src/HOL/Old_Number_Theory/BijectionRel.thy
author haftmann
Fri Nov 27 08:41:10 2009 +0100 (2009-11-27)
changeset 33963 977b94b64905
parent 32479 521cc9bf2958
child 38159 e9b4835a54ee
permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     1 (*  Author:     Thomas M. Rasmussen
     2     Copyright   2000  University of Cambridge
     3 *)
     4 
     5 header {* Bijections between sets *}
     6 
     7 theory BijectionRel imports Main begin
     8 
     9 text {*
    10   Inductive definitions of bijections between two different sets and
    11   between the same set.  Theorem for relating the two definitions.
    12 
    13   \bigskip
    14 *}
    15 
    16 inductive_set
    17   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
    18   for P :: "'a => 'b => bool"
    19 where
    20   empty [simp]: "({}, {}) \<in> bijR P"
    21 | insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P
    22     ==> (insert a A, insert b B) \<in> bijR P"
    23 
    24 text {*
    25   Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
    26   (and similar for @{term A}).
    27 *}
    28 
    29 definition
    30   bijP :: "('a => 'a => bool) => 'a set => bool" where
    31   "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
    32 
    33 definition
    34   uniqP :: "('a => 'a => bool) => bool" where
    35   "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
    36 
    37 definition
    38   symP :: "('a => 'a => bool) => bool" where
    39   "symP P = (\<forall>a b. P a b = P b a)"
    40 
    41 inductive_set
    42   bijER :: "('a => 'a => bool) => 'a set set"
    43   for P :: "'a => 'a => bool"
    44 where
    45   empty [simp]: "{} \<in> bijER P"
    46 | insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
    47 | insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
    48     ==> insert a (insert b A) \<in> bijER P"
    49 
    50 
    51 text {* \medskip @{term bijR} *}
    52 
    53 lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
    54   apply (erule bijR.induct)
    55   apply auto
    56   done
    57 
    58 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
    59   apply (erule bijR.induct)
    60   apply auto
    61   done
    62 
    63 lemma aux_induct:
    64   assumes major: "finite F"
    65     and subs: "F \<subseteq> A"
    66     and cases: "P {}"
    67       "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
    68   shows "P F"
    69   using major subs
    70   apply (induct set: finite)
    71    apply (blast intro: cases)+
    72   done
    73 
    74 
    75 lemma inj_func_bijR_aux1:
    76     "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
    77   apply (unfold inj_on_def)
    78   apply auto
    79   done
    80 
    81 lemma inj_func_bijR_aux2:
    82   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
    83     ==> (F, f ` F) \<in> bijR P"
    84   apply (rule_tac F = F and A = A in aux_induct)
    85      apply (rule finite_subset)
    86       apply auto
    87   apply (rule bijR.insert)
    88      apply (rule_tac [3] inj_func_bijR_aux1)
    89         apply auto
    90   done
    91 
    92 lemma inj_func_bijR:
    93   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
    94     ==> (A, f ` A) \<in> bijR P"
    95   apply (rule inj_func_bijR_aux2)
    96      apply auto
    97   done
    98 
    99 
   100 text {* \medskip @{term bijER} *}
   101 
   102 lemma fin_bijER: "A \<in> bijER P ==> finite A"
   103   apply (erule bijER.induct)
   104     apply auto
   105   done
   106 
   107 lemma aux1:
   108   "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
   109     ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
   110   apply (rule_tac x = "F - {a}" in exI)
   111   apply auto
   112   done
   113 
   114 lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
   115     ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
   116     ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
   117   apply (rule_tac x = "F - {a, b}" in exI)
   118   apply auto
   119   done
   120 
   121 lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
   122   apply (unfold uniqP_def)
   123   apply auto
   124   done
   125 
   126 lemma aux_sym: "symP P ==> P a b = P b a"
   127   apply (unfold symP_def)
   128   apply auto
   129   done
   130 
   131 lemma aux_in1:
   132     "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
   133   apply (unfold bijP_def)
   134   apply auto
   135   apply (subgoal_tac "b \<noteq> a")
   136    prefer 2
   137    apply clarify
   138   apply (simp add: aux_uniq)
   139   apply auto
   140   done
   141 
   142 lemma aux_in2:
   143   "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
   144     ==> bijP P (insert a (insert b C)) ==> bijP P C"
   145   apply (unfold bijP_def)
   146   apply auto
   147   apply (subgoal_tac "aa \<noteq> a")
   148    prefer 2
   149    apply clarify
   150   apply (subgoal_tac "aa \<noteq> b")
   151    prefer 2
   152    apply clarify
   153   apply (simp add: aux_uniq)
   154   apply (subgoal_tac "ba \<noteq> a")
   155    apply auto
   156   apply (subgoal_tac "P a aa")
   157    prefer 2
   158    apply (simp add: aux_sym)
   159   apply (subgoal_tac "b = aa")
   160    apply (rule_tac [2] iffD1)
   161     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
   162       apply auto
   163   done
   164 
   165 lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
   166   apply auto
   167   done
   168 
   169 lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
   170   apply (unfold bijP_def)
   171   apply (rule iffI)
   172   apply (erule_tac [!] aux_foo)
   173       apply simp_all
   174   apply (rule iffD2)
   175    apply (rule_tac P = P in aux_sym)
   176    apply simp_all
   177   done
   178 
   179 
   180 lemma aux_bijRER:
   181   "(A, B) \<in> bijR P ==> uniqP P ==> symP P
   182     ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
   183   apply (erule bijR.induct)
   184    apply simp
   185   apply (case_tac "a = b")
   186    apply clarify
   187    apply (case_tac "b \<in> F")
   188     prefer 2
   189     apply (simp add: subset_insert)
   190    apply (cut_tac F = F and a = b and A = A and B = B in aux1)
   191         prefer 6
   192         apply clarify
   193         apply (rule bijER.insert1)
   194           apply simp_all
   195    apply (subgoal_tac "bijP P C")
   196     apply simp
   197    apply (rule aux_in1)
   198       apply simp_all
   199   apply clarify
   200   apply (case_tac "a \<in> F")
   201    apply (case_tac [!] "b \<in> F")
   202      apply (cut_tac F = F and a = a and b = b and A = A and B = B
   203        in aux2)
   204             apply (simp_all add: subset_insert)
   205     apply clarify
   206     apply (rule bijER.insert2)
   207         apply simp_all
   208     apply (subgoal_tac "bijP P C")
   209      apply simp
   210     apply (rule aux_in2)
   211           apply simp_all
   212    apply (subgoal_tac "b \<in> F")
   213     apply (rule_tac [2] iffD1)
   214      apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
   215        apply (simp_all (no_asm_simp))
   216    apply (subgoal_tac [2] "a \<in> F")
   217     apply (rule_tac [3] iffD2)
   218      apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
   219        apply auto
   220   done
   221 
   222 lemma bijR_bijER:
   223   "(A, A) \<in> bijR P ==>
   224     bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
   225   apply (cut_tac A = A and B = A and P = P in aux_bijRER)
   226      apply auto
   227   done
   228 
   229 end