src/HOL/Library/Multiset.thy
author huffman
Tue, 29 May 2012 17:06:04 +0200
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(*  Title:      HOL/Library/Multiset.thy
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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*)
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header {* (Finite) multisets *}
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theory Multiset
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imports Main DAList
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begin
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subsection {* The type of multisets *}
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definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
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typedef (open) 'a multiset = "multiset :: ('a => nat) set"
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  morphisms count Abs_multiset
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  unfolding multiset_def
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proof
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  show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
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qed
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setup_lifting type_definition_multiset
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abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
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  "a :# M == 0 < count M a"
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notation (xsymbols)
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  Melem (infix "\<in>#" 50)
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lemma multiset_eq_iff:
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  "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
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  by (simp only: count_inject [symmetric] fun_eq_iff)
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lemma multiset_eqI:
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  "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
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  using multiset_eq_iff by auto
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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:
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  "(\<lambda>a. 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma only1_in_multiset:
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  "(\<lambda>b. if b = a then n else 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma union_preserves_multiset:
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  "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
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  by (simp add: multiset_def)
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lemma diff_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>a. M a - N a) \<in> multiset"
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proof -
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  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemma filter_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
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proof -
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  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemmas in_multiset = const0_in_multiset only1_in_multiset
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  union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
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subsection {* Representing multisets *}
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text {* Multiset enumeration *}
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instantiation multiset :: (type) cancel_comm_monoid_add
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begin
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lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
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by (rule const0_in_multiset)
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abbreviation Mempty :: "'a multiset" ("{#}") where
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  "Mempty \<equiv> 0"
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lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
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by (rule union_preserves_multiset)
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instance
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by default (transfer, simp add: fun_eq_iff)+
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end
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lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
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by (rule only1_in_multiset)
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syntax
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  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
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translations
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  "{#x, xs#}" == "{#x#} + {#xs#}"
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  "{#x#}" == "CONST single x"
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lemma count_empty [simp]: "count {#} a = 0"
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  by (simp add: zero_multiset.rep_eq)
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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  by (simp add: single.rep_eq)
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subsection {* Basic operations *}
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subsubsection {* Union *}
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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  by (simp add: plus_multiset.rep_eq)
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subsubsection {* Difference *}
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instantiation multiset :: (type) minus
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begin
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lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
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by (rule diff_preserves_multiset)
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instance ..
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end
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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  by (simp add: minus_multiset.rep_eq)
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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by(simp add: multiset_eq_iff)
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lemma diff_cancel[simp]: "A - A = {#}"
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by (rule multiset_eqI) simp
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lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
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by(simp add: multiset_eq_iff)
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lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
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by(simp add: multiset_eq_iff)
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lemma insert_DiffM:
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  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
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  by (clarsimp simp: multiset_eq_iff)
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lemma insert_DiffM2 [simp]:
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  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
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  by (clarsimp simp: multiset_eq_iff)
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lemma diff_right_commute:
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  "(M::'a multiset) - N - Q = M - Q - N"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_add:
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  "(M::'a multiset) - (N + Q) = M - N - Q"
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by (simp add: multiset_eq_iff)
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lemma diff_union_swap:
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  "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_union_single_conv:
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  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
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  by (simp add: multiset_eq_iff)
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subsubsection {* Equality of multisets *}
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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  by (simp add: multiset_eq_iff)
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lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
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  by (auto simp add: multiset_eq_iff)
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lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
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  by (auto simp add: multiset_eq_iff)
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lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
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  by (auto simp add: multiset_eq_iff)
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lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
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  by (auto simp add: multiset_eq_iff)
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lemma diff_single_trivial:
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  "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
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   194
  by (auto simp add: multiset_eq_iff)
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lemma diff_single_eq_union:
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  "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
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  by auto
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lemma union_single_eq_diff:
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  "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
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  by (auto dest: sym)
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lemma union_single_eq_member:
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  "M + {#x#} = N \<Longrightarrow> x \<in># N"
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  by auto
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lemma union_is_single:
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  "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
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   210
proof
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  assume ?rhs then show ?lhs by auto
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   212
