src/HOL/Library/Permutations.thy
author haftmann
Fri Mar 22 19:18:08 2019 +0000 (4 months ago)
changeset 69946 494934c30f38
parent 69895 6b03a8cf092d
permissions -rw-r--r--
improved code equations taken over from AFP
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(*  Title:      HOL/Library/Permutations.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section \<open>Permutations, both general and specifically on finite sets.\<close>
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theory Permutations
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  imports Multiset Disjoint_Sets
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begin
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subsection \<open>Transpositions\<close>
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lemma swap_id_idempotent [simp]: "Fun.swap a b id \<circ> Fun.swap a b id = id"
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  by (rule ext) (auto simp add: Fun.swap_def)
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lemma inv_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id"
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  by (rule inv_unique_comp) simp_all
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lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
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  by (simp add: Fun.swap_def)
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lemma bij_swap_comp:
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  assumes "bij p"
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  shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
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  using surj_f_inv_f[OF bij_is_surj[OF \<open>bij p\<close>]]
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  by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF \<open>bij p\<close>])
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lemma bij_swap_compose_bij:
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  assumes "bij p"
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  shows "bij (Fun.swap a b id \<circ> p)"
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  by (simp only: bij_swap_comp[OF \<open>bij p\<close>] bij_swap_iff \<open>bij p\<close>)
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subsection \<open>Basic consequences of the definition\<close>
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definition permutes  (infixr "permutes" 41)
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  where "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"
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lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
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  unfolding permutes_def by metis
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lemma permutes_not_in: "f permutes S \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = x"
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  by (auto simp: permutes_def)
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lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S"
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  unfolding permutes_def
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  apply (rule set_eqI)
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  apply (simp add: image_iff)
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  apply metis
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  done
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lemma permutes_inj: "p permutes S \<Longrightarrow> inj p"
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  unfolding permutes_def inj_def by blast
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lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A"
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  by (auto simp: permutes_def inj_on_def)
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lemma permutes_surj: "p permutes s \<Longrightarrow> surj p"
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  unfolding permutes_def surj_def by metis
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lemma permutes_bij: "p permutes s \<Longrightarrow> bij p"
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  unfolding bij_def by (metis permutes_inj permutes_surj)
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lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S"
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  by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI)
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lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S"
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  unfolding permutes_def bij_betw_def inj_on_def
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  by auto (metis image_iff)+
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lemma permutes_inv_o:
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  assumes permutes: "p permutes S"
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  shows "p \<circ> inv p = id"
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    and "inv p \<circ> p = id"
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  using permutes_inj[OF permutes] permutes_surj[OF permutes]
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  unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
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lemma permutes_inverses:
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  fixes p :: "'a \<Rightarrow> 'a"
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  assumes permutes: "p permutes S"
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  shows "p (inv p x) = x"
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    and "inv p (p x) = x"
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  using permutes_inv_o[OF permutes, unfolded fun_eq_iff o_def] by auto
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lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T"
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  unfolding permutes_def by blast
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lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
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  by (auto simp add: fun_eq_iff permutes_def)
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lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
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  by (simp add: fun_eq_iff permutes_def) metis  (*somewhat slow*)
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lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
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  by (simp add: permutes_def)
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lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y"
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  unfolding permutes_def inv_def
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  apply auto
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  apply (erule allE[where x=y])
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  apply (erule allE[where x=y])
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  apply (rule someI_ex)
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  apply blast
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  apply (rule some1_equality)
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  apply blast
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  apply blast
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  done
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lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S"
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  unfolding permutes_def Fun.swap_def fun_upd_def by auto metis
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lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
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  by (simp add: Ball_def permutes_def) metis
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lemma permutes_bij_inv_into: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
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  fixes A :: "'a set"
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    and B :: "'b set"
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  assumes "p permutes A"
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    and "bij_betw f A B"
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  shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
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proof (rule bij_imp_permutes)
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  from assms have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
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    by (auto simp add: permutes_imp_bij bij_betw_inv_into)
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  then have "bij_betw (f \<circ> p \<circ> inv_into A f) B B"
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    by (simp add: bij_betw_trans)
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  then show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B"
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    by (subst bij_betw_cong[where g="f \<circ> p \<circ> inv_into A f"]) auto
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next
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  fix x
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  assume "x \<notin> B"
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  then show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto
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qed
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lemma permutes_image_mset: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
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  assumes "p permutes A"
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  shows "image_mset p (mset_set A) = mset_set A"
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  using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image)
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lemma permutes_implies_image_mset_eq: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
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  assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)"
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  shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
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proof -
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  have "f x = f' (p x)" if "x \<in># mset_set A" for x
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    using assms(2)[of x] that by (cases "finite A") auto
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  with assms have "image_mset f (mset_set A) = image_mset (f' \<circ> p) (mset_set A)"
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    by (auto intro!: image_mset_cong)
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  also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
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    by (simp add: image_mset.compositionality)
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  also have "\<dots> = image_mset f' (mset_set A)"
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  proof -
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    from assms permutes_image_mset have "image_mset p (mset_set A) = mset_set A"
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      by blast
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    then show ?thesis by simp
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  qed
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  finally show ?thesis ..
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qed
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subsection \<open>Group properties\<close>
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lemma permutes_id: "id permutes S"
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  by (simp add: permutes_def)
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lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S"
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  unfolding permutes_def o_def by metis
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lemma permutes_inv:
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  assumes "p permutes S"
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  shows "inv p permutes S"
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  using assms unfolding permutes_def permutes_inv_eq[OF assms] by metis
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lemma permutes_inv_inv:
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  assumes "p permutes S"
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  shows "inv (inv p) = p"
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  unfolding fun_eq_iff permutes_inv_eq[OF assms] permutes_inv_eq[OF permutes_inv[OF assms]]
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  by blast
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lemma permutes_invI:
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  assumes perm: "p permutes S"
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    and inv: "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
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    and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
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  shows "inv p = p'"
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proof
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  show "inv p x = p' x" for x
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  proof (cases "x \<in> S")
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    case True
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    from assms have "p' x = p' (p (inv p x))"
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      by (simp add: permutes_inverses)
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    also from permutes_inv[OF perm] True have "\<dots> = inv p x"
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      by (subst inv) (simp_all add: permutes_in_image)
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    finally show ?thesis ..
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  next
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    case False
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    with permutes_inv[OF perm] show ?thesis
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      by (simp_all add: outside permutes_not_in)
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  qed
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qed
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lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
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  by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
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subsection \<open>Mapping permutations with bijections\<close>
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lemma bij_betw_permutations:
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  assumes "bij_betw f A B"
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  shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x) 
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             {\<pi>. \<pi> permutes A} {\<pi>. \<pi> permutes B}" (is "bij_betw ?f _ _")
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proof -
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  let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
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  show ?thesis
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  proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
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    case 3
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    show ?case using permutes_bij_inv_into[OF _ assms] by auto
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  next
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    case 4
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    have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
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    {
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      fix \<pi> assume "\<pi> permutes B"
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      from permutes_bij_inv_into[OF this bij_inv] and assms
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        have "(\<lambda>x. if x \<in> A then inv_into A f (\<pi> (f x)) else x) permutes A"
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        by (simp add: inv_into_inv_into_eq cong: if_cong)
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    }
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    from this show ?case by (auto simp: permutes_inv)
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  next
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    case 1
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    thus ?case using assms
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      by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
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               dest: bij_betwE)
