src/HOL/Library/Multiset.thy
author berghofe
Mon Sep 30 16:14:02 2002 +0200 (2002-09-30)
changeset 13601 fd3e3d6b37b2
parent 13596 ee5f79b210c1
child 14445 4392cb82018b
permissions -rw-r--r--
Adapted to new simplifier.
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(*  Title:      HOL/Library/Multiset.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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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 :: (type) plus ..
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instance multiset :: (type) minus ..
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instance multiset :: (type) 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_multiset_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 :: (type) 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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  done
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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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  done
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subsubsection {* Set of elements *}
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theorem set_of_empty [simp]: "set_of {#} = {}"
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  apply (simp add: set_of_def)
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  done
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theorem set_of_single [simp]: "set_of {#b#} = {b}"
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  apply (simp add: set_of_def)
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  done
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theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
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  apply (auto simp add: set_of_def)
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  done
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theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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  apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
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  done
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theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
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  apply (auto simp add: set_of_def)
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  done
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subsubsection {* Size *}
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theorem size_empty [simp]: "size {#} = 0"
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  apply (simp add: size_def)
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  done
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theorem size_single [simp]: "size {#b#} = 1"
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  apply (simp add: size_def)
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  done
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theorem finite_set_of [iff]: "finite (set_of M)"
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  apply (cut_tac x = M in Rep_multiset)
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  apply (simp add: multiset_def set_of_def count_def)
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  done
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theorem setsum_count_Int:
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    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
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  apply (erule finite_induct)
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   apply simp
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  apply (simp add: Int_insert_left set_of_def)
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  done
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theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
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  apply (unfold size_def)
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  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
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   prefer 2
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   apply (rule ext)
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   apply simp
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  apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
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  apply (subst Int_commute)
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  apply (simp (no_asm_simp) add: setsum_count_Int)
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  done
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theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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  apply (unfold size_def Mempty_def count_def)
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  apply auto
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  apply (simp add: set_of_def count_def expand_fun_eq)
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  done
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theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
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  apply (unfold size_def)
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  apply (drule setsum_SucD)
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  apply auto
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  done
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subsubsection {* Equality of multisets *}
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theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
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  apply (simp add: count_def expand_fun_eq)
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  done
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theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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  apply (simp add: single_def Mempty_def expand_fun_eq)
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  done
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theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
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  apply (auto simp add: single_def expand_fun_eq)
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  done
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theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
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  apply (auto simp add: union_def Mempty_def expand_fun_eq)
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  done
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theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
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  apply (auto simp add: union_def Mempty_def expand_fun_eq)
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  done
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theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
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  apply (simp add: union_def expand_fun_eq)
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  done
