src/HOL/Old_Number_Theory/BijectionRel.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 38159 e9b4835a54ee
child 61382 efac889fccbc
permissions -rw-r--r--
modernized header uniformly as section;
wenzelm@38159
     1
(*  Title:      HOL/Old_Number_Theory/BijectionRel.thy
wenzelm@38159
     2
    Author:     Thomas M. Rasmussen
wenzelm@11049
     3
    Copyright   2000  University of Cambridge
paulson@9508
     4
*)
paulson@9508
     5
wenzelm@58889
     6
section {* Bijections between sets *}
wenzelm@11049
     7
wenzelm@38159
     8
theory BijectionRel
wenzelm@38159
     9
imports Main
wenzelm@38159
    10
begin
wenzelm@11049
    11
wenzelm@11049
    12
text {*
wenzelm@11049
    13
  Inductive definitions of bijections between two different sets and
wenzelm@11049
    14
  between the same set.  Theorem for relating the two definitions.
wenzelm@11049
    15
wenzelm@11049
    16
  \bigskip
wenzelm@11049
    17
*}
paulson@9508
    18
berghofe@23755
    19
inductive_set
wenzelm@11049
    20
  bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
berghofe@23755
    21
  for P :: "'a => 'b => bool"
berghofe@23755
    22
where
wenzelm@11049
    23
  empty [simp]: "({}, {}) \<in> bijR P"
berghofe@23755
    24
| insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P
wenzelm@11049
    25
    ==> (insert a A, insert b B) \<in> bijR P"
wenzelm@11049
    26
wenzelm@11049
    27
text {*
wenzelm@11049
    28
  Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
wenzelm@11049
    29
  (and similar for @{term A}).
wenzelm@11049
    30
*}
paulson@9508
    31
wenzelm@19670
    32
definition
wenzelm@21404
    33
  bijP :: "('a => 'a => bool) => 'a set => bool" where
wenzelm@19670
    34
  "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
wenzelm@11049
    35
wenzelm@21404
    36
definition
wenzelm@21404
    37
  uniqP :: "('a => 'a => bool) => bool" where
wenzelm@19670
    38
  "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
wenzelm@11049
    39
wenzelm@21404
    40
definition
wenzelm@21404
    41
  symP :: "('a => 'a => bool) => bool" where
wenzelm@19670
    42
  "symP P = (\<forall>a b. P a b = P b a)"
paulson@9508
    43
berghofe@23755
    44
inductive_set
wenzelm@11049
    45
  bijER :: "('a => 'a => bool) => 'a set set"
berghofe@23755
    46
  for P :: "'a => 'a => bool"
berghofe@23755
    47
where
wenzelm@11049
    48
  empty [simp]: "{} \<in> bijER P"
berghofe@23755
    49
| insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
berghofe@23755
    50
| insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
wenzelm@11049
    51
    ==> insert a (insert b A) \<in> bijER P"
wenzelm@11049
    52
wenzelm@11049
    53
wenzelm@11049
    54
text {* \medskip @{term bijR} *}
wenzelm@11049
    55
wenzelm@11049
    56
lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
wenzelm@11049
    57
  apply (erule bijR.induct)
wenzelm@11049
    58
  apply auto
wenzelm@11049
    59
  done
wenzelm@11049
    60
wenzelm@11049
    61
lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
wenzelm@11049
    62
  apply (erule bijR.induct)
wenzelm@11049
    63
  apply auto
wenzelm@11049
    64
  done
wenzelm@11049
    65
wenzelm@11049
    66
lemma aux_induct:
wenzelm@18369
    67
  assumes major: "finite F"
wenzelm@11049
    68
    and subs: "F \<subseteq> A"
wenzelm@18369
    69
    and cases: "P {}"
wenzelm@18369
    70
      "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
wenzelm@18369
    71
  shows "P F"
wenzelm@18369
    72
  using major subs
berghofe@22274
    73
  apply (induct set: finite)
wenzelm@18369
    74
   apply (blast intro: cases)+
wenzelm@18369
