Theory Permutations

(*  Author:     Amine Chaieb, University of Cambridge
*)

section ‹Permutations, both general and specifically on finite sets.›

theory Permutations
  imports
    "HOL-Library.Multiset"
    "HOL-Library.Disjoint_Sets"
    Transposition
begin

subsection ‹Auxiliary›

abbreviation (input) fixpoints :: ‹('a ⇒ 'a) ⇒ 'a set›
  where ‹fixpoints f ≡ {x. f x = x}›

lemma inj_on_fixpoints:
  ‹inj_on f (fixpoints f)›
  by (rule inj_onI) simp

lemma bij_betw_fixpoints:
  ‹bij_betw f (fixpoints f) (fixpoints f)›
  using inj_on_fixpoints by (auto simp add: bij_betw_def)


subsection ‹Basic definition and consequences›

definition permutes :: ‹('a ⇒ 'a) ⇒ 'a set ⇒ bool›  (infixr ‹permutes› 41)
  where ‹p permutes S ⟷ (∀x. x ∉ S ⟶ p x = x) ∧ (∀y. ∃!x. p x = y)›

lemma bij_imp_permutes:
  ‹p permutes S› if ‹bij_betw p S S› and stable: ‹⋀x. x ∉ S ⟹ p x = x›
proof -
  note ‹bij_betw p S S›
  moreover have ‹bij_betw p (- S) (- S)›
    by (auto simp add: stable intro!: bij_betw_imageI inj_onI)
  ultimately have ‹bij_betw p (S ∪ - S) (S ∪ - S)›
    by (rule bij_betw_combine) simp
  then have ‹∃!x. p x = y› for y
    by (simp add: bij_iff)
  with stable show ?thesis
    by (simp add: permutes_def)
qed

context
  fixes p :: ‹'a ⇒ 'a› and S :: ‹'a set›
  assumes perm: ‹p permutes S›
begin

lemma permutes_inj:
  ‹inj p›
  using perm by (auto simp: permutes_def inj_on_def)

lemma permutes_image:
  ‹p ` S = S›
proof (rule set_eqI)
  fix x
  show ‹x ∈ p ` S ⟷ x ∈ S›
  proof
    assume ‹x ∈ p ` S›
    then obtain y where ‹y ∈ S› ‹p y = x›
      by blast
    with perm show ‹x ∈ S›
      by (cases ‹y = x›) (auto simp add: permutes_def)
  next
    assume ‹x ∈ S›
    with perm obtain y where ‹y ∈ S› ‹p y = x›
      by (metis permutes_def)
    then show ‹x ∈ p ` S›
      by blast
  qed
qed

lemma permutes_not_in:
  ‹x ∉ S ⟹ p x = x›
  using perm by (auto simp: permutes_def)

lemma permutes_image_complement:
  ‹p ` (- S) = - S›
  by (auto simp add: permutes_not_in)

lemma permutes_in_image:
  ‹p x ∈ S ⟷ x ∈ S›
  using permutes_image permutes_inj by (auto dest: inj_image_mem_iff)

lemma permutes_surj:
  ‹surj p›
proof -
  have ‹p ` (S ∪ - S) = p ` S ∪ p ` (- S)›
    by (rule image_Un)
  then show ?thesis
    by (simp add: permutes_image permutes_image_complement)
qed

lemma permutes_inv_o:
  shows "p ∘ inv p = id"
    and "inv p ∘ 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:
  ‹inv p y = x ⟷ p x = y›
  by (auto simp add: permutes_inverses)

lemma permutes_inj_on:
  ‹inj_on p A›
  by (rule inj_on_subset [of _ UNIV]) (auto intro: permutes_inj)

lemma permutes_bij:
  ‹bij p›
  unfolding bij_def by (metis permutes_inj permutes_surj)

lemma permutes_imp_bij:
  ‹bij_betw p S S›
  by (simp add: bij_betw_def permutes_image permutes_inj_on)

lemma permutes_subset:
  ‹p permutes T› if ‹S ⊆ T›
proof (rule bij_imp_permutes)
  define R where ‹R = T - S›
  with that have ‹T = R ∪ S› ‹R ∩ S = {}›
    by auto
  then have ‹p x = x› if ‹x ∈ R› for x
    using that by (auto intro: permutes_not_in)
  then have ‹p ` R = R›
    by simp
  with ‹T = R ∪ S› show ‹bij_betw p T T›
    by (simp add: bij_betw_def permutes_inj_on image_Un permutes_image)
  fix x
  assume ‹x ∉ T›
  with ‹T = R ∪ S› show ‹p x = x›
    by (simp add: permutes_not_in)
qed

lemma permutes_imp_permutes_insert:
  ‹p permutes insert x S›
  by (rule permutes_subset) auto

end

lemma permutes_id [simp]:
  ‹id permutes S›
  by (auto intro: bij_imp_permutes)

lemma permutes_empty [simp]:
  ‹p permutes {} ⟷ p = id›
proof
  assume ‹p permutes {}›
  then show ‹p = id›
    by (auto simp add: fun_eq_iff permutes_not_in)
next
  assume ‹p = id›
  then show ‹p permutes {}›
    by simp
qed

lemma permutes_sing [simp]:
  ‹p permutes {a} ⟷ p = id›
proof
  assume perm: ‹p permutes {a}›
  show ‹p = id›
  proof
    fix x
    from perm have ‹p ` {a} = {a}›
      by (rule permutes_image)
    with perm show ‹p x = id x›
      by (cases ‹x = a›) (auto simp add: permutes_not_in)
  qed
next
  assume ‹p = id›
  then show ‹p permutes {a}›
    by simp
qed

