src/HOL/Library/Permutation.thy
 author paulson Thu Jul 22 10:33:26 2004 +0200 (2004-07-22) changeset 15072 4861bf6af0b4 parent 15005 546c8e7e28d4 child 15131 c69542757a4d permissions -rw-r--r--
new material courtesy of Norbert Voelker
```     1 (*  Title:      HOL/Library/Permutation.thy
```
```     2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
```
```     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 = Multiset:
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```     8
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```     9 consts
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```    10   perm :: "('a list * 'a list) set"
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```    11
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```    12 syntax
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```    13   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
```
```    14 translations
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```    15   "x <~~> y" == "(x, y) \<in> perm"
```
```    16
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```    17 inductive perm
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```    18   intros
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```    19     Nil  [intro!]: "[] <~~> []"
```
```    20     swap [intro!]: "y # x # l <~~> x # y # l"
```
```    21     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
```
```    22     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
```
```    23
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```    24 lemma perm_refl [iff]: "l <~~> l"
```
```    25 by (induct l, auto)
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```    26
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```    27
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```    28 subsection {* Some examples of rule induction on permutations *}
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```    29
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```    30 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
```
```    31     -- {*the form of the premise lets the induction bind @{term xs}
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```    32          and @{term ys} *}
```
```    33   apply (erule perm.induct)
```
```    34      apply (simp_all (no_asm_simp))
```
```    35   done
```
```    36
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```    37 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
```
```    38 by (insert xperm_empty_imp_aux, blast)
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```    39
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```    40
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```    41 text {*
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```    42   \medskip This more general theorem is easier to understand!
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```    43   *}
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```    44
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```    45 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
```
```    46 by (erule perm.induct, simp_all)
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```    47
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```    48 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
```
```    49 by (drule perm_length, auto)
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```    50
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```    51 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
```
```    52 by (erule perm.induct, auto)
```
```    53
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```    54 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
```
```    55 by (erule perm.induct, auto)
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```    56
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```    57
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```    58 subsection {* Ways of making new permutations *}
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```    59
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```    60 text {*
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```    61   We can insert the head anywhere in the list.
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```    62 *}
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```    63
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```    64 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
```
```    65 by (induct xs, auto)
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```    66
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```    67 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
```
```    68   apply (induct xs, simp_all)
```
```    69   apply (blast intro: perm_append_Cons)
```
```    70   done
```
```    71
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```    72 lemma perm_append_single: "a # xs <~~> xs @ [a]"
```
```    73   by (rule perm.trans [OF _ perm_append_swap], simp)
```
```    74
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```    75 lemma perm_rev: "rev xs <~~> xs"
```
```    76   apply (induct xs, simp_all)
```
```    77   apply (blast intro!: perm_append_single intro: perm_sym)
```
```    78   done
```
```    79
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```    80 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
```
```    81 by (induct l, auto)
```
```    82
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```    83 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
```
```    84 by (blast intro!: perm_append_swap perm_append1)
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```    85
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```    86
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```    87 subsection {* Further results *}
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```    88
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```    89 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
```
```    90 by (blast intro: perm_empty_imp)
```
```    91
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```    92 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
```
```    93   apply auto
```
```    94   apply (erule perm_sym [THEN perm_empty_imp])
```
```    95   done
```
```    96
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```    97 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
```
```    98 by (erule perm.induct, auto)
```
```    99
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```   100 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
```
```   101 by (blast intro: perm_sing_imp)
```
```   102
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```   103 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
```
```   104 by (blast dest: perm_sym)
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```   105
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```   106
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```   107 subsection {* Removing elements *}
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```   108
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```   109 consts
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```   110   remove :: "'a => 'a list => 'a list"
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```   111 primrec
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```   112   "remove x [] = []"
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```   113   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
```
```   114
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```   115 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
```
```   116 by (induct ys, auto)
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```   117
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```   118 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
```
```   119 by (induct l, auto)
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```   120
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```   121 lemma multiset_of_remove[simp]:
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```   122   "multiset_of (remove a x) = multiset_of x - {#a#}"
```
```   123   by (induct_tac x, auto simp: multiset_eq_conv_count_eq)
```
```   124
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```   125
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```   126 text {* \medskip Congruence rule *}
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```   127
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```   128 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
```
```   129 by (erule perm.induct, auto)
```
```   130
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```   131 lemma remove_hd [simp]: "remove z (z # xs) = xs"
```
```   132   by auto
```
```   133
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```   134 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
```
```   135 by (drule_tac z = z in perm_remove_perm, auto)
```
```   136
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```   137 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
```
```   138 by (blast intro: cons_perm_imp_perm)
```
```   139
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```   140 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
```
```   141   apply (induct zs rule: rev_induct)
```
```   142    apply (simp_all (no_asm_use))
```
```   143   apply blast
```
```   144   done
```
```   145
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```   146 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
```
```   147 by (blast intro: append_perm_imp_perm perm_append1)
```
```   148
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```   149 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
```
```   150   apply (safe intro!: perm_append2)
```
```   151   apply (rule append_perm_imp_perm)
```
```   152   apply (rule perm_append_swap [THEN perm.trans])
```
```   153     -- {* the previous step helps this @{text blast} call succeed quickly *}
```
```   154   apply (blast intro: perm_append_swap)
```
```   155   done
```
```   156
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```   157 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
```
```   158   apply (rule iffI)
```
```   159   apply (erule_tac  perm.induct, simp_all add: union_ac)
```
```   160   apply (erule rev_mp, rule_tac x=ys in spec)
```
```   161   apply (induct_tac xs, auto)
```
```   162   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
```
```   163   apply (subgoal_tac "a \<in> set x")
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```   164   apply (drule_tac z=a in perm.Cons)
```
```   165   apply (erule perm.trans, rule perm_sym, erule perm_remove)
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```   166   apply (drule_tac f=set_of in arg_cong, simp)
```
```   167   done
```
```   168
```
```   169 lemma multiset_of_le_perm_append:
```
```   170   "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)";
```
```   171   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
```
```   172   apply (insert surj_multiset_of, drule surjD)
```
```   173   apply (blast intro: sym)+
```
```   174   done
```
```   175
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```   176 end
```