    75
  done
wenzelm@18369
    76
wenzelm@11049
    77
wenzelm@13524
    78
lemma inj_func_bijR_aux1:
wenzelm@13524
    79
    "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
wenzelm@11049
    80
  apply (unfold inj_on_def)
wenzelm@11049
    81
  apply auto
wenzelm@11049
    82
  done
wenzelm@11049
    83
wenzelm@13524
    84
lemma inj_func_bijR_aux2:
wenzelm@11049
    85
  "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
wenzelm@11049
    86
    ==> (F, f ` F) \<in> bijR P"
wenzelm@11049
    87
  apply (rule_tac F = F and A = A in aux_induct)
wenzelm@11049
    88
     apply (rule finite_subset)
wenzelm@11049
    89
      apply auto
wenzelm@11049
    90
  apply (rule bijR.insert)
wenzelm@13524
    91
     apply (rule_tac [3] inj_func_bijR_aux1)
wenzelm@11049
    92
        apply auto
wenzelm@11049
    93
  done
wenzelm@11049
    94
wenzelm@11049
    95
lemma inj_func_bijR:
wenzelm@11049
    96
  "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
wenzelm@11049
    97
    ==> (A, f ` A) \<in> bijR P"
wenzelm@13524
    98
  apply (rule inj_func_bijR_aux2)
wenzelm@11049
    99
     apply auto
wenzelm@11049
   100
  done
wenzelm@11049
   101
wenzelm@11049
   102
wenzelm@11049
   103
text {* \medskip @{term bijER} *}
wenzelm@11049
   104
wenzelm@11049
   105
lemma fin_bijER: "A \<in> bijER P ==> finite A"
wenzelm@11049
   106
  apply (erule bijER.induct)
wenzelm@11049
   107
    apply auto
wenzelm@11049
   108
  done
wenzelm@11049
   109
wenzelm@11049
   110
lemma aux1:
wenzelm@11049
   111
  "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
wenzelm@11049
   112
    ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
wenzelm@11049
   113
  apply (rule_tac x = "F - {a}" in exI)
wenzelm@11049
   114
  apply auto
wenzelm@11049
   115
  done
wenzelm@11049
   116
wenzelm@11049
   117
lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
wenzelm@11049
   118
    ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
wenzelm@11049
   119
    ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
wenzelm@11049
   120
  apply (rule_tac x = "F - {a, b}" in exI)
wenzelm@11049
   121
  apply auto
wenzelm@11049
   122
  done
wenzelm@11049
   123
wenzelm@11049
   124
lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
wenzelm@11049
   125
  apply (unfold uniqP_def)
wenzelm@11049
   126
  apply auto
wenzelm@11049
   127
  done
wenzelm@11049
   128
wenzelm@11049
   129
lemma aux_sym: "symP P ==> P a b = P b a"
wenzelm@11049
   130
  apply (unfold symP_def)
wenzelm@11049
   131
  apply auto
wenzelm@11049
   132
  done
wenzelm@11049
   133
wenzelm@11049
   134
lemma aux_in1:
wenzelm@11049
   135
    "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
wenzelm@11049
   136
  apply (unfold bijP_def)
wenzelm@11049
   137
  apply auto
wenzelm@11049
   138
  apply (subgoal_tac "b \<noteq> a")
wenzelm@11049
   139
   prefer 2
wenzelm@11049
   140
   apply clarify
wenzelm@11049
   141
  apply (simp add: aux_uniq)
wenzelm@11049
   142
  apply auto
wenzelm@11049
   143
  done
wenzelm@11049
   144
wenzelm@11049
   145
lemma aux_in2:
wenzelm@11049
   146
  "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
wenzelm@11049
   147
    ==> bijP P (insert a (insert b C)) ==> bijP P C"
wenzelm@11049
   148
  apply (unfold bijP_def)
wenzelm@11049
   149
  apply auto
wenzelm@11049
   150
  apply (subgoal_tac "aa \<noteq> a")
wenzelm@11049
   151
   prefer 2
wenzelm@11049
   152
   apply clarify
wenzelm@11049
   153
  apply (subgoal_tac "aa \<noteq> b")
wenzelm@11049
   154
   prefer 2
wenzelm@11049
   155
   apply clarify
wenzelm@11049
   156
  apply (simp add: aux_uniq)
