src/HOL/Old_Number_Theory/BijectionRel.thy
author wenzelm
Fri Aug 06 12:37:00 2010 +0200 (2010-08-06)
changeset 38159 e9b4835a54ee
parent 32479 521cc9bf2958
child 58889 5b7a9633cfa8
permissions -rw-r--r--
modernized specifications;
tuned headers;
     1 (*  Title:      HOL/Old_Number_Theory/BijectionRel.thy
     2     Author:     Thomas M. Rasmussen
     3     Copyright   2000  University of Cambridge
     4 *)
     5 
     6 header {* Bijections between sets *}
     7 
     8 theory BijectionRel
     9 imports Main
    10 begin
    11 
    12 text {*
    13   Inductive definitions of bijections between two different sets and
    14   between the same set.  Theorem for relating the two definitions.
    15 
    16   \bigskip
    17 *}
    18 
    19 inductive_set
    20   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
    21   for P :: "'a => 'b => bool"
    22 where
    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 definition
    33   bijP :: "('a => 'a => bool) => 'a set => bool" where
    34   "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
    35 
    36 definition
    37   uniqP :: "('a => 'a => bool) => bool" where
    38   "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
    39 
    40 definition
    41   symP :: "('a => 'a => bool) => bool" where
    42   "symP P = (\<forall>a b. P a b = P b a)"
    43 
    44 inductive_set
    45   bijER :: "('a => 'a => bool) => 'a set set"
    46   for P :: "'a => 'a => bool"
    47 where
    48   empty [simp]: "{} \<in> bijER P"
    49 | insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
    50 | insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
    51     ==> insert a (insert b A) \<in> bijER P"
    52 
    53 
    54 text {* \medskip @{term bijR} *}
    55 
    56 lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
    57   apply (erule bijR.induct)
    58   apply auto
    59   done
    60 
    61 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
    62   apply (erule bijR.induct)
    63   apply auto
    64   done
    65 
    66 lemma aux_induct:
    67   assumes major: "finite F"
    68     and subs: "F \<subseteq> A"
    69     and cases: "P {}"
    70       "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
    71   shows "P F"
    72   using major subs
    73   apply (induct set: finite)
    74    apply (blast intro: cases)+
    75   done
    76 
    77 
    78 lemma inj_func_bijR_aux1:
    79     "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
    80   apply (unfold inj_on_def)
    81   apply auto
    82   done
    83 
    84 lemma inj_func_bijR_aux2:
    85   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
    86     ==> (F, f ` F) \<in> bijR P"
    87   apply (rule_tac F = F and A = A in aux_induct)
    88      apply (rule finite_subset)
    89       apply auto
    90   apply (rule bijR.insert)
    91      apply (rule_tac [3] inj_func_bijR_aux1)
    92         apply auto
    93   done
    94 
    95 lemma inj_func_bijR:
    96   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
    97     ==> (A, f ` A) \<in> bijR P"
    98   apply (rule inj_func_bijR_aux2)
    99      apply auto
   100   done
   101 
   102 
   103 text {* \medskip @{term bijER} *}
   104 
   105 lemma fin_bijER: "A \<in> bijER P ==> finite A"
   106   apply (erule bijER.induct)
   107     apply auto
   108   done
   109 
   110 lemma aux1:
   111   "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
   112     ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
   113   apply (rule_tac x = "F - {a}" in exI)
   114   apply auto
   115   done
   116 
   117 lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
   118     ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
   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"
   120   apply (rule_tac x = "F - {a, b}" in exI)
   121   apply auto
   122   done
   123 
   124 lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
   125   apply (unfold uniqP_def)
   126   apply auto
   127   done
   128 
   129 lemma aux_sym: "symP P ==> P a b = P b a"
   130   apply (unfold symP_def)
   131   apply auto
   132   done
   133 
   134 lemma aux_in1:
   135     "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
   136   apply (unfold bijP_def)
   137   apply auto
   138   apply (subgoal_tac "b \<noteq> a")
   139    prefer 2
   140    apply clarify
   141   apply (simp add: aux_uniq)
   142   apply auto
   143   done
   144 
   145 lemma aux_in2:
   146   "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
   147     ==> bijP P (insert a (insert b C)) ==> bijP P C"
   148   apply (unfold bijP_def)
   149   apply auto
   150   apply (subgoal_tac "aa \<noteq> a")
   151    prefer 2
   152    apply clarify
   153   apply (subgoal_tac "aa \<noteq> b")
   154    prefer 2
   155    apply clarify
   156   apply (simp add: aux_uniq)
   157   apply (subgoal_tac "ba \<noteq> a")
   158    apply auto
   159   apply (subgoal_tac "P a aa")
   160    prefer 2
   161    apply (simp add: aux_sym)
   162   apply (subgoal_tac "b = aa")
   163    apply (rule_tac [2] iffD1)
   164     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
   165       apply auto
   166   done
   167 
   168 lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
   169   apply auto
   170   done
   171 
   172 lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
   173   apply (unfold bijP_def)
   174   apply (rule iffI)
   175   apply (erule_tac [!] aux_foo)
   176       apply simp_all
   177   apply (rule iffD2)
   178    apply (rule_tac P = P in aux_sym)
   179    apply simp_all
   180   done
   181 
   182 
   183 lemma aux_bijRER:
   184   "(A, B) \<in> bijR P ==> uniqP P ==> symP P
   185     ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
   186   apply (erule bijR.induct)
   187    apply simp
   188   apply (case_tac "a = b")
   189    apply clarify
   190    apply (case_tac "b \<in> F")
   191     prefer 2
   192     apply (simp add: subset_insert)
   193    apply (cut_tac F = F and a = b and A = A and B = B in aux1)
   194         prefer 6
   195         apply clarify
   196         apply (rule bijER.insert1)
   197           apply simp_all
   198    apply (subgoal_tac "bijP P C")
   199     apply simp
   200    apply (rule aux_in1)
   201       apply simp_all
   202   apply clarify
   203   apply (case_tac "a \<in> F")
   204    apply (case_tac [!] "b \<in> F")
   205      apply (cut_tac F = F and a = a and b = b and A = A and B = B
   206        in aux2)
   207             apply (simp_all add: subset_insert)
   208     apply clarify
   209     apply (rule bijER.insert2)
   210         apply simp_all
   211     apply (subgoal_tac "bijP P C")
   212      apply simp
   213     apply (rule aux_in2)
   214           apply simp_all
   215    apply (subgoal_tac "b \<in> F")
   216     apply (rule_tac [2] iffD1)
   217      apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
   218        apply (simp_all (no_asm_simp))
   219    apply (subgoal_tac [2] "a \<in> F")
   220     apply (rule_tac [3] iffD2)
   221      apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
   222        apply auto
   223   done
   224 
   225 lemma bijR_bijER:
   226   "(A, A) \<in> bijR P ==>
   227     bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
   228   apply (cut_tac A = A and B = A and P = P in aux_bijRER)
   229      apply auto
   230   done
   231 
   232 end