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theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
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  apply (simp add: union_def expand_fun_eq)
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  done
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theorem union_is_single:
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    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
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  apply (unfold Mempty_def single_def union_def)
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  apply (simp add: add_is_1 expand_fun_eq)
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  apply blast
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  done
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theorem single_is_union:
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  "({#a#} = M + N) =
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    ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
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  apply (unfold Mempty_def single_def union_def)
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  apply (simp add: add_is_1 one_is_add expand_fun_eq)
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  apply (blast dest: sym)
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  done
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theorem add_eq_conv_diff:
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  "(M + {#a#} = N + {#b#}) =
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    (M = N \<and> a = b \<or>
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      M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
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  apply (unfold single_def union_def diff_def)
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  apply (simp (no_asm) add: expand_fun_eq)
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  apply (rule conjI)
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   apply force
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  apply safe
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  apply simp_all
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  apply (simp add: eq_sym_conv)
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  done
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(*
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val prems = Goal
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 "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
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by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
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     measure_induct 1);
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by (Clarify_tac 1);
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by (resolve_tac prems 1);
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 by (assume_tac 1);
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by (Clarify_tac 1);
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by (subgoal_tac "finite G" 1);
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 by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
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by (etac allE 1);
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by (etac impE 1);
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 by (Blast_tac 2);
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by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
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no_qed();
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val lemma = result();
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val prems = Goal
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 "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
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by (rtac (lemma RS mp) 1);
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by (REPEAT(ares_tac prems 1));
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qed "finite_psubset_induct";
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Better: use wf_finite_psubset in WF_Rel
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*)
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subsection {* Induction over multisets *}
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lemma setsum_decr:
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  "finite F ==> (0::nat) < f a ==>
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    setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
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  apply (erule finite_induct)
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   apply auto
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  apply (drule_tac a = a in mk_disjoint_insert)
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  apply auto
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  done
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lemma rep_multiset_induct_aux:
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  "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
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    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
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proof -
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  case rule_context
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  note premises = this [unfolded multiset_def]
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  show ?thesis
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    apply (unfold multiset_def)
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    apply (induct_tac n)
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     apply simp
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     apply clarify
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     apply (subgoal_tac "f = (\<lambda>a.0)")
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      apply simp
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      apply (rule premises)
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     apply (rule ext)
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     apply force
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    apply clarify
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    apply (frule setsum_SucD)
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    apply clarify
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    apply (rename_tac a)
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    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
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     prefer 2
wenzelm@10249
   342
     apply (rule finite_subset)
wenzelm@10249
   343
      prefer 2
wenzelm@10249
   344
      apply assumption
wenzelm@10249
   345
     apply simp
wenzelm@10249
   346
     apply blast
wenzelm@11701
   347
    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
wenzelm@10249
   348
     prefer 2
wenzelm@10249
   349
     apply (rule ext)
wenzelm@10249
   350
     apply (simp (no_asm_simp))
wenzelm@11549
   351
     apply (erule ssubst, rule premises)