lemma permutes_univ: "p permutes UNIV ⟷ (∀y. ∃!x. p x = y)"
  by (simp add: permutes_def)

lemma permutes_swap_id: "a ∈ S ⟹ b ∈ S ⟹ transpose a b permutes S"
  by (rule bij_imp_permutes) (auto intro: transpose_apply_other)

lemma permutes_superset:
  ‹p permutes T› if ‹p permutes S› ‹⋀x. x ∈ S - T ⟹ p x = x›
proof -
  define R U where ‹R = T ∩ S› and ‹U = S - T›
  then have ‹T = R ∪ (T - S)› ‹S = R ∪ U› ‹R ∩ U = {}›
    by auto
  from that ‹U = S - T› have ‹p ` U = U›
    by simp
  from ‹p permutes S› have ‹bij_betw p (R ∪ U) (R ∪ U)›
    by (simp add: permutes_imp_bij ‹S = R ∪ U›)
  moreover have ‹bij_betw p U U›
    using that ‹U = S - T› by (simp add: bij_betw_def permutes_inj_on)
  ultimately have ‹bij_betw p R R›
    using ‹R ∩ U = {}› ‹R ∩ U = {}› by (rule bij_betw_partition)
  then have ‹p permutes R›
  proof (rule bij_imp_permutes)
    fix x
    assume ‹x ∉ R›
    with ‹R = T ∩ S› ‹p permutes S› show ‹p x = x›
      by (cases ‹x ∈ S›) (auto simp add: permutes_not_in that(2))
  qed
  then have ‹p permutes R ∪ (T - S)›
    by (rule permutes_subset) simp
  with ‹T = R ∪ (T - S)› show ?thesis
    by simp
qed

lemma permutes_bij_inv_into: ✐‹contributor ‹Lukas Bulwahn››
  fixes A :: "'a set"
    and B :: "'b set"
  assumes "p permutes A"
    and "bij_betw f A B"
  shows "(λx. if x ∈ 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 ∘ p ∘ inv_into A f) B B"
    by (simp add: bij_betw_trans)
  then show "bij_betw (λx. if x ∈ B then f (p (inv_into A f x)) else x) B B"
    by (subst bij_betw_cong[where g="f ∘ p ∘ inv_into A f"]) auto
next
  fix x
  assume "x ∉ B"
  then show "(if x ∈ B then f (p (inv_into A f x)) else x) = x" by auto
qed

lemma permutes_image_mset: ✐‹contributor ‹Lukas Bulwahn››
  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: ✐‹contributor ‹Lukas Bulwahn››
  assumes "p permutes A" "⋀x. x ∈ A ⟹ 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 ∈# 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' ∘ p) (mset_set A)"
    by (auto intro!: image_mset_cong)
  also have "… = image_mset f' (image_mset p (mset_set A))"
    by (simp add: image_mset.compositionality)
  also have "… = 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 ‹Group properties›

lemma permutes_compose: "p permutes S ⟹ q permutes S ⟹ q ∘ 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: "⋀x. x ∈ S ⟹ p' (p x) = x"
    and outside: "⋀x. x ∉ S ⟹ p' x = x"
  shows "inv p = p'"
proof
  show "inv p x = p' x" for x
  proof (cases "x ∈ 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 "… = 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 ⟹ f -` A = A"
  by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])


subsection ‹Mapping permutations with bijections›

lemma bij_betw_permutations:
  assumes "bij_betw f A B"
  shows   "bij_betw (λπ x. if x ∈ B then f (π (inv_into A f x)) else x) 
             {π. π permutes A} {π. π permutes B}" (is "bij_betw ?f _ _")
proof -
  let ?g = "(λπ x. if x ∈ A then inv_into A f (π (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 π assume "π permutes B"
      from permutes_bij_inv_into[OF this bij_inv] and assms
        have "(λx. if x ∈ A then inv_into A f (π (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 (λπ x. if x ∈ B then f (π (inv_into A f x)) else x) 
             {π. π permutes A ∧ (∀x∈A. π x ≠ x)} {π. π permutes B ∧ (∀x∈B. π x ≠ x)}" 
           (is "bij_betw ?f _ _")
proof -
  let ?g = "(λπ x. if x ∈ A then inv_into A f (π (f x)) else x)"
  show ?thesis
  proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
    case 3
    have "?f π x ≠ x" if "π permutes A" "⋀x. x ∈ A ⟹ π x ≠ x" "x ∈ B" for π 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 π permutes A" if "π permutes B" for π
      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 π x ≠ x" if "π permutes B" "⋀x. x ∈ B ⟹ π x ≠ x" "x ∈ A" for π 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 ‹The number of permutations on a finite set›

lemma permutes_insert_lemma:
  assumes "p permutes (insert a S)"
  shows "transpose a (p a) ∘ p permutes S"
proof (rule permutes_superset[where S = "insert a S"])
  show "Transposition.transpose a (p a) ∘ p permutes insert a S"
    by (meson assms insertI1 permutes_compose permutes_in_image permutes_swap_id)
qed auto

