wneuper/isa: src/HOL/Library/Multiset.thy@83f1a514dcb4

     1 (*  Title:      HOL/Library/Multiset.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson

     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)

     5 *)

     7 header {* Multisets *}

     9 theory Multiset = Accessible_Part:

    11 subsection {* The type of multisets *}

    13 typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"

    14 proof

    15   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp

    16 qed

    18 lemmas multiset_typedef [simp] =

    19     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset

    20   and [simp] = Rep_multiset_inject [symmetric]

    22 constdefs

    23   Mempty :: "'a multiset"    ("{#}")

    24   "{#} == Abs_multiset (\<lambda>a. 0)"

    26   single :: "'a => 'a multiset"    ("{#_#}")

    27   "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"

    29   count :: "'a multiset => 'a => nat"

    30   "count == Rep_multiset"

    32   MCollect :: "'a multiset => ('a => bool) => 'a multiset"

    33   "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"

    35 syntax

    36   "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)

    37   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")

    38 translations

    39   "a :# M" == "0 < count M a"

    40   "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"

    42 constdefs

    43   set_of :: "'a multiset => 'a set"

    44   "set_of M == {x. x :# M}"

    46 instance multiset :: (type) "{plus, minus, zero}" ..

    48 defs (overloaded)

    49   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"

    50   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"

    51   Zero_multiset_def [simp]: "0 == {#}"

    52   size_def: "size M == setsum (count M) (set_of M)"

    55 text {*

    56  \medskip Preservation of the representing set @{term multiset}.

    57 *}

    59 lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"

    60   apply (simp add: multiset_def)

    61   done

    63 lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"

    64   apply (simp add: multiset_def)

    65   done

    67 lemma union_preserves_multiset [simp]:

    68     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"

    69   apply (unfold multiset_def)

    70   apply simp

    71   apply (drule finite_UnI)

    72    apply assumption

    73   apply (simp del: finite_Un add: Un_def)

    74   done

    76 lemma diff_preserves_multiset [simp]:

    77     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"

    78   apply (unfold multiset_def)

    79   apply simp

    80   apply (rule finite_subset)

    81    prefer 2

    82    apply assumption

    83   apply auto

    84   done

    87 subsection {* Algebraic properties of multisets *}

    89 subsubsection {* Union *}

    91 theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"

    92   apply (simp add: union_def Mempty_def)

    93   done

    95 theorem union_commute: "M + N = N + (M::'a multiset)"

    96   apply (simp add: union_def add_ac)

    97   done

    99 theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"

   100   apply (simp add: union_def add_ac)

   101   done

   103 theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"

   104   apply (rule union_commute [THEN trans])

   105   apply (rule union_assoc [THEN trans])

   106   apply (rule union_commute [THEN arg_cong])

   107   done

   109 theorems union_ac = union_assoc union_commute union_lcomm

   111 instance multiset :: (type) comm_monoid_add

   112 proof

   113   fix a b c :: "'a multiset"

   114   show "(a + b) + c = a + (b + c)" by (rule union_assoc)

   115   show "a + b = b + a" by (rule union_commute)

   116   show "0 + a = a" by simp

   117 qed

   120 subsubsection {* Difference *}

   122 theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"

   123   apply (simp add: Mempty_def diff_def)

   124   done

   126 theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"

   127   apply (simp add: union_def diff_def)

   128   done

   131 subsubsection {* Count of elements *}

   133 theorem count_empty [simp]: "count {#} a = 0"

   134   apply (simp add: count_def Mempty_def)

   135   done

   137 theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"

   138   apply (simp add: count_def single_def)

   139   done

   141 theorem count_union [simp]: "count (M + N) a = count M a + count N a"

   142   apply (simp add: count_def union_def)

   143   done

   145 theorem count_diff [simp]: "count (M - N) a = count M a - count N a"

   146   apply (simp add: count_def diff_def)