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  next
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    case 2
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    moreover have "bij_betw (inv_into A f) B A"
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      by (intro bij_betw_inv_into assms)
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    ultimately show ?case using assms
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      by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right 
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               dest: bij_betwE)
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  qed
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qed
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lemma bij_betw_derangements:
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  assumes "bij_betw f A B"
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  shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x) 
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             {\<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)}" 
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           (is "bij_betw ?f _ _")
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proof -
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  let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
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  show ?thesis
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  proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
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    case 3
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    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
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      using that and assms by (metis bij_betwE bij_betw_imp_inj_on bij_betw_imp_surj_on
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                                     inv_into_f_f inv_into_into permutes_imp_bij)
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    with permutes_bij_inv_into[OF _ assms] show ?case by auto
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  next
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    case 4
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    have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
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    have "?g \<pi> permutes A" if "\<pi> permutes B" for \<pi>
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      using permutes_bij_inv_into[OF that bij_inv] and assms
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      by (simp add: inv_into_inv_into_eq cong: if_cong)
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    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
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      using that and assms by (metis bij_betwE bij_betw_imp_surj_on f_inv_into_f permutes_imp_bij)
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    ultimately show ?case by auto
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  next
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    case 1
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    thus ?case using assms
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      by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
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                dest: bij_betwE)
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  next
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    case 2
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    moreover have "bij_betw (inv_into A f) B A"
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      by (intro bij_betw_inv_into assms)
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    ultimately show ?case using assms
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      by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right 
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                dest: bij_betwE)
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  qed
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qed
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subsection \<open>The number of permutations on a finite set\<close>
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lemma permutes_insert_lemma:
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  assumes "p permutes (insert a S)"
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  shows "Fun.swap a (p a) id \<circ> p permutes S"
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  apply (rule permutes_superset[where S = "insert a S"])
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  apply (rule permutes_compose[OF assms])
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  apply (rule permutes_swap_id, simp)
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  using permutes_in_image[OF assms, of a]
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  apply simp
haftmann@56545
   289
  apply (auto simp add: Ball_def Fun.swap_def)
chaieb@29840
   290
  done
chaieb@29840
   291
chaieb@29840
   292
lemma permutes_insert: "{p. p permutes (insert a S)} =
wenzelm@65342
   293
  (\<lambda>(b, p). Fun.swap a b id \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
wenzelm@54681
   294
proof -
wenzelm@65342
   295
  have "p permutes insert a S \<longleftrightarrow>
wenzelm@65342
   296
    (\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)" for p
wenzelm@65342
   297
  proof -
wenzelm@65342
   298
    have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S"
wenzelm@65342
   299
      if p: "p permutes insert a S"
wenzelm@65342
   300
    proof -
chaieb@29840
   301
      let ?b = "p a"
wenzelm@54681
   302
      let ?q = "Fun.swap a (p a) id \<circ> p"
wenzelm@65342
   303
      have *: "p = Fun.swap a ?b id \<circ> ?q"
wenzelm@65342
   304
        by (simp add: fun_eq_iff o_assoc)
wenzelm@65342
   305
      have **: "?b \<in> insert a S"
wenzelm@65342
   306
        unfolding permutes_in_image[OF p] by simp
wenzelm@65342
   307
      from permutes_insert_lemma[OF p] * ** show ?thesis
wenzelm@65342
   308
       by blast
wenzelm@65342
   309
    qed
wenzelm@65342
   310
    moreover have "p permutes insert a S"
wenzelm@65342
   311
      if bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" for b q
wenzelm@65342
   312
    proof -
wenzelm@65342
   313
      from permutes_subset[OF bq(3), of "insert a S"] have q: "q permutes insert a S"
wenzelm@54681
   314
        by auto
wenzelm@65342
   315
      have a: "a \<in> insert a S"
wenzelm@54681
   316
        by simp
wenzelm@65342
   317
      from bq(1) permutes_compose[OF q permutes_swap_id[OF a bq(2)]] show ?thesis
wenzelm@54681
   318
        by simp
wenzelm@65342
   319
    qed
wenzelm@65342
   320
    ultimately show ?thesis by blast
wenzelm@65342
   321
  qed
wenzelm@65342
   322
  then show ?thesis by auto
chaieb@29840
   323
qed
chaieb@29840
   324
wenzelm@54681
   325
lemma card_permutations:
wenzelm@65342
   326
  assumes "card S = n"
wenzelm@65342
   327
    and "finite S"
hoelzl@33715
   328
  shows "card {p. p permutes S} = fact n"
wenzelm@65342
   329
  using assms(2,1)
wenzelm@54681
   330
proof (induct arbitrary: n)
wenzelm@54681
   331
  case empty
wenzelm@54681
   332
  then show ?case by simp
hoelzl@33715
   333
next
hoelzl@33715
   334
  case (insert x F)
wenzelm@54681
   335
  {
wenzelm@54681
   336
    fix n
wenzelm@65342
   337
    assume card_insert: "card (insert x F) = n"
hoelzl@33715
   338
    let ?xF = "{p. p permutes insert x F}"
hoelzl@33715
   339
    let ?pF = "{p. p permutes F}"
hoelzl@33715
   340
    let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
hoelzl@33715
   341
    let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
wenzelm@65342
   342
    have xfgpF': "?xF = ?g ` ?pF'"
wenzelm@65342
   343
      by (rule permutes_insert[of x F])
wenzelm@65342
   344
    from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have Fs: "card F = n - 1"
wenzelm@65342
   345
      by auto
wenzelm@65342
   346
    from \<open>finite F\<close> insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
wenzelm@65342
   347
      by auto
wenzelm@54681
   348
    then have "finite ?pF"
lp15@59730
   349
      by (auto intro: card_ge_0_finite)
wenzelm@65342
   350
    with \<open>finite F\<close> card_insert have pF'f: "finite ?pF'"
haftmann@61424
   351
      apply (simp only: Collect_case_prod Collect_mem_eq)
hoelzl@33715
   352
      apply (rule finite_cartesian_product)
hoelzl@33715
   353
      apply simp_all
hoelzl@33715
   354
      done
chaieb@29840
   355
hoelzl@33715
   356
    have ginj: "inj_on ?g ?pF'"
wenzelm@54681
   357
    proof -
hoelzl@33715
   358
      {
wenzelm@54681
   359
        fix b p c q
wenzelm@65342
   360
        assume bp: "(b, p) \<in> ?pF'"
wenzelm@65342
   361
        assume cq: "(c, q) \<in> ?pF'"
wenzelm@65342
   362
        assume eq: "?g (b, p) = ?g (c, q)"
wenzelm@65342
   363
        from bp cq have pF: "p permutes F" and qF: "q permutes F"
wenzelm@54681
   364
          by auto
wenzelm@65342
   365
        from pF \<open>x \<notin> F\<close> eq have "b = ?g (b, p) x"
wenzelm@65342
   366
          by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
wenzelm@65342
   367
        also from qF \<open>x \<notin> F\<close> eq have "\<dots> = ?g (c, q) x"
wenzelm@65342
   368
          by (auto simp: swap_def fun_upd_def fun_eq_iff)
wenzelm@65342
   369
        also from qF \<open>x \<notin> F\<close> have "\<dots> = c"
wenzelm@65342
   370
          by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
wenzelm@65342
   371
        finally have "b = c" .
wenzelm@54681
   372
        then have "Fun.swap x b id = Fun.swap x c id"
wenzelm@54681
   373
          by simp
wenzelm@54681
   374
        with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q"
wenzelm@54681
   375
          by simp
wenzelm@65342
   376
        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)"
wenzelm@54681
   377
          by simp
wenzelm@54681
   378
        then have "p = q"
wenzelm@54681
   379
          by (simp add: o_assoc)
wenzelm@65342
   380
        with \<open>b = c\<close> have "(b, p) = (c, q)"
wenzelm@54681
   381
          by simp
hoelzl@33715
   382
      }
wenzelm@54681
   383
      then show ?thesis
wenzelm@54681
   384
        unfolding inj_on_def by blast
hoelzl@33715
   385
    qed
wenzelm@65342
   386
    from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have "n \<noteq> 0"
wenzelm@65342
   387
      by auto
wenzelm@54681
   388
    then have "\<exists>m. n = Suc m"
wenzelm@54681
   389
      by presburger
wenzelm@65342
   390
    then obtain m where n: "n = Suc m"
wenzelm@54681
   391
      by blast
wenzelm@65342
   392
    from pFs card_insert have *: "card ?xF = fact n"
wenzelm@54681
   393
      unfolding xfgpF' card_image[OF ginj]
wenzelm@60500
   394
      using \<open>finite F\<close> \<open>finite ?pF\<close>
wenzelm@65342
   395
      by (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) (simp add: n)
wenzelm@54681
   396
    from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF"
wenzelm@65342
   397
      by (simp add: xfgpF' n)
wenzelm@65342
   398
    from * have "card ?xF = fact n"
wenzelm@65342
   399
      unfolding xFf by blast
hoelzl@33715
   400
  }
wenzelm@65342
   401
  with insert show ?case by simp
chaieb@29840
   402
qed
chaieb@29840
   403
wenzelm@54681
   404
lemma finite_permutations:
wenzelm@65342
   405
  assumes "finite S"
wenzelm@54681
   406
  shows "finite {p. p permutes S}"
wenzelm@65342
   407
  using card_permutations[OF refl assms] by (auto intro: card_ge_0_finite)
chaieb@29840
   408
wenzelm@54681
   409
wenzelm@60500
   410
subsection \<open>Permutations of index set for iterated operations\<close>
chaieb@29840
   411
haftmann@51489
   412
lemma (in comm_monoid_set) permute:
haftmann@51489
   413
  assumes "p permutes S"
wenzelm@54681
   414
  shows "F g S = F (g \<circ> p) S"
haftmann@51489
   415
proof -
wenzelm@60500
   416
  from \<open>p permutes S\<close> have "inj p"
wenzelm@54681
   417
    by (rule permutes_inj)
wenzelm@54681
   418
  then have "inj_on p S"
wenzelm@54681
   419
    by (auto intro: subset_inj_on)
wenzelm@54681
   420
  then have "F g (p ` S) = F (g \<circ> p) S"
wenzelm@54681
   421
    by (rule reindex)
wenzelm@60500
   422
  moreover from \<open>p permutes S\<close> have "p ` S = S"
wenzelm@54681
   423
    by (rule permutes_image)
wenzelm@54681
   424
  ultimately show ?thesis
wenzelm@54681
   425
    by simp
chaieb@29840
   426
qed
chaieb@29840
   427
wenzelm@54681
   428
wenzelm@60500
   429
subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close>
wenzelm@54681
   430
wenzelm@54681
   431
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>
wenzelm@54681
   432
  Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
haftmann@56545
   433
  by (simp add: fun_eq_iff Fun.swap_def)
chaieb@29840
   434
wenzelm@54681
   435
lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow>
wenzelm@54681
   436
  Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"
haftmann@56545
   437
  by (simp add: fun_eq_iff Fun.swap_def)
chaieb@29840
   438
wenzelm@54681
   439
lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow>
wenzelm@54681
   440
  Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id"
haftmann@56545
   441
  by (simp add: fun_eq_iff Fun.swap_def)
chaieb@29840
   442
wenzelm@54681
   443
wenzelm@60500
   444
subsection \<open>Permutations as transposition sequences\<close>
wenzelm@54681
   445
wenzelm@54681
   446
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
wenzelm@65342
   447
  where
wenzelm@65342
   448
    id[simp]: "swapidseq 0 id"
wenzelm@65342
   449
  | comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
wenzelm@54681
   450
wenzelm@54681
   451
declare id[unfolded id_def, simp]
wenzelm@54681
   452
wenzelm@54681
   453
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
chaieb@29840
   454
chaieb@29840
   455
wenzelm@60500
   456
subsection \<open>Some closure properties of the set of permutations, with lengths\<close>
chaieb@29840
   457
wenzelm@54681
   458
lemma permutation_id[simp]: "permutation id"
wenzelm@54681
   459
  unfolding permutation_def by (rule exI[where x=0]) simp
chaieb@29840
   460
chaieb@29840
   461
declare permutation_id[unfolded id_def, simp]
chaieb@29840
   462
chaieb@29840
   463
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
chaieb@29840
   464
  apply clarsimp
wenzelm@54681
   465
  using comp_Suc[of 0 id a b]
wenzelm@54681
   466
  apply simp
wenzelm@54681
   467
  done
chaieb@29840
   468
chaieb@29840
   469
lemma permutation_swap_id: "permutation (Fun.swap a b id)"
wenzelm@65342
   470
proof (cases "a = b")
wenzelm@65342
   471
  case True
wenzelm@65342
   472
  then show ?thesis by simp
wenzelm@65342
   473
next
wenzelm@65342
   474
  case False
wenzelm@65342
   475
  then show ?thesis
wenzelm@65342
   476
    unfolding permutation_def
wenzelm@65342
   477
    using swapidseq_swap[of a b] by blast
wenzelm@65342
   478
qed
wenzelm@65342
   479
chaieb@29840
   480
wenzelm@54681
   481
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)"
wenzelm@54681
   482
proof (induct n p arbitrary: m q rule: swapidseq.induct)
wenzelm@54681
   483
  case (id m q)
wenzelm@54681
   484
  then show ?case by simp
wenzelm@54681
   485
next
wenzelm@54681
   486
  case (comp_Suc n p a b m q)
wenzelm@65342
   487
  have eq: "Suc n + m = Suc (n + m)"
wenzelm@54681
   488
    by arith
wenzelm@54681
   489
  show ?case
wenzelm@65342
   490
    apply (simp only: eq comp_assoc)
wenzelm@54681
   491
    apply (rule swapidseq.comp_Suc)
wenzelm@54681
   492
    using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
wenzelm@65342
   493
     apply blast+
wenzelm@54681
   494
    done
chaieb@29840
   495
qed
chaieb@29840
   496
wenzelm@54681
   497
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)"
chaieb@29840
   498
  unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
chaieb@29840
   499
wenzelm@54681
   500
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
wenzelm@65342
   501
  by (induct n p rule: swapidseq.induct)
wenzelm@65342
   502
    (use swapidseq_swap[of a b] in \<open>auto simp add: comp_assoc intro: swapidseq.comp_Suc\<close>)
chaieb@29840
   503
wenzelm@54681
   504
lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
wenzelm@54681
   505
proof (induct n p rule: swapidseq.induct)
wenzelm@54681
   506
  case id
wenzelm@54681
   507
  then show ?case
wenzelm@54681
   508
    by (rule exI[where x=id]) simp
huffman@30488
   509
next
chaieb@29840
   510
  case (comp_Suc n p a b)
wenzelm@54681
   511
  from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
wenzelm@54681
   512
    by blast
wenzelm@54681
   513
  let ?q = "q \<circ> Fun.swap a b id"
chaieb@29840
   514
  note H = comp_Suc.hyps
wenzelm@65342
   515
  from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (Fun.swap a b id)"
wenzelm@54681
   516
    by simp
wenzelm@65342
   517
  from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q"
wenzelm@54681
   518
    by simp
wenzelm@54681
   519
  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"
wenzelm@54681
   520
    by (simp add: o_assoc)
wenzelm@54681
   521
  also have "\<dots> = id"
wenzelm@54681
   522
    by (simp add: q(2))
wenzelm@65342
   523
  finally have ***: "Fun.swap a b id \<circ> p \<circ> ?q = id" .
