(*  Title:      HOL/Library/Permutation.thy

     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker

 *)

 header {* Permutations *}

 theory Permutation

 imports Multiset

 begin

 inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)

 where

   Nil [intro!]: "[] <~~> []"

 | swap [intro!]: "y # x # l <~~> x # y # l"

 | Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"

 | trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"

 lemma perm_refl [iff]: "l <~~> l"

   by (induct l) auto

 subsection {* Some examples of rule induction on permutations *}

 lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"

   by (induct xs == "[]::'a list" ys pred: perm) simp_all

 text {*

   \medskip This more general theorem is easier to understand!

   *}

 lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"

   by (induct pred: perm) simp_all

 lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"

   by (drule perm_length) auto

 lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"

   by (induct pred: perm) auto

 subsection {* Ways of making new permutations *}

 text {*

   We can insert the head anywhere in the list.

 *}

 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"

   by (induct xs) auto

 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"

   apply (induct xs)

     apply simp_all

   apply (blast intro: perm_append_Cons)

   done

 lemma perm_append_single: "a # xs <~~> xs @ [a]"

   by (rule perm.trans [OF _ perm_append_swap]) simp

 lemma perm_rev: "rev xs <~~> xs"

   apply (induct xs)

    apply simp_all

   apply (blast intro!: perm_append_single intro: perm_sym)

   done

 lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"

   by (induct l) auto

 lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"

   by (blast intro!: perm_append_swap perm_append1)

 subsection {* Further results *}

 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"

   by (blast intro: perm_empty_imp)

 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"

   apply auto

   apply (erule perm_sym [THEN perm_empty_imp])

   done

 lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"

   by (induct pred: perm) auto

 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"

   by (blast intro: perm_sing_imp)

 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"

   by (blast dest: perm_sym)

 subsection {* Removing elements *}

 lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"

   by (induct ys) auto

 text {* \medskip Congruence rule *}

 lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"

   by (induct pred: perm) auto

 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"

   by auto

 lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"

   by (drule_tac z = z in perm_remove_perm) auto

 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"

   by (blast intro: cons_perm_imp_perm)

 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"

   by (induct zs arbitrary: xs ys rule: rev_induct) auto

 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"

   by (blast intro: append_perm_imp_perm perm_append1)

 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"

   apply (safe intro!: perm_append2)

   apply (rule append_perm_imp_perm)

   apply (rule perm_append_swap [THEN perm.trans])

     -- {* the previous step helps this @{text blast} call succeed quickly *}

   apply (blast intro: perm_append_swap)

   done

 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "

   apply (rule iffI)

   apply (erule_tac [2] perm.induct, simp_all add: union_ac)

   apply (erule rev_mp, rule_tac x=ys in spec)

   apply (induct_tac xs, auto)

   apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)

   apply (subgoal_tac "a \<in> set x")

   apply (drule_tac z = a in perm.Cons)

   apply (erule perm.trans, rule perm_sym, erule perm_remove)

   apply (drule_tac f=set_of in arg_cong, simp)

   done

 lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"

   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)

   apply (insert surj_multiset_of, drule surjD)

   apply (blast intro: sym)+

   done

 lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"

   by (metis multiset_of_eq_perm multiset_of_eq_setD)

 lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"

   apply (induct pred: perm)

      apply simp_all

    apply fastforce

   apply (metis perm_set_eq)

   done

 lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"

   apply (induct xs arbitrary: ys rule: length_induct)

   apply (case_tac "remdups xs")

    apply simp_all

   apply (subgoal_tac "a \<in> set (remdups ys)")

    prefer 2 apply (metis set.simps(2) insert_iff set_remdups)

   apply (drule split_list) apply(elim exE conjE)

   apply (drule_tac x=list in spec) apply(erule impE) prefer 2

    apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2

     apply simp

     apply (subgoal_tac "a # list <~~> a # ysa @ zs")

      apply (metis Cons_eq_appendI perm_append_Cons trans)

     apply (metis Cons Cons_eq_appendI distinct.simps(2)

       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)

    apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")

     apply (fastforce simp add: insert_ident)

    apply (metis distinct_remdups set_remdups)

    apply (subgoal_tac "length (remdups xs) < Suc (length xs)")

    apply simp

    apply (subgoal_tac "length (remdups xs) \<le> length xs")

    apply simp

    apply (rule length_remdups_leq)

   done

 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"

   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)

 lemma permutation_Ex_bij:

   assumes "xs <~~> ys"

   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"

 using assms proof induct

   case Nil

   then show ?case unfolding bij_betw_def by simp

 next

   case (swap y x l)

   show ?case

   proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)

     show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"

       by (auto simp: bij_betw_def)

     fix i

     assume "i < length(y#x#l)"

     show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"

       by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)

   qed

 next

   case (Cons xs ys z)

   then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and

     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast

   let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"

   show ?case

   proof (intro exI[of _ ?f] allI conjI impI)

     have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"

             "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"

       by (simp_all add: lessThan_Suc_eq_insert_0)

     show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"

       unfolding *

     proof (rule bij_betw_combine)

       show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"

         using bij unfolding bij_betw_def

         by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)

     qed (auto simp: bij_betw_def)

     fix i

     assume "i < length (z#xs)"

     then show "(z # xs) ! i = (z # ys) ! (?f i)"

       using perm by (cases i) auto

   qed

 next

   case (trans xs ys zs)

   then obtain f g where

     bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and

     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast

   show ?case

   proof (intro exI[of _ "g \<circ> f"] conjI allI impI)

     show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"

       using bij by (rule bij_betw_trans)

     fix i assume "i < length xs"

     with bij have "f i < length ys" unfolding bij_betw_def by force

     with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"

       using trans(1,3)[THEN perm_length] perm by auto

   qed

 qed

 end