src/HOL/NumberTheory/BijectionRel.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16417 9bc16273c2d4 child 18369 694ea14ab4f2 permissions -rw-r--r--
Constant "If" is now local
```     1 (*  Title:      HOL/NumberTheory/BijectionRel.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Thomas M. Rasmussen
```
```     4     Copyright   2000  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Bijections between sets *}
```
```     8
```
```     9 theory BijectionRel imports Main begin
```
```    10
```
```    11 text {*
```
```    12   Inductive definitions of bijections between two different sets and
```
```    13   between the same set.  Theorem for relating the two definitions.
```
```    14
```
```    15   \bigskip
```
```    16 *}
```
```    17
```
```    18 consts
```
```    19   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
```
```    20
```
```    21 inductive "bijR P"
```
```    22   intros
```
```    23   empty [simp]: "({}, {}) \<in> bijR P"
```
```    24   insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P
```
```    25     ==> (insert a A, insert b B) \<in> bijR P"
```
```    26
```
```    27 text {*
```
```    28   Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
```
```    29   (and similar for @{term A}).
```
```    30 *}
```
```    31
```
```    32 constdefs
```
```    33   bijP :: "('a => 'a => bool) => 'a set => bool"
```
```    34   "bijP P F == \<forall>a b. a \<in> F \<and> P a b --> b \<in> F"
```
```    35
```
```    36   uniqP :: "('a => 'a => bool) => bool"
```
```    37   "uniqP P == \<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d)"
```
```    38
```
```    39   symP :: "('a => 'a => bool) => bool"
```
```    40   "symP P == \<forall>a b. P a b = P b a"
```
```    41
```
```    42 consts
```
```    43   bijER :: "('a => 'a => bool) => 'a set set"
```
```    44
```
```    45 inductive "bijER P"
```
```    46   intros
```
```    47   empty [simp]: "{} \<in> bijER P"
```
```    48   insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
```
```    49   insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
```
```    50     ==> insert a (insert b A) \<in> bijER P"
```
```    51
```
```    52
```
```    53 text {* \medskip @{term bijR} *}
```
```    54
```
```    55 lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
```
```    56   apply (erule bijR.induct)
```
```    57   apply auto
```
```    58   done
```
```    59
```
```    60 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
```
```    61   apply (erule bijR.induct)
```
```    62   apply auto
```
```    63   done
```
```    64
```
```    65 lemma aux_induct:
```
```    66   "finite F ==> F \<subseteq> A ==> P {} ==>
```
```    67     (!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F))
```
```    68   ==> P F"
```
```    69 proof -
```
```    70   case rule_context
```
```    71   assume major: "finite F"
```
```    72     and subs: "F \<subseteq> A"
```
```    73   show ?thesis
```
```    74     apply (rule subs [THEN rev_mp])
```
```    75     apply (rule major [THEN finite_induct])
```
```    76      apply (blast intro: rule_context)+
```
```    77     done
```
```    78 qed
```
```    79
```
```    80 lemma inj_func_bijR_aux1:
```
```    81     "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
```
```    82   apply (unfold inj_on_def)
```
```    83   apply auto
```
```    84   done
```
```    85
```
```    86 lemma inj_func_bijR_aux2:
```
```    87   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
```
```    88     ==> (F, f ` F) \<in> bijR P"
```
```    89   apply (rule_tac F = F and A = A in aux_induct)
```
```    90      apply (rule finite_subset)
```
```    91       apply auto
```
```    92   apply (rule bijR.insert)
```
```    93      apply (rule_tac [3] inj_func_bijR_aux1)
```
```    94         apply auto
```
```    95   done
```
```    96
```
```    97 lemma inj_func_bijR:
```
```    98   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
```
```    99     ==> (A, f ` A) \<in> bijR P"
```
```   100   apply (rule inj_func_bijR_aux2)
```
```   101      apply auto
```
```   102   done
```
```   103
```
```   104
```
```   105 text {* \medskip @{term bijER} *}
```
```   106
```
```   107 lemma fin_bijER: "A \<in> bijER P ==> finite A"
```
```   108   apply (erule bijER.induct)
```
```   109     apply auto
```
```   110   done
```
```   111
```
```   112 lemma aux1:
```
```   113   "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
```
```   114     ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
```
```   115   apply (rule_tac x = "F - {a}" in exI)
```
```   116   apply auto
```
```   117   done
