
```     1
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```     2 (* Author: Tobias Nipkow, TU Muenchen *)
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```     3
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```     4 header {* Summation over lists *}
```
```     5
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```     6 theory Groups_List
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```     7 imports List
```
```     8 begin
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```     9
```
```    10 definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
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```    11   "listsum xs = foldr plus xs 0"
```
```    12
```
```    13 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
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```    14
```
```    15 lemma (in monoid_add) listsum_simps [simp]:
```
```    16   "listsum [] = 0"
```
```    17   "listsum (x # xs) = x + listsum xs"
```
```    18   by (simp_all add: listsum_def)
```
```    19
```
```    20 lemma (in monoid_add) listsum_append [simp]:
```
```    21   "listsum (xs @ ys) = listsum xs + listsum ys"
```
```    22   by (induct xs) (simp_all add: add.assoc)
```
```    23
```
```    24 lemma (in comm_monoid_add) listsum_rev [simp]:
```
```    25   "listsum (rev xs) = listsum xs"
```
```    26   by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
```
```    27
```
```    28 lemma (in monoid_add) fold_plus_listsum_rev:
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```    29   "fold plus xs = plus (listsum (rev xs))"
```
```    30 proof
```
```    31   fix x
```
```    32   have "fold plus xs x = fold plus xs (x + 0)" by simp
```
```    33   also have "\<dots> = fold plus (x # xs) 0" by simp
```
```    34   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
```
```    35   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
```
```    36   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
```
```    37   finally show "fold plus xs x = listsum (rev xs) + x" by simp
```
```    38 qed
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```    39
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```    40 text{* Some syntactic sugar for summing a function over a list: *}
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```    41
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```    42 syntax
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```    43   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
```
```    44 syntax (xsymbols)
```
```    45   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```    46 syntax (HTML output)
```
```    47   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
```
```    48
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```    49 translations -- {* Beware of argument permutation! *}
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```    50   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
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```    51   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
```
```    52
```
```    53 lemma (in comm_monoid_add) listsum_map_remove1:
```
```    54   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
```
```    55   by (induct xs) (auto simp add: ac_simps)
```
```    56
```
```    57 lemma (in monoid_add) size_list_conv_listsum:
```
```    58   "size_list f xs = listsum (map f xs) + size xs"
```
```    59   by (induct xs) auto
```
```    60
```
```    61 lemma (in monoid_add) length_concat:
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```    62   "length (concat xss) = listsum (map length xss)"
```
```    63   by (induct xss) simp_all
```
```    64
```
```    65 lemma (in monoid_add) length_product_lists:
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```    66   "length (product_lists xss) = foldr op * (map length xss) 1"
```
```    67 proof (induct xss)
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```    68   case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
```
```    69 qed simp
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```    70
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```    71 lemma (in monoid_add) listsum_map_filter:
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```    72   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
```
```    73   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
```
```    74   using assms by (induct xs) auto
```
```    75
```
```    76 lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
```
```    77   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
```
```    78   by (induct xs) simp_all
```
```    79
```
```    80 lemma listsum_eq_0_nat_iff_nat [simp]:
```
```    81   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
```
```    82   by (induct ns) simp_all
```
```    83
```
```    84 lemma member_le_listsum_nat:
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```    85   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
```
```    86   by (induct ns) auto
```
```    87
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```    88 lemma elem_le_listsum_nat:
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```    89   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
```
```    90   by (rule member_le_listsum_nat) simp
```
```    91
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```    92 lemma listsum_update_nat:
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```    93   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
```
```    94 apply(induct ns arbitrary:k)
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```    95  apply (auto split:nat.split)
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```    96 apply(drule elem_le_listsum_nat)
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```    97 apply arith
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```    98 done
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```    99
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```   100 lemma (in monoid_add) listsum_triv:
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```   101   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
```
```   102   by (induct xs) (simp_all add: distrib_right)
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```   103
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```   104 lemma (in monoid_add) listsum_0 [simp]:
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```   105   "(\<Sum>x\<leftarrow>xs. 0) = 0"
```
```   106   by (induct xs) (simp_all add: distrib_right)
```
```   107
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```   108 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
```
```   109 lemma (in ab_group_add) uminus_listsum_map:
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```   110   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
