tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/BNF/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
Basic_BNFs
"~~/src/HOL/Quotient_Examples/Lift_FSet"
"~~/src/HOL/Library/Multiset"
Countable_Type
begin
lemma option_rec_conv_option_case: "option_rec = option_case"
by (simp add: fun_eq_iff split: option.split)
bnf Option.map [Option.set] "\<lambda>_::'a option. natLeq" ["None"] option_rel
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 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
next
fix R
show "option_rel R =
(Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map fst))\<inverse>\<inverse> OO
Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map snd)"
unfolding option_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff prod.cases
by (auto simp: trans[OF eq_commute option_map_is_None] trans[OF eq_commute option_map_eq_Some]
split: option.splits)
qed
lemma wpull_map:
assumes "wpull A B1 B2 f1 f2 p1 p2"
shows "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 assms obtain z where "z \<in> A" "p1 z = a" "p2 z = b" unfolding wpull_def 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
bnf 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"
by (metis List.finite_set finite_iff_ordLess_natLeq ordLess_imp_ordLeq)
qed (simp add: wpull_map)+
(* Finite sets *)
definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" where
"fset_rel 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)"
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
by (rule f_the_inv_into_f[unfolded inj_on_def])
(transfer, simp,
transfer, metis Collect_finite_eq_lists lists_UNIV mem_Collect_eq)
lemma fset_rel_aux:
"(\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>u \<in> fset b. \<exists>t \<in> fset a. R t u) \<longleftrightarrow>
((Grp {a. fset a \<subseteq> {(a, b). R a b}} (fmap fst))\<inverse>\<inverse> OO
Grp {a. fset a \<subseteq> {(a, b). R a b}} (fmap snd)) a b" (is "?L = ?R")
proof
assume ?L
def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")
have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
show ?R unfolding Grp_def relcompp.simps conversep.simps
proof (intro CollectI prod_caseI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
from * show "a = fmap fst R'" using conjunct1[OF `?L`]
by (transfer, auto simp add: image_def Int_def split: prod.splits)
from * show "b = fmap snd R'" using conjunct2[OF `?L`]
by (transfer, auto simp add: image_def Int_def split: prod.splits)
qed (auto simp add: *)
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
apply (simp add: subset_eq Ball_def)
apply (rule conjI)
apply (transfer, clarsimp, metis snd_conv)
by (transfer, clarsimp, metis fst_conv)
qed
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 wpull_fmap:
assumes "wpull A B1 B2 f1 f2 p1 p2"
shows "wpull {x. fset x \<subseteq> A} {x. fset x \<subseteq> B1} {x. fset x \<subseteq> B2}
(fmap f1) (fmap f2) (fmap p1) (fmap 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 "fmap f1 y1 = fmap f2 y2"
hence EQ: "f1 ` (fset y1) = f2 ` (fset y2)" by transfer simp
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 assms] unfolding wpull_def Pow_def
by (auto elim!: allE[of _ "fset y1"] allE[of _ "fset y2"])
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 transfer simp
then obtain x where X'eq: "X' = fset x" by transfer (metis finite_list)
show "\<exists>x. fset x \<subseteq> A \<and> fmap p1 x = y1 \<and> fmap p2 x = y2"
using X' Y1 Y2 by (auto simp: X'eq intro!: exI[of _ "x"]) (transfer, simp)+
qed
bnf fmap [fset] "\<lambda>_::'a fset. natLeq" ["{||}"] fset_rel
apply -
apply transfer' apply simp
apply transfer' apply simp
apply transfer apply force
apply transfer apply force
apply (rule natLeq_card_order)
apply (rule natLeq_cinfinite)
apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite_set)
apply (erule wpull_fmap)
apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff fset_rel_def fset_rel_aux)
apply transfer apply simp
done
lemmas [simp] = fset.map_comp fset.map_id fset.set_map
lemma fset_rel_fset: "set_rel \<chi> (fset A1) (fset A2) = fset_rel \<chi> A1 A2"
unfolding fset_rel_def set_rel_def by auto
