wenzelm@10249: (*  Title:      HOL/Library/Multiset.thy
paulson@15072:     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
wenzelm@10249: *)
wenzelm@10249: 
haftmann@34943: header {* (Finite) multisets *}
wenzelm@10249: 
nipkow@15131: theory Multiset
bulwahn@46237: imports Main DAList
nipkow@15131: begin
wenzelm@10249: 
wenzelm@10249: subsection {* The type of multisets *}
wenzelm@10249: 
wenzelm@45694: definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
wenzelm@45694: 
wenzelm@45694: typedef (open) 'a multiset = "multiset :: ('a => nat) set"
haftmann@34943:   morphisms count Abs_multiset
wenzelm@45694:   unfolding multiset_def
wenzelm@10249: proof
wenzelm@45694:   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
wenzelm@10249: qed
wenzelm@10249: 
bulwahn@47429: setup_lifting type_definition_multiset
wenzelm@19086: 
haftmann@28708: abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
kleing@25610:   "a :# M == 0 < count M a"
kleing@25610: 
wenzelm@26145: notation (xsymbols)
wenzelm@26145:   Melem (infix "\<in>#" 50)
wenzelm@10249: 
nipkow@39302: lemma multiset_eq_iff:
haftmann@34943:   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
nipkow@39302:   by (simp only: count_inject [symmetric] fun_eq_iff)
haftmann@34943: 
nipkow@39302: lemma multiset_eqI:
haftmann@34943:   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
nipkow@39302:   using multiset_eq_iff by auto
haftmann@34943: 
haftmann@34943: text {*
haftmann@34943:  \medskip Preservation of the representing set @{term multiset}.
haftmann@34943: *}
haftmann@34943: 
haftmann@34943: lemma const0_in_multiset:
haftmann@34943:   "(\<lambda>a. 0) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
haftmann@34943: lemma only1_in_multiset:
haftmann@34943:   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
haftmann@34943: lemma union_preserves_multiset:
haftmann@34943:   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
haftmann@34943: lemma diff_preserves_multiset:
haftmann@34943:   assumes "M \<in> multiset"
haftmann@34943:   shows "(\<lambda>a. M a - N a) \<in> multiset"
haftmann@34943: proof -
haftmann@34943:   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
haftmann@34943:     by auto
haftmann@34943:   with assms show ?thesis
haftmann@34943:     by (auto simp add: multiset_def intro: finite_subset)
haftmann@34943: qed
haftmann@34943: 
haftmann@41069: lemma filter_preserves_multiset:
haftmann@34943:   assumes "M \<in> multiset"
haftmann@34943:   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
haftmann@34943: proof -
haftmann@34943:   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
haftmann@34943:     by auto
haftmann@34943:   with assms show ?thesis
haftmann@34943:     by (auto simp add: multiset_def intro: finite_subset)
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemmas in_multiset = const0_in_multiset only1_in_multiset
haftmann@41069:   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
haftmann@34943: 
haftmann@34943: 
haftmann@34943: subsection {* Representing multisets *}
haftmann@34943: 
haftmann@34943: text {* Multiset enumeration *}
haftmann@34943: 
huffman@48008: instantiation multiset :: (type) cancel_comm_monoid_add
haftmann@25571: begin
haftmann@25571: 
bulwahn@47429: lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
bulwahn@47429: by (rule const0_in_multiset)
haftmann@25571: 
haftmann@34943: abbreviation Mempty :: "'a multiset" ("{#}") where
haftmann@34943:   "Mempty \<equiv> 0"
haftmann@25571: 
bulwahn@47429: lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
bulwahn@47429: by (rule union_preserves_multiset)
haftmann@25571: 
huffman@48008: instance
huffman@48008: by default (transfer, simp add: fun_eq_iff)+
haftmann@25571: 
haftmann@25571: end
wenzelm@10249: 
bulwahn@47429: lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
bulwahn@47429: by (rule only1_in_multiset)
kleing@15869: 
wenzelm@26145: syntax
wenzelm@26176:   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
nipkow@25507: translations
nipkow@25507:   "{#x, xs#}" == "{#x#} + {#xs#}"
nipkow@25507:   "{#x#}" == "CONST single x"
nipkow@25507: 
haftmann@34943: lemma count_empty [simp]: "count {#} a = 0"
bulwahn@47429:   by (simp add: zero_multiset.rep_eq)
wenzelm@10249: 
haftmann@34943: lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
bulwahn@47429:   by (simp add: single.rep_eq)
nipkow@29901: 
wenzelm@10249: 
haftmann@34943: subsection {* Basic operations *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Union *}
wenzelm@10249: 
haftmann@34943: lemma count_union [simp]: "count (M + N) a = count M a + count N a"
bulwahn@47429:   by (simp add: plus_multiset.rep_eq)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Difference *}
wenzelm@10249: 
haftmann@49388: instantiation multiset :: (type) comm_monoid_diff
haftmann@34943: begin
haftmann@34943: 
bulwahn@47429: lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
bulwahn@47429: by (rule diff_preserves_multiset)
bulwahn@47429:  
haftmann@49388: instance
haftmann@49388: by default (transfer, simp add: fun_eq_iff)+
haftmann@34943: 
haftmann@34943: end
haftmann@34943: 
haftmann@34943: lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
bulwahn@47429:   by (simp add: minus_multiset.rep_eq)
haftmann@34943: 
wenzelm@17161: lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
nipkow@39302: by(simp add: multiset_eq_iff)
nipkow@36903: 
nipkow@36903: lemma diff_cancel[simp]: "A - A = {#}"
nipkow@39302: by (rule multiset_eqI) simp
wenzelm@10249: 
nipkow@36903: lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
nipkow@39302: by(simp add: multiset_eq_iff)
wenzelm@10249: 
nipkow@36903: lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
nipkow@39302: by(simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma insert_DiffM:
haftmann@34943:   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
nipkow@39302:   by (clarsimp simp: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma insert_DiffM2 [simp]:
haftmann@34943:   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
nipkow@39302:   by (clarsimp simp: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma diff_right_commute:
haftmann@34943:   "(M::'a multiset) - N - Q = M - Q - N"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
nipkow@36903: 
nipkow@36903: lemma diff_add:
nipkow@36903:   "(M::'a multiset) - (N + Q) = M - N - Q"
nipkow@39302: by (simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma diff_union_swap:
haftmann@34943:   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma diff_union_single_conv:
haftmann@34943:   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
nipkow@39302:   by (simp add: multiset_eq_iff)
bulwahn@26143: 
wenzelm@10249: 
haftmann@34943: subsubsection {* Equality of multisets *}
haftmann@34943: 
haftmann@34943: lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
nipkow@39302:   by (simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma diff_single_trivial:
haftmann@34943:   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma diff_single_eq_union:
haftmann@34943:   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
haftmann@34943:   by auto
haftmann@34943: 
haftmann@34943: lemma union_single_eq_diff:
haftmann@34943:   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
haftmann@34943:   by (auto dest: sym)
haftmann@34943: 
haftmann@34943: lemma union_single_eq_member:
haftmann@34943:   "M + {#x#} = N \<Longrightarrow> x \<in># N"
haftmann@34943:   by auto
haftmann@34943: 
haftmann@34943: lemma union_is_single:
wenzelm@46730:   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
wenzelm@46730: proof
haftmann@34943:   assume ?rhs then show ?lhs by auto
haftmann@34943: next
wenzelm@46730:   assume ?lhs then show ?rhs
wenzelm@46730:     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma single_is_union:
haftmann@34943:   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
haftmann@34943:   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
haftmann@34943: 
haftmann@34943: lemma add_eq_conv_diff:
haftmann@34943:   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
nipkow@44890: (* shorter: by (simp add: multiset_eq_iff) fastforce *)
haftmann@34943: proof
haftmann@34943:   assume ?rhs then show ?lhs
haftmann@34943:   by (auto simp add: add_assoc add_commute [of "{#b#}"])
haftmann@34943:     (drule sym, simp add: add_assoc [symmetric])
haftmann@34943: next
haftmann@34943:   assume ?lhs
haftmann@34943:   show ?rhs
haftmann@34943:   proof (cases "a = b")
haftmann@34943:     case True with `?lhs` show ?thesis by simp
haftmann@34943:   next
haftmann@34943:     case False
haftmann@34943:     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
haftmann@34943:     with False have "a \<in># N" by auto
haftmann@34943:     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
haftmann@34943:     moreover note False
haftmann@34943:     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
haftmann@34943:   qed
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma insert_noteq_member: 
haftmann@34943:   assumes BC: "B + {#b#} = C + {#c#}"
haftmann@34943:    and bnotc: "b \<noteq> c"
haftmann@34943:   shows "c \<in># B"
haftmann@34943: proof -
haftmann@34943:   have "c \<in># C + {#c#}" by simp
haftmann@34943:   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
haftmann@34943:   then have "c \<in># B + {#b#}" using BC by simp
haftmann@34943:   then show "c \<in># B" using nc by simp
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma add_eq_conv_ex:
haftmann@34943:   "(M + {#a#} = N + {#b#}) =
haftmann@34943:     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
haftmann@34943:   by (auto simp add: add_eq_conv_diff)
haftmann@34943: 