   147   done

   150 subsubsection {* Set of elements *}

   152 theorem set_of_empty [simp]: "set_of {#} = {}"

   153   apply (simp add: set_of_def)

   154   done

   156 theorem set_of_single [simp]: "set_of {#b#} = {b}"

   157   apply (simp add: set_of_def)

   158   done

   160 theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"

   161   apply (auto simp add: set_of_def)

   162   done

   164 theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"

   165   apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)

   166   done

   168 theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"

   169   apply (auto simp add: set_of_def)

   170   done

   173 subsubsection {* Size *}

   175 theorem size_empty [simp]: "size {#} = 0"

   176   apply (simp add: size_def)

   177   done

   179 theorem size_single [simp]: "size {#b#} = 1"

   180   apply (simp add: size_def)

   181   done

   183 theorem finite_set_of [iff]: "finite (set_of M)"

   184   apply (cut_tac x = M in Rep_multiset)

   185   apply (simp add: multiset_def set_of_def count_def)

   186   done

   188 theorem setsum_count_Int:

   189     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"

   190   apply (erule finite_induct)

   191    apply simp

   192   apply (simp add: Int_insert_left set_of_def)

   193   done

   195 theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"

   196   apply (unfold size_def)

   197   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")

   198    prefer 2

   199    apply (rule ext)

   200    apply simp

   201   apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)

   202   apply (subst Int_commute)

   203   apply (simp (no_asm_simp) add: setsum_count_Int)

   204   done

   206 theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"

   207   apply (unfold size_def Mempty_def count_def)

   208   apply auto

   209   apply (simp add: set_of_def count_def expand_fun_eq)

   210   done

   212 theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"

   213   apply (unfold size_def)

   214   apply (drule setsum_SucD)

   215   apply auto

   216   done

   219 subsubsection {* Equality of multisets *}

   221 theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"

   222   apply (simp add: count_def expand_fun_eq)

   223   done

   225 theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"

   226   apply (simp add: single_def Mempty_def expand_fun_eq)

   227   done

   229 theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"

   230   apply (auto simp add: single_def expand_fun_eq)

   231   done

   233 theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"

   234   apply (auto simp add: union_def Mempty_def expand_fun_eq)

   235   done

   237 theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"

   238   apply (auto simp add: union_def Mempty_def expand_fun_eq)

   239   done

   241 theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"

   242   apply (simp add: union_def expand_fun_eq)

   243   done

   245 theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"

   246   apply (simp add: union_def expand_fun_eq)

   247   done

   249 theorem union_is_single:

   250     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"

   251   apply (unfold Mempty_def single_def union_def)

   252   apply (simp add: add_is_1 expand_fun_eq)

   253   apply blast

   254   done

   256 theorem single_is_union:

   257   "({#a#} = M + N) =

   258     ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"

   259   apply (unfold Mempty_def single_def union_def)

   260   apply (simp add: add_is_1 one_is_add expand_fun_eq)

   261   apply (blast dest: sym)

   262   done

   264 theorem add_eq_conv_diff:

   265   "(M + {#a#} = N + {#b#}) =

   266     (M = N \<and> a = b \<or>

   267       M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"

   268   apply (unfold single_def union_def diff_def)

   269   apply (simp (no_asm) add: expand_fun_eq)

   270   apply (rule conjI)

   271    apply force

   272   apply safe

   273   apply simp_all

   274   apply (simp add: eq_sym_conv)

   275   done

   277 (*

   278 val prems = Goal

   279  "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";

   280 by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]

   281      measure_induct 1);

   282 by (Clarify_tac 1);

   283 by (resolve_tac prems 1);

   284  by (assume_tac 1);

   285 by (Clarify_tac 1);

   286 by (subgoal_tac "finite G" 1);

   287  by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);

   288 by (etac allE 1);

   289 by (etac impE 1);

   290  by (Blast_tac 2);

   291 by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);

   292 no_qed();

   293 val lemma = result();