(* 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 from_nat_into]) using inj_on_from_nat_into
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 rcset_to_rcset: "countable A \<Longrightarrow> rcset (the_inv rcset A) = A"
apply (rule f_the_inv_into_f[unfolded inj_on_def image_iff])
apply transfer' apply simp
apply transfer' apply simp
done
lemma Collect_Int_Times:
"{(x, y). R x y} \<inter> A \<times> B = {(x, y). R x y \<and> x \<in> A \<and> y \<in> B}"
by auto
definition cset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a cset \<Rightarrow> 'b cset \<Rightarrow> bool" where
"cset_rel R a b \<longleftrightarrow>
(\<forall>t \<in> rcset a. \<exists>u \<in> rcset b. R t u) \<and>
(\<forall>t \<in> rcset b. \<exists>u \<in> rcset a. R u t)"
lemma cset_rel_aux:
"(\<forall>t \<in> rcset a. \<exists>u \<in> rcset b. R t u) \<and> (\<forall>t \<in> rcset b. \<exists>u \<in> rcset a. R u t) \<longleftrightarrow>
((Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage fst))\<inverse>\<inverse> OO
Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage snd)) a b" (is "?L = ?R")
proof
assume ?L
def R' \<equiv> "the_inv rcset (Collect (split R) \<inter> (rcset a \<times> rcset b))"
(is "the_inv rcset ?L'")
have L: "countable ?L'" by auto
hence *: "rcset R' = ?L'" unfolding R'_def using fset_to_fset by (intro rcset_to_rcset)
thus ?R unfolding Grp_def relcompp.simps conversep.simps
proof (intro CollectI prod_caseI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
from * `?L` show "a = cimage fst R'" by transfer (auto simp: image_def Collect_Int_Times)
next
from * `?L` show "b = cimage snd R'" by transfer (auto simp: image_def Collect_Int_Times)
qed simp_all
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
by transfer force
qed
bnf cimage [rcset] "\<lambda>_::'a cset. natLeq" ["cempty"] cset_rel
proof -
show "cimage id = id" by transfer' simp
next
fix f g show "cimage (g \<circ> f) = cimage g \<circ> cimage f" by transfer' fastforce
next
fix C f g assume eq: "\<And>a. a \<in> rcset C \<Longrightarrow> f a = g a"
thus "cimage f C = cimage g C" by transfer force
next
fix f show "rcset \<circ> cimage f = op ` f \<circ> rcset" by transfer' fastforce
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" by transfer (unfold countable_card_le_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}
(cimage f1) (cimage f2) (cimage p1) (cimage p2)"
unfolding wpull_def proof safe
fix y1 y2
assume Y1: "rcset y1 \<subseteq> B1" and Y2: "rcset y2 \<subseteq> B2"
assume "cimage f1 y1 = cimage f2 y2"
hence EQ: "f1 ` (rcset y1) = f2 ` (rcset y2)" by transfer
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 Bex_def mem_Collect_eq
by (auto elim!: allE[of _ "rcset y1"] allE[of _ "rcset y2"])
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 transfer blast
show "\<exists>x\<in>{x. rcset x \<subseteq> A}. cimage p1 x = y1 \<and> cimage p2 x = y2"
using X' Y1 Y2 unfolding X'eq by (intro bexI[of _ "x"]) (transfer, auto)
qed
next
fix R
show "cset_rel R =
(Grp {x. rcset x \<subseteq> Collect (split R)} (cimage fst))\<inverse>\<inverse> OO
Grp {x. rcset x \<subseteq> Collect (split R)} (cimage snd)"
unfolding cset_rel_def[abs_def] cset_rel_aux by simp
qed (transfer, simp)
(* 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
lift_definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" is
"\<lambda>h f b. setsum f {a. h a = b \<and> f a > 0} :: nat"
unfolding multiset_def proof safe
fix h :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat"
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 fin by simp
hence B: "?B = {b. ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
hence "?B \<subseteq> h ` ?A" by auto
thus ?thesis using finite_surj[OF fin] by auto
qed
qed
lemma mmap_id0: "mmap id = id"
proof (intro ext multiset_eqI)
fix f a show "count (mmap id f) a = count (id f) a"
proof (cases "count f a = 0")
case False
hence 1: "{aa. aa = a \<and> aa \<in># f} = {a}" by auto
thus ?thesis by transfer auto
qed (transfer, simp)
qed
lemma inj_on_setsum_inv:
assumes 1: "(0::nat) < setsum (count f) {a. h a = b' \<and> a \<in># f}" (is "0 < setsum (count f) ?A'")