wenzelm@54681
   524
  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"
wenzelm@54681
   525
    by (simp only: o_assoc)
wenzelm@54681
   526
  then have "?q \<circ> (Fun.swap a b id \<circ> p) = id"
wenzelm@54681
   527
    by (simp add: q(3))
wenzelm@65342
   528
  with ** *** show ?case
wenzelm@54681
   529
    by blast
chaieb@29840
   530
qed
chaieb@29840
   531
wenzelm@54681
   532
lemma swapidseq_inverse:
wenzelm@65342
   533
  assumes "swapidseq n p"
wenzelm@54681
   534
  shows "swapidseq n (inv p)"
wenzelm@65342
   535
  using swapidseq_inverse_exists[OF assms] inv_unique_comp[of p] by auto
wenzelm@54681
   536
wenzelm@54681
   537
lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)"
wenzelm@54681
   538
  using permutation_def swapidseq_inverse by blast
wenzelm@54681
   539
chaieb@29840
   540
wenzelm@60500
   541
subsection \<open>The identity map only has even transposition sequences\<close>
chaieb@29840
   542
wenzelm@54681
   543
lemma symmetry_lemma:
wenzelm@54681
   544
  assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"
wenzelm@54681
   545
    and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
wenzelm@54681
   546
      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>
wenzelm@54681
   547
      P a b c d"
wenzelm@54681
   548
  shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>  P a b c d"
wenzelm@54681
   549
  using assms by metis
chaieb@29840
   550
wenzelm@54681
   551
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
wenzelm@54681
   552
  Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
wenzelm@54681
   553
  (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
wenzelm@54681
   554
    Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)"
wenzelm@54681
   555
proof -
wenzelm@65342
   556
  assume neq: "a \<noteq> b" "c \<noteq> d"
wenzelm@54681
   557
  have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
wenzelm@54681
   558
    (Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
wenzelm@54681
   559
      (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
wenzelm@54681
   560
        Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))"
wenzelm@54681
   561
    apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
wenzelm@65342
   562
     apply (simp_all only: swap_commute)
wenzelm@54681
   563
    apply (case_tac "a = c \<and> b = d")
wenzelm@65342
   564
     apply (clarsimp simp only: swap_commute swap_id_idempotent)
wenzelm@54681
   565
    apply (case_tac "a = c \<and> b \<noteq> d")
wenzelm@65342
   566
     apply (rule disjI2)
wenzelm@65342
   567
     apply (rule_tac x="b" in exI)
wenzelm@65342
   568
     apply (rule_tac x="d" in exI)
wenzelm@65342
   569
     apply (rule_tac x="b" in exI)
wenzelm@65342
   570
     apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
wenzelm@54681
   571
    apply (case_tac "a \<noteq> c \<and> b = d")
wenzelm@65342
   572
     apply (rule disjI2)
wenzelm@65342
   573
     apply (rule_tac x="c" in exI)
wenzelm@65342
   574
     apply (rule_tac x="d" in exI)
wenzelm@65342
   575
     apply (rule_tac x="c" in exI)
wenzelm@65342
   576
     apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
wenzelm@54681
   577
    apply (rule disjI2)
wenzelm@54681
   578
    apply (rule_tac x="c" in exI)
wenzelm@54681
   579
    apply (rule_tac x="d" in exI)
wenzelm@54681
   580
    apply (rule_tac x="b" in exI)
haftmann@56545
   581
    apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
wenzelm@54681
   582
    done
wenzelm@65342
   583
  with neq show ?thesis by metis
chaieb@29840
   584
qed
chaieb@29840
   585
chaieb@29840
   586
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
wenzelm@65342
   587
  using swapidseq.cases[of 0 p "p = id"] by auto
chaieb@29840
   588
wenzelm@54681
   589
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow>
wenzelm@65342
   590
    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)"
chaieb@29840
   591
  apply (rule iffI)
wenzelm@65342
   592
   apply (erule swapidseq.cases[of n p])
wenzelm@65342
   593
    apply simp
wenzelm@65342
   594
   apply (rule disjI2)
wenzelm@65342
   595
   apply (rule_tac x= "a" in exI)
wenzelm@65342
   596
   apply (rule_tac x= "b" in exI)
wenzelm@65342
   597
   apply (rule_tac x= "pa" in exI)
wenzelm@65342
   598
   apply (rule_tac x= "na" in exI)
wenzelm@65342
   599
   apply simp
chaieb@29840
   600
  apply auto
chaieb@29840
   601
  apply (rule comp_Suc, simp_all)
chaieb@29840
   602
  done
wenzelm@54681
   603
chaieb@29840
   604
lemma fixing_swapidseq_decrease:
wenzelm@65342
   605
  assumes "swapidseq n p"
wenzelm@65342
   606
    and "a \<noteq> b"
wenzelm@65342
   607
    and "(Fun.swap a b id \<circ> p) a = a"
wenzelm@54681
   608
  shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
wenzelm@65342
   609
  using assms
wenzelm@54681
   610
proof (induct n arbitrary: p a b)
wenzelm@54681
   611
  case 0
wenzelm@54681
   612
  then show ?case
haftmann@56545
   613
    by (auto simp add: Fun.swap_def fun_upd_def)
chaieb@29840
   614
next
chaieb@29840
   615
  case (Suc n p a b)
wenzelm@54681
   616
  from Suc.prems(1) swapidseq_cases[of "Suc n" p]
wenzelm@54681
   617
  obtain c d q m where
wenzelm@54681
   618
    cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m"
chaieb@29840
   619
    by auto
wenzelm@65342
   620
  consider "Fun.swap a b id \<circ> Fun.swap c d id = id"
wenzelm@65342
   621
    | x y z where "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y"
wenzelm@54681
   622
      "Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id"
wenzelm@65342
   623
    using swap_general[OF Suc.prems(2) cdqm(4)] by metis
wenzelm@65342
   624
  then show ?case
wenzelm@65342
   625
  proof cases
wenzelm@65342
   626
    case 1
wenzelm@65342
   627
    then show ?thesis
wenzelm@65342
   628
      by (simp only: cdqm o_assoc) (simp add: cdqm)
wenzelm@65342
   629
  next
wenzelm@65342
   630
    case prems: 2
wenzelm@65342
   631
    then have az: "a \<noteq> z"
wenzelm@54681
   632
      by simp
wenzelm@65342
   633
    from prems have *: "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" for h
wenzelm@65342
   634
      by (simp add: Fun.swap_def)
wenzelm@54681
   635
    from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)"
wenzelm@54681
   636
      by simp
wenzelm@54681
   637
    then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)"
wenzelm@65342
   638
      by (simp add: o_assoc prems)
wenzelm@54681
   639
    then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a"
wenzelm@54681
   640
      by simp
wenzelm@54681
   641
    then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a"
wenzelm@54681
   642
      unfolding Suc by metis
wenzelm@65342
   643
    then have "(Fun.swap a z id \<circ> q) a = a"
wenzelm@65342
   644
      by (simp only: *)
wenzelm@65342
   645
    from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this]
wenzelm@65342
   646
    have **: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0"
wenzelm@54681
   647
      by blast+
wenzelm@65342
   648
    from \<open>n \<noteq> 0\<close> have ***: "Suc n - 1 = Suc (n - 1)"
wenzelm@65342
   649
      by auto
wenzelm@65342
   650
    show ?thesis
wenzelm@65342
   651
      apply (simp only: cdqm(2) prems o_assoc ***)
haftmann@49739
   652
      apply (simp only: Suc_not_Zero simp_thms comp_assoc)
chaieb@29840
   653
      apply (rule comp_Suc)
wenzelm@65342
   654
      using ** prems
wenzelm@65342
   655
       apply blast+
wenzelm@54681
   656
      done
wenzelm@65342
   657
  qed
chaieb@29840
   658
qed
chaieb@29840
   659
huffman@30488
   660
lemma swapidseq_identity_even:
wenzelm@54681
   661
  assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)"
wenzelm@54681
   662
  shows "even n"
wenzelm@60500
   663
  using \<open>swapidseq n id\<close>
wenzelm@54681
   664
proof (induct n rule: nat_less_induct)
wenzelm@65342
   665
  case H: (1 n)
wenzelm@65342
   666
  consider "n = 0"
wenzelm@65342
   667
    | a b :: 'a and q m where "n = Suc m" "id = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
wenzelm@65342
   668
    using H(2)[unfolded swapidseq_cases[of n id]] by auto
wenzelm@65342
   669
  then show ?case
wenzelm@65342
   670
  proof cases
wenzelm@65342
   671
    case 1
wenzelm@65342
   672
    then show ?thesis by presburger
wenzelm@65342
   673
  next
wenzelm@65342
   674
    case h: 2
chaieb@29840
   675
    from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
wenzelm@54681
   676
    have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)"
