src/HOL/Library/DAList_Multiset.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 55887 25bd4745ee38 child 58806 bb5ab5fce93a permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/DAList_Multiset.thy
```
```     2     Author:     Lukas Bulwahn, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Multisets partially implemented by association lists *}
```
```     6
```
```     7 theory DAList_Multiset
```
```     8 imports Multiset DAList
```
```     9 begin
```
```    10
```
```    11 text {* Delete prexisting code equations *}
```
```    12
```
```    13 lemma [code, code del]:
```
```    14   "{#} = {#}"
```
```    15   ..
```
```    16
```
```    17 lemma [code, code del]:
```
```    18   "single = single"
```
```    19   ..
```
```    20
```
```    21 lemma [code, code del]:
```
```    22   "plus = (plus :: 'a multiset \<Rightarrow> _)"
```
```    23   ..
```
```    24
```
```    25 lemma [code, code del]:
```
```    26   "minus = (minus :: 'a multiset \<Rightarrow> _)"
```
```    27   ..
```
```    28
```
```    29 lemma [code, code del]:
```
```    30   "inf = (inf :: 'a multiset \<Rightarrow> _)"
```
```    31   ..
```
```    32
```
```    33 lemma [code, code del]:
```
```    34   "sup = (sup :: 'a multiset \<Rightarrow> _)"
```
```    35   ..
```
```    36
```
```    37 lemma [code, code del]:
```
```    38   "image_mset = image_mset"
```
```    39   ..
```
```    40
```
```    41 lemma [code, code del]:
```
```    42   "Multiset.filter = Multiset.filter"
```
```    43   ..
```
```    44
```
```    45 lemma [code, code del]:
```
```    46   "count = count"
```
```    47   ..
```
```    48
```
```    49 lemma [code, code del]:
```
```    50   "mcard = mcard"
```
```    51   ..
```
```    52
```
```    53 lemma [code, code del]:
```
```    54   "msetsum = msetsum"
```
```    55   ..
```
```    56
```
```    57 lemma [code, code del]:
```
```    58   "msetprod = msetprod"
```
```    59   ..
```
```    60
```
```    61 lemma [code, code del]:
```
```    62   "set_of = set_of"
```
```    63   ..
```
```    64
```
```    65 lemma [code, code del]:
```
```    66   "sorted_list_of_multiset = sorted_list_of_multiset"
```
```    67   ..
```
```    68
```
```    69 lemma [code, code del]:
```
```    70   "ord_multiset_inst.less_eq_multiset = ord_multiset_inst.less_eq_multiset"
```
```    71   ..
```
```    72
```
```    73 lemma [code, code del]:
```
```    74   "ord_multiset_inst.less_multiset = ord_multiset_inst.less_multiset"
```
```    75   ..
```
```    76
```
```    77 lemma [code, code del]:
```
```    78   "equal_multiset_inst.equal_multiset = equal_multiset_inst.equal_multiset"
```
```    79   ..
```
```    80
```
```    81
```
```    82 text {* Raw operations on lists *}
```
```    83
```
```    84 definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
```
```    85 where
```
```    86   "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
```
```    87
```
```    88 lemma join_raw_Nil [simp]:
```
```    89   "join_raw f xs [] = xs"
```
```    90 by (simp add: join_raw_def)
```
```    91
```
```    92 lemma join_raw_Cons [simp]:
```
```    93   "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
```
```    94 by (simp add: join_raw_def)
```
```    95
```
```    96 lemma map_of_join_raw:
```
```    97   assumes "distinct (map fst ys)"
```
```    98   shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
```
```    99     (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
```
```   100 using assms
```
```   101 apply (induct ys)
```
```   102 apply (auto simp add: map_of_map_default split: option.split)
```
```   103 apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
```
```   104 by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
```
```   105
```
```   106 lemma distinct_join_raw:
```
```   107   assumes "distinct (map fst xs)"
```
```   108   shows "distinct (map fst (join_raw f xs ys))"
```
```   109 using assms
```
```   110 proof (induct ys)
```
```   111   case (Cons y ys)
```
```   112   thus ?case by (cases y) (simp add: distinct_map_default)
```
```   113 qed auto
```
```   114
```
```   115 definition
```
```   116   "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
```
```   117
```
```   118 lemma map_of_subtract_entries_raw:
```
```   119   assumes "distinct (map fst ys)"
```
```   120   shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v =>
```
```   121     (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
```
```   122 using assms unfolding subtract_entries_raw_def
```
```   123 apply (induct ys)
```
```   124 apply auto
```
```   125 apply (simp split: option.split)
```
```   126 apply (simp add: map_of_map_entry)
```
```   127 apply (auto split: option.split)
```
```   128 apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
```
```   129 by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
```