lemma permutes_insert: "{p. p permutes (insert a S)} =
  (λ(b, p). transpose a b ∘ p) ` {(b, p). b ∈ insert a S ∧ p ∈ {p. p permutes S}}"
proof -
  have "p permutes insert a S ⟷
    (∃b q. p = transpose a b ∘ q ∧ b ∈ insert a S ∧ q permutes S)" for p
  proof -
    have "∃b q. p = transpose a b ∘ q ∧ b ∈ insert a S ∧ q permutes S"
      if p: "p permutes insert a S"
    proof -
      let ?b = "p a"
      let ?q = "transpose a (p a) ∘ p"
      have *: "p = transpose a ?b ∘ ?q"
        by (simp add: fun_eq_iff o_assoc)
      have **: "?b ∈ 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 = transpose a b ∘ q" "b ∈ 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 ∈ 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 ∈ insert x F ∧ p ∈ ?pF}"
    let ?g = "(λ(b, p). transpose x b ∘ p)"
    have xfgpF': "?xF = ?g ` ?pF'"
      by (rule permutes_insert[of x F])
    from ‹x ∉ F› ‹finite F› card_insert have Fs: "card F = n - 1"
      by auto
    from ‹finite F› insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
      by auto
    then have "finite ?pF"
      by (auto intro: card_ge_0_finite)
    with ‹finite F› card.insert_remove have pF'f: "finite ?pF'"
      by simp
    have ginj: "inj_on ?g ?pF'"
    proof -
      {
        fix b p c q
        assume bp: "(b, p) ∈ ?pF'"
        assume cq: "(c, q) ∈ ?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 ‹x ∉ F› eq have "b = ?g (b, p) x"
          by (auto simp: permutes_def fun_upd_def fun_eq_iff)
        also from qF ‹x ∉ F› eq have "… = ?g (c, q) x"
          by (auto simp: fun_upd_def fun_eq_iff)
        also from qF ‹x ∉ F› have "… = c"
          by (auto simp: permutes_def fun_upd_def fun_eq_iff)
        finally have "b = c" .
        then have "transpose x b = transpose x c"
          by simp
        with eq have "transpose x b ∘ p = transpose x b ∘ q"
          by simp
        then have "transpose x b ∘ (transpose x b ∘ p) = transpose x b ∘ (transpose x b ∘ q)"
          by simp
        then have "p = q"
          by (simp add: o_assoc)
        with ‹b = c› have "(b, p) = (c, q)"
          by simp
      }
      then show ?thesis
        unfolding inj_on_def by blast
    qed
    from ‹x ∉ F› ‹finite F› card_insert have "n ≠ 0"
      by auto
    then have "∃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 ‹finite F› ‹finite ?pF›
      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 ‹Hence a sort of induction principle composing by swaps›

lemma permutes_induct [consumes 2, case_names id swap]:
  ‹P p› if ‹p permutes S› ‹finite S›
  and id: ‹P id›
  and swap: ‹⋀a b p. a ∈ S ⟹ b ∈ S ⟹ p permutes S ⟹ P p ⟹ P (transpose a b ∘ p)›
using ‹finite S› ‹p permutes S› 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 ‹q = transpose x (p x) ∘ p›
  then have swap_q: ‹transpose x (p x) ∘ q = p›
    by (simp add: o_assoc)
  from ‹p permutes insert x S› have ‹q permutes S›
    by (simp add: q_def permutes_insert_lemma)
  then have ‹q permutes insert x S›
    by (simp add: permutes_imp_permutes_insert)
  from ‹q permutes S› have ‹P q›
    by (auto intro: insert.IH insert.prems(2) permutes_imp_permutes_insert)
  have ‹x ∈ insert x S›
    by simp
  moreover from ‹p permutes insert x S› have ‹p x ∈ insert x S›
    using permutes_in_image [of p ‹insert x S› x] by simp
  ultimately have ‹P (transpose x (p x) ∘ q)›
    using ‹q permutes insert x S› ‹P q›
    by (rule insert.prems(2))
  then show ?case
    by (simp add: swap_q)
qed

lemma permutes_rev_induct [consumes 2, case_names id swap]:
  ‹P p› if ‹p permutes S› ‹finite S›
  and id': ‹P id›
  and swap': ‹⋀a b p. a ∈ S ⟹ b ∈ S ⟹ p permutes S ⟹ P p ⟹ P (p ∘ transpose a b)›
using ‹p permutes S› ‹finite S› proof (induction rule: permutes_induct)
  case id
  from id' show ?case .
next
  case (swap a b p)
  then have ‹bij p›
    using permutes_bij by blast
  have ‹P (p ∘ transpose (inv p a) (inv p b))›
    by (rule swap') (auto simp add: swap permutes_in_image permutes_inv)
  also have ‹p ∘ transpose (inv p a) (inv p b) = transpose a b ∘ p›
    using ‹bij p› by (rule transpose_comp_eq [symmetric])
  finally show ?case .
qed


subsection ‹Permutations of index set for iterated operations›

lemma (in comm_monoid_set) permute:
  assumes "p permutes S"
  shows "F g S = F (g ∘ p) S"
proof -
  from ‹p permutes S› 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 ∘ p) S"
    by (rule reindex)
  moreover from ‹p permutes S› have "p ` S = S"
    by (rule permutes_image)
  ultimately show ?thesis
    by simp
qed


subsection ‹Permutations as transposition sequences›

inductive swapidseq :: "nat ⇒ ('a ⇒ 'a) ⇒ bool"
  where
    id[simp]: "swapidseq 0 id"
  | comp_Suc: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (transpose a b ∘ p)"

declare id[unfolded id_def, simp]

definition "permutation p ⟷ (∃n. swapidseq n p)"


subsection ‹Some closure properties of the set of permutations, with lengths›

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) (transpose a b)"
  using swapidseq.simps by fastforce

lemma permutation_swap_id: "permutation (transpose a b)"
  by (meson permutation_def swapidseq_swap)


