src/HOL/Library/Permutations.thy

-(*  Title:      HOL/Library/Permutations.thy
-    Author:     Amine Chaieb, University of Cambridge
-*)
-
-section \<open>Permutations, both general and specifically on finite sets.\<close>
-
-theory Permutations
-  imports Multiset 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_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>
-
-lemma sign_nz: "sign p \<noteq> 0"
-  by (simp add: sign_def)
-
-lemma sign_id: "sign id = 1"
-  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)"
-  by (simp add: sign_def evenperm_swap)
-
-lemma sign_idempotent: "sign p * sign p = 1"
-  by (simp add: sign_def)
-
-
-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>
-
-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
-
-
-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