```   130
```
```   131 lemma distinct_subtract_entries_raw:
```
```   132   assumes "distinct (map fst xs)"
```
```   133   shows "distinct (map fst (subtract_entries_raw xs ys))"
```
```   134 using assms
```
```   135 unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
```
```   136
```
```   137
```
```   138 text {* Operations on alists with distinct keys *}
```
```   139
```
```   140 lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```   141 is join_raw
```
```   142 by (simp add: distinct_join_raw)
```
```   143
```
```   144 lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```   145 is subtract_entries_raw
```
```   146 by (simp add: distinct_subtract_entries_raw)
```
```   147
```
```   148
```
```   149 text {* Implementing multisets by means of association lists *}
```
```   150
```
```   151 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
```
```   152   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
```
```   153
```
```   154 lemma count_of_multiset:
```
```   155   "count_of xs \<in> multiset"
```
```   156 proof -
```
```   157   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
```
```   158   have "?A \<subseteq> dom (map_of xs)"
```
```   159   proof
```
```   160     fix x
```
```   161     assume "x \<in> ?A"
```
```   162     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
```
```   163     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
```
```   164     then show "x \<in> dom (map_of xs)" by auto
```
```   165   qed
```
```   166   with finite_dom_map_of [of xs] have "finite ?A"
```
```   167     by (auto intro: finite_subset)
```
```   168   then show ?thesis
```
```   169     by (simp add: count_of_def fun_eq_iff multiset_def)
```
```   170 qed
```
```   171
```
```   172 lemma count_simps [simp]:
```
```   173   "count_of [] = (\<lambda>_. 0)"
```
```   174   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
```
```   175   by (simp_all add: count_of_def fun_eq_iff)
```
```   176
```
```   177 lemma count_of_empty:
```
```   178   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
```
```   179   by (induct xs) (simp_all add: count_of_def)
```
```   180
```
```   181 lemma count_of_filter:
```
```   182   "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
```
```   183   by (induct xs) auto
```
```   184
```
```   185 lemma count_of_map_default [simp]:
```
```   186   "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
```
```   187 unfolding count_of_def by (simp add: map_of_map_default split: option.split)
```
```   188
```
```   189 lemma count_of_join_raw:
```
```   190   "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
```
```   191 unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
```
```   192
```
```   193 lemma count_of_subtract_entries_raw:
```
```   194   "distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
```
```   195 unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
```
```   196
```
```   197
```
```   198 text {* Code equations for multiset operations *}
```
```   199
```
```   200 definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
```
```   201   "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
```
```   202
```
```   203 code_datatype Bag
```
```   204
```
```   205 lemma count_Bag [simp, code]:
```
```   206   "count (Bag xs) = count_of (DAList.impl_of xs)"
```
```   207   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
```
```   208
```
```   209 lemma Mempty_Bag [code]:
```
```   210   "{#} = Bag (DAList.empty)"
```
```   211   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
```
```   212
```
```   213 lemma single_Bag [code]:
```
```   214   "{#x#} = Bag (DAList.update x 1 DAList.empty)"
```
```   215   by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
```
```   216
```
```   217 lemma union_Bag [code]:
```
```   218   "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
```
```   219 by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
```
```   220
```
```   221 lemma minus_Bag [code]:
```
```   222   "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
```
```   223 by (rule multiset_eqI)
```
```   224   (simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
```
```   225
```
```   226 lemma filter_Bag [code]:
```
```   227   "Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
```
```   228 by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
```
```   229
```
```   230
```
```   231 lemma mset_eq [code]: "HOL.equal (m1::'a::equal multiset) m2 \<longleftrightarrow> m1 \<le> m2 \<and> m2 \<le> m1"
```
```   232 by (metis equal_multiset_def eq_iff)
```
```   233
```
```   234 text{* By default the code for @{text "<"} is @{prop"xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> xs = ys"}.
```
```   235 With equality implemented by @{text"\<le>"}, this leads to three calls of  @{text"\<le>"}.