lemma swapidseq_comp_add: "swapidseq n p ⟹ swapidseq m q ⟹ swapidseq (n + m) (p ∘ 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)
  then show ?case
    by (metis add_Suc comp_assoc swapidseq.comp_Suc)
qed

lemma permutation_compose: "permutation p ⟹ permutation q ⟹ permutation (p ∘ q)"
  unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis

lemma swapidseq_endswap: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (p ∘ transpose a b)"
  by (induct n p rule: swapidseq.induct)
    (use swapidseq_swap[of a b] in ‹auto simp add: comp_assoc intro: swapidseq.comp_Suc›)

lemma swapidseq_inverse_exists: "swapidseq n p ⟹ ∃q. swapidseq n q ∧ p ∘ q = id ∧ q ∘ 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 ∘ q = id" "q ∘ p = id"
    by blast
  let ?q = "q ∘ transpose a b"
  note H = comp_Suc.hyps
  from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (transpose a b)"
    by simp
  from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q"
    by simp
  have "transpose a b ∘ p ∘ ?q = transpose a b ∘ (p ∘ q) ∘ transpose a b"
    by (simp add: o_assoc)
  also have "… = id"
    by (simp add: q(2))
  finally have ***: "transpose a b ∘ p ∘ ?q = id" .
  have "?q ∘ (transpose a b ∘ p) = q ∘ (transpose a b ∘ transpose a b) ∘ p"
    by (simp only: o_assoc)
  then have "?q ∘ (transpose a b ∘ 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 ⟹ permutation (inv p)"
  using permutation_def swapidseq_inverse by blast



subsection ‹Various combinations of transpositions with 2, 1 and 0 common elements›

lemma swap_id_common:" a ≠ c ⟹ b ≠ c ⟹
  transpose a b ∘ transpose a c = transpose b c ∘ transpose a b"
  by (simp add: fun_eq_iff transpose_def)

lemma swap_id_common': "a ≠ b ⟹ a ≠ c ⟹
  transpose a c ∘ transpose b c = transpose b c ∘ transpose a b"
  by (simp add: fun_eq_iff transpose_def)

lemma swap_id_independent: "a ≠ c ⟹ a ≠ d ⟹ b ≠ c ⟹ b ≠ d ⟹
  transpose a b ∘ transpose c d = transpose c d ∘ transpose a b"
  by (simp add: fun_eq_iff transpose_def)


subsection ‹The identity map only has even transposition sequences›

lemma symmetry_lemma:
  assumes "⋀a b c d. P a b c d ⟹ P a b d c"
    and "⋀a b c d. a ≠ b ⟹ c ≠ d ⟹
      a = c ∧ b = d ∨ a = c ∧ b ≠ d ∨ a ≠ c ∧ b = d ∨ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ⟹
      P a b c d"
  shows "⋀a b c d. a ≠ b ⟶ c ≠ d ⟶  P a b c d"
  using assms by metis

lemma swap_general:
  assumes "a ≠ b" "c ≠ d"
  shows "transpose a b ∘ transpose c d = id ∨
  (∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧
    transpose a b ∘ transpose c d = transpose x y ∘ transpose a z)"
  by (metis assms swap_id_common' swap_id_independent transpose_commute transpose_comp_involutory)

lemma swapidseq_id_iff[simp]: "swapidseq 0 p ⟷ p = id"
  using swapidseq.cases[of 0 p "p = id"] by auto

lemma swapidseq_cases: "swapidseq n p ⟷
    n = 0 ∧ p = id ∨ (∃a b q m. n = Suc m ∧ p = transpose a b ∘ q ∧ swapidseq m q ∧ a ≠ b)"
  by (meson comp_Suc id swapidseq.cases)

lemma fixing_swapidseq_decrease:
  assumes "swapidseq n p"
    and "a ≠ b"
    and "(transpose a b ∘ p) a = a"
  shows "n ≠ 0 ∧ swapidseq (n - 1) (transpose a b ∘ p)"
  using assms
proof (induct n arbitrary: p a b)
  case 0
  then show ?case
    by (auto simp add: 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 = transpose c d ∘ q" "swapidseq m q" "c ≠ d" "n = m"
    by auto
  consider "transpose a b ∘ transpose c d = id"
    | x y z where "x ≠ a" "y ≠ a" "z ≠ a" "x ≠ y"
      "transpose a b ∘ transpose c d = transpose x y ∘ transpose a z"
    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 2
    then have az: "a ≠ z"
      by simp
    from 2 have *: "(transpose x y ∘ h) a = a ⟷ h a = a" for h
      by (simp add: transpose_def)
    from cdqm(2) have "transpose a b ∘ p = transpose a b ∘ (transpose c d ∘ q)"
      by simp
    then have §: "transpose a b ∘ p = transpose x y ∘ (transpose a z ∘ q)"
      by (simp add: o_assoc 2)
    obtain **: "swapidseq (n - 1) (transpose a z ∘ q)" and "n≠0"
      by (metis "*" "§" Suc.hyps Suc.prems(3) az cdqm(3,5))
    then have "Suc n - 1 = Suc (n - 1)"
      by auto
    with 2 show ?thesis
      using "**" § swapidseq.simps by blast
  qed
qed

lemma swapidseq_identity_even:
  assumes "swapidseq n (id :: 'a ⇒ 'a)"
  shows "even n"
  using ‹swapidseq n id›
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 = transpose a b ∘ q" "swapidseq m q" "a ≠ 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 ≠ 0" "swapidseq (m - 1) (id :: 'a ⇒ '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 ‹Therefore we have a welldefined notion of parity›

