diff --git a/src/HOL/Combinatorics/Permutations.thy b/src/HOL/Combinatorics/Permutations.thy
--- a/src/HOL/Combinatorics/Permutations.thy
+++ b/src/HOL/Combinatorics/Permutations.thy
@@ -1,1670 +1,1779 @@
 (*  Author:     Amine Chaieb, University of Cambridge
 *)
 
 section \<open>Permutations, both general and specifically on finite sets.\<close>
 
 theory Permutations
   imports
     "HOL-Library.Multiset"
     "HOL-Library.Disjoint_Sets"
 begin
 
 subsection \<open>Auxiliary\<close>
 
 abbreviation (input) fixpoints :: \<open>('a \<Rightarrow> 'a) \<Rightarrow> 'a set\<close>
   where \<open>fixpoints f \<equiv> {x. f x = x}\<close>
 
 lemma inj_on_fixpoints:
   \<open>inj_on f (fixpoints f)\<close>
   by (rule inj_onI) simp
 
 lemma bij_betw_fixpoints:
   \<open>bij_betw f (fixpoints f) (fixpoints f)\<close>
   using inj_on_fixpoints by (auto simp add: bij_betw_def)
 
 
 subsection \<open>Basic definition and consequences\<close>
 
 definition permutes :: \<open>('a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> bool\<close>  (infixr \<open>permutes\<close> 41)
   where \<open>p permutes S \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)\<close>
 
 lemma bij_imp_permutes:
   \<open>p permutes S\<close> if \<open>bij_betw p S S\<close> and stable: \<open>\<And>x. x \<notin> S \<Longrightarrow> p x = x\<close>
 proof -
   note \<open>bij_betw p S S\<close>
   moreover have \<open>bij_betw p (- S) (- S)\<close>
     by (auto simp add: stable intro!: bij_betw_imageI inj_onI)
   ultimately have \<open>bij_betw p (S \<union> - S) (S \<union> - S)\<close>
     by (rule bij_betw_combine) simp
   then have \<open>\<exists>!x. p x = y\<close> for y
     by (simp add: bij_iff)
   with stable show ?thesis
     by (simp add: permutes_def)
 qed
 
 context
   fixes p :: \<open>'a \<Rightarrow> 'a\<close> and S :: \<open>'a set\<close>
   assumes perm: \<open>p permutes S\<close>
 begin
 
 lemma permutes_inj:
   \<open>inj p\<close>
   using perm by (auto simp: permutes_def inj_on_def)
 
 lemma permutes_image:
   \<open>p ` S = S\<close>
 proof (rule set_eqI)
   fix x
   show \<open>x \<in> p ` S \<longleftrightarrow> x \<in> S\<close>
   proof
     assume \<open>x \<in> p ` S\<close>
     then obtain y where \<open>y \<in> S\<close> \<open>p y = x\<close>
       by blast
     with perm show \<open>x \<in> S\<close>
       by (cases \<open>y = x\<close>) (auto simp add: permutes_def)
   next
     assume \<open>x \<in> S\<close>
     with perm obtain y where \<open>y \<in> S\<close> \<open>p y = x\<close>
       by (metis permutes_def)
     then show \<open>x \<in> p ` S\<close>
       by blast
   qed
 qed
 
 lemma permutes_not_in:
   \<open>x \<notin> S \<Longrightarrow> p x = x\<close>
   using perm by (auto simp: permutes_def)
 
 lemma permutes_image_complement:
   \<open>p ` (- S) = - S\<close>
   by (auto simp add: permutes_not_in)
 
 lemma permutes_in_image:
   \<open>p x \<in> S \<longleftrightarrow> x \<in> S\<close>
   using permutes_image permutes_inj by (auto dest: inj_image_mem_iff)
 
 lemma permutes_surj:
   \<open>surj p\<close>
 proof -
   have \<open>p ` (S \<union> - S) = p ` S \<union> p ` (- S)\<close>
     by (rule image_Un)
   then show ?thesis
     by (simp add: permutes_image permutes_image_complement)
 qed
 
 lemma permutes_inv_o:
   shows "p \<circ> inv p = id"
     and "inv p \<circ> p = id"
   using permutes_inj permutes_surj
   unfolding inj_iff [symmetric] surj_iff [symmetric] by auto
 
 lemma permutes_inverses:
   shows "p (inv p x) = x"
     and "inv p (p x) = x"
   using permutes_inv_o [unfolded fun_eq_iff o_def] by auto
 
 lemma permutes_inv_eq:
   \<open>inv p y = x \<longleftrightarrow> p x = y\<close>
   by (auto simp add: permutes_inverses)
 
 lemma permutes_inj_on:
   \<open>inj_on p A\<close>
   by (rule inj_on_subset [of _ UNIV]) (auto intro: permutes_inj)
 
 lemma permutes_bij:
   \<open>bij p\<close>
   unfolding bij_def by (metis permutes_inj permutes_surj)
 
 lemma permutes_imp_bij:
   \<open>bij_betw p S S\<close>
   by (simp add: bij_betw_def permutes_image permutes_inj_on)
 
 lemma permutes_subset:
   \<open>p permutes T\<close> if \<open>S \<subseteq> T\<close>
 proof (rule bij_imp_permutes)
   define R where \<open>R = T - S\<close>
   with that have \<open>T = R \<union> S\<close> \<open>R \<inter> S = {}\<close>
     by auto
   then have \<open>p x = x\<close> if \<open>x \<in> R\<close> for x
     using that by (auto intro: permutes_not_in)
   then have \<open>p ` R = R\<close>
     by simp
   with \<open>T = R \<union> S\<close> show \<open>bij_betw p T T\<close>
     by (simp add: bij_betw_def permutes_inj_on image_Un permutes_image)
   fix x
   assume \<open>x \<notin> T\<close>
   with \<open>T = R \<union> S\<close> show \<open>p x = x\<close>
     by (simp add: permutes_not_in)
 qed
 
 lemma permutes_imp_permutes_insert:
   \<open>p permutes insert x S\<close>
   by (rule permutes_subset) auto
 
 end
 
 lemma permutes_id [simp]:
   \<open>id permutes S\<close>
   by (auto intro: bij_imp_permutes)
 
 lemma permutes_empty [simp]:
   \<open>p permutes {} \<longleftrightarrow> p = id\<close>
 proof
   assume \<open>p permutes {}\<close>
   then show \<open>p = id\<close>
     by (auto simp add: fun_eq_iff permutes_not_in)
 next
   assume \<open>p = id\<close>
   then show \<open>p permutes {}\<close>
     by simp
 qed
 
 lemma permutes_sing [simp]:
   \<open>p permutes {a} \<longleftrightarrow> p = id\<close>
 proof
   assume perm: \<open>p permutes {a}\<close>
   show \<open>p = id\<close>
   proof
     fix x
     from perm have \<open>p ` {a} = {a}\<close>
       by (rule permutes_image)
     with perm show \<open>p x = id x\<close>
       by (cases \<open>x = a\<close>) (auto simp add: permutes_not_in)
   qed
 next
   assume \<open>p = id\<close>
   then show \<open>p permutes {a}\<close>
     by simp
 qed
 
 lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
   by (simp add: permutes_def)
 
 lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S"
   by (rule bij_imp_permutes) (auto simp add: swap_id_eq)
 