```
```   236 Here is a more efficient version: *}
```
```   237 lemma mset_less[code]: "xs < (ys :: 'a multiset) \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
```
```   238 by (rule less_le_not_le)
```
```   239
```
```   240 lemma mset_less_eq_Bag0:
```
```   241   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
```
```   242     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   243 proof
```
```   244   assume ?lhs thus ?rhs by (auto simp add: mset_le_def)
```
```   245 next
```
```   246   assume ?rhs
```
```   247   show ?lhs
```
```   248   proof (rule mset_less_eqI)
```
```   249     fix x
```
```   250     from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
```
```   251       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
```
```   252     thus "count (Bag xs) x \<le> count A x" by (simp add: mset_le_def)
```
```   253   qed
```
```   254 qed
```
```   255
```
```   256 lemma mset_less_eq_Bag [code]:
```
```   257   "Bag xs \<le> (A :: 'a multiset) \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). n \<le> count A x)"
```
```   258 proof -
```
```   259   {
```
```   260     fix x n
```
```   261     assume "(x,n) \<in> set (DAList.impl_of xs)"
```
```   262     hence "count_of (DAList.impl_of xs) x = n"
```
```   263     proof (transfer)
```
```   264       fix x n and xs :: "('a \<times> nat) list"
```
```   265       show "(distinct \<circ> map fst) xs \<Longrightarrow> (x, n) \<in> set xs \<Longrightarrow> count_of xs x = n"
```
```   266       proof (induct xs)
```
```   267         case (Cons ym ys)
```
```   268         obtain y m where ym: "ym = (y,m)" by force
```
```   269         note Cons = Cons[unfolded ym]
```
```   270         show ?case
```
```   271         proof (cases "x = y")
```
```   272           case False
```
```   273           with Cons show ?thesis unfolding ym by auto
```
```   274         next
```
```   275           case True
```
```   276           with Cons(2-3) have "m = n" by force
```
```   277           with True show ?thesis unfolding ym by auto
```
```   278         qed
```
```   279       qed auto
```
```   280     qed
```
```   281   }
```
```   282   thus ?thesis unfolding mset_less_eq_Bag0 by auto
```
```   283 qed
```
```   284
```
```   285 declare multiset_inter_def [code]
```
```   286 declare sup_multiset_def [code]
```
```   287 declare multiset_of.simps [code]
```
```   288
```
```   289
```
```   290 fun fold_impl :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<times> nat)list \<Rightarrow> 'b" where
```
```   291   "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)"
```
```   292 | "fold_impl fn e [] = e"
```
```   293
```
```   294 definition fold :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a, nat)alist \<Rightarrow> 'b" where
```
```   295 "fold f e al = fold_impl f e (DAList.impl_of al)"
```
```   296
```
```   297 hide_const (open) fold
```
```   298
```
```   299 context comp_fun_commute
```
```   300 begin
```
```   301
```
```   302 lemma DAList_Multiset_fold: assumes fn: "\<And> a n x. fn a n x = (f a ^^ n) x"
```
```   303   shows "Multiset.fold f e (Bag al) = DAList_Multiset.fold fn e al"
```
```   304 unfolding DAList_Multiset.fold_def
```
```   305 proof (induct al)
```
```   306   fix ys
```
```   307   let ?inv = "{xs :: ('a \<times> nat)list. (distinct \<circ> map fst) xs}"
```
```   308   note cs[simp del] = count_simps
```
```   309   have count[simp]: "\<And> x. count (Abs_multiset (count_of x)) = count_of x"
```
```   310     by (rule Abs_multiset_inverse[OF count_of_multiset])
```
```   311   assume ys: "ys \<in> ?inv"
```
```   312   thus "Multiset.fold f e (Bag (Alist ys)) = fold_impl fn e (DAList.impl_of (Alist ys))"
```
```   313     unfolding Bag_def unfolding Alist_inverse[OF ys]
```
```   314   proof (induct ys arbitrary: e rule: list.induct)
```
```   315     case Nil
```
```   316     show ?case
```
```   317       by (rule trans[OF arg_cong[of _ "{#}" "Multiset.fold f e", OF multiset_eqI]])
```
```   318          (auto, simp add: cs)
```
```   319   next
```
```   320     case (Cons pair ys e)
```
```   321     obtain a n where pair: "pair = (a,n)" by force
```
```   322     from fn[of a n] have [simp]: "fn a n = (f a ^^ n)" by auto
```
```   323     have inv: "ys \<in> ?inv" using Cons(2) by auto
```
```   324     note IH = Cons(1)[OF inv]
```
```   325     def Ys \<equiv> "Abs_multiset (count_of ys)"
```
```   326     have id: "Abs_multiset (count_of ((a, n) # ys)) = ((op + {# a #}) ^^ n) Ys"
```
```   327       unfolding Ys_def
```
```   328     proof (rule multiset_eqI, unfold count)
```
```   329       fix c
```
```   330       show "count_of ((a, n) # ys) c = count ((op + {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r")
```
```   331       proof (cases "c = a")
```
```   332         case False thus ?thesis unfolding cs by (induct n) auto
```
```   333       next
```
```   334         case True
```
```   335         hence "?l = n" by (simp add: cs)
```
```   336         also have "n = ?r" unfolding True
```
```   337         proof (induct n)
```
```   338           case 0
```
```   339           from Cons(2)[unfolded pair] have "a \<notin> fst ` set ys" by auto
```
```   340           thus ?case by (induct ys) (simp, auto simp: cs)
```
```   341         qed auto
```
```   342         finally show ?thesis .