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 ⟷ even n"
proof -
  from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p ∘ q = id" "q ∘ 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 "swapidseq n p" and"even n = b"
  shows "evenperm p = b"
  by (metis evenperm_def assms someI swapidseq_even_even)


subsection ‹And it has the expected composition properties›

lemma evenperm_id[simp]: "evenperm id = True"
  by (rule evenperm_unique[where n = 0]) simp_all

lemma evenperm_identity [simp]:
  ‹evenperm (λx. x)›
  using evenperm_id by (simp add: id_def [abs_def])

lemma evenperm_swap: "evenperm (transpose a b) = (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 ∘ q) ⟷ 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) ⟷ (even n ⟷ 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 ‹A more abstract characterization of permutations›

lemma permutation_bijective:
  assumes "permutation p"
  shows "bij p"
  by (meson assms o_bij permutation_def swapidseq_inverse_exists)

lemma permutation_finite_support:
  assumes "permutation p"
  shows "finite {x. p x ≠ 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 ≠ x})"
    from comp_Suc.hyps(2) have *: "finite ?S"
      by simp
    from ‹a ≠ b› have **: "{x. (transpose a b ∘ p) x ≠ x} ⊆ ?S"
      by auto
    show ?case
      by (rule finite_subset[OF ** *])
  qed
qed

lemma permutation_lemma:
  assumes "finite S"
    and "bij p"
    and "∀x. x ∉ S ⟶ 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 = "transpose a (p a) ∘ p"
  let ?q = "transpose a (p a) ∘ ?r"
  have *: "?r a = a"
    by simp
  from insert * have **: "∀x. x ∉ F ⟶ ?r x = x"
    by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
  have "bij ?r"
    using insert by (simp add: bij_comp)
  have "permutation ?r"
    by (rule insert(3)[OF ‹bij ?r› **])
  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 ⟷ bij p ∧ finite {x. p x ≠ x}"
  using permutation_bijective permutation_finite_support permutation_lemma by auto

lemma permutation_inverse_works:
  assumes "permutation p"
  shows "inv p ∘ p = id"
    and "p ∘ 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 ∘ q) = inv q ∘ inv p"
  by (simp add: o_inv_distrib p permutation_bijective q)


subsection ‹Relation to ‹permutes››

lemma permutes_imp_permutation:
  ‹permutation p› if ‹finite S› ‹p permutes S›
proof -
  from ‹p permutes S› have ‹{x. p x ≠ x} ⊆ S›
    by (auto dest: permutes_not_in)
  then have ‹finite {x. p x ≠ x}›
    using ‹finite S› by (rule finite_subset)
  moreover from ‹p permutes S› have ‹bij p›
    by (auto dest: permutes_bij)
  ultimately show ?thesis
    by (simp add: permutation)
qed

lemma permutation_permutesE:
  assumes ‹permutation p›
  obtains S where ‹finite S› ‹p permutes S›
proof -
  from assms have fin: ‹finite {x. p x ≠ x}›
    by (simp add: permutation)
  from assms have ‹bij p›
    by (simp add: permutation)
  also have ‹UNIV = {x. p x ≠ x} ∪ {x. p x = x}›
    by auto
  finally have ‹bij_betw p {x. p x ≠ x} {x. p x ≠ x}›
    by (rule bij_betw_partition) (auto simp add: bij_betw_fixpoints)
  then have ‹p permutes {x. p x ≠ x}›
    by (auto intro: bij_imp_permutes)
  with fin show thesis ..
qed

lemma permutation_permutes: "permutation p ⟷ (∃S. finite S ∧ p permutes S)"
  by (auto elim: permutation_permutesE intro: permutes_imp_permutation)


subsection ‹Sign of a permutation as a real number›

definition sign :: ‹('a ⇒ 'a) ⇒ int› ― ‹TODO: prefer less generic name›
  where ‹sign p = (if evenperm p then 1 else - 1)›

lemma sign_cases [case_names even odd]:
  obtains ‹sign p = 1› | ‹sign p = - 1›
  by (cases ‹evenperm p›) (simp_all add: sign_def)

lemma sign_nz [simp]: "sign p ≠ 0"
  by (cases p rule: sign_cases) simp_all

lemma sign_id [simp]: "sign id = 1"
  by (simp add: sign_def)

lemma sign_identity [simp]:
  ‹sign (λx. x) = 1›
  by (simp add: sign_def)

lemma sign_inverse: "permutation p ⟹ sign (inv p) = sign p"
  by (simp add: sign_def evenperm_inv)

lemma sign_compose: "permutation p ⟹ permutation q ⟹ sign (p ∘ q) = sign p * sign q"
  by (simp add: sign_def evenperm_comp)

lemma sign_swap_id: "sign (transpose a b) = (if a = b then 1 else - 1)"
  by (simp add: sign_def evenperm_swap)

lemma sign_idempotent [simp]: "sign p * sign p = 1"
  by (simp add: sign_def)

lemma sign_left_idempotent [simp]:
  ‹sign p * (sign p * sign q) = sign q›
  by (simp add: sign_def)

term "(bij, bij_betw, permutation)"