wenzelm@10249
   352
     apply blast
wenzelm@10249
   353
    apply (erule allE, erule impE, erule_tac [2] mp)
wenzelm@10249
   354
     apply blast
wenzelm@11701
   355
    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@11464
   356
    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
wenzelm@10249
   357
     prefer 2
wenzelm@10249
   358
     apply blast
nipkow@11464
   359
    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
wenzelm@10249
   360
     prefer 2
wenzelm@10249
   361
     apply blast
wenzelm@10249
   362
    apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
wenzelm@10249
   363
    done
wenzelm@10249
   364
qed
wenzelm@10249
   365
wenzelm@10313
   366
theorem rep_multiset_induct:
nipkow@11464
   367
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701
   368
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
wenzelm@10313
   369
  apply (insert rep_multiset_induct_aux)
wenzelm@10249
   370
  apply blast
wenzelm@10249
   371
  done
wenzelm@10249
   372
wenzelm@10249
   373
theorem multiset_induct [induct type: multiset]:
wenzelm@10249
   374
  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
wenzelm@10249
   375
proof -
wenzelm@10249
   376
  note defns = union_def single_def Mempty_def
wenzelm@10249
   377
  assume prem1 [unfolded defns]: "P {#}"
wenzelm@10249
   378
  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
wenzelm@10249
   379
  show ?thesis
wenzelm@10249
   380
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   381
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@10249
   382
     apply (rule prem1)
wenzelm@11701
   383
    apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
wenzelm@10249
   384
     prefer 2
wenzelm@10249
   385
     apply (simp add: expand_fun_eq)
wenzelm@10249
   386
    apply (erule ssubst)
wenzelm@10249
   387
    apply (erule Abs_multiset_inverse [THEN subst])
wenzelm@10249
   388
    apply (erule prem2 [simplified])
wenzelm@10249
   389
    done
wenzelm@10249
   390
qed
wenzelm@10249
   391
wenzelm@10249
   392
wenzelm@10249
   393
lemma MCollect_preserves_multiset:
nipkow@11464
   394
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
wenzelm@10249
   395
  apply (simp add: multiset_def)
wenzelm@10249
   396
  apply (rule finite_subset)
wenzelm@10249
   397
   apply auto
wenzelm@10249
   398
  done
wenzelm@10249
   399
wenzelm@10249
   400
theorem count_MCollect [simp]:
wenzelm@10249
   401
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
wenzelm@10249
   402
  apply (unfold count_def MCollect_def)
wenzelm@10249
   403
  apply (simp add: MCollect_preserves_multiset)
wenzelm@10249
   404
  done
wenzelm@10249
   405
nipkow@11464
   406
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
wenzelm@10249
   407
  apply (auto simp add: set_of_def)
wenzelm@10249
   408
  done
wenzelm@10249
   409
nipkow@11464
   410
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
wenzelm@10249
   411
  apply (subst multiset_eq_conv_count_eq)
wenzelm@10249
   412
  apply auto
wenzelm@10249
   413
  done
wenzelm@10249
   414
wenzelm@10277
   415
declare Rep_multiset_inject [symmetric, simp del]
wenzelm@10249
   416
declare multiset_typedef [simp del]
wenzelm@10249
   417
wenzelm@10249
   418
theorem add_eq_conv_ex:
wenzelm@10249
   419
  "(M + {#a#} = N + {#b#}) =
nipkow@11464
   420
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
wenzelm@10249
   421
  apply (auto simp add: add_eq_conv_diff)
wenzelm@10249
   422
  done
wenzelm@10249
   423
wenzelm@10249
   424
wenzelm@10249
   425
subsection {* Multiset orderings *}
wenzelm@10249
   426
wenzelm@10249
   427
subsubsection {* Well-foundedness *}
wenzelm@10249
   428
wenzelm@10249
   429
constdefs
nipkow@11464
   430
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10249
   431
  "mult1 r ==
nipkow@11464
   432
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
nipkow@11464
   433
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   434
nipkow@11464
   435
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10392
   436
  "mult r == (mult1 r)\<^sup>+"
wenzelm@10249
   437
nipkow@11464
   438
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
wenzelm@10277
   439
  by (simp add: mult1_def)
wenzelm@10249
   440
nipkow@11464
   441
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
nipkow@11464
   442
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
nipkow@11464
   443
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
nipkow@11464
   444
  (concl is "?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   445
proof (unfold mult1_def)
nipkow@11464
   446
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   447
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@10249
   448
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   449
nipkow@11464
   450
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
nipkow@11464
   451
  hence "\<exists>a' M0' K.
nipkow@11464
   452
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
nipkow@11464
   453
  thus "?case1 \<or> ?case2"
wenzelm@10249
   454
  proof (elim exE conjE)
wenzelm@10249
   455
    fix a' M0' K
wenzelm@10249
   456
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   457
    assume "M0 + {#a#} = M0' + {#a'#}"
nipkow@11464
   458
    hence "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   459
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   460
      by (simp only: add_eq_conv_ex)
wenzelm@10249
   461
    thus ?thesis
wenzelm@10249
   462
    proof (elim disjE conjE exE)
wenzelm@10249
   463
      assume "M0 = M0'" "a = a'"
nipkow@11464
   464
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@10249
   465
      hence ?case2 .. thus ?thesis ..
wenzelm@10249
   466
    next
wenzelm@10249
   467
      fix K'
wenzelm@10249
   468
      assume "M0' = K' + {#a#}"
wenzelm@10249
   469
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   470
wenzelm@10249
   471
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   472
      with r have "?R (K' + K) M0" by blast
wenzelm@10249
   473
      with n have ?case1 by simp thus ?thesis ..
wenzelm@10249
   474
    qed
wenzelm@10249
   475
  qed
wenzelm@10249
   476
qed
wenzelm@10249
   477
nipkow@11464
   478
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   479
proof
wenzelm@10249
   480
  let ?R = "mult1 r"
wenzelm@10249
   481
  let ?W = "acc ?R"
wenzelm@10249
   482
  {
wenzelm@10249
   483
    fix M M0 a
nipkow@11464
   484
    assume M0: "M0 \<in> ?W"
wenzelm@12399
   485
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   486
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
nipkow@11464
   487
    have "M0 + {#a#} \<in> ?W"
wenzelm@10249
   488
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   489
      fix N
nipkow@11464
   490
      assume "(N, M0 + {#a#}) \<in> ?R"
nipkow@11464
   491
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
nipkow@11464
   492
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   493
        by (rule less_add)
nipkow@11464
   494
      thus "N \<in> ?W"
wenzelm@10249
   495
      proof (elim exE disjE conjE)
nipkow@11464
   496
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
nipkow@11464
   497
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
nipkow@11464
   498
        hence "M + {#a#} \<in> ?W" ..
