(*  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