subsection ‹Permuting a list›

text ‹This function permutes a list by applying a permutation to the indices.›

definition permute_list :: "(nat ⇒ nat) ⇒ 'a list ⇒ 'a list"
  where "permute_list f xs = map (λi. 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 ∘ 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 (λx. 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 ⟷ x < length xs" for x
    using permutes_in_image[OF assms] by auto
  have "count (mset (permute_list f xs)) y = card ((λi. xs ! f i) -` {y} ∩ {..<length xs})"
    by (simp add: permute_list_def count_image_mset atLeast0LessThan)
  also have "(λi. xs ! f i) -` {y} ∩ {..<length xs} = f -` {i. i < length xs ∧ y = xs ! i}"
    by auto
  also from assms have "card … = card {i. i < length xs ∧ y = xs ! i}"
    by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
  also have "… = 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 ⟷ i < length ys" for i
    by simp
  have "permute_list f (zip xs ys) = map (λi. zip xs ys ! f i) [0..<length ys]"
    by (simp_all add: permute_list_def zip_map_map)
  also have "… = map (λ(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
    by (intro nth_equalityI) (simp_all add: *)
  also have "… = 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 "σ permutes fst ` set xs"
  shows "map_of xs ∘ σ = map_of (map (λ(x,y). (inv σ x, y)) xs)"
    (is "_ = map_of (map ?f _)")
proof
  from assms have "inj σ" "surj σ"
    by (simp_all add: permutes_inj permutes_surj)
  then show "(map_of xs ∘ σ) x = map_of (map ?f xs) x" for x
    by (induct xs) (auto simp: inv_f_f surj_f_inv_f)
qed

lemma list_all2_permute_list_iff:
  ‹list_all2 P (permute_list p xs) (permute_list p ys) ⟷ list_all2 P xs ys›
  if ‹p permutes {..<length xs}›
  using that by (auto simp add: list_all2_iff simp flip: permute_list_zip)


subsection ‹More lemmas about permutations›

lemma permutes_in_funpow_image: ✐‹contributor ‹Lars Noschinski››
  assumes "f permutes S" "x ∈ S"
  shows "(f ^^ n) x ∈ S"
  using assms by (induction n) (auto simp: permutes_in_image)

lemma permutation_self: ✐‹contributor ‹Lars Noschinski››
  assumes ‹permutation p›
  obtains n where ‹n > 0› ‹(p ^^ n) x = x›
proof (cases ‹p x = x›)
  case True
  with that [of 1] show thesis by simp
next
  case False
  from ‹permutation p› have ‹inj p›
    by (intro permutation_bijective bij_is_inj)
  moreover from ‹p x ≠ x› have ‹(p ^^ Suc n) x ≠ (p ^^ n) x› for n
  proof (induction n arbitrary: x)
    case 0 then show ?case by simp
  next
    case (Suc n)
    have "p (p x) ≠ p x"
    proof (rule notI)
      assume "p (p x) = p x"
      then show False using ‹p x ≠ x› ‹inj p› by (simp add: inj_eq)
    qed
    have "(p ^^ Suc (Suc n)) x = (p ^^ Suc n) (p x)"
      by (simp add: funpow_swap1)
    also have "… ≠ (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. ∃n. y = (p ^^ n) x} ⊆ {x. p x ≠ x}"
    by auto
  then have "finite {y. ∃n. y = (p ^^ n) x}"
    using permutation_finite_support[OF assms] by (rule finite_subset)
  ultimately obtain n where ‹n > 0› ‹(p ^^ n) x = x›
    by (rule funpow_inj_finite)
  with that [of n] show thesis by blast
qed

text ‹The following few lemmas were contributed by Lukas Bulwahn.›

lemma count_image_mset_eq_card_vimage:
  assumes "finite A"
  shows "count (image_mset f (mset_set A)) b = card {a ∈ 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 ∈ F. f a = f x})"
      by auto
    also from insert.hyps(1,2) have "… = card (insert x {a ∈ F. f a = f x})"
      by simp
    also from ‹f x = b› have "card (insert x {a ∈ F. f a = f x}) = card {a ∈ 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 ∈ F. f a = b} = {a ∈ insert x F. f a = b}"
      by auto
    with insert False show ?thesis
      by simp
  qed
qed

― ‹Prove ‹image_mset_eq_implies_permutes› ...›
lemma image_mset_eq_implies_permutes:
  fixes f :: "'a ⇒ '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 "∀x∈A. f x = f' (p x)"
proof -
  from ‹finite A› have [simp]: "finite {a ∈ A. f a = (b::'b)}" for f b by auto
  have "f ` A = f' ` A"
  proof -
    from ‹finite A› have "f ` A = f ` (set_mset (mset_set A))"
      by simp
    also have "… = f' ` set_mset (mset_set A)"
      by (metis mset_eq multiset.set_map)
    also from ‹finite A› have "… = f' ` A"
      by simp
    finally show ?thesis .
  qed
  have "∀b∈(f ` A). ∃p. bij_betw p {a ∈ A. f a = b} {a ∈ 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 ‹finite A› have "card {a ∈ A. f a = b} = card {a ∈ A. f' a = b}"
      by (simp add: count_image_mset_eq_card_vimage)
    then show "∃p. bij_betw p {a∈A. f a = b} {a ∈ A. f' a = b}"
      by (intro finite_same_card_bij) simp_all
  qed
  then have "∃p. ∀b∈f ` A. bij_betw (p b) {a ∈ A. f a = b} {a ∈ A. f' a = b}"
    by (rule bchoice)
  then obtain p where p: "∀b∈f ` A. bij_betw (p b) {a ∈ A. f a = b} {a ∈ A. f' a = b}" ..
  define p' where "p' = (λa. if a ∈ A then p (f a) a else a)"
  have "p' permutes A"
  proof (rule bij_imp_permutes)
    have "disjoint_family_on (λi. {a ∈ A. f' a = i}) (f ` A)"
      by (auto simp: disjoint_family_on_def)
    moreover
    have "bij_betw (λa. p (f a) a) {a ∈ A. f a = b} {a ∈ A. f' a = b}" if "b ∈ f ` A" for b
      using p that by (subst bij_betw_cong[where g="p b"]) auto
    ultimately
    have "bij_betw (λa. p (f a) a) (⋃b∈f ` A. {a ∈ A. f a = b}) (⋃b∈f ` A. {a ∈ A. f' a = b})"
      by (rule bij_betw_UNION_disjoint)
    moreover have "(⋃b∈f ` A. {a ∈ A. f a = b}) = A"
      by auto
    moreover from ‹f ` A = f' ` A› have "(⋃b∈f ` A. {a ∈ A. f' a = b}) = A"
      by auto
    ultimately show "bij_betw p' A A"
      unfolding p'_def by (subst bij_betw_cong[where g="(λa. p (f a) a)"]) auto
  next
    show "⋀x. x ∉ A ⟹ p' x = x"
      by (simp add: p'_def)
  qed
  moreover from p have "∀x∈A. f x = f' (p' x)"
    unfolding p'_def using bij_betwE by fastforce
  ultimately show ?thesis ..
qed