```
```   111   by (induct xs) simp_all
```
```   112
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```   113 lemma (in comm_monoid_add) listsum_addf:
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```   114   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
```
```   115   by (induct xs) (simp_all add: algebra_simps)
```
```   116
```
```   117 lemma (in ab_group_add) listsum_subtractf:
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```   118   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
```
```   119   by (induct xs) (simp_all add: algebra_simps)
```
```   120
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```   121 lemma (in semiring_0) listsum_const_mult:
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```   122   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
```
```   123   by (induct xs) (simp_all add: algebra_simps)
```
```   124
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```   125 lemma (in semiring_0) listsum_mult_const:
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```   126   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
```
```   127   by (induct xs) (simp_all add: algebra_simps)
```
```   128
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```   129 lemma (in ordered_ab_group_add_abs) listsum_abs:
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```   130   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
```
```   131   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
```
```   132
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```   133 lemma listsum_mono:
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```   134   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
```
```   135   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)"
```
```   136   by (induct xs) (simp, simp add: add_mono)
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```   137
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```   138 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
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```   139   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
```
```   140   by (induct xs) simp_all
```
```   141
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```   142 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
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```   143   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
```
```   144   by (simp add: listsum_distinct_conv_setsum_set)
```
```   145
```
```   146 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
```
```   147   "listsum (map f [k..l]) = setsum f (set [k..l])"
```
```   148   by (simp add: listsum_distinct_conv_setsum_set)
```
```   149
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```   150 text {* General equivalence between @{const listsum} and @{const setsum} *}
```
```   151 lemma (in monoid_add) listsum_setsum_nth:
```
```   152   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
```
```   153   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
```
```   154
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```   155
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```   156 subsection {* Further facts about @{const List.n_lists} *}
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```   157
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```   158 lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
```
```   159   by (induct n) (auto simp add: comp_def length_concat listsum_triv)
```
```   160
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```   161 lemma distinct_n_lists:
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```   162   assumes "distinct xs"
```
```   163   shows "distinct (List.n_lists n xs)"
```
```   164 proof (rule card_distinct)
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```   165   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   166   have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
```
```   167   proof (induct n)
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```   168     case 0 then show ?case by simp
```
```   169   next
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```   170     case (Suc n)
```
```   171     moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   172       = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   173       by (rule card_UN_disjoint) auto
```
```   174     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   175       by (rule card_image) (simp add: inj_on_def)
```
```   176     ultimately show ?case by auto
```
```   177   qed
```
```   178   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   179   finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
```
```   180     by (simp add: length_n_lists)
```
```   181 qed
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```   182
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```   183
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```   184 subsection {* Tools setup *}
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```   185
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```   186 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
```
```   187   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
```
```   188   by (simp add: interv_listsum_conv_setsum_set_int)
```
```   189
```
```   190 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
```
```   191   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
```
```   192   by (simp add: interv_listsum_conv_setsum_set_nat)
```
```   193
```
```   194 lemma setsum_code [code]:
```
```   195   "setsum f (set xs) = listsum (map f (remdups xs))"
```
```   196   by (simp add: listsum_distinct_conv_setsum_set)
```
```   197
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```   198 context
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```   199 begin
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```   200
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```   201 interpretation lifting_syntax .
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```   202
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```   203 lemma listsum_transfer[transfer_rule]:
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```   204   assumes [transfer_rule]: "A 0 0"
```
```   205   assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
```
```   206   shows "(list_all2 A ===> A) listsum listsum"
```
```   207   unfolding listsum_def[abs_def]
```
```   208   by transfer_prover
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```   209
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```   210 end
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```   211
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`   212 end`