haftmann@34943: 
haftmann@34943: subsubsection {* Pointwise ordering induced by count *}
haftmann@34943: 
haftmann@35268: instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
haftmann@35268: begin
haftmann@35268: 
bulwahn@47429: lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)"
bulwahn@47429: by simp
bulwahn@47429: lemmas mset_le_def = less_eq_multiset_def
haftmann@34943: 
haftmann@35268: definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
haftmann@35268:   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
haftmann@34943: 
wenzelm@46921: instance
wenzelm@46921:   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
haftmann@35268: 
haftmann@35268: end
haftmann@34943: 
haftmann@34943: lemma mset_less_eqI:
haftmann@35268:   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
haftmann@34943:   by (simp add: mset_le_def)
haftmann@34943: 
haftmann@35268: lemma mset_le_exists_conv:
haftmann@35268:   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
haftmann@34943: apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow@39302: apply (auto intro: multiset_eq_iff [THEN iffD2])
haftmann@34943: done
haftmann@34943: 
haftmann@35268: lemma mset_le_mono_add_right_cancel [simp]:
haftmann@35268:   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
haftmann@35268:   by (fact add_le_cancel_right)
haftmann@34943: 
haftmann@35268: lemma mset_le_mono_add_left_cancel [simp]:
haftmann@35268:   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
haftmann@35268:   by (fact add_le_cancel_left)
haftmann@35268: 
haftmann@35268: lemma mset_le_mono_add:
haftmann@35268:   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
haftmann@35268:   by (fact add_mono)
haftmann@34943: 
haftmann@35268: lemma mset_le_add_left [simp]:
haftmann@35268:   "(A::'a multiset) \<le> A + B"
haftmann@35268:   unfolding mset_le_def by auto
haftmann@35268: 
haftmann@35268: lemma mset_le_add_right [simp]:
haftmann@35268:   "B \<le> (A::'a multiset) + B"
haftmann@35268:   unfolding mset_le_def by auto
haftmann@34943: 
haftmann@35268: lemma mset_le_single:
haftmann@35268:   "a :# B \<Longrightarrow> {#a#} \<le> B"
haftmann@35268:   by (simp add: mset_le_def)
haftmann@34943: 
haftmann@35268: lemma multiset_diff_union_assoc:
haftmann@35268:   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
nipkow@39302:   by (simp add: multiset_eq_iff mset_le_def)
haftmann@34943: 
haftmann@34943: lemma mset_le_multiset_union_diff_commute:
nipkow@36867:   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
nipkow@39302: by (simp add: multiset_eq_iff mset_le_def)
haftmann@34943: 
nipkow@39301: lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
nipkow@39301: by(simp add: mset_le_def)
nipkow@39301: 
haftmann@35268: lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
haftmann@34943: apply (clarsimp simp: mset_le_def mset_less_def)
haftmann@34943: apply (erule_tac x=x in allE)
haftmann@34943: apply auto
haftmann@34943: done
haftmann@34943: 
haftmann@35268: lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
haftmann@34943: apply (clarsimp simp: mset_le_def mset_less_def)
haftmann@34943: apply (erule_tac x = x in allE)
haftmann@34943: apply auto
haftmann@34943: done
haftmann@34943:   
haftmann@35268: lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
haftmann@34943: apply (rule conjI)
haftmann@34943:  apply (simp add: mset_lessD)
haftmann@34943: apply (clarsimp simp: mset_le_def mset_less_def)
haftmann@34943: apply safe
haftmann@34943:  apply (erule_tac x = a in allE)
haftmann@34943:  apply (auto split: split_if_asm)
haftmann@34943: done
haftmann@34943: 
haftmann@35268: lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
haftmann@34943: apply (rule conjI)
haftmann@34943:  apply (simp add: mset_leD)
haftmann@34943: apply (force simp: mset_le_def mset_less_def split: split_if_asm)
haftmann@34943: done
haftmann@34943: 
haftmann@35268: lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
nipkow@39302:   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
haftmann@34943: 
haftmann@35268: lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
haftmann@35268:   by (auto simp: mset_le_def mset_less_def)
haftmann@34943: 
haftmann@35268: lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
haftmann@35268:   by simp
haftmann@34943: 
haftmann@34943: lemma mset_less_add_bothsides:
haftmann@35268:   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
haftmann@35268:   by (fact add_less_imp_less_right)
haftmann@35268: 
haftmann@35268: lemma mset_less_empty_nonempty:
haftmann@35268:   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
haftmann@35268:   by (auto simp: mset_le_def mset_less_def)
haftmann@35268: 
haftmann@35268: lemma mset_less_diff_self:
haftmann@35268:   "c \<in># B \<Longrightarrow> B - {#c#} < B"
nipkow@39302:   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
haftmann@35268: 
haftmann@35268: 
haftmann@35268: subsubsection {* Intersection *}
haftmann@35268: 
haftmann@35268: instantiation multiset :: (type) semilattice_inf
haftmann@35268: begin
haftmann@35268: 
haftmann@35268: definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
haftmann@35268:   multiset_inter_def: "inf_multiset A B = A - (A - B)"
haftmann@35268: 
wenzelm@46921: instance
wenzelm@46921: proof -
haftmann@35268:   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
wenzelm@46921:   show "OFCLASS('a multiset, semilattice_inf_class)"
wenzelm@46921:     by default (auto simp add: multiset_inter_def mset_le_def aux)
haftmann@35268: qed
haftmann@35268: 
haftmann@35268: end
haftmann@35268: 
haftmann@35268: abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
haftmann@35268:   "multiset_inter \<equiv> inf"
haftmann@34943: 
haftmann@41069: lemma multiset_inter_count [simp]:
haftmann@35268:   "count (A #\<inter> B) x = min (count A x) (count B x)"
bulwahn@47429:   by (simp add: multiset_inter_def)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
wenzelm@46730:   by (rule multiset_eqI) auto
haftmann@34943: 
haftmann@35268: lemma multiset_union_diff_commute:
haftmann@35268:   assumes "B #\<inter> C = {#}"
haftmann@35268:   shows "A + B - C = A - C + B"
nipkow@39302: proof (rule multiset_eqI)
haftmann@35268:   fix x
haftmann@35268:   from assms have "min (count B x) (count C x) = 0"
wenzelm@46730:     by (auto simp add: multiset_eq_iff)
haftmann@35268:   then have "count B x = 0 \<or> count C x = 0"
haftmann@35268:     by auto
haftmann@35268:   then show "count (A + B - C) x = count (A - C + B) x"
haftmann@35268:     by auto
haftmann@35268: qed
haftmann@35268: 
haftmann@35268: 
haftmann@41069: subsubsection {* Filter (with comprehension syntax) *}
haftmann@41069: 
haftmann@41069: text {* Multiset comprehension *}
haftmann@41069: 
bulwahn@47429: 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"
bulwahn@47429: by (rule filter_preserves_multiset)
haftmann@35268: 
haftmann@41069: hide_const (open) filter
haftmann@35268: 
haftmann@41069: lemma count_filter [simp]:
haftmann@41069:   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
bulwahn@47429:   by (simp add: filter.rep_eq)
haftmann@41069: 
haftmann@41069: lemma filter_empty [simp]:
haftmann@41069:   "Multiset.filter P {#} = {#}"
nipkow@39302:   by (rule multiset_eqI) simp
haftmann@35268: 
haftmann@41069: lemma filter_single [simp]:
haftmann@41069:   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
haftmann@41069:   by (rule multiset_eqI) simp
haftmann@41069: 
haftmann@41069: lemma filter_union [simp]:
haftmann@41069:   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
nipkow@39302:   by (rule multiset_eqI) simp
haftmann@35268: 
haftmann@41069: lemma filter_diff [simp]:
haftmann@41069:   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
haftmann@41069:   by (rule multiset_eqI) simp
haftmann@41069: 
haftmann@41069: lemma filter_inter [simp]:
haftmann@41069:   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
nipkow@39302:   by (rule multiset_eqI) simp
wenzelm@10249: 
haftmann@41069: syntax
haftmann@41069:   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
haftmann@41069: syntax (xsymbol)
haftmann@41069:   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
haftmann@41069: translations
haftmann@41069:   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
haftmann@41069: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Set of elements *}
wenzelm@10249: 
haftmann@34943: definition set_of :: "'a multiset => 'a set" where
haftmann@34943:   "set_of M = {x. x :# M}"
haftmann@34943: 
wenzelm@17161: lemma set_of_empty [simp]: "set_of {#} = {}"
nipkow@26178: by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_single [simp]: "set_of {#b#} = {b}"
nipkow@26178: by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
nipkow@26178: by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
nipkow@39302: by (auto simp add: set_of_def multiset_eq_iff)
wenzelm@10249: 
wenzelm@17161: lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
nipkow@26178: by (auto simp add: set_of_def)
nipkow@26016: 
haftmann@41069: lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
nipkow@26178: by (auto simp add: set_of_def)
wenzelm@10249: 
haftmann@34943: lemma finite_set_of [iff]: "finite (set_of M)"
haftmann@34943:   using count [of M] by (simp add: multiset_def set_of_def)
haftmann@34943: 
bulwahn@46756: lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
bulwahn@46756:   unfolding set_of_def[symmetric] by simp
wenzelm@10249: 
wenzelm@10249: subsubsection {* Size *}
wenzelm@10249: 
haftmann@34943: instantiation multiset :: (type) size
haftmann@34943: begin
haftmann@34943: 
haftmann@34943: definition size_def:
haftmann@34943:   "size M = setsum (count M) (set_of M)"
haftmann@34943: 
haftmann@34943: instance ..
