(*  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/Library/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 "'a option"

   map: Option.map

   sets: Option.set

   bd: natLeq 

   wits: None

   rel: 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 "'a list"

   map: map

   sets: set

   bd: natLeq

   wits: Nil

   rel: list_all2

 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)

 next

   fix R

   show "list_all2 R =

          (Grp {x. set x \<subseteq> {(x, y). R x y}} (map fst))\<inverse>\<inverse> OO

          Grp {x. set x \<subseteq> {(x, y). R x y}} (map snd)"

     unfolding list_all2_def[abs_def] Grp_def fun_eq_iff relcompp.simps conversep.simps

     by (force simp: zip_map_fst_snd)

 qed (simp add: wpull_map)+

 (* Finite sets *)

 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

 context

 includes fset.lifting

 begin

 lemma fset_rel_alt: "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)"

   by transfer (simp add: set_rel_def)

 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"

   apply (rule f_the_inv_into_f[unfolded inj_on_def])

   apply (simp add: fset_inject) apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+

   .

 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}} (fimage fst))\<inverse>\<inverse> OO

   Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage 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 = fimage fst R'" using conjunct1[OF `?L`]

       by (transfer, auto simp add: image_def Int_def split: prod.splits)

     from * show "b = fimage 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_fimage:

   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}

               (fimage f1) (fimage f2) (fimage p1) (fimage 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 "fimage f1 y1 = fimage 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 simp

   show "\<exists>x. fset x \<subseteq> A \<and> fimage p1 x = y1 \<and> fimage p2 x = y2"

      using X' Y1 Y2 by (auto simp: X'eq intro!: exI[of _ "x"]) (transfer, blast)+

 qed

 bnf "'a fset"

   map: fimage

   sets: fset 

   bd: natLeq

   wits: "{||}"

   rel: fset_rel

 apply -

           apply transfer' apply simp

          apply transfer' apply force

         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)

   apply (erule wpull_fimage)

  apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff fset_rel_alt fset_rel_aux) 

 apply transfer apply simp

 done

 lemma fset_rel_fset: "set_rel \<chi> (fset A1) (fset A2) = fset_rel \<chi> A1 A2"

   by transfer (rule refl)

 end

 lemmas [simp] = fset.map_comp fset.map_id fset.set_map

 (* 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 "'a cset"

   map: cimage

   sets: rcset

   bd: natLeq

   wits: "cempty"

   rel: 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 "'a multiset"

   map: mmap

   sets: set_of 

   bd: natLeq

   wits: "{#}"

 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 rel_multiset' where

 Zero: "rel_multiset' R {#} {#}"

 |

 Plus: "\<lbrakk>R a b; rel_multiset' R M N\<rbrakk> \<Longrightarrow> rel_multiset' R (M + {#a#}) (N + {#b#})"

 lemma map_multiset_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"

 by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)

 lemma map_multiset_Zero[simp]: "mmap f {#} = {#}" by simp

 lemma rel_multiset_Zero: "rel_multiset R {#} {#}"

 unfolding rel_multiset_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 map_multiset_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 map_multiset_singl[simp]: "mmap f {#a#} = {#f a#}"

   by transfer auto

 lemma rel_multiset_Plus:

 assumes ab: "R a b" and MN: "rel_multiset R M N"

 shows "rel_multiset 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 rel_multiset_def Grp_def by force

 qed

 lemma rel_multiset'_imp_rel_multiset:

 "rel_multiset' R M N \<Longrightarrow> rel_multiset R M N"

 apply(induct rule: rel_multiset'.induct)

 using rel_multiset_Zero rel_multiset_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, full_types) mem_Collect_eq neq0_conv setsum.neutral)

   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 rel_multiset_mcard:

 assumes "rel_multiset R M N"

 shows "mcard M = mcard N"

 using assms unfolding rel_multiset_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 "rel_multiset R (M + {#a#}) N"

 shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_multiset 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 rel_multiset_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 "rel_multiset R M N1" using sK K1M K1N1

   unfolding K rel_multiset_def Grp_def by auto

   thus ?thesis using N Rab by auto

 qed

 lemma msed_rel_invR:

 assumes "rel_multiset R M (N + {#b#})"

 shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_multiset 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 rel_multiset_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 "rel_multiset R M1 N" using sK K1N K1M1

   unfolding K rel_multiset_def Grp_def by auto

   thus ?thesis using M Rab by auto

 qed

 lemma rel_multiset_imp_rel_multiset':

 assumes "rel_multiset R M N"

 shows "rel_multiset' 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 rel_multiset_mcard[OF less.prems] .

   show ?case

   proof(cases "M = {#}")

     case True hence "N = {#}" using c by simp

     thus ?thesis using True rel_multiset'.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: "rel_multiset R M1 N1"

     using msed_rel_invL[OF less.prems[unfolded M]] by auto

     have "rel_multiset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp

     thus ?thesis using rel_multiset'.Plus[of R a b, OF R] unfolding M N by simp

   qed

 qed

 lemma rel_multiset_rel_multiset':

 "rel_multiset R M N = rel_multiset' R M N"

 using  rel_multiset_imp_rel_multiset' rel_multiset'_imp_rel_multiset by auto

 (* The main end product for rel_multiset: inductive characterization *)

 theorems rel_multiset_induct[case_names empty add, induct pred: rel_multiset] =

          rel_multiset'.induct[unfolded rel_multiset_rel_multiset'[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