src/HOL/Library/Multiset.thy
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(*  Title:      HOL/Library/Multiset.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, TU Muenchen
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    Author:     Markus Wenzel, TU Muenchen
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header {*
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 \title{Multisets}
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 \author{Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson}
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*}
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theory Multiset = Accessible_Part:
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subsection {* The type of multisets *}
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typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
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proof
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  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
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qed
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lemmas multiset_typedef [simp] =
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    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
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  and [simp] = Rep_multiset_inject [symmetric]
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constdefs
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  Mempty :: "'a multiset"    ("{#}")
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  "{#} == Abs_multiset (\<lambda>a. 0)"
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  single :: "'a => 'a multiset"    ("{#_#}")
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  "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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  count :: "'a multiset => 'a => nat"
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  "count == Rep_multiset"
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  MCollect :: "'a multiset => ('a => bool) => 'a multiset"
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  "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
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syntax
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  "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
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  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
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translations
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  "a :# M" == "0 < count M a"
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  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
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constdefs
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  set_of :: "'a multiset => 'a set"
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  "set_of M == {x. x :# M}"
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instance multiset :: ("term") plus ..
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instance multiset :: ("term") minus ..
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instance multiset :: ("term") zero ..
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defs (overloaded)
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  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
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  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
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  Zero_def [simp]: "0 == {#}"
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  size_def: "size M == setsum (count M) (set_of M)"
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text {*
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 \medskip Preservation of the representing set @{term multiset}.
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*}
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lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
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  apply (simp add: multiset_def)
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  done
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lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
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  apply (simp add: multiset_def)
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  done
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lemma union_preserves_multiset [simp]:
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    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
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  apply (unfold multiset_def)
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  apply simp
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  apply (drule finite_UnI)
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   apply assumption
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  apply (simp del: finite_Un add: Un_def)
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  done
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lemma diff_preserves_multiset [simp]:
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    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
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  apply (unfold multiset_def)
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  apply simp
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  apply (rule finite_subset)
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   prefer 2
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   apply assumption
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  apply auto
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  done
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subsection {* Algebraic properties of multisets *}
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subsubsection {* Union *}
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theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
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  apply (simp add: union_def Mempty_def)
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  done
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theorem union_commute: "M + N = N + (M::'a multiset)"
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  apply (simp add: union_def add_ac)
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  done
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theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
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  apply (simp add: union_def add_ac)
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  done
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theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
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  apply (rule union_commute [THEN trans])
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  apply (rule union_assoc [THEN trans])
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  apply (rule union_commute [THEN arg_cong])
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  done
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theorems union_ac = union_assoc union_commute union_lcomm
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instance multiset :: ("term") plus_ac0
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  apply intro_classes
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    apply (rule union_commute)
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   apply (rule union_assoc)
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  apply simp
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  done
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subsubsection {* Difference *}
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theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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  apply (simp add: Mempty_def diff_def)
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  done
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theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
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  apply (simp add: union_def diff_def)
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  done
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subsubsection {* Count of elements *}
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theorem count_empty [simp]: "count {#} a = 0"
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  apply (simp add: count_def Mempty_def)
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  done
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theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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  apply (simp add: count_def single_def)
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  done
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theorem count_union [simp]: "count (M + N) a = count M a + count N a"
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  apply (simp add: count_def union_def)
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   148
  done
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   149
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
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  apply (simp add: count_def diff_def)
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   152
  done
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   153
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   154
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   155
subsubsection {* Set of elements *}
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   156
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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theorem set_of_empty [simp]: "set_of {#} = {}"
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   158
  apply (simp add: set_of_def)
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   159
  done
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   160
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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theorem set_of_single [simp]: "set_of {#b#} = {b}"
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   162
  apply (simp add: set_of_def)
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   163
  done
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   164
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   165
theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
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   166
  apply (auto simp add: set_of_def)
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   167
  done
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   168
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   169
theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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   170
  apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
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   171
  done
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   172
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   173
theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
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   174
  apply (auto simp add: set_of_def)
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   175
  done
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   176
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   177
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   178
subsubsection {* Size *}
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   179
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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theorem size_empty [simp]: "size {#} = 0"
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   181