haftmann@34943: 
haftmann@34943: end
haftmann@34943: 
haftmann@28708: lemma size_empty [simp]: "size {#} = 0"
nipkow@26178: by (simp add: size_def)
wenzelm@10249: 
haftmann@28708: lemma size_single [simp]: "size {#b#} = 1"
nipkow@26178: by (simp add: size_def)
wenzelm@10249: 
wenzelm@17161: lemma setsum_count_Int:
nipkow@26178:   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow@26178: apply (induct rule: finite_induct)
nipkow@26178:  apply simp
nipkow@26178: apply (simp add: Int_insert_left set_of_def)
nipkow@26178: done
wenzelm@10249: 
haftmann@28708: lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
nipkow@26178: apply (unfold size_def)
nipkow@26178: apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow@26178:  prefer 2
nipkow@26178:  apply (rule ext, simp)
nipkow@26178: apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow@26178: apply (subst Int_commute)
nipkow@26178: apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
nipkow@39302: by (auto simp add: size_def multiset_eq_iff)
nipkow@26016: 
nipkow@26016: lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
nipkow@26178: by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
wenzelm@10249: 
wenzelm@17161: lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
nipkow@26178: apply (unfold size_def)
nipkow@26178: apply (drule setsum_SucD)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
haftmann@34943: lemma size_eq_Suc_imp_eq_union:
haftmann@34943:   assumes "size M = Suc n"
haftmann@34943:   shows "\<exists>a N. M = N + {#a#}"
haftmann@34943: proof -
haftmann@34943:   from assms obtain a where "a \<in># M"
haftmann@34943:     by (erule size_eq_Suc_imp_elem [THEN exE])
haftmann@34943:   then have "M = M - {#a#} + {#a#}" by simp
haftmann@34943:   then show ?thesis by blast
nipkow@23611: qed
kleing@15869: 
nipkow@26016: 
nipkow@26016: subsection {* Induction and case splits *}
wenzelm@10249: 
wenzelm@18258: theorem multiset_induct [case_names empty add, induct type: multiset]:
huffman@48009:   assumes empty: "P {#}"
huffman@48009:   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
huffman@48009:   shows "P M"
huffman@48009: proof (induct n \<equiv> "size M" arbitrary: M)
huffman@48009:   case 0 thus "P M" by (simp add: empty)
huffman@48009: next
huffman@48009:   case (Suc k)
huffman@48009:   obtain N x where "M = N + {#x#}"
huffman@48009:     using `Suc k = size M` [symmetric]
huffman@48009:     using size_eq_Suc_imp_eq_union by fast
huffman@48009:   with Suc add show "P M" by simp
wenzelm@10249: qed
wenzelm@10249: 
kleing@25610: lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178: by (induct M) auto
kleing@25610: 
kleing@25610: lemma multiset_cases [cases type, case_names empty add]:
nipkow@26178: assumes em:  "M = {#} \<Longrightarrow> P"
nipkow@26178: assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow@26178: shows "P"
huffman@48009: using assms by (induct M) simp_all
kleing@25610: 
kleing@25610: lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
huffman@48009: by (rule_tac x="M - {#x#}" in exI, simp)
kleing@25610: 
haftmann@34943: lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
haftmann@34943: by (cases "B = {#}") (auto dest: multi_member_split)
haftmann@34943: 
nipkow@26033: lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
nipkow@39302: apply (subst multiset_eq_iff)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
haftmann@35268: lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
haftmann@34943: proof (induct A arbitrary: B)
haftmann@34943:   case (empty M)
haftmann@34943:   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
haftmann@34943:   then obtain M' x where "M = M' + {#x#}" 
haftmann@34943:     by (blast dest: multi_nonempty_split)
haftmann@34943:   then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (add S x T)
haftmann@35268:   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
haftmann@35268:   have SxsubT: "S + {#x#} < T" by fact
haftmann@35268:   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
haftmann@34943:   then obtain T' where T: "T = T' + {#x#}" 
haftmann@34943:     by (blast dest: multi_member_split)
haftmann@35268:   then have "S < T'" using SxsubT 
haftmann@34943:     by (blast intro: mset_less_add_bothsides)
haftmann@34943:   then have "size S < size T'" using IH by simp
haftmann@34943:   then show ?case using T by simp
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: 
haftmann@34943: subsubsection {* Strong induction and subset induction for multisets *}
haftmann@34943: 
haftmann@34943: text {* Well-foundedness of proper subset operator: *}
haftmann@34943: 
haftmann@34943: text {* proper multiset subset *}
haftmann@34943: 
haftmann@34943: definition
haftmann@34943:   mset_less_rel :: "('a multiset * 'a multiset) set" where
haftmann@35268:   "mset_less_rel = {(A,B). A < B}"
wenzelm@10249: 
haftmann@34943: lemma multiset_add_sub_el_shuffle: 
haftmann@34943:   assumes "c \<in># B" and "b \<noteq> c" 
haftmann@34943:   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
haftmann@34943: proof -
haftmann@34943:   from `c \<in># B` obtain A where B: "B = A + {#c#}" 
haftmann@34943:     by (blast dest: multi_member_split)
haftmann@34943:   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
haftmann@34943:   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
haftmann@34943:     by (simp add: add_ac)
haftmann@34943:   then show ?thesis using B by simp
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma wf_mset_less_rel: "wf mset_less_rel"
haftmann@34943: apply (unfold mset_less_rel_def)
haftmann@34943: apply (rule wf_measure [THEN wf_subset, where f1=size])
haftmann@34943: apply (clarsimp simp: measure_def inv_image_def mset_less_size)
haftmann@34943: done
haftmann@34943: 
haftmann@34943: text {* The induction rules: *}
haftmann@34943: 
haftmann@34943: lemma full_multiset_induct [case_names less]:
haftmann@35268: assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
haftmann@34943: shows "P B"
haftmann@34943: apply (rule wf_mset_less_rel [THEN wf_induct])
haftmann@34943: apply (rule ih, auto simp: mset_less_rel_def)
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma multi_subset_induct [consumes 2, case_names empty add]:
haftmann@35268: assumes "F \<le> A"
haftmann@34943:   and empty: "P {#}"
haftmann@34943:   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
haftmann@34943: shows "P F"
haftmann@34943: proof -
haftmann@35268:   from `F \<le> A`
haftmann@34943:   show ?thesis
haftmann@34943:   proof (induct F)
haftmann@34943:     show "P {#}" by fact
haftmann@34943:   next
haftmann@34943:     fix x F
haftmann@35268:     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
haftmann@34943:     show "P (F + {#x#})"
haftmann@34943:     proof (rule insert)
haftmann@34943:       from i show "x \<in># A" by (auto dest: mset_le_insertD)
haftmann@35268:       from i have "F \<le> A" by (auto dest: mset_le_insertD)
haftmann@34943:       with P show "P F" .
haftmann@34943:     qed
haftmann@34943:   qed
haftmann@34943: qed
wenzelm@26145: 
wenzelm@17161: 
huffman@48023: subsection {* The fold combinator *}
huffman@48023: 
haftmann@49822: definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
huffman@48023: where
haftmann@49822:   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
huffman@48023: 
haftmann@49822: lemma fold_mset_empty [simp]:
haftmann@49822:   "fold f s {#} = s"
haftmann@49822:   by (simp add: fold_def)
huffman@48023: 
huffman@48023: context comp_fun_commute
huffman@48023: begin
huffman@48023: 
haftmann@49822: lemma fold_mset_insert:
haftmann@49822:   "fold f s (M + {#x#}) = f x (fold f s M)"
haftmann@49822: proof -
haftmann@49822:   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
haftmann@49822:     by (fact comp_fun_commute_funpow)
haftmann@49822:   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
haftmann@49822:     by (fact comp_fun_commute_funpow)
haftmann@49822:   show ?thesis
haftmann@49822:   proof (cases "x \<in> set_of M")
haftmann@49822:     case False
haftmann@49822:     then have *: "count (M + {#x#}) x = 1" by simp
haftmann@49822:     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
haftmann@49822:       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
haftmann@49822:       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
haftmann@49822:     with False * show ?thesis
haftmann@49822:       by (simp add: fold_def del: count_union)
huffman@48023:   next
haftmann@49822:     case True
haftmann@49822:     def N \<equiv> "set_of M - {x}"
haftmann@49822:     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
haftmann@49822:     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
haftmann@49822:       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
haftmann@49822:       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
haftmann@49822:     with * show ?thesis by (simp add: fold_def del: count_union) simp
huffman@48023:   qed
huffman@48023: qed
huffman@48023: 
haftmann@49822: corollary fold_mset_single [simp]:
haftmann@49822:   "fold f s {#x#} = f x s"
haftmann@49822: proof -
haftmann@49822:   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
haftmann@49822:   then show ?thesis by simp
haftmann@49822: qed
huffman@48023: 
haftmann@49822: lemma fold_mset_fun_comm:
haftmann@49822:   "f x (fold f s M) = fold f (f x s) M"
haftmann@49822:   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
huffman@48023: 
huffman@48023: lemma fold_mset_union [simp]:
haftmann@49822:   "fold f s (M + N) = fold f (fold f s M) N"
haftmann@49822: proof (induct M)
huffman@48023:   case empty then show ?case by simp
huffman@48023: next
haftmann@49822:   case (add M x)
haftmann@49822:   have "M + {#x#} + N = (M + N) + {#x#}"
haftmann@49822:     by (simp add: add_ac)
haftmann@49822:   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_comm)
huffman@48023: qed
huffman@48023: 
huffman@48023: lemma fold_mset_fusion:
huffman@48023:   assumes "comp_fun_commute g"
haftmann@49822:   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold g w A) = fold f (h w) A" (is "PROP ?P")
huffman@48023: proof -
huffman@48023:   interpret comp_fun_commute g by (fact assms)
huffman@48023:   show "PROP ?P" by (induct A) auto
huffman@48023: qed
huffman@48023: 
huffman@48023: end
huffman@48023: 
huffman@48023: text {*
huffman@48023:   A note on code generation: When defining some function containing a
haftmann@49822:   subterm @{term "fold F"}, code generation is not automatic. When
huffman@48023:   interpreting locale @{text left_commutative} with @{text F}, the
haftmann@49822:   would be code thms for @{const fold} become thms like
haftmann@49822:   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
huffman@48023:   contains defined symbols, i.e.\ is not a code thm. Hence a separate
huffman@48023:   constant with its own code thms needs to be introduced for @{text
huffman@48023:   F}. See the image operator below.