nipkow@11464
   499
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   500
      next
wenzelm@10249
   501
        fix K
wenzelm@10249
   502
        assume N: "N = M0 + K"
nipkow@11464
   503
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   504
        have "?this --> M0 + K \<in> ?W" (is "?P K")
wenzelm@10249
   505
        proof (induct K)
nipkow@11464
   506
          from M0 have "M0 + {#} \<in> ?W" by simp
wenzelm@10249
   507
          thus "?P {#}" ..
wenzelm@10249
   508
wenzelm@10249
   509
          fix K x assume hyp: "?P K"
wenzelm@10249
   510
          show "?P (K + {#x#})"
wenzelm@10249
   511
          proof
nipkow@11464
   512
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
nipkow@11464
   513
            hence "(x, a) \<in> r" by simp
nipkow@11464
   514
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
wenzelm@10249
   515
nipkow@11464
   516
            from a hyp have "M0 + K \<in> ?W" by simp
nipkow@11464
   517
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
nipkow@11464
   518
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   519
          qed
wenzelm@10249
   520
        qed
nipkow@11464
   521
        hence "M0 + K \<in> ?W" ..
nipkow@11464
   522
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   523
      qed
wenzelm@10249
   524
    qed
wenzelm@10249
   525
  } note tedious_reasoning = this
wenzelm@10249
   526
wenzelm@10249
   527
  assume wf: "wf r"
wenzelm@10249
   528
  fix M
nipkow@11464
   529
  show "M \<in> ?W"
wenzelm@10249
   530
  proof (induct M)
nipkow@11464
   531
    show "{#} \<in> ?W"
wenzelm@10249
   532
    proof (rule accI)
nipkow@11464
   533
      fix b assume "(b, {#}) \<in> ?R"
nipkow@11464
   534
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   535
    qed
wenzelm@10249
   536
nipkow@11464
   537
    fix M a assume "M \<in> ?W"
nipkow@11464
   538
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   539
    proof induct
wenzelm@10249
   540
      fix a
wenzelm@12399
   541
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   542
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   543
      proof
nipkow@11464
   544
        fix M assume "M \<in> ?W"
nipkow@11464
   545
        thus "M + {#a#} \<in> ?W"
wenzelm@10249
   546
          by (rule acc_induct) (rule tedious_reasoning)
wenzelm@10249
   547
      qed
wenzelm@10249
   548
    qed
nipkow@11464
   549
    thus "M + {#a#} \<in> ?W" ..