next
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   213
  assume ?lhs then show ?rhs
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    by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
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   215
qed
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lemma single_is_union:
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  "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
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   219
  by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
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lemma add_eq_conv_diff:
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   222
  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
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   223
(* shorter: by (simp add: multiset_eq_iff) fastforce *)
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   224
proof
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  assume ?rhs then show ?lhs
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   226
  by (auto simp add: add_assoc add_commute [of "{#b#}"])
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    (drule sym, simp add: add_assoc [symmetric])
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   228
next
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   229
  assume ?lhs
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   230
  show ?rhs
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   231
  proof (cases "a = b")
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   232
    case True with `?lhs` show ?thesis by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   233
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   234
    case False
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   235
    from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   236
    with False have "a \<in># N" by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   237
    moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   238
    moreover note False
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   239
    ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   240
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   241
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   242
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   243
lemma insert_noteq_member: 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   244
  assumes BC: "B + {#b#} = C + {#c#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   245
   and bnotc: "b \<noteq> c"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   246
  shows "c \<in># B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   247
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   248
  have "c \<in># C + {#c#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   249
  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   250
  then have "c \<in># B + {#b#}" using BC by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   251
  then show "c \<in># B" using nc by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   252
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   253
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   254
lemma add_eq_conv_ex:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   255
  "(M + {#a#} = N + {#b#}) =
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   256
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   257
  by (auto simp add: add_eq_conv_diff)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   258
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   259
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   260
subsubsection {* Pointwise ordering induced by count *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   261
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   262
instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   263
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   264
47429
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   265
lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)"
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   266
by simp
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   267
lemmas mset_le_def = less_eq_multiset_def
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   268
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   269
definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   270
  mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   271
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   272
instance
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   273
  by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   274
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   275
end
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   276
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   277
lemma mset_less_eqI:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   278
  "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   279
  by (simp add: mset_le_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   280
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   281
lemma mset_le_exists_conv:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   282
  "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   283
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   284
apply (auto intro: multiset_eq_iff [THEN iffD2])
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   285
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   286
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   287
lemma mset_le_mono_add_right_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   288
  "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   289
  by (fact add_le_cancel_right)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   290
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   291
lemma mset_le_mono_add_left_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   292
  "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   293
  by (fact add_le_cancel_left)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   294
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   295
lemma mset_le_mono_add:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   296
  "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   297
  by (fact add_mono)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   298
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   299
lemma mset_le_add_left [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   300
  "(A::'a multiset) \<le> A + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   301
  unfolding mset_le_def by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   302
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   303
lemma mset_le_add_right [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   304
  "B \<le> (A::'a multiset) + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   305
  unfolding mset_le_def by auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   306
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   307
lemma mset_le_single:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   308
  "a :# B \<Longrightarrow> {#a#} \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   309
  by (simp add: mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   310
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   311
lemma multiset_diff_union_assoc:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   312
  "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   313
  by (simp add: multiset_eq_iff mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   314
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   315
lemma mset_le_multiset_union_diff_commute:
36867
6c28c702ed22 simplified proof
nipkow
parents: 36635
diff changeset
   316
  "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   317
by (simp add: multiset_eq_iff mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   318
39301
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   319
lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   320
by(simp add: mset_le_def)
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   321
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   322
lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   323
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   324
apply (erule_tac x=x in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   325
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   326
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   327
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   328
lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   329
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   330
apply (erule_tac x = x in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   331
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   332
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   333
  
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   334
lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   335
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   336
 apply (simp add: mset_lessD)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   337
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   338
apply safe
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   339
 apply (erule_tac x = a in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   340
 apply (auto split: split_if_asm)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   341
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   342
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   343
lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   344
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   345
 apply (simp add: mset_leD)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   346
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   347
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   348
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   349
lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   350
  by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   351
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   352
lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   353
  by (auto simp: mset_le_def mset_less_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   354
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   355
lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   356
  by simp
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   357
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   358
lemma mset_less_add_bothsides:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   359
  "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   360
  by (fact add_less_imp_less_right)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   361
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   362
lemma mset_less_empty_nonempty:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   363