   295 val prems = Goal

   296  "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";

   297 by (rtac (lemma RS mp) 1);

   298 by (REPEAT(ares_tac prems 1));

   299 qed "finite_psubset_induct";

   301 Better: use wf_finite_psubset in WF_Rel

   302 *)

   305 subsection {* Induction over multisets *}

   307 lemma setsum_decr:

   308   "finite F ==> (0::nat) < f a ==>

   309     setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"

   310   apply (erule finite_induct)

   311    apply auto

   312   apply (drule_tac a = a in mk_disjoint_insert)

   313   apply auto

   314   done

   316 lemma rep_multiset_induct_aux:

   317   "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))

   318     ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"

   319 proof -

   320   case rule_context

   321   note premises = this [unfolded multiset_def]

   322   show ?thesis

   323     apply (unfold multiset_def)

   324     apply (induct_tac n)

   325      apply simp

   326      apply clarify

   327      apply (subgoal_tac "f = (\<lambda>a.0)")

   328       apply simp

   329       apply (rule premises)

   330      apply (rule ext)

   331      apply force

   332     apply clarify

   333     apply (frule setsum_SucD)

   334     apply clarify

   335     apply (rename_tac a)

   336     apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")

   337      prefer 2

   338      apply (rule finite_subset)

   339       prefer 2

   340       apply assumption

   341      apply simp

   342      apply blast

   343     apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")

   344      prefer 2

   345      apply (rule ext)

   346      apply (simp (no_asm_simp))

   347      apply (erule ssubst, rule premises)

   348      apply blast

   349     apply (erule allE, erule impE, erule_tac [2] mp)

   350      apply blast

   351     apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)

   352     apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")

   353      prefer 2

   354      apply blast

   355     apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")

   356      prefer 2

   357      apply blast

   358     apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)

   359     done

   360 qed

   362 theorem rep_multiset_induct:

   363   "f \<in> multiset ==> P (\<lambda>a. 0) ==>

   364     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"

   365   apply (insert rep_multiset_induct_aux)

   366   apply blast

   367   done

   369 theorem multiset_induct [induct type: multiset]:

   370   "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"

   371 proof -

   372   note defns = union_def single_def Mempty_def

   373   assume prem1 [unfolded defns]: "P {#}"

   374   assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"

   375   show ?thesis

   376     apply (rule Rep_multiset_inverse [THEN subst])

   377     apply (rule Rep_multiset [THEN rep_multiset_induct])

   378      apply (rule prem1)

   379     apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")

   380      prefer 2

   381      apply (simp add: expand_fun_eq)

   382     apply (erule ssubst)

   383     apply (erule Abs_multiset_inverse [THEN subst])

   384     apply (erule prem2 [simplified])

   385     done

   386 qed

   389 lemma MCollect_preserves_multiset:

   390     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"

   391   apply (simp add: multiset_def)

   392   apply (rule finite_subset)

   393    apply auto

   394   done

   396 theorem count_MCollect [simp]:

   397     "count {# x:M. P x #} a = (if P a then count M a else 0)"

   398   apply (unfold count_def MCollect_def)

   399   apply (simp add: MCollect_preserves_multiset)

   400   done

   402 theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"

   403   apply (auto simp add: set_of_def)

   404   done

   406 theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"

   407   apply (subst multiset_eq_conv_count_eq)

   408   apply auto

   409   done

   411 declare Rep_multiset_inject [symmetric, simp del]

   412 declare multiset_typedef [simp del]

   414 theorem add_eq_conv_ex:

   415   "(M + {#a#} = N + {#b#}) =

   416     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"

   417   apply (auto simp add: add_eq_conv_diff)

   418   done

   421 subsection {* Multiset orderings *}

   423 subsubsection {* Well-foundedness *}

   425 constdefs

   426   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"

   427   "mult1 r ==

   428     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>

   429       (\<forall>b. b :# K --> (b, a) \<in> r)}"

   431   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"

   432   "mult r == (mult1 r)\<^sup>+"

   434 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"

   435   by (simp add: mult1_def)

   437 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>

   438     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>

   439     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"

   440   (concl is "?case1 (mult1 r) \<or> ?case2")

   441 proof (unfold mult1_def)

   442   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"

   443   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"

   444   let ?case1 = "?case1 {(N, M). ?R N M}"

   446   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"

   447   hence "\<exists>a' M0' K.