wenzelm@10249
   550
  qed
wenzelm@10249
   551
qed
wenzelm@10249
   552
wenzelm@10249
   553
theorem wf_mult1: "wf r ==> wf (mult1 r)"
wenzelm@10249
   554
  by (rule acc_wfI, rule all_accessible)
wenzelm@10249
   555
wenzelm@10249
   556
theorem wf_mult: "wf r ==> wf (mult r)"
wenzelm@10249
   557
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
wenzelm@10249
   558
wenzelm@10249
   559
wenzelm@10249
   560
subsubsection {* Closure-free presentation *}
wenzelm@10249
   561
wenzelm@10249
   562
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   563
wenzelm@10249
   564
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
wenzelm@10249
   565
  apply (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   566
  done
wenzelm@10249
   567
wenzelm@10249
   568
text {* One direction. *}
wenzelm@10249
   569
wenzelm@10249
   570
lemma mult_implies_one_step:
nipkow@11464
   571
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   572
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@11464
   573
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
wenzelm@10249
   574
  apply (unfold mult_def mult1_def set_of_def)
wenzelm@10249
   575
  apply (erule converse_trancl_induct)
wenzelm@10249
   576
  apply clarify
wenzelm@10249
   577
   apply (rule_tac x = M0 in exI)
wenzelm@10249
   578
   apply simp
wenzelm@10249
   579
  apply clarify
wenzelm@10249
   580
  apply (case_tac "a :# K")
wenzelm@10249
   581
   apply (rule_tac x = I in exI)
wenzelm@10249
   582
   apply (simp (no_asm))
wenzelm@10249
   583
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
wenzelm@10249
   584
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@11464
   585
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249
   586
   apply (simp add: diff_union_single_conv)
wenzelm@10249
   587
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   588
   apply blast
wenzelm@10249
   589
  apply (subgoal_tac "a :# I")
wenzelm@10249
   590
   apply (rule_tac x = "I - {#a#}" in exI)
wenzelm@10249
   591
   apply (rule_tac x = "J + {#a#}" in exI)
wenzelm@10249
   592
   apply (rule_tac x = "K + Ka" in exI)
wenzelm@10249
   593
   apply (rule conjI)
wenzelm@10249
   594
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   595
   apply (rule conjI)
nipkow@11464
   596
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249
   597
    apply simp
wenzelm@10249
   598
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   599
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   600
   apply blast
wenzelm@10277
   601
  apply (subgoal_tac "a :# (M0 + {#a#})")
wenzelm@10249
   602
   apply simp
wenzelm@10249
   603
  apply (simp (no_asm))
wenzelm@10249
   604
  done
wenzelm@10249
   605
wenzelm@10249
   606
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
wenzelm@10249
   607
  apply (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   608
  done
wenzelm@10249
   609
nipkow@11464
   610
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
wenzelm@10249
   611
  apply (erule size_eq_Suc_imp_elem [THEN exE])
wenzelm@10249
   612
  apply (drule elem_imp_eq_diff_union)
wenzelm@10249
   613
  apply auto
wenzelm@10249
   614
  done
wenzelm@10249
   615
wenzelm@10249
   616
lemma one_step_implies_mult_aux:
wenzelm@10249
   617
  "trans r ==>
nipkow@11464
   618
    \<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))
nipkow@11464
   619
      --> (I + K, I + J) \<in> mult r"
wenzelm@10249
   620
  apply (induct_tac n)
wenzelm@10249
   621
   apply auto
wenzelm@10249
   622
  apply (frule size_eq_Suc_imp_eq_union)
wenzelm@10249
   623
  apply clarify
wenzelm@10249
   624
  apply (rename_tac "J'")
wenzelm@10249
   625
  apply simp
wenzelm@10249
   626
  apply (erule notE)
wenzelm@10249
   627
   apply auto
wenzelm@10249
   628
  apply (case_tac "J' = {#}")
wenzelm@10249
   629
   apply (simp add: mult_def)
wenzelm@10249
   630
   apply (rule r_into_trancl)
wenzelm@10249
   631
   apply (simp add: mult1_def set_of_def)
wenzelm@10249
   632
   apply blast
nipkow@11464
   633
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@11464
   634
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@11464
   635
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
wenzelm@10249
   636
  apply (erule ssubst)
wenzelm@10249
   637
  apply (simp add: Ball_def)
wenzelm@10249
   638
  apply auto
wenzelm@10249
   639
  apply (subgoal_tac
nipkow@11464
   640
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
nipkow@11464
   641
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
wenzelm@10249
   642
   prefer 2
wenzelm@10249
   643
   apply force
wenzelm@10249
   644
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
wenzelm@10249
   645
  apply (erule trancl_trans)
wenzelm@10249
   646
  apply (rule r_into_trancl)
wenzelm@10249
   647
  apply (simp add: mult1_def set_of_def)
wenzelm@10249
   648
  apply (rule_tac x = a in exI)
wenzelm@10249
   649
  apply (rule_tac x = "I + J'" in exI)
wenzelm@10249
   650
  apply (simp add: union_ac)
wenzelm@10249
   651
  done
wenzelm@10249
   652
wenzelm@10249
   653
theorem one_step_implies_mult:
nipkow@11464
   654
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
nipkow@11464
   655
    ==> (I + K, I + J) \<in> mult r"
wenzelm@10249
   656
  apply (insert one_step_implies_mult_aux)
wenzelm@10249
   657
  apply blast
wenzelm@10249
   658
  done
wenzelm@10249
   659
wenzelm@10249
   660
wenzelm@10249
   661
subsubsection {* Partial-order properties *}
wenzelm@10249
   662
wenzelm@12338
   663
instance multiset :: (type) ord ..