 lemma permutes_superset:
   \<open>p permutes T\<close> if \<open>p permutes S\<close> \<open>\<And>x. x \<in> S - T \<Longrightarrow> p x = x\<close>
 proof -
   define R U where \<open>R = T \<inter> S\<close> and \<open>U = S - T\<close>
   then have \<open>T = R \<union> (T - S)\<close> \<open>S = R \<union> U\<close> \<open>R \<inter> U = {}\<close>
     by auto
   from that \<open>U = S - T\<close> have \<open>p ` U = U\<close>
     by simp
   from \<open>p permutes S\<close> have \<open>bij_betw p (R \<union> U) (R \<union> U)\<close>
     by (simp add: permutes_imp_bij \<open>S = R \<union> U\<close>)
   moreover have \<open>bij_betw p U U\<close>
     using that \<open>U = S - T\<close> by (simp add: bij_betw_def permutes_inj_on)
   ultimately have \<open>bij_betw p R R\<close>
     using \<open>R \<inter> U = {}\<close> \<open>R \<inter> U = {}\<close> by (rule bij_betw_partition)
   then have \<open>p permutes R\<close>
   proof (rule bij_imp_permutes)
     fix x
     assume \<open>x \<notin> R\<close>
     with \<open>R = T \<inter> S\<close> \<open>p permutes S\<close> show \<open>p x = x\<close>
       by (cases \<open>x \<in> S\<close>) (auto simp add: permutes_not_in that(2))
   qed
   then have \<open>p permutes R \<union> (T - S)\<close>
     by (rule permutes_subset) simp
   with \<open>T = R \<union> (T - S)\<close> show ?thesis
     by simp
 qed
 
 lemma permutes_bij_inv_into: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
   fixes A :: "'a set"
     and B :: "'b set"
   assumes "p permutes A"
     and "bij_betw f A B"
   shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
 proof (rule bij_imp_permutes)
   from assms have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
     by (auto simp add: permutes_imp_bij bij_betw_inv_into)
   then have "bij_betw (f \<circ> p \<circ> inv_into A f) B B"
     by (simp add: bij_betw_trans)
   then show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B"
     by (subst bij_betw_cong[where g="f \<circ> p \<circ> inv_into A f"]) auto
 next
   fix x
   assume "x \<notin> B"
   then show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto
 qed
 
 lemma permutes_image_mset: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
   assumes "p permutes A"
   shows "image_mset p (mset_set A) = mset_set A"
   using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image)
 
 lemma permutes_implies_image_mset_eq: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
   assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)"
   shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
 proof -
   have "f x = f' (p x)" if "x \<in># mset_set A" for x
     using assms(2)[of x] that by (cases "finite A") auto
   with assms have "image_mset f (mset_set A) = image_mset (f' \<circ> p) (mset_set A)"
     by (auto intro!: image_mset_cong)
   also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
     by (simp add: image_mset.compositionality)
   also have "\<dots> = image_mset f' (mset_set A)"
   proof -
     from assms permutes_image_mset have "image_mset p (mset_set A) = mset_set A"
       by blast
     then show ?thesis by simp
   qed
   finally show ?thesis ..
 qed
 
 
 subsection \<open>Group properties\<close>
 
 lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S"
   unfolding permutes_def o_def by metis
 
 lemma permutes_inv:
   assumes "p permutes S"
   shows "inv p permutes S"
   using assms unfolding permutes_def permutes_inv_eq[OF assms] by metis
 
 lemma permutes_inv_inv:
   assumes "p permutes S"
   shows "inv (inv p) = p"
   unfolding fun_eq_iff permutes_inv_eq[OF assms] permutes_inv_eq[OF permutes_inv[OF assms]]
   by blast
 
 lemma permutes_invI:
   assumes perm: "p permutes S"
     and inv: "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
     and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
   shows "inv p = p'"
 proof
   show "inv p x = p' x" for x
   proof (cases "x \<in> S")
     case True
     from assms have "p' x = p' (p (inv p x))"
       by (simp add: permutes_inverses)
     also from permutes_inv[OF perm] True have "\<dots> = inv p x"
       by (subst inv) (simp_all add: permutes_in_image)
     finally show ?thesis ..
   next
     case False
     with permutes_inv[OF perm] show ?thesis
       by (simp_all add: outside permutes_not_in)
   qed
 qed
 
 lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
   by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
 
 
 subsection \<open>Mapping permutations with bijections\<close>
 
 lemma bij_betw_permutations:
   assumes "bij_betw f A B"
   shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x) 
              {\<pi>. \<pi> permutes A} {\<pi>. \<pi> permutes B}" (is "bij_betw ?f _ _")
 proof -
   let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
   show ?thesis
   proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
     case 3
     show ?case using permutes_bij_inv_into[OF _ assms] by auto
   next
     case 4
     have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
     {
       fix \<pi> assume "\<pi> permutes B"
       from permutes_bij_inv_into[OF this bij_inv] and assms
         have "(\<lambda>x. if x \<in> A then inv_into A f (\<pi> (f x)) else x) permutes A"
         by (simp add: inv_into_inv_into_eq cong: if_cong)
     }
     from this show ?case by (auto simp: permutes_inv)
   next
     case 1
     thus ?case using assms
       by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
                dest: bij_betwE)
   next
     case 2
     moreover have "bij_betw (inv_into A f) B A"
       by (intro bij_betw_inv_into assms)
     ultimately show ?case using assms
       by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right 
                dest: bij_betwE)
   qed
 qed
 
 lemma bij_betw_derangements:
   assumes "bij_betw f A B"
   shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x) 
              {\<pi>. \<pi> permutes A \<and> (\<forall>x\<in>A. \<pi> x \<noteq> x)} {\<pi>. \<pi> permutes B \<and> (\<forall>x\<in>B. \<pi> x \<noteq> x)}" 
            (is "bij_betw ?f _ _")
 proof -
   let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
   show ?thesis
   proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
     case 3
     have "?f \<pi> x \<noteq> x" if "\<pi> permutes A" "\<And>x. x \<in> A \<Longrightarrow> \<pi> x \<noteq> x" "x \<in> B" for \<pi> x
       using that and assms by (metis bij_betwE bij_betw_imp_inj_on bij_betw_imp_surj_on
                                      inv_into_f_f inv_into_into permutes_imp_bij)
     with permutes_bij_inv_into[OF _ assms] show ?case by auto
   next
     case 4
     have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
     have "?g \<pi> permutes A" if "\<pi> permutes B" for \<pi>
       using permutes_bij_inv_into[OF that bij_inv] and assms
       by (simp add: inv_into_inv_into_eq cong: if_cong)
     moreover have "?g \<pi> x \<noteq> x" if "\<pi> permutes B" "\<And>x. x \<in> B \<Longrightarrow> \<pi> x \<noteq> x" "x \<in> A" for \<pi> x
       using that and assms by (metis bij_betwE bij_betw_imp_surj_on f_inv_into_f permutes_imp_bij)
     ultimately show ?case by auto
   next
     case 1
     thus ?case using assms
       by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
                 dest: bij_betwE)
   next
     case 2
     moreover have "bij_betw (inv_into A f) B A"
       by (intro bij_betw_inv_into assms)
     ultimately show ?case using assms
       by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right 
                 dest: bij_betwE)
   qed
 qed
 
 
 subsection \<open>The number of permutations on a finite set\<close>
 
 lemma permutes_insert_lemma:
   assumes "p permutes (insert a S)"
   shows "Fun.swap a (p a) id \<circ> p permutes S"
   apply (rule permutes_superset[where S = "insert a S"])
   apply (rule permutes_compose[OF assms])
   apply (rule permutes_swap_id, simp)
   using permutes_in_image[OF assms, of a]
   apply simp
   apply (auto simp add: Ball_def Fun.swap_def)
   done
 
 lemma permutes_insert: "{p. p permutes (insert a S)} =
   (\<lambda>(b, p). Fun.swap a b id \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
 proof -
   have "p permutes insert a S \<longleftrightarrow>
     (\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)" for p
   proof -
     have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S"
       if p: "p permutes insert a S"
     proof -
       let ?b = "p a"
       let ?q = "Fun.swap a (p a) id \<circ> p"
       have *: "p = Fun.swap a ?b id \<circ> ?q"
         by (simp add: fun_eq_iff o_assoc)
       have **: "?b \<in> insert a S"
         unfolding permutes_in_image[OF p] by simp
       from permutes_insert_lemma[OF p] * ** show ?thesis
        by blast
     qed
     moreover have "p permutes insert a S"
       if bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" for b q
     proof -
       from permutes_subset[OF bq(3), of "insert a S"] have q: "q permutes insert a S"
         by auto
       have a: "a \<in> insert a S"
         by simp
       from bq(1) permutes_compose[OF q permutes_swap_id[OF a bq(2)]] show ?thesis
         by simp
     qed
     ultimately show ?thesis by blast
   qed
   then show ?thesis by auto
 qed
 