  apply (simp add: size_def)
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   182
  done
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   183
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   184
theorem size_single [simp]: "size {#b#} = 1"
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   185
  apply (simp add: size_def)
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   186
  done
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   187
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   188
theorem finite_set_of [iff]: "finite (set_of M)"
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   189
  apply (cut_tac x = M in Rep_multiset)
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   190
  apply (simp add: multiset_def set_of_def count_def)
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   191
  done
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   192
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   193
theorem setsum_count_Int:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   194
    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
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   195
  apply (erule finite_induct)
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   196
   apply simp
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   197
  apply (simp add: Int_insert_left set_of_def)
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   198
  done
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   199
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   200
theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
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   201
  apply (unfold size_def)
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   202
  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
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   203
   prefer 2
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   204
   apply (rule ext)
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   205
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   206
  apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
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   207
  apply (subst Int_commute)
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   208
  apply (simp (no_asm_simp) add: setsum_count_Int)
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   209
  done
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   210
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   211
theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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   212
  apply (unfold size_def Mempty_def count_def)
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   213
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   214
  apply (simp add: set_of_def count_def expand_fun_eq)
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   215
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   216
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   217
theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
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   218
  apply (unfold size_def)
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   219
  apply (drule setsum_SucD)
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   220
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   221
  done
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   222
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   223
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   224
subsubsection {* Equality of multisets *}
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   225
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   226
theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
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   227
  apply (simp add: count_def expand_fun_eq)
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   228
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   229
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   230
theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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   231
  apply (simp add: single_def Mempty_def expand_fun_eq)
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   232
  done
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   233
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   234
theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
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   235
  apply (auto simp add: single_def expand_fun_eq)
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diff changeset
   236
  done
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diff changeset
   237
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   238
theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   239
  apply (auto simp add: union_def Mempty_def expand_fun_eq)
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   240
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   241
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   242
theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   243
  apply (auto simp add: union_def Mempty_def expand_fun_eq)
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diff changeset
   244
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   245
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   246
theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   247
  apply (simp add: union_def expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   248
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   250
theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
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diff changeset
   251
  apply (simp add: union_def expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   252
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   253
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   254
theorem union_is_single:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   255
    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   256
  apply (unfold Mempty_def single_def union_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   257
  apply (simp add: add_is_1 expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   258
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   259
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   260
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   261
theorem single_is_union:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   262
  "({#a#} = M + N) =
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   263
    ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   264
  apply (unfold Mempty_def single_def union_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   265
  apply (simp add: one_is_add expand_fun_eq)
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   266
  apply (blast dest: sym)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   267
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   268
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   269
theorem add_eq_conv_diff:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   270
  "(M + {#a#} = N + {#b#}) =
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   271
    (M = N \<and> a = b \<or>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   272
      M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   273
  apply (unfold single_def union_def diff_def)
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diff changeset
   274
  apply (simp (no_asm) add: expand_fun_eq)
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   275
  apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   276
   apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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   277
  apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   278
  apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   279
   apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   280
  apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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parents:
diff changeset
   281
  apply (rule iffI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   282
   apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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parents:
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   283
    apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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parents:
diff changeset
   284
    apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   285
     apply (simp add: eq_sym_conv)   (* FIXME blast fails !? *)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   286
    apply fast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
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diff changeset
   287
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   288
  apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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parents:
diff changeset
   289
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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diff changeset
   290
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   291
(*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   292
val prems = Goal
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   293
 "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   294
by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   295
     measure_induct 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   296
by (Clarify_tac 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   297
by (resolve_tac prems 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   298
 by (assume_tac 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   299
by (Clarify_tac 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   300
by (subgoal_tac "finite G" 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   301
 by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   302
by (etac allE 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   303
by (etac impE 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   304
 by (Blast_tac 2);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   305
by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   306
no_qed();
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   307
val lemma = result();
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   308
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   309
val prems = Goal
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   310
 "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   311
by (rtac (lemma RS mp) 1);
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   312
by (REPEAT(ares_tac prems 1));
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   313
qed "finite_psubset_induct";
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   314
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   315
Better: use wf_finite_psubset in WF_Rel
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   316
*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   317
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   318
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   319
subsection {* Induction over multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   320
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   321
lemma setsum_decr:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   322
  "finite F ==> 0 < f a ==>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   323
    setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   324
  apply (erule finite_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   325
   apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   326
  apply (drule_tac a = a in mk_disjoint_insert)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   327