and 2: "{a. h a = b \<and> a \<in># f} = {a. h a = b' \<and> a \<in># f}" (is "?A = ?A'")
shows "b = b'"
using assms by (auto simp add: setsum_gt_0_iff)
lemma mmap_comp:
fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
shows "mmap (h2 o h1) = mmap h2 o mmap h1"
proof (intro ext multiset_eqI)
fix f :: "'a multiset" fix c :: 'c
let ?A = "{a. h2 (h1 a) = c \<and> a \<in># f}"
let ?As = "\<lambda> b. {a. h1 a = b \<and> a \<in># f}"
let ?B = "{b. h2 b = c \<and> 0 < setsum (count f) (?As b)}"
have 0: "{?As b | b. b \<in> ?B} = ?As ` ?B" by auto
have "\<And> b. finite (?As b)" by transfer (simp add: multiset_def)
hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
hence A: "?A = \<Union> {?As b | b. b \<in> ?B}" by auto
have "setsum (count f) ?A = setsum (setsum (count f)) {?As b | b. b \<in> ?B}"
unfolding A by transfer (intro setsum_Union_disjoint, auto simp: multiset_def)
also have "... = setsum (setsum (count f)) (?As ` ?B)" unfolding 0 ..
also have "... = setsum (setsum (count f) o ?As) ?B"
by(intro setsum_reindex) (auto simp add: setsum_gt_0_iff inj_on_def)
also have "... = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" unfolding comp_def ..
finally have "setsum (count f) ?A = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" .
thus "count (mmap (h2 \<circ> h1) f) c = count ((mmap h2 \<circ> mmap h1) f) c"
by transfer (unfold o_apply, blast)
qed
lemma mmap_cong:
assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
shows "mmap f M = mmap g M"
using assms by transfer (auto intro!: setsum_cong)
context
begin
interpretation lifting_syntax .
lemma set_of_transfer[transfer_rule]: "(pcr_multiset op = ===> op =) (\<lambda>f. {a. 0 < f a}) set_of"
unfolding set_of_def pcr_multiset_def cr_multiset_def fun_rel_def by auto
end
lemma set_of_mmap: "set_of o mmap h = image h o set_of"
proof (rule ext, unfold o_apply)
fix M show "set_of (mmap h M) = h ` set_of M"
by transfer (auto simp add: multiset_def setsum_gt_0_iff)
qed
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_setsum_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 image_eqI 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 image_eqI 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: "set_of M \<subseteq> f -` (set_of (mmap f M))"
by transfer (auto simp: multiset_def setsum_gt_0_iff)
lemma mmap_ge_0: "b \<in># mmap f M \<longleftrightarrow> (\<exists>a. a \<in># M \<and> f a = b)"
by transfer (auto simp: multiset_def setsum_gt_0_iff)
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 wpull_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. set_of M \<subseteq> A}
{N1. set_of N1 \<subseteq> B1} {N2. set_of 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 multiset" and N2 :: "'b2 multiset"
assume mmap': "mmap f1 N1 = mmap f2 N2"
and N1[simp]: "set_of N1 \<subseteq> B1"
and N2[simp]: "set_of N2 \<subseteq> B2"
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 fin_N1[simp]: "finite (set_of N1)"
and fin_N2[simp]: "finite (set_of N2)"
and fin_P[simp]: "finite (set_of P)" by auto
(* *)
def set1 \<equiv> "\<lambda> c. {b1 \<in> set_of 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> set_of P \<Longrightarrow> finite (set1 c)"
using N1(1) unfolding set1_def multiset_def by auto
have set1_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<noteq> {}"
unfolding set1_def set_of_def P mmap_ge_0 by auto
have supp_N1_set1: "set_of N1 = (\<Union> c \<in> set_of P. set1 c)"
using supp_vimage_mmap[of N1 f1] unfolding set1_def P1 by auto
hence set1_inclN1: "\<And>c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> set_of N1" by auto
hence set1_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> B1" using N1 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 (count N1) (set1 c) = count P c"
unfolding P1 set1_def by transfer (auto intro: setsum_cong)
(* *)
def set2 \<equiv> "\<lambda> c. {b2 \<in> set_of 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> set_of P \<Longrightarrow> finite (set2 c)"
using N2(1) unfolding set2_def multiset_def by auto
have set2_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<noteq> {}"
unfolding set2_def P2 mmap_ge_0 set_of_def by auto
