--- a/src/HOL/NumberTheory/BijectionRel.thy Tue Sep 01 14:13:34 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,231 +0,0 @@
-(* Title: HOL/NumberTheory/BijectionRel.thy
- ID: $Id$
- 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