src/HOL/Library/Permutation.thy
 author nipkow Mon Nov 05 18:18:39 2007 +0100 (2007-11-05) changeset 25287 094dab519ff5 parent 25277 95128fcdd7e8 child 25379 12bcf37252b1 permissions -rw-r--r--
```     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_aux: "xs <~~> ys ==> xs = [] --> ys = []"
```
```    26     -- {*the form of the premise lets the induction bind @{term xs}
```
```    27          and @{term ys} *}
```
```    28   apply (erule perm.induct)
```
```    29      apply (simp_all (no_asm_simp))
```
```    30   done
```
```    31
```
```    32 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
```
```    33   using xperm_empty_imp_aux by blast
```
```    34
```
```    35
```
```    36 text {*
```
```    37   \medskip This more general theorem is easier to understand!
```
```    38   *}
```
```    39
```
```    40 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
```
```    41   by (erule perm.induct) simp_all
```
```    42
```
```    43 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
```
```    44   by (drule perm_length) auto
```
```    45
```
```    46 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
```
```    47   by (erule perm.induct) auto
```
```    48
```
```    49
```
```    50 subsection {* Ways of making new permutations *}
```
```    51
```
```    52 text {*
```
```    53   We can insert the head anywhere in the list.
```
```    54 *}
```
```    55
```
```    56 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
```
```    57   by (induct xs) auto
```
```    58
```
```    59 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
```
```    60   apply (induct xs)
```
```    61     apply simp_all
```
```    62   apply (blast intro: perm_append_Cons)
```
```    63   done
```
```    64
```
```    65 lemma perm_append_single: "a # xs <~~> xs @ [a]"
```
```    66   by (rule perm.trans [OF _ perm_append_swap]) simp
```
```    67
```
```    68 lemma perm_rev: "rev xs <~~> xs"
```
```    69   apply (induct xs)
```
```    70    apply simp_all
```
```    71   apply (blast intro!: perm_append_single intro: perm_sym)
```
```    72   done
```
```    73
```
```    74 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
```
```    75   by (induct l) auto
```
```    76
```
```    77 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
```
```    78   by (blast intro!: perm_append_swap perm_append1)
```
```    79
```
```    80
```
```    81 subsection {* Further results *}
```
```    82
```
```    83 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
```
```    84   by (blast intro: perm_empty_imp)
```
```    85
```
```    86 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
```
```    87   apply auto
```
```    88   apply (erule perm_sym [THEN perm_empty_imp])
```
```    89   done
```
```    90
```
```    91 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
```
```    92   by (erule perm.induct) auto
```
```    93
```
```    94 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
```
```    95   by (blast intro: perm_sing_imp)
```
```    96
```
```    97 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
```
```    98   by (blast dest: perm_sym)
```
```    99
```
```   100
```
```   101 subsection {* Removing elements *}
```
```   102
```
```   103 consts
```
```   104   remove :: "'a => 'a list => 'a list"
```
```   105 primrec
```
```   106   "remove x [] = []"
```
```   107   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
```
```   108
```
```   109 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
```
```   110   by (induct ys) auto
```
```   111
```
```   112 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
```
```   113   by (induct l) auto
```
```   114
```
```   115 lemma multiset_of_remove[simp]:
```
```   116     "multiset_of (remove a x) = multiset_of x - {#a#}"
```
```   117   apply (induct x)
```
```   118    apply (auto simp: multiset_eq_conv_count_eq)
```
```   119   done
```
```   120
```
```   121
```
```   122 text {* \medskip Congruence rule *}
```
```   123
```
```   124 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
```
```   125   by (erule perm.induct) auto
```
```   126
```
```   127 lemma remove_hd [simp]: "remove z (z # xs) = xs"
```
```   128   by auto
```
```   129
```
```   130 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
```
```   131   by (drule_tac z = z in perm_remove_perm) auto
```
```   132
```
```   133 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
```
```   134   by (blast intro: cons_perm_imp_perm)
```
```   135
```
```   136 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
```
```   137   apply (induct zs rule: rev_induct)
```
```   138    apply (simp_all (no_asm_use))
```
```   139   apply blast
```
```   140   done
```
```   141
```
```   142 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
```
```   143   by (blast intro: append_perm_imp_perm perm_append1)
```
```   144
```
```   145 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
```
```   146   apply (safe intro!: perm_append2)
```
```   147   apply (rule append_perm_imp_perm)
```
```   148   apply (rule perm_append_swap [THEN perm.trans])
```
```   149     -- {* the previous step helps this @{text blast} call succeed quickly *}
```
```   150   apply (blast intro: perm_append_swap)
```
```   151   done
```
```   152
```
```   153 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
```
```   154   apply (rule iffI)
```
```   155   apply (erule_tac  perm.induct, simp_all add: union_ac)
```
```   156   apply (erule rev_mp, rule_tac x=ys in spec)
```
```   157   apply (induct_tac xs, auto)
```
```   158   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
```
```   159   apply (subgoal_tac "a \<in> set x")
```
```   160   apply (drule_tac z=a in perm.Cons)
```
```   161   apply (erule perm.trans, rule perm_sym, erule perm_remove)
```
```   162   apply (drule_tac f=set_of in arg_cong, simp)
```
```   163   done
```
```   164
```
```   165 lemma multiset_of_le_perm_append:
```
```   166     "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)";
```
```   167   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
```
```   168   apply (insert surj_multiset_of, drule surjD)
```
```   169   apply (blast intro: sym)+
```
```   170   done
```
```   171
```
```   172 lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
```
```   173 by (metis multiset_of_eq_perm multiset_of_eq_setD)
```
```   174
```
```   175 lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
```
```   176 apply(induct rule:perm.induct)
```
```   177    apply simp_all
```
```   178  apply fastsimp
```
```   179 apply (metis perm_set_eq)
```
```   180 done
```
```   181
```
```   182 lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
```
```   183 apply(induct xs arbitrary: ys rule:length_induct)
```
```   184 apply (case_tac "remdups xs", simp, simp)
```
```   185 apply(subgoal_tac "a : set (remdups ys)")
```
```   186  prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
```
```   187 apply(drule split_list) apply(elim exE conjE)
```
```   188 apply(drule_tac x=list in spec) apply(erule impE) prefer 2
```
```   189  apply(drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
```
```   190   apply simp
```
```   191   apply(subgoal_tac "a#list <~~> a#ysa@zs")
```
```   192    apply (metis Cons_eq_appendI perm_append_Cons trans)
```
```   193   apply (metis Cons Cons_eq_appendI distinct.simps(2) distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
```
```   194  apply(subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
```
```   195   apply(fastsimp simp add: insert_ident)
```
```   196  apply (metis distinct_remdups set_remdups)
```
```   197 apply (metis Nat.le_less_trans Suc_length_conv le_def length_remdups_leq less_Suc_eq)
```
```   198 done
```
```   199
```
```   200 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
```
```   201 by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
```
```   202
```
```   203 end
```