src/HOL/Library/Multiset.thy
changeset 10249 e4d13d8a9011
child 10277 081c8641aa11
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Library/Multiset.thy	Wed Oct 18 23:28:33 2000 +0200
     1.3 @@ -0,0 +1,854 @@
     1.4 +(*  Title:      HOL/Library/Multiset.thy
     1.5 +    ID:         $Id$
     1.6 +    Author:     Tobias Nipkow, TU Muenchen
     1.7 +    Author:     Markus Wenzel, TU Muenchen
     1.8 +    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     1.9 +*)
    1.10 +
    1.11 +header {*
    1.12 + \title{Multisets}
    1.13 + \author{Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson}
    1.14 +*}
    1.15 +
    1.16 +theory Multiset = Accessible_Part:
    1.17 +
    1.18 +subsection {* The type of multisets *}
    1.19 +
    1.20 +typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
    1.21 +proof
    1.22 +  show "(\<lambda>x. 0::nat) \<in> {f. finite {x. 0 < f x}}"
    1.23 +    by simp
    1.24 +qed
    1.25 +
    1.26 +lemmas multiset_typedef [simp] =
    1.27 +  Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
    1.28 +
    1.29 +constdefs
    1.30 +  Mempty :: "'a multiset"    ("{#}")
    1.31 +  "{#} == Abs_multiset (\<lambda>a. 0)"
    1.32 +
    1.33 +  single :: "'a => 'a multiset"    ("{#_#}")
    1.34 +  "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
    1.35 +
    1.36 +  count :: "'a multiset => 'a => nat"
    1.37 +  "count == Rep_multiset"
    1.38 +
    1.39 +  MCollect :: "'a multiset => ('a => bool) => 'a multiset"
    1.40 +  "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
    1.41 +
    1.42 +syntax
    1.43 +  "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
    1.44 +  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
    1.45 +translations
    1.46 +  "a :# M" == "0 < count M a"
    1.47 +  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
    1.48 +
    1.49 +constdefs
    1.50 +  set_of :: "'a multiset => 'a set"
    1.51 +  "set_of M == {x. x :# M}"
    1.52 +
    1.53 +instance multiset :: ("term") plus by intro_classes
    1.54 +instance multiset :: ("term") minus by intro_classes
    1.55 +instance multiset :: ("term") zero by intro_classes
    1.56 +
    1.57 +defs (overloaded)
    1.58 +  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
    1.59 +  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
    1.60 +  Zero_def [simp]: "0 == {#}"
    1.61 +  size_def: "size M == setsum (count M) (set_of M)"
    1.62 +
    1.63 +
    1.64 +text {*
    1.65 + \medskip Preservation of the representing set @{term multiset}.
    1.66 +*}
    1.67 +
    1.68 +lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
    1.69 +  apply (simp add: multiset_def)
    1.70 +  done
    1.71 +
    1.72 +lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
    1.73 +  apply (simp add: multiset_def)
    1.74 +  done
    1.75 +
    1.76 +lemma union_preserves_multiset [simp]:
    1.77 +    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
    1.78 +  apply (unfold multiset_def)
    1.79 +  apply simp
    1.80 +  apply (drule finite_UnI)
    1.81 +   apply assumption
    1.82 +  apply (simp del: finite_Un add: Un_def)
    1.83 +  done
    1.84 +
    1.85 +lemma diff_preserves_multiset [simp]:
    1.86 +    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
    1.87 +  apply (unfold multiset_def)
    1.88 +  apply simp
    1.89 +  apply (rule finite_subset)
    1.90 +   prefer 2
    1.91 +   apply assumption
    1.92 +  apply auto
    1.93 +  done
    1.94 +
    1.95 +text {*
    1.96 + \medskip Injectivity of @{term Rep_multiset}.