huffman@48023: *}
huffman@48023: 
huffman@48023: 
huffman@48023: subsection {* Image *}
huffman@48023: 
huffman@48023: definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
haftmann@49822:   "image_mset f = fold (plus o single o f) {#}"
huffman@48023: 
huffman@48023: interpretation image_fun_commute: comp_fun_commute "op + o single o f" for f
huffman@48023: proof qed (simp add: add_ac fun_eq_iff)
huffman@48023: 
huffman@48023: lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
huffman@48023: by (simp add: image_mset_def)
huffman@48023: 
huffman@48023: lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
huffman@48023: by (simp add: image_mset_def)
huffman@48023: 
huffman@48023: lemma image_mset_insert:
huffman@48023:   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
huffman@48023: by (simp add: image_mset_def add_ac)
huffman@48023: 
huffman@48023: lemma image_mset_union [simp]:
huffman@48023:   "image_mset f (M+N) = image_mset f M + image_mset f N"
huffman@48023: apply (induct N)
huffman@48023:  apply simp
huffman@48023: apply (simp add: add_assoc [symmetric] image_mset_insert)
huffman@48023: done
huffman@48023: 
huffman@48040: lemma set_of_image_mset [simp]: "set_of (image_mset f M) = image f (set_of M)"
huffman@48040: by (induct M) simp_all
huffman@48040: 
huffman@48023: lemma size_image_mset [simp]: "size (image_mset f M) = size M"
huffman@48023: by (induct M) simp_all
huffman@48023: 
huffman@48023: lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
huffman@48023: by (cases M) auto
huffman@48023: 
huffman@48023: syntax
huffman@48023:   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
huffman@48023:       ("({#_/. _ :# _#})")
huffman@48023: translations
huffman@48023:   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
huffman@48023: 
huffman@48023: syntax
huffman@48023:   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
huffman@48023:       ("({#_/ | _ :# _./ _#})")
huffman@48023: translations
huffman@48023:   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
huffman@48023: 
huffman@48023: text {*
huffman@48023:   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
huffman@48023:   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
huffman@48023:   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
huffman@48023:   @{term "{#x+x|x:#M. x<c#}"}.
huffman@48023: *}
huffman@48023: 
huffman@48023: enriched_type image_mset: image_mset
huffman@48023: proof -
huffman@48023:   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
huffman@48023:   proof
huffman@48023:     fix A
huffman@48023:     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
huffman@48023:       by (induct A) simp_all
huffman@48023:   qed
huffman@48023:   show "image_mset id = id"
huffman@48023:   proof
huffman@48023:     fix A
huffman@48023:     show "image_mset id A = id A"
huffman@48023:       by (induct A) simp_all
huffman@48023:   qed
huffman@48023: qed
huffman@48023: 
haftmann@49717: declare image_mset.identity [simp]
haftmann@49717: 
huffman@48023: 
haftmann@34943: subsection {* Alternative representations *}
haftmann@34943: 
haftmann@34943: subsubsection {* Lists *}
haftmann@34943: 
haftmann@34943: primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
haftmann@34943:   "multiset_of [] = {#}" |
haftmann@34943:   "multiset_of (a # x) = multiset_of x + {# a #}"
haftmann@34943: 
haftmann@37107: lemma in_multiset_in_set:
haftmann@37107:   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
haftmann@37107:   by (induct xs) simp_all
haftmann@37107: 
haftmann@37107: lemma count_multiset_of:
haftmann@37107:   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
haftmann@37107:   by (induct xs) simp_all
haftmann@37107: 
haftmann@34943: lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
haftmann@34943: by (induct x) auto
haftmann@34943: 
haftmann@34943: lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
haftmann@34943: by (induct x) auto
haftmann@34943: 
haftmann@40950: lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
haftmann@34943: by (induct x) auto
haftmann@34943: 
haftmann@34943: lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
haftmann@34943: by (induct xs) auto
haftmann@34943: 
huffman@48012: lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
huffman@48012:   by (induct xs) simp_all
huffman@48012: 
haftmann@34943: lemma multiset_of_append [simp]:
haftmann@34943:   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
haftmann@34943:   by (induct xs arbitrary: ys) (auto simp: add_ac)
haftmann@34943: 
haftmann@40303: lemma multiset_of_filter:
haftmann@40303:   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
haftmann@40303:   by (induct xs) simp_all
haftmann@40303: 
haftmann@40950: lemma multiset_of_rev [simp]:
haftmann@40950:   "multiset_of (rev xs) = multiset_of xs"
haftmann@40950:   by (induct xs) simp_all
haftmann@40950: 
haftmann@34943: lemma surj_multiset_of: "surj multiset_of"
haftmann@34943: apply (unfold surj_def)
haftmann@34943: apply (rule allI)
haftmann@34943: apply (rule_tac M = y in multiset_induct)
haftmann@34943:  apply auto
haftmann@34943: apply (rule_tac x = "x # xa" in exI)
haftmann@34943: apply auto
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
haftmann@34943: by (induct x) auto
haftmann@34943: 
haftmann@34943: lemma distinct_count_atmost_1:
haftmann@34943:   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
haftmann@34943: apply (induct x, simp, rule iffI, simp_all)
haftmann@34943: apply (rule conjI)
haftmann@34943: apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
haftmann@34943: apply (erule_tac x = a in allE, simp, clarify)
haftmann@34943: apply (erule_tac x = aa in allE, simp)
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma multiset_of_eq_setD:
haftmann@34943:   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
nipkow@39302: by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
haftmann@34943: 
haftmann@34943: lemma set_eq_iff_multiset_of_eq_distinct:
haftmann@34943:   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
haftmann@34943:     (set x = set y) = (multiset_of x = multiset_of y)"
nipkow@39302: by (auto simp: multiset_eq_iff distinct_count_atmost_1)
haftmann@34943: 
haftmann@34943: lemma set_eq_iff_multiset_of_remdups_eq:
haftmann@34943:    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
haftmann@34943: apply (rule iffI)
haftmann@34943: apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
haftmann@34943: apply (drule distinct_remdups [THEN distinct_remdups
haftmann@34943:       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
haftmann@34943: apply simp
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma multiset_of_compl_union [simp]:
haftmann@34943:   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
haftmann@34943:   by (induct xs) (auto simp: add_ac)
haftmann@34943: 
haftmann@41069: lemma count_multiset_of_length_filter:
haftmann@39533:   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
haftmann@39533:   by (induct xs) auto
haftmann@34943: 
haftmann@34943: lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
haftmann@34943: apply (induct ls arbitrary: i)
haftmann@34943:  apply simp
haftmann@34943: apply (case_tac i)
haftmann@34943:  apply auto
haftmann@34943: done
haftmann@34943: 
nipkow@36903: lemma multiset_of_remove1[simp]:
nipkow@36903:   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
nipkow@39302: by (induct xs) (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma multiset_of_eq_length:
haftmann@37107:   assumes "multiset_of xs = multiset_of ys"
haftmann@37107:   shows "length xs = length ys"
huffman@48012:   using assms by (metis size_multiset_of)
haftmann@34943: 
haftmann@39533: lemma multiset_of_eq_length_filter:
haftmann@39533:   assumes "multiset_of xs = multiset_of ys"
haftmann@39533:   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
huffman@48012:   using assms by (metis count_multiset_of)
haftmann@39533: 
haftmann@45989: lemma fold_multiset_equiv:
haftmann@45989:   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"
haftmann@45989:     and equiv: "multiset_of xs = multiset_of ys"
haftmann@49822:   shows "List.fold f xs = List.fold f ys"
wenzelm@46921: using f equiv [symmetric]
wenzelm@46921: proof (induct xs arbitrary: ys)
haftmann@45989:   case Nil then show ?case by simp
haftmann@45989: next
haftmann@45989:   case (Cons x xs)
haftmann@45989:   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
haftmann@45989:   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" 
haftmann@45989:     by (rule Cons.prems(1)) (simp_all add: *)
haftmann@45989:   moreover from * have "x \<in> set ys" by simp
haftmann@49822:   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
haftmann@49822:   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
haftmann@45989:   ultimately show ?case by simp
haftmann@45989: qed
haftmann@45989: 
haftmann@39533: context linorder
haftmann@39533: begin
haftmann@39533: 
haftmann@40210: lemma multiset_of_insort [simp]:
haftmann@39533:   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
haftmann@37107:   by (induct xs) (simp_all add: ac_simps)
haftmann@39533:  
haftmann@40210: lemma multiset_of_sort [simp]:
haftmann@39533:   "multiset_of (sort_key k xs) = multiset_of xs"
haftmann@37107:   by (induct xs) (simp_all add: ac_simps)
haftmann@37107: 
haftmann@34943: text {*
haftmann@34943:   This lemma shows which properties suffice to show that a function
haftmann@34943:   @{text "f"} with @{text "f xs = ys"} behaves like sort.