```
```   343       qed
```
```   344     qed
```
```   345     show ?case unfolding pair
```
```   346       by (simp add: IH[symmetric], unfold id Ys_def[symmetric],
```
```   347       induct n, auto simp: fold_mset_fun_left_comm[symmetric])
```
```   348   qed
```
```   349 qed
```
```   350
```
```   351 end
```
```   352
```
```   353 lift_definition single_alist_entry :: "'a \<Rightarrow> 'b \<Rightarrow> ('a,'b)alist" is "\<lambda> a b. [(a,b)]" by auto
```
```   354
```
```   355 lemma image_mset_Bag[code]:
```
```   356   "image_mset f (Bag ms) =
```
```   357    DAList_Multiset.fold (\<lambda> a n m. Bag (single_alist_entry (f a) n) + m) {#} ms"
```
```   358 unfolding image_mset_def
```
```   359 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps))
```
```   360   fix a n m
```
```   361   show "Bag (single_alist_entry (f a) n) + m = ((op + \<circ> single \<circ> f) a ^^ n) m" (is "?l = ?r")
```
```   362   proof (rule multiset_eqI)
```
```   363     fix x
```
```   364     have "count ?r x = (if x = f a then n + count m x else count m x)"
```
```   365       by (induct n, auto)
```
```   366     also have "\<dots> = count ?l x" by (simp add: single_alist_entry.rep_eq)
```
```   367     finally show "count ?l x = count ?r x" ..
```
```   368   qed
```
```   369 qed
```
```   370
```
```   371 hide_const single_alist_entry
```
```   372
```
```   373 (* we cannot use (\<lambda> a n. op + (a * n)) for folding, since * is not defined
```
```   374    in comm_monoid_add *)
```
```   375 lemma msetsum_Bag[code]:
```
```   376   "msetsum (Bag ms) = DAList_Multiset.fold (\<lambda> a n. ((op + a) ^^ n)) 0 ms"
```
```   377 unfolding msetsum.eq_fold
```
```   378 by (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, auto simp: ac_simps)
```
```   379
```
```   380 (* we cannot use (\<lambda> a n. op * (a ^ n)) for folding, since ^ is not defined
```
```   381    in comm_monoid_mult *)
```
```   382 lemma msetprod_Bag[code]:
```
```   383   "msetprod (Bag ms) = DAList_Multiset.fold (\<lambda> a n. ((op * a) ^^ n)) 1 ms"
```
```   384 unfolding msetprod.eq_fold
```
```   385 by (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, auto simp: ac_simps)
```
```   386
```
```   387 lemma mcard_fold: "mcard A = Multiset.fold (\<lambda> _. Suc) 0 A" (is "_ = Multiset.fold ?f _ _")
```
```   388 proof -
```
```   389   interpret comp_fun_commute ?f by (default, auto)
```
```   390   show ?thesis by (induct A) auto
```
```   391 qed
```
```   392
```
```   393 lemma mcard_Bag[code]:
```
```   394   "mcard (Bag ms) = DAList_Multiset.fold (\<lambda> a n. op + n) 0 ms"
```
```   395 unfolding mcard_fold
```
```   396 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, simp)
```
```   397   fix a n x
```
```   398   show "n + x = (Suc ^^ n) x" by (induct n) auto
```
```   399 qed
```
```   400
```
```   401
```
```   402 lemma set_of_fold: "set_of A = Multiset.fold insert {} A" (is "_ = Multiset.fold ?f _ _")
```
```   403 proof -
```
```   404   interpret comp_fun_commute ?f by (default, auto)
```
```   405   show ?thesis by (induct A, auto)
```
```   406 qed
```
```   407
```
```   408 lemma set_of_Bag[code]:
```
```   409   "set_of (Bag ms) = DAList_Multiset.fold (\<lambda> a n. (if n = 0 then (\<lambda> m. m) else insert a)) {} ms"
```
```   410 unfolding set_of_fold
```
```   411 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps))
```
```   412   fix a n x
```
```   413   show "(if n = 0 then \<lambda>m. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n")
```
```   414   proof (cases n)
```
```   415     case (Suc m)
```
```   416     hence "?l n = insert a x" by simp
```
```   417     moreover have "?r n = insert a x" unfolding Suc by (induct m) auto
```
```   418     ultimately show ?thesis by auto
```
```   419   qed auto
```
```   420 qed
```
```   421
```
```   422
```
```   423 instantiation multiset :: (exhaustive) exhaustive
```
```   424 begin
```
```   425
```
```   426 definition exhaustive_multiset :: "('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool * term list) option"
```
```   427 where
```
```   428   "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i"
```
```   429
```
```   430 instance ..
```
```   431
```
```   432 end
```
```   433
```
```   434 end
```
```   435
```