    1.97 +*}  (* FIXME typedef package (!?) *)
    1.98 +
    1.99 +lemma multiset_eq_conv_Rep_eq [simp]:
   1.100 +    "(M = N) = (Rep_multiset M = Rep_multiset N)"
   1.101 +  apply (rule iffI)
   1.102 +   apply simp
   1.103 +  apply (drule_tac f = Abs_multiset in arg_cong)
   1.104 +  apply simp
   1.105 +  done
   1.106 +
   1.107 +(* FIXME
   1.108 +Goal
   1.109 + "[| f : multiset; g : multiset |] ==> \
   1.110 +\ (Abs_multiset f = Abs_multiset g) = (!x. f x = g x)";
   1.111 +by (rtac iffI 1);
   1.112 + by (dres_inst_tac [("f","Rep_multiset")] arg_cong 1);
   1.113 + by (Asm_full_simp_tac 1);
   1.114 +by (subgoal_tac "f = g" 1);
   1.115 + by (Asm_simp_tac 1);
   1.116 +by (rtac ext 1);
   1.117 +by (Blast_tac 1);
   1.118 +qed "Abs_multiset_eq";
   1.119 +Addsimps [Abs_multiset_eq];
   1.120 +*)
   1.121 +
   1.122 +
   1.123 +subsection {* Algebraic properties of multisets *}
   1.124 +
   1.125 +subsubsection {* Union *}
   1.126 +
   1.127 +theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
   1.128 +  apply (simp add: union_def Mempty_def)
   1.129 +  done
   1.130 +
   1.131 +theorem union_commute: "M + N = N + (M::'a multiset)"
   1.132 +  apply (simp add: union_def add_ac)
   1.133 +  done
   1.134 +
   1.135 +theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
   1.136 +  apply (simp add: union_def add_ac)
   1.137 +  done
   1.138 +
   1.139 +theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
   1.140 +  apply (rule union_commute [THEN trans])
   1.141 +  apply (rule union_assoc [THEN trans])
   1.142 +  apply (rule union_commute [THEN arg_cong])
   1.143 +  done
   1.144 +
   1.145 +theorems union_ac = union_assoc union_commute union_lcomm
   1.146 +
   1.147 +
   1.148 +subsubsection {* Difference *}
   1.149 +
   1.150 +theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
   1.151 +  apply (simp add: Mempty_def diff_def)
   1.152 +  done
   1.153 +
   1.154 +theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
   1.155 +  apply (simp add: union_def diff_def)
   1.156 +  done
   1.157 +
   1.158 +
   1.159 +subsubsection {* Count of elements *}
   1.160 +
   1.161 +theorem count_empty [simp]: "count {#} a = 0"
   1.162 +  apply (simp add: count_def Mempty_def)
   1.163 +  done
   1.164 +
   1.165 +theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
   1.166 +  apply (simp add: count_def single_def)
   1.167 +  done
   1.168 +
   1.169 +theorem count_union [simp]: "count (M + N) a = count M a + count N a"
   1.170 +  apply (simp add: count_def union_def)
   1.171 +  done
   1.172 +
   1.173 +theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
   1.174 +  apply (simp add: count_def diff_def)
   1.175 +  done
   1.176 +
   1.177 +
   1.178 +subsubsection {* Set of elements *}
   1.179 +
   1.180 +theorem set_of_empty [simp]: "set_of {#} = {}"
   1.181 +  apply (simp add: set_of_def)
   1.182 +  done
   1.183 +
   1.184 +theorem set_of_single [simp]: "set_of {#b#} = {b}"
   1.185 +  apply (simp add: set_of_def)
   1.186 +  done
   1.187 +
   1.188 +theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
   1.189 +  apply (auto simp add: set_of_def)
   1.190 +  done
   1.191 +
   1.192 +theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
   1.193 +  apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
   1.194 +  done
   1.195 +
   1.196 +theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
   1.197 +  apply (auto simp add: set_of_def)
   1.198 +  done
   1.199 +
   1.200 +
   1.201 +subsubsection {* Size *}
   1.202 +
   1.203 +theorem size_empty [simp]: "size {#} = 0"
   1.204 +  apply (simp add: size_def)
   1.205 +  done
   1.206 +
   1.207 +theorem size_single [simp]: "size {#b#} = 1"
   1.208 +  apply (simp add: size_def)
   1.209 +  done
   1.210 +
   1.211 +theorem finite_set_of [iff]: "finite (set_of M)"
   1.212 +  apply (cut_tac x = M in Rep_multiset)
   1.213 +  apply (simp add: multiset_def set_of_def count_def)
   1.214 +  done
   1.215 +
   1.216 +theorem setsum_count_Int:
   1.217 +    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
   1.218 +  apply (erule finite_induct)
   1.219 +   apply simp
   1.220 +  apply (simp add: Int_insert_left set_of_def)
   1.221 +  done
   1.222 +
   1.223 +theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
   1.224 +  apply (unfold size_def)
   1.225 +  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
   1.226 +   prefer 2
   1.227 +   apply (rule ext)
   1.228 +   apply simp
   1.229 +  apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
   1.230 +  apply (subst Int_commute)
   1.231 +  apply (simp (no_asm_simp) add: setsum_count_Int)
   1.232 +  done
   1.233 +
   1.234 +theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
   1.235 +  apply (unfold size_def Mempty_def count_def)
   1.236 +  apply auto
   1.237 +  apply (simp add: set_of_def count_def expand_fun_eq)
   1.238 +  done
   1.239 +
   1.240 +theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
   1.241 +  apply (unfold size_def)
   1.242 +  apply (drule setsum_SucD)
   1.243 +  apply auto
   1.244 +  done
   1.245 +
   1.246 +
   1.247 +subsubsection {* Equality of multisets *}
   1.248 +
   1.249 +theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
   1.250 +  apply (simp add: count_def expand_fun_eq)
   1.251 +  done
   1.252 +
   1.253 +theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
   1.254 +  apply (simp add: single_def Mempty_def expand_fun_eq)
   1.255 +  done
   1.256 +
   1.257 +theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
   1.258 +  apply (auto simp add: single_def expand_fun_eq)
   1.259 +  done
   1.260 +
   1.261 +theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
   1.262 +  apply (auto simp add: union_def Mempty_def expand_fun_eq)
   1.263 +  done
   1.264 +
   1.265 +theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
   1.266 +  apply (auto simp add: union_def Mempty_def expand_fun_eq)
   1.267 +  done
   1.268 +
   1.269 +theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
   1.270 +  apply (simp add: union_def expand_fun_eq)
   1.271 +  done
   1.272 +
   1.273 +theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
   1.274 +  apply (simp add: union_def expand_fun_eq)
   1.275 +  done
   1.276 +
   1.277 +theorem union_is_single:
   1.278 +    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
   1.279 +  apply (unfold Mempty_def single_def union_def)
   1.280 +  apply (simp add: add_is_1 expand_fun_eq)
   1.281 +  apply blast
   1.282 +  done
   1.283 +
   1.284 +theorem single_is_union:
   1.285 +  "({#a#} = M + N) =
   1.286 +    ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
   1.287 +  apply (unfold Mempty_def single_def union_def)
   1.288 +  apply (simp add: one_is_add expand_fun_eq)
   1.289 +  apply (blast dest: sym)
   1.290 +  done
   1.291 +
   1.292 +theorem add_eq_conv_diff:
   1.293 +  "(M + {#a#} = N + {#b#}) =
   1.294 +    (M = N \<and> a = b \<or>
   1.295 +      M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
   1.296 +  apply (unfold single_def union_def diff_def)
   1.297 +  apply (simp (no_asm) add: expand_fun_eq)
   1.298 +  apply (rule conjI)
   1.299 +   apply force
   1.300 +  apply clarify
   1.301 +  apply (rule conjI)
   1.302 +   apply blast
   1.303 +  apply clarify
   1.304 +  apply (rule iffI)
   1.305 +   apply (rule conjI)
   1.306 +    apply clarify
   1.307 +    apply (rule conjI)
   1.308 +     apply (simp add: eq_sym_conv)   (* FIXME blast fails !? *)
   1.309 +    apply fast
   1.310 +   apply simp
   1.311 +  apply force
   1.312 +  done
   1.313 +
   1.314 +(*
   1.315 +val prems = Goal
   1.316 + "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
   1.317 +by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
   1.318 +     measure_induct 1);
   1.319 +by (Clarify_tac 1);
   1.320 +by (resolve_tac prems 1);
   1.321 + by (assume_tac 1);
   1.322 +by (Clarify_tac 1);
   1.323 +by (subgoal_tac "finite G" 1);
   1.324 + by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
   1.325 +by (etac allE 1);
   1.326 +by (etac impE 1);
   1.327 + by (Blast_tac 2);
   1.328 +by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
   1.329 +no_qed();
   1.330 +val lemma = result();
   1.331 +
   1.332 +val prems = Goal
   1.333 + "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
   1.334 +by (rtac (lemma RS mp) 1);
   1.335 +by (REPEAT(ares_tac prems 1));
   1.336 +qed "finite_psubset_induct";
   1.337 +
   1.338 +Better: use wf_finite_psubset in WF_Rel
   1.339 +*)
   1.340 +
   1.341 +
   1.342 +subsection {* Induction over multisets *}
   1.343 +
   1.344 +lemma setsum_decr:
   1.345 +  "finite F ==> 0 < f a ==>
   1.346 +    setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
   1.347 +  apply (erule finite_induct)
   1.348 +   apply auto
   1.349 +  apply (drule_tac a = a in mk_disjoint_insert)
   1.350 +  apply auto
   1.351 +  done
   1.352 +
   1.353 +lemma Rep_multiset_induct_aux:
   1.354 +  "P (\<lambda>a. 0) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
   1.355 +    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
   1.356 +proof -
   1.357 +  case antecedent
   1.358 +  note prems = this [unfolded multiset_def]
   1.359 +  show ?thesis
   1.360 +    apply (unfold multiset_def)
   1.361 +    apply (induct_tac n)
   1.362 +     apply simp
   1.363 +     apply clarify
   1.364 +     apply (subgoal_tac "f = (\<lambda>a.0)")
   1.365 +      apply simp
   1.366 +      apply (rule prems)
   1.367 +     apply (rule ext)
   1.368 +     apply force
   1.369 +    apply clarify
   1.370 +    apply (frule setsum_SucD)
   1.371 +    apply clarify
   1.372 +    apply (rename_tac a)
   1.373 +    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
   1.374 +     prefer 2
   1.375 +     apply (rule finite_subset)
   1.376 +      prefer 2
   1.377 +      apply assumption
   1.378 +     apply simp
   1.379 +     apply blast
   1.380 +    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
   1.381 +     prefer 2
   1.382 +     apply (rule ext)
   1.383 +     apply (simp (no_asm_simp))
   1.384 +     apply (erule ssubst, rule prems)
   1.385 +     apply blast
   1.386 +    apply (erule allE, erule impE, erule_tac [2] mp)
   1.387 +     apply blast
   1.388 +    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply)
   1.389 +    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
   1.390 +     prefer 2
   1.391 +     apply blast
   1.392 +    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
   1.393 +     prefer 2
   1.394 +     apply blast
   1.395 +    apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
   1.396 +    done
   1.397 +qed
   1.398 +
   1.399 +theorem Rep_multiset_induct:
   1.400 +  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
   1.401 +    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
   1.402 +  apply (insert Rep_multiset_induct_aux)
   1.403 +  apply blast
   1.404 +  done
   1.405 +
   1.406 +theorem multiset_induct [induct type: multiset]:
   1.407 +  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
   1.408 +proof -
   1.409 +  note defns = union_def single_def Mempty_def
   1.410 +  assume prem1 [unfolded defns]: "P {#}"
   1.411 +  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
   1.412 +  show ?thesis
   1.413 +    apply (rule Rep_multiset_inverse [THEN subst])
   1.414 +    apply (rule Rep_multiset [THEN Rep_multiset_induct])
   1.415 +     apply (rule prem1)
   1.416 +    apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
   1.417 +     prefer 2
   1.418 +     apply (simp add: expand_fun_eq)
   1.419 +    apply (erule ssubst)
   1.420 +    apply (erule Abs_multiset_inverse [THEN subst])
   1.421 +    apply (erule prem2 [simplified])
   1.422 +    done
   1.423 +qed
   1.424 +
   1.425 +
   1.426 +lemma MCollect_preserves_multiset:
   1.427 +    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
   1.428 +  apply (simp add: multiset_def)
   1.429 +  apply (rule finite_subset)
   1.430 +   apply auto
   1.431 +  done
   1.432 +
   1.433 +theorem count_MCollect [simp]:
   1.434 +    "count {# x:M. P x #} a = (if P a then count M a else 0)"
   1.435 +  apply (unfold count_def MCollect_def)
   1.436 +  apply (simp add: MCollect_preserves_multiset)
   1.437 +  done
   1.438 +
   1.439 +theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
   1.440 +  apply (auto simp add: set_of_def)
   1.441 +  done
   1.442 +
   1.443 +theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
   1.444 +  apply (subst multiset_eq_conv_count_eq)
   1.445 +  apply auto
   1.446 +  done
   1.447 +
   1.448 +declare multiset_eq_conv_Rep_eq [simp del]
   1.449 +declare multiset_typedef [simp del]
   1.450 +
   1.451 +theorem add_eq_conv_ex:
   1.452 +  "(M + {#a#} = N + {#b#}) =
   1.453 +    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
   1.454 +  apply (auto simp add: add_eq_conv_diff)
   1.455 +  done
   1.456 +
   1.457 +
   1.458 +subsection {* Multiset orderings *}
   1.459 +
   1.460 +subsubsection {* Well-foundedness *}
   1.461 +
   1.462 +constdefs
   1.463 +  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
   1.464 +  "mult1 r ==
   1.465 +    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
   1.466 +      (\<forall>b. b :# K --> (b, a) \<in> r)}"
   1.467 +
   1.468 +  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
   1.469 +  "mult r == (mult1 r)^+"
   1.470 +
   1.471 +lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
   1.472 +  apply (simp add: mult1_def)
   1.473 +  done
   1.474 +
   1.475 +lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
   1.476 +    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
   1.477 +    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
   1.478 +  (concl is "?case1 (mult1 r) \<or> ?case2")
   1.479 +proof (unfold mult1_def)
   1.480 +  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
   1.481 +  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
   1.482 +  let ?case1 = "?case1 {(N, M). ?R N M}"
   1.483 +
   1.484 +  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
   1.485 +  hence "\<exists>a' M0' K.
   1.486 +      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
   1.487 +  thus "?case1 \<or> ?case2"
   1.488 +  proof (elim exE conjE)
   1.489 +    fix a' M0' K
   1.490 +    assume N: "N = M0' + K" and r: "?r K a'"
   1.491 +    assume "M0 + {#a#} = M0' + {#a'#}"
   1.492 +    hence "M0 = M0' \<and> a = a' \<or>
   1.493 +        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
   1.494 +      by (simp only: add_eq_conv_ex)
   1.495 +    thus ?thesis
   1.496 +    proof (elim disjE conjE exE)
   1.497 +      assume "M0 = M0'" "a = a'"
   1.498 +      with N r have "?r K a \<and> N = M0 + K" by simp
   1.499 +      hence ?case2 .. thus ?thesis ..
   1.500 +    next
   1.501 +      fix K'
   1.502 +      assume "M0' = K' + {#a#}"
   1.503 +      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
   1.504 +
   1.505 +      assume "M0 = K' + {#a'#}"
   1.506 +      with r have "?R (K' + K) M0" by blast
   1.507 +      with n have ?case1 by simp thus ?thesis ..
   1.508 +    qed
   1.509 +  qed
   1.510 +qed
   1.511 +
   1.512 +lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
   1.513 +proof
   1.514 +  let ?R = "mult1 r"
   1.515 +  let ?W = "acc ?R"
   1.516 +  {
   1.517 +    fix M M0 a
   1.518 +    assume M0: "M0 \<in> ?W"
   1.519 +      and wf_hyp: "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
   1.520 +      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
   1.521 +    have "M0 + {#a#} \<in> ?W"
   1.522 +    proof (rule accI [of "M0 + {#a#}"])
   1.523 +      fix N
   1.524 +      assume "(N, M0 + {#a#}) \<in> ?R"
   1.525 +      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
   1.526 +          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
   1.527 +        by (rule less_add)
   1.528 +      thus "N \<in> ?W"
   1.529 +      proof (elim exE disjE conjE)
   1.530 +        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
   1.531 +        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
   1.532 +        hence "M + {#a#} \<in> ?W" ..
   1.533 +        thus "N \<in> ?W" by (simp only: N)
   1.534 +      next
   1.535 +        fix K
   1.536 +        assume N: "N = M0 + K"
   1.537 +        assume "\<forall>b. b :# K --> (b, a) \<in> r"
   1.538 +        have "?this --> M0 + K \<in> ?W" (is "?P K")
   1.539 +        proof (induct K)
   1.540 +          from M0 have "M0 + {#} \<in> ?W" by simp
   1.541 +          thus "?P {#}" ..
   1.542 +
   1.543 +          fix K x assume hyp: "?P K"
   1.544 +          show "?P (K + {#x#})"
   1.545 +          proof
   1.546 +            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
   1.547 +            hence "(x, a) \<in> r" by simp
   1.548 +            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
   1.549 +
   1.550 +            from a hyp have "M0 + K \<in> ?W" by simp
   1.551 +            with b have "(M0 + K) + {#x#} \<in> ?W" ..
   1.552 +            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
   1.553 +          qed
   1.554 +        qed
   1.555 +        hence "M0 + K \<in> ?W" ..
   1.556 +        thus "N \<in> ?W" by (simp only: N)
   1.557 +      qed
   1.558 +    qed
   1.559 +  } note tedious_reasoning = this
   1.560 +
   1.561 +  assume wf: "wf r"
   1.562 +  fix M
   1.563 +  show "M \<in> ?W"
   1.564 +  proof (induct M)
   1.565 +    show "{#} \<in> ?W"
   1.566 +    proof (rule accI)
   1.567 +      fix b assume "(b, {#}) \<in> ?R"
   1.568 +      with not_less_empty show "b \<in> ?W" by contradiction
   1.569 +    qed
   1.570 +
   1.571 +    fix M a assume "M \<in> ?W"
   1.572 +    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
   1.573 +    proof induct
   1.574 +      fix a
   1.575 +      assume "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
   1.576 +      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
   1.577 +      proof
   1.578 +        fix M assume "M \<in> ?W"
   1.579 +        thus "M + {#a#} \<in> ?W"
   1.580 +          by (rule acc_induct) (rule tedious_reasoning)
   1.581 +      qed
   1.582 +    qed
   1.583 +    thus "M + {#a#} \<in> ?W" ..
   1.584 +  qed
   1.585 +qed
   1.586 +
   1.587 +theorem wf_mult1: "wf r ==> wf (mult1 r)"
   1.588 +  by (rule acc_wfI, rule all_accessible)
   1.589 +
   1.590 +theorem wf_mult: "wf r ==> wf (mult r)"
   1.591 +  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
   1.592 +
   1.593 +
   1.594 +subsubsection {* Closure-free presentation *}
   1.595 +
   1.596 +(*Badly needed: a linear arithmetic procedure for multisets*)
   1.597 +
   1.598 +lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
   1.599 +  apply (simp add: multiset_eq_conv_count_eq)
   1.600 +  done
   1.601 +
   1.602 +text {* One direction. *}
   1.603 +
   1.604 +lemma mult_implies_one_step:
   1.605 +  "trans r ==> (M, N) \<in> mult r ==>
   1.606 +    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
   1.607 +    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
   1.608 +  apply (unfold mult_def mult1_def set_of_def)
   1.609 +  apply (erule converse_trancl_induct)
   1.610 +  apply clarify
   1.611 +   apply (rule_tac x = M0 in exI)
   1.612 +   apply simp
   1.613 +  apply clarify
   1.614 +  apply (case_tac "a :# K")
   1.615 +   apply (rule_tac x = I in exI)
   1.616 +   apply (simp (no_asm))
   1.617 +   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
   1.618 +   apply (simp (no_asm_simp) add: union_assoc [symmetric])
   1.619 +   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
   1.620 +   apply (simp add: diff_union_single_conv)
   1.621 +   apply (simp (no_asm_use) add: trans_def)
   1.622 +   apply blast
   1.623 +  apply (subgoal_tac "a :# I")
   1.624 +   apply (rule_tac x = "I - {#a#}" in exI)
   1.625 +   apply (rule_tac x = "J + {#a#}" in exI)
   1.626 +   apply (rule_tac x = "K + Ka" in exI)
   1.627 +   apply (rule conjI)
   1.628 +    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
   1.629 +   apply (rule conjI)
   1.630 +    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
   1.631 +    apply simp
   1.632 +    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
   1.633 +   apply (simp (no_asm_use) add: trans_def)
   1.634 +   apply blast
   1.635 +  apply (subgoal_tac "a :# (M0 +{#a#})")
   1.636 +   apply simp
   1.637 +  apply (simp (no_asm))
   1.638 +  done
   1.639 +
   1.640 +lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
   1.641 +  apply (simp add: multiset_eq_conv_count_eq)
   1.642 +  done
   1.643 +
   1.644 +lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
   1.645 +  apply (erule size_eq_Suc_imp_elem [THEN exE])
   1.646 +  apply (drule elem_imp_eq_diff_union)
   1.647 +  apply auto
   1.648 +  done
   1.649 +
   1.650 +lemma one_step_implies_mult_aux:
   1.651 +  "trans r ==>
   1.652 +    \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
   1.653 +      --> (I + K, I + J) \<in> mult r"
   1.654 +  apply (induct_tac n)
   1.655 +   apply auto
   1.656 +  apply (frule size_eq_Suc_imp_eq_union)
   1.657 +  apply clarify
   1.658 +  apply (rename_tac "J'")
   1.659 +  apply simp
   1.660 +  apply (erule notE)
   1.661 +   apply auto
   1.662 +  apply (case_tac "J' = {#}")
   1.663 +   apply (simp add: mult_def)
   1.664 +   apply (rule r_into_trancl)
   1.665 +   apply (simp add: mult1_def set_of_def)
   1.666 +   apply blast
   1.667 +  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
   1.668 +  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
   1.669 +  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
   1.670 +  apply (erule ssubst)
   1.671 +  apply (simp add: Ball_def)
   1.672 +  apply auto
   1.673 +  apply (subgoal_tac
   1.674 +    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
   1.675 +      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
   1.676 +   prefer 2
   1.677 +   apply force
   1.678 +  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
   1.679 +  apply (erule trancl_trans)
   1.680 +  apply (rule r_into_trancl)
   1.681 +  apply (simp add: mult1_def set_of_def)
   1.682 +  apply (rule_tac x = a in exI)
   1.683 +  apply (rule_tac x = "I + J'" in exI)
   1.684 +  apply (simp add: union_ac)
   1.685 +  done
   1.686 +
   1.687 +theorem one_step_implies_mult:
   1.688 +  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
   1.689 +    ==> (I + K, I + J) \<in> mult r"
   1.690 +  apply (insert one_step_implies_mult_aux)
   1.691 +  apply blast
   1.692 +  done
   1.693 +
   1.694 +
   1.695 +subsubsection {* Partial-order properties *}
   1.696 +
   1.697 +instance multiset :: ("term") ord by intro_classes
   1.698 +
   1.699 +defs (overloaded)
   1.700 +  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
   1.701 +  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
   1.702 +
   1.703 +lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
   1.704 +  apply (unfold trans_def)
   1.705 +  apply (blast intro: order_less_trans)
   1.706 +  done
   1.707 +
   1.708 +text {*
   1.709 + \medskip Irreflexivity.
   1.710 +*}
   1.711 +
   1.712 +lemma mult_irrefl_aux:
   1.713 +    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
   1.714 +  apply (erule finite_induct)
   1.715 +   apply (auto intro: order_less_trans)
   1.716 +  done
   1.717 +
   1.718 +theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
   1.719 +  apply (unfold less_multiset_def)
   1.720 +  apply auto
   1.721 +  apply (drule trans_base_order [THEN mult_implies_one_step])
   1.722 +  apply auto
   1.723 +  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
   1.724 +  apply (simp add: set_of_eq_empty_iff)
   1.725 +  done
   1.726 +
   1.727 +lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
   1.728 +  apply (insert mult_less_not_refl)
   1.729 +  apply blast
   1.730 +  done
   1.731 +
   1.732 +
   1.733 +text {* Transitivity. *}
   1.734 +
   1.735 +theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
   1.736 +  apply (unfold less_multiset_def mult_def)
   1.737 +  apply (blast intro: trancl_trans)
   1.738 +  done
   1.739 +
   1.740 +text {* Asymmetry. *}
   1.741 +
   1.742 +theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
   1.743 +  apply auto
   1.744 +  apply (rule mult_less_not_refl [THEN notE])
   1.745 +  apply (erule mult_less_trans)
   1.746 +  apply assumption
   1.747 +  done
   1.748 +
   1.749 +theorem mult_less_asym:
   1.750 +    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
   1.751 +  apply (insert mult_less_not_sym)
   1.752 +  apply blast
   1.753 +  done
   1.754 +
   1.755 +theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
   1.756 +  apply (unfold le_multiset_def)
   1.757 +  apply auto
   1.758 +  done
   1.759 +
   1.760 +text {* Anti-symmetry. *}
   1.761 +
   1.762 +theorem mult_le_antisym:
   1.763 +    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
   1.764 +  apply (unfold le_multiset_def)
   1.765 +  apply (blast dest: mult_less_not_sym)
   1.766 +  done
   1.767 +
   1.768 +text {* Transitivity. *}
   1.769 +
   1.770 +theorem mult_le_trans:
   1.771 +    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
   1.772 +  apply (unfold le_multiset_def)
   1.773 +  apply (blast intro: mult_less_trans)
   1.774 +  done
   1.775 +
   1.776 +theorem mult_less_le: "M < N = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
   1.777 +  apply (unfold le_multiset_def)
   1.778 +  apply auto
   1.779 +  done
   1.780 +
   1.781 +
   1.782 +subsubsection {* Monotonicity of multiset union *}
   1.783 +
   1.784 +theorem mult1_union:
   1.785 +    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
   1.786 +  apply (unfold mult1_def)
   1.787 +  apply auto
   1.788 +  apply (rule_tac x = a in exI)
   1.789 +  apply (rule_tac x = "C + M0" in exI)
   1.790 +  apply (simp add: union_assoc)
   1.791 +  done
   1.792 +
   1.793 +lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
   1.794 +  apply (unfold less_multiset_def mult_def)
   1.795 +  apply (erule trancl_induct)
   1.796 +   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
   1.797 +  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
   1.798 +  done
   1.799 +
   1.800 +lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
   1.801 +  apply (subst union_commute [of B C])
   1.802 +  apply (subst union_commute [of D C])
   1.803 +  apply (erule union_less_mono2)
   1.804 +  done
   1.805 +
   1.806 +theorem union_less_mono:
   1.807 +    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
   1.808 +  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
   1.809 +  done
   1.810 +
   1.811 +theorem union_le_mono:
   1.812 +    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
   1.813 +  apply (unfold le_multiset_def)
   1.814 +  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
   1.815 +  done
   1.816 +
   1.817 +theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
   1.818 +  apply (unfold le_multiset_def less_multiset_def)
   1.819 +  apply (case_tac "M = {#}")
   1.820 +   prefer 2
   1.821 +   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
   1.822 +    prefer 2
   1.823 +    apply (rule one_step_implies_mult)
   1.824 +      apply (simp only: trans_def)
   1.825 +      apply auto
   1.826 +  apply (blast intro: order_less_trans)
   1.827 +  done
   1.828 +
   1.829 +theorem union_upper1: "A <= A + (B::'a::order multiset)"
   1.830 +  apply (subgoal_tac "A + {#} <= A + B")
   1.831 +   prefer 2
   1.832 +   apply (rule union_le_mono)
   1.833 +    apply auto
   1.834 +  done
   1.835 +
   1.836 +theorem union_upper2: "B <= A + (B::'a::order multiset)"
   1.837 +  apply (subst union_commute, rule union_upper1)
   1.838 +  done
   1.839 +
   1.840 +instance multiset :: (order) order
   1.841 +  apply intro_classes
   1.842 +     apply (rule mult_le_refl)
   1.843 +    apply (erule mult_le_trans)
   1.844 +    apply assumption
   1.845 +   apply (erule mult_le_antisym)
   1.846 +   apply assumption
   1.847 +  apply (rule mult_less_le)
   1.848 +  done
   1.849 +
   1.850 +instance multiset :: ("term") plus_ac0
   1.851 +  apply intro_classes
   1.852 +    apply (rule union_commute)
   1.853 +   apply (rule union_assoc)
   1.854 +  apply simp
   1.855 +  done
   1.856 +
   1.857 +end