# HG changeset patch
# User blanchet
# Date 1408307278 -7200
# Node ID 6fab7e95587d0911ae07ae542f71d6af6a399f7f
# Parent  a18a351132b7669e7efa9bec086297d6766a961f
use 'image_mset' as BNF map function

diff -r a18a351132b7 -r 6fab7e95587d src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy	Sun Aug 17 16:24:04 2014 +0200
+++ b/src/HOL/Library/Multiset.thy	Sun Aug 17 22:27:58 2014 +0200
@@ -2224,697 +2224,214 @@
 
 subsection {* BNF setup *}
 
-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 [simplified], auto simp: multiset_def setsum.Union_disjoint)
-  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 comp_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 rel_fun_def by auto
-
-end
-
-lemma set_of_mmap: "set_of o mmap h = image h o set_of"
-proof (rule ext, unfold comp_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
+definition rel_mset where
+  "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. multiset_of xs = X \<and> multiset_of ys = Y \<and> list_all2 R xs ys)"
+
+lemma multiset_of_zip_take_Cons_drop_twice:
+  assumes "length xs = length ys" "j \<le> length xs"
+  shows "multiset_of (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
+    multiset_of (zip xs ys) + {#(x, y)#}"
+using assms
+proof (induct xs ys arbitrary: x y j rule: list_induct2)
+  case Nil
+  thus ?case
+    by simp
 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:
-"(card_of {M. set_of M \<subseteq> A}, card_of {as. set as \<subseteq> A}) \<in> ordLeq"
-apply(rule surj_imp_ordLeq[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
+  case (Cons x xs y ys)
+  thus ?case
+  proof (cases "j = 0")
+    case True
+    thus ?thesis
+      by simp
   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
+    case False
+    then obtain k where k: "j = Suc k"
+      by (case_tac j) simp
+    hence "k \<le> length xs"
+      using Cons.prems by auto
+    hence "multiset_of (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
+      multiset_of (zip xs ys) + {#(x, y)#}"
+      by (rule Cons.hyps(2))
+    thus ?thesis
+      unfolding k by (auto simp: add.commute union_lcomm)
+  qed      
 qed
 
