(* Title: HOL/Old_Number_Theory/BijectionRel.thy
Author: Thomas M. Rasmussen
Copyright 2000 University of Cambridge
*)
section \<open>Bijections between sets\<close>
theory BijectionRel
imports Main
begin
text \<open>
Inductive definitions of bijections between two different sets and
between the same set. Theorem for relating the two definitions.
\bigskip
\<close>
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 \<open>
Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
(and similar for @{term A}).
\<close>
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 \<open>\medskip @{term bijR}\<close>
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 \<open>\medskip @{term bijER}\<close>
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