multiset as equivalence class of permuted lists

--- a/src/HOL/Library/List_Permutation.thy	Wed Feb 24 13:31:28 2021 +0000
+++ b/src/HOL/Library/List_Permutation.thy	Wed Feb 24 13:31:33 2021 +0000
@@ -52,30 +52,6 @@
     by induction simp_all
 qed
 
-lemma list_permuted_induct [consumes 1, case_names Nil swap Cons trans]:
-  \<open>P xs ys\<close>
-  if \<open>mset xs = mset ys\<close>
-  \<open>P [] []\<close>
-  \<open>\<And>y x zs. P (y # x # zs) (x # y # zs)\<close>
-  \<open>\<And>xs ys z. mset xs = mset ys \<Longrightarrow> P xs ys \<Longrightarrow> P (z # xs) (z # ys)\<close>
-  \<open>\<And>xs ys zs. mset xs = mset ys \<Longrightarrow> mset ys = mset zs \<Longrightarrow> P xs ys \<Longrightarrow> P ys zs \<Longrightarrow> P xs zs\<close>
-proof -
-  from \<open>mset xs = mset ys\<close> have \<open>xs <~~> ys\<close>
-    by (simp add: perm_iff_eq_mset)
-  then show ?thesis
-    using that(2-3) apply (rule perm.induct)
-     apply (simp_all add: perm_iff_eq_mset)
-     apply (fact that(4))
-    subgoal for xs ys zs
-      apply (rule that(5) [of xs ys zs])
-         apply simp_all
-      done
-    done
-qed
-
-
-subsection \<open>\<dots>that is equivalent to an already existing notion:\<close>
-
 theorem mset_eq_perm: \<open>mset xs = mset ys \<longleftrightarrow> xs <~~> ys\<close>
   by (simp add: perm_iff_eq_mset)
 
@@ -85,7 +61,7 @@
 proposition perm_swap:
   \<open>xs[i := xs ! j, j := xs ! i] <~~> xs\<close>
   if \<open>i < length xs\<close> \<open>j < length xs\<close>
-  using that by (cases \<open>i = j\<close>) (simp_all add: perm_iff_eq_mset mset_update)
+  using that by (simp add: perm_iff_eq_mset mset_swap)
 
 proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   by (auto simp add: perm_iff_eq_mset mset_subset_eq_exists_conv ex_mset dest: sym)
@@ -93,8 +69,8 @@
 proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
   by (rule mset_eq_setD) (simp add: perm_iff_eq_mset)
 
-proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
-  by (auto simp add: perm_iff_eq_mset distinct_count_atmost_1 dest: mset_eq_setD)
+proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs \<longleftrightarrow> distinct ys"
+  by (rule mset_eq_imp_distinct_iff) (simp add: perm_iff_eq_mset)
 
 theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
   by (simp add: perm_iff_eq_mset set_eq_iff_mset_remdups_eq)
@@ -105,72 +81,17 @@
 theorem permutation_Ex_bij:
   assumes "xs <~~> ys"
   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
-  using assms
-proof induct
-  case Nil
-  then show ?case
-    unfolding bij_betw_def by simp
-next
-  case (swap y x l)
-  show ?case
-  proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
-    show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
-      by (auto simp: bij_betw_def)
-    fix i
-    assume "i < length (y # x # l)"
-    show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
-      by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
-  qed
-next
-  case (Cons xs ys z)
-  then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
-    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
-    by blast
-  let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
-  show ?case
-  proof (intro exI[of _ ?f] allI conjI impI)
-    have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
-            "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
-      by (simp_all add: lessThan_Suc_eq_insert_0)
-    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
-      unfolding *
-    proof (rule bij_betw_combine)
-      show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
-        using bij unfolding bij_betw_def
-        by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
-    qed (auto simp: bij_betw_def)
-    fix i
-    assume "i < length (z # xs)"
-    then show "(z # xs) ! i = (z # ys) ! (?f i)"
-      using perm by (cases i) auto
-  qed
-next
-  case (trans xs ys zs)
-  then obtain f g
-    where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
-    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
-    by blast
-  show ?case
-  proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
-    show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
-      using bij by (rule bij_betw_trans)
-    fix i
-    assume "i < length xs"
-    with bij have "f i < length ys"
-      unfolding bij_betw_def by force
-    with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i"
-      using trans(1,3) perm by auto
-  qed
+proof -
+  from assms have \<open>mset ys = mset xs\<close>
+    by (simp add: perm_iff_eq_mset)
+  then obtain f where \<open>bij_betw f {..<length ys} {..<length xs}\<close>
+    \<open>xs = map (\<lambda>n. ys ! f n) [0..<length ys]\<close>
+    by (rule mset_eq_permutation)
+  then show ?thesis by auto
 qed
 