haftmann@34943: *}
haftmann@37074: 
haftmann@39533: lemma properties_for_sort_key:
haftmann@39533:   assumes "multiset_of ys = multiset_of xs"
haftmann@40305:   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
haftmann@39533:   and "sorted (map f ys)"
haftmann@39533:   shows "sort_key f xs = ys"
wenzelm@46921: using assms
wenzelm@46921: proof (induct xs arbitrary: ys)
haftmann@34943:   case Nil then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (Cons x xs)
haftmann@39533:   from Cons.prems(2) have
haftmann@40305:     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
haftmann@39533:     by (simp add: filter_remove1)
haftmann@39533:   with Cons.prems have "sort_key f xs = remove1 x ys"
haftmann@39533:     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
haftmann@39533:   moreover from Cons.prems have "x \<in> set ys"
haftmann@39533:     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
haftmann@39533:   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
haftmann@34943: qed
haftmann@34943: 
haftmann@39533: lemma properties_for_sort:
haftmann@39533:   assumes multiset: "multiset_of ys = multiset_of xs"
haftmann@39533:   and "sorted ys"
haftmann@39533:   shows "sort xs = ys"
haftmann@39533: proof (rule properties_for_sort_key)
haftmann@39533:   from multiset show "multiset_of ys = multiset_of xs" .
haftmann@39533:   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
haftmann@39533:   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
haftmann@39533:     by (rule multiset_of_eq_length_filter)
haftmann@39533:   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
haftmann@39533:     by simp
haftmann@40305:   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
haftmann@39533:     by (simp add: replicate_length_filter)
haftmann@39533: qed
haftmann@39533: 
haftmann@40303: lemma sort_key_by_quicksort:
haftmann@40303:   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
haftmann@40303:     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
haftmann@40303:     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
haftmann@40303: proof (rule properties_for_sort_key)
haftmann@40303:   show "multiset_of ?rhs = multiset_of ?lhs"
haftmann@40303:     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
haftmann@40303: next
haftmann@40303:   show "sorted (map f ?rhs)"
haftmann@40303:     by (auto simp add: sorted_append intro: sorted_map_same)
haftmann@40303: next
haftmann@40305:   fix l
haftmann@40305:   assume "l \<in> set ?rhs"
haftmann@40346:   let ?pivot = "f (xs ! (length xs div 2))"
haftmann@40346:   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
haftmann@40306:   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
haftmann@40305:     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
haftmann@40346:   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
haftmann@40346:   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
haftmann@40346:   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
haftmann@40346:     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
haftmann@40346:   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
haftmann@40306:   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
haftmann@40305:   proof (cases "f l" ?pivot rule: linorder_cases)
wenzelm@46730:     case less
wenzelm@46730:     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
wenzelm@46730:     with less show ?thesis
haftmann@40346:       by (simp add: filter_sort [symmetric] ** ***)
haftmann@40305:   next
haftmann@40306:     case equal then show ?thesis
haftmann@40346:       by (simp add: * less_le)
haftmann@40305:   next
wenzelm@46730:     case greater
wenzelm@46730:     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
wenzelm@46730:     with greater show ?thesis
haftmann@40346:       by (simp add: filter_sort [symmetric] ** ***)
haftmann@40306:   qed
haftmann@40303: qed
haftmann@40303: 
haftmann@40303: lemma sort_by_quicksort:
haftmann@40303:   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
haftmann@40303:     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
haftmann@40303:     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
haftmann@40303:   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
haftmann@40303: 
haftmann@40347: text {* A stable parametrized quicksort *}
haftmann@40347: 
haftmann@40347: definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
haftmann@40347:   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
haftmann@40347: 
haftmann@40347: lemma part_code [code]:
haftmann@40347:   "part f pivot [] = ([], [], [])"
haftmann@40347:   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
haftmann@40347:      if x' < pivot then (x # lts, eqs, gts)
haftmann@40347:      else if x' > pivot then (lts, eqs, x # gts)
haftmann@40347:      else (lts, x # eqs, gts))"
haftmann@40347:   by (auto simp add: part_def Let_def split_def)
haftmann@40347: 
haftmann@40347: lemma sort_key_by_quicksort_code [code]:
haftmann@40347:   "sort_key f xs = (case xs of [] \<Rightarrow> []
haftmann@40347:     | [x] \<Rightarrow> xs
haftmann@40347:     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
haftmann@40347:     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
haftmann@40347:        in sort_key f lts @ eqs @ sort_key f gts))"
haftmann@40347: proof (cases xs)
haftmann@40347:   case Nil then show ?thesis by simp
haftmann@40347: next
wenzelm@46921:   case (Cons _ ys) note hyps = Cons show ?thesis
wenzelm@46921:   proof (cases ys)
haftmann@40347:     case Nil with hyps show ?thesis by simp
haftmann@40347:   next
wenzelm@46921:     case (Cons _ zs) note hyps = hyps Cons show ?thesis
wenzelm@46921:     proof (cases zs)
haftmann@40347:       case Nil with hyps show ?thesis by auto
haftmann@40347:     next
haftmann@40347:       case Cons 
haftmann@40347:       from sort_key_by_quicksort [of f xs]
haftmann@40347:       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
haftmann@40347:         in sort_key f lts @ eqs @ sort_key f gts)"
haftmann@40347:       by (simp only: split_def Let_def part_def fst_conv snd_conv)
haftmann@40347:       with hyps Cons show ?thesis by (simp only: list.cases)
haftmann@40347:     qed
haftmann@40347:   qed
haftmann@40347: qed
haftmann@40347: 
haftmann@39533: end
haftmann@39533: 
haftmann@40347: hide_const (open) part
haftmann@40347: 
haftmann@35268: lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
haftmann@35268:   by (induct xs) (auto intro: order_trans)
haftmann@34943: 
haftmann@34943: lemma multiset_of_update:
haftmann@34943:   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
haftmann@34943: proof (induct ls arbitrary: i)
haftmann@34943:   case Nil then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (Cons x xs)
haftmann@34943:   show ?case
haftmann@34943:   proof (cases i)
haftmann@34943:     case 0 then show ?thesis by simp
haftmann@34943:   next
haftmann@34943:     case (Suc i')
haftmann@34943:     with Cons show ?thesis
haftmann@34943:       apply simp
haftmann@34943:       apply (subst add_assoc)
haftmann@34943:       apply (subst add_commute [of "{#v#}" "{#x#}"])
haftmann@34943:       apply (subst add_assoc [symmetric])
haftmann@34943:       apply simp
haftmann@34943:       apply (rule mset_le_multiset_union_diff_commute)
haftmann@34943:       apply (simp add: mset_le_single nth_mem_multiset_of)
haftmann@34943:       done
haftmann@34943:   qed
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma multiset_of_swap:
haftmann@34943:   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
haftmann@34943:     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
haftmann@34943:   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
haftmann@34943: 
haftmann@34943: 
bulwahn@46168: subsubsection {* Association lists -- including code generation *}
bulwahn@46168: 
bulwahn@46168: text {* Preliminaries *}
bulwahn@46168: 
bulwahn@46168: text {* Raw operations on lists *}
bulwahn@46168: 
bulwahn@46168: 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"
bulwahn@46168: where
bulwahn@46168:   "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
bulwahn@46168: 
bulwahn@46168: lemma join_raw_Nil [simp]:
bulwahn@46168:   "join_raw f xs [] = xs"
bulwahn@46168: by (simp add: join_raw_def)
bulwahn@46168: 
bulwahn@46168: lemma join_raw_Cons [simp]:
bulwahn@46168:   "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
bulwahn@46168: by (simp add: join_raw_def)
bulwahn@46168: 
bulwahn@46168: lemma map_of_join_raw:
bulwahn@46168:   assumes "distinct (map fst ys)"
bulwahn@47429:   shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
bulwahn@47429:     (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
bulwahn@46168: using assms
bulwahn@46168: apply (induct ys)
bulwahn@46168: apply (auto simp add: map_of_map_default split: option.split)
bulwahn@46168: apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
bulwahn@46168: by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
bulwahn@46168: 
bulwahn@46168: lemma distinct_join_raw:
bulwahn@46168:   assumes "distinct (map fst xs)"
bulwahn@46168:   shows "distinct (map fst (join_raw f xs ys))"
bulwahn@46168: using assms
bulwahn@46168: proof (induct ys)
bulwahn@46168:   case (Cons y ys)
bulwahn@46168:   thus ?case by (cases y) (simp add: distinct_map_default)
bulwahn@46168: qed auto
bulwahn@46168: 
bulwahn@46168: definition
bulwahn@46238:   "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
bulwahn@46168: 
bulwahn@46168: lemma map_of_subtract_entries_raw:
bulwahn@47429:   assumes "distinct (map fst ys)"
bulwahn@47429:   shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v =>
bulwahn@47429:     (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
bulwahn@47429: using assms unfolding subtract_entries_raw_def
bulwahn@46168: apply (induct ys)
bulwahn@46168: apply auto
