src/HOL/Old_Number_Theory/BijectionRel.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 61382 efac889fccbc permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Old_Number_Theory/BijectionRel.thy
```
```     2     Author:     Thomas M. Rasmussen
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```     3     Copyright   2000  University of Cambridge
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```     4 *)
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```     5
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```     6 section \<open>Bijections between sets\<close>
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```     7
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```     8 theory BijectionRel
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```     9 imports Main
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```    10 begin
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```    11
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```    12 text \<open>
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```    13   Inductive definitions of bijections between two different sets and
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```    14   between the same set.  Theorem for relating the two definitions.
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```    15
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```    16   \bigskip
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```    17 \<close>
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```    18
```
```    19 inductive_set
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```    20   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
```
```    21   for P :: "'a => 'b => bool"
```
```    22 where
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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
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```    27 text \<open>
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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}).
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```    30 \<close>
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```    31
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```    32 definition
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```    33   bijP :: "('a => 'a => bool) => 'a set => bool" where
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```    34   "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
```
```    35
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```    36 definition
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```    37   uniqP :: "('a => 'a => bool) => bool" where
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```    38   "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
```
```    39
```
```    40 definition
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```    41   symP :: "('a => 'a => bool) => bool" where
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```    42   "symP P = (\<forall>a b. P a b = P b a)"
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```    43
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```    44 inductive_set
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```    45   bijER :: "('a => 'a => bool) => 'a set set"
```
```    46   for P :: "'a => 'a => bool"
```
```    47 where
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```    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
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```    53
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```    54 text \<open>\medskip @{term bijR}\<close>
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```    55
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```    56 lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
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```    57   apply (erule bijR.induct)
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```    58   apply auto
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```    59   done
```
```    60
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```    61 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
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```    62   apply (erule bijR.induct)
```
```    63   apply auto
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```    64   done
```
```    65
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```    66 lemma aux_induct:
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```    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)
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```    74    apply (blast intro: cases)+
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```    75   done
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```    76
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```    77
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```    78 lemma inj_func_bijR_aux1:
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```    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
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```    82   done
```
```    83
```
```    84 lemma inj_func_bijR_aux2:
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```    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)
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```    88      apply (rule finite_subset)
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```    89       apply auto
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```    90   apply (rule bijR.insert)
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```    91      apply (rule_tac [3] inj_func_bijR_aux1)
```
```    92         apply auto
```
```    93   done
```
```    94
```
```    95 lemma inj_func_bijR:
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```    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)
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```    99      apply auto
```
```   100   done
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```   101
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```   102
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```   103 text \<open>\medskip @{term bijER}\<close>
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```   104
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```   105 lemma fin_bijER: "A \<in> bijER P ==> finite A"
```
```   106   apply (erule bijER.induct)
```
```   107     apply auto
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```   108   done
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```   109
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```   110 lemma aux1:
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```   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
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```   115   done
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```   116
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```   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
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```   122   done
```
```   123
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```   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
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```   129 lemma aux_sym: "symP P ==> P a b = P b a"
```
```   130   apply (unfold symP_def)
```
```   131   apply auto
```
```   132   done
```
```   133
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```   134 lemma aux_in1:
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```   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
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```   140    apply clarify
```
```   141   apply (simp add: aux_uniq)
```
```   142   apply auto
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```   143   done
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```   144
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```   145 lemma aux_in2:
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```   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
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```   150   apply (subgoal_tac "aa \<noteq> a")
```
```   151    prefer 2
```
```   152    apply clarify
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```   153   apply (subgoal_tac "aa \<noteq> b")
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```   154    prefer 2
```
```   155    apply clarify
```
```   156   apply (simp add: aux_uniq)
```
```   157   apply (subgoal_tac "ba \<noteq> a")
```
```   158    apply auto
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```   159   apply (subgoal_tac "P a aa")
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```   160    prefer 2
```
```   161    apply (simp add: aux_sym)
```
```   162   apply (subgoal_tac "b = aa")
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```   163    apply (rule_tac [2] iffD1)
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```   164     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
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```   165       apply auto
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```   166   done
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```   167
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```   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
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```   170   done
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```   171
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```   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)
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```   175   apply (erule_tac [!] aux_foo)
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```   176       apply simp_all
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```   177   apply (rule iffD2)
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```   178    apply (rule_tac P = P in aux_sym)
```
```   179    apply simp_all
```
```   180   done
```
```   181
```
```   182
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```   183 lemma aux_bijRER:
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```   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
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```   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
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```   198    apply (subgoal_tac "bijP P C")
```
```   199     apply simp
```
```   200    apply (rule aux_in1)
```
```   201       apply simp_all
```
```   202   apply clarify
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```   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
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```   206        in aux2)
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```   207             apply (simp_all add: subset_insert)
```
```   208     apply clarify
```
```   209     apply (rule bijER.insert2)
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```   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
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```   230   done
```
```   231
```
```   232 end
```