wenzelm@54681
   677
      by auto
wenzelm@54681
   678
    from h m have mn: "m - 1 < n"
wenzelm@54681
   679
      by arith
wenzelm@65342
   680
    from H(1)[rule_format, OF mn m(2)] h(1) m(1) show ?thesis
wenzelm@54681
   681
      by presburger
wenzelm@65342
   682
  qed
chaieb@29840
   683
qed
chaieb@29840
   684
wenzelm@54681
   685
wenzelm@60500
   686
subsection \<open>Therefore we have a welldefined notion of parity\<close>
chaieb@29840
   687
chaieb@29840
   688
definition "evenperm p = even (SOME n. swapidseq n p)"
chaieb@29840
   689
wenzelm@54681
   690
lemma swapidseq_even_even:
wenzelm@54681
   691
  assumes m: "swapidseq m p"
wenzelm@54681
   692
    and n: "swapidseq n p"
chaieb@29840
   693
  shows "even m \<longleftrightarrow> even n"
wenzelm@54681
   694
proof -
wenzelm@65342
   695
  from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
wenzelm@54681
   696
    by blast
wenzelm@65342
   697
  from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] show ?thesis
wenzelm@54681
   698
    by arith
chaieb@29840
   699
qed
chaieb@29840
   700
wenzelm@54681
   701
lemma evenperm_unique:
wenzelm@54681
   702
  assumes p: "swapidseq n p"
wenzelm@54681
   703
    and n:"even n = b"
chaieb@29840
   704
  shows "evenperm p = b"
chaieb@29840
   705
  unfolding n[symmetric] evenperm_def
chaieb@29840
   706
  apply (rule swapidseq_even_even[where p = p])
wenzelm@65342
   707
   apply (rule someI[where x = n])
wenzelm@54681
   708
  using p
wenzelm@65342
   709
   apply blast+
wenzelm@54681
   710
  done
chaieb@29840
   711
wenzelm@54681
   712
wenzelm@60500
   713
subsection \<open>And it has the expected composition properties\<close>
chaieb@29840
   714
chaieb@29840
   715
lemma evenperm_id[simp]: "evenperm id = True"
wenzelm@54681
   716
  by (rule evenperm_unique[where n = 0]) simp_all
chaieb@29840
   717
chaieb@29840
   718
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
wenzelm@54681
   719
  by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap)
chaieb@29840
   720
huffman@30488
   721
lemma evenperm_comp:
wenzelm@65342
   722
  assumes "permutation p" "permutation q"
wenzelm@65342
   723
  shows "evenperm (p \<circ> q) \<longleftrightarrow> evenperm p = evenperm q"
wenzelm@54681
   724
proof -
wenzelm@65342
   725
  from assms obtain n m where n: "swapidseq n p" and m: "swapidseq m q"
chaieb@29840
   726
    unfolding permutation_def by blast
wenzelm@65342
   727
  have "even (n + m) \<longleftrightarrow> (even n \<longleftrightarrow> even m)"
wenzelm@54681
   728
    by arith
chaieb@29840
   729
  from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
wenzelm@65342
   730
    and evenperm_unique[OF swapidseq_comp_add[OF n m] this] show ?thesis
wenzelm@54681
   731
    by blast
chaieb@29840
   732
qed
chaieb@29840
   733
wenzelm@54681
   734
lemma evenperm_inv:
wenzelm@65342
   735
  assumes "permutation p"
chaieb@29840
   736
  shows "evenperm (inv p) = evenperm p"
wenzelm@54681
   737
proof -
wenzelm@65342
   738
  from assms obtain n where n: "swapidseq n p"
wenzelm@54681
   739
    unfolding permutation_def by blast
wenzelm@65342
   740
  show ?thesis
wenzelm@65342
   741
    by (rule evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]])
chaieb@29840
   742
qed
chaieb@29840
   743
chaieb@29840
   744
wenzelm@60500
   745
subsection \<open>A more abstract characterization of permutations\<close>
chaieb@29840
   746
chaieb@29840
   747
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
wenzelm@64966
   748
  unfolding bij_def inj_def surj_def
chaieb@29840
   749
  apply auto
wenzelm@65342
   750
   apply metis
chaieb@29840
   751
  apply metis
chaieb@29840
   752
  done
chaieb@29840
   753
huffman@30488
   754
lemma permutation_bijective:
wenzelm@65342
   755
  assumes "permutation p"
chaieb@29840
   756
  shows "bij p"
wenzelm@54681
   757
proof -
wenzelm@65342
   758
  from assms obtain n where n: "swapidseq n p"
wenzelm@54681
   759
    unfolding permutation_def by blast
wenzelm@65342
   760
  from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
wenzelm@54681
   761
    by blast
wenzelm@65342
   762
  then show ?thesis
wenzelm@65342
   763
    unfolding bij_iff
wenzelm@54681
   764
    apply (auto simp add: fun_eq_iff)
wenzelm@54681
   765
    apply metis
wenzelm@54681
   766
    done
huffman@30488
   767
qed
chaieb@29840
   768
wenzelm@54681
   769
lemma permutation_finite_support:
wenzelm@65342
   770
  assumes "permutation p"
chaieb@29840
   771
  shows "finite {x. p x \<noteq> x}"
wenzelm@54681
   772
proof -
wenzelm@65342
   773
  from assms obtain n where "swapidseq n p"
wenzelm@54681
   774
    unfolding permutation_def by blast
wenzelm@65342
   775
  then show ?thesis
wenzelm@54681
   776
  proof (induct n p rule: swapidseq.induct)
wenzelm@54681
   777
    case id
wenzelm@54681
   778
    then show ?case by simp
chaieb@29840
   779
  next
chaieb@29840
   780
    case (comp_Suc n p a b)
chaieb@29840
   781
    let ?S = "insert a (insert b {x. p x \<noteq> x})"
wenzelm@65342
   782
    from comp_Suc.hyps(2) have *: "finite ?S"
wenzelm@54681
   783
      by simp
wenzelm@65342
   784
    from \<open>a \<noteq> b\<close> have **: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
wenzelm@65342
   785
      by (auto simp: Fun.swap_def)
wenzelm@65342
   786
    show ?case
wenzelm@65342
   787
      by (rule finite_subset[OF ** *])
wenzelm@54681
   788
  qed
chaieb@29840
   789
qed
chaieb@29840
   790
huffman@30488
   791
lemma permutation_lemma:
wenzelm@65342
   792
  assumes "finite S"
wenzelm@65342
   793
    and "bij p"
wenzelm@65342
   794
    and "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
chaieb@29840
   795
  shows "permutation p"
wenzelm@65342
   796
  using assms
wenzelm@54681
   797
proof (induct S arbitrary: p rule: finite_induct)
wenzelm@65342
   798
  case empty
wenzelm@65342
   799
  then show ?case
wenzelm@65342
   800
    by simp
chaieb@29840
   801
next
chaieb@29840
   802
  case (insert a F p)
wenzelm@54681
   803
  let ?r = "Fun.swap a (p a) id \<circ> p"
wenzelm@54681
   804
  let ?q = "Fun.swap a (p a) id \<circ> ?r"
wenzelm@65342
   805
  have *: "?r a = a"
haftmann@56545
   806
    by (simp add: Fun.swap_def)
wenzelm@65342
   807
  from insert * have **: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
wenzelm@64966
   808
    by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
wenzelm@65342
   809
  have "bij ?r"
wenzelm@65342
   810
    by (rule bij_swap_compose_bij[OF insert(4)])
wenzelm@65342
   811
  have "permutation ?r"
wenzelm@65342
   812
    by (rule insert(3)[OF \<open>bij ?r\<close> **])
wenzelm@65342
   813
  then have "permutation ?q"
wenzelm@65342
   814
    by (simp add: permutation_compose permutation_swap_id)
wenzelm@54681
   815
  then show ?case
wenzelm@54681
   816
    by (simp add: o_assoc)
chaieb@29840
   817
qed
chaieb@29840
   818
huffman@30488
   819
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
chaieb@29840
   820
  (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
chaieb@29840
   821
proof
wenzelm@65342
   822
  assume ?lhs
wenzelm@65342
   823
  with permutation_bijective permutation_finite_support show "?b \<and> ?f"
wenzelm@54681
   824
    by auto
chaieb@29840
   825
next
wenzelm@54681
   826
  assume "?b \<and> ?f"
wenzelm@54681
   827
  then have "?f" "?b" by blast+
wenzelm@54681
   828
  from permutation_lemma[OF this] show ?lhs
wenzelm@54681
   829
    by blast
chaieb@29840
   830
qed
chaieb@29840
   831
wenzelm@54681
   832
lemma permutation_inverse_works:
wenzelm@65342
   833
  assumes "permutation p"
wenzelm@54681
   834
  shows "inv p \<circ> p = id"
wenzelm@54681
   835
    and "p \<circ> inv p = id"
wenzelm@65342
   836
  using permutation_bijective [OF assms] by (auto simp: bij_def inj_iff surj_iff)
chaieb@29840
   837
chaieb@29840
   838
lemma permutation_inverse_compose:
wenzelm@54681
   839
  assumes p: "permutation p"
wenzelm@54681
   840
    and q: "permutation q"
wenzelm@54681
   841
  shows "inv (p \<circ> q) = inv q \<circ> inv p"
wenzelm@54681
   842
proof -
chaieb@29840
   843
  note ps = permutation_inverse_works[OF p]
chaieb@29840
   844
  note qs = permutation_inverse_works[OF q]
wenzelm@54681
   845
  have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p"
wenzelm@54681
   846
    by (simp add: o_assoc)
wenzelm@54681
   847
  also have "\<dots> = id"
wenzelm@54681
   848
    by (simp add: ps qs)
wenzelm@65342
   849
  finally have *: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" .