  "{#} < S \<longleftrightarrow> S \<noteq> {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   364
  by (auto simp: mset_le_def mset_less_def)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   365
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   366
lemma mset_less_diff_self:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   367
  "c \<in># B \<Longrightarrow> B - {#c#} < B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   368
  by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   369
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   370
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   371
subsubsection {* Intersection *}
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   372
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   373
instantiation multiset :: (type) semilattice_inf
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   374
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   375
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   376
definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   377
  multiset_inter_def: "inf_multiset A B = A - (A - B)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   378
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   379
instance
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   380
proof -
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   381
  have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   382
  show "OFCLASS('a multiset, semilattice_inf_class)"
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   383
    by default (auto simp add: multiset_inter_def mset_le_def aux)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   384
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   385
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   386
end
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   387
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   388
abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   389
  "multiset_inter \<equiv> inf"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   390
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   391
lemma multiset_inter_count [simp]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   392
  "count (A #\<inter> B) x = min (count A x) (count B x)"
47429
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   393
  by (simp add: multiset_inter_def)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   394
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   395
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   396
  by (rule multiset_eqI) auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   397
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   398
lemma multiset_union_diff_commute:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   399
  assumes "B #\<inter> C = {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   400
  shows "A + B - C = A - C + B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   401
proof (rule multiset_eqI)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   402
  fix x
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   403
  from assms have "min (count B x) (count C x) = 0"
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   404
    by (auto simp add: multiset_eq_iff)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   405
  then have "count B x = 0 \<or> count C x = 0"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   406
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   407
  then show "count (A + B - C) x = count (A - C + B) x"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   408
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   409
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   410
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   411
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   412
subsubsection {* Filter (with comprehension syntax) *}
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   413
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   414
text {* Multiset comprehension *}
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   415
47429
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   416
lift_definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   417
by (rule filter_preserves_multiset)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   418
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   419
hide_const (open) filter
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   420
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   421
lemma count_filter [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   422
  "count (Multiset.filter P M) a = (if P a then count M a else 0)"
47429
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
   423
  by (simp add: filter.rep_eq)
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   424
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   425
lemma filter_empty [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   426
  "Multiset.filter P {#} = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   427
  by (rule multiset_eqI) simp
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   428
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   429
lemma filter_single [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   430
  "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   431
  by (rule multiset_eqI) simp
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   432
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   433
lemma filter_union [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   434
  "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   435
  by (rule multiset_eqI) simp
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   436
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   437
lemma filter_diff [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   438
  "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   439
  by (rule multiset_eqI) simp
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   440
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   441
lemma filter_inter [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   442
  "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   443
  by (rule multiset_eqI) simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   444
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   445
syntax
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   446
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   447
syntax (xsymbol)
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   448
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   449
translations
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   450
  "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   451
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   452
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   453
subsubsection {* Set of elements *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   454
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   455
definition set_of :: "'a multiset => 'a set" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   456
  "set_of M = {x. x :# M}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   457
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   458
lemma set_of_empty [simp]: "set_of {#} = {}"
26178
nipkow
parents: 26176
diff changeset
   459
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   460
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   461
lemma set_of_single [simp]: "set_of {#b#} = {b}"
26178
nipkow
parents: 26176
diff changeset
   462
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   463
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   464
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
26178
nipkow
parents: 26176
diff changeset
   465
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   466
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   467
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   468
by (auto simp add: set_of_def multiset_eq_iff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   469
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   470
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
26178
nipkow
parents: 26176
diff changeset
   471
by (auto simp add: set_of_def)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   472
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   473
lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
26178
nipkow
parents: 26176
diff changeset
   474
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   475
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   476
lemma finite_set_of [iff]: "finite (set_of M)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   477
  using count [of M] by (simp add: multiset_def set_of_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   478
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46730
diff changeset
   479
lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46730
diff changeset
   480
  unfolding set_of_def[symmetric] by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   481
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   482
subsubsection {* Size *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   483
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   484
instantiation multiset :: (type) size
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   485
begin
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   486
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   487
definition size_def:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   488
  "size M = setsum (count M) (set_of M)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   489
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   490
instance ..
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   491
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   492
end
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   493
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   494
lemma size_empty [simp]: "size {#} = 0"
26178
nipkow
parents: 26176
diff changeset
   495
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   496
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   497
lemma size_single [simp]: "size {#b#} = 1"
26178
nipkow
parents: 26176
diff changeset
   498
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   499
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   500
lemma setsum_count_Int:
26178
nipkow
parents: 26176
diff changeset
   501
  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow
parents: 26176
diff changeset
   502
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   503
 apply simp
nipkow
parents: 26176
diff changeset
   504
apply (simp add: Int_insert_left set_of_def)
nipkow
parents: 26176
diff changeset
   505
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   506
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   507
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
26178
nipkow
parents: 26176
diff changeset
   508
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   509