-(* Weak pullbacks: *)
-definition wpull where
-"wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
- (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow> (\<exists> a \<in> A. p1 a = b1 \<and> p2 a = b2))"
-
-(* Weak pseudo-pullbacks *)
-definition wppull where
-"wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
- (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
-           (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
-
-
-(* The pullback of sets *)
-definition thePull where
-"thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
-
-lemma wpull_thePull:
-"wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
-unfolding wpull_def thePull_def by auto
-
-lemma wppull_thePull:
-assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
-shows
-"\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
-   j a' \<in> A \<and>
-   e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
-(is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
-proof(rule bchoice[of ?A' ?phi], default)
-  fix a' assume a': "a' \<in> ?A'"
-  hence "fst a' \<in> B1" unfolding thePull_def by auto
-  moreover
-  from a' have "snd a' \<in> B2" unfolding thePull_def by auto
-  moreover have "f1 (fst a') = f2 (snd a')"
-  using a' unfolding csquare_def thePull_def by auto
-  ultimately show "\<exists> ja'. ?phi a' ja'"
-  using assms unfolding wppull_def by blast
-qed
-
-lemma wpull_wppull:
-assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
-1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')"
-shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
-unfolding wppull_def proof safe
-  fix b1 b2
-  assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
-  then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
-  using wp unfolding wpull_def by blast
-  show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
-  apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
+lemma ex_multiset_of_zip_left:
+  assumes "length xs = length ys" "multiset_of xs' = multiset_of xs"
+  shows "\<exists>ys'. length ys' = length xs' \<and> multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
+using assms 
+proof (induct xs ys arbitrary: xs' rule: list_induct2)
+  case Nil
+  thus ?case
+    by auto
+next
+  case (Cons x xs y ys xs')
+  obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
+  proof -
+    assume "\<And>j. \<lbrakk>j < length xs'; xs' ! j = x\<rbrakk> \<Longrightarrow> ?thesis"
+    moreover have "\<And>k m n. (m\<Colon>nat) + n < m + k \<or> \<not> n < k" by linarith
+    moreover have "\<And>n a as. n - n < length (a # as) \<or> n < n"
+      by (metis Nat.add_diff_inverse diff_add_inverse2 impossible_Cons le_add1
+        less_diff_conv not_add_less2)
+    moreover have "\<not> length xs' < length xs'" by blast
+    ultimately show ?thesis
+      by (metis (no_types) Cons.prems Nat.add_diff_inverse diff_add_inverse2 length_append
+        less_diff_conv list.set_intros(1) multiset_of_eq_setD nth_append_length split_list)
+  qed
+
+  def xsa \<equiv> "take j xs' @ drop (Suc j) xs'" 
+  have "multiset_of xs' = {#x#} + multiset_of xsa"
+    unfolding xsa_def using j_len nth_j
+    by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id drop_Suc_conv_tl
+      multiset_of.simps(2) union_code union_commute)
+  hence ms_x: "multiset_of xsa = multiset_of xs"
+    by (metis Cons.prems add.commute add_right_imp_eq multiset_of.simps(2))
+  then obtain ysa where
+    len_a: "length ysa = length xsa" and ms_a: "multiset_of (zip xsa ysa) = multiset_of (zip xs ys)"
+    using Cons.hyps(2) by blast
+
+  def ys' \<equiv> "take j ysa @ y # drop j ysa"
+  have xs': "xs' = take j xsa @ x # drop j xsa"
+    using ms_x j_len nth_j Cons.prems xsa_def
+    by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc drop_Suc_conv_tl length_Cons
+      length_drop mcard_multiset_of)
+  have j_len': "j \<le> length xsa"
+    using j_len xs' xsa_def
+    by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
+  have "length ys' = length xs'"
+    unfolding ys'_def using Cons.prems len_a ms_x
+    by (metis add_Suc_right append_take_drop_id length_Cons length_append multiset_of_eq_length)
+  moreover have "multiset_of (zip xs' ys') = multiset_of (zip (x # xs) (y # ys))"
+    unfolding xs' ys'_def
+    by (rule trans[OF multiset_of_zip_take_Cons_drop_twice])
+      (auto simp: len_a ms_a j_len' add.commute)
+  ultimately show ?case
+    by blast
 qed
 
-lemma wppull_fstOp_sndOp:
-shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
-  snd fst fst snd (BNF_Def.fstOp P Q) (BNF_Def.sndOp P Q)"
-using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
-
-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: comp_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_neutral_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]
-            [[hypsubst_thin = true]]
-          by fastforce
-        finally show ?thesis .
-      qed
-      thus "count (mmap p1 M) b1 = count N1 b1" by transfer
-    qed
-  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: comp_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_neutral_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] comp_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
-            [[hypsubst_thin = true]]
-          by fastforce
-        finally show ?thesis .
-      qed
-      thus "count (mmap p2 M) b2 = count N2 b2" by transfer
-    qed
-  qed
+lemma list_all2_reorder_left_invariance:
+  assumes rel: "list_all2 R xs ys" and ms_x: "multiset_of xs' = multiset_of xs"
+  shows "\<exists>ys'. list_all2 R xs' ys' \<and> multiset_of ys' = multiset_of ys"
+proof -
+  have len: "length xs = length ys"
+    using rel list_all2_conv_all_nth by auto
+  obtain ys' where
+    len': "length xs' = length ys'" and ms_xy: "multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
+    using len ms_x by (metis ex_multiset_of_zip_left)
+  have "list_all2 R xs' ys'"
+    using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: multiset_of_eq_setD)
+  moreover have "multiset_of ys' = multiset_of ys"
+    using len len' ms_xy map_snd_zip multiset_of_map by metis
+  ultimately show ?thesis
+    by blast
 qed
 