bulwahn@46168: apply (simp split: option.split)
bulwahn@46168: apply (simp add: map_of_map_entry)
bulwahn@46168: apply (auto split: option.split)
bulwahn@46168: apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
bulwahn@46168: by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
bulwahn@46168: 
bulwahn@46168: lemma distinct_subtract_entries_raw:
bulwahn@46168:   assumes "distinct (map fst xs)"
bulwahn@46168:   shows "distinct (map fst (subtract_entries_raw xs ys))"
bulwahn@46168: using assms
bulwahn@46168: unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
bulwahn@46168: 
bulwahn@47179: text {* Operations on alists with distinct keys *}
bulwahn@46168: 
kuncar@47308: lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist" 
kuncar@47308: is join_raw
bulwahn@47179: by (simp add: distinct_join_raw)
bulwahn@46168: 
kuncar@47308: lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
kuncar@47308: is subtract_entries_raw 
bulwahn@47179: by (simp add: distinct_subtract_entries_raw)
bulwahn@46168: 
bulwahn@46168: text {* Implementing multisets by means of association lists *}
haftmann@34943: 
haftmann@34943: definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
haftmann@34943:   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
haftmann@34943: 
haftmann@34943: lemma count_of_multiset:
haftmann@34943:   "count_of xs \<in> multiset"
haftmann@34943: proof -
haftmann@34943:   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
haftmann@34943:   have "?A \<subseteq> dom (map_of xs)"
haftmann@34943:   proof
haftmann@34943:     fix x
haftmann@34943:     assume "x \<in> ?A"
haftmann@34943:     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
haftmann@34943:     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
haftmann@34943:     then show "x \<in> dom (map_of xs)" by auto
haftmann@34943:   qed
haftmann@34943:   with finite_dom_map_of [of xs] have "finite ?A"
haftmann@34943:     by (auto intro: finite_subset)
haftmann@34943:   then show ?thesis
nipkow@39302:     by (simp add: count_of_def fun_eq_iff multiset_def)
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma count_simps [simp]:
haftmann@34943:   "count_of [] = (\<lambda>_. 0)"
haftmann@34943:   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
nipkow@39302:   by (simp_all add: count_of_def fun_eq_iff)
haftmann@34943: 
haftmann@34943: lemma count_of_empty:
haftmann@34943:   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
haftmann@34943:   by (induct xs) (simp_all add: count_of_def)
haftmann@34943: 
haftmann@34943: lemma count_of_filter:
bulwahn@46168:   "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
haftmann@34943:   by (induct xs) auto
haftmann@34943: 
bulwahn@46168: lemma count_of_map_default [simp]:
bulwahn@46168:   "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
bulwahn@46168: unfolding count_of_def by (simp add: map_of_map_default split: option.split)
bulwahn@46168: 
bulwahn@46168: lemma count_of_join_raw:
bulwahn@46168:   "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
bulwahn@46168: unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
bulwahn@46168: 
bulwahn@46168: lemma count_of_subtract_entries_raw:
bulwahn@46168:   "distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
bulwahn@46168: unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
bulwahn@46168: 
bulwahn@46168: text {* Code equations for multiset operations *}
bulwahn@46168: 
bulwahn@46168: definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
bulwahn@46237:   "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
haftmann@34943: 
haftmann@34943: code_datatype Bag
haftmann@34943: 
haftmann@34943: lemma count_Bag [simp, code]:
bulwahn@46237:   "count (Bag xs) = count_of (DAList.impl_of xs)"
haftmann@34943:   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
haftmann@34943: 
haftmann@34943: lemma Mempty_Bag [code]:
bulwahn@46394:   "{#} = Bag (DAList.empty)"
bulwahn@46394:   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
bulwahn@47143: 
haftmann@34943: lemma single_Bag [code]:
bulwahn@46394:   "{#x#} = Bag (DAList.update x 1 DAList.empty)"
kuncar@47198:   by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
bulwahn@46168: 
bulwahn@46168: lemma union_Bag [code]:
bulwahn@46168:   "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
bulwahn@46168: by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
bulwahn@46168: 
bulwahn@46168: lemma minus_Bag [code]:
bulwahn@46168:   "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
bulwahn@46168: by (rule multiset_eqI)
bulwahn@46168:   (simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
haftmann@34943: 
haftmann@41069: lemma filter_Bag [code]:
bulwahn@46237:   "Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
bulwahn@47429: by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
haftmann@34943: 
haftmann@34943: lemma mset_less_eq_Bag [code]:
bulwahn@46237:   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
haftmann@34943:     (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@34943: proof
haftmann@34943:   assume ?lhs then show ?rhs
wenzelm@46730:     by (auto simp add: mset_le_def)
haftmann@34943: next
haftmann@34943:   assume ?rhs
haftmann@34943:   show ?lhs
haftmann@34943:   proof (rule mset_less_eqI)
haftmann@34943:     fix x
bulwahn@46237:     from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
bulwahn@46237:       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
haftmann@34943:     then show "count (Bag xs) x \<le> count A x"
wenzelm@46730:       by (simp add: mset_le_def)
haftmann@34943:   qed
haftmann@34943: qed
haftmann@34943: 
haftmann@38857: instantiation multiset :: (equal) equal
haftmann@34943: begin
haftmann@34943: 
haftmann@34943: definition
bulwahn@45866:   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
haftmann@34943: 
wenzelm@46921: instance
wenzelm@46921:   by default (simp add: equal_multiset_def eq_iff)
haftmann@34943: 
haftmann@34943: end
haftmann@34943: 
bulwahn@46168: text {* Quickcheck generators *}
haftmann@38857: 
haftmann@34943: definition (in term_syntax)
bulwahn@46168:   bagify :: "('a\<Colon>typerep, nat) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)
haftmann@34943:     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
haftmann@34943:   [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"
haftmann@34943: 
haftmann@37751: notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@34943: 
haftmann@34943: instantiation multiset :: (random) random
haftmann@34943: begin
haftmann@34943: 
haftmann@34943: definition
haftmann@37751:   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"
haftmann@34943: 
haftmann@34943: instance ..
haftmann@34943: 
haftmann@34943: end
haftmann@34943: 
haftmann@37751: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@34943: 
bulwahn@46168: instantiation multiset :: (exhaustive) exhaustive
bulwahn@46168: begin
bulwahn@46168: 
bulwahn@46168: definition exhaustive_multiset :: "('a multiset => (bool * term list) option) => code_numeral => (bool * term list) option"
bulwahn@46168: where
bulwahn@46168:   "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (%xs. f (Bag xs)) i"
bulwahn@46168: 
bulwahn@46168: instance ..
bulwahn@46168: 
bulwahn@46168: end
bulwahn@46168: 
bulwahn@46168: instantiation multiset :: (full_exhaustive) full_exhaustive
bulwahn@46168: begin
bulwahn@46168: 
bulwahn@46168: definition full_exhaustive_multiset :: "('a multiset * (unit => term) => (bool * term list) option) => code_numeral => (bool * term list) option"
bulwahn@46168: where
bulwahn@46168:   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (%xs. f (bagify xs)) i"
bulwahn@46168: 
bulwahn@46168: instance ..
bulwahn@46168: 
bulwahn@46168: end
bulwahn@46168: 
wenzelm@36176: hide_const (open) bagify
haftmann@34943: 
haftmann@34943: 
haftmann@34943: subsection {* The multiset order *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Well-foundedness *}
wenzelm@10249: 
haftmann@28708: definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
haftmann@37765:   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
berghofe@23751:       (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249: 
haftmann@28708: definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
haftmann@37765:   "mult r = (mult1 r)\<^sup>+"
wenzelm@10249: 
berghofe@23751: lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178: by (simp add: mult1_def)
wenzelm@10249: 
berghofe@23751: lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
berghofe@23751:     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
berghofe@23751:     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@19582:   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
wenzelm@10249: proof (unfold mult1_def)
berghofe@23751:   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464:   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
berghofe@23751:   let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249: 
berghofe@23751:   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
wenzelm@18258:   then have "\<exists>a' M0' K.
nipkow@11464:       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
wenzelm@18258:   then show "?case1 \<or> ?case2"
wenzelm@10249:   proof (elim exE conjE)
wenzelm@10249:     fix a' M0' K
wenzelm@10249:     assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249:     assume "M0 + {#a#} = M0' + {#a'#}"
wenzelm@18258:     then have "M0 = M0' \<and> a = a' \<or>
nipkow@11464:         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249:       by (simp only: add_eq_conv_ex)
wenzelm@18258:     then show ?thesis
wenzelm@10249:     proof (elim disjE conjE exE)
wenzelm@10249:       assume "M0 = M0'" "a = a'"
nipkow@11464:       with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@18258:       then have ?case2 .. then show ?thesis ..
