src/HOL/Library/Permutation.thy
 author paulson Fri Feb 16 13:37:21 2001 +0100 (2001-02-16) changeset 11153 950ede59c05a parent 11054 a5404c70982f child 14706 71590b7733b7 permissions -rw-r--r--
Blast bug fix made old proof too slow
```     1 (*  Title:      HOL/Library/Permutation.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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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 {*
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```    11  \title{Permutations}
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```    12  \author{Lawrence C Paulson and Thomas M Rasmussen}
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```    13 *}
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```    14
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```    15 theory Permutation = Main:
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```    16
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```    17 consts
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```    18   perm :: "('a list * 'a list) set"
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```    19
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```    20 syntax
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```    21   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
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```    22 translations
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```    23   "x <~~> y" == "(x, y) \<in> perm"
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```    24
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```    25 inductive perm
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```    26   intros
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```    27     Nil  [intro!]: "[] <~~> []"
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```    28     swap [intro!]: "y # x # l <~~> x # y # l"
```
```    29     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
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```    30     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
```
```    31
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```    32 lemma perm_refl [iff]: "l <~~> l"
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```    33   apply (induct l)
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```    34    apply auto
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```    35   done
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```    36
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```    37
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```    38 subsection {* Some examples of rule induction on permutations *}
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```    39
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```    40 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
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```    41     -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}
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```    42   apply (erule perm.induct)
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```    43      apply (simp_all (no_asm_simp))
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```    44   done
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```    45
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```    46 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
```
```    47   apply (insert xperm_empty_imp_aux)
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```    48   apply blast
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```    49   done
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```    50
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```    51
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```    52 text {*
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```    53   \medskip This more general theorem is easier to understand!
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```    54   *}
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```    55
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```    56 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
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```    57   apply (erule perm.induct)
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```    58      apply simp_all
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```    59   done
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```    60
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```    61 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
```
```    62   apply (drule perm_length)
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```    63   apply auto
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```    64   done
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```    65
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```    66 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
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```    67   apply (erule perm.induct)
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```    68      apply auto
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```    69   done
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```    70
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```    71 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
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```    72   apply (erule perm.induct)
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```    73      apply auto
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```    74   done
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```    75
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```    76
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```    77 subsection {* Ways of making new permutations *}
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```    78
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```    79 text {*
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```    80   We can insert the head anywhere in the list.
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```    81 *}
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```    82
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```    83 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
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```    84   apply (induct xs)
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```    85    apply auto
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```    86   done
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```    87
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```    88 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
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```    89   apply (induct xs)
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```    90     apply simp_all
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```    91   apply (blast intro: perm_append_Cons)
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```    92   done
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```    93
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```    94 lemma perm_append_single: "a # xs <~~> xs @ [a]"
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```    95   apply (rule perm.trans)
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```    96    prefer 2
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```    97    apply (rule perm_append_swap)
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```    98   apply simp
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```    99   done
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```   100
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```   101 lemma perm_rev: "rev xs <~~> xs"
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```   102   apply (induct xs)
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```   103    apply simp_all
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```   104   apply (blast intro!: perm_append_single intro: perm_sym)
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```   105   done
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```   106
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```   107 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
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```   108   apply (induct l)
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```   109    apply auto
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```   110   done
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```   111
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```   112 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
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```   113   apply (blast intro!: perm_append_swap perm_append1)
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```   114   done
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```   115
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```   116
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```   117 subsection {* Further results *}
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```   118
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```   119 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
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```   120   apply (blast intro: perm_empty_imp)
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```   121   done
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```   122
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```   123 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
```
```   124   apply auto
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```   125   apply (erule perm_sym [THEN perm_empty_imp])
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```   126   done
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```   127
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```   128 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
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```   129   apply (erule perm.induct)
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```   130      apply auto
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```   131   done
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```   132
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```   133 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
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```   134   apply (blast intro: perm_sing_imp)
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```   135   done
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```   136
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```   137 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
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```   138   apply (blast dest: perm_sym)
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```   139   done
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```   140
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```   141
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```   142 subsection {* Removing elements *}
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```   143
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```   144 consts
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```   145   remove :: "'a => 'a list => 'a list"
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```   146 primrec
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```   147   "remove x [] = []"
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```   148   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
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```   149
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```   150 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
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```   151   apply (induct ys)
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```   152    apply auto
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```   153   done
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```   154
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```   155 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
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```   156   apply (induct l)
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```   157    apply auto
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```   158   done
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```   159
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```   160
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```   161 text {* \medskip Congruence rule *}
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```   162
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```   163 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
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```   164   apply (erule perm.induct)
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```   165      apply auto
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```   166   done
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```   167
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```   168 lemma remove_hd [simp]: "remove z (z # xs) = xs"
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```   169   apply auto
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```   170   done
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```   171
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```   172 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
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```   173   apply (drule_tac z = z in perm_remove_perm)
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```   174   apply auto
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```   175   done
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```   176
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```   177 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
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```   178   apply (blast intro: cons_perm_imp_perm)
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```   179   done
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```   180
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```   181 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
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```   182   apply (induct zs rule: rev_induct)
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```   183    apply (simp_all (no_asm_use))
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```   184   apply blast
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```   185   done
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```   186
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```   187 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
```
```   188   apply (blast intro: append_perm_imp_perm perm_append1)
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```   189   done
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```   190
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```   191 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
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```   192   apply (safe intro!: perm_append2)
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```   193   apply (rule append_perm_imp_perm)
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```   194   apply (rule perm_append_swap [THEN perm.trans])
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```   195     -- {* the previous step helps this @{text blast} call succeed quickly *}
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```   196   apply (blast intro: perm_append_swap)
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```   197   done
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```   198
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```   199 end
```