 proposition perm_finite: "finite {B. B <~~> A}"
-proof (rule finite_subset [where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
-  show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
-    using finite_lists_length_le by blast
-next
-  show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
-    by (auto simp add: perm_iff_eq_mset dest: mset_eq_setD mset_eq_length)
-qed
+  using mset_eq_finite by (auto simp add: perm_iff_eq_mset)
 
 
 subsection \<open>Trivial conclusions:\<close>

--- a/src/HOL/Library/Multiset.thy	Wed Feb 24 13:31:28 2021 +0000
+++ b/src/HOL/Library/Multiset.thy	Wed Feb 24 13:31:33 2021 +0000
@@ -1863,12 +1863,11 @@
 qed
 
 lemma set_eq_iff_mset_eq_distinct:
-  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
-    (set x = set y) = (mset x = mset y)"
-by (auto simp: multiset_eq_iff distinct_count_atmost_1)
+  \<open>distinct x \<Longrightarrow> distinct y \<Longrightarrow> set x = set y \<longleftrightarrow> mset x = mset y\<close>
+  by (auto simp: multiset_eq_iff distinct_count_atmost_1)
 
 lemma set_eq_iff_mset_remdups_eq:
-   "(set x = set y) = (mset (remdups x) = mset (remdups y))"
+  \<open>set x = set y \<longleftrightarrow> mset (remdups x) = mset (remdups y)\<close>
 apply (rule iffI)
 apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
 apply (drule distinct_remdups [THEN distinct_remdups
@@ -1876,6 +1875,10 @@
 apply simp
 done
 
+lemma mset_eq_imp_distinct_iff:
+  \<open>distinct xs \<longleftrightarrow> distinct ys\<close> if \<open>mset xs = mset ys\<close>
+  using that by (auto simp add: distinct_count_atmost_1 dest: mset_eq_setD)
+
 lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
 proof (induct ls arbitrary: i)
   case Nil
@@ -2509,9 +2512,7 @@
 qed
 
 
-subsection \<open>Alternative representations\<close>
-
-subsubsection \<open>Lists\<close>
+subsection \<open>Multiset as order-ignorant lists\<close>
 
 context linorder
 begin
@@ -2719,6 +2720,119 @@
     mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
   by (cases "i = j") (simp_all add: mset_update nth_mem_mset)
 
