(* Title: HOL/Codatatype/More_BNFs.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Author: Andreas Lochbihler, Karlsruhe Institute of Technology
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Registration of various types as bounded natural functors.
*)
header {* Registration of Various Types as Bounded Natural Functors *}
theory More_BNFs
imports
BNF_LFP
BNF_GFP
"~~/src/HOL/Quotient_Examples/FSet"
"~~/src/HOL/Library/Multiset"
Countable_Set
begin
lemma option_rec_conv_option_case: "option_rec = option_case"
by (simp add: fun_eq_iff split: option.split)
bnf_def option = Option.map [Option.set] "\<lambda>_::'a option. natLeq" ["None"]
proof -
show "Option.map id = id" by (simp add: fun_eq_iff Option.map_def split: option.split)
next
fix f g
show "Option.map (g \<circ> f) = Option.map g \<circ> Option.map f"
by (auto simp add: fun_eq_iff Option.map_def split: option.split)
next
fix f g x
assume "\<And>z. z \<in> Option.set x \<Longrightarrow> f z = g z"
thus "Option.map f x = Option.map g x"
by (simp cong: Option.map_cong)
next
fix f
show "Option.set \<circ> Option.map f = op ` f \<circ> Option.set"
by fastforce
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix x
show "|Option.set x| \<le>o natLeq"
by (cases x) (simp_all add: ordLess_imp_ordLeq finite_iff_ordLess_natLeq[symmetric])
next
fix A
have unfold: "{x. Option.set x \<subseteq> A} = Some ` A \<union> {None}"
by (auto simp add: option_rec_conv_option_case Option.set_def split: option.split_asm)
show "|{x. Option.set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
apply (rule ordIso_ordLeq_trans)
apply (rule card_of_ordIso_subst[OF unfold])
apply (rule ordLeq_transitive)
apply (rule Un_csum)
apply (rule ordLeq_transitive)
apply (rule csum_mono)
apply (rule card_of_image)
apply (rule ordIso_ordLeq_trans)
apply (rule single_cone)
apply (rule cone_ordLeq_ctwo)
apply (rule ordLeq_cexp1)
apply (simp_all add: natLeq_cinfinite natLeq_Card_order cinfinite_not_czero Card_order_csum)
done
next
fix A B1 B2 f1 f2 p1 p2
assume wpull: "wpull A B1 B2 f1 f2 p1 p2"
show "wpull {x. Option.set x \<subseteq> A} {x. Option.set x \<subseteq> B1} {x. Option.set x \<subseteq> B2}
(Option.map f1) (Option.map f2) (Option.map p1) (Option.map p2)"
(is "wpull ?A ?B1 ?B2 ?f1 ?f2 ?p1 ?p2")
unfolding wpull_def
proof (intro strip, elim conjE)
fix b1 b2
assume "b1 \<in> ?B1" "b2 \<in> ?B2" "?f1 b1 = ?f2 b2"
thus "\<exists>a \<in> ?A. ?p1 a = b1 \<and> ?p2 a = b2" using wpull
unfolding wpull_def by (cases b2) (auto 4 5)
qed
next
fix z
assume "z \<in> Option.set None"
thus False by simp
qed
lemma card_of_list_in:
"|{xs. set xs \<subseteq> A}| \<le>o |Pfunc (UNIV :: nat set) A|" (is "|?LHS| \<le>o |?RHS|")
proof -
let ?f = "%xs. %i. if i < length xs \<and> set xs \<subseteq> A then Some (nth xs i) else None"
have "inj_on ?f ?LHS" unfolding inj_on_def fun_eq_iff
proof safe
fix xs :: "'a list" and ys :: "'a list"
assume su: "set xs \<subseteq> A" "set ys \<subseteq> A" and eq: "\<forall>i. ?f xs i = ?f ys i"
hence *: "length xs = length ys"
by (metis linorder_cases option.simps(2) order_less_irrefl)
thus "xs = ys" by (rule nth_equalityI) (metis * eq su option.inject)
qed
moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Pfunc_def by fastforce
ultimately show ?thesis using card_of_ordLeq by blast
qed
lemma list_in_empty: "A = {} \<Longrightarrow> {x. set x \<subseteq> A} = {[]}"
by simp
lemma card_of_Func: "|Func A B| =o |B| ^c |A|"
unfolding cexp_def Field_card_of by (rule card_of_refl)
lemma not_emp_czero_notIn_ordIso_Card_order:
"A \<noteq> {} \<Longrightarrow> ( |A|, czero) \<notin> ordIso \<and> Card_order |A|"
apply (rule conjI)
apply (metis Field_card_of czeroE)
by (rule card_of_Card_order)
lemma list_in_bd: "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
proof -
fix A :: "'a set"
show "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
proof (cases "A = {}")
case False thus ?thesis
apply -
apply (rule ordLeq_transitive)
apply (rule card_of_list_in)
apply (rule ordLeq_transitive)
apply (erule card_of_Pfunc_Pow_Func)
apply (rule ordIso_ordLeq_trans)
apply (rule Times_cprod)
apply (rule cprod_cinfinite_bound)
apply (rule ordIso_ordLeq_trans)
apply (rule Pow_cexp_ctwo)
apply (rule ordIso_ordLeq_trans)
apply (rule cexp_cong2)
apply (rule card_of_nat)
apply (rule Card_order_ctwo)
apply (rule card_of_Card_order)
apply (rule natLeq_Card_order)
apply (rule disjI1)
apply (rule ctwo_Cnotzero)
apply (rule cexp_mono1)
apply (rule ordLeq_csum2)
apply (rule Card_order_ctwo)
apply (rule disjI1)
apply (rule ctwo_Cnotzero)
apply (rule natLeq_Card_order)
apply (rule ordIso_ordLeq_trans)
apply (rule card_of_Func)
apply (rule ordIso_ordLeq_trans)
apply (rule cexp_cong2)
apply (rule card_of_nat)
apply (rule card_of_Card_order)
apply (rule card_of_Card_order)
apply (rule natLeq_Card_order)
apply (rule disjI1)
apply (erule not_emp_czero_notIn_ordIso_Card_order)
apply (rule cexp_mono1)
apply (rule ordLeq_csum1)
apply (rule card_of_Card_order)
apply (rule disjI1)
apply (erule not_emp_czero_notIn_ordIso_Card_order)
apply (rule natLeq_Card_order)
apply (rule card_of_Card_order)
apply (rule card_of_Card_order)
apply (rule Cinfinite_cexp)
apply (rule ordLeq_csum2)
apply (rule Card_order_ctwo)
apply (rule conjI)
apply (rule natLeq_cinfinite)
by (rule natLeq_Card_order)
next
case True thus ?thesis
apply -
apply (rule ordIso_ordLeq_trans)
apply (rule card_of_ordIso_subst)
apply (erule list_in_empty)
apply (rule ordIso_ordLeq_trans)
apply (rule single_cone)
apply (rule cone_ordLeq_cexp)
apply (rule ordLeq_transitive)
apply (rule cone_ordLeq_ctwo)
apply (rule ordLeq_csum2)
by (rule Card_order_ctwo)
qed
qed
bnf_def list = map [set] "\<lambda>_::'a list. natLeq" ["[]"]
proof -
show "map id = id" by (rule List.map.id)
next
fix f g
show "map (g o f) = map g o map f" by (rule List.map.comp[symmetric])
next
fix x f g
assume "\<And>z. z \<in> set x \<Longrightarrow> f z = g z"
