--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Old_Number_Theory/BijectionRel.thy Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,229 @@
+(* Author: Thomas M. Rasmussen
+ Copyright 2000 University of Cambridge
+*)
+
+header {* Bijections between sets *}
+
+theory BijectionRel imports Main begin
+
+text {*
+ Inductive definitions of bijections between two different sets and
+ between the same set. Theorem for relating the two definitions.
+
+ \bigskip
+*}
+
+inductive_set
+ bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
+ for P :: "'a => 'b => bool"
+where
+ 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}).
+*}
+
+definition
+ bijP :: "('a => 'a => bool) => 'a set => bool" where
+ "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
+
+definition
+ uniqP :: "('a => 'a => bool) => bool" where
+ "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
+
+definition
+ symP :: "('a => 'a => bool) => bool" where
+ "symP P = (\<forall>a b. P a b = P b a)"
+
+inductive_set
+ bijER :: "('a => 'a => bool) => 'a set set"
+ for P :: "'a => 'a => bool"
+where
+ 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:
+ assumes major: "finite F"
+ and subs: "F \<subseteq> A"
+ and cases: "P {}"
+ "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
+ shows "P F"
+ using major subs
+ apply (induct set: finite)
+ apply (blast intro: cases)+
+ done
+
+
+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