src/HOL/Library/Permutation.thy
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 15131 c69542757a4d child 17200 3a4d03d1a31b permissions -rw-r--r--
import -> imports
```     1 (*  Title:      HOL/Library/Permutation.thy
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```     2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
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```     3 *)
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```     4
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```     5 header {* Permutations *}
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```     6
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```     7 theory Permutation
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```     8 imports Multiset
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```     9 begin
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```    10
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```    11 consts
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```    12   perm :: "('a list * 'a list) set"
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```    13
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```    14 syntax
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```    15   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
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```    16 translations
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```    17   "x <~~> y" == "(x, y) \<in> perm"
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```    18
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```    19 inductive perm
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```    20   intros
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```    21     Nil  [intro!]: "[] <~~> []"
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```    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
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```    26 lemma perm_refl [iff]: "l <~~> l"
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```    27 by (induct l, auto)
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```    28
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```    29
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```    30 subsection {* Some examples of rule induction on permutations *}
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```    31
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```    32 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
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```    33     -- {*the form of the premise lets the induction bind @{term xs}
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```    34          and @{term ys} *}
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```    35   apply (erule perm.induct)
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```    36      apply (simp_all (no_asm_simp))
```
```    37   done
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```    38
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```    39 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
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```    40 by (insert xperm_empty_imp_aux, blast)
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```    41
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```    42
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```    43 text {*
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```    44   \medskip This more general theorem is easier to understand!
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```    45   *}
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```    46
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```    47 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
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```    48 by (erule perm.induct, simp_all)
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```    49
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```    50 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
```
```    51 by (drule perm_length, auto)
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```    52
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```    53 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
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```    54 by (erule perm.induct, auto)
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```    55
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```    56 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
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```    57 by (erule perm.induct, auto)
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```    58
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```    59
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```    60 subsection {* Ways of making new permutations *}
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```    61
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```    62 text {*
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```    63   We can insert the head anywhere in the list.
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```    64 *}
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```    65
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```    66 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
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```    67 by (induct xs, auto)
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```    68
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```    69 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
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```    70   apply (induct xs, simp_all)
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```    71   apply (blast intro: perm_append_Cons)
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```    72   done
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```    73
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```    74 lemma perm_append_single: "a # xs <~~> xs @ [a]"
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```    75   by (rule perm.trans [OF _ perm_append_swap], simp)
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```    76
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```    77 lemma perm_rev: "rev xs <~~> xs"
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```    78   apply (induct xs, simp_all)
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```    79   apply (blast intro!: perm_append_single intro: perm_sym)
```
```    80   done
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```    81
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```    82 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
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```    83 by (induct l, auto)
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```    84
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```    85 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
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```    86 by (blast intro!: perm_append_swap perm_append1)
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```    87
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```    88
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```    89 subsection {* Further results *}
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```    90
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```    91 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
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```    92 by (blast intro: perm_empty_imp)
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```    93
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```    94 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
```
```    95   apply auto
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```    96   apply (erule perm_sym [THEN perm_empty_imp])
```
```    97   done
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```    98
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```    99 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
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```   100 by (erule perm.induct, auto)
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```   101
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```   102 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
```
```   103 by (blast intro: perm_sing_imp)
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```   104
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```   105 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
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```   106 by (blast dest: perm_sym)
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```   107
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```   108
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```   109 subsection {* Removing elements *}
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```   110
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```   111 consts
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```   112   remove :: "'a => 'a list => 'a list"
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```   113 primrec
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```   114   "remove x [] = []"
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```   115   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
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```   116
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```   117 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
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```   118 by (induct ys, auto)
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```   119
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```   120 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
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```   121 by (induct l, auto)
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```   122
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```   123 lemma multiset_of_remove[simp]:
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```   124   "multiset_of (remove a x) = multiset_of x - {#a#}"
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```   125   by (induct_tac x, auto simp: multiset_eq_conv_count_eq)
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```   126
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```   127
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```   128 text {* \medskip Congruence rule *}
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```   129
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```   130 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
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```   131 by (erule perm.induct, auto)
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```   132
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```   133 lemma remove_hd [simp]: "remove z (z # xs) = xs"
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```   134   by auto
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```   135
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```   136 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
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```   137 by (drule_tac z = z in perm_remove_perm, auto)
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```   138
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```   139 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
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```   140 by (blast intro: cons_perm_imp_perm)
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```   141
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```   142 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
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```   143   apply (induct zs rule: rev_induct)
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```   144    apply (simp_all (no_asm_use))
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```   145   apply blast
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```   146   done
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```   147
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```   148 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
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```   149 by (blast intro: append_perm_imp_perm perm_append1)
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```   150
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```   151 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
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```   152   apply (safe intro!: perm_append2)
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```   153   apply (rule append_perm_imp_perm)
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```   154   apply (rule perm_append_swap [THEN perm.trans])
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```   155     -- {* the previous step helps this @{text blast} call succeed quickly *}
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```   156   apply (blast intro: perm_append_swap)
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```   157   done
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```   158
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```   159 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
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```   160   apply (rule iffI)
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```   161   apply (erule_tac [2] perm.induct, simp_all add: union_ac)
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```   162   apply (erule rev_mp, rule_tac x=ys in spec)
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```   163   apply (induct_tac xs, auto)
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```   164   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
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```   165   apply (subgoal_tac "a \<in> set x")
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```   166   apply (drule_tac z=a in perm.Cons)
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```   167   apply (erule perm.trans, rule perm_sym, erule perm_remove)
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```   168   apply (drule_tac f=set_of in arg_cong, simp)
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```   169   done
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```   170
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```   171 lemma multiset_of_le_perm_append:
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```   172   "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)";
```
```   173   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
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```   174   apply (insert surj_multiset_of, drule surjD)
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```   175   apply (blast intro: sym)+
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```   176   done
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```   177
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```   178 end
```