wenzelm@11049
   157
  apply (subgoal_tac "ba \<noteq> a")
wenzelm@11049
   158
   apply auto
wenzelm@11049
   159
  apply (subgoal_tac "P a aa")
wenzelm@11049
   160
   prefer 2
wenzelm@11049
   161
   apply (simp add: aux_sym)
wenzelm@11049
   162
  apply (subgoal_tac "b = aa")
wenzelm@11049
   163
   apply (rule_tac [2] iffD1)
wenzelm@11049
   164
    apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
wenzelm@11049
   165
      apply auto
wenzelm@11049
   166
  done
wenzelm@11049
   167
wenzelm@13524
   168
lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
wenzelm@11049
   169
  apply auto
wenzelm@11049
   170
  done
wenzelm@11049
   171
wenzelm@11049
   172
lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
wenzelm@11049
   173
  apply (unfold bijP_def)
wenzelm@11049
   174
  apply (rule iffI)
wenzelm@13524
   175
  apply (erule_tac [!] aux_foo)
wenzelm@11049
   176
      apply simp_all
wenzelm@11049
   177
  apply (rule iffD2)
wenzelm@11049
   178
   apply (rule_tac P = P in aux_sym)
wenzelm@11049
   179
   apply simp_all
wenzelm@11049
   180
  done
wenzelm@11049
   181
wenzelm@11049
   182
wenzelm@11049
   183
lemma aux_bijRER:
wenzelm@11049
   184
  "(A, B) \<in> bijR P ==> uniqP P ==> symP P
wenzelm@11049
   185
    ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
wenzelm@11049
   186
  apply (erule bijR.induct)
wenzelm@11049
   187
   apply simp
wenzelm@11049
   188
  apply (case_tac "a = b")
wenzelm@11049
   189
   apply clarify
wenzelm@11049
   190
   apply (case_tac "b \<in> F")
wenzelm@11049
   191
    prefer 2
wenzelm@11049
   192
    apply (simp add: subset_insert)
wenzelm@11049
   193
   apply (cut_tac F = F and a = b and A = A and B = B in aux1)
wenzelm@11049
   194
        prefer 6
wenzelm@11049
   195
        apply clarify
wenzelm@11049
   196
        apply (rule bijER.insert1)
wenzelm@11049
   197
          apply simp_all
wenzelm@11049
   198
   apply (subgoal_tac "bijP P C")
wenzelm@11049
   199
    apply simp
wenzelm@11049
   200
   apply (rule aux_in1)
wenzelm@11049
   201
      apply simp_all
wenzelm@11049
   202
  apply clarify
wenzelm@11049
   203
  apply (case_tac "a \<in> F")
wenzelm@11049
   204
   apply (case_tac [!] "b \<in> F")
wenzelm@11049
   205
     apply (cut_tac F = F and a = a and b = b and A = A and B = B
wenzelm@11049
   206
       in aux2)
wenzelm@11049
   207
            apply (simp_all add: subset_insert)
wenzelm@11049
   208
    apply clarify
wenzelm@11049
   209
    apply (rule bijER.insert2)
wenzelm@11049
   210
        apply simp_all
wenzelm@11049
   211
    apply (subgoal_tac "bijP P C")
wenzelm@11049
   212
     apply simp
wenzelm@11049
   213
    apply (rule aux_in2)
wenzelm@11049
   214
          apply simp_all
wenzelm@11049
   215
   apply (subgoal_tac "b \<in> F")
wenzelm@11049
   216
    apply (rule_tac [2] iffD1)
wenzelm@11049
   217
     apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
wenzelm@11049
   218
       apply (simp_all (no_asm_simp))
wenzelm@11049
   219
   apply (subgoal_tac [2] "a \<in> F")
wenzelm@11049
   220
    apply (rule_tac [3] iffD2)
wenzelm@11049
   221
     apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
wenzelm@11049
   222
       apply auto
wenzelm@11049
   223
  done
wenzelm@11049
   224
wenzelm@11049
   225
lemma bijR_bijER:
wenzelm@11049
   226
  "(A, A) \<in> bijR P ==>
wenzelm@11049
   227
    bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
wenzelm@11049
   228
  apply (cut_tac A = A and B = A and P = P in aux_bijRER)
wenzelm@11049
   229
     apply auto
wenzelm@11049
   230
  done
paulson@9508
   231
paulson@9508
   232
end