thus "map f x = map g x" by simp
next
fix f
show "set o map f = image f o set" by (rule ext, unfold o_apply, rule set_map)
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix x
show "|set x| \<le>o natLeq"
apply (rule ordLess_imp_ordLeq)
apply (rule finite_ordLess_infinite[OF _ natLeq_Well_order])
unfolding Field_natLeq Field_card_of by (auto simp: card_of_well_order_on)
next
fix A :: "'a set"
show "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" by (rule list_in_bd)
next
fix A B1 B2 f1 f2 p1 p2
assume "wpull A B1 B2 f1 f2 p1 p2"
hence pull: "\<And>b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<Longrightarrow> \<exists>a \<in> A. p1 a = b1 \<and> p2 a = b2"
unfolding wpull_def by auto
show "wpull {x. set x \<subseteq> A} {x. set x \<subseteq> B1} {x. set x \<subseteq> B2} (map f1) (map f2) (map p1) (map p2)"
(is "wpull ?A ?B1 ?B2 _ _ _ _")
proof (unfold wpull_def)
{ fix as bs assume *: "as \<in> ?B1" "bs \<in> ?B2" "map f1 as = map f2 bs"
hence "length as = length bs" by (metis length_map)
hence "\<exists>zs \<in> ?A. map p1 zs = as \<and> map p2 zs = bs" using *
proof (induct as bs rule: list_induct2)
case (Cons a as b bs)
hence "a \<in> B1" "b \<in> B2" "f1 a = f2 b" by auto
with pull obtain z where "z \<in> A" "p1 z = a" "p2 z = b" by blast
moreover
from Cons obtain zs where "zs \<in> ?A" "map p1 zs = as" "map p2 zs = bs" by auto
ultimately have "z # zs \<in> ?A" "map p1 (z # zs) = a # as \<and> map p2 (z # zs) = b # bs" by auto
thus ?case by (rule_tac x = "z # zs" in bexI)
qed simp
}
thus "\<forall>as bs. as \<in> ?B1 \<and> bs \<in> ?B2 \<and> map f1 as = map f2 bs \<longrightarrow>
(\<exists>zs \<in> ?A. map p1 zs = as \<and> map p2 zs = bs)" by blast
qed
qed auto
(* Finite sets *)
abbreviation afset where "afset \<equiv> abs_fset"
abbreviation rfset where "rfset \<equiv> rep_fset"
lemma fset_fset_member:
"fset A = {a. a |\<in>| A}"
unfolding fset_def fset_member_def by auto
lemma afset_rfset:
"afset (rfset x) = x"
by (rule Quotient_fset[unfolded Quotient_def, THEN conjunct1, rule_format])
lemma afset_rfset_id:
"afset o rfset = id"
unfolding comp_def afset_rfset id_def ..
lemma rfset:
"rfset A = rfset B \<longleftrightarrow> A = B"
by (metis afset_rfset)
lemma afset_set:
"afset as = afset bs \<longleftrightarrow> set as = set bs"
using Quotient_fset unfolding Quotient_def list_eq_def by auto
lemma surj_afset:
"\<exists> as. A = afset as"
by (metis afset_rfset)
lemma fset_def2:
"fset = set o rfset"
unfolding fset_def map_fun_def[abs_def] by simp
lemma fset_def2_raw:
"fset A = set (rfset A)"
unfolding fset_def2 by simp
lemma fset_comp_afset:
"fset o afset = set"
unfolding fset_def2 comp_def apply(rule ext)
unfolding afset_set[symmetric] afset_rfset ..
lemma fset_afset:
"fset (afset as) = set as"
unfolding fset_comp_afset[symmetric] by simp
lemma set_rfset_afset:
"set (rfset (afset as)) = set as"
unfolding afset_set[symmetric] afset_rfset ..
lemma map_fset_comp_afset:
"(map_fset f) o afset = afset o (map f)"
unfolding map_fset_def map_fun_def[abs_def] comp_def apply(rule ext)
unfolding afset_set set_map set_rfset_afset id_apply ..
lemma map_fset_afset:
"(map_fset f) (afset as) = afset (map f as)"
using map_fset_comp_afset unfolding comp_def fun_eq_iff by auto
lemma fset_map_fset:
"fset (map_fset f A) = (image f) (fset A)"
apply(subst afset_rfset[symmetric, of A])
unfolding map_fset_afset fset_afset set_map
unfolding fset_def2_raw ..
lemma map_fset_def2:
"map_fset f = afset o (map f) o rfset"
unfolding map_fset_def map_fun_def[abs_def] by simp
lemma map_fset_def2_raw:
"map_fset f A = afset (map f (rfset A))"
unfolding map_fset_def2 by simp
lemma finite_ex_fset:
assumes "finite A"
shows "\<exists> B. fset B = A"
by (metis assms finite_list fset_afset)
lemma wpull_image:
assumes "wpull A B1 B2 f1 f2 p1 p2"
shows "wpull (Pow A) (Pow B1) (Pow B2) (image f1) (image f2) (image p1) (image p2)"
unfolding wpull_def Pow_def Bex_def mem_Collect_eq proof clarify
fix Y1 Y2 assume Y1: "Y1 \<subseteq> B1" and Y2: "Y2 \<subseteq> B2" and EQ: "f1 ` Y1 = f2 ` Y2"
def X \<equiv> "{a \<in> A. p1 a \<in> Y1 \<and> p2 a \<in> Y2}"
show "\<exists>X\<subseteq>A. p1 ` X = Y1 \<and> p2 ` X = Y2"
proof (rule exI[of _ X], intro conjI)
show "p1 ` X = Y1"
proof
show "Y1 \<subseteq> p1 ` X"
proof safe
fix y1 assume y1: "y1 \<in> Y1"
then obtain y2 where y2: "y2 \<in> Y2" and eq: "f1 y1 = f2 y2" using EQ by auto
then obtain x where "x \<in> A" and "p1 x = y1" and "p2 x = y2"
using assms y1 Y1 Y2 unfolding wpull_def by blast
thus "y1 \<in> p1 ` X" unfolding X_def using y1 y2 by auto
qed
qed(unfold X_def, auto)
show "p2 ` X = Y2"
proof
show "Y2 \<subseteq> p2 ` X"
proof safe
fix y2 assume y2: "y2 \<in> Y2"
then obtain y1 where y1: "y1 \<in> Y1" and eq: "f1 y1 = f2 y2" using EQ by force
then obtain x where "x \<in> A" and "p1 x = y1" and "p2 x = y2"
using assms y2 Y1 Y2 unfolding wpull_def by blast
thus "y2 \<in> p2 ` X" unfolding X_def using y1 y2 by auto
qed
qed(unfold X_def, auto)
qed(unfold X_def, auto)
qed
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
by (rule f_the_inv_into_f) (auto simp: inj_on_def fset_cong dest!: finite_ex_fset)
bnf_def fset = map_fset [fset] "\<lambda>_::'a fset. natLeq" ["{||}"]
proof -
show "map_fset id = id"
unfolding map_fset_def2 map_id o_id afset_rfset_id ..
next
fix f g
show "map_fset (g o f) = map_fset g o map_fset f"
unfolding map_fset_def2 map.comp[symmetric] comp_def apply(rule ext)
unfolding afset_set set_map fset_def2_raw[symmetric] image_image[symmetric]
unfolding map_fset_afset[symmetric] map_fset_image afset_rfset
by (rule refl)
next
fix x f g
assume "\<And>z. z \<in> fset x \<Longrightarrow> f z = g z"
hence "map f (rfset x) = map g (rfset x)"
apply(intro map_cong) unfolding fset_def2_raw by auto
thus "map_fset f x = map_fset g x" unfolding map_fset_def2_raw
by (rule arg_cong)
next
fix f
show "fset o map_fset f = image f o fset"
unfolding comp_def fset_map_fset ..
