src/HOL/NumberTheory/BijectionRel.thy
 author wenzelm Sun, 04 Feb 2001 19:31:13 +0100 changeset 11049 7eef34adb852 parent 9508 4d01dbf6ded7 child 11549 e7265e70fd7c permissions -rw-r--r--
HOL-NumberTheory: converted to new-style format and proper document setup;
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(*  Title:      HOL/NumberTheory/BijectionRel.thy
ID:         \$Id\$
Author:     Thomas M. Rasmussen
*)

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 antecedent
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: antecedent)+
done
qed

lemma aux: "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 aux:
"\<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] aux)
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 aux)
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 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 (subgoal_tac "ba \<noteq> a")
apply auto
apply (subgoal_tac "P a aa")
prefer 2
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: "\<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)
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 (rotate_tac -1)
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 (rotate_tac [2-4] -2)
apply (cut_tac F = F and a = a and b = b and A = A and B = B
in aux2)
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
```