src/HOL/Library/Permutation.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15140 322485b816ac
child 17200 3a4d03d1a31b
permissions -rw-r--r--
Constant "If" is now local
     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 consts
    12   perm :: "('a list * 'a list) set"
    13 
    14 syntax
    15   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
    16 translations
    17   "x <~~> y" == "(x, y) \<in> perm"
    18 
    19 inductive perm
    20   intros
    21     Nil  [intro!]: "[] <~~> []"
    22     swap [intro!]: "y # x # l <~~> x # y # l"
    23     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
    24     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    25 
    26 lemma perm_refl [iff]: "l <~~> l"
    27 by (induct l, auto)
    28 
    29 
    30 subsection {* Some examples of rule induction on permutations *}
    31 
    32 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
    33     -- {*the form of the premise lets the induction bind @{term xs} 
    34          and @{term ys} *}
    35   apply (erule perm.induct)
    36      apply (simp_all (no_asm_simp))
    37   done
    38 
    39 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    40 by (insert xperm_empty_imp_aux, blast)
    41 
    42 
    43 text {*
    44   \medskip This more general theorem is easier to understand!
    45   *}
    46 
    47 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    48 by (erule perm.induct, simp_all)
    49 
    50 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    51 by (drule perm_length, auto)
    52 
    53 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    54 by (erule perm.induct, auto)
    55 
    56 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
    57 by (erule perm.induct, auto)
    58 
    59 
    60 subsection {* Ways of making new permutations *}
    61 
    62 text {*
    63   We can insert the head anywhere in the list.
    64 *}
    65 
    66 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    67 by (induct xs, auto)
    68 
    69 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    70   apply (induct xs, simp_all)
    71   apply (blast intro: perm_append_Cons)
    72   done
    73 
    74 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    75   by (rule perm.trans [OF _ perm_append_swap], simp)
    76 
    77 lemma perm_rev: "rev xs <~~> xs"
    78   apply (induct xs, simp_all)
    79   apply (blast intro!: perm_append_single intro: perm_sym)
    80   done
    81 
    82 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
    83 by (induct l, auto)
    84 
    85 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
    86 by (blast intro!: perm_append_swap perm_append1)
    87 
    88 
    89 subsection {* Further results *}
    90 
    91 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
    92 by (blast intro: perm_empty_imp)
    93 
    94 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
    95   apply auto
    96   apply (erule perm_sym [THEN perm_empty_imp])
    97   done
    98 
    99 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
   100 by (erule perm.induct, auto)
   101 
   102 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
   103 by (blast intro: perm_sing_imp)
   104 
   105 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
   106 by (blast dest: perm_sym)
   107 
   108 
   109 subsection {* Removing elements *}
   110 
   111 consts
   112   remove :: "'a => 'a list => 'a list"
   113 primrec
   114   "remove x [] = []"
   115   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
   116 
   117 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
   118 by (induct ys, auto)
   119 
   120 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
   121 by (induct l, auto)
   122 
   123 lemma multiset_of_remove[simp]: 
   124   "multiset_of (remove a x) = multiset_of x - {#a#}"
   125   by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 
   126 
   127 
   128 text {* \medskip Congruence rule *}
   129 
   130 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   131 by (erule perm.induct, auto)
   132 
   133 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   134   by auto
   135 
   136 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   137 by (drule_tac z = z in perm_remove_perm, auto)
   138 
   139 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   140 by (blast intro: cons_perm_imp_perm)
   141 
   142 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   143   apply (induct zs rule: rev_induct)
   144    apply (simp_all (no_asm_use))
   145   apply blast
   146   done
   147 
   148 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   149 by (blast intro: append_perm_imp_perm perm_append1)
   150 
   151 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   152   apply (safe intro!: perm_append2)
   153   apply (rule append_perm_imp_perm)
   154   apply (rule perm_append_swap [THEN perm.trans])
   155     -- {* the previous step helps this @{text blast} call succeed quickly *}
   156   apply (blast intro: perm_append_swap)
   157   done
   158 
   159 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   160   apply (rule iffI) 
   161   apply (erule_tac [2] perm.induct, simp_all add: union_ac) 
   162   apply (erule rev_mp, rule_tac x=ys in spec) 
   163   apply (induct_tac xs, auto) 
   164   apply (erule_tac x = "remove a x" in allE, drule sym, simp) 
   165   apply (subgoal_tac "a \<in> set x") 
   166   apply (drule_tac z=a in perm.Cons) 
   167   apply (erule perm.trans, rule perm_sym, erule perm_remove) 
   168   apply (drule_tac f=set_of in arg_cong, simp)
   169   done
   170 
   171 lemma multiset_of_le_perm_append: 
   172   "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)"; 
   173   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) 
   174   apply (insert surj_multiset_of, drule surjD)
   175   apply (blast intro: sym)+
   176   done
   177 
   178 end