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix x
show "|fset x| \<le>o natLeq"
unfolding fset_def2_raw
apply (rule ordLess_imp_ordLeq)
apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
by (rule finite_set)
next
fix A :: "'a set"
have "|{x. fset x \<subseteq> A}| \<le>o |afset ` {as. set as \<subseteq> A}|"
apply(rule card_of_mono1) unfolding fset_def2_raw apply auto
apply (rule image_eqI)
by (auto simp: afset_rfset)
also have "|afset ` {as. set as \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|" using card_of_image .
also have "|{as. set as \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" by (rule list_in_bd)
finally show "|{x. fset x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
next
fix A B1 B2 f1 f2 p1 p2
assume wp: "wpull A B1 B2 f1 f2 p1 p2"
hence "wpull (Pow A) (Pow B1) (Pow B2) (image f1) (image f2) (image p1) (image p2)"
by (rule wpull_image)
show "wpull {x. fset x \<subseteq> A} {x. fset x \<subseteq> B1} {x. fset x \<subseteq> B2}
(map_fset f1) (map_fset f2) (map_fset p1) (map_fset p2)"
unfolding wpull_def Pow_def Bex_def mem_Collect_eq proof clarify
fix y1 y2
assume Y1: "fset y1 \<subseteq> B1" and Y2: "fset y2 \<subseteq> B2"
assume "map_fset f1 y1 = map_fset f2 y2"
hence EQ: "f1 ` (fset y1) = f2 ` (fset y2)" unfolding map_fset_def2_raw
unfolding afset_set set_map fset_def2_raw .
with Y1 Y2 obtain X where X: "X \<subseteq> A"
and Y1: "p1 ` X = fset y1" and Y2: "p2 ` X = fset y2"
using wpull_image[OF wp] unfolding wpull_def Pow_def
unfolding Bex_def mem_Collect_eq apply -
apply(erule allE[of _ "fset y1"], erule allE[of _ "fset y2"]) by auto
have "\<forall> y1' \<in> fset y1. \<exists> x. x \<in> X \<and> y1' = p1 x" using Y1 by auto
then obtain q1 where q1: "\<forall> y1' \<in> fset y1. q1 y1' \<in> X \<and> y1' = p1 (q1 y1')" by metis
have "\<forall> y2' \<in> fset y2. \<exists> x. x \<in> X \<and> y2' = p2 x" using Y2 by auto
then obtain q2 where q2: "\<forall> y2' \<in> fset y2. q2 y2' \<in> X \<and> y2' = p2 (q2 y2')" by metis
def X' \<equiv> "q1 ` (fset y1) \<union> q2 ` (fset y2)"
have X': "X' \<subseteq> A" and Y1: "p1 ` X' = fset y1" and Y2: "p2 ` X' = fset y2"
using X Y1 Y2 q1 q2 unfolding X'_def by auto
have fX': "finite X'" unfolding X'_def by simp
then obtain x where X'eq: "X' = fset x" by (auto dest: finite_ex_fset)
show "\<exists>x. fset x \<subseteq> A \<and> map_fset p1 x = y1 \<and> map_fset p2 x = y2"
apply(intro exI[of _ "x"]) using X' Y1 Y2
unfolding X'eq map_fset_def2_raw fset_def2_raw set_map[symmetric]
afset_set[symmetric] afset_rfset by simp
qed
qed auto
lemma fset_pred[simp]: "fset_pred R a b \<longleftrightarrow>
((\<forall>t \<in> fset a. (\<exists>u \<in> fset b. R t u)) \<and>
(\<forall>t \<in> fset b. (\<exists>u \<in> fset a. R u t)))" (is "?L = ?R")
proof
assume ?L thus ?R unfolding fset_rel_def fset_pred_def
Gr_def relcomp_unfold converse_unfold
apply (simp add: subset_eq Ball_def)
apply (rule conjI)
apply (clarsimp, metis snd_conv)
by (clarsimp, metis fst_conv)
next
assume ?R
def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?R'")
have "finite ?R'" by (intro finite_Int[OF disjI2] finite_cartesian_product) auto
hence *: "fset R' = ?R'" unfolding R'_def by (intro fset_to_fset)
show ?L unfolding fset_rel_def fset_pred_def Gr_def relcomp_unfold converse_unfold
proof (intro CollectI prod_caseI exI conjI)
from * show "(R', a) = (R', map_fset fst R')" using conjunct1[OF `?R`]
by (auto simp add: fset_cong[symmetric] image_def Int_def split: prod.splits)
from * show "(R', b) = (R', map_fset snd R')" using conjunct2[OF `?R`]
by (auto simp add: fset_cong[symmetric] image_def Int_def split: prod.splits)
qed (auto simp add: *)
qed
(* Countable sets *)
lemma card_of_countable_sets_range:
fixes A :: "'a set"
shows "|{X. X \<subseteq> A \<and> countable X \<and> X \<noteq> {}}| \<le>o |{f::nat \<Rightarrow> 'a. range f \<subseteq> A}|"
apply(rule card_of_ordLeqI[of fromNat]) using inj_on_fromNat
unfolding inj_on_def by auto
lemma card_of_countable_sets_Func:
"|{X. X \<subseteq> A \<and> countable X \<and> X \<noteq> {}}| \<le>o |A| ^c natLeq"
using card_of_countable_sets_range card_of_Func_UNIV[THEN ordIso_symmetric]
unfolding cexp_def Field_natLeq Field_card_of
by (rule ordLeq_ordIso_trans)
lemma ordLeq_countable_subsets:
"|A| \<le>o |{X. X \<subseteq> A \<and> countable X}|"
apply (rule card_of_ordLeqI[of "\<lambda> a. {a}"]) unfolding inj_on_def by auto
lemma finite_countable_subset:
"finite {X. X \<subseteq> A \<and> countable X} \<longleftrightarrow> finite A"
apply default
apply (erule contrapos_pp)
apply (rule card_of_ordLeq_infinite)
apply (rule ordLeq_countable_subsets)
apply assumption
apply (rule finite_Collect_conjI)
apply (rule disjI1)
by (erule finite_Collect_subsets)
lemma card_of_countable_sets:
"|{X. X \<subseteq> A \<and> countable X}| \<le>o ( |A| +c ctwo) ^c natLeq"
(is "|?L| \<le>o _")
proof(cases "finite A")
let ?R = "Func (UNIV::nat set) (A <+> (UNIV::bool set))"
case True hence "finite ?L" by simp
moreover have "infinite ?R"
apply(rule infinite_Func[of _ "Inr True" "Inr False"]) by auto
ultimately show ?thesis unfolding cexp_def csum_def ctwo_def Field_natLeq Field_card_of
apply(intro ordLess_imp_ordLeq) by (rule finite_ordLess_infinite2)
next
case False
hence "|{X. X \<subseteq> A \<and> countable X}| =o |{X. X \<subseteq> A \<and> countable X} - {{}}|"
by (intro card_of_infinite_diff_finitte finite.emptyI finite.insertI ordIso_symmetric)
(unfold finite_countable_subset)
also have "|{X. X \<subseteq> A \<and> countable X} - {{}}| \<le>o |A| ^c natLeq"
using card_of_countable_sets_Func[of A] unfolding set_diff_eq by auto
also have "|A| ^c natLeq \<le>o ( |A| +c ctwo) ^c natLeq"
apply(rule cexp_mono1_cone_ordLeq)
apply(rule ordLeq_csum1, rule card_of_Card_order)
apply (rule cone_ordLeq_cexp)
apply (rule cone_ordLeq_Cnotzero)
using csum_Cnotzero2 ctwo_Cnotzero apply blast
by (rule natLeq_Card_order)
finally show ?thesis .
qed
bnf_def cset = cIm [rcset] "\<lambda>_::'a cset. natLeq" ["cEmp"]
proof -
show "cIm id = id" unfolding cIm_def[abs_def] id_def by auto
next
fix f g show "cIm (g \<circ> f) = cIm g \<circ> cIm f"
unfolding cIm_def[abs_def] apply(rule ext) unfolding comp_def by auto
next
fix C f g assume eq: "\<And>a. a \<in> rcset C \<Longrightarrow> f a = g a"
thus "cIm f C = cIm g C"
unfolding cIm_def[abs_def] unfolding image_def by auto
next
fix f show "rcset \<circ> cIm f = op ` f \<circ> rcset" unfolding cIm_def[abs_def] by auto
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix C show "|rcset C| \<le>o natLeq" using rcset unfolding countable_def .
next
fix A :: "'a set"
have "|{Z. rcset Z \<subseteq> A}| \<le>o |acset ` {X. X \<subseteq> A \<and> countable X}|"
apply(rule card_of_mono1) unfolding Pow_def image_def
proof (rule Collect_mono, clarsimp)
fix x
assume "rcset x \<subseteq> A"
hence "rcset x \<subseteq> A \<and> countable (rcset x) \<and> x = acset (rcset x)"
using acset_rcset[of x] rcset[of x] by force
thus "\<exists>y \<subseteq> A. countable y \<and> x = acset y" by blast
qed
also have "|acset ` {X. X \<subseteq> A \<and> countable X}| \<le>o |{X. X \<subseteq> A \<and> countable X}|"
using card_of_image .
also have "|{X. X \<subseteq> A \<and> countable X}| \<le>o ( |A| +c ctwo) ^c natLeq"
using card_of_countable_sets .
finally show "|{Z. rcset Z \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
next
fix A B1 B2 f1 f2 p1 p2
assume wp: "wpull A B1 B2 f1 f2 p1 p2"
show "wpull {x. rcset x \<subseteq> A} {x. rcset x \<subseteq> B1} {x. rcset x \<subseteq> B2}
(cIm f1) (cIm f2) (cIm p1) (cIm p2)"
unfolding wpull_def proof safe
fix y1 y2
assume Y1: "rcset y1 \<subseteq> B1" and Y2: "rcset y2 \<subseteq> B2"
assume "cIm f1 y1 = cIm f2 y2"
hence EQ: "f1 ` (rcset y1) = f2 ` (rcset y2)"
unfolding cIm_def by auto
with Y1 Y2 obtain X where X: "X \<subseteq> A"
and Y1: "p1 ` X = rcset y1" and Y2: "p2 ` X = rcset y2"
using wpull_image[OF wp] unfolding wpull_def Pow_def
unfolding Bex_def mem_Collect_eq apply -
apply(erule allE[of _ "rcset y1"], erule allE[of _ "rcset y2"]) by auto
have "\<forall> y1' \<in> rcset y1. \<exists> x. x \<in> X \<and> y1' = p1 x" using Y1 by auto
then obtain q1 where q1: "\<forall> y1' \<in> rcset y1. q1 y1' \<in> X \<and> y1' = p1 (q1 y1')" by metis
have "\<forall> y2' \<in> rcset y2. \<exists> x. x \<in> X \<and> y2' = p2 x" using Y2 by auto
then obtain q2 where q2: "\<forall> y2' \<in> rcset y2. q2 y2' \<in> X \<and> y2' = p2 (q2 y2')" by metis
def X' \<equiv> "q1 ` (rcset y1) \<union> q2 ` (rcset y2)"
have X': "X' \<subseteq> A" and Y1: "p1 ` X' = rcset y1" and Y2: "p2 ` X' = rcset y2"
using X Y1 Y2 q1 q2 unfolding X'_def by fast+
have fX': "countable X'" unfolding X'_def by simp
then obtain x where X'eq: "X' = rcset x" by (metis rcset_acset)
show "\<exists>x\<in>{x. rcset x \<subseteq> A}. cIm p1 x = y1 \<and> cIm p2 x = y2"
apply(intro bexI[of _ "x"]) using X' Y1 Y2 unfolding X'eq cIm_def by auto
qed
qed (unfold cEmp_def, auto)
(* Multisets *)
lemma setsum_gt_0_iff:
fixes f :: "'a \<Rightarrow> nat" assumes "finite A"
shows "setsum f A > 0 \<longleftrightarrow> (\<exists> a \<in> A. f a > 0)"
(is "?L \<longleftrightarrow> ?R")
proof-
have "?L \<longleftrightarrow> \<not> setsum f A = 0" by fast
also have "... \<longleftrightarrow> (\<exists> a \<in> A. f a \<noteq> 0)" using assms by simp
also have "... \<longleftrightarrow> ?R" by simp
finally show ?thesis .
qed
(* *)
definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> 'b \<Rightarrow> nat" where
"mmap h f b = setsum f {a. h a = b \<and> f a > 0}"
lemma mmap_id: "mmap id = id"
proof (rule ext)+
fix f a show "mmap id f a = id f a"
proof(cases "f a = 0")
case False
hence 1: "{aa. aa = a \<and> 0 < f aa} = {a}" by auto
show ?thesis by (simp add: mmap_def id_apply 1)
qed(unfold mmap_def, auto)
qed
lemma inj_on_setsum_inv:
assumes f: "f \<in> multiset"
and 1: "(0::nat) < setsum f {a. h a = b' \<and> 0 < f a}" (is "0 < setsum f ?A'")
and 2: "{a. h a = b \<and> 0 < f a} = {a. h a = b' \<and> 0 < f a}" (is "?A = ?A'")
shows "b = b'"
proof-
have "finite ?A'" using f unfolding multiset_def by auto
hence "?A' \<noteq> {}" using 1 setsum_gt_0_iff by auto
thus ?thesis using 2 by auto
qed
lemma mmap_comp:
fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
assumes f: "f \<in> multiset"
shows "mmap (h2 o h1) f = (mmap h2 o mmap h1) f"
unfolding mmap_def[abs_def] comp_def proof(rule ext)+
fix c :: 'c
let ?A = "{a. h2 (h1 a) = c \<and> 0 < f a}"
let ?As = "\<lambda> b. {a. h1 a = b \<and> 0 < f a}"
let ?B = "{b. h2 b = c \<and> 0 < setsum f (?As b)}"
have 0: "{?As b | b. b \<in> ?B} = ?As ` ?B" by auto
have "\<And> b. finite (?As b)" using f unfolding multiset_def by simp
hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" using setsum_gt_0_iff by auto
hence A: "?A = \<Union> {?As b | b. b \<in> ?B}" by auto
have "setsum f ?A = setsum (setsum f) {?As b | b. b \<in> ?B}"
unfolding A apply(rule setsum_Union_disjoint)
using f unfolding multiset_def by auto
also have "... = setsum (setsum f) (?As ` ?B)" unfolding 0 ..
also have "... = setsum (setsum f o ?As) ?B" apply(rule setsum_reindex)
unfolding inj_on_def apply auto using inj_on_setsum_inv[OF f, of h1] by blast
also have "... = setsum (\<lambda> b. setsum f (?As b)) ?B" unfolding comp_def ..
finally show "setsum f ?A = setsum (\<lambda> b. setsum f (?As b)) ?B" .
qed
lemma mmap_comp1:
fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
assumes "f \<in> multiset"
shows "mmap (\<lambda> a. h2 (h1 a)) f = mmap h2 (mmap h1 f)"
using mmap_comp[OF assms] unfolding comp_def by auto
lemma mmap:
assumes "f \<in> multiset"
shows "mmap h f \<in> multiset"
using assms unfolding mmap_def[abs_def] multiset_def proof safe
assume fin: "finite {a. 0 < f a}" (is "finite ?A")
show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
(is "finite {b. 0 < setsum f (?As b)}")
proof- let ?B = "{b. 0 < setsum f (?As b)}"
have "\<And> b. finite (?As b)" using assms unfolding multiset_def by simp
hence B: "?B = {b. ?As b \<noteq> {}}" using setsum_gt_0_iff by auto
hence "?B \<subseteq> h ` ?A" by auto
thus ?thesis using finite_surj[OF fin] by auto
qed
qed
lemma mmap_cong:
assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
shows "mmap f (count M) = mmap g (count M)"
using assms unfolding mmap_def[abs_def] by (intro ext, intro setsum_cong) auto
abbreviation supp where "supp f \<equiv> {a. f a > 0}"
lemma mmap_image_comp:
assumes f: "f \<in> multiset"
shows "(supp o mmap h) f = (image h o supp) f"
unfolding mmap_def[abs_def] comp_def proof-
have "\<And> b. finite {a. h a = b \<and> 0 < f a}" (is "\<And> b. finite (?As b)")
using f unfolding multiset_def by auto
thus "{b. 0 < setsum f (?As b)} = h ` {a. 0 < f a}"
using setsum_gt_0_iff by auto
qed
lemma mmap_image:
assumes f: "f \<in> multiset"
shows "supp (mmap h f) = h ` (supp f)"
using mmap_image_comp[OF assms] unfolding comp_def .
lemma set_of_Abs_multiset:
assumes f: "f \<in> multiset"
shows "set_of (Abs_multiset f) = supp f"
using assms unfolding set_of_def by (auto simp: Abs_multiset_inverse)
lemma supp_count:
"supp (count M) = set_of M"
using assms unfolding set_of_def by auto
lemma multiset_of_surj:
"multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
proof safe
fix M assume M: "set_of M \<subseteq> A"
obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
hence "set as \<subseteq> A" using M by auto
thus "M \<in> multiset_of ` {as. set as \<subseteq> A}" using eq by auto
next
show "\<And>x xa xb. \<lbrakk>set xa \<subseteq> A; xb \<in> set_of (multiset_of xa)\<rbrakk> \<Longrightarrow> xb \<in> A"
by (erule set_mp) (unfold set_of_multiset_of)
qed
lemma card_of_set_of:
"|{M. set_of M \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|"
apply(rule card_of_ordLeqI2[of _ multiset_of]) using multiset_of_surj by auto
lemma nat_sum_induct:
assumes "\<And>n1 n2. (\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow> phi m1 m2) \<Longrightarrow> phi n1 n2"
shows "phi (n1::nat) (n2::nat)"
proof-
let ?chi = "\<lambda> n1n2 :: nat * nat. phi (fst n1n2) (snd n1n2)"
have "?chi (n1,n2)"
apply(induct rule: measure_induct[of "\<lambda> n1n2. fst n1n2 + snd n1n2" ?chi])
using assms by (metis fstI sndI)
thus ?thesis by simp
qed
lemma matrix_count:
fixes ct1 ct2 :: "nat \<Rightarrow> nat"
assumes "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
shows
"\<exists> ct. (\<forall> i1 \<le> n1. setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ct1 i1) \<and>
(\<forall> i2 \<le> n2. setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ct2 i2)"
(is "?phi ct1 ct2 n1 n2")
proof-
have "\<forall> ct1 ct2 :: nat \<Rightarrow> nat.
setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"
proof(induct rule: nat_sum_induct[of
"\<lambda> n1 n2. \<forall> ct1 ct2 :: nat \<Rightarrow> nat.
setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"],
clarify)
fix n1 n2 :: nat and ct1 ct2 :: "nat \<Rightarrow> nat"
assume IH: "\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow>
\<forall> dt1 dt2 :: nat \<Rightarrow> nat.
setsum dt1 {..<Suc m1} = setsum dt2 {..<Suc m2} \<longrightarrow> ?phi dt1 dt2 m1 m2"
and ss: "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
show "?phi ct1 ct2 n1 n2"
proof(cases n1)
case 0 note n1 = 0
show ?thesis
proof(cases n2)
case 0 note n2 = 0
let ?ct = "\<lambda> i1 i2. ct2 0"
show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by simp
next
case (Suc m2) note n2 = Suc
let ?ct = "\<lambda> i1 i2. ct2 i2"
show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
qed
next
case (Suc m1) note n1 = Suc
show ?thesis
proof(cases n2)
case 0 note n2 = 0
let ?ct = "\<lambda> i1 i2. ct1 i1"
show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
next
case (Suc m2) note n2 = Suc
show ?thesis
proof(cases "ct1 n1 \<le> ct2 n2")
case True
def dt2 \<equiv> "\<lambda> i2. if i2 = n2 then ct2 i2 - ct1 n1 else ct2 i2"
have "setsum ct1 {..<Suc m1} = setsum dt2 {..<Suc n2}"
unfolding dt2_def using ss n1 True by auto
hence "?phi ct1 dt2 m1 n2" using IH[of m1 n2] n1 by simp
then obtain dt where
1: "\<And> i1. i1 \<le> m1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc n2} = ct1 i1" and
2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc m1} = dt2 i2" by auto
let ?ct = "\<lambda> i1 i2. if i1 = n1 then (if i2 = n2 then ct1 n1 else 0)
else dt i1 i2"
show ?thesis apply(rule exI[of _ ?ct])
using n1 n2 1 2 True unfolding dt2_def by simp
next
case False
hence False: "ct2 n2 < ct1 n1" by simp
def dt1 \<equiv> "\<lambda> i1. if i1 = n1 then ct1 i1 - ct2 n2 else ct1 i1"
have "setsum dt1 {..<Suc n1} = setsum ct2 {..<Suc m2}"
unfolding dt1_def using ss n2 False by auto
hence "?phi dt1 ct2 n1 m2" using IH[of n1 m2] n2 by simp
then obtain dt where
1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc m2} = dt1 i1" and
2: "\<And> i2. i2 \<le> m2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc n1} = ct2 i2" by force
let ?ct = "\<lambda> i1 i2. if i2 = n2 then (if i1 = n1 then ct2 n2 else 0)
else dt i1 i2"
show ?thesis apply(rule exI[of _ ?ct])
using n1 n2 1 2 False unfolding dt1_def by simp
qed
qed
qed
qed
thus ?thesis using assms by auto
qed
definition
"inj2 u B1 B2 \<equiv>
\<forall> b1 b1' b2 b2'. {b1,b1'} \<subseteq> B1 \<and> {b2,b2'} \<subseteq> B2 \<and> u b1 b2 = u b1' b2'
\<longrightarrow> b1 = b1' \<and> b2 = b2'"
lemma matrix_count_finite:
assumes B1: "B1 \<noteq> {}" "finite B1" and B2: "B2 \<noteq> {}" "finite B2" and u: "inj2 u B1 B2"
and ss: "setsum N1 B1 = setsum N2 B2"
shows "\<exists> M :: 'a \<Rightarrow> nat.
(\<forall> b1 \<in> B1. setsum (\<lambda> b2. M (u b1 b2)) B2 = N1 b1) \<and>
(\<forall> b2 \<in> B2. setsum (\<lambda> b1. M (u b1 b2)) B1 = N2 b2)"
proof-
obtain n1 where "card B1 = Suc n1" using B1 by (metis card_insert finite.simps)
then obtain e1 where e1: "bij_betw e1 {..<Suc n1} B1"
using ex_bij_betw_finite_nat[OF B1(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
hence e1_inj: "inj_on e1 {..<Suc n1}" and e1_surj: "e1 ` {..<Suc n1} = B1"
unfolding bij_betw_def by auto
def f1 \<equiv> "inv_into {..<Suc n1} e1"
have f1: "bij_betw f1 B1 {..<Suc n1}"
and f1e1[simp]: "\<And> i1. i1 < Suc n1 \<Longrightarrow> f1 (e1 i1) = i1"
and e1f1[simp]: "\<And> b1. b1 \<in> B1 \<Longrightarrow> e1 (f1 b1) = b1" unfolding f1_def
apply (metis bij_betw_inv_into e1, metis bij_betw_inv_into_left e1 lessThan_iff)
by (metis e1_surj f_inv_into_f)
(* *)
obtain n2 where "card B2 = Suc n2" using B2 by (metis card_insert finite.simps)
then obtain e2 where e2: "bij_betw e2 {..<Suc n2} B2"
using ex_bij_betw_finite_nat[OF B2(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
hence e2_inj: "inj_on e2 {..<Suc n2}" and e2_surj: "e2 ` {..<Suc n2} = B2"
unfolding bij_betw_def by auto
def f2 \<equiv> "inv_into {..<Suc n2} e2"
have f2: "bij_betw f2 B2 {..<Suc n2}"
and f2e2[simp]: "\<And> i2. i2 < Suc n2 \<Longrightarrow> f2 (e2 i2) = i2"
and e2f2[simp]: "\<And> b2. b2 \<in> B2 \<Longrightarrow> e2 (f2 b2) = b2" unfolding f2_def
apply (metis bij_betw_inv_into e2, metis bij_betw_inv_into_left e2 lessThan_iff)
by (metis e2_surj f_inv_into_f)
(* *)
let ?ct1 = "N1 o e1" let ?ct2 = "N2 o e2"
have ss: "setsum ?ct1 {..<Suc n1} = setsum ?ct2 {..<Suc n2}"
unfolding setsum_reindex[OF e1_inj, symmetric] setsum_reindex[OF e2_inj, symmetric]
e1_surj e2_surj using ss .
obtain ct where
ct1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ?ct1 i1" and
ct2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ?ct2 i2"
using matrix_count[OF ss] by blast
(* *)
def A \<equiv> "{u b1 b2 | b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"
have "\<forall> a \<in> A. \<exists> b1b2 \<in> B1 <*> B2. u (fst b1b2) (snd b1b2) = a"
unfolding A_def Ball_def mem_Collect_eq by auto
then obtain h1h2 where h12:
"\<And>a. a \<in> A \<Longrightarrow> u (fst (h1h2 a)) (snd (h1h2 a)) = a \<and> h1h2 a \<in> B1 <*> B2" by metis
def h1 \<equiv> "fst o h1h2" def h2 \<equiv> "snd o h1h2"
have h12[simp]: "\<And>a. a \<in> A \<Longrightarrow> u (h1 a) (h2 a) = a"
"\<And> a. a \<in> A \<Longrightarrow> h1 a \<in> B1" "\<And> a. a \<in> A \<Longrightarrow> h2 a \<in> B2"
using h12 unfolding h1_def h2_def by force+
{fix b1 b2 assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2"
hence inA: "u b1 b2 \<in> A" unfolding A_def by auto
hence "u b1 b2 = u (h1 (u b1 b2)) (h2 (u b1 b2))" by auto
moreover have "h1 (u b1 b2) \<in> B1" "h2 (u b1 b2) \<in> B2" using inA by auto
ultimately have "h1 (u b1 b2) = b1 \<and> h2 (u b1 b2) = b2"
using u b1 b2 unfolding inj2_def by fastforce
}
hence h1[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h1 (u b1 b2) = b1" and
h2[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h2 (u b1 b2) = b2" by auto
def M \<equiv> "\<lambda> a. ct (f1 (h1 a)) (f2 (h2 a))"
show ?thesis
apply(rule exI[of _ M]) proof safe
fix b1 assume b1: "b1 \<in> B1"
hence f1b1: "f1 b1 \<le> n1" using f1 unfolding bij_betw_def
by (metis bij_betwE f1 lessThan_iff less_Suc_eq_le)
have "(\<Sum>b2\<in>B2. M (u b1 b2)) = (\<Sum>i2<Suc n2. ct (f1 b1) (f2 (e2 i2)))"
unfolding e2_surj[symmetric] setsum_reindex[OF e2_inj]
unfolding M_def comp_def apply(intro setsum_cong) apply force
by (metis e2_surj b1 h1 h2 imageI)
also have "... = N1 b1" using b1 ct1[OF f1b1] by simp
finally show "(\<Sum>b2\<in>B2. M (u b1 b2)) = N1 b1" .
next
fix b2 assume b2: "b2 \<in> B2"
hence f2b2: "f2 b2 \<le> n2" using f2 unfolding bij_betw_def
by (metis bij_betwE f2 lessThan_iff less_Suc_eq_le)
have "(\<Sum>b1\<in>B1. M (u b1 b2)) = (\<Sum>i1<Suc n1. ct (f1 (e1 i1)) (f2 b2))"
unfolding e1_surj[symmetric] setsum_reindex[OF e1_inj]
unfolding M_def comp_def apply(intro setsum_cong) apply force
by (metis e1_surj b2 h1 h2 imageI)
also have "... = N2 b2" using b2 ct2[OF f2b2] by simp
finally show "(\<Sum>b1\<in>B1. M (u b1 b2)) = N2 b2" .
qed
qed
lemma supp_vimage_mmap:
assumes "M \<in> multiset"
shows "supp M \<subseteq> f -` (supp (mmap f M))"
using assms by (auto simp: mmap_image)
lemma mmap_ge_0:
assumes "M \<in> multiset"
shows "0 < mmap f M b \<longleftrightarrow> (\<exists>a. 0 < M a \<and> f a = b)"
proof-
have f: "finite {a. f a = b \<and> 0 < M a}" using assms unfolding multiset_def by auto
show ?thesis unfolding mmap_def setsum_gt_0_iff[OF f] by auto
qed
lemma finite_twosets:
assumes "finite B1" and "finite B2"
shows "finite {u b1 b2 |b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}" (is "finite ?A")
proof-
have A: "?A = (\<lambda> b1b2. u (fst b1b2) (snd b1b2)) ` (B1 <*> B2)" by force
show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
qed
lemma wp_mmap:
fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
shows
"wpull {M. M \<in> multiset \<and> supp M \<subseteq> A}
{N1. N1 \<in> multiset \<and> supp N1 \<subseteq> B1} {N2. N2 \<in> multiset \<and> supp N2 \<subseteq> B2}
(mmap f1) (mmap f2) (mmap p1) (mmap p2)"
unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
fix N1 :: "'b1 \<Rightarrow> nat" and N2 :: "'b2 \<Rightarrow> nat"
assume mmap': "mmap f1 N1 = mmap f2 N2"
and N1[simp]: "N1 \<in> multiset" "supp N1 \<subseteq> B1"
and N2[simp]: "N2 \<in> multiset" "supp N2 \<subseteq> B2"
have mN1[simp]: "mmap f1 N1 \<in> multiset" using N1 by (auto simp: mmap)
have mN2[simp]: "mmap f2 N2 \<in> multiset" using N2 by (auto simp: mmap)
def P \<equiv> "mmap f1 N1"
have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
note P = P1 P2
have P_mult[simp]: "P \<in> multiset" unfolding P_def using N1 by auto
have fin_N1[simp]: "finite (supp N1)" using N1(1) unfolding multiset_def by auto
have fin_N2[simp]: "finite (supp N2)" using N2(1) unfolding multiset_def by auto
have fin_P[simp]: "finite (supp P)" using P_mult unfolding multiset_def by auto
(* *)
def set1 \<equiv> "\<lambda> c. {b1 \<in> supp N1. f1 b1 = c}"
have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
have fin_set1: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set1 c)"
using N1(1) unfolding set1_def multiset_def by auto
have set1_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<noteq> {}"
unfolding set1_def P1 mmap_ge_0[OF N1(1)] by auto
have supp_N1_set1: "supp N1 = (\<Union> c \<in> supp P. set1 c)"
using supp_vimage_mmap[OF N1(1), of f1] unfolding set1_def P1 by auto
hence set1_inclN1: "\<And>c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> supp N1" by auto
hence set1_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> B1" using N1(2) by blast
have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
unfolding set1_def by auto
have setsum_set1: "\<And> c. setsum N1 (set1 c) = P c"
unfolding P1 set1_def mmap_def apply(rule setsum_cong) by auto
(* *)
def set2 \<equiv> "\<lambda> c. {b2 \<in> supp N2. f2 b2 = c}"
have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
have fin_set2: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set2 c)"
using N2(1) unfolding set2_def multiset_def by auto
have set2_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<noteq> {}"
unfolding set2_def P2 mmap_ge_0[OF N2(1)] by auto
have supp_N2_set2: "supp N2 = (\<Union> c \<in> supp P. set2 c)"
using supp_vimage_mmap[OF N2(1), of f2] unfolding set2_def P2 by auto
hence set2_inclN2: "\<And>c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> supp N2" by auto
hence set2_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> B2" using N2(2) by blast
have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
unfolding set2_def by auto
have setsum_set2: "\<And> c. setsum N2 (set2 c) = P c"
unfolding P2 set2_def mmap_def apply(rule setsum_cong) by auto
(* *)
have ss: "\<And> c. c \<in> supp P \<Longrightarrow> setsum N1 (set1 c) = setsum N2 (set2 c)"
unfolding setsum_set1 setsum_set2 ..
have "\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
\<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
by simp (metis set1 set2 set_rev_mp)
then obtain uu where uu:
"\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
have u[simp]:
"\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
"\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
"\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
using uu unfolding u_def by auto
{fix c assume c: "c \<in> supp P"
have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
fix b1 b1' b2 b2'
assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
hence "p1 (u c b1 b2) = b1 \<and> p2 (u c b1 b2) = b2 \<and>
p1 (u c b1' b2') = b1' \<and> p2 (u c b1' b2') = b2'"
using u(2)[OF c] u(3)[OF c] by simp metis
thus "b1 = b1' \<and> b2 = b2'" using 0 by auto
qed
} note inj = this
def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
have fin_sset[simp]: "\<And> c. c \<in> supp P \<Longrightarrow> finite (sset c)" unfolding sset_def
using fin_set1 fin_set2 finite_twosets by blast
have sset_A: "\<And> c. c \<in> supp P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
{fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
and a: "a = u c b1 b2" unfolding sset_def by auto
have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
using ac a b1 b2 c u(2) u(3) by simp+
hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
unfolding inj2_def by (metis c u(2) u(3))
} note u_p12[simp] = this
{fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
hence "p1 a \<in> set1 c" unfolding sset_def by auto
}note p1[simp] = this
{fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
hence "p2 a \<in> set2 c" unfolding sset_def by auto
}note p2[simp] = this
(* *)
{fix c assume c: "c \<in> supp P"
hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = N1 b1) \<and>
(\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = N2 b2)"
unfolding sset_def
using matrix_count_finite[OF set1_NE[OF c] fin_set1[OF c]
set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
}
then obtain Ms where
ss1: "\<And> c b1. \<lbrakk>c \<in> supp P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = N1 b1" and
ss2: "\<And> c b2. \<lbrakk>c \<in> supp P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = N2 b2"
by metis
def SET \<equiv> "\<Union> c \<in> supp P. sset c"
have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by auto
have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
unfolding SET_def sset_def by blast
{fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
unfolding SET_def by auto
hence "p1 a \<in> set1 c'" unfolding sset_def by auto
hence eq: "c = c'" using p1a c c' set1_disj by auto
hence "a \<in> sset c" using ac' by simp
} note p1_rev = this
{fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
unfolding SET_def by auto
hence "p2 a \<in> set2 c'" unfolding sset_def by auto
hence eq: "c = c'" using p2a c c' set2_disj by auto
hence "a \<in> sset c" using ac' by simp
} note p2_rev = this
(* *)
have "\<forall> a \<in> SET. \<exists> c \<in> supp P. a \<in> sset c" unfolding SET_def by auto
then obtain h where h: "\<forall> a \<in> SET. h a \<in> supp P \<and> a \<in> sset (h a)" by metis
have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = c"
by (metis h p2 set2 u(3) u_SET)
have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = f1 b1"
using h unfolding sset_def by auto
have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = f2 b2"
using h unfolding sset_def by auto
def M \<equiv> "\<lambda> a. if a \<in> SET \<and> p1 a \<in> supp N1 \<and> p2 a \<in> supp N2 then Ms (h a) a else 0"
have sM: "supp M \<subseteq> SET" "supp M \<subseteq> p1 -` (supp N1)" "supp M \<subseteq> p2 -` (supp N2)"
unfolding M_def by auto
show "\<exists>M. (M \<in> multiset \<and> supp M \<subseteq> A) \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
proof(rule exI[of _ M], safe)
show "M \<in> multiset"
unfolding multiset_def using finite_subset[OF sM(1) fin_SET] by simp
next
fix a assume "0 < M a"
thus "a \<in> A" unfolding M_def using SET_A by (cases "a \<in> SET") auto
next
show "mmap p1 M = N1"
unfolding mmap_def[abs_def] proof(rule ext)
fix b1
let ?K = "{a. p1 a = b1 \<and> 0 < M a}"
show "setsum M ?K = N1 b1"
proof(cases "b1 \<in> supp N1")
case False
hence "?K = {}" using sM(2) by auto
thus ?thesis using False by auto
next
case True
def c \<equiv> "f1 b1"
have c: "c \<in> supp P" and b1: "b1 \<in> set1 c"
unfolding set1_def c_def P1 using True by (auto simp: mmap_image)
have "setsum M ?K = setsum M {a. p1 a = b1 \<and> a \<in> SET}"
apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
also have "... = setsum M ((\<lambda> b2. u c b1 b2) ` (set2 c))"
apply(rule setsum_cong) using c b1 proof safe
fix a assume p1a: "p1 a \<in> set1 c" and "0 < P c" and "a \<in> SET"
hence ac: "a \<in> sset c" using p1_rev by auto
hence "a = u c (p1 a) (p2 a)" using c by auto
moreover have "p2 a \<in> set2 c" using ac c by auto
ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
next
fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
hence "u c b1 b2 \<in> SET" using c by auto
qed auto
also have "... = setsum (\<lambda> b2. M (u c b1 b2)) (set2 c)"
unfolding comp_def[symmetric] apply(rule setsum_reindex)
using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
also have "... = N1 b1" unfolding ss1[OF c b1, symmetric]
apply(rule setsum_cong[OF refl]) unfolding M_def
using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1] by fastforce
finally show ?thesis .
qed
qed
next
show "mmap p2 M = N2"
unfolding mmap_def[abs_def] proof(rule ext)
fix b2
let ?K = "{a. p2 a = b2 \<and> 0 < M a}"
show "setsum M ?K = N2 b2"
proof(cases "b2 \<in> supp N2")
case False
hence "?K = {}" using sM(3) by auto
thus ?thesis using False by auto
next
case True
def c \<equiv> "f2 b2"
have c: "c \<in> supp P" and b2: "b2 \<in> set2 c"
unfolding set2_def c_def P2 using True by (auto simp: mmap_image)
have "setsum M ?K = setsum M {a. p2 a = b2 \<and> a \<in> SET}"
apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
also have "... = setsum M ((\<lambda> b1. u c b1 b2) ` (set1 c))"
apply(rule setsum_cong) using c b2 proof safe
fix a assume p2a: "p2 a \<in> set2 c" and "0 < P c" and "a \<in> SET"
hence ac: "a \<in> sset c" using p2_rev by auto
hence "a = u c (p1 a) (p2 a)" using c by auto
moreover have "p1 a \<in> set1 c" using ac c by auto
ultimately show "a \<in> (\<lambda>b1. u c b1 (p2 a)) ` set1 c" by auto
next
fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
hence "u c b1 b2 \<in> SET" using c by auto
qed auto
also have "... = setsum (M o (\<lambda> b1. u c b1 b2)) (set1 c)"
apply(rule setsum_reindex)
using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
also have "... = setsum (\<lambda> b1. M (u c b1 b2)) (set1 c)"
unfolding comp_def[symmetric] by simp
also have "... = N2 b2" unfolding ss2[OF c b2, symmetric]
apply(rule setsum_cong[OF refl]) unfolding M_def set2_def
using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2]
unfolding set1_def by fastforce
finally show ?thesis .
qed
qed
qed
qed
definition mset_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
"mset_map h = Abs_multiset \<circ> mmap h \<circ> count"
bnf_def mset = mset_map [set_of] "\<lambda>_::'a multiset. natLeq" ["{#}"]
unfolding mset_map_def
proof -
show "Abs_multiset \<circ> mmap id \<circ> count = id" unfolding mmap_id by (auto simp: count_inverse)
next
fix f g
show "Abs_multiset \<circ> mmap (g \<circ> f) \<circ> count =
Abs_multiset \<circ> mmap g \<circ> count \<circ> (Abs_multiset \<circ> mmap f \<circ> count)"
unfolding comp_def apply(rule ext)
by (auto simp: Abs_multiset_inverse count mmap_comp1 mmap)
next
fix M f g assume eq: "\<And>a. a \<in> set_of M \<Longrightarrow> f a = g a"
thus "(Abs_multiset \<circ> mmap f \<circ> count) M = (Abs_multiset \<circ> mmap g \<circ> count) M" apply auto
unfolding cIm_def[abs_def] image_def
by (auto intro!: mmap_cong simp: Abs_multiset_inject count mmap)
next
fix f show "set_of \<circ> (Abs_multiset \<circ> mmap f \<circ> count) = op ` f \<circ> set_of"
by (auto simp: count mmap mmap_image set_of_Abs_multiset supp_count)
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix M show "|set_of M| \<le>o natLeq"
apply(rule ordLess_imp_ordLeq)
unfolding finite_iff_ordLess_natLeq[symmetric] using finite_set_of .
next
fix A :: "'a set"
have "|{M. set_of M \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|" using card_of_set_of .
also have "|{as. set as \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
by (rule list_in_bd)
finally show "|{M. set_of M \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
next
fix A B1 B2 f1 f2 p1 p2
let ?map = "\<lambda> f. Abs_multiset \<circ> mmap f \<circ> count"
assume wp: "wpull A B1 B2 f1 f2 p1 p2"
show "wpull {x. set_of x \<subseteq> A} {x. set_of x \<subseteq> B1} {x. set_of x \<subseteq> B2}
(?map f1) (?map f2) (?map p1) (?map p2)"
unfolding wpull_def proof safe
fix y1 y2
assume y1: "set_of y1 \<subseteq> B1" and y2: "set_of y2 \<subseteq> B2"
and m: "?map f1 y1 = ?map f2 y2"
def N1 \<equiv> "count y1" def N2 \<equiv> "count y2"
have "N1 \<in> multiset \<and> supp N1 \<subseteq> B1" and "N2 \<in> multiset \<and> supp N2 \<subseteq> B2"
and "mmap f1 N1 = mmap f2 N2"
using y1 y2 m unfolding N1_def N2_def
by (auto simp: Abs_multiset_inject count mmap)
then obtain M where M: "M \<in> multiset \<and> supp M \<subseteq> A"
and N1: "mmap p1 M = N1" and N2: "mmap p2 M = N2"
using wp_mmap[OF wp] unfolding wpull_def by auto
def x \<equiv> "Abs_multiset M"
show "\<exists>x\<in>{x. set_of x \<subseteq> A}. ?map p1 x = y1 \<and> ?map p2 x = y2"
apply(intro bexI[of _ x]) using M N1 N2 unfolding N1_def N2_def x_def
by (auto simp: count_inverse Abs_multiset_inverse)
qed
qed (unfold set_of_empty, auto)
end