+lemma mset_eq_permutation:
+  assumes \<open>mset xs = mset ys\<close>
+  obtains f where 
+    \<open>bij_betw f {..<length xs} {..<length ys}\<close>
+    \<open>ys = map (\<lambda>n. xs ! f n) [0..<length xs]\<close>
+proof -
+  from assms have \<open>length ys = length xs\<close>
+    by (auto dest: mset_eq_length)
+  from assms have \<open>\<exists>f. f ` {..<length xs} = {..<length xs} \<and> ys = map (\<lambda>n. xs ! f n) [0..<length xs]\<close>
+  proof (induction xs arbitrary: ys rule: rev_induct)
+    case Nil
+    then show ?case by simp
+  next
+    case (snoc x xs)
+    from snoc.prems have \<open>x \<in> set ys\<close>
+      by (auto dest: union_single_eq_member)
+    then obtain zs ws where split: \<open>ys = zs @ x # ws\<close> and \<open>x \<notin> set zs\<close>
+      by (auto dest: split_list_first)
+    then have \<open>remove1 x ys = zs @ ws\<close>
+      by (simp add: remove1_append)
+    moreover from snoc.prems [symmetric] have \<open>mset xs = mset (remove1 x ys)\<close>
+      by simp
+    ultimately have \<open>mset xs = mset (zs @ ws)\<close>
+      by simp
+    then have \<open>\<exists>f. f ` {..<length xs} = {..<length xs} \<and> zs @ ws = map (\<lambda>n. xs ! f n) [0..<length xs]\<close>
+      by (rule snoc.IH)
+    then obtain f where
+      raw_surj: \<open>f ` {..<length xs} = {..<length xs}\<close>
+      and hyp: \<open>zs @ ws = map (\<lambda>n. xs ! f n) [0..<length xs]\<close> by blast
+    define l and k where \<open>l = length zs\<close> and \<open>k = length ws\<close>
+    then have [simp]: \<open>length zs = l\<close> \<open>length ws = k\<close>
+      by simp_all
+    from \<open>mset xs = mset (zs @ ws)\<close> have \<open>length xs = length (zs @ ws)\<close>
+      by (rule mset_eq_length)
+    then have [simp]: \<open>length xs = l + k\<close>
+      by simp
+    from raw_surj have f_surj: \<open>f ` {..<l + k} = {..<l + k}\<close>
+      by simp
+    have [simp]: \<open>[0..<l + k] = [0..<l] @ [l..<l + k]\<close>
+      by (rule nth_equalityI) (simp_all add: nth_append)
+    have [simp]: \<open>[l..<l + k] @ [l + k] = [l] @ [Suc l..<Suc (l + k)]\<close>
+      by (rule nth_equalityI)
+        (auto simp add: nth_append nth_Cons split: nat.split)
+    define g :: \<open>nat \<Rightarrow> nat\<close>
+      where \<open>g n = (if n < l then f n
+        else if n = l then l + k
+        else f (n - 1))\<close> for n
+    have \<open>{..<Suc (l + k)} = {..<l} \<union> {l} \<union> {Suc l..<Suc (l + k)}\<close>
+      by auto
+    then have \<open>g ` {..<Suc (l + k)} = g ` {..<l} \<union> {g l} \<union> g ` {Suc l..<Suc (l + k)}\<close>
+      by auto
+    also have \<open>g ` {Suc l..<Suc (l + k)} = f ` {l..<l + k}\<close>
+      apply (auto simp add: g_def Suc_le_lessD)
+      apply (auto simp add: image_def)
+      apply (metis Suc_le_mono atLeastLessThan_iff diff_Suc_Suc diff_zero lessI less_trans_Suc)
+      done
+    finally have \<open>g ` {..<Suc (l + k)} = f ` {..<l} \<union> {l + k} \<union> f ` {l..<l + k}\<close>
+      by (simp add: g_def)
+    also have \<open>\<dots> = {..<Suc (l + k)}\<close>
+      by simp (metis atLeastLessThan_add_Un f_surj image_Un le_add1 le_add_same_cancel1 lessThan_Suc lessThan_atLeast0)
+    finally have g_surj: \<open>g ` {..<Suc (l + k)} = {..<Suc (l + k)}\<close> .
+    from hyp have zs_f: \<open>zs = map (\<lambda>n. xs ! f n) [0..<l]\<close>
+      and ws_f: \<open>ws = map (\<lambda>n. xs ! f n) [l..<l + k]\<close>
+      by simp_all
+    have \<open>zs = map (\<lambda>n. (xs @ [x]) ! g n) [0..<l]\<close>
+    proof (rule sym, rule map_upt_eqI)
+      fix n
+      assume \<open>n < length zs\<close>
+      then have \<open>n < l\<close>
+        by simp
+      with f_surj have \<open>f n < l + k\<close>
+        by auto
+      with \<open>n < l\<close> show \<open>zs ! n = (xs @ [x]) ! g (0 + n)\<close>
+        by (simp add: zs_f g_def nth_append)
+    qed simp
+    moreover have \<open>x = (xs @ [x]) ! g l\<close>
+      by (simp add: g_def nth_append)
+    moreover have \<open>ws = map (\<lambda>n. (xs @ [x]) ! g n) [Suc l..<Suc (l + k)]\<close>
+    proof (rule sym, rule map_upt_eqI)
+      fix n
+      assume \<open>n < length ws\<close>
+      then have \<open>n < k\<close>
+        by simp
+      with f_surj have \<open>f (l + n) < l + k\<close>
+        by auto
+      with \<open>n < k\<close> show \<open>ws ! n = (xs @ [x]) ! g (Suc l + n)\<close>
+        by (simp add: ws_f g_def nth_append)
+    qed simp
+    ultimately have \<open>zs @ x # ws = map (\<lambda>n. (xs @ [x]) ! g n) [0..<length (xs @ [x])]\<close>
+      by simp
+    with g_surj show ?case
+      by (auto simp add: split)
+  qed
+  then obtain f where
+    surj: \<open>f ` {..<length xs} = {..<length xs}\<close>
+    and hyp: \<open>ys = map (\<lambda>n. xs ! f n) [0..<length xs]\<close> by blast
+  from surj have \<open>bij_betw f {..<length xs} {..<length ys}\<close>
+    by (simp add: bij_betw_def \<open>length ys = length xs\<close> eq_card_imp_inj_on)
+  then show thesis
+    using hyp ..
+qed
+
+proposition mset_eq_finite:
+  \<open>finite {ys. mset ys = mset xs}\<close>
+proof -
+  have \<open>{ys. mset ys = mset xs} \<subseteq> {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
+    by (auto simp add: dest: mset_eq_setD mset_eq_length)
+  moreover have \<open>finite {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
+    using finite_lists_length_le by blast
+  ultimately show ?thesis
+    by (rule finite_subset)
+qed
+
 
