src/HOL/Library/Permutation.thy
author bulwahn
Thu, 08 Nov 2012 20:02:41 +0100
changeset 50037 f2a32197a33a
parent 44890 22f665a2e91c
child 51542 738598beeb26
permissions -rw-r--r--
tuned proofs
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
     1
(*  Title:      HOL/Library/Permutation.thy
15005
546c8e7e28d4 Norbert Voelker
paulson
parents: 14706
diff changeset
     2
    Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
     3
*)
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
     4
14706
71590b7733b7 tuned document;
wenzelm
parents: 11153
diff changeset
     5
header {* Permutations *}
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
     7
theory Permutation
30738
0842e906300c normalized imports
haftmann
parents: 27368
diff changeset
     8
imports Main Multiset
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
     9
begin
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    10
23755
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    11
inductive
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    12
  perm :: "'a list => 'a list => bool"  ("_ <~~> _"  [50, 50] 50)
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    13
  where
11153
950ede59c05a Blast bug fix made old proof too slow
paulson
parents: 11054
diff changeset
    14
    Nil  [intro!]: "[] <~~> []"
23755
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    15
  | swap [intro!]: "y # x # l <~~> x # y # l"
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    16
  | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
1c4672d130b1 Adapted to new inductive definition package.
berghofe
parents: 21404
diff changeset
    17
  | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    18
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    19
lemma perm_refl [iff]: "l <~~> l"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    20
  by (induct l) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    21
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    22
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    23
subsection {* Some examples of rule induction on permutations *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    24
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    25
lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
    26
  by (induct xs == "[]::'a list" ys pred: perm) simp_all
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    27
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    28
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    29
text {*
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    30
  \medskip This more general theorem is easier to understand!
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    31
  *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    32
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    33
lemma perm_length: "xs <~~> ys ==> length xs = length ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
    34
  by (induct pred: perm) simp_all
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    35
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    36
lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    37
  by (drule perm_length) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    38
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    39
lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
    40
  by (induct pred: perm) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    41
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    42
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    43
subsection {* Ways of making new permutations *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    44
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    45
text {*
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    46
  We can insert the head anywhere in the list.
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    47
*}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    48
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    49
lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    50
  by (induct xs) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    51
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    52
lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    53
  apply (induct xs)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    54
    apply simp_all
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    55
  apply (blast intro: perm_append_Cons)
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    56
  done
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    57
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    58
lemma perm_append_single: "a # xs <~~> xs @ [a]"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    59
  by (rule perm.trans [OF _ perm_append_swap]) simp
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    60
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    61
lemma perm_rev: "rev xs <~~> xs"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    62
  apply (induct xs)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    63
   apply simp_all
11153
950ede59c05a Blast bug fix made old proof too slow
paulson
parents: 11054
diff changeset
    64
  apply (blast intro!: perm_append_single intro: perm_sym)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    65
  done
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    66
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    67
lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    68
  by (induct l) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    69
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    70
lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    71
  by (blast intro!: perm_append_swap perm_append1)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    72
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    73
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    74
subsection {* Further results *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    75
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    76
lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    77
  by (blast intro: perm_empty_imp)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    78
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    79
lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    80
  apply auto
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    81
  apply (erule perm_sym [THEN perm_empty_imp])
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    82
  done
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    83
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
    84
lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
    85
  by (induct pred: perm) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    86
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    87
lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    88
  by (blast intro: perm_sing_imp)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    89
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    90
lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    91
  by (blast dest: perm_sym)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    92
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    93
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    94
subsection {* Removing elements *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    95
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 35272
diff changeset
    96
lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
    97
  by (induct ys) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    98
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
    99
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   100
text {* \medskip Congruence rule *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   101
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 35272
diff changeset
   102
lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   103
  by (induct pred: perm) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   104
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 35272
diff changeset
   105
lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 15005
diff changeset
   106
  by auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   107
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   108
lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   109
  by (drule_tac z = z in perm_remove_perm) auto
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   110
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   111
lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   112
  by (blast intro: cons_perm_imp_perm)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   113
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   114
lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   115
  apply (induct zs arbitrary: xs ys rule: rev_induct)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   116
   apply (simp_all (no_asm_use))
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   117
  apply blast
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   118
  done
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   119
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   120
lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   121
  by (blast intro: append_perm_imp_perm perm_append1)
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   122
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   123
lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   124
  apply (safe intro!: perm_append2)
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   125
  apply (rule append_perm_imp_perm)