have supp_N2_set2: "set_of N2 = (\<Union> c \<in> set_of P. set2 c)"
using supp_vimage_mmap[of N2 f2] unfolding set2_def P2 by auto
hence set2_inclN2: "\<And>c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> set_of N2" by auto
hence set2_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> B2" using N2 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 (count N2) (set2 c) = count P c"
unfolding P2 set2_def by transfer (auto intro: setsum_cong)
(* *)
have ss: "\<And> c. c \<in> set_of P \<Longrightarrow> setsum (count N1) (set1 c) = setsum (count N2) (set2 c)"
unfolding setsum_set1 setsum_set2 ..
have "\<forall> c \<in> set_of 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> set_of 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> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
"\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
"\<And> c b1 b2. \<lbrakk>c \<in> set_of 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> set_of 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> set_of P \<Longrightarrow> finite (sset c)" unfolding sset_def
using fin_set1 fin_set2 finite_twosets by blast
have sset_A: "\<And> c. c \<in> set_of P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
{fix c a assume c: "c \<in> set_of 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> set_of 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> set_of 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> set_of P"
hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = count N1 b1) \<and>
(\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = count N2 b2)"
unfolding sset_def
using matrix_setsum_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> set_of P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = count N1 b1" and
ss2: "\<And> c b2. \<lbrakk>c \<in> set_of P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = count N2 b2"
by metis
def SET \<equiv> "\<Union> c \<in> set_of 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 blast
have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of 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> set_of P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
then obtain c' where c': "c' \<in> set_of 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> set_of P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
then obtain c' where c': "c' \<in> set_of 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> set_of P. a \<in> sset c" unfolding SET_def by auto
then obtain h where h: "\<forall> a \<in> SET. h a \<in> set_of P \<and> a \<in> sset (h a)" by metis
have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of 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> set_of 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> set_of 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>
"Abs_multiset (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0)"
have "(\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) \<in> multiset"
unfolding multiset_def by auto
hence [transfer_rule]: "pcr_multiset op = (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) M"
unfolding M_def pcr_multiset_def cr_multiset_def by (auto simp: Abs_multiset_inverse)
have sM: "set_of M \<subseteq> SET" "set_of M \<subseteq> p1 -` (set_of N1)" "set_of M \<subseteq> p2 -` set_of N2"
by (transfer, auto split: split_if_asm)+
show "\<exists>M. set_of M \<subseteq> A \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
proof(rule exI[of _ M], safe)
fix a assume *: "a \<in> set_of M"
from SET_A show "a \<in> A"
proof (cases "a \<in> SET")
case False thus ?thesis using * by transfer' auto
qed blast
next
show "mmap p1 M = N1"
proof(intro multiset_eqI)
fix b1
let ?K = "{a. p1 a = b1 \<and> a \<in># M}"
have "setsum (count M) ?K = count N1 b1"
proof(cases "b1 \<in> set_of 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> set_of P" and b1: "b1 \<in> set1 c"
unfolding set1_def c_def P1 using True by (auto simp: o_eq_dest[OF set_of_mmap])
with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p1 a = b1 \<and> a \<in> SET}"
by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
