(* Title: HOL/Library/Multiset.thy
Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
*)
header {* Multisets *}
theory Multiset
imports List Main
begin
subsection {* The type of multisets *}
typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
proof
show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
qed
lemmas multiset_typedef [simp] =
Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
and [simp] = Rep_multiset_inject [symmetric]
definition Mempty :: "'a multiset" ("{#}") where
[code del]: "{#} = Abs_multiset (\<lambda>a. 0)"
definition single :: "'a => 'a multiset" where
[code del]: "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
definition count :: "'a multiset => 'a => nat" where
"count = Rep_multiset"
definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
"MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
abbreviation Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where
"a :# M == 0 < count M a"
notation (xsymbols)
Melem (infix "\<in>#" 50)
syntax
"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})")
translations
"{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
definition set_of :: "'a multiset => 'a set" where
"set_of M = {x. x :# M}"
instantiation multiset :: (type) "{plus, minus, zero, size}"
begin
definition union_def [code del]:
"M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
definition diff_def [code del]:
"M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
definition Zero_multiset_def [simp]:
"0 = {#}"
definition size_def:
"size M = setsum (count M) (set_of M)"
instance ..
end
definition
multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
"multiset_inter A B = A - (A - B)"
text {* Multiset Enumeration *}
syntax
"_multiset" :: "args => 'a multiset" ("{#(_)#}")
translations
"{#x, xs#}" == "{#x#} + {#xs#}"
"{#x#}" == "CONST single x"
text {*
\medskip Preservation of the representing set @{term multiset}.
*}
lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
by (simp add: multiset_def)
lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
by (simp add: multiset_def)
lemma union_preserves_multiset:
"M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
by (simp add: multiset_def)
lemma diff_preserves_multiset:
"M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
apply (simp add: multiset_def)
apply (rule finite_subset)
apply auto
done
lemma MCollect_preserves_multiset:
"M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
apply (simp add: multiset_def)
apply (rule finite_subset, auto)
done
lemmas in_multiset = const0_in_multiset only1_in_multiset
union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
subsection {* Algebraic properties *}
subsubsection {* Union *}
lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
by (simp add: union_def Mempty_def in_multiset)
lemma union_commute: "M + N = N + (M::'a multiset)"
by (simp add: union_def add_ac in_multiset)
lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
by (simp add: union_def add_ac in_multiset)
lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
proof -
have "M + (N + K) = (N + K) + M" by (rule union_commute)
also have "\<dots> = N + (K + M)" by (rule union_assoc)
also have "K + M = M + K" by (rule union_commute)
finally show ?thesis .
qed
lemmas union_ac = union_assoc union_commute union_lcomm
instance multiset :: (type) comm_monoid_add
proof
fix a b c :: "'a multiset"
show "(a + b) + c = a + (b + c)" by (rule union_assoc)
show "a + b = b + a" by (rule union_commute)
show "0 + a = a" by simp
qed
subsubsection {* Difference *}
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
by (simp add: Mempty_def diff_def in_multiset)
lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
by (simp add: union_def diff_def in_multiset)
lemma diff_cancel: "A - A = {#}"
by (simp add: diff_def Mempty_def)
subsubsection {* Count of elements *}
lemma count_empty [simp]: "count {#} a = 0"
by (simp add: count_def Mempty_def in_multiset)
lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
by (simp add: count_def single_def in_multiset)
lemma count_union [simp]: "count (M + N) a = count M a + count N a"
by (simp add: count_def union_def in_multiset)
lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
by (simp add: count_def diff_def in_multiset)
lemma count_MCollect [simp]:
"count {# x:#M. P x #} a = (if P a then count M a else 0)"
by (simp add: count_def MCollect_def in_multiset)
subsubsection {* Set of elements *}
lemma set_of_empty [simp]: "set_of {#} = {}"
by (simp add: set_of_def)
lemma set_of_single [simp]: "set_of {#b#} = {b}"
by (simp add: set_of_def)
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
by (auto simp add: set_of_def)
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
by (auto simp add: set_of_def)
lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
by (auto simp add: set_of_def)
subsubsection {* Size *}
lemma size_empty [simp]: "size {#} = 0"
by (simp add: size_def)
lemma size_single [simp]: "size {#b#} = 1"
by (simp add: size_def)
lemma finite_set_of [iff]: "finite (set_of M)"
using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