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   328
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   329
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   330
lemma rep_multiset_induct_aux:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   331
  "P (\<lambda>a. 0) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   332
    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   333
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   334
  case antecedent
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   335
  note prems = this [unfolded multiset_def]
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   336
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   337
    apply (unfold multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   338
    apply (induct_tac n)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   339
     apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   340
     apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   341
     apply (subgoal_tac "f = (\<lambda>a.0)")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   342
      apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   343
      apply (rule prems)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   344
     apply (rule ext)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   345
     apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   346
    apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   347
    apply (frule setsum_SucD)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   348
    apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   349
    apply (rename_tac a)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   350
    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   351
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   352
     apply (rule finite_subset)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   353
      prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   354
      apply assumption
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   355
     apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   356
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   357
    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   358
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   359
     apply (rule ext)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   360
     apply (simp (no_asm_simp))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   361
     apply (erule ssubst, rule prems)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   362
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   363
    apply (erule allE, erule impE, erule_tac [2] mp)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   364
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   365
    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   366
    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   367
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   368
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   369
    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   370
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   371
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   372
    apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   373
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   374
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   375
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   376
theorem rep_multiset_induct:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   377
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   378
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   379
  apply (insert rep_multiset_induct_aux)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   380
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   381
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   382
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   383
theorem multiset_induct [induct type: multiset]:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   384
  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   385
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   386
  note defns = union_def single_def Mempty_def
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   387
  assume prem1 [unfolded defns]: "P {#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   388
  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   389
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   390
    apply (rule Rep_multiset_inverse [THEN subst])
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   391
    apply (rule Rep_multiset [THEN rep_multiset_induct])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   392
     apply (rule prem1)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   393
    apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   394
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   395
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   396
    apply (erule ssubst)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   397
    apply (erule Abs_multiset_inverse [THEN subst])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   398
    apply (erule prem2 [simplified])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   399
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   400
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   401
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   402
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   403
lemma MCollect_preserves_multiset:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   404
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   405
  apply (simp add: multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   406
  apply (rule finite_subset)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   407
   apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   408
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   409
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   410
theorem count_MCollect [simp]:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   411
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   412
  apply (unfold count_def MCollect_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   413
  apply (simp add: MCollect_preserves_multiset)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   414
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   415
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   416
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   417
  apply (auto simp add: set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   418
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   419
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   420
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   421
  apply (subst multiset_eq_conv_count_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   422
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   423
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   424
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   425
declare Rep_multiset_inject [symmetric, simp del]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   426
declare multiset_typedef [simp del]
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   427
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   428
theorem add_eq_conv_ex:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   429
  "(M + {#a#} = N + {#b#}) =
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   430
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   431
  apply (auto simp add: add_eq_conv_diff)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   432
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   433
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   434
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   435
subsection {* Multiset orderings *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   436
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   437
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   438
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   439
constdefs
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   440
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   441
  "mult1 r ==
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   442
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   443
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   444
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   445
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   446
  "mult r == (mult1 r)^+"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   447
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   448
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   449
  by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   450
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   451
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   452
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   453
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   454
  (concl is "?case1 (mult1 r) \<or> ?case2")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   455
proof (unfold mult1_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   456
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   457
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   458
  let ?case1 = "?case1 {(N, M). ?R N M}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   459
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   460
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   461
  hence "\<exists>a' M0' K.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   462
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   463
  thus "?case1 \<or> ?case2"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   464
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   465
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   466
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   467
    assume "M0 + {#a#} = M0' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   468
    hence "M0 = M0' \<and> a = a' \<or>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   469
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   470
      by (simp only: add_eq_conv_ex)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   471
    thus ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   472
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   473
      assume "M0 = M0'" "a = a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   474
      with N r have "?r K a \<and> N = M0 + K" by simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   475
      hence ?case2 .. thus ?thesis ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   476
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   477
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   478
      assume "M0' = K' + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   479
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   480
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   481
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   482
      with r have "?R (K' + K) M0" by blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   483
      with n have ?case1 by simp thus ?thesis ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   484
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   485
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   486
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   487
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   488
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   489
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   490
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   491
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   492
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   493
    fix M M0 a
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   494