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow
parents: 26176
diff changeset
   510
 prefer 2
nipkow
parents: 26176
diff changeset
   511
 apply (rule ext, simp)
nipkow
parents: 26176
diff changeset
   512
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow
parents: 26176
diff changeset
   513
apply (subst Int_commute)
nipkow
parents: 26176
diff changeset
   514
apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow
parents: 26176
diff changeset
   515
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   516
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   517
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   518
by (auto simp add: size_def multiset_eq_iff)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   519
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   520
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
26178
nipkow
parents: 26176
diff changeset
   521
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   522
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   523
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
26178
nipkow
parents: 26176
diff changeset
   524
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   525
apply (drule setsum_SucD)
nipkow
parents: 26176
diff changeset
   526
apply auto
nipkow
parents: 26176
diff changeset
   527
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   528
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   529
lemma size_eq_Suc_imp_eq_union:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   530
  assumes "size M = Suc n"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   531
  shows "\<exists>a N. M = N + {#a#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   532
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   533
  from assms obtain a where "a \<in># M"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   534
    by (erule size_eq_Suc_imp_elem [THEN exE])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   535
  then have "M = M - {#a#} + {#a#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   536
  then show ?thesis by blast
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   537
qed
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   538
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   539
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   540
subsection {* Induction and case splits *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   541
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   542
theorem multiset_induct [case_names empty add, induct type: multiset]:
48009
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   543
  assumes empty: "P {#}"
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   544
  assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   545
  shows "P M"
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   546
proof (induct n \<equiv> "size M" arbitrary: M)
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   547
  case 0 thus "P M" by (simp add: empty)
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   548
next
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   549
  case (Suc k)
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   550
  obtain N x where "M = N + {#x#}"
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   551
    using `Suc k = size M` [symmetric]
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   552
    using size_eq_Suc_imp_eq_union by fast
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   553
  with Suc add show "P M" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   554
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   555
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   556
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   557
by (induct M) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   558
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   559
lemma multiset_cases [cases type, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
   560
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   561
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   562
shows "P"
48009
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   563
using assms by (induct M) simp_all
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   564
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   565
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
48009
9b9150033b5a shortened some proofs
huffman
parents: 48008
diff changeset
   566
by (rule_tac x="M - {#x#}" in exI, simp)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   567
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   568
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   569
by (cases "B = {#}") (auto dest: multi_member_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   570
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   571
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   572
apply (subst multiset_eq_iff)
26178
nipkow
parents: 26176
diff changeset
   573
apply auto
nipkow
parents: 26176
diff changeset
   574
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   575
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   576
lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   577
proof (induct A arbitrary: B)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   578
  case (empty M)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   579
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   580
  then obtain M' x where "M = M' + {#x#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   581
    by (blast dest: multi_nonempty_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   582
  then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   583
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   584
  case (add S x T)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   585
  have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   586
  have SxsubT: "S + {#x#} < T" by fact
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   587
  then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   588
  then obtain T' where T: "T = T' + {#x#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   589
    by (blast dest: multi_member_split)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   590
  then have "S < T'" using SxsubT 
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   591
    by (blast intro: mset_less_add_bothsides)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   592
  then have "size S < size T'" using IH by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   593
  then show ?case using T by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   594
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   595
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   596
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   597
subsubsection {* Strong induction and subset induction for multisets *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   598
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   599
text {* Well-foundedness of proper subset operator: *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   600
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   601
text {* proper multiset subset *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   602
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   603
definition
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   604
  mset_less_rel :: "('a multiset * 'a multiset) set" where
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   605
  "mset_less_rel = {(A,B). A < B}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   606
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   607
lemma multiset_add_sub_el_shuffle: 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   608
  assumes "c \<in># B" and "b \<noteq> c" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   609
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   610
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   611
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   612
    by (blast dest: multi_member_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   613
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   614
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   615
    by (simp add: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   616
  then show ?thesis using B by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   617
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   618
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   619
lemma wf_mset_less_rel: "wf mset_less_rel"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   620
apply (unfold mset_less_rel_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   621
apply (rule wf_measure [THEN wf_subset, where f1=size])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   622
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   623
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   624
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   625
text {* The induction rules: *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   626
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   627
lemma full_multiset_induct [case_names less]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   628
assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   629
shows "P B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   630
apply (rule wf_mset_less_rel [THEN wf_induct])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   631
apply (rule ih, auto simp: mset_less_rel_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   632
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   633
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   634
lemma multi_subset_induct [consumes 2, case_names empty add]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   635
assumes "F \<le> A"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   636
  and empty: "P {#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   637
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   638
shows "P F"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   639
proof -
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   640
  from `F \<le> A`
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   641
  show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   642
  proof (induct F)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   643
    show "P {#}" by fact
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   644
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   645
    fix x F
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   646
    assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   647
    show "P (F + {#x#})"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   648
    proof (rule insert)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   649
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   650
      from i have "F \<le> A" by (auto dest: mset_le_insertD)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   651
      with P show "P F" .
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   652
    qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   653
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   654
qed
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   655
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   656
48023
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   657
subsection {* The fold combinator *}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   658
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   659
text {*
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   660
  The intended behaviour is
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   661
  @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   662
  if @{text f} is associative-commutative. 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   663
*}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   664
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   665
text {*
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   666
  The graph of @{text "fold_mset"}, @{text "z"}: the start element,
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   667
  @{text "f"}: folding function, @{text "A"}: the multiset, @{text
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   668
  "y"}: the result.
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   669
*}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   670
inductive 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   671
  fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   672
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   673
  and z :: 'b
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   674
where
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   675
  emptyI [intro]:  "fold_msetG f z {#} z"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   676
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   677
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   678
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   679
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   680
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   681
definition
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   682
  fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   683
  "fold_mset f z A = (THE x. fold_msetG f z A x)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   684
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   685
lemma Diff1_fold_msetG:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   686
  "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   687
apply (frule_tac x = x in fold_msetG.insertI)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   688
apply auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   689
done
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   690
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   691
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   692
apply (induct A)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   693
 apply blast
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   694
apply clarsimp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   695
apply (drule_tac x = x in fold_msetG.insertI)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   696
apply auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   697
done
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   698
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   699
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   700
unfolding fold_mset_def by blast
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   701
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   702
context comp_fun_commute
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   703
begin
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   704
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   705
lemma fold_msetG_insertE_aux:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   706
  "fold_msetG f z A y \<Longrightarrow> a \<in># A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_msetG f z (A - {#a#}) y'"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   707
proof (induct set: fold_msetG)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   708
  case (insertI A y x) show ?case
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   709
  proof (cases "x = a")
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   710
    assume "x = a" with insertI show ?case by auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   711
  next
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   712
    assume "x \<noteq> a"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   713
    then obtain y' where y: "y = f a y'" and y': "fold_msetG f z (A - {#a#}) y'"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   714
      using insertI by auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   715
    have "f x y = f a (f x y')"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   716
      unfolding y by (rule fun_left_comm)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   717
    moreover have "fold_msetG f z (A + {#x#} - {#a#}) (f x y')"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   718
      using y' and `x \<noteq> a`
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   719
      by (simp add: diff_union_swap [symmetric] fold_msetG.insertI)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   720
    ultimately show ?case by fast
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   721
  qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   722
qed simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   723
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   724
lemma fold_msetG_insertE:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   725
  assumes "fold_msetG f z (A + {#x#}) v"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   726
  obtains y where "v = f x y" and "fold_msetG f z A y"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   727
using assms by (auto dest: fold_msetG_insertE_aux [where a=x])
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   728
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   729
lemma fold_msetG_determ:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   730
  "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   731
proof (induct arbitrary: y set: fold_msetG)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   732
  case (insertI A y x v)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   733
  from `fold_msetG f z (A + {#x#}) v`
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   734
  obtain y' where "v = f x y'" and "fold_msetG f z A y'"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   735
    by (rule fold_msetG_insertE)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   736
  from `fold_msetG f z A y'` have "y' = y" by (rule insertI)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   737
  with `v = f x y'` show "v = f x y" by simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   738
qed fast
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   739
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   740
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   741
unfolding fold_mset_def by (blast intro: fold_msetG_determ)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   742
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   743
lemma fold_msetG_fold_mset: "fold_msetG f z A (fold_mset f z A)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   744
proof -
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   745
  from fold_msetG_nonempty fold_msetG_determ
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   746
  have "\<exists>!x. fold_msetG f z A x" by (rule ex_ex1I)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   747
  then show ?thesis unfolding fold_mset_def by (rule theI')
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   748
qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   749
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   750
lemma fold_mset_insert:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   751
  "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   752
by (intro fold_mset_equality fold_msetG.insertI fold_msetG_fold_mset)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   753
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   754
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   755
by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x])
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   756
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   757
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   758
using fold_mset_insert [of z "{#}"] by simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   759
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   760
lemma fold_mset_union [simp]:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   761
  "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   762
proof (induct A)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   763
  case empty then show ?case by simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   764
next
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   765
  case (add A x)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   766
  have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   767
  then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   768
    by (simp add: fold_mset_insert)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   769
  also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   770
    by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   771
  finally show ?case .
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   772
qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   773
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   774
lemma fold_mset_fusion:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   775
  assumes "comp_fun_commute g"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   776
  shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   777
proof -
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   778
  interpret comp_fun_commute g by (fact assms)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   779
  show "PROP ?P" by (induct A) auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   780
qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   781
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   782
lemma fold_mset_rec:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   783
  assumes "a \<in># A" 
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   784
  shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   785
proof -
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   786
  from assms obtain A' where "A = A' + {#a#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   787
    by (blast dest: multi_member_split)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   788
  then show ?thesis by simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   789
qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   790
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   791
end
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   792
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   793
text {*
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   794
  A note on code generation: When defining some function containing a
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   795
  subterm @{term"fold_mset F"}, code generation is not automatic. When
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   796
  interpreting locale @{text left_commutative} with @{text F}, the
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   797
  would be code thms for @{const fold_mset} become thms like
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   798
  @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   799
  contains defined symbols, i.e.\ is not a code thm. Hence a separate
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   800
  constant with its own code thms needs to be introduced for @{text
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   801
  F}. See the image operator below.
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   802
*}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   803
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   804
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   805
subsection {* Image *}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   806
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   807
definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   808
  "image_mset f = fold_mset (op + o single o f) {#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   809
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   810
interpretation image_fun_commute: comp_fun_commute "op + o single o f" for f
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   811
proof qed (simp add: add_ac fun_eq_iff)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   812
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   813
lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   814
by (simp add: image_mset_def)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   815
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   816
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   817
by (simp add: image_mset_def)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   818
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   819
lemma image_mset_insert:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   820
  "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   821
by (simp add: image_mset_def add_ac)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   822
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   823
lemma image_mset_union [simp]:
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   824
  "image_mset f (M+N) = image_mset f M + image_mset f N"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   825
apply (induct N)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   826
 apply simp
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   827
apply (simp add: add_assoc [symmetric] image_mset_insert)
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   828
done
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   829
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   830
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   831
by (induct M) simp_all
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   832
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   833
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   834
by (cases M) auto
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   835
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   836
syntax
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   837
  "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   838
      ("({#_/. _ :# _#})")
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   839
translations
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   840
  "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   841
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   842
syntax
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   843
  "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   844
      ("({#_/ | _ :# _./ _#})")
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   845
translations
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   846
  "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   847
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   848
text {*
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   849
  This allows to write not just filters like @{term "{#x:#M. x<c#}"}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   850
  but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   851
  "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   852
  @{term "{#x+x|x:#M. x<c#}"}.
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   853
*}
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   854
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   855
enriched_type image_mset: image_mset
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   856
proof -
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   857
  fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   858
  proof
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   859
    fix A
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   860
    show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   861
      by (induct A) simp_all
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   862
  qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   863
  show "image_mset id = id"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   864
  proof
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   865
    fix A
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   866
    show "image_mset id A = id A"
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   867
      by (induct A) simp_all
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   868
  qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   869
qed
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   870
6dfe5e774012 reordered sections
huffman
parents: 48012
diff changeset
   871
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   872
subsection {* Alternative representations *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   873
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   874
subsubsection {* Lists *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   875
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   876
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   877
  "multiset_of [] = {#}" |
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   878
  "multiset_of (a # x) = multiset_of x + {# a #}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   879
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   880
lemma in_multiset_in_set:
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   881
  "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   882
  by (induct xs) simp_all
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   883
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   884
lemma count_multiset_of:
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   885
  "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   886
  by (induct xs) simp_all
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   887
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   888
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   889
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   890
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   891
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   892
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   893
40950
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   894
lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   895
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   896
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   897
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   898
by (induct xs) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   899
48012
b6e5e86a7303 shortened yet more multiset proofs;
huffman
parents: 48011
diff changeset
   900
lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
b6e5e86a7303 shortened yet more multiset proofs;
huffman
parents: 48011
diff changeset
   901
  by (induct xs) simp_all
b6e5e86a7303 shortened yet more multiset proofs;
huffman
parents: 48011
diff changeset
   902
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   903
lemma multiset_of_append [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   904
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   905
  by (induct xs arbitrary: ys) (auto simp: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   906
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   907
lemma multiset_of_filter:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   908
  "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   909
  by (induct xs) simp_all
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   910
40950