lemma setsum_count_Int:
"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
apply (induct rule: finite_induct)
apply simp
apply (simp add: Int_insert_left set_of_def)
done
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
apply (unfold size_def)
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
prefer 2
apply (rule ext, simp)
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
apply (subst Int_commute)
apply (simp (no_asm_simp) add: setsum_count_Int)
done
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
done
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
apply (unfold size_def)
apply (drule setsum_SucD)
apply auto
done
subsubsection {* Equality of multisets *}
lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
by (simp add: count_def expand_fun_eq)
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
by (auto simp add: single_def in_multiset expand_fun_eq)
lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
by (simp add: union_def in_multiset expand_fun_eq)
lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
by (simp add: union_def in_multiset expand_fun_eq)
lemma union_is_single:
"(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
apply blast
done
lemma single_is_union:
"({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
apply (unfold Mempty_def single_def union_def)
apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
apply (blast dest: sym)
done
lemma add_eq_conv_diff:
"(M + {#a#} = N + {#b#}) =
(M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
using [[simproc del: neq]]
apply (unfold single_def union_def diff_def)
apply (simp (no_asm) add: in_multiset expand_fun_eq)
apply (rule conjI, force, safe, simp_all)
apply (simp add: eq_sym_conv)
done
declare Rep_multiset_inject [symmetric, simp del]
instance multiset :: (type) cancel_ab_semigroup_add
proof
fix a b c :: "'a multiset"
show "a + b = a + c \<Longrightarrow> b = c" by simp
qed
lemma insert_DiffM:
"x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
by (clarsimp simp: multiset_eq_conv_count_eq)
lemma insert_DiffM2[simp]:
"x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
by (clarsimp simp: multiset_eq_conv_count_eq)
lemma multi_union_self_other_eq:
"(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
by (induct A arbitrary: X Y) auto
lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
by (metis single_not_empty union_empty union_left_cancel)
lemma insert_noteq_member:
assumes BC: "B + {#b#} = C + {#c#}"
and bnotc: "b \<noteq> c"
shows "c \<in># B"
proof -
have "c \<in># C + {#c#}" by simp
have nc: "\<not> c \<in># {#b#}" using bnotc by simp
then have "c \<in># B + {#b#}" using BC by simp
then show "c \<in># B" using nc by simp
qed
lemma add_eq_conv_ex:
"(M + {#a#} = N + {#b#}) =
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
by (auto simp add: add_eq_conv_diff)
lemma empty_multiset_count:
"(\<forall>x. count A x = 0) = (A = {#})"
by (metis count_empty multiset_eq_conv_count_eq)
subsubsection {* Intersection *}
lemma multiset_inter_count:
"count (A #\<inter> B) x = min (count A x) (count B x)"
by (simp add: multiset_inter_def)
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
min_max.inf_commute)
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
min_max.inf_assoc)
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
lemmas multiset_inter_ac =
multiset_inter_commute
multiset_inter_assoc
multiset_inter_left_commute
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count
split: split_if_asm)
apply clarsimp
apply (erule_tac x = a in allE)
apply auto
done
subsubsection {* Comprehension (filter) *}
lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
in_multiset expand_fun_eq)
lemma MCollect_single [simp]:
"MCollect {#x#} P = (if P x then {#x#} else {#})"
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
in_multiset expand_fun_eq)
lemma MCollect_union [simp]:
"MCollect (M+N) f = MCollect M f + MCollect N f"
by (simp add: MCollect_def union_def Abs_multiset_inject
in_multiset expand_fun_eq)
subsection {* Induction and case splits *}
lemma setsum_decr:
"finite F ==> (0::nat) < f a ==>
setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
apply (induct rule: finite_induct)
apply auto
apply (drule_tac a = a in mk_disjoint_insert, auto)
done
lemma rep_multiset_induct_aux:
assumes 1: "P (\<lambda>a. (0::nat))"
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
apply (unfold multiset_def)
apply (induct_tac n, simp, clarify)
apply (subgoal_tac "f = (\<lambda>a.0)")
apply simp
apply (rule 1)
apply (rule ext, force, clarify)
apply (frule setsum_SucD, clarify)
apply (rename_tac a)
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
prefer 2
apply (rule finite_subset)
prefer 2
apply assumption
apply simp
apply blast
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
prefer 2
apply (rule ext)
apply (simp (no_asm_simp))
apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
apply (erule allE, erule impE, erule_tac [2] mp, blast)
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
prefer 2
apply blast
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