wenzelm@54681
   850
  have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q"
wenzelm@54681
   851
    by (simp add: o_assoc)
wenzelm@54681
   852
  also have "\<dots> = id"
wenzelm@54681
   853
    by (simp add: ps qs)
wenzelm@65342
   854
  finally have **: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
wenzelm@65342
   855
  show ?thesis
wenzelm@65342
   856
    by (rule inv_unique_comp[OF * **])
chaieb@29840
   857
qed
chaieb@29840
   858
wenzelm@54681
   859
wenzelm@65342
   860
subsection \<open>Relation to \<open>permutes\<close>\<close>
chaieb@29840
   861
chaieb@29840
   862
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
wenzelm@54681
   863
  unfolding permutation permutes_def bij_iff[symmetric]
wenzelm@54681
   864
  apply (rule iffI, clarify)
wenzelm@65342
   865
   apply (rule exI[where x="{x. p x \<noteq> x}"])
wenzelm@65342
   866
   apply simp
wenzelm@54681
   867
  apply clarsimp
wenzelm@54681
   868
  apply (rule_tac B="S" in finite_subset)
wenzelm@65342
   869
   apply auto
wenzelm@54681
   870
  done
chaieb@29840
   871
wenzelm@54681
   872
wenzelm@60500
   873
subsection \<open>Hence a sort of induction principle composing by swaps\<close>
chaieb@29840
   874
wenzelm@54681
   875
lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow>
wenzelm@65342
   876
  (\<And>a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p \<Longrightarrow> P (Fun.swap a b id \<circ> p)) \<Longrightarrow>
wenzelm@54681
   877
  (\<And>p. p permutes S \<Longrightarrow> P p)"
wenzelm@54681
   878
proof (induct S rule: finite_induct)
wenzelm@54681
   879
  case empty
wenzelm@54681
   880
  then show ?case by auto
huffman@30488
   881
next
chaieb@29840
   882
  case (insert x F p)
wenzelm@54681
   883
  let ?r = "Fun.swap x (p x) id \<circ> p"
wenzelm@54681
   884
  let ?q = "Fun.swap x (p x) id \<circ> ?r"
wenzelm@54681
   885
  have qp: "?q = p"
wenzelm@54681
   886
    by (simp add: o_assoc)
wenzelm@54681
   887
  from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r"
wenzelm@54681
   888
    by blast
huffman@30488
   889
  from permutes_in_image[OF insert.prems(3), of x]
wenzelm@54681
   890
  have pxF: "p x \<in> insert x F"
wenzelm@54681
   891
    by simp
wenzelm@54681
   892
  have xF: "x \<in> insert x F"
wenzelm@54681
   893
    by simp
chaieb@29840
   894
  have rp: "permutation ?r"
wenzelm@65342
   895
    unfolding permutation_permutes
wenzelm@65342
   896
    using insert.hyps(1) permutes_insert_lemma[OF insert.prems(3)]
wenzelm@54681
   897
    by blast
wenzelm@65342
   898
  from insert.prems(2)[OF xF pxF Pr Pr rp] qp show ?case
wenzelm@65342
   899
    by (simp only:)
chaieb@29840
   900
qed
chaieb@29840
   901
wenzelm@54681
   902
wenzelm@60500
   903
subsection \<open>Sign of a permutation as a real number\<close>
chaieb@29840
   904
chaieb@29840
   905
definition "sign p = (if evenperm p then (1::int) else -1)"
chaieb@29840
   906
wenzelm@54681
   907
lemma sign_nz: "sign p \<noteq> 0"
wenzelm@54681
   908
  by (simp add: sign_def)
wenzelm@54681
   909
wenzelm@54681
   910
lemma sign_id: "sign id = 1"
wenzelm@54681
   911
  by (simp add: sign_def)
wenzelm@54681
   912
wenzelm@54681
   913
lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p"
chaieb@29840
   914
  by (simp add: sign_def evenperm_inv)
wenzelm@54681
   915
wenzelm@54681
   916
lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q"
wenzelm@54681
   917
  by (simp add: sign_def evenperm_comp)
wenzelm@54681
   918
chaieb@29840
   919
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
chaieb@29840
   920
  by (simp add: sign_def evenperm_swap)
chaieb@29840
   921
wenzelm@54681
   922
lemma sign_idempotent: "sign p * sign p = 1"
wenzelm@54681
   923
  by (simp add: sign_def)
wenzelm@54681
   924
hoelzl@64284
   925
eberlm@63099
   926
subsection \<open>Permuting a list\<close>
eberlm@63099
   927
eberlm@63099
   928
text \<open>This function permutes a list by applying a permutation to the indices.\<close>
eberlm@63099
   929
wenzelm@65342
   930
definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list"
wenzelm@65342
   931
  where "permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"
eberlm@63099
   932
hoelzl@64284
   933
lemma permute_list_map:
eberlm@63099
   934
  assumes "f permutes {..<length xs}"
wenzelm@65342
   935
  shows "permute_list f (map g xs) = map g (permute_list f xs)"
eberlm@63099
   936
  using permutes_in_image[OF assms] by (auto simp: permute_list_def)
eberlm@63099
   937
eberlm@63099
   938
lemma permute_list_nth:
eberlm@63099
   939
  assumes "f permutes {..<length xs}" "i < length xs"
wenzelm@65342
   940
  shows "permute_list f xs ! i = xs ! f i"
hoelzl@64284
   941
  using permutes_in_image[OF assms(1)] assms(2)
eberlm@63099
   942
  by (simp add: permute_list_def)
eberlm@63099
   943
eberlm@63099
   944
lemma permute_list_Nil [simp]: "permute_list f [] = []"
eberlm@63099
   945
  by (simp add: permute_list_def)
eberlm@63099
   946
eberlm@63099
   947
lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"
eberlm@63099
   948
  by (simp add: permute_list_def)
eberlm@63099
   949
hoelzl@64284
   950
lemma permute_list_compose:
eberlm@63099
   951
  assumes "g permutes {..<length xs}"
wenzelm@65342
   952
  shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
eberlm@63099
   953
  using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)
eberlm@63099
   954
eberlm@63099
   955
lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs"
eberlm@63099
   956
  by (simp add: permute_list_def map_nth)
eberlm@63099
   957
eberlm@63099
   958
lemma permute_list_id [simp]: "permute_list id xs = xs"
eberlm@63099
   959
  by (simp add: id_def)
eberlm@63099
   960
eberlm@63099
   961
lemma mset_permute_list [simp]:
wenzelm@65342
   962
  fixes xs :: "'a list"
wenzelm@65342
   963
  assumes "f permutes {..<length xs}"
wenzelm@65342
   964
  shows "mset (permute_list f xs) = mset xs"
eberlm@63099
   965
proof (rule multiset_eqI)
eberlm@63099
   966
  fix y :: 'a
eberlm@63099
   967
  from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
eberlm@63099
   968
    using permutes_in_image[OF assms] by auto
wenzelm@65342
   969
  have "count (mset (permute_list f xs)) y = card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})"
bulwahn@64543
   970
    by (simp add: permute_list_def count_image_mset atLeast0LessThan)
eberlm@63099
   971
  also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}"
eberlm@63099
   972
    by auto
eberlm@63099
   973
  also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}"
eberlm@63099
   974
    by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
wenzelm@65342
   975
  also have "\<dots> = count (mset xs) y"
wenzelm@65342
   976
    by (simp add: count_mset length_filter_conv_card)
wenzelm@65342
   977
  finally show "count (mset (permute_list f xs)) y = count (mset xs) y"
wenzelm@65342
   978
    by simp
eberlm@63099
   979
qed
eberlm@63099
   980
hoelzl@64284
   981
lemma set_permute_list [simp]:
eberlm@63099
   982
  assumes "f permutes {..<length xs}"
wenzelm@65342
   983
  shows "set (permute_list f xs) = set xs"
eberlm@63099
   984
  by (rule mset_eq_setD[OF mset_permute_list]) fact
eberlm@63099
   985
eberlm@63099
   986
lemma distinct_permute_list [simp]:
eberlm@63099
   987
  assumes "f permutes {..<length xs}"
wenzelm@65342
   988
  shows "distinct (permute_list f xs) = distinct xs"
eberlm@63099
   989
  by (simp add: distinct_count_atmost_1 assms)
eberlm@63099
   990
hoelzl@64284
   991
lemma permute_list_zip:
eberlm@63099
   992
  assumes "f permutes A" "A = {..<length xs}"
eberlm@63099
   993
  assumes [simp]: "length xs = length ys"
wenzelm@65342
   994
  shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
eberlm@63099
   995
proof -
wenzelm@65342
   996
  from permutes_in_image[OF assms(1)] assms(2) have *: "f i < length ys \<longleftrightarrow> i < length ys" for i
wenzelm@65342
   997
    by simp
eberlm@63099
   998
  have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]"
eberlm@63099
   999
    by (simp_all add: permute_list_def zip_map_map)
eberlm@63099
  1000
  also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
wenzelm@65342
  1001
    by (intro nth_equalityI) (simp_all add: *)
eberlm@63099
  1002
  also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
eberlm@63099
  1003
    by (simp_all add: permute_list_def zip_map_map)
eberlm@63099
  1004
  finally show ?thesis .
