src/HOL/NumberTheory/BijectionRel.thy
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(*  Title:      HOL/NumberTheory/BijectionRel.thy
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    ID:         $Id$
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    Author:     Thomas M. Rasmussen
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    Copyright   2000  University of Cambridge
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*)
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header {* Bijections between sets *}
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theory BijectionRel imports Main begin
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text {*
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  Inductive definitions of bijections between two different sets and
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  between the same set.  Theorem for relating the two definitions.
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  \bigskip
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*}
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consts
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  bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
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inductive "bijR P"
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  intros
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  empty [simp]: "({}, {}) \<in> bijR P"
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  insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P
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    ==> (insert a A, insert b B) \<in> bijR P"
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text {*
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  Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
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  (and similar for @{term A}).
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*}
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constdefs
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  bijP :: "('a => 'a => bool) => 'a set => bool"
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  "bijP P F == \<forall>a b. a \<in> F \<and> P a b --> b \<in> F"
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  uniqP :: "('a => 'a => bool) => bool"
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  "uniqP P == \<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d)"
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  symP :: "('a => 'a => bool) => bool"
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  "symP P == \<forall>a b. P a b = P b a"
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consts
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  bijER :: "('a => 'a => bool) => 'a set set"
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inductive "bijER P"
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  intros
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  empty [simp]: "{} \<in> bijER P"
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  insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
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  insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
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    ==> insert a (insert b A) \<in> bijER P"
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text {* \medskip @{term bijR} *}
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lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
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  apply (erule bijR.induct)
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  apply auto
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  done
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lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
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  apply (erule bijR.induct)
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  apply auto
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  done
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lemma aux_induct:
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  assumes major: "finite F"
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    and subs: "F \<subseteq> A"
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    and cases: "P {}"
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      "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  shows "P F"
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  using major subs
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  apply (induct set: Finites)
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   apply (blast intro: cases)+
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  done
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lemma inj_func_bijR_aux1:
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    "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
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  apply (unfold inj_on_def)
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  apply auto
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  done
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lemma inj_func_bijR_aux2:
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  "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
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    ==> (F, f ` F) \<in> bijR P"
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  apply (rule_tac F = F and A = A in aux_induct)
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     apply (rule finite_subset)
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      apply auto
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  apply (rule bijR.insert)
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     apply (rule_tac [3] inj_func_bijR_aux1)
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        apply auto
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  done
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lemma inj_func_bijR:
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  "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
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    ==> (A, f ` A) \<in> bijR P"
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  apply (rule inj_func_bijR_aux2)
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     apply auto
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  done
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text {* \medskip @{term bijER} *}
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lemma fin_bijER: "A \<in> bijER P ==> finite A"
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  apply (erule bijER.induct)
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    apply auto
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  done
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lemma aux1:
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  "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
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    ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
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  apply (rule_tac x = "F - {a}" in exI)
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  apply auto
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  done
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lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
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    ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
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    ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
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  apply (rule_tac x = "F - {a, b}" in exI)
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  apply auto
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  done
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lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
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  apply (unfold uniqP_def)
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  apply auto
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  done
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lemma aux_sym: "symP P ==> P a b = P b a"
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  apply (unfold symP_def)
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  apply auto
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  done
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lemma aux_in1:
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    "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
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  apply (unfold bijP_def)
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  apply auto
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  apply (subgoal_tac "b \<noteq> a")
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   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   139
   apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   140
  apply (simp add: aux_uniq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   141
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   142
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   143
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   144
lemma aux_in2:
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   145
  "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   146
    ==> bijP P (insert a (insert b C)) ==> bijP P C"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   147
  apply (unfold bijP_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   148
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   149
  apply (subgoal_tac "aa \<noteq> a")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   150
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   151
   apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   152
  apply (subgoal_tac "aa \<noteq> b")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   153
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   154
   apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   155
  apply (simp add: aux_uniq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   156
  apply (subgoal_tac "ba \<noteq> a")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   157
   apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   158
  apply (subgoal_tac "P a aa")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   159
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   160
   apply (simp add: aux_sym)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   161
  apply (subgoal_tac "b = aa")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   162
   apply (rule_tac [2] iffD1)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   163
    apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   164
      apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   165
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   166
13524
604d0f3622d6 *** empty log message ***
wenzelm
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diff changeset
   167
lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   168
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   169
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   170
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   171
lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   172
  apply (unfold bijP_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   173
  apply (rule iffI)
13524
604d0f3622d6 *** empty log message ***
wenzelm
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diff changeset
   174
  apply (erule_tac [!] aux_foo)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   175
      apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   176
  apply (rule iffD2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   177
   apply (rule_tac P = P in aux_sym)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   178
   apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   179
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   180
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   181
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   182
lemma aux_bijRER:
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
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diff changeset
   183
  "(A, B) \<in> bijR P ==> uniqP P ==> symP P
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   184
    ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   185
  apply (erule bijR.induct)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   186
   apply simp
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   187
  apply (case_tac "a = b")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   188
   apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   189
   apply (case_tac "b \<in> F")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   190
    prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   191
    apply (simp add: subset_insert)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   192
   apply (cut_tac F = F and a = b and A = A and B = B in aux1)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   193
        prefer 6
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   194
        apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   195
        apply (rule bijER.insert1)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   196
          apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   197
   apply (subgoal_tac "bijP P C")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   198
    apply simp
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   199
   apply (rule aux_in1)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   200
      apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   201
  apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   202
  apply (case_tac "a \<in> F")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   203
   apply (case_tac [!] "b \<in> F")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   204
     apply (cut_tac F = F and a = a and b = b and A = A and B = B
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   205
       in aux2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   206
            apply (simp_all add: subset_insert)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   207
    apply clarify
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   208
    apply (rule bijER.insert2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   209
        apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   210
    apply (subgoal_tac "bijP P C")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   211
     apply simp
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   212
    apply (rule aux_in2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   213
          apply simp_all
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   214
   apply (subgoal_tac "b \<in> F")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   215
    apply (rule_tac [2] iffD1)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   216
     apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   217
       apply (simp_all (no_asm_simp))
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   218
   apply (subgoal_tac [2] "a \<in> F")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   219
    apply (rule_tac [3] iffD2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   220
     apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   221
       apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   222
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   223
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   224
lemma bijR_bijER:
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   225
  "(A, A) \<in> bijR P ==>
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
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diff changeset
   226
    bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   227
  apply (cut_tac A = A and B = A and P = P in aux_bijRER)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   228
     apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 9508
diff changeset
   229
  done
9508
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   230
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   231
end