wenzelm@10249
   664
wenzelm@10249
   665
defs (overloaded)
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   666
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
nipkow@11464
   667
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   668
wenzelm@10249
   669
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
wenzelm@10249
   670
  apply (unfold trans_def)
wenzelm@10249
   671
  apply (blast intro: order_less_trans)
wenzelm@10249
   672
  done
wenzelm@10249
   673
wenzelm@10249
   674
text {*
wenzelm@10249
   675
 \medskip Irreflexivity.
wenzelm@10249
   676
*}
wenzelm@10249
   677
wenzelm@10249
   678
lemma mult_irrefl_aux:
nipkow@11464
   679
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
wenzelm@10249
   680
  apply (erule finite_induct)
wenzelm@10249
   681
   apply (auto intro: order_less_trans)
wenzelm@10249
   682
  done
wenzelm@10249
   683
nipkow@11464
   684
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
wenzelm@10249
   685
  apply (unfold less_multiset_def)
wenzelm@10249
   686
  apply auto
wenzelm@10249
   687
  apply (drule trans_base_order [THEN mult_implies_one_step])
wenzelm@10249
   688
  apply auto
wenzelm@10249
   689
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
wenzelm@10249
   690
  apply (simp add: set_of_eq_empty_iff)
wenzelm@10249
   691
  done
wenzelm@10249
   692
wenzelm@10249
   693
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
wenzelm@10249
   694
  apply (insert mult_less_not_refl)
nipkow@13596
   695
  apply fast
wenzelm@10249
   696
  done
wenzelm@10249
   697
wenzelm@10249
   698
wenzelm@10249
   699
text {* Transitivity. *}
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   700
wenzelm@10249
   701
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
wenzelm@10249
   702
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   703
  apply (blast intro: trancl_trans)
wenzelm@10249
   704
  done
wenzelm@10249
   705
wenzelm@10249
   706
text {* Asymmetry. *}
wenzelm@10249
   707
nipkow@11464
   708
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
wenzelm@10249
   709
  apply auto
wenzelm@10249
   710
  apply (rule mult_less_not_refl [THEN notE])
wenzelm@10249
   711
  apply (erule mult_less_trans)
wenzelm@10249
   712
  apply assumption
wenzelm@10249
   713
  done
wenzelm@10249
   714
wenzelm@10249
   715
theorem mult_less_asym:
nipkow@11464
   716
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
wenzelm@10249
   717
  apply (insert mult_less_not_sym)
wenzelm@10249
   718
  apply blast
wenzelm@10249
   719
  done
wenzelm@10249
   720
wenzelm@10249
   721
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
wenzelm@10249
   722
  apply (unfold le_multiset_def)
wenzelm@10249
   723
  apply auto
wenzelm@10249
   724
  done
wenzelm@10249
   725
wenzelm@10249
   726
text {* Anti-symmetry. *}
wenzelm@10249
   727
wenzelm@10249
   728
theorem mult_le_antisym:
wenzelm@10249
   729
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
wenzelm@10249
   730
  apply (unfold le_multiset_def)
wenzelm@10249
   731
  apply (blast dest: mult_less_not_sym)
wenzelm@10249
   732
  done
wenzelm@10249
   733
wenzelm@10249
   734
text {* Transitivity. *}
wenzelm@10249
   735
wenzelm@10249
   736
theorem mult_le_trans:
wenzelm@10249
   737
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
wenzelm@10249
   738
  apply (unfold le_multiset_def)
wenzelm@10249
   739
  apply (blast intro: mult_less_trans)
wenzelm@10249
   740
  done
wenzelm@10249
   741
wenzelm@11655
   742
