src/HOL/Library/Permutation.thy
author haftmann
Fri Feb 15 16:09:12 2008 +0100 (2008-02-15)
changeset 26072 f65a7fa2da6c
parent 25379 12bcf37252b1
child 26316 9e9e67e33557
permissions -rw-r--r--
<= and < on nat no longer depend on wellfounded relations
     1 (*  Title:      HOL/Library/Permutation.thy
     2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
     3 *)
     4 
     5 header {* Permutations *}
     6 
     7 theory Permutation
     8 imports Multiset
     9 begin
    10 
    11 inductive
    12   perm :: "'a list => 'a list => bool"  ("_ <~~> _"  [50, 50] 50)
    13   where
    14     Nil  [intro!]: "[] <~~> []"
    15   | swap [intro!]: "y # x # l <~~> x # y # l"
    16   | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
    17   | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    18 
    19 lemma perm_refl [iff]: "l <~~> l"
    20   by (induct l) auto
    21 
    22 
    23 subsection {* Some examples of rule induction on permutations *}
    24 
    25 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    26   by (induct xs == "[]::'a list" ys pred: perm) simp_all
    27 
    28 
    29 text {*
    30   \medskip This more general theorem is easier to understand!
    31   *}
    32 
    33 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    34   by (induct pred: perm) simp_all
    35 
    36 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    37   by (drule perm_length) auto
    38 
    39 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    40   by (induct pred: perm) auto
    41 
    42 
    43 subsection {* Ways of making new permutations *}
    44 
    45 text {*
    46   We can insert the head anywhere in the list.
    47 *}
    48 
    49 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    50   by (induct xs) auto
    51 
    52 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    53   apply (induct xs)
    54     apply simp_all
    55   apply (blast intro: perm_append_Cons)
    56   done
    57 
    58 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    59   by (rule perm.trans [OF _ perm_append_swap]) simp
    60 
    61 lemma perm_rev: "rev xs <~~> xs"
    62   apply (induct xs)
    63    apply simp_all
    64   apply (blast intro!: perm_append_single intro: perm_sym)
    65   done
    66 
    67 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
    68   by (induct l) auto
    69 
    70 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
    71   by (blast intro!: perm_append_swap perm_append1)
    72 
    73 
    74 subsection {* Further results *}
    75 
    76 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
    77   by (blast intro: perm_empty_imp)
    78 
    79 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
    80   apply auto
    81   apply (erule perm_sym [THEN perm_empty_imp])
    82   done
    83 
    84 lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
    85   by (induct pred: perm) auto
    86 
    87 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
    88   by (blast intro: perm_sing_imp)
    89 
    90 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
    91   by (blast dest: perm_sym)
    92 
    93 
    94 subsection {* Removing elements *}
    95 
    96 consts
    97   remove :: "'a => 'a list => 'a list"
    98 primrec
    99   "remove x [] = []"
   100   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
   101 
   102 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
   103   by (induct ys) auto
   104 
   105 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
   106   by (induct l) auto
   107 
   108 lemma multiset_of_remove [simp]:
   109     "multiset_of (remove a x) = multiset_of x - {#a#}"
   110   apply (induct x)
   111    apply (auto simp: multiset_eq_conv_count_eq)
   112   done
   113 
   114 
   115 text {* \medskip Congruence rule *}
   116 
   117 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   118   by (induct pred: perm) auto
   119 
   120 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   121   by auto
   122 
   123 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   124   by (drule_tac z = z in perm_remove_perm) auto
   125 
   126 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   127   by (blast intro: cons_perm_imp_perm)
   128 
   129 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   130   apply (induct zs arbitrary: xs ys rule: rev_induct)
   131    apply (simp_all (no_asm_use))
   132   apply blast
   133   done
   134 
   135 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   136   by (blast intro: append_perm_imp_perm perm_append1)
   137 
   138 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   139   apply (safe intro!: perm_append2)
   140   apply (rule append_perm_imp_perm)
   141   apply (rule perm_append_swap [THEN perm.trans])
   142     -- {* the previous step helps this @{text blast} call succeed quickly *}
   143   apply (blast intro: perm_append_swap)
   144   done
   145 
   146 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   147   apply (rule iffI)
   148   apply (erule_tac [2] perm.induct, simp_all add: union_ac)
   149   apply (erule rev_mp, rule_tac x=ys in spec)
   150   apply (induct_tac xs, auto)
   151   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
   152   apply (subgoal_tac "a \<in> set x")
   153   apply (drule_tac z=a in perm.Cons)
   154   apply (erule perm.trans, rule perm_sym, erule perm_remove)
   155   apply (drule_tac f=set_of in arg_cong, simp)
   156   done
   157 
   158 lemma multiset_of_le_perm_append:
   159     "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)";
   160   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   161   apply (insert surj_multiset_of, drule surjD)
   162   apply (blast intro: sym)+
   163   done
   164 
   165 lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
   166   by (metis multiset_of_eq_perm multiset_of_eq_setD)
   167 
   168 lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
   169   apply (induct pred: perm)
   170      apply simp_all
   171    apply fastsimp
   172   apply (metis perm_set_eq)
   173   done
   174 
   175 lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
   176   apply (induct xs arbitrary: ys rule: length_induct)
   177   apply (case_tac "remdups xs", simp, simp)
   178   apply (subgoal_tac "a : set (remdups ys)")
   179    prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
   180   apply (drule split_list) apply(elim exE conjE)
   181   apply (drule_tac x=list in spec) apply(erule impE) prefer 2
   182    apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
   183     apply simp
   184     apply (subgoal_tac "a#list <~~> a#ysa@zs")
   185      apply (metis Cons_eq_appendI perm_append_Cons trans)
   186     apply (metis Cons Cons_eq_appendI distinct.simps(2)
   187       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
   188    apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
   189     apply (fastsimp simp add: insert_ident)
   190    apply (metis distinct_remdups set_remdups)
   191   apply (metis Nat.le_less_trans Suc_length_conv length_remdups_leq less_Suc_eq nat_less_le)
   192   done
   193 
   194 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
   195   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
   196 
   197 end