― ‹... and derive the existing property:›
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 (λi. xs ! i) (mset_set {..<length ys}) =
    image_mset (λi. 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 "∀i∈{..<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 "∀i ∈ S. p i ≤ 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 ∈ S")
      case False
      with less(2) show ?thesis
        unfolding permutes_def by metis
    next
      case True
      with less(3) have "p n < n ∨ 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 ‹p n < n› 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: "∀i ∈ S. p i ≥ i"
  shows "p = id"
proof -
  have "i ≥ inv p i" if "i ∈ S" for i
  proof -
    from that permutes_in_image[OF permutes_inv[OF p]] have "inv p i ∈ S"
      by simp
    with le have "p (inv p i) ≥ inv p i"
      by blast
    with permutes_inverses[OF p] show ?thesis
      by simp
  qed
  then have "∀i∈S. inv p i ≤ i"
    by blast
  from permutes_natset_le[OF permutes_inv[OF p] this] have "inv p = inv id"
    by simp
  then show ?thesis
    using p permutes_inv_inv by fastforce
qed

lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
  using permutes_inv permutes_inv_inv by force

lemma image_compose_permutations_left:
  assumes "q permutes S"
  shows "{q ∘ p |p. p permutes S} = {p. p permutes S}"
proof -
  have "⋀p. p permutes S ⟹ q ∘ p permutes S"
    by (simp add: assms permutes_compose)
  moreover have "⋀x. x permutes S ⟹ ∃p. x = q ∘ p ∧ p permutes S"
    by (metis assms id_comp o_assoc permutes_compose permutes_inv permutes_inv_o(1))
  ultimately show ?thesis
    by auto
qed

lemma image_compose_permutations_right:
  assumes "q permutes S"
  shows "{p ∘ q | p. p permutes S} = {p . p permutes S}"
  by (metis (no_types, opaque_lifting) assms comp_id fun.map_comp permutes_compose permutes_inv permutes_inv_o(2))

lemma permutes_in_seg: "p permutes {1 ..n} ⟹ i ∈ {1..n} ⟹ 1 ≤ p i ∧ p i ≤ n"
  by (simp add: permutes_def) metis

lemma sum_permutations_inverse: "sum f {p. p permutes S} = sum (λp. 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 ∘ 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 (λp. f(q ∘ p)) {p. p permutes S}"
  (is "?lhs = ?rhs")
proof -
  let ?S = "{p. p permutes S}"
  have *: "?rhs = sum (f ∘ ((∘) q)) ?S"
    by (simp add: o_def)
  have **: "inj_on ((∘) q) ?S"
  proof (auto simp add: inj_on_def)
    fix p r
    assume "p permutes S"
      and r: "r permutes S"
      and rp: "q ∘ p = q ∘ r"
    then have "inv q ∘ q ∘ p = inv q ∘ q ∘ r"
      by (simp add: comp_assoc)
    with permutes_inj[OF q, unfolded inj_iff] show "p = r"
      by simp
  qed
  have "((∘) 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 (λp. f(p ∘ q)) {p. p permutes S}"
  (is "?lhs = ?rhs")
proof -
  let ?S = "{p. p permutes S}"
  have *: "?rhs = sum (f ∘ (λp. p ∘ q)) ?S"
    by (simp add: o_def)
  have **: "inj_on (λp. p ∘ q) ?S"
  proof (auto simp add: inj_on_def)
    fix p r
    assume "p permutes S"
      and r: "r permutes S"
      and rp: "p ∘ q = r ∘ q"
    then have "p ∘ (q ∘ inv q) = r ∘ (q ∘ 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 "(λp. p ∘ q) ` ?S = ?S"
    by auto
  with * sum.reindex[OF **, of f] show ?thesis
    by (simp only:)
qed

lemma inv_inj_on_permutes:
  ‹inj_on inv {p. p permutes S}›
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:
  ‹{(p s, s) |s. s ∈ S} = {(s, inv p s) |s. s ∈ S}› (is ‹?L = ?R›) if ‹p permutes S›
proof
  show "?L ⊆ ?R"
  proof
    fix x assume "x ∈ ?L"
    then obtain s where x: "x = (p s, s)" and s: "s ∈ 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 "... ∈ ?R" using s permutes_in_image[OF that] by auto
    finally show "x ∈ ?R".
  qed
  show "?R ⊆ ?L"
  proof
    fix x assume "x ∈ ?R"
    then obtain s
      where x: "x = (s, Hilbert_Choice.inv p s)" (is "_ = (s, ?ips)")
        and s: "s ∈ 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 "... ∈ ?L"
      using s permutes_in_image[OF permutes_inv[OF that]] by auto
    finally show "x ∈ ?L".
  qed
qed