eberlm@63099
  1005
qed
eberlm@63099
  1006
hoelzl@64284
  1007
lemma map_of_permute:
eberlm@63099
  1008
  assumes "\<sigma> permutes fst ` set xs"
wenzelm@65342
  1009
  shows "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)"
wenzelm@65342
  1010
    (is "_ = map_of (map ?f _)")
eberlm@63099
  1011
proof
wenzelm@65342
  1012
  from assms have "inj \<sigma>" "surj \<sigma>"
wenzelm@65342
  1013
    by (simp_all add: permutes_inj permutes_surj)
wenzelm@65342
  1014
  then show "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x" for x
wenzelm@65342
  1015
    by (induct xs) (auto simp: inv_f_f surj_f_inv_f)
eberlm@63099
  1016
qed
eberlm@63099
  1017
wenzelm@54681
  1018
wenzelm@60500
  1019
subsection \<open>More lemmas about permutations\<close>
chaieb@29840
  1020
wenzelm@65342
  1021
text \<open>The following few lemmas were contributed by Lukas Bulwahn.\<close>
eberlm@63921
  1022
eberlm@63921
  1023
lemma count_image_mset_eq_card_vimage:
eberlm@63921
  1024
  assumes "finite A"
eberlm@63921
  1025
  shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
eberlm@63921
  1026
  using assms
eberlm@63921
  1027
proof (induct A)
eberlm@63921
  1028
  case empty
eberlm@63921
  1029
  show ?case by simp
eberlm@63921
  1030
next
eberlm@63921
  1031
  case (insert x F)
eberlm@63921
  1032
  show ?case
wenzelm@65342
  1033
  proof (cases "f x = b")
wenzelm@65342
  1034
    case True
wenzelm@65342
  1035
    with insert.hyps
wenzelm@65342
  1036
    have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
wenzelm@65342
  1037
      by auto
wenzelm@65342
  1038
    also from insert.hyps(1,2) have "\<dots> = card (insert x {a \<in> F. f a = f x})"
wenzelm@65342
  1039
      by simp
wenzelm@65342
  1040
    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}"
wenzelm@65342
  1041
      by (auto intro: arg_cong[where f="card"])
wenzelm@65342
  1042
    finally show ?thesis
wenzelm@65342
  1043
      using insert by auto
eberlm@63921
  1044
  next
wenzelm@65342
  1045
    case False
wenzelm@65342
  1046
    then have "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}"
wenzelm@65342
  1047
      by auto
wenzelm@65342
  1048
    with insert False show ?thesis
wenzelm@65342
  1049
      by simp
eberlm@63921
  1050
  qed
eberlm@63921
  1051
qed
hoelzl@64284
  1052
wenzelm@67408
  1053
\<comment> \<open>Prove \<open>image_mset_eq_implies_permutes\<close> ...\<close>
eberlm@63921
  1054
lemma image_mset_eq_implies_permutes:
eberlm@63921
  1055
  fixes f :: "'a \<Rightarrow> 'b"
eberlm@63921
  1056
  assumes "finite A"
wenzelm@65342
  1057
    and mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)"
eberlm@63921
  1058
  obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
eberlm@63099
  1059
proof -
eberlm@63921
  1060
  from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto
eberlm@63921
  1061
  have "f ` A = f' ` A"
eberlm@63921
  1062
  proof -
wenzelm@65342
  1063
    from \<open>finite A\<close> have "f ` A = f ` (set_mset (mset_set A))"
wenzelm@65342
  1064
      by simp
wenzelm@65342
  1065
    also have "\<dots> = f' ` set_mset (mset_set A)"
eberlm@63921
  1066
      by (metis mset_eq multiset.set_map)
wenzelm@65342
  1067
    also from \<open>finite A\<close> have "\<dots> = f' ` A"
wenzelm@65342
  1068
      by simp
eberlm@63921
  1069
    finally show ?thesis .
eberlm@63921
  1070
  qed
eberlm@63921
  1071
  have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}"
eberlm@63099
  1072
  proof
eberlm@63921
  1073
    fix b
wenzelm@65342
  1074
    from mset_eq have "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b"
wenzelm@65342
  1075
      by simp
wenzelm@65342
  1076
    with \<open>finite A\<close> have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
eberlm@63921
  1077
      by (simp add: count_image_mset_eq_card_vimage)
wenzelm@65342
  1078
    then show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
eberlm@63099
  1079
      by (intro finite_same_card_bij) simp_all
eberlm@63099
  1080
  qed
wenzelm@65342
  1081
  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}"
eberlm@63099
  1082
    by (rule bchoice)
wenzelm@65342
  1083
  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}" ..
eberlm@63921
  1084
  define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
eberlm@63921
  1085
  have "p' permutes A"
eberlm@63921
  1086
  proof (rule bij_imp_permutes)
eberlm@63921
  1087
    have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)"
wenzelm@65342
  1088
      by (auto simp: disjoint_family_on_def)
wenzelm@65342
  1089
    moreover
wenzelm@65342
  1090
    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
wenzelm@65342
  1091
      using p that by (subst bij_betw_cong[where g="p b"]) auto
wenzelm@65342
  1092
    ultimately
wenzelm@65342
  1093
    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})"
eberlm@63921
  1094
      by (rule bij_betw_UNION_disjoint)
wenzelm@65342
  1095
    moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A"
wenzelm@65342
  1096
      by auto
wenzelm@65342
  1097
    moreover from \<open>f ` A = f' ` A\<close> have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A"
wenzelm@65342
  1098
      by auto
eberlm@63921
  1099
    ultimately show "bij_betw p' A A"
eberlm@63921
  1100
      unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto
eberlm@63921
  1101
  next
wenzelm@65342
  1102
    show "\<And>x. x \<notin> A \<Longrightarrow> p' x = x"
wenzelm@65342
  1103
      by (simp add: p'_def)
eberlm@63099
  1104
  qed
eberlm@63921
  1105
  moreover from p have "\<forall>x\<in>A. f x = f' (p' x)"
eberlm@63921
  1106
    unfolding p'_def using bij_betwE by fastforce
wenzelm@65342
  1107
  ultimately show ?thesis ..
eberlm@63921
  1108
qed
eberlm@63099
  1109
wenzelm@65342
  1110
lemma mset_set_upto_eq_mset_upto: "mset_set {..<n} = mset [0..<n]"
wenzelm@65342
  1111
  by (induct n) (auto simp: add.commute lessThan_Suc)
eberlm@63099
  1112
wenzelm@67408
  1113
\<comment> \<open>... and derive the existing property:\<close>
eberlm@63921
  1114
lemma mset_eq_permutation:
wenzelm@65342
  1115
  fixes xs ys :: "'a list"
wenzelm@65342
  1116
  assumes mset_eq: "mset xs = mset ys"
eberlm@63921
  1117
  obtains p where "p permutes {..<length ys}" "permute_list p ys = xs"
eberlm@63921
  1118
proof -
eberlm@63921
  1119
  from mset_eq have length_eq: "length xs = length ys"
wenzelm@65342
  1120
    by (rule mset_eq_length)
eberlm@63921
  1121
  have "mset_set {..<length ys} = mset [0..<length ys]"
wenzelm@65342
  1122
    by (rule mset_set_upto_eq_mset_upto)
wenzelm@65342
  1123
  with mset_eq length_eq have "image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) =
wenzelm@65342
  1124
    image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})"
eberlm@63921
  1125
    by (metis map_nth mset_map)
eberlm@63921
  1126
  from image_mset_eq_implies_permutes[OF _ this]
wenzelm@65342
  1127
  obtain p where p: "p permutes {..<length ys}" and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)"
wenzelm@65342
  1128
    by auto
wenzelm@65342
  1129
  with length_eq have "permute_list p ys = xs"
wenzelm@65342
  1130
    by (auto intro!: nth_equalityI simp: permute_list_nth)
wenzelm@65342
  1131
  with p show thesis ..
eberlm@63099
  1132
qed
eberlm@63099
  1133
chaieb@29840
  1134
lemma permutes_natset_le:
wenzelm@54681
  1135
  fixes S :: "'a::wellorder set"
wenzelm@65342
  1136
  assumes "p permutes S"
wenzelm@65342
  1137
    and "\<forall>i \<in> S. p i \<le> i"
wenzelm@54681
  1138
  shows "p = id"
wenzelm@54681
  1139
proof -
wenzelm@65342
  1140
  have "p n = n" for n
wenzelm@65342
  1141
    using assms
wenzelm@65342
  1142
  proof (induct n arbitrary: S rule: less_induct)
wenzelm@65342
  1143
    case (less n)
wenzelm@65342
  1144
    show ?case
wenzelm@65342
  1145
    proof (cases "n \<in> S")
wenzelm@65342
  1146
      case False
wenzelm@65342
  1147
      with less(2) show ?thesis
wenzelm@65342
  1148
        unfolding permutes_def by metis
wenzelm@65342
  1149
    next
wenzelm@65342
  1150
      case True
wenzelm@65342
  1151
      with less(3) have "p n < n \<or> p n = n"
wenzelm@65342
  1152
        by auto
wenzelm@65342
  1153
      then show ?thesis
wenzelm@65342
  1154
      proof
wenzelm@65342
  1155
        assume "p n < n"
wenzelm@65342
  1156
        with less have "p (p n) = p n"
wenzelm@65342
  1157
          by metis
wenzelm@65342
  1158
        with permutes_inj[OF less(2)] have "p n = n"
wenzelm@65342
  1159
          unfolding inj_def by blast
wenzelm@65342
  1160
        with \<open>p n < n\<close> have False
wenzelm@65342
  1161
          by simp
wenzelm@65342
  1162
        then show ?thesis ..
