src/HOL/Library/Permutation.thy
author paulson
Thu Jun 24 17:54:53 2004 +0200 (2004-06-24)
changeset 15005 546c8e7e28d4
parent 14706 71590b7733b7
child 15072 4861bf6af0b4
permissions -rw-r--r--
Norbert Voelker
     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 = Multiset:
     8 
     9 consts
    10   perm :: "('a list * 'a list) set"
    11 
    12 syntax
    13   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
    14 translations
    15   "x <~~> y" == "(x, y) \<in> perm"
    16 
    17 inductive perm
    18   intros
    19     Nil  [intro!]: "[] <~~> []"
    20     swap [intro!]: "y # x # l <~~> x # y # l"
    21     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
    22     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    23 
    24 lemma perm_refl [iff]: "l <~~> l"
    25 by (induct l, auto)
    26 
    27 
    28 subsection {* Some examples of rule induction on permutations *}
    29 
    30 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
    31     -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}
    32   apply (erule perm.induct)
    33      apply (simp_all (no_asm_simp))
    34   done
    35 
    36 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    37 by (insert xperm_empty_imp_aux, blast)
    38 
    39 
    40 text {*
    41   \medskip This more general theorem is easier to understand!
    42   *}
    43 
    44 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    45 by (erule perm.induct, simp_all)
    46 
    47 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    48 by (drule perm_length, auto)
    49 
    50 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    51 by (erule perm.induct, auto)
    52 
    53 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
    54 by (erule perm.induct, auto)
    55 
    56 
    57 subsection {* Ways of making new permutations *}
    58 
    59 text {*
    60   We can insert the head anywhere in the list.
    61 *}
    62 
    63 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    64 by (induct xs, auto)
    65 
    66 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    67   apply (induct xs, simp_all)
    68   apply (blast intro: perm_append_Cons)
    69   done
    70 
    71 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    72   apply (rule perm.trans)
    73    prefer 2
    74    apply (rule perm_append_swap, simp)
    75   done
    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 
   124 text {* \medskip Congruence rule *}
   125 
   126 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   127 by (erule perm.induct, auto)
   128 
   129 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   130   apply auto
   131   done
   132 
   133 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   134 by (drule_tac z = z in perm_remove_perm, auto)
   135 
   136 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   137 by (blast intro: cons_perm_imp_perm)
   138 
   139 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   140   apply (induct zs rule: rev_induct)
   141    apply (simp_all (no_asm_use))
   142   apply blast
   143   done
   144 
   145 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   146 by (blast intro: append_perm_imp_perm perm_append1)
   147 
   148 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   149   apply (safe intro!: perm_append2)
   150   apply (rule append_perm_imp_perm)
   151   apply (rule perm_append_swap [THEN perm.trans])
   152     -- {* the previous step helps this @{text blast} call succeed quickly *}
   153   apply (blast intro: perm_append_swap)
   154   done
   155 
   156 (****************** Norbert Voelker 17 June 2004 **************) 
   157 
   158 consts 
   159   multiset_of :: "'a list \<Rightarrow> 'a multiset"
   160 primrec
   161   "multiset_of [] = {#}"
   162   "multiset_of (a # x) = multiset_of x + {# a #}"
   163 
   164 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
   165   by (induct_tac x, auto) 
   166 
   167 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
   168   by (induct_tac x, auto)
   169 
   170 lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
   171  by (induct_tac x, auto) 
   172 
   173 lemma multiset_of_remove[simp]: 
   174   "multiset_of (remove a x) = multiset_of x - {#a#}"
   175   by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 
   176 
   177 lemma multiset_of_eq_perm:  "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   178   apply (rule iffI) 
   179   apply (erule_tac [2] perm.induct, simp_all add: union_ac) 
   180   apply (erule rev_mp, rule_tac x=ys in spec, induct_tac xs, auto) 
   181   apply (erule_tac x = "remove a x" in allE, drule sym, simp) 
   182   apply (subgoal_tac "a \<in> set x") 
   183   apply (drule_tac z=a in perm.Cons) 
   184   apply (erule perm.trans, rule perm_sym, erule perm_remove) 
   185   apply (drule_tac f=set_of in arg_cong, simp)
   186   done
   187 
   188 lemma set_count_multiset_of: "set x = {a. 0 < count (multiset_of x) a}"
   189   by (induct_tac x, auto)  
   190 
   191 lemma distinct_count_multiset_of: 
   192    "distinct x \<Longrightarrow> count (multiset_of x) a = (if a \<in> set x then 1 else 0)"
   193   by (erule rev_mp, induct_tac x, auto) 
   194 
   195 lemma distinct_set_eq_iff_multiset_of_eq: 
   196   "\<lbrakk>distinct x; distinct y\<rbrakk> \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
   197   by (auto simp: multiset_eq_conv_count_eq distinct_count_multiset_of) 
   198 
   199 end