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 *)
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```     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  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
```