 lemma card_permutations:
   assumes "card S = n"
     and "finite S"
   shows "card {p. p permutes S} = fact n"
   using assms(2,1)
 proof (induct arbitrary: n)
   case empty
   then show ?case by simp
 next
   case (insert x F)
   {
     fix n
     assume card_insert: "card (insert x F) = n"
     let ?xF = "{p. p permutes insert x F}"
     let ?pF = "{p. p permutes F}"
     let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
     let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
     have xfgpF': "?xF = ?g ` ?pF'"
       by (rule permutes_insert[of x F])
     from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have Fs: "card F = n - 1"
       by auto
     from \<open>finite F\<close> insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
       by auto
     then have "finite ?pF"
       by (auto intro: card_ge_0_finite)
     with \<open>finite F\<close> card.insert_remove have pF'f: "finite ?pF'"
       apply (simp only: Collect_case_prod Collect_mem_eq)
       apply (rule finite_cartesian_product)
       apply simp_all
       done
 
     have ginj: "inj_on ?g ?pF'"
     proof -
       {
         fix b p c q
         assume bp: "(b, p) \<in> ?pF'"
         assume cq: "(c, q) \<in> ?pF'"
         assume eq: "?g (b, p) = ?g (c, q)"
         from bp cq have pF: "p permutes F" and qF: "q permutes F"
           by auto
         from pF \<open>x \<notin> F\<close> eq have "b = ?g (b, p) x"
           by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
         also from qF \<open>x \<notin> F\<close> eq have "\<dots> = ?g (c, q) x"
           by (auto simp: fun_upd_def fun_eq_iff)
         also from qF \<open>x \<notin> F\<close> have "\<dots> = c"
           by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
         finally have "b = c" .
         then have "Fun.swap x b id = Fun.swap x c id"
           by simp
         with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q"
           by simp
         then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) = Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)"
           by simp
         then have "p = q"
           by (simp add: o_assoc)
         with \<open>b = c\<close> have "(b, p) = (c, q)"
           by simp
       }
       then show ?thesis
         unfolding inj_on_def by blast
     qed
     from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have "n \<noteq> 0"
       by auto
     then have "\<exists>m. n = Suc m"
       by presburger
     then obtain m where n: "n = Suc m"
       by blast
     from pFs card_insert have *: "card ?xF = fact n"
       unfolding xfgpF' card_image[OF ginj]
       using \<open>finite F\<close> \<open>finite ?pF\<close>
       by (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) (simp add: n)
     from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF"
       by (simp add: xfgpF' n)
     from * have "card ?xF = fact n"
       unfolding xFf by blast
   }
   with insert show ?case by simp
 qed
 
 lemma finite_permutations:
   assumes "finite S"
   shows "finite {p. p permutes S}"
   using card_permutations[OF refl assms] by (auto intro: card_ge_0_finite)
 
 
 subsection \<open>Hence a sort of induction principle composing by swaps\<close>
 
 lemma permutes_induct [consumes 2, case_names id swap]:
   \<open>P p\<close> if \<open>p permutes S\<close> \<open>finite S\<close>
   and id: \<open>P id\<close>
   and swap: \<open>\<And>a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> p permutes S \<Longrightarrow> P p \<Longrightarrow> P (Fun.swap a b id \<circ> p)\<close>
 using \<open>finite S\<close> \<open>p permutes S\<close> swap proof (induction S arbitrary: p)
   case empty
   with id show ?case
     by (simp only: permutes_empty)
 next
   case (insert x S p)
   define q where \<open>q = Fun.swap x (p x) id \<circ> p\<close>
   then have swap_q: \<open>Fun.swap x (p x) id \<circ> q = p\<close>
     by (simp add: o_assoc)
   from \<open>p permutes insert x S\<close> have \<open>q permutes S\<close>
     by (simp add: q_def permutes_insert_lemma)
   then have \<open>q permutes insert x S\<close>
     by (simp add: permutes_imp_permutes_insert)
   from \<open>q permutes S\<close> have \<open>P q\<close>
     by (auto intro: insert.IH insert.prems(2) permutes_imp_permutes_insert)
   have \<open>x \<in> insert x S\<close>
     by simp
   moreover from \<open>p permutes insert x S\<close> have \<open>p x \<in> insert x S\<close>
     using permutes_in_image [of p \<open>insert x S\<close> x] by simp
   ultimately have \<open>P (Fun.swap x (p x) id \<circ> q)\<close>
     using \<open>q permutes insert x S\<close> \<open>P q\<close>
     by (rule insert.prems(2))
   then show ?case
     by (simp add: swap_q)
 qed
 
 lemma permutes_rev_induct [consumes 2, case_names id swap]:
   \<open>P p\<close> if \<open>p permutes S\<close> \<open>finite S\<close>
   and id': \<open>P id\<close>
   and swap': \<open>\<And>a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> p permutes S \<Longrightarrow> P p \<Longrightarrow> P (Fun.swap a b p)\<close>
 using \<open>p permutes S\<close> \<open>finite S\<close> proof (induction rule: permutes_induct)
   case id
   from id' show ?case .
 next
   case (swap a b p)
   have \<open>P (Fun.swap (inv p a) (inv p b) p)\<close>
     by (rule swap') (auto simp add: swap permutes_in_image permutes_inv)
   also have \<open>Fun.swap (inv p a) (inv p b) p = Fun.swap a b id \<circ> p\<close>
     by (rule bij_swap_comp [symmetric]) (rule permutes_bij, rule swap)
   finally show ?case .
 qed
 
 
 subsection \<open>Permutations of index set for iterated operations\<close>
 
 lemma (in comm_monoid_set) permute:
   assumes "p permutes S"
   shows "F g S = F (g \<circ> p) S"
 proof -
   from \<open>p permutes S\<close> have "inj p"
     by (rule permutes_inj)
   then have "inj_on p S"
     by (auto intro: subset_inj_on)
   then have "F g (p ` S) = F (g \<circ> p) S"
     by (rule reindex)
   moreover from \<open>p permutes S\<close> have "p ` S = S"
     by (rule permutes_image)
   ultimately show ?thesis
     by simp
 qed
 
 
 subsection \<open>Permutations as transposition sequences\<close>
 
 inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
   where
     id[simp]: "swapidseq 0 id"
   | comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
 
 declare id[unfolded id_def, simp]
 
 definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
 
 
 subsection \<open>Some closure properties of the set of permutations, with lengths\<close>
 
 lemma permutation_id[simp]: "permutation id"
   unfolding permutation_def by (rule exI[where x=0]) simp
 
 declare permutation_id[unfolded id_def, simp]
 
 lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
   apply clarsimp
   using comp_Suc[of 0 id a b]
   apply simp
   done
 
 lemma permutation_swap_id: "permutation (Fun.swap a b id)"
 proof (cases "a = b")
   case True
   then show ?thesis by simp
 next
   case False
   then show ?thesis
     unfolding permutation_def
     using swapidseq_swap[of a b] by blast
 qed
 
 
 lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)"
 proof (induct n p arbitrary: m q rule: swapidseq.induct)
   case (id m q)
   then show ?case by simp
 next
   case (comp_Suc n p a b m q)
   have eq: "Suc n + m = Suc (n + m)"
     by arith
   show ?case
     apply (simp only: eq comp_assoc)
     apply (rule swapidseq.comp_Suc)
     using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
      apply blast+
     done
 qed
 
 lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)"
   unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
 
 lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
   by (induct n p rule: swapidseq.induct)
     (use swapidseq_swap[of a b] in \<open>auto simp add: comp_assoc intro: swapidseq.comp_Suc\<close>)
 
 lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
 proof (induct n p rule: swapidseq.induct)
   case id
   then show ?case
     by (rule exI[where x=id]) simp
 next
   case (comp_Suc n p a b)
   from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
     by blast
   let ?q = "q \<circ> Fun.swap a b id"
   note H = comp_Suc.hyps
   from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (Fun.swap a b id)"
     by simp
   from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q"
     by simp
   have "Fun.swap a b id \<circ> p \<circ> ?q = Fun.swap a b id \<circ> (p \<circ> q) \<circ> Fun.swap a b id"
     by (simp add: o_assoc)
   also have "\<dots> = id"
     by (simp add: q(2))
   finally have ***: "Fun.swap a b id \<circ> p \<circ> ?q = id" .
   have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id \<circ> Fun.swap a b id) \<circ> p"
     by (simp only: o_assoc)
   then have "?q \<circ> (Fun.swap a b id \<circ> p) = id"
     by (simp add: q(3))
   with ** *** show ?case
     by blast
 qed
 
 lemma swapidseq_inverse:
   assumes "swapidseq n p"
   shows "swapidseq n (inv p)"
   using swapidseq_inverse_exists[OF assms] inv_unique_comp[of p] by auto
 
 lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)"
   using permutation_def swapidseq_inverse by blast
 