    assume M0: "M0 \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   495
      and wf_hyp: "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   496
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   497
    have "M0 + {#a#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   498
    proof (rule accI [of "M0 + {#a#}"])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   499
      fix N
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   500
      assume "(N, M0 + {#a#}) \<in> ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   501
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   502
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   503
        by (rule less_add)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   504
      thus "N \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   505
      proof (elim exE disjE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   506
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   507
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   508
        hence "M + {#a#} \<in> ?W" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   509
        thus "N \<in> ?W" by (simp only: N)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   510
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   511
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   512
        assume N: "N = M0 + K"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   513
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   514
        have "?this --> M0 + K \<in> ?W" (is "?P K")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   515
        proof (induct K)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   516
          from M0 have "M0 + {#} \<in> ?W" by simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   517
          thus "?P {#}" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   518
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   519
          fix K x assume hyp: "?P K"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   520
          show "?P (K + {#x#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   521
          proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   522
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   523
            hence "(x, a) \<in> r" by simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   524
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   525
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   526
            from a hyp have "M0 + K \<in> ?W" by simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   527
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   528
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
          qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   530
        qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   531
        hence "M0 + K \<in> ?W" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   532
        thus "N \<in> ?W" by (simp only: N)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   533
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   534
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   535
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   536
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   537
  assume wf: "wf r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   538
  fix M
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   539
  show "M \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   540
  proof (induct M)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   541
    show "{#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   542
    proof (rule accI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   543
      fix b assume "(b, {#}) \<in> ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   544
      with not_less_empty show "b \<in> ?W" by contradiction
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   545
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   546
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   547
    fix M a assume "M \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   548
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   549
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   550
      fix a
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   551
      assume "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   552
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   553
      proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   554
        fix M assume "M \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   555
        thus "M + {#a#} \<in> ?W"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   556
          by (rule acc_induct) (rule tedious_reasoning)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   557
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   558
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   559
    thus "M + {#a#} \<in> ?W" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   560
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   561
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   562
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   563
theorem wf_mult1: "wf r ==> wf (mult1 r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   564
  by (rule acc_wfI, rule all_accessible)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   565
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   566
theorem wf_mult: "wf r ==> wf (mult r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   567
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   568
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   569
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   570
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   571
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   572
(*Badly needed: a linear arithmetic procedure for multisets*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   573
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   574
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   575
  apply (simp add: multiset_eq_conv_count_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   576
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   577
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   578
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   579
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   580
lemma mult_implies_one_step:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   581
  "trans r ==> (M, N) \<in> mult r ==>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   582
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   583
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   584
  apply (unfold mult_def mult1_def set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   585
  apply (erule converse_trancl_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   586
  apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   587
   apply (rule_tac x = M0 in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   588
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   589
  apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   590
  apply (case_tac "a :# K")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   591
   apply (rule_tac x = I in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   592
   apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   593
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   594
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   595
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   596
   apply (simp add: diff_union_single_conv)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   597
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   598
   apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   599
  apply (subgoal_tac "a :# I")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   600
   apply (rule_tac x = "I - {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   601
   apply (rule_tac x = "J + {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   602
   apply (rule_tac x = "K + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   603
   apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   604
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   605
   apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   606
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   607
    apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   608
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   609
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   610
   apply blast
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   611
  apply (subgoal_tac "a :# (M0 + {#a#})")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   612
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   613
  apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   615
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   616
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   617
  apply (simp add: multiset_eq_conv_count_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   618
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   619
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   620
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   621
  apply (erule size_eq_Suc_imp_elem [THEN exE])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   622
  apply (drule elem_imp_eq_diff_union)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   623
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   624
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   625
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   626
lemma one_step_implies_mult_aux:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   627
  "trans r ==>
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   628
    \<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))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   629
      --> (I + K, I + J) \<in> mult r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   630
  apply (induct_tac n)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   631
   apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   632
  apply (frule size_eq_Suc_imp_eq_union)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   633
  apply clarify
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   634
  apply (rename_tac "J'")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   635
  apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   636
  apply (erule notE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   637
   apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   638
  apply (case_tac "J' = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   639
   apply (simp add: mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   640
   apply (rule r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   641
   apply (simp add: mult1_def set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   642
   apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   643
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   644
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   645
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   646
  apply (erule ssubst)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   647
  apply (simp add: Ball_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   648
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   649
  apply (subgoal_tac
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   650
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   651
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   652
   prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   653
   apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   654
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   655
  apply (erule trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   656
  apply (rule r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   657