   448       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp

   449   thus "?case1 \<or> ?case2"

   450   proof (elim exE conjE)

   451     fix a' M0' K

   452     assume N: "N = M0' + K" and r: "?r K a'"

   453     assume "M0 + {#a#} = M0' + {#a'#}"

   454     hence "M0 = M0' \<and> a = a' \<or>

   455         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"

   456       by (simp only: add_eq_conv_ex)

   457     thus ?thesis

   458     proof (elim disjE conjE exE)

   459       assume "M0 = M0'" "a = a'"

   460       with N r have "?r K a \<and> N = M0 + K" by simp

   461       hence ?case2 .. thus ?thesis ..

   462     next

   463       fix K'

   464       assume "M0' = K' + {#a#}"

   465       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)

   467       assume "M0 = K' + {#a'#}"

   468       with r have "?R (K' + K) M0" by blast

   469       with n have ?case1 by simp thus ?thesis ..

   470     qed

   471   qed

   472 qed

   474 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"

   475 proof

   476   let ?R = "mult1 r"

   477   let ?W = "acc ?R"

   478   {

   479     fix M M0 a

   480     assume M0: "M0 \<in> ?W"

   481       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"

   482       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"

   483     have "M0 + {#a#} \<in> ?W"

   484     proof (rule accI [of "M0 + {#a#}"])

   485       fix N

   486       assume "(N, M0 + {#a#}) \<in> ?R"

   487       hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>

   488           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"

   489         by (rule less_add)

   490       thus "N \<in> ?W"

   491       proof (elim exE disjE conjE)

   492         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"

   493         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..

   494         hence "M + {#a#} \<in> ?W" ..

   495         thus "N \<in> ?W" by (simp only: N)

   496       next

   497         fix K

   498         assume N: "N = M0 + K"

   499         assume "\<forall>b. b :# K --> (b, a) \<in> r"

   500         have "?this --> M0 + K \<in> ?W" (is "?P K")

   501         proof (induct K)

   502           from M0 have "M0 + {#} \<in> ?W" by simp

   503           thus "?P {#}" ..

   505           fix K x assume hyp: "?P K"

   506           show "?P (K + {#x#})"

   507           proof

   508             assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"

   509             hence "(x, a) \<in> r" by simp

   510             with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast

   512             from a hyp have "M0 + K \<in> ?W" by simp

   513             with b have "(M0 + K) + {#x#} \<in> ?W" ..

   514             thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)

   515           qed

   516         qed

   517         hence "M0 + K \<in> ?W" ..

   518         thus "N \<in> ?W" by (simp only: N)

   519       qed

   520     qed

   521   } note tedious_reasoning = this

   523   assume wf: "wf r"

   524   fix M

   525   show "M \<in> ?W"

   526   proof (induct M)

   527     show "{#} \<in> ?W"

   528     proof (rule accI)

   529       fix b assume "(b, {#}) \<in> ?R"

   530       with not_less_empty show "b \<in> ?W" by contradiction

   531     qed

   533     fix M a assume "M \<in> ?W"

   534     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"

   535     proof induct

   536       fix a

   537       assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"

   538       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"

   539       proof

   540         fix M assume "M \<in> ?W"

   541         thus "M + {#a#} \<in> ?W"

   542           by (rule acc_induct) (rule tedious_reasoning)

   543       qed

   544     qed

   545     thus "M + {#a#} \<in> ?W" ..