 
 
 subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close>
 
 lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>
   Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
   by (simp add: fun_eq_iff Fun.swap_def)
 
 lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow>
   Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"
   by (simp add: fun_eq_iff Fun.swap_def)
 
 lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow>
   Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id"
   by (simp add: fun_eq_iff Fun.swap_def)
 
 
 subsection \<open>The identity map only has even transposition sequences\<close>
 
 lemma symmetry_lemma:
   assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"
     and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
       a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<Longrightarrow>
       P a b c d"
   shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>  P a b c d"
   using assms by metis
 
 lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
   Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
     Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)"
 proof -
   assume neq: "a \<noteq> b" "c \<noteq> d"
   have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
     (Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
       (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
         Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))"
     apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
      apply (simp_all only: swap_commute)
     apply (case_tac "a = c \<and> b = d")
      apply (clarsimp simp only: swap_commute swap_id_idempotent)
     apply (case_tac "a = c \<and> b \<noteq> d")
      apply (rule disjI2)
      apply (rule_tac x="b" in exI)
      apply (rule_tac x="d" in exI)
      apply (rule_tac x="b" in exI)
      apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
     apply (case_tac "a \<noteq> c \<and> b = d")
      apply (rule disjI2)
      apply (rule_tac x="c" in exI)
      apply (rule_tac x="d" in exI)
      apply (rule_tac x="c" in exI)
      apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
     apply (rule disjI2)
     apply (rule_tac x="c" in exI)
     apply (rule_tac x="d" in exI)
     apply (rule_tac x="b" in exI)
     apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
     done
   with neq show ?thesis by metis
 qed
 
 lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
   using swapidseq.cases[of 0 p "p = id"] by auto
 
 lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow>
     n = 0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id \<circ> q \<and> swapidseq m q \<and> a \<noteq> b)"
   apply (rule iffI)
    apply (erule swapidseq.cases[of n p])
     apply simp
    apply (rule disjI2)
    apply (rule_tac x= "a" in exI)
    apply (rule_tac x= "b" in exI)
    apply (rule_tac x= "pa" in exI)
    apply (rule_tac x= "na" in exI)
    apply simp
   apply auto
   apply (rule comp_Suc, simp_all)
   done
 
 lemma fixing_swapidseq_decrease:
   assumes "swapidseq n p"
     and "a \<noteq> b"
     and "(Fun.swap a b id \<circ> p) a = a"
   shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
   using assms
 proof (induct n arbitrary: p a b)
   case 0
   then show ?case
     by (auto simp add: Fun.swap_def fun_upd_def)
 next
   case (Suc n p a b)
   from Suc.prems(1) swapidseq_cases[of "Suc n" p]
   obtain c d q m where
     cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m"
     by auto
   consider "Fun.swap a b id \<circ> Fun.swap c d id = id"
     | x y z where "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y"
       "Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id"
     using swap_general[OF Suc.prems(2) cdqm(4)] by metis
   then show ?case
   proof cases
     case 1
     then show ?thesis
       by (simp only: cdqm o_assoc) (simp add: cdqm)
   next
     case prems: 2
     then have az: "a \<noteq> z"
       by simp
     from prems have *: "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" for h
       by (simp add: Fun.swap_def)
     from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)"
       by simp
     then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)"
       by (simp add: o_assoc prems)
     then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a"
       by simp
     then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a"
       unfolding Suc by metis
     then have "(Fun.swap a z id \<circ> q) a = a"
       by (simp only: *)
     from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this]
     have **: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0"
       by blast+
     from \<open>n \<noteq> 0\<close> have ***: "Suc n - 1 = Suc (n - 1)"
       by auto
     show ?thesis
       apply (simp only: cdqm(2) prems o_assoc ***)
       apply (simp only: Suc_not_Zero simp_thms comp_assoc)
       apply (rule comp_Suc)
       using ** prems
        apply blast+
       done
   qed
 qed
 
 lemma swapidseq_identity_even:
   assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)"
   shows "even n"
   using \<open>swapidseq n id\<close>
 proof (induct n rule: nat_less_induct)
   case H: (1 n)
   consider "n = 0"
     | a b :: 'a and q m where "n = Suc m" "id = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
     using H(2)[unfolded swapidseq_cases[of n id]] by auto
   then show ?case
   proof cases
     case 1
     then show ?thesis by presburger
   next
     case h: 2
     from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
     have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)"
       by auto
     from h m have mn: "m - 1 < n"
       by arith
     from H(1)[rule_format, OF mn m(2)] h(1) m(1) show ?thesis
       by presburger
   qed
 qed
 
 
 subsection \<open>Therefore we have a welldefined notion of parity\<close>
 
 definition "evenperm p = even (SOME n. swapidseq n p)"
 
 lemma swapidseq_even_even:
   assumes m: "swapidseq m p"
     and n: "swapidseq n p"
   shows "even m \<longleftrightarrow> even n"
 proof -
   from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
     by blast
   from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] show ?thesis
     by arith
 qed
 
 lemma evenperm_unique:
   assumes p: "swapidseq n p"
     and n:"even n = b"
   shows "evenperm p = b"
   unfolding n[symmetric] evenperm_def
   apply (rule swapidseq_even_even[where p = p])
    apply (rule someI[where x = n])
   using p
    apply blast+
   done
 
 
 subsection \<open>And it has the expected composition properties\<close>
 
 lemma evenperm_id[simp]: "evenperm id = True"
   by (rule evenperm_unique[where n = 0]) simp_all
 
+lemma evenperm_identity [simp]:
+  \<open>evenperm (\<lambda>x. x)\<close>
+  using evenperm_id by (simp add: id_def [abs_def])
+
 lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
   by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap)
 
 lemma evenperm_comp:
   assumes "permutation p" "permutation q"
   shows "evenperm (p \<circ> q) \<longleftrightarrow> evenperm p = evenperm q"
 proof -
   from assms obtain n m where n: "swapidseq n p" and m: "swapidseq m q"
     unfolding permutation_def by blast
   have "even (n + m) \<longleftrightarrow> (even n \<longleftrightarrow> even m)"
     by arith
   from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
     and evenperm_unique[OF swapidseq_comp_add[OF n m] this] show ?thesis
     by blast
 qed
 
 lemma evenperm_inv:
   assumes "permutation p"
   shows "evenperm (inv p) = evenperm p"
 proof -
   from assms obtain n where n: "swapidseq n p"
     unfolding permutation_def by blast
   show ?thesis
     by (rule evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]])
 qed
 
 
 subsection \<open>A more abstract characterization of permutations\<close>
 
 lemma permutation_bijective:
   assumes "permutation p"
   shows "bij p"
 proof -
   from assms obtain n where n: "swapidseq n p"
     unfolding permutation_def by blast
   from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
     by blast
   then show ?thesis
     unfolding bij_iff
     apply (auto simp add: fun_eq_iff)
     apply metis
     done
 qed
 
