(*  Title:      HOL/NumberTheory/BijectionRel.thy

     ID:         $Id$

     Author:     Thomas M. Rasmussen

     Copyright   2000  University of Cambridge

 *)

 header {* Bijections between sets *}

 theory BijectionRel = Main:

 text {*

   Inductive definitions of bijections between two different sets and

   between the same set.  Theorem for relating the two definitions.

   \bigskip

 *}

 consts

   bijR :: "('a => 'b => bool) => ('a set * 'b set) set"

 inductive "bijR P"

   intros

   empty [simp]: "({}, {}) \<in> bijR P"

   insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P

     ==> (insert a A, insert b B) \<in> bijR P"

 text {*

   Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}

   (and similar for @{term A}).

 *}

 constdefs

   bijP :: "('a => 'a => bool) => 'a set => bool"

   "bijP P F == \<forall>a b. a \<in> F \<and> P a b --> b \<in> F"

   uniqP :: "('a => 'a => bool) => bool"

   "uniqP P == \<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d)"

   symP :: "('a => 'a => bool) => bool"

   "symP P == \<forall>a b. P a b = P b a"

 consts

   bijER :: "('a => 'a => bool) => 'a set set"

 inductive "bijER P"

   intros

   empty [simp]: "{} \<in> bijER P"

   insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"

   insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P

     ==> insert a (insert b A) \<in> bijER P"

 text {* \medskip @{term bijR} *}

 lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"

   apply (erule bijR.induct)

   apply auto

   done

 lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"

   apply (erule bijR.induct)

   apply auto

   done

 lemma aux_induct:

   "finite F ==> F \<subseteq> A ==> P {} ==>

     (!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F))

   ==> P F"

 proof -

   case rule_context

   assume major: "finite F"

     and subs: "F \<subseteq> A"

   show ?thesis

     apply (rule subs [THEN rev_mp])

     apply (rule major [THEN finite_induct])

      apply (blast intro: rule_context)+

     done

 qed

 lemma inj_func_bijR_aux1:

     "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"

   apply (unfold inj_on_def)

   apply auto

   done

 lemma inj_func_bijR_aux2:

   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A

     ==> (F, f ` F) \<in> bijR P"

   apply (rule_tac F = F and A = A in aux_induct)

      apply (rule finite_subset)

       apply auto

   apply (rule bijR.insert)

      apply (rule_tac [3] inj_func_bijR_aux1)

         apply auto

   done

 lemma inj_func_bijR:

   "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A

     ==> (A, f ` A) \<in> bijR P"

   apply (rule inj_func_bijR_aux2)

      apply auto

   done

 text {* \medskip @{term bijER} *}

 lemma fin_bijER: "A \<in> bijER P ==> finite A"

   apply (erule bijER.induct)

     apply auto

   done

 lemma aux1:

   "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F

     ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"

   apply (rule_tac x = "F - {a}" in exI)

   apply auto

   done

 lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F

     ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B

     ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"

   apply (rule_tac x = "F - {a, b}" in exI)

   apply auto

   done

 lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"

   apply (unfold uniqP_def)

   apply auto

   done

 lemma aux_sym: "symP P ==> P a b = P b a"

   apply (unfold symP_def)

   apply auto

   done

 lemma aux_in1:

     "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"

   apply (unfold bijP_def)

   apply auto

   apply (subgoal_tac "b \<noteq> a")

    prefer 2

    apply clarify

   apply (simp add: aux_uniq)

   apply auto

   done

 lemma aux_in2:

   "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b

     ==> bijP P (insert a (insert b C)) ==> bijP P C"

   apply (unfold bijP_def)

   apply auto

   apply (subgoal_tac "aa \<noteq> a")

    prefer 2

    apply clarify

   apply (subgoal_tac "aa \<noteq> b")

    prefer 2

    apply clarify

   apply (simp add: aux_uniq)

   apply (subgoal_tac "ba \<noteq> a")

    apply auto

   apply (subgoal_tac "P a aa")

    prefer 2

    apply (simp add: aux_sym)

   apply (subgoal_tac "b = aa")

    apply (rule_tac [2] iffD1)

     apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)

       apply auto

   done

 lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"

   apply auto

   done

 lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"

   apply (unfold bijP_def)

   apply (rule iffI)

   apply (erule_tac [!] aux_foo)

       apply simp_all

   apply (rule iffD2)

    apply (rule_tac P = P in aux_sym)

    apply simp_all

   done

 lemma aux_bijRER:

   "(A, B) \<in> bijR P ==> uniqP P ==> symP P

     ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"

   apply (erule bijR.induct)

    apply simp

   apply (case_tac "a = b")

    apply clarify

    apply (case_tac "b \<in> F")

     prefer 2

     apply (simp add: subset_insert)

    apply (cut_tac F = F and a = b and A = A and B = B in aux1)

         prefer 6

         apply clarify

         apply (rule bijER.insert1)

           apply simp_all

    apply (subgoal_tac "bijP P C")

     apply simp

    apply (rule aux_in1)

       apply simp_all

   apply clarify

   apply (case_tac "a \<in> F")

    apply (case_tac [!] "b \<in> F")

      apply (cut_tac F = F and a = a and b = b and A = A and B = B

        in aux2)

             apply (simp_all add: subset_insert)

     apply clarify

     apply (rule bijER.insert2)

         apply simp_all

     apply (subgoal_tac "bijP P C")

      apply simp

     apply (rule aux_in2)

           apply simp_all

    apply (subgoal_tac "b \<in> F")

     apply (rule_tac [2] iffD1)

      apply (rule_tac [2] a = a and F = F and P = P in aux_bij)

        apply (simp_all (no_asm_simp))

    apply (subgoal_tac [2] "a \<in> F")

     apply (rule_tac [3] iffD2)

      apply (rule_tac [3] b = b and F = F and P = P in aux_bij)

        apply auto

   done

 lemma bijR_bijER:

   "(A, A) \<in> bijR P ==>

     bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"

   apply (cut_tac A = A and B = A and P = P in aux_bijRER)

      apply auto

   done

 end