src/HOL/Library/Permutation.thy
 author wenzelm Thu May 06 14:14:18 2004 +0200 (2004-05-06) changeset 14706 71590b7733b7 parent 11153 950ede59c05a child 15005 546c8e7e28d4 permissions -rw-r--r--
tuned document;
```     1 (*  Title:      HOL/Library/Permutation.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson and Thomas M Rasmussen
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```     4     Copyright   1995  University of Cambridge
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```     5
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```     6 TODO: it would be nice to prove (for "multiset", defined on
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```     7 HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)
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```     8 *)
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```     9
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```    10 header {* Permutations *}
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```    11
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```    12 theory Permutation = Main:
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```    13
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```    14 consts
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```    15   perm :: "('a list * 'a list) set"
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```    16
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```    17 syntax
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```    18   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
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```    19 translations
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```    20   "x <~~> y" == "(x, y) \<in> perm"
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```    21
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```    22 inductive perm
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```    23   intros
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```    24     Nil  [intro!]: "[] <~~> []"
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```    25     swap [intro!]: "y # x # l <~~> x # y # l"
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```    26     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
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```    27     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
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```    28
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```    29 lemma perm_refl [iff]: "l <~~> l"
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```    30   apply (induct l)
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```    31    apply auto
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```    32   done
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```    33
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```    34
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```    35 subsection {* Some examples of rule induction on permutations *}
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```    36
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```    37 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
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```    38     -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}
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```    39   apply (erule perm.induct)
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```    40      apply (simp_all (no_asm_simp))
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```    41   done
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```    42
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```    43 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
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```    44   apply (insert xperm_empty_imp_aux)
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```    45   apply blast
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```    46   done
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```    47
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```    48
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```    49 text {*
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```    50   \medskip This more general theorem is easier to understand!
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```    51   *}
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```    52
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```    53 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
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```    54   apply (erule perm.induct)
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```    55      apply simp_all
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```    56   done
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```    57
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```    58 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
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```    59   apply (drule perm_length)
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```    60   apply auto
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```    61   done
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```    62
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```    63 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
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```    64   apply (erule perm.induct)
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```    65      apply auto
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```    66   done
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```    67
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```    68 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
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```    69   apply (erule perm.induct)
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```    70      apply auto
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```    71   done
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```    72
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```    73
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```    74 subsection {* Ways of making new permutations *}
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```    75
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```    76 text {*
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```    77   We can insert the head anywhere in the list.
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```    78 *}
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```    79
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```    80 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
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```    81   apply (induct xs)
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```    82    apply auto
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```    83   done
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```    84
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```    85 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
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```    86   apply (induct xs)
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```    87     apply simp_all
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```    88   apply (blast intro: perm_append_Cons)
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```    89   done
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```    90
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```    91 lemma perm_append_single: "a # xs <~~> xs @ [a]"
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```    92   apply (rule perm.trans)
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```    93    prefer 2
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```    94    apply (rule perm_append_swap)
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```    95   apply simp
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```    96   done
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```    97
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```    98 lemma perm_rev: "rev xs <~~> xs"
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```    99   apply (induct xs)
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```   100    apply simp_all
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```   101   apply (blast intro!: perm_append_single intro: perm_sym)
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```   102   done
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```   103
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```   104 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
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```   105   apply (induct l)
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```   106    apply auto
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```   107   done
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```   108
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```   109 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
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```   110   apply (blast intro!: perm_append_swap perm_append1)
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```   111   done
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```   112
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```   113
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```   114 subsection {* Further results *}
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```   115
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```   116 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
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```   117   apply (blast intro: perm_empty_imp)
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```   118   done
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```   119
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```   120 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
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```   121   apply auto
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```   122   apply (erule perm_sym [THEN perm_empty_imp])
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```   123   done
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```   124
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```   125 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
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```   126   apply (erule perm.induct)
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```   127      apply auto
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```   128   done
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```   129
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```   130 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
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```   131   apply (blast intro: perm_sing_imp)
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```   132   done
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```   133
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```   134 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
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```   135   apply (blast dest: perm_sym)
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```   136   done
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```   137
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```   138
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```   139 subsection {* Removing elements *}
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```   140
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```   141 consts
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```   142   remove :: "'a => 'a list => 'a list"
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```   143 primrec
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```   144   "remove x [] = []"
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```   145   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
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```   146
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```   147 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
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```   148   apply (induct ys)
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```   149    apply auto
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```   150   done
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```   151
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```   152 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
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```   153   apply (induct l)
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```   154    apply auto
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```   155   done
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```   156
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```   157
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```   158 text {* \medskip Congruence rule *}
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```   159
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```   160 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
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```   161   apply (erule perm.induct)
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```   162      apply auto
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```   163   done
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```   164
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```   165 lemma remove_hd [simp]: "remove z (z # xs) = xs"
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```   166   apply auto
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```   167   done
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```   168
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```   169 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
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```   170   apply (drule_tac z = z in perm_remove_perm)
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```   171   apply auto
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```   172   done
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```   173
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```   174 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
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```   175   apply (blast intro: cons_perm_imp_perm)
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```   176   done
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```   177
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```   178 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
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```   179   apply (induct zs rule: rev_induct)
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```   180    apply (simp_all (no_asm_use))
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```   181   apply blast
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```   182   done
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```   183
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```   184 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
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```   185   apply (blast intro: append_perm_imp_perm perm_append1)
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```   186   done
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```   187
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```   188 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
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```   189   apply (safe intro!: perm_append2)
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```   190   apply (rule append_perm_imp_perm)
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```   191   apply (rule perm_append_swap [THEN perm.trans])
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```   192     -- {* the previous step helps this @{text blast} call succeed quickly *}
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```   193   apply (blast intro: perm_append_swap)
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```   194   done
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```   195
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```   196 end
```