 lemma permutation_finite_support:
   assumes "permutation p"
   shows "finite {x. p x \<noteq> x}"
 proof -
   from assms obtain n where "swapidseq n p"
     unfolding permutation_def by blast
   then show ?thesis
   proof (induct n p rule: swapidseq.induct)
     case id
     then show ?case by simp
   next
     case (comp_Suc n p a b)
     let ?S = "insert a (insert b {x. p x \<noteq> x})"
     from comp_Suc.hyps(2) have *: "finite ?S"
       by simp
     from \<open>a \<noteq> b\<close> have **: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
       by (auto simp: Fun.swap_def)
     show ?case
       by (rule finite_subset[OF ** *])
   qed
 qed
 
 lemma permutation_lemma:
   assumes "finite S"
     and "bij p"
     and "\<forall>x. x \<notin> S \<longrightarrow> p x = x"
   shows "permutation p"
   using assms
 proof (induct S arbitrary: p rule: finite_induct)
   case empty
   then show ?case
     by simp
 next
   case (insert a F p)
   let ?r = "Fun.swap a (p a) id \<circ> p"
   let ?q = "Fun.swap a (p a) id \<circ> ?r"
   have *: "?r a = a"
     by (simp add: Fun.swap_def)
   from insert * have **: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
     by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
   have "bij ?r"
     by (rule bij_swap_compose_bij[OF insert(4)])
   have "permutation ?r"
     by (rule insert(3)[OF \<open>bij ?r\<close> **])
   then have "permutation ?q"
     by (simp add: permutation_compose permutation_swap_id)
   then show ?case
     by (simp add: o_assoc)
 qed
 
 lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
   (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
 proof
   assume ?lhs
   with permutation_bijective permutation_finite_support show "?b \<and> ?f"
     by auto
 next
   assume "?b \<and> ?f"
   then have "?f" "?b" by blast+
   from permutation_lemma[OF this] show ?lhs
     by blast
 qed
 
 lemma permutation_inverse_works:
   assumes "permutation p"
   shows "inv p \<circ> p = id"
     and "p \<circ> inv p = id"
   using permutation_bijective [OF assms] by (auto simp: bij_def inj_iff surj_iff)
 
 lemma permutation_inverse_compose:
   assumes p: "permutation p"
     and q: "permutation q"
   shows "inv (p \<circ> q) = inv q \<circ> inv p"
 proof -
   note ps = permutation_inverse_works[OF p]
   note qs = permutation_inverse_works[OF q]
   have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p"
     by (simp add: o_assoc)
   also have "\<dots> = id"
     by (simp add: ps qs)
   finally have *: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" .
   have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q"
     by (simp add: o_assoc)
   also have "\<dots> = id"
     by (simp add: ps qs)
   finally have **: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
   show ?thesis
     by (rule inv_unique_comp[OF * **])
 qed
 
 
 subsection \<open>Relation to \<open>permutes\<close>\<close>
 
 lemma permutes_imp_permutation:
   \<open>permutation p\<close> if \<open>finite S\<close> \<open>p permutes S\<close>
 proof -
   from \<open>p permutes S\<close> have \<open>{x. p x \<noteq> x} \<subseteq> S\<close>
     by (auto dest: permutes_not_in)
   then have \<open>finite {x. p x \<noteq> x}\<close>
     using \<open>finite S\<close> by (rule finite_subset)
   moreover from \<open>p permutes S\<close> have \<open>bij p\<close>
     by (auto dest: permutes_bij)
   ultimately show ?thesis
     by (simp add: permutation)
 qed
 
 lemma permutation_permutesE:
   assumes \<open>permutation p\<close>
   obtains S where \<open>finite S\<close> \<open>p permutes S\<close>
 proof -
   from assms have fin: \<open>finite {x. p x \<noteq> x}\<close>
     by (simp add: permutation)
   from assms have \<open>bij p\<close>
     by (simp add: permutation)
   also have \<open>UNIV = {x. p x \<noteq> x} \<union> {x. p x = x}\<close>
     by auto
   finally have \<open>bij_betw p {x. p x \<noteq> x} {x. p x \<noteq> x}\<close>
     by (rule bij_betw_partition) (auto simp add: bij_betw_fixpoints)
   then have \<open>p permutes {x. p x \<noteq> x}\<close>
     by (auto intro: bij_imp_permutes)
   with fin show thesis ..
 qed
 
 lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
   by (auto elim: permutation_permutesE intro: permutes_imp_permutation)
 
 
 subsection \<open>Sign of a permutation as a real number\<close>
 
 definition sign :: \<open>('a \<Rightarrow> 'a) \<Rightarrow> int\<close> \<comment> \<open>TODO: prefer less generic name\<close>
-  where \<open>sign p = (if evenperm p then (1::int) else -1)\<close>
+  where \<open>sign p = (if evenperm p then 1 else - 1)\<close>
 
-lemma sign_nz: "sign p \<noteq> 0"
+lemma sign_cases [case_names even odd]:
+  obtains \<open>sign p = 1\<close> | \<open>sign p = - 1\<close>
+  by (cases \<open>evenperm p\<close>) (simp_all add: sign_def)
+
+lemma sign_nz [simp]: "sign p \<noteq> 0"
+  by (cases p rule: sign_cases) simp_all
+
+lemma sign_id [simp]: "sign id = 1"
   by (simp add: sign_def)
 
-lemma sign_id: "sign id = 1"
+lemma sign_identity [simp]:
+  \<open>sign (\<lambda>x. x) = 1\<close>
   by (simp add: sign_def)
 
 lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p"
   by (simp add: sign_def evenperm_inv)
 
 lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q"
   by (simp add: sign_def evenperm_comp)
 
-lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
+lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else - 1)"
   by (simp add: sign_def evenperm_swap)
 
-lemma sign_idempotent: "sign p * sign p = 1"
+lemma sign_idempotent [simp]: "sign p * sign p = 1"
   by (simp add: sign_def)
 
+lemma sign_left_idempotent [simp]:
+  \<open>sign p * (sign p * sign q) = sign q\<close>
+  by (simp add: sign_def)
+
+term "(bij, bij_betw, permutation)"
+
 
 subsection \<open>Permuting a list\<close>
 
 text \<open>This function permutes a list by applying a permutation to the indices.\<close>
 
 definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list"
   where "permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"
 
 lemma permute_list_map:
   assumes "f permutes {..<length xs}"
   shows "permute_list f (map g xs) = map g (permute_list f xs)"
   using permutes_in_image[OF assms] by (auto simp: permute_list_def)
 
 lemma permute_list_nth:
   assumes "f permutes {..<length xs}" "i < length xs"
   shows "permute_list f xs ! i = xs ! f i"
   using permutes_in_image[OF assms(1)] assms(2)
   by (simp add: permute_list_def)
 
 lemma permute_list_Nil [simp]: "permute_list f [] = []"
   by (simp add: permute_list_def)
 
 lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"
   by (simp add: permute_list_def)
 
 lemma permute_list_compose:
   assumes "g permutes {..<length xs}"
   shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
   using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)
 
 lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs"
   by (simp add: permute_list_def map_nth)
 
 lemma permute_list_id [simp]: "permute_list id xs = xs"
   by (simp add: id_def)
 
 lemma mset_permute_list [simp]:
   fixes xs :: "'a list"
   assumes "f permutes {..<length xs}"
   shows "mset (permute_list f xs) = mset xs"
 proof (rule multiset_eqI)
   fix y :: 'a
   from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
     using permutes_in_image[OF assms] by auto
   have "count (mset (permute_list f xs)) y = card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})"
     by (simp add: permute_list_def count_image_mset atLeast0LessThan)
   also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}"
     by auto
   also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}"
     by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
   also have "\<dots> = count (mset xs) y"
     by (simp add: count_mset length_filter_conv_card)
   finally show "count (mset (permute_list f xs)) y = count (mset xs) y"
     by simp
 qed
 
 lemma set_permute_list [simp]:
   assumes "f permutes {..<length xs}"
   shows "set (permute_list f xs) = set xs"
   by (rule mset_eq_setD[OF mset_permute_list]) fact
 