wenzelm@10249:     next
wenzelm@10249:       fix K'
wenzelm@10249:       assume "M0' = K' + {#a#}"
haftmann@34943:       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
wenzelm@10249: 
wenzelm@10249:       assume "M0 = K' + {#a'#}"
wenzelm@10249:       with r have "?R (K' + K) M0" by blast
wenzelm@18258:       with n have ?case1 by simp then show ?thesis ..
wenzelm@10249:     qed
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249: proof
wenzelm@10249:   let ?R = "mult1 r"
wenzelm@10249:   let ?W = "acc ?R"
wenzelm@10249:   {
wenzelm@10249:     fix M M0 a
berghofe@23751:     assume M0: "M0 \<in> ?W"
berghofe@23751:       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
berghofe@23751:     have "M0 + {#a#} \<in> ?W"
berghofe@23751:     proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249:       fix N
berghofe@23751:       assume "(N, M0 + {#a#}) \<in> ?R"
berghofe@23751:       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
berghofe@23751:           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249:         by (rule less_add)
berghofe@23751:       then show "N \<in> ?W"
wenzelm@10249:       proof (elim exE disjE conjE)
berghofe@23751:         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
berghofe@23751:         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
berghofe@23751:         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       next
wenzelm@10249:         fix K
wenzelm@10249:         assume N: "N = M0 + K"
berghofe@23751:         assume "\<forall>b. b :# K --> (b, a) \<in> r"
berghofe@23751:         then have "M0 + K \<in> ?W"
wenzelm@10249:         proof (induct K)
wenzelm@18730:           case empty
berghofe@23751:           from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730:         next
wenzelm@18730:           case (add K x)
berghofe@23751:           from add.prems have "(x, a) \<in> r" by simp
berghofe@23751:           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751:           moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751:           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
haftmann@34943:           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
wenzelm@10249:         qed
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       qed
wenzelm@10249:     qed
wenzelm@10249:   } note tedious_reasoning = this
wenzelm@10249: 
berghofe@23751:   assume wf: "wf r"
wenzelm@10249:   fix M
berghofe@23751:   show "M \<in> ?W"
wenzelm@10249:   proof (induct M)
berghofe@23751:     show "{#} \<in> ?W"
wenzelm@10249:     proof (rule accI)
berghofe@23751:       fix b assume "(b, {#}) \<in> ?R"
berghofe@23751:       with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249:     qed
wenzelm@10249: 
berghofe@23751:     fix M a assume "M \<in> ?W"
berghofe@23751:     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:     proof induct
wenzelm@10249:       fix a
berghofe@23751:       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:       proof
berghofe@23751:         fix M assume "M \<in> ?W"
berghofe@23751:         then show "M + {#a#} \<in> ?W"
wenzelm@23373:           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249:       qed
wenzelm@10249:     qed
berghofe@23751:     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: theorem wf_mult1: "wf r ==> wf (mult1 r)"
nipkow@26178: by (rule acc_wfI) (rule all_accessible)
wenzelm@10249: 
berghofe@23751: theorem wf_mult: "wf r ==> wf (mult r)"
nipkow@26178: unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Closure-free presentation *}
wenzelm@10249: 
wenzelm@10249: text {* One direction. *}
wenzelm@10249: 
wenzelm@10249: lemma mult_implies_one_step:
berghofe@23751:   "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464:     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
berghofe@23751:     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
nipkow@26178: apply (unfold mult_def mult1_def set_of_def)
nipkow@26178: apply (erule converse_trancl_induct, clarify)
nipkow@26178:  apply (rule_tac x = M0 in exI, simp, clarify)
nipkow@26178: apply (case_tac "a :# K")
nipkow@26178:  apply (rule_tac x = I in exI)
nipkow@26178:  apply (simp (no_asm))
nipkow@26178:  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
haftmann@34943:  apply (simp (no_asm_simp) add: add_assoc [symmetric])
nipkow@26178:  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow@26178:  apply (simp add: diff_union_single_conv)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "a :# I")
nipkow@26178:  apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "K + Ka" in exI)
nipkow@26178:  apply (rule conjI)
nipkow@39302:   apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178:  apply (rule conjI)
nipkow@26178:   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow@39302:   apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow@26178:  apply simp
nipkow@26178: apply (simp (no_asm))
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma one_step_implies_mult_aux:
berghofe@23751:   "trans r ==>
berghofe@23751:     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
berghofe@23751:       --> (I + K, I + J) \<in> mult r"
nipkow@26178: apply (induct_tac n, auto)
nipkow@26178: apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178: apply (rename_tac "J'", simp)
nipkow@26178: apply (erule notE, auto)
nipkow@26178: apply (case_tac "J' = {#}")
nipkow@26178:  apply (simp add: mult_def)
nipkow@26178:  apply (rule r_into_trancl)
nipkow@26178:  apply (simp add: mult1_def set_of_def, blast)
nipkow@26178: txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@26178: apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@26178: apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow@26178: apply (erule ssubst)
nipkow@26178: apply (simp add: Ball_def, auto)
nipkow@26178: apply (subgoal_tac
nipkow@26178:   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow@26178:     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178:  prefer 2
nipkow@26178:  apply force
haftmann@34943: apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
nipkow@26178: apply (erule trancl_trans)
nipkow@26178: apply (rule r_into_trancl)
nipkow@26178: apply (simp add: mult1_def set_of_def)
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "I + J'" in exI)
haftmann@34943: apply (simp add: add_ac)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma one_step_implies_mult:
berghofe@23751:   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
berghofe@23751:     ==> (I + K, I + J) \<in> mult r"
nipkow@26178: using one_step_implies_mult_aux by blast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Partial-order properties *}
wenzelm@10249: 
haftmann@35273: definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
haftmann@35273:   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
wenzelm@10249: 
haftmann@35273: definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
haftmann@35273:   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
haftmann@35273: 
haftmann@35308: notation (xsymbols) less_multiset (infix "\<subset>#" 50)
haftmann@35308: notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
wenzelm@10249: 
haftmann@35268: interpretation multiset_order: order le_multiset less_multiset
haftmann@35268: proof -
haftmann@35268:   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
haftmann@35268:   proof
haftmann@35268:     fix M :: "'a multiset"
haftmann@35268:     assume "M \<subset># M"
haftmann@35268:     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
haftmann@35268:     have "trans {(x'::'a, x). x' < x}"
haftmann@35268:       by (rule transI) simp
haftmann@35268:     moreover note MM
haftmann@35268:     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
haftmann@35268:       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
haftmann@35268:       by (rule mult_implies_one_step)
haftmann@35268:     then obtain I J K where "M = I + J" and "M = I + K"
haftmann@35268:       and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast
haftmann@35268:     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
haftmann@35268:     have "finite (set_of K)" by simp
haftmann@35268:     moreover note aux2
haftmann@35268:     ultimately have "set_of K = {}"
haftmann@35268:       by (induct rule: finite_induct) (auto intro: order_less_trans)
haftmann@35268:     with aux1 show False by simp
haftmann@35268:   qed
haftmann@35268:   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
haftmann@35268:     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@46921:   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
wenzelm@46921:     by default (auto simp add: le_multiset_def irrefl dest: trans)
haftmann@35268: qed
wenzelm@10249: 
wenzelm@46730: lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
wenzelm@46730:   by simp
haftmann@26567: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Monotonicity of multiset union *}
wenzelm@10249: 
wenzelm@46730: lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
nipkow@26178: apply (unfold mult1_def)
nipkow@26178: apply auto
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "C + M0" in exI)
haftmann@34943: apply (simp add: add_assoc)
nipkow@26178: done
wenzelm@10249: 
haftmann@35268: lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
nipkow@26178: apply (unfold less_multiset_def mult_def)
nipkow@26178: apply (erule trancl_induct)
noschinl@40249:  apply (blast intro: mult1_union)
noschinl@40249: apply (blast intro: mult1_union trancl_trans)
nipkow@26178: done
wenzelm@10249: 
haftmann@35268: lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
haftmann@34943: apply (subst add_commute [of B C])
haftmann@34943: apply (subst add_commute [of D C])
nipkow@26178: apply (erule union_less_mono2)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma union_less_mono:
haftmann@35268:   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
haftmann@35268:   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
wenzelm@10249: 
haftmann@35268: interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
haftmann@35268: proof
haftmann@35268: qed (auto simp add: le_multiset_def intro: union_less_mono2)
wenzelm@26145: 
paulson@15072: 
krauss@29125: subsection {* Termination proofs with multiset orders *}
krauss@29125: 
krauss@29125: lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
krauss@29125:   and multi_member_this: "x \<in># {# x #} + XS"
krauss@29125:   and multi_member_last: "x \<in># {# x #}"
krauss@29125:   by auto