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   911
lemma multiset_of_rev [simp]:
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   912
  "multiset_of (rev xs) = multiset_of xs"
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   913
  by (induct xs) simp_all
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   914
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   915
lemma surj_multiset_of: "surj multiset_of"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   916
apply (unfold surj_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   917
apply (rule allI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   918
apply (rule_tac M = y in multiset_induct)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   919
 apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   920
apply (rule_tac x = "x # xa" in exI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   921
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   922
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   923
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   924
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   925
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   926
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   927
lemma distinct_count_atmost_1:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   928
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   929
apply (induct x, simp, rule iffI, simp_all)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   930
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   931
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   932
apply (erule_tac x = a in allE, simp, clarify)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   933
apply (erule_tac x = aa in allE, simp)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   934
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   935
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   936
lemma multiset_of_eq_setD:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   937
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   938
by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   939
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   940
lemma set_eq_iff_multiset_of_eq_distinct:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   941
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   942
    (set x = set y) = (multiset_of x = multiset_of y)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   943
by (auto simp: multiset_eq_iff distinct_count_atmost_1)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   944
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   945
lemma set_eq_iff_multiset_of_remdups_eq:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   946
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   947
apply (rule iffI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   948
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   949
apply (drule distinct_remdups [THEN distinct_remdups
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   950
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   951
apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   952
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   953
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   954
lemma multiset_of_compl_union [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   955
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   956
  by (induct xs) (auto simp: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   957
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   958
lemma count_multiset_of_length_filter:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   959
  "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   960
  by (induct xs) auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   961
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   962
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   963
apply (induct ls arbitrary: i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   964
 apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   965
apply (case_tac i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   966
 apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   967
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   968
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   969
lemma multiset_of_remove1[simp]:
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   970
  "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   971
by (induct xs) (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   972
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   973
lemma multiset_of_eq_length:
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   974
  assumes "multiset_of xs = multiset_of ys"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   975
  shows "length xs = length ys"
48012
b6e5e86a7303 shortened yet more multiset proofs;
huffman
parents: 48011
diff changeset
   976
  using assms by (metis size_multiset_of)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   977
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   978
lemma multiset_of_eq_length_filter:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   979
  assumes "multiset_of xs = multiset_of ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   980
  shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
48012
b6e5e86a7303 shortened yet more multiset proofs;
huffman
parents: 48011
diff changeset
   981
  using assms by (metis count_multiset_of)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   982
45989
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   983
lemma fold_multiset_equiv:
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   984
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   985
    and equiv: "multiset_of xs = multiset_of ys"
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   986
  shows "fold f xs = fold f ys"
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   987
using f equiv [symmetric]
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   988
proof (induct xs arbitrary: ys)
45989
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   989
  case Nil then show ?case by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   990
next
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   991
  case (Cons x xs)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   992
  then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   993
  have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" 
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   994
    by (rule Cons.prems(1)) (simp_all add: *)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   995
  moreover from * have "x \<in> set ys" by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   996
  ultimately have "fold f ys = fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   997
  moreover from Cons.prems have "fold f xs = fold f (remove1 x ys)" by (auto intro: Cons.hyps)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   998
  ultimately show ?case by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   999
qed
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
  1000
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1001
context linorder
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1002
begin
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1003
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 39533
diff changeset
  1004
lemma multiset_of_insort [simp]:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1005
  "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
  1006
  by (induct xs) (simp_all add: ac_simps)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1007
 
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 39533
diff changeset
  1008
lemma multiset_of_sort [simp]:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1009
  "multiset_of (sort_key k xs) = multiset_of xs"
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
  1010
  by (induct xs) (simp_all add: ac_simps)
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
  1011
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1012
text {*
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1013
  This lemma shows which properties suffice to show that a function
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1014
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1015
*}
37074
322d065ebef7 localized properties_for_sort
haftmann
parents: 36903
diff changeset
  1016
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1017
lemma properties_for_sort_key:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1018
  assumes "multiset_of ys = multiset_of xs"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1019
  and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1020
  and "sorted (map f ys)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1021
  shows "sort_key f xs = ys"
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1022
using assms
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1023
proof (induct xs arbitrary: ys)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1024
  case Nil then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1025
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1026
  case (Cons x xs)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1027
  from Cons.prems(2) have
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1028
    "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1029
    by (simp add: filter_remove1)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1030
  with Cons.prems have "sort_key f xs = remove1 x ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1031
    by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1032
  moreover from Cons.prems have "x \<in> set ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1033
    by (auto simp add: mem_set_multiset_eq intro!: ccontr)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1034
  ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1035
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1036
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1037
lemma properties_for_sort:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1038
  assumes multiset: "multiset_of ys = multiset_of xs"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1039
  and "sorted ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1040
  shows "sort xs = ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1041
proof (rule properties_for_sort_key)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1042
  from multiset show "multiset_of ys = multiset_of xs" .