   546   qed

   547 qed

   549 theorem wf_mult1: "wf r ==> wf (mult1 r)"

   550   by (rule acc_wfI, rule all_accessible)

   552 theorem wf_mult: "wf r ==> wf (mult r)"

   553   by (unfold mult_def, rule wf_trancl, rule wf_mult1)

   556 subsubsection {* Closure-free presentation *}

   558 (*Badly needed: a linear arithmetic procedure for multisets*)

   560 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"

   561   apply (simp add: multiset_eq_conv_count_eq)

   562   done

   564 text {* One direction. *}

   566 lemma mult_implies_one_step:

   567   "trans r ==> (M, N) \<in> mult r ==>

   568     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>

   569     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"

   570   apply (unfold mult_def mult1_def set_of_def)

   571   apply (erule converse_trancl_induct)

   572   apply clarify

   573    apply (rule_tac x = M0 in exI)

   574    apply simp

   575   apply clarify

   576   apply (case_tac "a :# K")

   577    apply (rule_tac x = I in exI)

   578    apply (simp (no_asm))

   579    apply (rule_tac x = "(K - {#a#}) + Ka" in exI)

   580    apply (simp (no_asm_simp) add: union_assoc [symmetric])

   581    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)

   582    apply (simp add: diff_union_single_conv)

   583    apply (simp (no_asm_use) add: trans_def)

   584    apply blast

   585   apply (subgoal_tac "a :# I")

   586    apply (rule_tac x = "I - {#a#}" in exI)

   587    apply (rule_tac x = "J + {#a#}" in exI)

   588    apply (rule_tac x = "K + Ka" in exI)

   589    apply (rule conjI)

   590     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)

   591    apply (rule conjI)

   592     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)

   593     apply simp

   594     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)

   595    apply (simp (no_asm_use) add: trans_def)

   596    apply blast

   597   apply (subgoal_tac "a :# (M0 + {#a#})")

   598    apply simp

   599   apply (simp (no_asm))

   600   done

   602 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"

   603   apply (simp add: multiset_eq_conv_count_eq)

   604   done

   606 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"

   607   apply (erule size_eq_Suc_imp_elem [THEN exE])

   608   apply (drule elem_imp_eq_diff_union)

   609   apply auto

   610   done

   612 lemma one_step_implies_mult_aux:

   613   "trans r ==>

   614     \<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))

   615       --> (I + K, I + J) \<in> mult r"

   616   apply (induct_tac n)

   617    apply auto

   618   apply (frule size_eq_Suc_imp_eq_union)

   619   apply clarify

   620   apply (rename_tac "J'")

   621   apply simp

   622   apply (erule notE)

   623    apply auto

   624   apply (case_tac "J' = {#}")

   625    apply (simp add: mult_def)

   626    apply (rule r_into_trancl)

   627    apply (simp add: mult1_def set_of_def)

   628    apply blast

   629   txt {* Now we know @{term "J' \<noteq> {#}"}. *}

   630   apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)

   631   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)

   632   apply (erule ssubst)

   633   apply (simp add: Ball_def)

   634   apply auto

   635   apply (subgoal_tac

   636     "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},

   637       (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")

   638    prefer 2

   639    apply force

   640   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)

   641   apply (erule trancl_trans)

   642   apply (rule r_into_trancl)

   643   apply (simp add: mult1_def set_of_def)

   644   apply (rule_tac x = a in exI)

   645   apply (rule_tac x = "I + J'" in exI)

   646   apply (simp add: union_ac)

   647   done

   649 theorem one_step_implies_mult:

   650   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r

   651     ==> (I + K, I + J) \<in> mult r"

   652   apply (insert one_step_implies_mult_aux)

   653   apply blast

   654   done

   657 subsubsection {* Partial-order properties *}

   659 instance multiset :: (type) ord ..

   661 defs (overloaded)

   662   less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"

   663   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"

   665 lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"

   666   apply (unfold trans_def)

   667   apply (blast intro: order_less_trans)

   668   done

   670 text {*

   671  \medskip Irreflexivity.