krauss@29125: 
krauss@29125: definition "ms_strict = mult pair_less"
haftmann@37765: definition "ms_weak = ms_strict \<union> Id"
krauss@29125: 
krauss@29125: lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
krauss@29125: unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
krauss@29125: by (auto intro: wf_mult1 wf_trancl simp: mult_def)
krauss@29125: 
krauss@29125: lemma smsI:
krauss@29125:   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
krauss@29125:   unfolding ms_strict_def
krauss@29125: by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
krauss@29125: 
krauss@29125: lemma wmsI:
krauss@29125:   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
krauss@29125:   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
krauss@29125: unfolding ms_weak_def ms_strict_def
krauss@29125: by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
krauss@29125: 
krauss@29125: inductive pw_leq
krauss@29125: where
krauss@29125:   pw_leq_empty: "pw_leq {#} {#}"
krauss@29125: | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
krauss@29125: 
krauss@29125: lemma pw_leq_lstep:
krauss@29125:   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
krauss@29125: by (drule pw_leq_step) (rule pw_leq_empty, simp)
krauss@29125: 
krauss@29125: lemma pw_leq_split:
krauss@29125:   assumes "pw_leq X Y"
krauss@29125:   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125:   using assms
krauss@29125: proof (induct)
krauss@29125:   case pw_leq_empty thus ?case by auto
krauss@29125: next
krauss@29125:   case (pw_leq_step x y X Y)
krauss@29125:   then obtain A B Z where
krauss@29125:     [simp]: "X = A + Z" "Y = B + Z" 
krauss@29125:       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" 
krauss@29125:     by auto
krauss@29125:   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less" 
krauss@29125:     unfolding pair_leq_def by auto
krauss@29125:   thus ?case
krauss@29125:   proof
krauss@29125:     assume [simp]: "x = y"
krauss@29125:     have
krauss@29125:       "{#x#} + X = A + ({#y#}+Z) 
krauss@29125:       \<and> {#y#} + Y = B + ({#y#}+Z)
krauss@29125:       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125:       by (auto simp: add_ac)
krauss@29125:     thus ?case by (intro exI)
krauss@29125:   next
krauss@29125:     assume A: "(x, y) \<in> pair_less"
krauss@29125:     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
krauss@29125:     have "{#x#} + X = ?A' + Z"
krauss@29125:       "{#y#} + Y = ?B' + Z"
krauss@29125:       by (auto simp add: add_ac)
krauss@29125:     moreover have 
krauss@29125:       "(set_of ?A', set_of ?B') \<in> max_strict"
krauss@29125:       using 1 A unfolding max_strict_def 
krauss@29125:       by (auto elim!: max_ext.cases)
krauss@29125:     ultimately show ?thesis by blast
krauss@29125:   qed
krauss@29125: qed
krauss@29125: 
krauss@29125: lemma 
krauss@29125:   assumes pwleq: "pw_leq Z Z'"
krauss@29125:   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
krauss@29125:   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
krauss@29125:   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
krauss@29125: proof -
krauss@29125:   from pw_leq_split[OF pwleq] 
krauss@29125:   obtain A' B' Z''
krauss@29125:     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
krauss@29125:     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
krauss@29125:     by blast
krauss@29125:   {
krauss@29125:     assume max: "(set_of A, set_of B) \<in> max_strict"
krauss@29125:     from mx_or_empty
krauss@29125:     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
krauss@29125:     proof
krauss@29125:       assume max': "(set_of A', set_of B') \<in> max_strict"
krauss@29125:       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
krauss@29125:         by (auto simp: max_strict_def intro: max_ext_additive)
krauss@29125:       thus ?thesis by (rule smsI) 
krauss@29125:     next
krauss@29125:       assume [simp]: "A' = {#} \<and> B' = {#}"
krauss@29125:       show ?thesis by (rule smsI) (auto intro: max)
krauss@29125:     qed
krauss@29125:     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
krauss@29125:     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
krauss@29125:   }
krauss@29125:   from mx_or_empty
krauss@29125:   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
krauss@29125:   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
krauss@29125: qed
krauss@29125: 
nipkow@39301: lemma empty_neutral: "{#} + x = x" "x + {#} = x"
krauss@29125: and nonempty_plus: "{# x #} + rs \<noteq> {#}"
krauss@29125: and nonempty_single: "{# x #} \<noteq> {#}"
krauss@29125: by auto
krauss@29125: 
krauss@29125: setup {*
krauss@29125: let
wenzelm@35402:   fun msetT T = Type (@{type_name multiset}, [T]);
krauss@29125: 
wenzelm@35402:   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
krauss@29125:     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
krauss@29125:     | mk_mset T (x :: xs) =
krauss@29125:           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
krauss@29125:                 mk_mset T [x] $ mk_mset T xs
krauss@29125: 
krauss@29125:   fun mset_member_tac m i =
krauss@29125:       (if m <= 0 then
krauss@29125:            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
krauss@29125:        else
krauss@29125:            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
krauss@29125: 
krauss@29125:   val mset_nonempty_tac =
krauss@29125:       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
krauss@29125: 
krauss@29125:   val regroup_munion_conv =
wenzelm@35402:       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
nipkow@39301:         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
krauss@29125: 
krauss@29125:   fun unfold_pwleq_tac i =
krauss@29125:     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
krauss@29125:       ORELSE (rtac @{thm pw_leq_lstep} i)
krauss@29125:       ORELSE (rtac @{thm pw_leq_empty} i)
krauss@29125: 
krauss@29125:   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
krauss@29125:                       @{thm Un_insert_left}, @{thm Un_empty_left}]
krauss@29125: in
krauss@29125:   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset 
krauss@29125:   {
krauss@29125:     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
krauss@29125:     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
krauss@29125:     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
wenzelm@30595:     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
wenzelm@30595:     reduction_pair= @{thm ms_reduction_pair}
krauss@29125:   })
wenzelm@10249: end
krauss@29125: *}
krauss@29125: 
haftmann@34943: 
haftmann@34943: subsection {* Legacy theorem bindings *}
haftmann@34943: 
nipkow@39302: lemmas multi_count_eq = multiset_eq_iff [symmetric]
haftmann@34943: 
haftmann@34943: lemma union_commute: "M + N = N + (M::'a multiset)"
haftmann@34943:   by (fact add_commute)
haftmann@34943: 
haftmann@34943: lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
haftmann@34943:   by (fact add_assoc)
haftmann@34943: 
haftmann@34943: lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
haftmann@34943:   by (fact add_left_commute)
haftmann@34943: 
haftmann@34943: lemmas union_ac = union_assoc union_commute union_lcomm
haftmann@34943: 
haftmann@34943: lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943:   by (fact add_right_cancel)
haftmann@34943: 
haftmann@34943: lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943:   by (fact add_left_cancel)
haftmann@34943: 
haftmann@34943: lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
haftmann@34943:   by (fact add_imp_eq)
haftmann@34943: 
haftmann@35268: lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
haftmann@35268:   by (fact order_less_trans)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
haftmann@35268:   by (fact inf.commute)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
haftmann@35268:   by (fact inf.assoc [symmetric])
haftmann@35268: 
haftmann@35268: lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
haftmann@35268:   by (fact inf.left_commute)
haftmann@35268: 
haftmann@35268: lemmas multiset_inter_ac =
haftmann@35268:   multiset_inter_commute
haftmann@35268:   multiset_inter_assoc
haftmann@35268:   multiset_inter_left_commute
haftmann@35268: 
haftmann@35268: lemma mult_less_not_refl:
haftmann@35268:   "\<not> M \<subset># (M::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_irrefl)
haftmann@35268: 
haftmann@35268: lemma mult_less_trans:
haftmann@35268:   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_trans)
haftmann@35268:     
haftmann@35268: lemma mult_less_not_sym:
haftmann@35268:   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_not_sym)
haftmann@35268: 
haftmann@35268: lemma mult_less_asym:
haftmann@35268:   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
haftmann@35268:   by (fact multiset_order.less_asym)
haftmann@34943: 
blanchet@35712: ML {*
blanchet@35712: fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
blanchet@35712:                       (Const _ $ t') =
blanchet@35712:     let
blanchet@35712:       val (maybe_opt, ps) =
blanchet@35712:         Nitpick_Model.dest_plain_fun t' ||> op ~~
blanchet@35712:         ||> map (apsnd (snd o HOLogic.dest_number))
blanchet@35712:       fun elems_for t =
blanchet@35712:         case AList.lookup (op =) ps t of
blanchet@35712:           SOME n => replicate n t
blanchet@35712:         | NONE => [Const (maybe_name, elem_T --> elem_T) $ t]
blanchet@35712:     in
blanchet@35712:       case maps elems_for (all_values elem_T) @
blanchet@37261:            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
blanchet@37261:             else []) of
blanchet@35712:         [] => Const (@{const_name zero_class.zero}, T)
blanchet@35712:       | ts => foldl1 (fn (t1, t2) =>
blanchet@35712:                          Const (@{const_name plus_class.plus}, T --> T --> T)
blanchet@35712:                          $ t1 $ t2)
blanchet@35712:                      (map (curry (op $) (Const (@{const_name single},
blanchet@35712:                                                 elem_T --> T))) ts)
blanchet@35712:     end
blanchet@35712:   | multiset_postproc _ _ _ _ t = t
blanchet@35712: *}
blanchet@35712: 
blanchet@38287: declaration {*
blanchet@38287: Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
blanchet@38242:     multiset_postproc
blanchet@35712: *}
blanchet@35712: 
haftmann@49822: hide_const (open) fold
haftmann@49822: 
blanchet@37169: end
haftmann@49388: