src/HOL/Library/Permutation.thy
author wenzelm
Wed Aug 31 15:46:37 2005 +0200 (2005-08-31)
changeset 17200 3a4d03d1a31b
parent 15140 322485b816ac
child 19380 b808efaa5828
permissions -rw-r--r--
tuned presentation;
     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   using xperm_empty_imp_aux by 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)
    71     apply simp_all
    72   apply (blast intro: perm_append_Cons)
    73   done
    74 
    75 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    76   by (rule perm.trans [OF _ perm_append_swap]) simp
    77 
    78 lemma perm_rev: "rev xs <~~> xs"
    79   apply (induct xs)
    80    apply simp_all
    81   apply (blast intro!: perm_append_single intro: perm_sym)
    82   done
    83 
    84 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
    85   by (induct l) auto
    86 
    87 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
    88   by (blast intro!: perm_append_swap perm_append1)
    89 
    90 
    91 subsection {* Further results *}
    92 
    93 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
    94   by (blast intro: perm_empty_imp)
    95 
    96 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
    97   apply auto
    98   apply (erule perm_sym [THEN perm_empty_imp])
    99   done
   100 
   101 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
   102   by (erule perm.induct) auto
   103 
   104 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
   105   by (blast intro: perm_sing_imp)
   106 
   107 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
   108   by (blast dest: perm_sym)
   109 
   110 
   111 subsection {* Removing elements *}
   112 
   113 consts
   114   remove :: "'a => 'a list => 'a list"
   115 primrec
   116   "remove x [] = []"
   117   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
   118 
   119 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
   120   by (induct ys) auto
   121 
   122 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
   123   by (induct l) auto
   124 
   125 lemma multiset_of_remove[simp]:
   126     "multiset_of (remove a x) = multiset_of x - {#a#}"
   127   apply (induct x)
   128    apply (auto simp: multiset_eq_conv_count_eq)
   129   done
   130 
   131 
   132 text {* \medskip Congruence rule *}
   133 
   134 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   135   by (erule perm.induct) auto
   136 
   137 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   138   by auto
   139 
   140 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   141   by (drule_tac z = z in perm_remove_perm) auto
   142 
   143 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   144   by (blast intro: cons_perm_imp_perm)
   145 
   146 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   147   apply (induct zs rule: rev_induct)
   148    apply (simp_all (no_asm_use))
   149   apply blast
   150   done
   151 
   152 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   153   by (blast intro: append_perm_imp_perm perm_append1)
   154 
   155 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   156   apply (safe intro!: perm_append2)
   157   apply (rule append_perm_imp_perm)
   158   apply (rule perm_append_swap [THEN perm.trans])
   159     -- {* the previous step helps this @{text blast} call succeed quickly *}
   160   apply (blast intro: perm_append_swap)
   161   done
   162 
   163 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   164   apply (rule iffI)
   165   apply (erule_tac [2] perm.induct, simp_all add: union_ac)
   166   apply (erule rev_mp, rule_tac x=ys in spec)
   167   apply (induct_tac xs, auto)
   168   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
   169   apply (subgoal_tac "a \<in> set x")
   170   apply (drule_tac z=a in perm.Cons)
   171   apply (erule perm.trans, rule perm_sym, erule perm_remove)
   172   apply (drule_tac f=set_of in arg_cong, simp)
   173   done
   174 
   175 lemma multiset_of_le_perm_append:
   176     "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)";
   177   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   178   apply (insert surj_multiset_of, drule surjD)
   179   apply (blast intro: sym)+
   180   done
   181 
   182 end