also have "... = setsum (count 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 "c \<in> set_of P" 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
qed auto
also have "... = setsum (\<lambda> b2. count 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 "... = count N1 b1" unfolding ss1[OF c b1, symmetric]
apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b1 set2)
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
thus "count (mmap p1 M) b1 = count N1 b1" by transfer
qed
next
next
show "mmap p2 M = N2"
proof(intro multiset_eqI)
fix b2
let ?K = "{a. p2 a = b2 \<and> a \<in># M}"
have "setsum (count M) ?K = count N2 b2"
proof(cases "b2 \<in> set_of 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> set_of P" and b2: "b2 \<in> set2 c"
unfolding set2_def c_def P2 using True by (auto simp: o_eq_dest[OF set_of_mmap])
with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p2 a = b2 \<and> a \<in> SET}"
by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
also have "... = setsum (count 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 "c \<in> set_of P" 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>x. u c x (p2 a)) ` set1 c" by auto
qed auto
also have "... = setsum (count 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. count M (u c b1 b2)) (set1 c)" by simp
also have "... = count N2 b2" unfolding ss2[OF c b2, symmetric] o_def
apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b2 set1)
using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2] set1_def by fastforce
finally show ?thesis .
qed
thus "count (mmap p2 M) b2 = count N2 b2" by transfer
qed
qed
qed
lemma set_of_bd: "|set_of x| \<le>o natLeq"
by transfer
(auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
bnf mmap [set_of] "\<lambda>_::'a multiset. natLeq" ["{#}"]
by (auto simp add: mmap_id0 mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
intro: mmap_cong wpull_mmap)
inductive multiset_rel' where
Zero: "multiset_rel' R {#} {#}"
|
Plus: "\<lbrakk>R a b; multiset_rel' R M N\<rbrakk> \<Longrightarrow> multiset_rel' R (M + {#a#}) (N + {#b#})"
lemma multiset_map_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)
lemma multiset_map_Zero[simp]: "mmap f {#} = {#}" by simp
lemma multiset_rel_Zero: "multiset_rel R {#} {#}"
unfolding multiset_rel_def Grp_def by auto
declare multiset.count[simp]
declare Abs_multiset_inverse[simp]
declare multiset.count_inverse[simp]
declare union_preserves_multiset[simp]
lemma multiset_map_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
proof (intro multiset_eqI, transfer fixing: f)
fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
assume "M1 \<in> multiset" "M2 \<in> multiset"
hence "setsum M1 {a. f a = x \<and> 0 < M1 a} = setsum M1 {a. f a = x \<and> 0 < M1 a + M2 a}"
"setsum M2 {a. f a = x \<and> 0 < M2 a} = setsum M2 {a. f a = x \<and> 0 < M1 a + M2 a}"
by (auto simp: multiset_def intro!: setsum_mono_zero_cong_left)
then show "(\<Sum>a | f a = x \<and> 0 < M1 a + M2 a. M1 a + M2 a) =
setsum M1 {a. f a = x \<and> 0 < M1 a} +
setsum M2 {a. f a = x \<and> 0 < M2 a}"
by (auto simp: setsum.distrib[symmetric])
qed
lemma multiset_map_singl[simp]: "mmap f {#a#} = {#f a#}"
by transfer auto
lemma multiset_rel_Plus:
assumes ab: "R a b" and MN: "multiset_rel R M N"
shows "multiset_rel R (M + {#a#}) (N + {#b#})"
proof-
{fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
mmap snd y + {#b#} = mmap snd ya \<and>
set_of ya \<subseteq> {(x, y). R x y}"
apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
}
thus ?thesis
using assms
unfolding multiset_rel_def Grp_def by force
qed
lemma multiset_rel'_imp_multiset_rel:
"multiset_rel' R M N \<Longrightarrow> multiset_rel R M N"
apply(induct rule: multiset_rel'.induct)
using multiset_rel_Zero multiset_rel_Plus by auto
lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
proof -
def A \<equiv> "\<lambda> b. {a. f a = b \<and> a \<in># M}"
let ?B = "{b. 0 < setsum (count M) (A b)}"
have "{b. \<exists>a. f a = b \<and> a \<in># M} \<subseteq> f ` {a. a \<in># M}" by auto
moreover have "finite (f ` {a. a \<in># M})" apply(rule finite_imageI)
using finite_Collect_mem .