 subsection \<open>The multiset order\<close>
 
@@ -3653,11 +3767,22 @@
 lemma ex_mset: "\<exists>xs. mset xs = X"
   by (induct X) (simp, metis mset.simps(2))
 
-inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool"
+inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool" 
 where
   "pred_mset P {#}"
 | "\<lbrakk>P a; pred_mset P M\<rbrakk> \<Longrightarrow> pred_mset P (add_mset a M)"
 
+lemma pred_mset_iff: \<comment> \<open>TODO: alias for \<^const>\<open>Multiset.Ball\<close>\<close>
+  \<open>pred_mset P M \<longleftrightarrow> Multiset.Ball M P\<close>  (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+proof
+  assume ?P
+  then show ?Q by induction simp_all
+next
+  assume ?Q
+  then show ?P
+    by (induction M) (auto intro: pred_mset.intros)
+qed
+
 bnf "'a multiset"
   map: image_mset
   sets: set_mset
@@ -3709,18 +3834,10 @@
   show "z \<in> set_mset {#} \<Longrightarrow> False" for z
     by auto
   show "pred_mset P = (\<lambda>x. Ball (set_mset x) P)" for P
-  proof (intro ext iffI)
-    fix x
-    assume "pred_mset P x"
-    then show "Ball (set_mset x) P" by (induct pred: pred_mset; simp)
-  next
-    fix x
-    assume "Ball (set_mset x) P"
-    then show "pred_mset P x" by (induct x; auto intro: pred_mset.intros)
-  qed
+    by (simp add: fun_eq_iff pred_mset_iff)
 qed
 
-inductive rel_mset'
+inductive rel_mset' :: \<open>('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset \<Rightarrow> bool\<close>
 where
   Zero[intro]: "rel_mset' R {#} {#}"
 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (add_mset a M) (add_mset b N)"

--- a/src/HOL/List.thy	Wed Feb 24 13:31:28 2021 +0000
+++ b/src/HOL/List.thy	Wed Feb 24 13:31:33 2021 +0000
@@ -3389,7 +3389,24 @@
 
 lemma nth_Cons_numeral [simp]:
   "(x # xs) ! numeral v = xs ! (numeral v - 1)"
-by (simp add: nth_Cons')
+  by (simp add: nth_Cons')
+
+lemma map_upt_eqI:
+  \<open>map f [m..<n] = xs\<close> if \<open>length xs = n - m\<close>
+    \<open>\<And>i. i < length xs \<Longrightarrow> xs ! i = f (m + i)\<close>
+proof (rule nth_equalityI)
+  from \<open>length xs = n - m\<close> show \<open>length (map f [m..<n]) = length xs\<close>
+    by simp
+next
+  fix i
+  assume \<open>i < length (map f [m..<n])\<close>
+  then have \<open>i < n - m\<close>
+    by simp
+  with that have \<open>xs ! i = f (m + i)\<close>
+    by simp
+  with \<open>i < n - m\<close> show \<open>map f [m..<n] ! i = xs ! i\<close>
+    by simp
+qed
 
 
 subsubsection \<open>\<open>upto\<close>: interval-list on \<^typ>\<open>int\<close>\<close>