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
wenzelm@10249
   743
  apply (unfold le_multiset_def)
wenzelm@10249
   744
  apply auto
wenzelm@10249
   745
  done
wenzelm@10249
   746
wenzelm@10277
   747
text {* Partial order. *}
wenzelm@10277
   748
wenzelm@10277
   749
instance multiset :: (order) order
wenzelm@10277
   750
  apply intro_classes
wenzelm@10277
   751
     apply (rule mult_le_refl)
wenzelm@10277
   752
    apply (erule mult_le_trans)
wenzelm@10277
   753
    apply assumption
wenzelm@10277
   754
   apply (erule mult_le_antisym)
wenzelm@10277
   755
   apply assumption
wenzelm@10277
   756
  apply (rule mult_less_le)
wenzelm@10277
   757
  done
wenzelm@10277
   758
wenzelm@10249
   759
wenzelm@10249
   760
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   761
wenzelm@10249
   762
theorem mult1_union:
nipkow@11464
   763
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
wenzelm@10249
   764
  apply (unfold mult1_def)
wenzelm@10249
   765
  apply auto
wenzelm@10249
   766
  apply (rule_tac x = a in exI)
wenzelm@10249
   767
  apply (rule_tac x = "C + M0" in exI)
wenzelm@10249
   768
  apply (simp add: union_assoc)
wenzelm@10249
   769
  done
wenzelm@10249
   770
wenzelm@10249
   771
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
wenzelm@10249
   772
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   773
  apply (erule trancl_induct)
wenzelm@10249
   774
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
wenzelm@10249
   775
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
wenzelm@10249
   776
  done
wenzelm@10249
   777
wenzelm@10249
   778
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
wenzelm@10249
   779
  apply (subst union_commute [of B C])
wenzelm@10249
   780
  apply (subst union_commute [of D C])
wenzelm@10249
   781
  apply (erule union_less_mono2)
wenzelm@10249
   782
  done
wenzelm@10249
   783
wenzelm@10249
   784
theorem union_less_mono:
wenzelm@10249
   785
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
wenzelm@10249
   786
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   787
  done
wenzelm@10249
   788
wenzelm@10249
   789
theorem union_le_mono:
wenzelm@10249
   790
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
wenzelm@10249
   791
  apply (unfold le_multiset_def)
wenzelm@10249
   792
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   793
  done
wenzelm@10249
   794
wenzelm@10249
   795
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
wenzelm@10249
   796
  apply (unfold le_multiset_def less_multiset_def)
wenzelm@10249
   797
  apply (case_tac "M = {#}")
wenzelm@10249
   798
   prefer 2
nipkow@11464
   799
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
wenzelm@10249
   800
    prefer 2
wenzelm@10249
   801
    apply (rule one_step_implies_mult)
wenzelm@10249
   802
      apply (simp only: trans_def)
wenzelm@10249
   803
      apply auto
wenzelm@10249
   804
  apply (blast intro: order_less_trans)
wenzelm@10249
   805
  done
wenzelm@10249
   806
wenzelm@10249
   807
theorem union_upper1: "A <= A + (B::'a::order multiset)"
wenzelm@10249
   808
  apply (subgoal_tac "A + {#} <= A + B")
wenzelm@10249
   809
   prefer 2
wenzelm@10249
   810
   apply (rule union_le_mono)
wenzelm@10249
   811
    apply auto
wenzelm@10249
   812
  done
wenzelm@10249
   813
wenzelm@10249
   814
theorem union_upper2: "B <= A + (B::'a::order multiset)"
wenzelm@10249
   815
  apply (subst union_commute, rule union_upper1)
wenzelm@10249
   816
  done
wenzelm@10249
   817
wenzelm@10249
   818
end