wenzelm@65342
  1163
      qed
wenzelm@54681
  1164
    qed
wenzelm@65342
  1165
  qed
wenzelm@65342
  1166
  then show ?thesis by (auto simp: fun_eq_iff)
chaieb@29840
  1167
qed
chaieb@29840
  1168
chaieb@29840
  1169
lemma permutes_natset_ge:
wenzelm@54681
  1170
  fixes S :: "'a::wellorder set"
wenzelm@54681
  1171
  assumes p: "p permutes S"
wenzelm@54681
  1172
    and le: "\<forall>i \<in> S. p i \<ge> i"
wenzelm@54681
  1173
  shows "p = id"
wenzelm@54681
  1174
proof -
wenzelm@65342
  1175
  have "i \<ge> inv p i" if "i \<in> S" for i
wenzelm@65342
  1176
  proof -
wenzelm@65342
  1177
    from that permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S"
wenzelm@54681
  1178
      by simp
wenzelm@54681
  1179
    with le have "p (inv p i) \<ge> inv p i"
wenzelm@54681
  1180
      by blast
wenzelm@65342
  1181
    with permutes_inverses[OF p] show ?thesis
wenzelm@54681
  1182
      by simp
wenzelm@65342
  1183
  qed
wenzelm@65342
  1184
  then have "\<forall>i\<in>S. inv p i \<le> i"
wenzelm@54681
  1185
    by blast
wenzelm@65342
  1186
  from permutes_natset_le[OF permutes_inv[OF p] this] have "inv p = inv id"
wenzelm@54681
  1187
    by simp
huffman@30488
  1188
  then show ?thesis
chaieb@29840
  1189
    apply (subst permutes_inv_inv[OF p, symmetric])
chaieb@29840
  1190
    apply (rule inv_unique_comp)
wenzelm@65342
  1191
     apply simp_all
chaieb@29840
  1192
    done
chaieb@29840
  1193
qed
chaieb@29840
  1194
chaieb@29840
  1195
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
wenzelm@54681
  1196
  apply (rule set_eqI)
wenzelm@54681
  1197
  apply auto
wenzelm@54681
  1198
  using permutes_inv_inv permutes_inv
wenzelm@65342
  1199
   apply auto
chaieb@29840
  1200
  apply (rule_tac x="inv x" in exI)
chaieb@29840
  1201
  apply auto
chaieb@29840
  1202
  done
chaieb@29840
  1203
huffman@30488
  1204
lemma image_compose_permutations_left:
wenzelm@65342
  1205
  assumes "q permutes S"
wenzelm@65342
  1206
  shows "{q \<circ> p |p. p permutes S} = {p. p permutes S}"
wenzelm@54681
  1207
  apply (rule set_eqI)
wenzelm@54681
  1208
  apply auto
wenzelm@65342
  1209
   apply (rule permutes_compose)
wenzelm@65342
  1210
  using assms
wenzelm@65342
  1211
    apply auto
wenzelm@54681
  1212
  apply (rule_tac x = "inv q \<circ> x" in exI)
wenzelm@54681
  1213
  apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
wenzelm@54681
  1214
  done
chaieb@29840
  1215
chaieb@29840
  1216
lemma image_compose_permutations_right:
wenzelm@65342
  1217
  assumes "q permutes S"
wenzelm@54681
  1218
  shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}"
wenzelm@54681
  1219
  apply (rule set_eqI)
wenzelm@54681
  1220
  apply auto
wenzelm@65342
  1221
   apply (rule permutes_compose)
wenzelm@65342
  1222
  using assms
wenzelm@65342
  1223
    apply auto
wenzelm@54681
  1224
  apply (rule_tac x = "x \<circ> inv q" in exI)
wenzelm@54681
  1225
  apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc)
wenzelm@54681
  1226
  done
chaieb@29840
  1227
wenzelm@54681
  1228
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n"
wenzelm@54681
  1229
  by (simp add: permutes_def) metis
chaieb@29840
  1230
wenzelm@65342
  1231
lemma sum_permutations_inverse: "sum f {p. p permutes S} = sum (\<lambda>p. f(inv p)) {p. p permutes S}"
wenzelm@54681
  1232
  (is "?lhs = ?rhs")
wenzelm@54681
  1233
proof -
huffman@30036
  1234
  let ?S = "{p . p permutes S}"
wenzelm@65342
  1235
  have *: "inj_on inv ?S"
wenzelm@54681
  1236
  proof (auto simp add: inj_on_def)
wenzelm@54681
  1237
    fix q r
wenzelm@54681
  1238
    assume q: "q permutes S"
wenzelm@54681
  1239
      and r: "r permutes S"
wenzelm@54681
  1240
      and qr: "inv q = inv r"
wenzelm@54681
  1241
    then have "inv (inv q) = inv (inv r)"
wenzelm@54681
  1242
      by simp
wenzelm@54681
  1243
    with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r"
wenzelm@54681
  1244
      by metis
wenzelm@54681
  1245
  qed
wenzelm@65342
  1246
  have **: "inv ` ?S = ?S"
wenzelm@54681
  1247
    using image_inverse_permutations by blast
wenzelm@65342
  1248
  have ***: "?rhs = sum (f \<circ> inv) ?S"
wenzelm@54681
  1249
    by (simp add: o_def)
wenzelm@65342
  1250
  from sum.reindex[OF *, of f] show ?thesis
wenzelm@65342
  1251
    by (simp only: ** ***)
chaieb@29840
  1252
qed
chaieb@29840
  1253
chaieb@29840
  1254
lemma setum_permutations_compose_left:
huffman@30036
  1255
  assumes q: "q permutes S"
nipkow@64267
  1256
  shows "sum f {p. p permutes S} = sum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
wenzelm@54681
  1257
  (is "?lhs = ?rhs")
wenzelm@54681
  1258
proof -
huffman@30036
  1259
  let ?S = "{p. p permutes S}"
nipkow@67399
  1260
  have *: "?rhs = sum (f \<circ> ((\<circ>) q)) ?S"
wenzelm@54681
  1261
    by (simp add: o_def)
nipkow@67399
  1262
  have **: "inj_on ((\<circ>) q) ?S"
wenzelm@54681
  1263
  proof (auto simp add: inj_on_def)
chaieb@29840
  1264
    fix p r
wenzelm@54681
  1265
    assume "p permutes S"
wenzelm@54681
  1266
      and r: "r permutes S"
wenzelm@54681
  1267
      and rp: "q \<circ> p = q \<circ> r"
wenzelm@54681
  1268
    then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r"
wenzelm@54681
  1269
      by (simp add: comp_assoc)
wenzelm@54681
  1270
    with permutes_inj[OF q, unfolded inj_iff] show "p = r"
wenzelm@54681
  1271
      by simp
chaieb@29840
  1272
  qed
nipkow@67399
  1273
  have "((\<circ>) q) ` ?S = ?S"
wenzelm@54681
  1274
    using image_compose_permutations_left[OF q] by auto
wenzelm@65342
  1275
  with * sum.reindex[OF **, of f] show ?thesis
wenzelm@65342
  1276
    by (simp only:)
chaieb@29840
  1277
qed
chaieb@29840
  1278
chaieb@29840
  1279
lemma sum_permutations_compose_right:
huffman@30036
  1280
  assumes q: "q permutes S"
nipkow@64267
  1281
  shows "sum f {p. p permutes S} = sum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
wenzelm@54681
  1282
  (is "?lhs = ?rhs")
wenzelm@54681
  1283
proof -
huffman@30036
  1284
  let ?S = "{p. p permutes S}"
wenzelm@65342
  1285
  have *: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
wenzelm@54681
  1286
    by (simp add: o_def)
wenzelm@65342
  1287
  have **: "inj_on (\<lambda>p. p \<circ> q) ?S"
wenzelm@54681
  1288
  proof (auto simp add: inj_on_def)
chaieb@29840
  1289
    fix p r
wenzelm@54681
  1290
    assume "p permutes S"
wenzelm@54681
  1291
      and r: "r permutes S"
wenzelm@54681
  1292
      and rp: "p \<circ> q = r \<circ> q"
wenzelm@54681
  1293
    then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)"
wenzelm@54681
  1294
      by (simp add: o_assoc)
wenzelm@54681
  1295
    with permutes_surj[OF q, unfolded surj_iff] show "p = r"
wenzelm@54681
  1296
      by simp
chaieb@29840
  1297
  qed
wenzelm@65342
  1298
  from image_compose_permutations_right[OF q] have "(\<lambda>p. p \<circ> q) ` ?S = ?S"
wenzelm@65342
  1299
    by auto
wenzelm@65342
  1300
  with * sum.reindex[OF **, of f] show ?thesis
wenzelm@65342
  1301
    by (simp only:)
chaieb@29840
  1302
qed
chaieb@29840
  1303
wenzelm@54681
  1304
wenzelm@60500
  1305
subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close>
chaieb@29840
  1306
nipkow@64267
  1307
lemma sum_over_permutations_insert:
wenzelm@54681
  1308
  assumes fS: "finite S"
wenzelm@54681
  1309
    and aS: "a \<notin> S"
nipkow@64267
  1310
  shows "sum f {p. p permutes (insert a S)} =
nipkow@64267
  1311
    sum (\<lambda>b. sum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)"
wenzelm@54681
  1312
proof -
wenzelm@65342
  1313
  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)"
nipkow@39302
  1314
    by (simp add: fun_eq_iff)
wenzelm@65342
  1315
  have **: "\<And>P Q. {(a, b). a \<in> P \<and> b \<in> Q} = P \<times> Q"
wenzelm@54681
  1316
    by blast
huffman@30488
  1317
  show ?thesis
wenzelm@65342
  1318
    unfolding * ** sum.cartesian_product permutes_insert
nipkow@64267
  1319
  proof (rule sum.reindex)
chaieb@29840
  1320
    let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
chaieb@29840
  1321
    let ?P = "{p. p permutes S}"
wenzelm@54681
  1322
    {
wenzelm@54681
  1323
      fix b c p q
wenzelm@54681
  1324
      assume b: "b \<in> insert a S"
wenzelm@54681
  1325
      assume c: "c \<in> insert a S"
wenzelm@54681
  1326
      assume p: "p permutes S"
wenzelm@54681
  1327
      assume q: "q permutes S"
wenzelm@54681
  1328
      assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> q"
chaieb@29840
  1329
      from p q aS have pa: "p a = a" and qa: "q a = a"
wenzelm@32960
  1330
        unfolding permutes_def by metis+
wenzelm@54681
  1331
      from eq have "(Fun.swap a b id \<circ> p) a  = (Fun.swap a c id \<circ> q) a"
wenzelm@54681
  1332
        by simp
wenzelm@54681
  1333
      then have bc: "b = c"
haftmann@56545
  1334
        by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def
nipkow@62390
  1335
            cong del: if_weak_cong split: if_split_asm)
wenzelm@54681
  1336
      from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) =
wenzelm@54681
  1337
        (\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp
wenzelm@54681
  1338
      then have "p = q"
wenzelm@65342
  1339
        unfolding o_assoc swap_id_idempotent by simp
wenzelm@54681
  1340
      with bc have "b = c \<and> p = q"
wenzelm@54681
  1341
        by blast
chaieb@29840
  1342
    }
huffman@30488
  1343
    then show "inj_on ?f (insert a S \<times> ?P)"