ultimately have fin: "finite {b. \<exists>a. f a = b \<and> a \<in># M}" by(rule finite_subset)
have i: "inj_on A ?B" unfolding inj_on_def A_def apply clarsimp
by (metis (lifting, mono_tags) mem_Collect_eq rel_simps(54)
setsum_gt_0_iff setsum_infinite)
have 0: "\<And> b. 0 < setsum (count M) (A b) \<longleftrightarrow> (\<exists> a \<in> A b. count M a > 0)"
apply safe
apply (metis less_not_refl setsum_gt_0_iff setsum_infinite)
by (metis A_def finite_Collect_conjI finite_Collect_mem setsum_gt_0_iff)
hence AB: "A ` ?B = {A b | b. \<exists> a \<in> A b. count M a > 0}" by auto
have "setsum (\<lambda> x. setsum (count M) (A x)) ?B = setsum (setsum (count M) o A) ?B"
unfolding comp_def ..
also have "... = (\<Sum>x\<in> A ` ?B. setsum (count M) x)"
unfolding setsum.reindex [OF i, symmetric] ..
also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
(is "_ = setsum (count M) ?J")
apply(rule setsum.UNION_disjoint[symmetric])
using 0 fin unfolding A_def by auto
also have "?J = {a. a \<in># M}" unfolding AB unfolding A_def by auto
finally have "setsum (\<lambda> x. setsum (count M) (A x)) ?B =
setsum (count M) {a. a \<in># M}" .
then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
qed
lemma multiset_rel_mcard:
assumes "multiset_rel R M N"
shows "mcard M = mcard N"
using assms unfolding multiset_rel_def Grp_def by auto
lemma multiset_induct2[case_names empty addL addR]:
assumes empty: "P {#} {#}"
and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
shows "P M N"
apply(induct N rule: multiset_induct)
apply(induct M rule: multiset_induct, rule empty, erule addL)
apply(induct M rule: multiset_induct, erule addR, erule addR)
done
lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
assumes c: "mcard M = mcard N"
and empty: "P {#} {#}"
and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
shows "P M N"
using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
case (less M) show ?case
proof(cases "M = {#}")
case True hence "N = {#}" using less.prems by auto
thus ?thesis using True empty by auto
next
case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
have "N \<noteq> {#}" using False less.prems by auto
then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
thus ?thesis using M N less.hyps add by auto
qed
qed
lemma msed_map_invL:
assumes "mmap f (M + {#a#}) = N"
shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
proof-
have "f a \<in># N"
using assms multiset.set_map[of f "M + {#a#}"] by auto
then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
have "mmap f M = N1" using assms unfolding N by simp
thus ?thesis using N by blast
qed
lemma msed_map_invR:
assumes "mmap f M = N + {#b#}"
shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> mmap f M1 = N"
proof-
obtain a where a: "a \<in># M" and fa: "f a = b"
using multiset.set_map[of f M] unfolding assms
by (metis image_iff mem_set_of_iff union_single_eq_member)
then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
have "mmap f M1 = N" using assms unfolding M fa[symmetric] by simp
thus ?thesis using M fa by blast
qed
lemma msed_rel_invL:
assumes "multiset_rel R (M + {#a#}) N"
shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> multiset_rel R M N1"
proof-
obtain K where KM: "mmap fst K = M + {#a#}"
and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
unfolding multiset_rel_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
using msed_map_invL[OF KN[unfolded K]] by auto
have Rab: "R a (snd ab)" using sK a unfolding K by auto
have "multiset_rel R M N1" using sK K1M K1N1
unfolding K multiset_rel_def Grp_def by auto
thus ?thesis using N Rab by auto
qed
lemma msed_rel_invR:
assumes "multiset_rel R M (N + {#b#})"
shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> multiset_rel R M1 N"
proof-
obtain K where KN: "mmap snd K = N + {#b#}"
and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
unfolding multiset_rel_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
using msed_map_invL[OF KM[unfolded K]] by auto
have Rab: "R (fst ab) b" using sK b unfolding K by auto
have "multiset_rel R M1 N" using sK K1N K1M1
unfolding K multiset_rel_def Grp_def by auto
thus ?thesis using M Rab by auto
qed
lemma multiset_rel_imp_multiset_rel':
assumes "multiset_rel R M N"
shows "multiset_rel' R M N"
using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
case (less M)
have c: "mcard M = mcard N" using multiset_rel_mcard[OF less.prems] .