```
```   118
```
```   119 lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
```
```   120     ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
```
```   121     ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
```
```   122   apply (rule_tac x = "F - {a, b}" in exI)
```
```   123   apply auto
```
```   124   done
```
```   125
```
```   126 lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
```
```   127   apply (unfold uniqP_def)
```
```   128   apply auto
```
```   129   done
```
```   130
```
```   131 lemma aux_sym: "symP P ==> P a b = P b a"
```
```   132   apply (unfold symP_def)
```
```   133   apply auto
```
```   134   done
```
```   135
```
```   136 lemma aux_in1:
```
```   137     "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
```
```   138   apply (unfold bijP_def)
```
```   139   apply auto
```
```   140   apply (subgoal_tac "b \<noteq> a")
```
```   141    prefer 2
```
```   142    apply clarify
```
```   143   apply (simp add: aux_uniq)
```
```   144   apply auto
```
```   145   done
```
```   146
```
```   147 lemma aux_in2:
```
```   148   "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
```
```   149     ==> bijP P (insert a (insert b C)) ==> bijP P C"
```
```   150   apply (unfold bijP_def)
```
```   151   apply auto
```
```   152   apply (subgoal_tac "aa \<noteq> a")
```
```   153    prefer 2
```
```   154    apply clarify
```
```   155   apply (subgoal_tac "aa \<noteq> b")
```
```   156    prefer 2
```
```   157    apply clarify
```
```   158   apply (simp add: aux_uniq)
```
```   159   apply (subgoal_tac "ba \<noteq> a")
```
```   160    apply auto
```
```   161   apply (subgoal_tac "P a aa")
```
```   162    prefer 2
```
```   163    apply (simp add: aux_sym)
```
```   164   apply (subgoal_tac "b = aa")
```
```   165    apply (rule_tac [2] iffD1)
```
```   166     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
```
```   167       apply auto
```
```   168   done
```
```   169
```
```   170 lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
```
```   171   apply auto
```
```   172   done
```
```   173
```
```   174 lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
```
```   175   apply (unfold bijP_def)
```
```   176   apply (rule iffI)
```
```   177   apply (erule_tac [!] aux_foo)
```
```   178       apply simp_all
```
```   179   apply (rule iffD2)
```
```   180    apply (rule_tac P = P in aux_sym)
```
```   181    apply simp_all
```
```   182   done
```
```   183
```
```   184
```
```   185 lemma aux_bijRER:
```
```   186   "(A, B) \<in> bijR P ==> uniqP P ==> symP P
```
```   187     ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
```
```   188   apply (erule bijR.induct)
```
```   189    apply simp
```
```   190   apply (case_tac "a = b")
```
```   191    apply clarify
```
```   192    apply (case_tac "b \<in> F")
```
```   193     prefer 2
```
```   194     apply (simp add: subset_insert)
```
```   195    apply (cut_tac F = F and a = b and A = A and B = B in aux1)
```
```   196         prefer 6
```
```   197         apply clarify
```
```   198         apply (rule bijER.insert1)
```
```   199           apply simp_all
```
```   200    apply (subgoal_tac "bijP P C")
```
```   201     apply simp
```
```   202    apply (rule aux_in1)
```
```   203       apply simp_all
```
```   204   apply clarify
```
```   205   apply (case_tac "a \<in> F")
```
```   206    apply (case_tac [!] "b \<in> F")
```
```   207      apply (cut_tac F = F and a = a and b = b and A = A and B = B
```
```   208        in aux2)
```
```   209             apply (simp_all add: subset_insert)
```
```   210     apply clarify
```
```   211     apply (rule bijER.insert2)
```
```   212         apply simp_all
```
```   213     apply (subgoal_tac "bijP P C")
```
```   214      apply simp
```
```   215     apply (rule aux_in2)
```
```   216           apply simp_all
```
```   217    apply (subgoal_tac "b \<in> F")
```
```   218     apply (rule_tac [2] iffD1)
```
```   219      apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
```
```   220        apply (simp_all (no_asm_simp))
```
```   221    apply (subgoal_tac [2] "a \<in> F")
```
```   222     apply (rule_tac [3] iffD2)
```
```   223      apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
```
```   224        apply auto
```
```   225   done
```
```   226
```
```   227 lemma bijR_bijER:
```
```   228   "(A, A) \<in> bijR P ==>
```
```   229     bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
```
```   230   apply (cut_tac A = A and B = A and P = P in aux_bijRER)
```
```   231      apply auto
```
```   232   done
```
```   233
```
```   234 end
```