prefer 2
apply blast
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
done
theorem rep_multiset_induct:
"f \<in> multiset ==> P (\<lambda>a. 0) ==>
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
using rep_multiset_induct_aux by blast
theorem multiset_induct [case_names empty add, induct type: multiset]:
assumes empty: "P {#}"
and add: "!!M x. P M ==> P (M + {#x#})"
shows "P M"
proof -
note defns = union_def single_def Mempty_def
show ?thesis
apply (rule Rep_multiset_inverse [THEN subst])
apply (rule Rep_multiset [THEN rep_multiset_induct])
apply (rule empty [unfolded defns])
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
prefer 2
apply (simp add: expand_fun_eq)
apply (erule ssubst)
apply (erule Abs_multiset_inverse [THEN subst])
apply (drule add [unfolded defns, simplified])
apply(simp add:in_multiset)
done
qed
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
by (induct M) auto
lemma multiset_cases [cases type, case_names empty add]:
assumes em: "M = {#} \<Longrightarrow> P"
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
shows "P"
proof (cases "M = {#}")
assume "M = {#}" then show ?thesis using em by simp
next
assume "M \<noteq> {#}"
then obtain M' m where "M = M' + {#m#}"
by (blast dest: multi_nonempty_split)
then show ?thesis using add by simp
qed
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
apply (cases M)
apply simp
apply (rule_tac x="M - {#x#}" in exI, simp)
done
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
apply (subst multiset_eq_conv_count_eq)
apply auto
done
declare multiset_typedef [simp del]
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
by (cases "B = {#}") (auto dest: multi_member_split)
subsection {* Orderings *}
subsubsection {* Well-foundedness *}
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
[code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
(\<forall>b. b :# K --> (b, a) \<in> r)}"
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
"mult r = (mult1 r)\<^sup>+"
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
by (simp add: mult1_def)
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
proof (unfold mult1_def)
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
let ?case1 = "?case1 {(N, M). ?R N M}"
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
then have "\<exists>a' M0' K.
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
then show "?case1 \<or> ?case2"
proof (elim exE conjE)
fix a' M0' K
assume N: "N = M0' + K" and r: "?r K a'"
assume "M0 + {#a#} = M0' + {#a'#}"
then have "M0 = M0' \<and> a = a' \<or>
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
by (simp only: add_eq_conv_ex)
then show ?thesis
proof (elim disjE conjE exE)
assume "M0 = M0'" "a = a'"
with N r have "?r K a \<and> N = M0 + K" by simp
then have ?case2 .. then show ?thesis ..
next
fix K'
assume "M0' = K' + {#a#}"
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
assume "M0 = K' + {#a'#}"
with r have "?R (K' + K) M0" by blast
with n have ?case1 by simp then show ?thesis ..
qed
qed
qed
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "acc ?R"
{
fix M M0 a
assume M0: "M0 \<in> ?W"
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
have "M0 + {#a#} \<in> ?W"
proof (rule accI [of "M0 + {#a#}"])
fix N
assume "(N, M0 + {#a#}) \<in> ?R"
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
by (rule less_add)
then show "N \<in> ?W"
proof (elim exE disjE conjE)
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
then show "N \<in> ?W" by (simp only: N)
next
fix K
assume N: "N = M0 + K"
assume "\<forall>b. b :# K --> (b, a) \<in> r"
then have "M0 + K \<in> ?W"
proof (induct K)
case empty
from M0 show "M0 + {#} \<in> ?W" by simp
next
case (add K x)
from add.prems have "(x, a) \<in> r" by simp
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
moreover from add have "M0 + K \<in> ?W" by simp
ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
qed
then show "N \<in> ?W" by (simp only: N)
qed
qed
} note tedious_reasoning = this
assume wf: "wf r"
fix M
show "M \<in> ?W"
proof (induct M)
show "{#} \<in> ?W"
proof (rule accI)
fix b assume "(b, {#}) \<in> ?R"
with not_less_empty show "b \<in> ?W" by contradiction
qed
fix M a assume "M \<in> ?W"
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof induct
fix a
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof
fix M assume "M \<in> ?W"
then show "M + {#a#} \<in> ?W"
by (rule acc_induct) (rule tedious_reasoning [OF _ r])
qed
qed
from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
qed
qed
theorem wf_mult1: "wf r ==> wf (mult1 r)"
by (rule acc_wfI) (rule all_accessible)
theorem wf_mult: "wf r ==> wf (mult r)"
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
subsubsection {* Closure-free presentation *}
(*Badly needed: a linear arithmetic procedure for multisets*)
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
by (simp add: multiset_eq_conv_count_eq)
text {* One direction. *}
lemma mult_implies_one_step:
"trans r ==> (M, N) \<in> mult r ==>
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
apply (unfold mult_def mult1_def set_of_def)
apply (erule converse_trancl_induct, clarify)
apply (rule_tac x = M0 in exI, simp, clarify)
apply (case_tac "a :# K")
apply (rule_tac x = I in exI)
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (simp (no_asm_simp) add: union_assoc [symmetric])
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
apply (simp add: diff_union_single_conv)
apply (simp (no_asm_use) add: trans_def)
apply blast
apply (subgoal_tac "a :# I")
apply (rule_tac x = "I - {#a#}" in exI)
apply (rule_tac x = "J + {#a#}" in exI)
apply (rule_tac x = "K + Ka" in exI)
apply (rule conjI)
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply (rule conjI)
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply (simp (no_asm_use) add: trans_def)
apply blast
apply (subgoal_tac "a :# (M0 + {#a#})")
apply simp
apply (simp (no_asm))
done
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
by (simp add: multiset_eq_conv_count_eq)
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
apply (erule size_eq_Suc_imp_elem [THEN exE])
apply (drule elem_imp_eq_diff_union, auto)
done
lemma one_step_implies_mult_aux:
"trans r ==>
\<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))
--> (I + K, I + J) \<in> mult r"
apply (induct_tac n, auto)
apply (frule size_eq_Suc_imp_eq_union, clarify)
apply (rename_tac "J'", simp)
apply (erule notE, auto)
apply (case_tac "J' = {#}")
apply (simp add: mult_def)
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def, blast)
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
apply (erule ssubst)
apply (simp add: Ball_def, auto)
apply (subgoal_tac
"((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
(I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
prefer 2
apply force
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
apply (erule trancl_trans)
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
apply (simp add: union_ac)
done
lemma one_step_implies_mult:
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
==> (I + K, I + J) \<in> mult r"
using one_step_implies_mult_aux by blast
subsubsection {* Partial-order properties *}
instantiation multiset :: (order) order
begin
definition less_multiset_def [code del]:
"M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
definition le_multiset_def [code del]:
"M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
unfolding trans_def by (blast intro: order_less_trans)
text {*
\medskip Irreflexivity.
*}
lemma mult_irrefl_aux:
"finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
by (induct rule: finite_induct) (auto intro: order_less_trans)
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
apply (unfold less_multiset_def, auto)
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
apply (simp add: set_of_eq_empty_iff)
done
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
using insert mult_less_not_refl by fast
text {* Transitivity. *}
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
text {* Asymmetry. *}
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
apply auto
apply (rule mult_less_not_refl [THEN notE])
apply (erule mult_less_trans, assumption)
done
theorem mult_less_asym:
"M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
using mult_less_not_sym by blast
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
unfolding le_multiset_def by auto
text {* Anti-symmetry. *}
theorem mult_le_antisym:
"M <= N ==> N <= M ==> M = (N::'a::order multiset)"
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
text {* Transitivity. *}
theorem mult_le_trans:
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
unfolding le_multiset_def by (blast intro: mult_less_trans)
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
unfolding le_multiset_def by auto
instance proof
qed (auto simp add: mult_less_le dest: mult_le_antisym elim: mult_le_trans)
end
subsubsection {* Monotonicity of multiset union *}
lemma mult1_union:
"(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
apply (unfold mult1_def)
apply auto
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
apply (simp add: union_assoc)
done
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
apply (erule trancl_induct)
apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
done
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
apply (subst union_commute [of B C])
apply (subst union_commute [of D C])
apply (erule union_less_mono2)
done
lemma union_less_mono:
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
lemma union_le_mono:
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
unfolding le_multiset_def
by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
apply (unfold le_multiset_def less_multiset_def)
apply (case_tac "M = {#}")
prefer 2
apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
prefer 2
apply (rule one_step_implies_mult)
apply (simp only: trans_def)
apply auto
done
lemma union_upper1: "A <= A + (B::'a::order multiset)"
proof -
have "A + {#} <= A + B" by (blast intro: union_le_mono)
then show ?thesis by simp
qed
lemma union_upper2: "B <= A + (B::'a::order multiset)"
by (subst union_commute) (rule union_upper1)
instance multiset :: (order) pordered_ab_semigroup_add
apply intro_classes
apply (erule union_le_mono[OF mult_le_refl])
done
subsection {* Link with lists *}
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
"multiset_of [] = {#}" |
"multiset_of (a # x) = multiset_of x + {# a #}"
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
by (induct x) auto
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
by (induct x) auto
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
by (induct x) auto
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
by (induct xs) auto
lemma multiset_of_append [simp]:
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
by (induct xs arbitrary: ys) (auto simp: union_ac)
lemma surj_multiset_of: "surj multiset_of"
apply (unfold surj_def)
apply (rule allI)
apply (rule_tac M = y in multiset_induct)
apply auto
apply (rule_tac x = "x # xa" in exI)
apply auto
done
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
by (induct x) auto
lemma distinct_count_atmost_1:
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
apply (induct x, simp, rule iffI, simp_all)
apply (rule conjI)
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
apply (erule_tac x = a in allE, simp, clarify)
apply (erule_tac x = aa in allE, simp)
done
lemma multiset_of_eq_setD:
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
lemma set_eq_iff_multiset_of_eq_distinct:
"distinct x \<Longrightarrow> distinct y \<Longrightarrow>
(set x = set y) = (multiset_of x = multiset_of y)"
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
lemma set_eq_iff_multiset_of_remdups_eq:
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
apply (rule iffI)
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
apply (drule distinct_remdups [THEN distinct_remdups
[THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
apply simp
done
lemma multiset_of_compl_union [simp]:
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
by (induct xs) (auto simp: union_ac)
lemma count_filter:
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
by (induct xs) auto
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
apply (induct ls arbitrary: i)
apply simp
apply (case_tac i)
apply auto
done
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
lemma multiset_of_eq_length:
assumes "multiset_of xs = multiset_of ys"
shows "length xs = length ys"
using assms
proof (induct arbitrary: ys rule: length_induct)
case (1 xs ys)
show ?case
proof (cases xs)
case Nil with "1.prems" show ?thesis by simp
next
case (Cons x xs')
note xCons = Cons
show ?thesis
proof (cases ys)
case Nil
with "1.prems" Cons show ?thesis by simp
next
case (Cons y ys')
have x_in_ys: "x = y \<or> x \<in> set ys'"
proof (cases "x = y")
case True then show ?thesis ..
next
case False
from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
with False show ?thesis by (simp add: mem_set_multiset_eq)
qed
from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
apply -
apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
apply fastsimp
done
with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
qed
qed
qed
text {*
This lemma shows which properties suffice to show that a function
@{text "f"} with @{text "f xs = ys"} behaves like sort.
*}
lemma properties_for_sort:
"multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
proof (induct xs arbitrary: ys)
case Nil then show ?case by simp
next
case (Cons x xs)
then have "x \<in> set ys"
by (auto simp add: mem_set_multiset_eq intro!: ccontr)
with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
qed
subsection {* Pointwise ordering induced by count *}
definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where
[code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
[code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
notation mset_le (infix "\<subseteq>#" 50)
notation mset_less (infix "\<subset>#" 50)
lemma mset_le_refl[simp]: "A \<le># A"
unfolding mset_le_def by auto
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
unfolding mset_le_def by (fast intro: order_trans)
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
apply (unfold mset_le_def)
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
apply (blast intro: order_antisym)
done
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
done
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
unfolding mset_le_def by auto
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
unfolding mset_le_def by auto
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
apply (unfold mset_le_def)
apply auto
apply (erule_tac x = a in allE)+
apply auto
done
lemma mset_le_add_left[simp]: "A \<le># A + B"
unfolding mset_le_def by auto
lemma mset_le_add_right[simp]: "B \<le># A + B"
unfolding mset_le_def by auto
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
by (simp add: mset_le_def)
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
by (simp add: multiset_eq_conv_count_eq mset_le_def)
lemma mset_le_multiset_union_diff_commute:
assumes "B \<le># A"
shows "A - B + C = A + C - B"
proof -
from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
from this obtain D where "A = B + D" ..
then show ?thesis
apply simp
apply (subst union_commute)
apply (subst multiset_diff_union_assoc)
apply simp
apply (simp add: diff_cancel)
apply (subst union_assoc)
apply (subst union_commute[of "B" _])
apply (subst multiset_diff_union_assoc)
apply simp
apply (simp add: diff_cancel)
done
qed
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
apply (induct xs)
apply auto
apply (rule mset_le_trans)
apply auto
done
lemma multiset_of_update:
"i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
proof (induct ls arbitrary: i)
case Nil then show ?case by simp
next
case (Cons x xs)
show ?case
proof (cases i)
case 0 then show ?thesis by simp
next
case (Suc i')
with Cons show ?thesis
apply simp
apply (subst union_assoc)
apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
apply (subst union_assoc [symmetric])
apply simp
apply (rule mset_le_multiset_union_diff_commute)
apply (simp add: mset_le_single nth_mem_multiset_of)
done
qed
qed
lemma multiset_of_swap:
"i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
apply (case_tac "i = j")
apply simp
apply (simp add: multiset_of_update)
apply (subst elem_imp_eq_diff_union[symmetric])
apply (simp add: nth_mem_multiset_of)
apply simp
done
interpretation mset_order: order "op \<le>#" "op <#"
proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
mset_le_trans simp: mset_less_def)
interpretation mset_order_cancel_semigroup:
pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
proof qed (erule mset_le_mono_add [OF mset_le_refl])
interpretation mset_order_semigroup_cancel:
pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
proof qed simp
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
apply (clarsimp simp: mset_le_def mset_less_def)
apply (erule_tac x=x in allE)
apply auto
done
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
apply (clarsimp simp: mset_le_def mset_less_def)
apply (erule_tac x = x in allE)
apply auto
done
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
apply (rule conjI)
apply (simp add: mset_lessD)
apply (clarsimp simp: mset_le_def mset_less_def)
apply safe
apply (erule_tac x = a in allE)
apply (auto split: split_if_asm)
done
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
apply (rule conjI)
apply (simp add: mset_leD)
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
done
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False"
by (induct A) (auto simp: mset_le_def mset_less_def)
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
by (auto simp: mset_le_def mset_less_def)
lemma multi_psub_self[simp]: "A \<subset># A = False"
by (auto simp: mset_le_def mset_less_def)
lemma mset_less_add_bothsides:
"T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
by (auto simp: mset_le_def mset_less_def)
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
by (auto simp: mset_le_def mset_less_def)
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
proof (induct A arbitrary: B)
case (empty M)
then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
then obtain M' x where "M = M' + {#x#}"
by (blast dest: multi_nonempty_split)
then show ?case by simp
next
case (add S x T)
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
have SxsubT: "S + {#x#} \<subset># T" by fact
then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
then obtain T' where T: "T = T' + {#x#}"
by (blast dest: multi_member_split)
then have "S \<subset># T'" using SxsubT
by (blast intro: mset_less_add_bothsides)
then have "size S < size T'" using IH by simp
then show ?case using T by simp
qed
lemmas mset_less_trans = mset_order.less_trans
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
subsection {* Strong induction and subset induction for multisets *}
text {* Well-foundedness of proper subset operator: *}
text {* proper multiset subset *}
definition
mset_less_rel :: "('a multiset * 'a multiset) set" where
"mset_less_rel = {(A,B). A \<subset># B}"
lemma multiset_add_sub_el_shuffle:
assumes "c \<in># B" and "b \<noteq> c"
shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
proof -
from `c \<in># B` obtain A where B: "B = A + {#c#}"
by (blast dest: multi_member_split)
have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
by (simp add: union_ac)
then show ?thesis using B by simp
qed
lemma wf_mset_less_rel: "wf mset_less_rel"
apply (unfold mset_less_rel_def)
apply (rule wf_measure [THEN wf_subset, where f1=size])
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
done
text {* The induction rules: *}
lemma full_multiset_induct [case_names less]:
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
shows "P B"
apply (rule wf_mset_less_rel [THEN wf_induct])
apply (rule ih, auto simp: mset_less_rel_def)
done
lemma multi_subset_induct [consumes 2, case_names empty add]:
assumes "F \<subseteq># A"
and empty: "P {#}"
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
shows "P F"
proof -
from `F \<subseteq># A`
show ?thesis
proof (induct F)
show "P {#}" by fact
next
fix x F
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
show "P (F + {#x#})"
proof (rule insert)
from i show "x \<in># A" by (auto dest: mset_le_insertD)
from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
with P show "P F" .
qed
qed
qed
text{* A consequence: Extensionality. *}
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
apply (rule iffI)
prefer 2
apply clarsimp
apply (induct A arbitrary: B rule: full_multiset_induct)
apply (rename_tac C)
apply (case_tac B rule: multiset_cases)
apply (simp add: empty_multiset_count)
apply simp
apply (case_tac "x \<in># C")
apply (force dest: multi_member_split)
apply (erule_tac x = x in allE)
apply simp
done
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
subsection {* The fold combinator *}
text {*
The intended behaviour is
@{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
if @{text f} is associative-commutative.
*}
text {*
The graph of @{text "fold_mset"}, @{text "z"}: the start element,
@{text "f"}: folding function, @{text "A"}: the multiset, @{text
"y"}: the result.
*}
inductive
fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool"
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
and z :: 'b
where
emptyI [intro]: "fold_msetG f z {#} z"
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y"
definition
fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
"fold_mset f z A = (THE x. fold_msetG f z A x)"
lemma Diff1_fold_msetG:
"fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
apply (frule_tac x = x in fold_msetG.insertI)
apply auto
done
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
apply (induct A)
apply blast
apply clarsimp
apply (drule_tac x = x in fold_msetG.insertI)
apply auto
done
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
unfolding fold_mset_def by blast
locale left_commutative =
fixes f :: "'a => 'b => 'b"
assumes left_commute: "f x (f y z) = f y (f x z)"
begin
lemma fold_msetG_determ:
"fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
proof (induct arbitrary: x y z rule: full_multiset_induct)
case (less M x\<^isub>1 x\<^isub>2 Z)
have IH: "\<forall>A. A \<subset># M \<longrightarrow>
(\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
\<longrightarrow> x' = x)" by fact
have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+
show ?case
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
assume "M = {#}" and "x\<^isub>1 = Z"
then show ?case using Mfoldx\<^isub>2 by auto
next
fix B b u
assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
show ?case
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
assume "M = {#}" "x\<^isub>2 = Z"
then show ?case using Mfoldx\<^isub>1 by auto
next
fix C c v
assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
then have CsubM: "C \<subset># M" by simp
from MBb have BsubM: "B \<subset># M" by simp
show ?case
proof cases
assume "b=c"
then moreover have "B = C" using MBb MCc by auto
ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
next
assume diff: "b \<noteq> c"
let ?D = "B - {#c#}"
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
by (auto intro: insert_noteq_member dest: sym)
have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
from MBb MCc have "B + {#b#} = C + {#c#}" by blast
then have [simp]: "B + {#b#} - {#c#} = C"
using MBb MCc binC cinB by auto
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
using MBb MCc diff binC cinB
by (auto simp: multiset_add_sub_el_shuffle)
then obtain d where Dfoldd: "fold_msetG f Z ?D d"
using fold_msetG_nonempty by iprover
then have "fold_msetG f Z B (f c d)" using cinB
by (rule Diff1_fold_msetG)
then have "f c d = u" using IH BsubM Bu by blast
moreover
have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
by (auto simp: multiset_add_sub_el_shuffle
dest: fold_msetG.insertI [where x=b])
then have "f b d = v" using IH CsubM Cv by blast
ultimately show ?thesis using x\<^isub>1 x\<^isub>2
by (auto simp: left_commute)
qed
qed
qed
qed
lemma fold_mset_insert_aux:
"(fold_msetG f z (A + {#x#}) v) =
(\<exists>y. fold_msetG f z A y \<and> v = f x y)"
apply (rule iffI)
prefer 2
apply blast
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
apply (blast intro: fold_msetG_determ)
done
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
unfolding fold_mset_def by (blast intro: fold_msetG_determ)
lemma fold_mset_insert:
"fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)
apply (rule the_equality)
apply (auto cong add: conj_cong
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
done
lemma fold_mset_insert_idem:
"fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
apply (simp add: fold_mset_def fold_mset_insert_aux)
apply (rule the_equality)
apply (auto cong add: conj_cong
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
done
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
by (induct A) (auto simp: fold_mset_insert left_commute [of x])
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
using fold_mset_insert [of z "{#}"] by simp
lemma fold_mset_union [simp]:
"fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
proof (induct A)
case empty then show ?case by simp
next
case (add A x)
have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
by (simp add: fold_mset_insert)
also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
finally show ?case .
qed
lemma fold_mset_fusion:
assumes "left_commutative g"
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")
proof -
interpret left_commutative g by fact
show "PROP ?P" by (induct A) auto
qed
lemma fold_mset_rec:
assumes "a \<in># A"
shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
proof -
from assms obtain A' where "A = A' + {#a#}"
by (blast dest: multi_member_split)
then show ?thesis by simp
qed
end
text {*
A note on code generation: When defining some function containing a
subterm @{term"fold_mset F"}, code generation is not automatic. When
interpreting locale @{text left_commutative} with @{text F}, the
would be code thms for @{const fold_mset} become thms like
@{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
contains defined symbols, i.e.\ is not a code thm. Hence a separate
constant with its own code thms needs to be introduced for @{text
F}. See the image operator below.
*}
subsection {* Image *}
definition [code del]:
"image_mset f = fold_mset (op + o single o f) {#}"
interpretation image_left_comm: left_commutative "op + o single o f"
proof qed (simp add:union_ac)
lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
by (simp add: image_mset_def)
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
by (simp add: image_mset_def)
lemma image_mset_insert:
"image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
by (simp add: image_mset_def add_ac)
lemma image_mset_union [simp]:
"image_mset f (M+N) = image_mset f M + image_mset f N"
apply (induct N)
apply simp
apply (simp add: union_assoc [symmetric] image_mset_insert)
done
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
by (induct M) simp_all
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
by (cases M) auto
syntax
comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
("({#_/. _ :# _#})")
translations
"{#e. x:#M#}" == "CONST image_mset (%x. e) M"
syntax
comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
("({#_/ | _ :# _./ _#})")
translations
"{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
text {*
This allows to write not just filters like @{term "{#x:#M. x<c#}"}
but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
"{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
@{term "{#x+x|x:#M. x<c#}"}.
*}
subsection {* Termination proofs with multiset orders *}
lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
and multi_member_this: "x \<in># {# x #} + XS"
and multi_member_last: "x \<in># {# x #}"
by auto
definition "ms_strict = mult pair_less"
definition [code del]: "ms_weak = ms_strict \<union> Id"
lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
by (auto intro: wf_mult1 wf_trancl simp: mult_def)
lemma smsI:
"(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
unfolding ms_strict_def
by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
lemma wmsI:
"(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
\<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
unfolding ms_weak_def ms_strict_def
by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
inductive pw_leq
where
pw_leq_empty: "pw_leq {#} {#}"
| pw_leq_step: "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
lemma pw_leq_lstep:
"(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
by (drule pw_leq_step) (rule pw_leq_empty, simp)
lemma pw_leq_split:
assumes "pw_leq X Y"
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 = {#}))"
using assms
proof (induct)
case pw_leq_empty thus ?case by auto
next
case (pw_leq_step x y X Y)
then obtain A B Z where
[simp]: "X = A + Z" "Y = B + Z"
and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
by auto
from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
unfolding pair_leq_def by auto
thus ?case
proof
assume [simp]: "x = y"
have
"{#x#} + X = A + ({#y#}+Z)
\<and> {#y#} + Y = B + ({#y#}+Z)
\<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
by (auto simp: add_ac)
thus ?case by (intro exI)
next
assume A: "(x, y) \<in> pair_less"
let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
have "{#x#} + X = ?A' + Z"
"{#y#} + Y = ?B' + Z"
by (auto simp add: add_ac)
moreover have
"(set_of ?A', set_of ?B') \<in> max_strict"
using 1 A unfolding max_strict_def
by (auto elim!: max_ext.cases)
ultimately show ?thesis by blast
qed
qed
lemma
assumes pwleq: "pw_leq Z Z'"
shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
and ms_weakI1: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
and ms_weakI2: "(Z + {#}, Z' + {#}) \<in> ms_weak"
proof -
from pw_leq_split[OF pwleq]
obtain A' B' Z''
where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
by blast
{
assume max: "(set_of A, set_of B) \<in> max_strict"
from mx_or_empty
have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
proof
assume max': "(set_of A', set_of B') \<in> max_strict"
with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
by (auto simp: max_strict_def intro: max_ext_additive)
thus ?thesis by (rule smsI)
next
assume [simp]: "A' = {#} \<and> B' = {#}"
show ?thesis by (rule smsI) (auto intro: max)
qed
thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
}
from mx_or_empty
have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
qed
lemma empty_idemp: "{#} + x = x" "x + {#} = x"
and nonempty_plus: "{# x #} + rs \<noteq> {#}"
and nonempty_single: "{# x #} \<noteq> {#}"
by auto
setup {*
let
fun msetT T = Type ("Multiset.multiset", [T]);
fun mk_mset T [] = Const (@{const_name Mempty}, msetT T)
| mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
| mk_mset T (x :: xs) =
Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
mk_mset T [x] $ mk_mset T xs
fun mset_member_tac m i =
(if m <= 0 then
rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
else
rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
val mset_nonempty_tac =
rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
val regroup_munion_conv =
FundefLib.regroup_conv @{const_name Multiset.Mempty} @{const_name plus}
(map (fn t => t RS eq_reflection) (@{thms union_ac} @ @{thms empty_idemp}))
fun unfold_pwleq_tac i =
(rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
ORELSE (rtac @{thm pw_leq_lstep} i)
ORELSE (rtac @{thm pw_leq_empty} i)
val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
@{thm Un_insert_left}, @{thm Un_empty_left}]
in
ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
{
msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
reduction_pair= @{thm ms_reduction_pair}
})
end
*}
end