show ?case
proof(cases "M = {#}")
case True hence "N = {#}" using c by simp
thus ?thesis using True multiset_rel'.Zero by auto
next
case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "multiset_rel R M1 N1"
using msed_rel_invL[OF less.prems[unfolded M]] by auto
have "multiset_rel' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
thus ?thesis using multiset_rel'.Plus[of R a b, OF R] unfolding M N by simp
qed
qed
lemma multiset_rel_multiset_rel':
"multiset_rel R M N = multiset_rel' R M N"
using multiset_rel_imp_multiset_rel' multiset_rel'_imp_multiset_rel by auto
(* The main end product for multiset_rel: inductive characterization *)
theorems multiset_rel_induct[case_names empty add, induct pred: multiset_rel] =
multiset_rel'.induct[unfolded multiset_rel_multiset_rel'[symmetric]]
(* Advanced relator customization *)
(* Set vs. sum relators: *)
(* FIXME: All such facts should be declared as simps: *)
declare sum_rel_simps[simp]
lemma set_rel_sum_rel[simp]:
"set_rel (sum_rel \<chi> \<phi>) A1 A2 \<longleftrightarrow>
set_rel \<chi> (Inl -` A1) (Inl -` A2) \<and> set_rel \<phi> (Inr -` A1) (Inr -` A2)"
(is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
proof safe
assume L: "?L"
show ?Rl unfolding set_rel_def Bex_def vimage_eq proof safe
fix l1 assume "Inl l1 \<in> A1"
then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inl l1) a2"
using L unfolding set_rel_def by auto
then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
next
fix l2 assume "Inl l2 \<in> A2"
then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inl l2)"
using L unfolding set_rel_def by auto
then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
qed
show ?Rr unfolding set_rel_def Bex_def vimage_eq proof safe
fix r1 assume "Inr r1 \<in> A1"
then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inr r1) a2"
using L unfolding set_rel_def by auto
then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
next
fix r2 assume "Inr r2 \<in> A2"
then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inr r2)"
using L unfolding set_rel_def by auto
then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
qed
next
assume Rl: "?Rl" and Rr: "?Rr"
show ?L unfolding set_rel_def Bex_def vimage_eq proof safe
fix a1 assume a1: "a1 \<in> A1"
show "\<exists> a2. a2 \<in> A2 \<and> sum_rel \<chi> \<phi> a1 a2"
proof(cases a1)
case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
using Rl a1 unfolding set_rel_def by blast
thus ?thesis unfolding Inl by auto
next
case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
using Rr a1 unfolding set_rel_def by blast
thus ?thesis unfolding Inr by auto
qed
next
fix a2 assume a2: "a2 \<in> A2"
show "\<exists> a1. a1 \<in> A1 \<and> sum_rel \<chi> \<phi> a1 a2"
proof(cases a2)
case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
using Rl a2 unfolding set_rel_def by blast
thus ?thesis unfolding Inl by auto
next
case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
using Rr a2 unfolding set_rel_def by blast
thus ?thesis unfolding Inr by auto
qed
qed
qed
end