   672 *}

   674 lemma mult_irrefl_aux:

   675     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"

   676   apply (erule finite_induct)

   677    apply (auto intro: order_less_trans)

   678   done

   680 theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"

   681   apply (unfold less_multiset_def)

   682   apply auto

   683   apply (drule trans_base_order [THEN mult_implies_one_step])

   684   apply auto

   685   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])

   686   apply (simp add: set_of_eq_empty_iff)

   687   done

   689 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"

   690   apply (insert mult_less_not_refl)

   691   apply fast

   692   done

   695 text {* Transitivity. *}

   697 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"

   698   apply (unfold less_multiset_def mult_def)

   699   apply (blast intro: trancl_trans)

   700   done

   702 text {* Asymmetry. *}

   704 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"

   705   apply auto

   706   apply (rule mult_less_not_refl [THEN notE])

   707   apply (erule mult_less_trans)

   708   apply assumption

   709   done

   711 theorem mult_less_asym:

   712     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"

   713   apply (insert mult_less_not_sym)

   714   apply blast

   715   done

   717 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"

   718   apply (unfold le_multiset_def)

   719   apply auto

   720   done

   722 text {* Anti-symmetry. *}

   724 theorem mult_le_antisym:

   725     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"

   726   apply (unfold le_multiset_def)

   727   apply (blast dest: mult_less_not_sym)

   728   done

   730 text {* Transitivity. *}

   732 theorem mult_le_trans:

   733     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"

   734   apply (unfold le_multiset_def)

   735   apply (blast intro: mult_less_trans)

   736   done

   738 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"

   739   apply (unfold le_multiset_def)

   740   apply auto

   741   done

   743 text {* Partial order. *}

   745 instance multiset :: (order) order

   746   apply intro_classes

   747      apply (rule mult_le_refl)

   748     apply (erule mult_le_trans)

   749     apply assumption

   750    apply (erule mult_le_antisym)

   751    apply assumption

   752   apply (rule mult_less_le)

   753   done

   756 subsubsection {* Monotonicity of multiset union *}

   758 theorem mult1_union:

   759     "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"

   760   apply (unfold mult1_def)

   761   apply auto

   762   apply (rule_tac x = a in exI)

   763   apply (rule_tac x = "C + M0" in exI)

   764   apply (simp add: union_assoc)

   765   done

   767 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"

   768   apply (unfold less_multiset_def mult_def)

   769   apply (erule trancl_induct)

   770    apply (blast intro: mult1_union transI order_less_trans r_into_trancl)

   771   apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)

   772   done

   774 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"

   775   apply (subst union_commute [of B C])

   776   apply (subst union_commute [of D C])

   777   apply (erule union_less_mono2)

   778   done

   780 theorem union_less_mono:

   781     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"

   782   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)

   783   done

   785 theorem union_le_mono:

   786     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"

   787   apply (unfold le_multiset_def)

   788   apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)

   789   done

   791 theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"

   792   apply (unfold le_multiset_def less_multiset_def)

   793   apply (case_tac "M = {#}")

   794    prefer 2

   795    apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")

   796     prefer 2

   797     apply (rule one_step_implies_mult)

   798       apply (simp only: trans_def)

   799       apply auto

   800   done

   802 theorem union_upper1: "A <= A + (B::'a::order multiset)"

   803   apply (subgoal_tac "A + {#} <= A + B")

   804    prefer 2

   805    apply (rule union_le_mono)

   806     apply auto

   807   done

   809 theorem union_upper2: "B <= A + (B::'a::order multiset)"

   810   apply (subst union_commute, rule union_upper1)

   811   done

   813 end

author	obua
	Tue, 11 May 2004 20:11:08 +0200
changeset 14738	83f1a514dcb4
parent 14722	8e739a6eaf11
child 14981	e73f8140af78
permissions	-rw-r--r--