  apply (simp add: mult1_def set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   658
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   659
  apply (rule_tac x = "I + J'" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   660
  apply (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   661
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   662
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   663
theorem one_step_implies_mult:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   664
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   665
    ==> (I + K, I + J) \<in> mult r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   666
  apply (insert one_step_implies_mult_aux)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   667
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   668
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   669
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   670
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   671
subsubsection {* Partial-order properties *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   672
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   673
instance multiset :: ("term") ord ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   674
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   675
defs (overloaded)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   676
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   677
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   678
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   679
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   680
  apply (unfold trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   681
  apply (blast intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   682
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   683
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   684
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   685
 \medskip Irreflexivity.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   686
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   687
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   688
lemma mult_irrefl_aux:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   689
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   690
  apply (erule finite_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   691
   apply (auto intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   692
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   693
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   694
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   695
  apply (unfold less_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   696
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   697
  apply (drule trans_base_order [THEN mult_implies_one_step])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   698
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   699
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   700
  apply (simp add: set_of_eq_empty_iff)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   701
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   702
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   703
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   704
  apply (insert mult_less_not_refl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   705
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   706
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   707
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   708
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   709
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   710
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   711
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   712
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   713
  apply (blast intro: trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   714
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   715
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   716
text {* Asymmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   717
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   718
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   719
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   720
  apply (rule mult_less_not_refl [THEN notE])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   721
  apply (erule mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   722
  apply assumption
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   723
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   724
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   725
theorem mult_less_asym:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   726
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   727
  apply (insert mult_less_not_sym)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   728
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   729
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   730
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   731
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   732
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   733
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   734
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   735
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   736
text {* Anti-symmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   737
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   738
theorem mult_le_antisym:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   739
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   740
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   741
  apply (blast dest: mult_less_not_sym)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   742
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   743
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   744
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   745
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   746
theorem mult_le_trans:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   747
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   748
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   749
  apply (blast intro: mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   750
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   751
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   752
theorem mult_less_le: "M < N = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   753
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   754
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   755
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   756
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   757
text {* Partial order. *}
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   758
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   759
instance multiset :: (order) order
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   760
  apply intro_classes
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   761
     apply (rule mult_le_refl)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   762
    apply (erule mult_le_trans)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   763
    apply assumption
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   764
   apply (erule mult_le_antisym)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   765
   apply assumption
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   766
  apply (rule mult_less_le)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   767
  done
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   768
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   769
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   770
subsubsection {* Monotonicity of multiset union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   771
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   772
theorem mult1_union:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   773
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   774
  apply (unfold mult1_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   775
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   776
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   777
  apply (rule_tac x = "C + M0" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   778
  apply (simp add: union_assoc)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   779
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   780
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   781
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   782
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   783
  apply (erule trancl_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   784
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   785
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   786
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   787
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   788
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   789
  apply (subst union_commute [of B C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   790
  apply (subst union_commute [of D C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   791
  apply (erule union_less_mono2)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   792
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   793
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   794
theorem union_less_mono:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   795
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   796
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   797
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   798
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   799
theorem union_le_mono:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   800
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   801
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   802
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   803
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   804
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   805
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   806
  apply (unfold le_multiset_def less_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   807
  apply (case_tac "M = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   808
   prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   809
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   810
    prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   811
    apply (rule one_step_implies_mult)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   812
      apply (simp only: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   813
      apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   814
  apply (blast intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   815
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   816
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   817
theorem union_upper1: "A <= A + (B::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   818
  apply (subgoal_tac "A + {#} <= A + B")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   819
   prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   820
   apply (rule union_le_mono)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   821
    apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   822
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   823
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   824
theorem union_upper2: "B <= A + (B::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   825
  apply (subst union_commute, rule union_upper1)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   826
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   827
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   828
end