src/HOL/NumberTheory/BijectionRel.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 18369 694ea14ab4f2 child 19670 2e4a143c73c5 permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     1 (*  Title:      HOL/NumberTheory/BijectionRel.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Thomas M. Rasmussen
```
```     4     Copyright   2000  University of Cambridge
```
```     5 *)
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```     6
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```     7 header {* Bijections between sets *}
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```     8
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```     9 theory BijectionRel imports Main begin
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```    10
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```    11 text {*
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```    12   Inductive definitions of bijections between two different sets and
```
```    13   between the same set.  Theorem for relating the two definitions.
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```    14
```
```    15   \bigskip
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```    16 *}
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```    17
```
```    18 consts
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```    19   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
```
```    20
```
```    21 inductive "bijR P"
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```    22   intros
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```    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 {*
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```    28   Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
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```    29   (and similar for @{term A}).
```
```    30 *}
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```    31
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```    32 constdefs
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```    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
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```    42 consts
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```    43   bijER :: "('a => 'a => bool) => 'a set set"
```
```    44
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```    45 inductive "bijER P"
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```    46   intros
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```    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
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```    60 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
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```    61   apply (erule bijR.induct)
```
```    62   apply auto
```
```    63   done
```
```    64
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```    65 lemma aux_induct:
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```    66   assumes major: "finite F"
```
```    67     and subs: "F \<subseteq> A"
```
```    68     and cases: "P {}"
```
```    69       "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
```
```    70   shows "P F"
```
```    71   using major subs
```
```    72   apply (induct set: Finites)
```
```    73    apply (blast intro: cases)+
```
```    74   done
```
```    75
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```    76
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```    77 lemma inj_func_bijR_aux1:
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```    78     "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
```
```    79   apply (unfold inj_on_def)
```
```    80   apply auto
```
```    81   done
```
```    82
```
```    83 lemma inj_func_bijR_aux2:
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```    84   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
```
```    85     ==> (F, f ` F) \<in> bijR P"
```
```    86   apply (rule_tac F = F and A = A in aux_induct)
```
```    87      apply (rule finite_subset)
```
```    88       apply auto
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```    89   apply (rule bijR.insert)
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```    90      apply (rule_tac [3] inj_func_bijR_aux1)
```
```    91         apply auto
```
```    92   done
```
```    93
```
```    94 lemma inj_func_bijR:
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```    95   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
```
```    96     ==> (A, f ` A) \<in> bijR P"
```
```    97   apply (rule inj_func_bijR_aux2)
```
```    98      apply auto
```
```    99   done
```
```   100
```
```   101
```
```   102 text {* \medskip @{term bijER} *}
```
```   103
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```   104 lemma fin_bijER: "A \<in> bijER P ==> finite A"
```
```   105   apply (erule bijER.induct)
```
```   106     apply auto
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```   107   done
```
```   108
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```   109 lemma aux1:
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```   110   "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
```
```   111     ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
```
```   112   apply (rule_tac x = "F - {a}" in exI)
```
```   113   apply auto
```
```   114   done
```
```   115
```
```   116 lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
```
```   117     ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
```
```   118     ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
```
```   119   apply (rule_tac x = "F - {a, b}" in exI)
```
```   120   apply auto
```
```   121   done
```
```   122
```
```   123 lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
```
```   124   apply (unfold uniqP_def)
```
```   125   apply auto
```
```   126   done
```
```   127
```
```   128 lemma aux_sym: "symP P ==> P a b = P b a"
```
```   129   apply (unfold symP_def)
```
```   130   apply auto
```
```   131   done
```
```   132
```
```   133 lemma aux_in1:
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```   134     "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
```
```   135   apply (unfold bijP_def)
```
```   136   apply auto
```
```   137   apply (subgoal_tac "b \<noteq> a")
```
```   138    prefer 2
```
```   139    apply clarify
```
```   140   apply (simp add: aux_uniq)
```
```   141   apply auto
```
```   142   done
```
```   143
```
```   144 lemma aux_in2:
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```   145   "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
```
```   146     ==> bijP P (insert a (insert b C)) ==> bijP P C"
```
```   147   apply (unfold bijP_def)
```
```   148   apply auto
```
```   149   apply (subgoal_tac "aa \<noteq> a")
```
```   150    prefer 2
```
```   151    apply clarify
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```   152   apply (subgoal_tac "aa \<noteq> b")
```
```   153    prefer 2
```
```   154    apply clarify
```
```   155   apply (simp add: aux_uniq)
```
```   156   apply (subgoal_tac "ba \<noteq> a")
```
```   157    apply auto
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```   158   apply (subgoal_tac "P a aa")
```
```   159    prefer 2
```
```   160    apply (simp add: aux_sym)
```
```   161   apply (subgoal_tac "b = aa")
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```   162    apply (rule_tac [2] iffD1)
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```   163     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
```
```   164       apply auto
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```   165   done
```
```   166
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```   167 lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
```
```   168   apply auto
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```   169   done
```
```   170
```
```   171 lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
```
```   172   apply (unfold bijP_def)
```
```   173   apply (rule iffI)
```
```   174   apply (erule_tac [!] aux_foo)
```
```   175       apply simp_all
```
```   176   apply (rule iffD2)
```
```   177    apply (rule_tac P = P in aux_sym)
```
```   178    apply simp_all
```
```   179   done
```
```   180
```
```   181
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```   182 lemma aux_bijRER:
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```   183   "(A, B) \<in> bijR P ==> uniqP P ==> symP P
```
```   184     ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
```
```   185   apply (erule bijR.induct)
```
```   186    apply simp
```
```   187   apply (case_tac "a = b")
```
```   188    apply clarify
```
```   189    apply (case_tac "b \<in> F")
```
```   190     prefer 2
```
```   191     apply (simp add: subset_insert)
```
```   192    apply (cut_tac F = F and a = b and A = A and B = B in aux1)
```
```   193         prefer 6
```
```   194         apply clarify
```
```   195         apply (rule bijER.insert1)
```
```   196           apply simp_all
```
```   197    apply (subgoal_tac "bijP P C")
```
```   198     apply simp
```
```   199    apply (rule aux_in1)
```
```   200       apply simp_all
```
```   201   apply clarify
```
```   202   apply (case_tac "a \<in> F")
```
```   203    apply (case_tac [!] "b \<in> F")
```
```   204      apply (cut_tac F = F and a = a and b = b and A = A and B = B
```
```   205        in aux2)
```
```   206             apply (simp_all add: subset_insert)
```
```   207     apply clarify
```
```   208     apply (rule bijER.insert2)
```
```   209         apply simp_all
```
```   210     apply (subgoal_tac "bijP P C")
```
```   211      apply simp
```
```   212     apply (rule aux_in2)
```
```   213           apply simp_all
```
```   214    apply (subgoal_tac "b \<in> F")
```
```   215     apply (rule_tac [2] iffD1)
```
```   216      apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
```
```   217        apply (simp_all (no_asm_simp))
```
```   218    apply (subgoal_tac [2] "a \<in> F")
```
```   219     apply (rule_tac [3] iffD2)
```
```   220      apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
```
```   221        apply auto
```
```   222   done
```
```   223
```
```   224 lemma bijR_bijER:
```
```   225   "(A, A) \<in> bijR P ==>
```
```   226     bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
```
```   227   apply (cut_tac A = A and B = A and P = P in aux_bijRER)
```
```   228      apply auto
```
```   229   done
```
```   230
```
```   231 end
```