 lemma distinct_permute_list [simp]:
   assumes "f permutes {..<length xs}"
   shows "distinct (permute_list f xs) = distinct xs"
   by (simp add: distinct_count_atmost_1 assms)
 
 lemma permute_list_zip:
   assumes "f permutes A" "A = {..<length xs}"
   assumes [simp]: "length xs = length ys"
   shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
 proof -
   from permutes_in_image[OF assms(1)] assms(2) have *: "f i < length ys \<longleftrightarrow> i < length ys" for i
     by simp
   have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]"
     by (simp_all add: permute_list_def zip_map_map)
   also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
     by (intro nth_equalityI) (simp_all add: *)
   also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
     by (simp_all add: permute_list_def zip_map_map)
   finally show ?thesis .
 qed
 
 lemma map_of_permute:
   assumes "\<sigma> permutes fst ` set xs"
   shows "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)"
     (is "_ = map_of (map ?f _)")
 proof
   from assms have "inj \<sigma>" "surj \<sigma>"
     by (simp_all add: permutes_inj permutes_surj)
   then show "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x" for x
     by (induct xs) (auto simp: inv_f_f surj_f_inv_f)
 qed
 
 
 subsection \<open>More lemmas about permutations\<close>
 
 lemma permutes_in_funpow_image: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
   assumes "f permutes S" "x \<in> S"
   shows "(f ^^ n) x \<in> S"
   using assms by (induction n) (auto simp: permutes_in_image)
 
 lemma permutation_self: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
   assumes \<open>permutation p\<close>
   obtains n where \<open>n > 0\<close> \<open>(p ^^ n) x = x\<close>
 proof (cases \<open>p x = x\<close>)
   case True
   with that [of 1] show thesis by simp
 next
   case False
   from \<open>permutation p\<close> have \<open>inj p\<close>
     by (intro permutation_bijective bij_is_inj)
   moreover from \<open>p x \<noteq> x\<close> have \<open>(p ^^ Suc n) x \<noteq> (p ^^ n) x\<close> for n
   proof (induction n arbitrary: x)
     case 0 then show ?case by simp
   next
     case (Suc n)
     have "p (p x) \<noteq> p x"
     proof (rule notI)
       assume "p (p x) = p x"
       then show False using \<open>p x \<noteq> x\<close> \<open>inj p\<close> by (simp add: inj_eq)
     qed
     have "(p ^^ Suc (Suc n)) x = (p ^^ Suc n) (p x)"
       by (simp add: funpow_swap1)
     also have "\<dots> \<noteq> (p ^^ n) (p x)"
       by (rule Suc) fact
     also have "(p ^^ n) (p x) = (p ^^ Suc n) x"
       by (simp add: funpow_swap1)
     finally show ?case by simp
   qed
   then have "{y. \<exists>n. y = (p ^^ n) x} \<subseteq> {x. p x \<noteq> x}"
     by auto
   then have "finite {y. \<exists>n. y = (p ^^ n) x}"
     using permutation_finite_support[OF assms] by (rule finite_subset)
   ultimately obtain n where \<open>n > 0\<close> \<open>(p ^^ n) x = x\<close>
     by (rule funpow_inj_finite)
   with that [of n] show thesis by blast
 qed
 
 text \<open>The following few lemmas were contributed by Lukas Bulwahn.\<close>
 
 lemma count_image_mset_eq_card_vimage:
   assumes "finite A"
   shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
   using assms
 proof (induct A)
   case empty
   show ?case by simp
 next
   case (insert x F)
   show ?case
   proof (cases "f x = b")
     case True
     with insert.hyps
     have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
       by auto
     also from insert.hyps(1,2) have "\<dots> = card (insert x {a \<in> F. f a = f x})"
       by simp
     also from \<open>f x = b\<close> have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
       by (auto intro: arg_cong[where f="card"])
     finally show ?thesis
       using insert by auto
   next
     case False
     then have "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}"
       by auto
     with insert False show ?thesis
       by simp
   qed
 qed
 
 \<comment> \<open>Prove \<open>image_mset_eq_implies_permutes\<close> ...\<close>
 lemma image_mset_eq_implies_permutes:
   fixes f :: "'a \<Rightarrow> 'b"
   assumes "finite A"
     and mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)"
   obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
 proof -
   from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto
   have "f ` A = f' ` A"
   proof -
     from \<open>finite A\<close> have "f ` A = f ` (set_mset (mset_set A))"
       by simp
     also have "\<dots> = f' ` set_mset (mset_set A)"
       by (metis mset_eq multiset.set_map)
     also from \<open>finite A\<close> have "\<dots> = f' ` A"
       by simp
     finally show ?thesis .
   qed
   have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}"
   proof
     fix b
     from mset_eq have "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b"
       by simp
     with \<open>finite A\<close> have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
       by (simp add: count_image_mset_eq_card_vimage)
     then show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
       by (intro finite_same_card_bij) simp_all
   qed
   then have "\<exists>p. \<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}"
     by (rule bchoice)
   then obtain p where p: "\<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}" ..
   define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
   have "p' permutes A"
   proof (rule bij_imp_permutes)
     have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)"
       by (auto simp: disjoint_family_on_def)
     moreover
     have "bij_betw (\<lambda>a. p (f a) a) {a \<in> A. f a = b} {a \<in> A. f' a = b}" if "b \<in> f ` A" for b
       using p that by (subst bij_betw_cong[where g="p b"]) auto
     ultimately
     have "bij_betw (\<lambda>a. p (f a) a) (\<Union>b\<in>f ` A. {a \<in> A. f a = b}) (\<Union>b\<in>f ` A. {a \<in> A. f' a = b})"
       by (rule bij_betw_UNION_disjoint)
     moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A"
       by auto
     moreover from \<open>f ` A = f' ` A\<close> have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A"
       by auto
     ultimately show "bij_betw p' A A"
       unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto
   next
     show "\<And>x. x \<notin> A \<Longrightarrow> p' x = x"
       by (simp add: p'_def)
   qed
   moreover from p have "\<forall>x\<in>A. f x = f' (p' x)"
     unfolding p'_def using bij_betwE by fastforce
   ultimately show ?thesis ..
 qed
 
 \<comment> \<open>... and derive the existing property:\<close>
 lemma mset_eq_permutation:
   fixes xs ys :: "'a list"
   assumes mset_eq: "mset xs = mset ys"
   obtains p where "p permutes {..<length ys}" "permute_list p ys = xs"
 proof -
   from mset_eq have length_eq: "length xs = length ys"
     by (rule mset_eq_length)
   have "mset_set {..<length ys} = mset [0..<length ys]"
     by (rule mset_set_upto_eq_mset_upto)
   with mset_eq length_eq have "image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) =
     image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})"
     by (metis map_nth mset_map)
   from image_mset_eq_implies_permutes[OF _ this]
   obtain p where p: "p permutes {..<length ys}" and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)"
     by auto
   with length_eq have "permute_list p ys = xs"
     by (auto intro!: nth_equalityI simp: permute_list_nth)
   with p show thesis ..
 qed
 
 lemma permutes_natset_le:
   fixes S :: "'a::wellorder set"
   assumes "p permutes S"
     and "\<forall>i \<in> S. p i \<le> i"
   shows "p = id"
 proof -
   have "p n = n" for n
     using assms
   proof (induct n arbitrary: S rule: less_induct)
     case (less n)
     show ?case
     proof (cases "n \<in> S")
       case False
       with less(2) show ?thesis
         unfolding permutes_def by metis
     next
       case True
       with less(3) have "p n < n \<or> p n = n"
         by auto
       then show ?thesis
       proof
         assume "p n < n"
         with less have "p (p n) = p n"
           by metis
         with permutes_inj[OF less(2)] have "p n = n"
           unfolding inj_def by blast
         with \<open>p n < n\<close> have False
           by simp
         then show ?thesis ..
       qed
     qed
   qed
   then show ?thesis by (auto simp: fun_eq_iff)
 qed
 