wenzelm@54681
  1344
      unfolding inj_on_def by clarify metis
chaieb@29840
  1345
  qed
chaieb@29840
  1346
qed
chaieb@29840
  1347
eberlm@63099
  1348
eberlm@63099
  1349
subsection \<open>Constructing permutations from association lists\<close>
eberlm@63099
  1350
wenzelm@65342
  1351
definition list_permutes :: "('a \<times> 'a) list \<Rightarrow> 'a set \<Rightarrow> bool"
wenzelm@65342
  1352
  where "list_permutes xs A \<longleftrightarrow>
wenzelm@65342
  1353
    set (map fst xs) \<subseteq> A \<and>
wenzelm@65342
  1354
    set (map snd xs) = set (map fst xs) \<and>
wenzelm@65342
  1355
    distinct (map fst xs) \<and>
wenzelm@65342
  1356
    distinct (map snd xs)"
eberlm@63099
  1357
eberlm@63099
  1358
lemma list_permutesI [simp]:
eberlm@63099
  1359
  assumes "set (map fst xs) \<subseteq> A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)"
wenzelm@65342
  1360
  shows "list_permutes xs A"
eberlm@63099
  1361
proof -
eberlm@63099
  1362
  from assms(2,3) have "distinct (map snd xs)"
eberlm@63099
  1363
    by (intro card_distinct) (simp_all add: distinct_card del: set_map)
wenzelm@65342
  1364
  with assms show ?thesis
wenzelm@65342
  1365
    by (simp add: list_permutes_def)
eberlm@63099
  1366
qed
eberlm@63099
  1367
wenzelm@65342
  1368
definition permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@65342
  1369
  where "permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)"
eberlm@63099
  1370
eberlm@63099
  1371
lemma permutation_of_list_Cons:
wenzelm@65342
  1372
  "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"
eberlm@63099
  1373
  by (simp add: permutation_of_list_def)
eberlm@63099
  1374
wenzelm@65342
  1375
fun inverse_permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@65342
  1376
  where
wenzelm@65342
  1377
    "inverse_permutation_of_list [] x = x"
wenzelm@65342
  1378
  | "inverse_permutation_of_list ((y, x') # xs) x =
wenzelm@65342
  1379
      (if x = x' then y else inverse_permutation_of_list xs x)"
eberlm@63099
  1380
eberlm@63099
  1381
declare inverse_permutation_of_list.simps [simp del]
eberlm@63099
  1382
eberlm@63099
  1383
lemma inj_on_map_of:
eberlm@63099
  1384
  assumes "distinct (map snd xs)"
wenzelm@65342
  1385
  shows "inj_on (map_of xs) (set (map fst xs))"
eberlm@63099
  1386
proof (rule inj_onI)
wenzelm@65342
  1387
  fix x y
wenzelm@65342
  1388
  assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)"
eberlm@63099
  1389
  assume eq: "map_of xs x = map_of xs y"
wenzelm@65342
  1390
  from xy obtain x' y' where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
wenzelm@65342
  1391
    by (cases "map_of xs x"; cases "map_of xs y") (simp_all add: map_of_eq_None_iff)
wenzelm@65342
  1392
  moreover from x'y' have *: "(x, x') \<in> set xs" "(y, y') \<in> set xs"
eberlm@63099
  1393
    by (force dest: map_of_SomeD)+
wenzelm@65342
  1394
  moreover from * eq x'y' have "x' = y'"
wenzelm@65342
  1395
    by simp
wenzelm@65342
  1396
  ultimately show "x = y"
wenzelm@65342
  1397
    using assms by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"])
eberlm@63099
  1398
qed
eberlm@63099
  1399
eberlm@63099
  1400
lemma inj_on_the: "None \<notin> A \<Longrightarrow> inj_on the A"
eberlm@63099
  1401
  by (auto simp: inj_on_def option.the_def split: option.splits)
eberlm@63099
  1402
eberlm@63099
  1403
lemma inj_on_map_of':
eberlm@63099
  1404
  assumes "distinct (map snd xs)"
wenzelm@65342
  1405
  shows "inj_on (the \<circ> map_of xs) (set (map fst xs))"
eberlm@63099
  1406
  by (intro comp_inj_on inj_on_map_of assms inj_on_the)
wenzelm@65342
  1407
    (force simp: eq_commute[of None] map_of_eq_None_iff)
eberlm@63099
  1408
eberlm@63099
  1409
lemma image_map_of:
eberlm@63099
  1410
  assumes "distinct (map fst xs)"
wenzelm@65342
  1411
  shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)"
eberlm@63099
  1412
  using assms by (auto simp: rev_image_eqI)
eberlm@63099
  1413
eberlm@63099
  1414
lemma the_Some_image [simp]: "the ` Some ` A = A"
eberlm@63099
  1415
  by (subst image_image) simp
eberlm@63099
  1416
eberlm@63099
  1417
lemma image_map_of':
eberlm@63099
  1418
  assumes "distinct (map fst xs)"
wenzelm@65342
  1419
  shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)"
eberlm@63099
  1420
  by (simp only: image_comp [symmetric] image_map_of assms the_Some_image)
eberlm@63099
  1421
eberlm@63099
  1422
lemma permutation_of_list_permutes [simp]:
eberlm@63099
  1423
  assumes "list_permutes xs A"
wenzelm@65342
  1424
  shows "permutation_of_list xs permutes A"
wenzelm@65342
  1425
    (is "?f permutes _")
eberlm@63099
  1426
proof (rule permutes_subset[OF bij_imp_permutes])
eberlm@63099
  1427
  from assms show "set (map fst xs) \<subseteq> A"
eberlm@63099
  1428
    by (simp add: list_permutes_def)
eberlm@63099
  1429
  from assms have "inj_on (the \<circ> map_of xs) (set (map fst xs))" (is ?P)
eberlm@63099
  1430
    by (intro inj_on_map_of') (simp_all add: list_permutes_def)
eberlm@63099
  1431
  also have "?P \<longleftrightarrow> inj_on ?f (set (map fst xs))"
eberlm@63099
  1432
    by (intro inj_on_cong)
wenzelm@65342
  1433
      (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
eberlm@63099
  1434
  finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))"
eberlm@63099
  1435
    by (rule inj_on_imp_bij_betw)
eberlm@63099
  1436
  also from assms have "?f ` set (map fst xs) = (the \<circ> map_of xs) ` set (map fst xs)"
eberlm@63099
  1437
    by (intro image_cong refl)
wenzelm@65342
  1438
      (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
hoelzl@64284
  1439
  also from assms have "\<dots> = set (map fst xs)"
eberlm@63099
  1440
    by (subst image_map_of') (simp_all add: list_permutes_def)
eberlm@63099
  1441
  finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" .
eberlm@63099
  1442
qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+
eberlm@63099
  1443
eberlm@63099
  1444
lemma eval_permutation_of_list [simp]:
eberlm@63099
  1445
  "permutation_of_list [] x = x"
eberlm@63099
  1446
  "x = x' \<Longrightarrow> permutation_of_list ((x',y)#xs) x = y"
eberlm@63099
  1447
  "x \<noteq> x' \<Longrightarrow> permutation_of_list ((x',y')#xs) x = permutation_of_list xs x"
eberlm@63099
  1448
  by (simp_all add: permutation_of_list_def)
eberlm@63099
  1449
eberlm@63099
  1450
lemma eval_inverse_permutation_of_list [simp]:
eberlm@63099
  1451
  "inverse_permutation_of_list [] x = x"
eberlm@63099
  1452
  "x = x' \<Longrightarrow> inverse_permutation_of_list ((y,x')#xs) x = y"
eberlm@63099
  1453
  "x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x"
eberlm@63099
  1454
  by (simp_all add: inverse_permutation_of_list.simps)
eberlm@63099
  1455
wenzelm@65342
  1456
lemma permutation_of_list_id: "x \<notin> set (map fst xs) \<Longrightarrow> permutation_of_list xs x = x"
wenzelm@65342
  1457
  by (induct xs) (auto simp: permutation_of_list_Cons)
eberlm@63099
  1458
eberlm@63099
  1459
lemma permutation_of_list_unique':
wenzelm@65342
  1460
  "distinct (map fst xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
wenzelm@65342
  1461
  by (induct xs) (force simp: permutation_of_list_Cons)+
eberlm@63099
  1462
eberlm@63099
  1463
lemma permutation_of_list_unique:
wenzelm@65342
  1464
  "list_permutes xs A \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
wenzelm@65342
  1465
  by (intro permutation_of_list_unique') (simp_all add: list_permutes_def)
eberlm@63099
  1466
eberlm@63099
  1467
lemma inverse_permutation_of_list_id:
wenzelm@65342
  1468
  "x \<notin> set (map snd xs) \<Longrightarrow> inverse_permutation_of_list xs x = x"
wenzelm@65342
  1469
  by (induct xs) auto
eberlm@63099
  1470
eberlm@63099
  1471
lemma inverse_permutation_of_list_unique':
wenzelm@65342
  1472
  "distinct (map snd xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
wenzelm@65342
  1473
  by (induct xs) (force simp: inverse_permutation_of_list.simps)+
eberlm@63099
  1474
eberlm@63099
  1475
lemma inverse_permutation_of_list_unique:
wenzelm@65342
  1476
  "list_permutes xs A \<Longrightarrow> (x,y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
wenzelm@65342
  1477
  by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def)
eberlm@63099
  1478
eberlm@63099
  1479
lemma inverse_permutation_of_list_correct:
wenzelm@65342
  1480
  fixes A :: "'a set"
wenzelm@65342
  1481
  assumes "list_permutes xs A"
wenzelm@65342
  1482
  shows "inverse_permutation_of_list xs = inv (permutation_of_list xs)"
eberlm@63099
  1483
proof (rule ext, rule sym, subst permutes_inv_eq)
wenzelm@65342
  1484
  from assms show "permutation_of_list xs permutes A"
wenzelm@65342
  1485
    by simp
wenzelm@65342
  1486
  show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" for x
eberlm@63099
  1487
  proof (cases "x \<in> set (map snd xs)")
eberlm@63099
  1488
    case True
wenzelm@65342
  1489
    then obtain y where "(y, x) \<in> set xs" by auto
eberlm@63099
  1490
    with assms show ?thesis
eberlm@63099
  1491
      by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique)
wenzelm@65342
  1492
  next
wenzelm@65342
  1493
    case False
wenzelm@65342
  1494
    with assms show ?thesis
wenzelm@65342
  1495
      by (auto simp: list_permutes_def inverse_permutation_of_list_id permutation_of_list_id)
wenzelm@65342
  1496
  qed
eberlm@63099
  1497
qed
eberlm@63099
  1498
chaieb@29840
  1499
end