-lemma set_of_bd: "(card_of (set_of x), natLeq) \<in> ordLeq"
-  by transfer
-    (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
-
-lemma wppull_mmap:
-  assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
-  shows "wppull {M. set_of M \<subseteq> A} {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
-    (mmap f1) (mmap f2) (mmap e1) (mmap e2) (mmap p1) (mmap p2)"
-proof -
-  from assms obtain j where j: "\<forall>a'\<in>thePull B1 B2 f1 f2.
-    j a' \<in> A \<and> e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')" 
-    by (blast dest: wppull_thePull)
-  then show ?thesis
-    by (intro wpull_wppull[OF wpull_mmap[OF wpull_thePull], of _ _ _ _ "mmap j"])
-      (auto simp: comp_eq_dest_lhs[OF mmap_comp[symmetric]] comp_eq_dest[OF set_of_mmap]
-        intro!: mmap_cong simp del: mem_set_of_iff simp: mem_set_of_iff[symmetric])
-qed
+lemma ex_multiset_of: "\<exists>xs. multiset_of xs = X"
+  by (induct X) (simp, metis multiset_of.simps(2))
 
 bnf "'a multiset"
-  map: mmap
+  map: image_mset
   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
-  Grp_def relcompp.simps intro: mmap_cong)
-  (metis wppull_mmap[OF wppull_fstOp_sndOp, unfolded wppull_def
-    o_eq_dest_lhs[OF mmap_comp[symmetric]] fstOp_def sndOp_def comp_def, simplified])
-
-inductive rel_multiset' where
-  Zero[intro]: "rel_multiset' R {#} {#}"
-| Plus[intro]: "\<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
+  rel: rel_mset
+proof -
+  show "image_mset id = id"
+    by (rule image_mset.id)
+next
+  show "\<And>f g. image_mset (g \<circ> f) = image_mset g \<circ> image_mset f"
+    unfolding comp_def by (rule ext) (simp add: image_mset.compositionality comp_def)
+next
+  fix X :: "'a multiset"
+  show "\<And>f g. (\<And>z. z \<in> set_of X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X"
+    by (induct X, (simp (no_asm))+,
+      metis One_nat_def Un_iff count_single mem_set_of_iff set_of_union zero_less_Suc)
+next
+  show "\<And>f. set_of \<circ> image_mset f = op ` f \<circ> set_of"
+    by auto
+next
+  show "card_order natLeq"
+    by (rule natLeq_card_order)
+next
+  show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
+    by (rule natLeq_cinfinite)
+next
+  show "\<And>X. ordLeq3 (card_of (set_of X)) natLeq"
+    by transfer
+      (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
+next
+  show "\<And>R S. rel_mset R OO rel_mset S \<le> rel_mset (R OO S)"
+    unfolding rel_mset_def[abs_def] OO_def
+    apply clarify
+    apply (rename_tac X Z Y xs ys' ys zs)
+    apply (drule_tac xs = ys' and ys = zs and xs' = ys in list_all2_reorder_left_invariance)
+    by (auto intro: list_all2_trans)
+next
+  show "\<And>R. rel_mset R =
+    (BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
+    BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset snd)"
+    unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
+    apply (rule ext)+
+    apply auto
+     apply (rule_tac x = "multiset_of (zip xs ys)" in exI)
+     apply auto[1]
+        apply (metis list_all2_lengthD map_fst_zip multiset_of_map)
+       apply (auto simp: list_all2_iff)[1]
+      apply (metis list_all2_lengthD map_snd_zip multiset_of_map)
+     apply (auto simp: list_all2_iff)[1]
+    apply (rename_tac XY)
+    apply (cut_tac X = XY in ex_multiset_of)
+    apply (erule exE)
+    apply (rename_tac xys)
+    apply (rule_tac x = "map fst xys" in exI)
+    apply (auto simp: multiset_of_map)
+    apply (rule_tac x = "map snd xys" in exI)
+    by (auto simp: multiset_of_map list_all2I subset_eq zip_map_fst_snd)
+next
+  show "\<And>z. z \<in> set_of {#} \<Longrightarrow> False"
+    by auto
+qed
+
+inductive rel_mset' where
+  Zero[intro]: "rel_mset' R {#} {#}"
+| Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
+
+lemma rel_mset_Zero: "rel_mset R {#} {#}"
+unfolding rel_mset_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_neutral_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_single[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#})"
+lemma rel_mset_Plus:
+assumes ab: "R a b" and MN: "rel_mset R M N"
+shows "rel_mset 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>
+   hence "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
+               image_mset snd y + {#b#} = image_mset 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
+  unfolding multiset.rel_compp_Grp Grp_def by blast
 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 rel_mset'_imp_rel_mset:
+"rel_mset' R M N \<Longrightarrow> rel_mset R M N"
+apply(induct rule: rel_mset'.induct)
+using rel_mset_Zero rel_mset_Plus by auto
+
+lemma mcard_image_mset[simp]: "mcard (image_mset f M) = mcard M"
+  unfolding size_eq_mcard[symmetric] by (rule size_image_mset)
+
+lemma rel_mset_mcard:
+  assumes "rel_mset R M N"
+  shows "mcard M = mcard N"
+using assms unfolding multiset.rel_compp_Grp Grp_def by auto
 