context
  fixes p and n i :: nat
  assumes p: ‹p permutes {0..<n}› and i: ‹i < n›
begin

lemma permutes_nat_less:
  ‹p i < n›
proof -
  have ‹?thesis ⟷ p i ∈ {0..<n}›
    by simp
  also from p have ‹p i ∈ {0..<n} ⟷ i ∈ {0..<n}›
    by (rule permutes_in_image)
  finally show ?thesis
    using i by simp
qed

lemma permutes_nat_inv_less:
  ‹inv p i < n›
proof -
  from p have ‹inv p permutes {0..<n}›
    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:
  ‹F (λs. g (p s) s) S = F (λs. g s (inv p s)) S› (is ‹?l = ?r›)
  if ‹p permutes S›
proof -
  let ?g = "λ(x, y). g x y"
  let ?ps = "λs. (p s, s)"
  let ?ips = "λs. (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 ∈ S}" by auto
  also have "... = {(s, inv p s) |s. s ∈ 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 ‹Sum over a set of permutations (could generalize to iteration)›

lemma sum_over_permutations_insert:
  assumes fS: "finite S"
    and aS: "a ∉ S"
  shows "sum f {p. p permutes (insert a S)} =
    sum (λb. sum (λq. f (transpose a b ∘ q)) {p. p permutes S}) (insert a S)"
proof -
  have *: "⋀f a b. (λ(b, p). f (transpose a b ∘ p)) = f ∘ (λ(b,p). transpose a b ∘ p)"
    by (simp add: fun_eq_iff)
  have **: "⋀P Q. {(a, b). a ∈ P ∧ b ∈ Q} = P × Q"
    by blast
  show ?thesis
    unfolding * ** sum.cartesian_product permutes_insert
  proof (rule sum.reindex)
    let ?f = "(λ(b, y). transpose a b ∘ y)"
    let ?P = "{p. p permutes S}"
    {
      fix b c p q
      assume b: "b ∈ insert a S"
      assume c: "c ∈ insert a S"
      assume p: "p permutes S"
      assume q: "q permutes S"
      assume eq: "transpose a b ∘ p = transpose a c ∘ q"
      from p q aS have pa: "p a = a" and qa: "q a = a"
        unfolding permutes_def by metis+
      from eq have "(transpose a b ∘ p) a  = (transpose a c ∘ q) a"
        by simp
      then have bc: "b = c"
        by (simp add: permutes_def pa qa o_def fun_upd_def id_def
            cong del: if_weak_cong split: if_split_asm)
      from eq[unfolded bc] have "(λp. transpose a c ∘ p) (transpose a c ∘ p) =
        (λp. transpose a c ∘ p) (transpose a c ∘ q)" by simp
      then have "p = q"
        unfolding o_assoc swap_id_idempotent by simp
      with bc have "b = c ∧ p = q"
        by blast
    }
    then show "inj_on ?f (insert a S × ?P)"
      unfolding inj_on_def by clarify metis
  qed
qed


subsection ‹Constructing permutations from association lists›

definition list_permutes :: "('a × 'a) list ⇒ 'a set ⇒ bool"
  where "list_permutes xs A ⟷
    set (map fst xs) ⊆ A ∧
    set (map snd xs) = set (map fst xs) ∧
    distinct (map fst xs) ∧
    distinct (map snd xs)"

lemma list_permutesI [simp]:
  assumes "set (map fst xs) ⊆ 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 × 'a) list ⇒ 'a ⇒ 'a"
  where "permutation_of_list xs x = (case map_of xs x of None ⇒ x | Some y ⇒ 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 × 'a) list ⇒ 'a ⇒ '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 ∈ set (map fst xs)" "y ∈ 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') ∈ set xs" "(y, y') ∈ 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 ∉ A ⟹ 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 ∘ 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 ∘ 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) ⊆ A"
    by (simp add: list_permutes_def)
  from assms have "inj_on (the ∘ map_of xs) (set (map fst xs))" (is ?P)
    by (intro inj_on_map_of') (simp_all add: list_permutes_def)
  also have "?P ⟷ 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 ∘ 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 "… = 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' ⟹ permutation_of_list ((x',y)#xs) x = y"
  "x ≠ x' ⟹ 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' ⟹ inverse_permutation_of_list ((y,x')#xs) x = y"
  "x ≠ x' ⟹ 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 ∉ set (map fst xs) ⟹ permutation_of_list xs x = x"
  by (induct xs) (auto simp: permutation_of_list_Cons)

lemma permutation_of_list_unique':
  "distinct (map fst xs) ⟹ (x, y) ∈ set xs ⟹ permutation_of_list xs x = y"
  by (induct xs) (force simp: permutation_of_list_Cons)+

lemma permutation_of_list_unique:
  "list_permutes xs A ⟹ (x, y) ∈ set xs ⟹ 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 ∉ set (map snd xs) ⟹ inverse_permutation_of_list xs x = x"
  by (induct xs) auto

lemma inverse_permutation_of_list_unique':
  "distinct (map snd xs) ⟹ (x, y) ∈ set xs ⟹ 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 ⟹ (x,y) ∈ set xs ⟹ 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 ∈ set (map snd xs)")
    case True
    then obtain y where "(y, x) ∈ 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