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   126
  apply (rule perm_append_swap [THEN perm.trans])
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   127
    -- {* the previous step helps this @{text blast} call succeed quickly *}
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   128
  apply (blast intro: perm_append_swap)
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   129
  done
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   130
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 15005
diff changeset
   131
lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   132
  apply (rule iffI)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   133
  apply (erule_tac [2] perm.induct, simp_all add: union_ac)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   134
  apply (erule rev_mp, rule_tac x=ys in spec)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   135
  apply (induct_tac xs, auto)
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 35272
diff changeset
   136
  apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   137
  apply (subgoal_tac "a \<in> set x")
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   138
  apply (drule_tac z=a in perm.Cons)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   139
  apply (erule perm.trans, rule perm_sym, erule perm_remove)
15005
546c8e7e28d4 Norbert Voelker
paulson
parents: 14706
diff changeset
   140
  apply (drule_tac f=set_of in arg_cong, simp)
546c8e7e28d4 Norbert Voelker
paulson
parents: 14706
diff changeset
   141
  done
546c8e7e28d4 Norbert Voelker
paulson
parents: 14706
diff changeset
   142
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   143
lemma multiset_of_le_perm_append:
35272
c283ae736bea switched notations for pointwise and multiset order
haftmann
parents: 33498
diff changeset
   144
    "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 15140
diff changeset
   145
  apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 15005
diff changeset
   146
  apply (insert surj_multiset_of, drule surjD)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 15005
diff changeset
   147
  apply (blast intro: sym)+
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 15005
diff changeset
   148
  done
15005
546c8e7e28d4 Norbert Voelker
paulson
parents: 14706
diff changeset
   149
25277
95128fcdd7e8 added lemmas
nipkow
parents: 23755
diff changeset
   150
lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   151
  by (metis multiset_of_eq_perm multiset_of_eq_setD)
25277
95128fcdd7e8 added lemmas
nipkow
parents: 23755
diff changeset
   152
95128fcdd7e8 added lemmas
nipkow
parents: 23755
diff changeset
   153
lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   154
  apply (induct pred: perm)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   155
     apply simp_all
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 40122
diff changeset
   156
   apply fastforce
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   157
  apply (metis perm_set_eq)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   158
  done
25277
95128fcdd7e8 added lemmas
nipkow
parents: 23755
diff changeset
   159
25287
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   160
lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   161
  apply (induct xs arbitrary: ys rule: length_induct)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   162
  apply (case_tac "remdups xs", simp, simp)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   163
  apply (subgoal_tac "a : set (remdups ys)")
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   164
   prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   165
  apply (drule split_list) apply(elim exE conjE)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   166
  apply (drule_tac x=list in spec) apply(erule impE) prefer 2
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   167
   apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   168
    apply simp
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   169
    apply (subgoal_tac "a#list <~~> a#ysa@zs")
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   170
     apply (metis Cons_eq_appendI perm_append_Cons trans)
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 39916
diff changeset
   171
    apply (metis Cons Cons_eq_appendI distinct.simps(2)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   172
      distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   173
   apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 40122
diff changeset
   174
    apply (fastforce simp add: insert_ident)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   175
   apply (metis distinct_remdups set_remdups)
30742
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   176
   apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   177
   apply simp
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   178
   apply (subgoal_tac "length (remdups xs) \<le> length xs")
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   179
   apply simp
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   180
   apply (rule length_remdups_leq)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   181
  done
25287
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   182
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   183
lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   184
  by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
25287
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   185
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   186
lemma permutation_Ex_bij:
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   187
  assumes "xs <~~> ys"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   188
  shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   189
using assms proof induct
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   190
  case Nil then show ?case unfolding bij_betw_def by simp
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   191
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   192
  case (swap y x l)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   193
  show ?case
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   194
  proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   195
    show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
50037
f2a32197a33a tuned proofs
bulwahn
parents: 44890
diff changeset
   196
      by (auto simp: bij_betw_def)
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   197
    fix i assume "i < length(y#x#l)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   198
    show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   199
      by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   200
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   201
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   202
  case (Cons xs ys z)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   203
  then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   204
    perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   205
  let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   206
  show ?case
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   207
  proof (intro exI[of _ ?f] allI conjI impI)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   208
    have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   209
            "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
39078
39f8f6d1eb74 Fixes lemma names
hoelzl
parents: 39075
diff changeset
   210
      by (simp_all add: lessThan_Suc_eq_insert_0)
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   211
    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   212
    proof (rule bij_betw_combine)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   213
      show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   214
        using bij unfolding bij_betw_def
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   215
        by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   216
    qed (auto simp: bij_betw_def)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   217
    fix i assume "i < length (z#xs)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   218
    then show "(z # xs) ! i = (z # ys) ! (?f i)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   219
      using perm by (cases i) auto
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   220
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   221
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   222
  case (trans xs ys zs)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   223
  then obtain f g where
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   224
    bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   225
    perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   226
  show ?case
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   227
  proof (intro exI[of _ "g\<circ>f"] conjI allI impI)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   228
    show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   229
      using bij by (rule bij_betw_trans)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   230
    fix i assume "i < length xs"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   231
    with bij have "f i < length ys" unfolding bij_betw_def by force
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   232
    with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   233
      using trans(1,3)[THEN perm_length] perm by force
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   234
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   235
qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   236
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   237
end