 lemma multiset_induct2[case_names empty addL addR]:
 assumes empty: "P {#} {#}"
@@ -2946,100 +2463,96 @@
 qed
 
 lemma msed_map_invL:
-assumes "mmap f (M + {#a#}) = N"
-shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
+assumes "image_mset f (M + {#a#}) = N"
+shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset 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
+  have "image_mset 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"
+assumes "image_mset f M = N + {#b#}"
+shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset 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
+  have "image_mset 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"
+assumes "rel_mset R (M + {#a#}) N"
+shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset 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}"
+  obtain K where KM: "image_mset fst K = M + {#a#}"
+  and KN: "image_mset snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
   using assms
-  unfolding rel_multiset_def Grp_def by auto
+  unfolding multiset.rel_compp_Grp 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"
+  and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
+  obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset 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
+  have "rel_mset R M N1" using sK K1M K1N1
+  unfolding K multiset.rel_compp_Grp 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"
+assumes "rel_mset R M (N + {#b#})"
+shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset 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}"
+  obtain K where KN: "image_mset snd K = N + {#b#}"
+  and KM: "image_mset fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
   using assms
-  unfolding rel_multiset_def Grp_def by auto
+  unfolding multiset.rel_compp_Grp 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"
+  and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
+  obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset 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
+  have "rel_mset R M1 N" using sK K1N K1M1
+  unfolding K multiset.rel_compp_Grp 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"
+lemma rel_mset_imp_rel_mset':
+assumes "rel_mset R M N"
+shows "rel_mset' 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] .
+  have c: "mcard M = mcard N" using rel_mset_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
+    thus ?thesis using True rel_mset'.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"
+    obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset 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
+    have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
+    thus ?thesis using rel_mset'.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]]
+lemma rel_mset_rel_mset':
+"rel_mset R M N = rel_mset' R M N"
+using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
+
+(* The main end product for rel_mset: inductive characterization *)
+theorems rel_mset_induct[case_names empty add, induct pred: rel_mset] =
+         rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
 
 
 subsection {* Size setup *}
 
-lemma multiset_size_o_map: "size_multiset g \<circ> mmap f = size_multiset (g \<circ> f)"
-apply (rule ext)
-apply (unfold o_apply)
-apply (induct_tac x)
-apply auto
-done
+lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
+  unfolding o_apply by (rule ext) (induct_tac, auto)
 
 setup {*
 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}