src/HOL/Library/Permutation.thy
 author chaieb Mon Jun 11 11:06:04 2007 +0200 (2007-06-11) changeset 23315 df3a7e9ebadb parent 21404 eb85850d3eb7 child 23755 1c4672d130b1 permissions -rw-r--r--
tuned Proof
```     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 abbreviation
```
```    15   perm_rel :: "'a list => 'a list => bool"  ("_ <~~> _"  [50, 50] 50) where
```
```    16   "x <~~> y == (x, y) \<in> perm"
```
```    17
```
```    18 inductive perm
```
```    19   intros
```
```    20     Nil  [intro!]: "[] <~~> []"
```
```    21     swap [intro!]: "y # x # l <~~> x # y # l"
```
```    22     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
```
```    23     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
```
```    24
```
```    25 lemma perm_refl [iff]: "l <~~> l"
```
```    26   by (induct l) auto
```
```    27
```
```    28
```
```    29 subsection {* Some examples of rule induction on permutations *}
```
```    30
```
```    31 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
```
```    32     -- {*the form of the premise lets the induction bind @{term xs}
```
```    33          and @{term ys} *}
```
```    34   apply (erule perm.induct)
```
```    35      apply (simp_all (no_asm_simp))
```
```    36   done
```
```    37
```
```    38 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
```
```    39   using xperm_empty_imp_aux by blast
```
```    40
```
```    41
```
```    42 text {*
```
```    43   \medskip This more general theorem is easier to understand!
```
```    44   *}
```
```    45
```
```    46 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
```
```    47   by (erule perm.induct) simp_all
```
```    48
```
```    49 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
```
```    50   by (drule perm_length) auto
```
```    51
```
```    52 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
```
```    53   by (erule perm.induct) auto
```
```    54
```
```    55 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
```
```    56   by (erule perm.induct) auto
```
```    57
```
```    58
```
```    59 subsection {* Ways of making new permutations *}
```
```    60
```
```    61 text {*
```
```    62   We can insert the head anywhere in the list.
```
```    63 *}
```
```    64
```
```    65 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
```
```    66   by (induct xs) auto
```
```    67
```
```    68 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
```
```    69   apply (induct xs)
```
```    70     apply 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)
```
```    79    apply simp_all
```
```    80   apply (blast intro!: perm_append_single intro: perm_sym)
```
```    81   done
```
```    82
```
```    83 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
```
```    84   by (induct l) auto
```
```    85
```
```    86 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
```
```    87   by (blast intro!: perm_append_swap perm_append1)
```
```    88
```
```    89
```
```    90 subsection {* Further results *}
```
```    91
```
```    92 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
```
```    93   by (blast intro: perm_empty_imp)
```
```    94
```
```    95 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
```
```    96   apply auto
```
```    97   apply (erule perm_sym [THEN perm_empty_imp])
```
```    98   done
```
```    99
```
```   100 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
```
```   101   by (erule perm.induct) auto
```
```   102
```
```   103 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
```
```   104   by (blast intro: perm_sing_imp)
```
```   105
```
```   106 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
```
```   107   by (blast dest: perm_sym)
```
```   108
```
```   109
```
```   110 subsection {* Removing elements *}
```
```   111
```
```   112 consts
```
```   113   remove :: "'a => 'a list => 'a list"
```
```   114 primrec
```
```   115   "remove x [] = []"
```
```   116   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
```
```   117
```
```   118 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
```
```   119   by (induct ys) auto
```
```   120
```
```   121 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
```
```   122   by (induct l) auto
```
```   123
```
```   124 lemma multiset_of_remove[simp]:
```
```   125     "multiset_of (remove a x) = multiset_of x - {#a#}"
```
```   126   apply (induct x)
```
```   127    apply (auto simp: multiset_eq_conv_count_eq)
```
```   128   done
```
```   129
```
```   130
```
```   131 text {* \medskip Congruence rule *}
```
```   132
```
```   133 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
```
```   134   by (erule perm.induct) auto
```
```   135
```
```   136 lemma remove_hd [simp]: "remove z (z # xs) = xs"
```
```   137   by auto
```
```   138
```
```   139 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
```
```   140   by (drule_tac z = z in perm_remove_perm) auto
```
```   141
```
```   142 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
```
```   143   by (blast intro: cons_perm_imp_perm)
```
```   144
```
```   145 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
```
```   146   apply (induct zs rule: rev_induct)
```
```   147    apply (simp_all (no_asm_use))
```
```   148   apply blast
```
```   149   done
```
```   150
```
```   151 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
```
```   152   by (blast intro: append_perm_imp_perm perm_append1)
```
```   153
```
```   154 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
```
```   155   apply (safe intro!: perm_append2)
```
```   156   apply (rule append_perm_imp_perm)
```
```   157   apply (rule perm_append_swap [THEN perm.trans])
```
```   158     -- {* the previous step helps this @{text blast} call succeed quickly *}
```
```   159   apply (blast intro: perm_append_swap)
```
```   160   done
```
```   161
```
```   162 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
```
```   163   apply (rule iffI)
```
```   164   apply (erule_tac [2] perm.induct, simp_all add: union_ac)
```
```   165   apply (erule rev_mp, rule_tac x=ys in spec)
```
```   166   apply (induct_tac xs, auto)
```
```   167   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
```
```   168   apply (subgoal_tac "a \<in> set x")
```
```   169   apply (drule_tac z=a in perm.Cons)
```
```   170   apply (erule perm.trans, rule perm_sym, erule perm_remove)
```
```   171   apply (drule_tac f=set_of in arg_cong, simp)
```
```   172   done
```
```   173
```
```   174 lemma multiset_of_le_perm_append:
```
```   175     "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)";
```
```   176   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
```
```   177   apply (insert surj_multiset_of, drule surjD)
```
```   178   apply (blast intro: sym)+
```
```   179   done
```
```   180
```
```   181 end
```