 lemma permutes_natset_ge:
   fixes S :: "'a::wellorder set"
   assumes p: "p permutes S"
     and le: "\<forall>i \<in> S. p i \<ge> i"
   shows "p = id"
 proof -
   have "i \<ge> inv p i" if "i \<in> S" for i
   proof -
     from that permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S"
       by simp
     with le have "p (inv p i) \<ge> inv p i"
       by blast
     with permutes_inverses[OF p] show ?thesis
       by simp
   qed
   then have "\<forall>i\<in>S. inv p i \<le> i"
     by blast
   from permutes_natset_le[OF permutes_inv[OF p] this] have "inv p = inv id"
     by simp
   then show ?thesis
     apply (subst permutes_inv_inv[OF p, symmetric])
     apply (rule inv_unique_comp)
      apply simp_all
     done
 qed
 
 lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
   apply (rule set_eqI)
   apply auto
   using permutes_inv_inv permutes_inv
    apply auto
   apply (rule_tac x="inv x" in exI)
   apply auto
   done
 
 lemma image_compose_permutations_left:
   assumes "q permutes S"
   shows "{q \<circ> p |p. p permutes S} = {p. p permutes S}"
   apply (rule set_eqI)
   apply auto
    apply (rule permutes_compose)
   using assms
     apply auto
   apply (rule_tac x = "inv q \<circ> x" in exI)
   apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
   done
 
 lemma image_compose_permutations_right:
   assumes "q permutes S"
   shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}"
   apply (rule set_eqI)
   apply auto
    apply (rule permutes_compose)
   using assms
     apply auto
   apply (rule_tac x = "x \<circ> inv q" in exI)
   apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc)
   done
 
 lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n"
   by (simp add: permutes_def) metis
 
 lemma sum_permutations_inverse: "sum f {p. p permutes S} = sum (\<lambda>p. f(inv p)) {p. p permutes S}"
   (is "?lhs = ?rhs")
 proof -
   let ?S = "{p . p permutes S}"
   have *: "inj_on inv ?S"
   proof (auto simp add: inj_on_def)
     fix q r
     assume q: "q permutes S"
       and r: "r permutes S"
       and qr: "inv q = inv r"
     then have "inv (inv q) = inv (inv r)"
       by simp
     with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r"
       by metis
   qed
   have **: "inv ` ?S = ?S"
     using image_inverse_permutations by blast
   have ***: "?rhs = sum (f \<circ> inv) ?S"
     by (simp add: o_def)
   from sum.reindex[OF *, of f] show ?thesis
     by (simp only: ** ***)
 qed
 
 lemma setum_permutations_compose_left:
   assumes q: "q permutes S"
   shows "sum f {p. p permutes S} = sum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
   (is "?lhs = ?rhs")
 proof -
   let ?S = "{p. p permutes S}"
   have *: "?rhs = sum (f \<circ> ((\<circ>) q)) ?S"
     by (simp add: o_def)
   have **: "inj_on ((\<circ>) q) ?S"
   proof (auto simp add: inj_on_def)
     fix p r
     assume "p permutes S"
       and r: "r permutes S"
       and rp: "q \<circ> p = q \<circ> r"
     then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r"
       by (simp add: comp_assoc)
     with permutes_inj[OF q, unfolded inj_iff] show "p = r"
       by simp
   qed
   have "((\<circ>) q) ` ?S = ?S"
     using image_compose_permutations_left[OF q] by auto
   with * sum.reindex[OF **, of f] show ?thesis
     by (simp only:)
 qed
 
 lemma sum_permutations_compose_right:
   assumes q: "q permutes S"
   shows "sum f {p. p permutes S} = sum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
   (is "?lhs = ?rhs")
 proof -
   let ?S = "{p. p permutes S}"
   have *: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
     by (simp add: o_def)
   have **: "inj_on (\<lambda>p. p \<circ> q) ?S"
   proof (auto simp add: inj_on_def)
     fix p r
     assume "p permutes S"
       and r: "r permutes S"
       and rp: "p \<circ> q = r \<circ> q"
     then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)"
       by (simp add: o_assoc)
     with permutes_surj[OF q, unfolded surj_iff] show "p = r"
       by simp
   qed
   from image_compose_permutations_right[OF q] have "(\<lambda>p. p \<circ> q) ` ?S = ?S"
     by auto
   with * sum.reindex[OF **, of f] show ?thesis
     by (simp only:)
 qed
 
+lemma inv_inj_on_permutes:
+  \<open>inj_on inv {p. p permutes S}\<close>
+proof (intro inj_onI, unfold mem_Collect_eq)
+  fix p q
+  assume p: "p permutes S" and q: "q permutes S" and eq: "inv p = inv q"
+  have "inv (inv p) = inv (inv q)" using eq by simp
+  thus "p = q"
+    using inv_inv_eq[OF permutes_bij] p q by metis
+qed
+
+lemma permutes_pair_eq:
+  \<open>{(p s, s) |s. s \<in> S} = {(s, inv p s) |s. s \<in> S}\<close> (is \<open>?L = ?R\<close>) if \<open>p permutes S\<close>
+proof
+  show "?L \<subseteq> ?R"
+  proof
+    fix x assume "x \<in> ?L"
+    then obtain s where x: "x = (p s, s)" and s: "s \<in> S" by auto
+    note x
+    also have "(p s, s) = (p s, Hilbert_Choice.inv p (p s))"
+      using permutes_inj [OF that] inv_f_f by auto
+    also have "... \<in> ?R" using s permutes_in_image[OF that] by auto
+    finally show "x \<in> ?R".
+  qed
+  show "?R \<subseteq> ?L"
+  proof
+    fix x assume "x \<in> ?R"
+    then obtain s
+      where x: "x = (s, Hilbert_Choice.inv p s)" (is "_ = (s, ?ips)")
+        and s: "s \<in> S" by auto
+    note x
+    also have "(s, ?ips) = (p ?ips, ?ips)"
+      using inv_f_f[OF permutes_inj[OF permutes_inv[OF that]]]
+      using inv_inv_eq[OF permutes_bij[OF that]] by auto
+    also have "... \<in> ?L"
+      using s permutes_in_image[OF permutes_inv[OF that]] by auto
+    finally show "x \<in> ?L".
+  qed
+qed
+
+context
+  fixes p and n i :: nat
+  assumes p: \<open>p permutes {0..<n}\<close> and i: \<open>i < n\<close>
+begin
+
+lemma permutes_nat_less:
+  \<open>p i < n\<close>
+proof -
+  have \<open>?thesis \<longleftrightarrow> p i \<in> {0..<n}\<close>
+    by simp
+  also from p have \<open>p i \<in> {0..<n} \<longleftrightarrow> i \<in> {0..<n}\<close>
+    by (rule permutes_in_image)
+  finally show ?thesis
+    using i by simp
+qed
+
+lemma permutes_nat_inv_less:
+  \<open>inv p i < n\<close>
+proof -
+  from p have \<open>inv p permutes {0..<n}\<close>
+    by (rule permutes_inv)
+  then show ?thesis
+    using i by (rule Permutations.permutes_nat_less)
+qed
+
+end
+
+context comm_monoid_set
+begin
+
+lemma permutes_inv:
+  \<open>F (\<lambda>s. g (p s) s) S = F (\<lambda>s. g s (inv p s)) S\<close> (is \<open>?l = ?r\<close>)
+  if \<open>p permutes S\<close>
+proof -
+  let ?g = "\<lambda>(x, y). g x y"
+  let ?ps = "\<lambda>s. (p s, s)"
+  let ?ips = "\<lambda>s. (s, inv p s)"
+  have inj1: "inj_on ?ps S" by (rule inj_onI) auto
+  have inj2: "inj_on ?ips S" by (rule inj_onI) auto
+  have "?l = F ?g (?ps ` S)"
+    using reindex [OF inj1, of ?g] by simp
+  also have "?ps ` S = {(p s, s) |s. s \<in> S}" by auto
+  also have "... = {(s, inv p s) |s. s \<in> S}"
+    unfolding permutes_pair_eq [OF that] by simp
+  also have "... = ?ips ` S" by auto
+  also have "F ?g ... = ?r"
+    using reindex [OF inj2, of ?g] by simp
+  finally show ?thesis.
+qed
+
+end
+
 
 subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close>
 
 lemma sum_over_permutations_insert:
   assumes fS: "finite S"
     and aS: "a \<notin> S"
   shows "sum f {p. p permutes (insert a S)} =
     sum (\<lambda>b. sum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)"
 proof -
   have *: "\<And>f a b. (\<lambda>(b, p). f (Fun.swap a b id \<circ> p)) = f \<circ> (\<lambda>(b,p). Fun.swap a b id \<circ> p)"
     by (simp add: fun_eq_iff)
   have **: "\<And>P Q. {(a, b). a \<in> P \<and> b \<in> Q} = P \<times> Q"
     by blast
   show ?thesis
     unfolding * ** sum.cartesian_product permutes_insert
   proof (rule sum.reindex)
     let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
     let ?P = "{p. p permutes S}"
     {
       fix b c p q
       assume b: "b \<in> insert a S"
       assume c: "c \<in> insert a S"
       assume p: "p permutes S"
       assume q: "q permutes S"
       assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> q"
       from p q aS have pa: "p a = a" and qa: "q a = a"
         unfolding permutes_def by metis+
       from eq have "(Fun.swap a b id \<circ> p) a  = (Fun.swap a c id \<circ> q) a"
         by simp
       then have bc: "b = c"
         by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def
             cong del: if_weak_cong split: if_split_asm)
       from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) =
         (\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp
       then have "p = q"
         unfolding o_assoc swap_id_idempotent by simp
       with bc have "b = c \<and> p = q"
         by blast
     }
     then show "inj_on ?f (insert a S \<times> ?P)"
       unfolding inj_on_def by clarify metis
   qed
 qed
 
 
 subsection \<open>Constructing permutations from association lists\<close>
 
 definition list_permutes :: "('a \<times> 'a) list \<Rightarrow> 'a set \<Rightarrow> bool"
   where "list_permutes xs A \<longleftrightarrow>
     set (map fst xs) \<subseteq> A \<and>
     set (map snd xs) = set (map fst xs) \<and>
     distinct (map fst xs) \<and>
     distinct (map snd xs)"
 
 lemma list_permutesI [simp]:
   assumes "set (map fst xs) \<subseteq> A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)"
   shows "list_permutes xs A"
 proof -
   from assms(2,3) have "distinct (map snd xs)"
     by (intro card_distinct) (simp_all add: distinct_card del: set_map)
   with assms show ?thesis
     by (simp add: list_permutes_def)
 qed
 
 definition permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
   where "permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)"
 
 lemma permutation_of_list_Cons:
   "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"
   by (simp add: permutation_of_list_def)
 
 fun inverse_permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
   where
     "inverse_permutation_of_list [] x = x"
   | "inverse_permutation_of_list ((y, x') # xs) x =
       (if x = x' then y else inverse_permutation_of_list xs x)"
 
 declare inverse_permutation_of_list.simps [simp del]
 
 lemma inj_on_map_of:
   assumes "distinct (map snd xs)"
   shows "inj_on (map_of xs) (set (map fst xs))"
 proof (rule inj_onI)
   fix x y
   assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)"
   assume eq: "map_of xs x = map_of xs y"
   from xy obtain x' y' where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
     by (cases "map_of xs x"; cases "map_of xs y") (simp_all add: map_of_eq_None_iff)
   moreover from x'y' have *: "(x, x') \<in> set xs" "(y, y') \<in> set xs"
     by (force dest: map_of_SomeD)+
   moreover from * eq x'y' have "x' = y'"
     by simp
   ultimately show "x = y"
     using assms by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"])
 qed
 
 lemma inj_on_the: "None \<notin> A \<Longrightarrow> inj_on the A"
   by (auto simp: inj_on_def option.the_def split: option.splits)
 
 lemma inj_on_map_of':
   assumes "distinct (map snd xs)"
   shows "inj_on (the \<circ> map_of xs) (set (map fst xs))"
   by (intro comp_inj_on inj_on_map_of assms inj_on_the)
     (force simp: eq_commute[of None] map_of_eq_None_iff)
 
 lemma image_map_of:
   assumes "distinct (map fst xs)"
   shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)"
   using assms by (auto simp: rev_image_eqI)
 
 lemma the_Some_image [simp]: "the ` Some ` A = A"
   by (subst image_image) simp
 
 lemma image_map_of':
   assumes "distinct (map fst xs)"
   shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)"
   by (simp only: image_comp [symmetric] image_map_of assms the_Some_image)
 
 lemma permutation_of_list_permutes [simp]:
   assumes "list_permutes xs A"
   shows "permutation_of_list xs permutes A"
     (is "?f permutes _")
 proof (rule permutes_subset[OF bij_imp_permutes])
   from assms show "set (map fst xs) \<subseteq> A"
     by (simp add: list_permutes_def)
   from assms have "inj_on (the \<circ> map_of xs) (set (map fst xs))" (is ?P)
     by (intro inj_on_map_of') (simp_all add: list_permutes_def)
   also have "?P \<longleftrightarrow> inj_on ?f (set (map fst xs))"
     by (intro inj_on_cong)
       (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
   finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))"
     by (rule inj_on_imp_bij_betw)
   also from assms have "?f ` set (map fst xs) = (the \<circ> map_of xs) ` set (map fst xs)"
     by (intro image_cong refl)
       (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
   also from assms have "\<dots> = set (map fst xs)"
     by (subst image_map_of') (simp_all add: list_permutes_def)
   finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" .
 qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+
 
 lemma eval_permutation_of_list [simp]:
   "permutation_of_list [] x = x"
   "x = x' \<Longrightarrow> permutation_of_list ((x',y)#xs) x = y"
   "x \<noteq> x' \<Longrightarrow> permutation_of_list ((x',y')#xs) x = permutation_of_list xs x"
   by (simp_all add: permutation_of_list_def)
 
 lemma eval_inverse_permutation_of_list [simp]:
   "inverse_permutation_of_list [] x = x"
   "x = x' \<Longrightarrow> inverse_permutation_of_list ((y,x')#xs) x = y"
   "x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x"
   by (simp_all add: inverse_permutation_of_list.simps)
 
 lemma permutation_of_list_id: "x \<notin> set (map fst xs) \<Longrightarrow> permutation_of_list xs x = x"
   by (induct xs) (auto simp: permutation_of_list_Cons)
 
 lemma permutation_of_list_unique':
   "distinct (map fst xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
   by (induct xs) (force simp: permutation_of_list_Cons)+
 
 lemma permutation_of_list_unique:
   "list_permutes xs A \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
   by (intro permutation_of_list_unique') (simp_all add: list_permutes_def)
 
 lemma inverse_permutation_of_list_id:
   "x \<notin> set (map snd xs) \<Longrightarrow> inverse_permutation_of_list xs x = x"
   by (induct xs) auto
 
 lemma inverse_permutation_of_list_unique':
   "distinct (map snd xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
   by (induct xs) (force simp: inverse_permutation_of_list.simps(2))+
 
 lemma inverse_permutation_of_list_unique:
   "list_permutes xs A \<Longrightarrow> (x,y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
   by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def)
 
 lemma inverse_permutation_of_list_correct:
   fixes A :: "'a set"
   assumes "list_permutes xs A"
   shows "inverse_permutation_of_list xs = inv (permutation_of_list xs)"
 proof (rule ext, rule sym, subst permutes_inv_eq)
   from assms show "permutation_of_list xs permutes A"
     by simp
   show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" for x
   proof (cases "x \<in> set (map snd xs)")
     case True
     then obtain y where "(y, x) \<in> set xs" by auto
     with assms show ?thesis
       by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique)
   next
     case False
     with assms show ?thesis
       by (auto simp: list_permutes_def inverse_permutation_of_list_id permutation_of_list_id)
   qed
 qed
 
 end