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1043
  from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1044
  from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1045
    by (rule multiset_of_eq_length_filter)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1046
  then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1047
    by simp
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1048
  then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1049
    by (simp add: replicate_length_filter)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1050
qed
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1051
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1052
lemma sort_key_by_quicksort:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1053
  "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1054
    @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1055
    @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1056
proof (rule properties_for_sort_key)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1057
  show "multiset_of ?rhs = multiset_of ?lhs"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1058
    by (rule multiset_eqI) (auto simp add: multiset_of_filter)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1059
next
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1060
  show "sorted (map f ?rhs)"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1061
    by (auto simp add: sorted_append intro: sorted_map_same)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1062
next
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1063
  fix l
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1064
  assume "l \<in> set ?rhs"
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1065
  let ?pivot = "f (xs ! (length xs div 2))"
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1066
  have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
  1067
  have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1068
    unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1069
  with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1070
  have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1071
  then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1072
    [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1073
  note *** = this [of "op <"] this [of "op >"] this [of "op ="]
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
  1074
  show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1075
  proof (cases "f l" ?pivot rule: linorder_cases)
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1076
    case less
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1077
    then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1078
    with less show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1079
      by (simp add: filter_sort [symmetric] ** ***)
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1080
  next
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
  1081
    case equal then show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1082
      by (simp add: * less_le)
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
  1083
  next
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1084
    case greater
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1085
    then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
  1086
    with greater show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
  1087
      by (simp add: filter_sort [symmetric] ** ***)
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
  1088
  qed
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1089
qed
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1090
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1091
lemma sort_by_quicksort:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1092
  "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1093
    @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1094
    @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1095
  using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
  1096
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1097
text {* A stable parametrized quicksort *}
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1098
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1099
definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1100
  "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1101
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1102
lemma part_code [code]:
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1103
  "part f pivot [] = ([], [], [])"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1104
  "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1105
     if x' < pivot then (x # lts, eqs, gts)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1106
     else if x' > pivot then (lts, eqs, x # gts)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1107
     else (lts, x # eqs, gts))"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1108
  by (auto simp add: part_def Let_def split_def)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1109
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1110
lemma sort_key_by_quicksort_code [code]:
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1111
  "sort_key f xs = (case xs of [] \<Rightarrow> []
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1112
    | [x] \<Rightarrow> xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1113
    | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1114
    | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1115
       in sort_key f lts @ eqs @ sort_key f gts))"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1116
proof (cases xs)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1117
  case Nil then show ?thesis by simp
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1118
next
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1119
  case (Cons _ ys) note hyps = Cons show ?thesis
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1120
  proof (cases ys)
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1121
    case Nil with hyps show ?thesis by simp
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1122
  next
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1123
    case (Cons _ zs) note hyps = hyps Cons show ?thesis
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1124
    proof (cases zs)
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1125
      case Nil with hyps show ?thesis by auto
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1126
    next
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1127
      case Cons 
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1128
      from sort_key_by_quicksort [of f xs]
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1129
      have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1130
        in sort_key f lts @ eqs @ sort_key f gts)"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1131
      by (simp only: split_def Let_def part_def fst_conv snd_conv)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1132
      with hyps Cons show ?thesis by (simp only: list.cases)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1133
    qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1134
  qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1135
qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1136
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1137
end
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1138
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1139
hide_const (open) part
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1140
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
  1141
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
  1142
  by (induct xs) (auto intro: order_trans)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1143
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1144
lemma multiset_of_update:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1145
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1146
proof (induct ls arbitrary: i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1147
  case Nil then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1148
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1149
  case (Cons x xs)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1150
  show ?case
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1151
  proof (cases i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1152
    case 0 then show ?thesis by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1153
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1154
    case (Suc i')
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1155
    with Cons show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1156
      apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1157
      apply (subst add_assoc)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1158
      apply (subst add_commute [of "{#v#}" "{#x#}"])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1159
      apply (subst add_assoc [symmetric])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1160
      apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1161
      apply (rule mset_le_multiset_union_diff_commute)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1162
      apply (simp add: mset_le_single nth_mem_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1163
      done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1164
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1165
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1166
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1167
lemma multiset_of_swap:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1168
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1169
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1170
  by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1171
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1172
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1173
subsubsection {* Association lists -- including code generation *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1174
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1175
text {* Preliminaries *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1176
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1177
text {* Raw operations on lists *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1178
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1179
definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1180
where
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1181
  "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1182
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1183
lemma join_raw_Nil [simp]:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1184
  "join_raw f xs [] = xs"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1185
by (simp add: join_raw_def)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1186
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1187
lemma join_raw_Cons [simp]:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1188
  "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1189
by (simp add: join_raw_def)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1190
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1191
lemma map_of_join_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1192
  assumes "distinct (map fst ys)"
47429
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
  1193
  shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
ec64d94cbf9c multiset operations are defined with lift_definitions;
bulwahn
parents: 47308
diff changeset
  1194
    (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1195
using assms
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1196
apply (induct ys)