# Theory BijectionRel

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theory BijectionRel = Main:

```(*  Title:      HOL/NumberTheory/BijectionRel.thy
ID:         \$Id: BijectionRel.thy,v 1.5 2002/10/08 06:20:17 nipkow Exp \$
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]: "({}, {}) ∈ bijR P"
insert: "P a b ==> a ∉ A ==> b ∉ B ==> (A, B) ∈ bijR P
==> (insert a A, insert b B) ∈ bijR P"

text {*
Add extra condition to @{term insert}: @{term "∀b ∈ B. ¬ P a b"}
(and similar for @{term A}).
*}

constdefs
bijP :: "('a => 'a => bool) => 'a set => bool"
"bijP P F == ∀a b. a ∈ F ∧ P a b --> b ∈ F"

uniqP :: "('a => 'a => bool) => bool"
"uniqP P == ∀a b c d. P a b ∧ P c d --> (a = c) = (b = d)"

symP :: "('a => 'a => bool) => bool"
"symP P == ∀a b. P a b = P b a"

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

inductive "bijER P"
intros
empty [simp]: "{} ∈ bijER P"
insert1: "P a a ==> a ∉ A ==> A ∈ bijER P ==> insert a A ∈ bijER P"
insert2: "P a b ==> a ≠ b ==> a ∉ A ==> b ∉ A ==> A ∈ bijER P
==> insert a (insert b A) ∈ bijER P"

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

lemma fin_bijRl: "(A, B) ∈ bijR P ==> finite A"
apply (erule bijR.induct)
apply auto
done

lemma fin_bijRr: "(A, B) ∈ bijR P ==> finite B"
apply (erule bijR.induct)
apply auto
done

lemma aux_induct:
"finite F ==> F ⊆ A ==> P {} ==>
(!!F a. F ⊆ A ==> a ∈ A ==> a ∉ F ==> P F ==> P (insert a F))
==> P F"
proof -
case rule_context
assume major: "finite F"
and subs: "F ⊆ 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 ⊆ B ==> a ∉ A ==> a ∈ B ==> inj_on f B ==> f a ∉ f ` A"
apply (unfold inj_on_def)
apply auto
done

lemma inj_func_bijR_aux2:
"∀a. a ∈ A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
==> (F, f ` F) ∈ 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:
"∀a. a ∈ A --> P a (f a) ==> inj_on f A ==> finite A
==> (A, f ` A) ∈ bijR P"
apply (rule inj_func_bijR_aux2)
apply auto
done

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

lemma fin_bijER: "A ∈ bijER P ==> finite A"
apply (erule bijER.induct)
apply auto
done

lemma aux1:
"a ∉ A ==> a ∉ B ==> F ⊆ insert a A ==> F ⊆ insert a B ==> a ∈ F
==> ∃C. F = insert a C ∧ a ∉ C ∧ C <= A ∧ C <= B"
apply (rule_tac x = "F - {a}" in exI)
apply auto
done

lemma aux2: "a ≠ b ==> a ∉ A ==> b ∉ B ==> a ∈ F ==> b ∈ F
==> F ⊆ insert a A ==> F ⊆ insert b B
==> ∃C. F = insert a (insert b C) ∧ a ∉ C ∧ b ∉ C ∧ C ⊆ A ∧ C ⊆ 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 ∉ C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
apply (unfold bijP_def)
apply auto
apply (subgoal_tac "b ≠ a")
prefer 2
apply clarify
apply (simp add: aux_uniq)
apply auto
done

lemma aux_in2:
"symP P ==> uniqP P ==> a ∉ C ==> b ∉ C ==> a ≠ b ==> P a b
==> bijP P (insert a (insert b C)) ==> bijP P C"
apply (unfold bijP_def)
apply auto
apply (subgoal_tac "aa ≠ a")
prefer 2
apply clarify
apply (subgoal_tac "aa ≠ b")
prefer 2
apply clarify
apply (simp add: aux_uniq)
apply (subgoal_tac "ba ≠ 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: "∀a b. Q a ∧ 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 ∈ F) = (b ∈ 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) ∈ bijR P ==> uniqP P ==> symP P
==> ∀F. bijP P F ∧ F ⊆ A ∧ F ⊆ B --> F ∈ bijER P"
apply (erule bijR.induct)
apply simp
apply (case_tac "a = b")
apply clarify
apply (case_tac "b ∈ 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 ∈ F")
apply (case_tac [!] "b ∈ 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 ∈ 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 ∈ 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) ∈ bijR P ==>
bijP P A ==> uniqP P ==> symP P ==> A ∈ bijER P"
apply (cut_tac A = A and B = A and P = P in aux_bijRER)
apply auto
done

end
```

lemma fin_bijRl:

`  (A, B) ∈ bijR P ==> finite A`

lemma fin_bijRr:

`  (A, B) ∈ bijR P ==> finite B`

lemma aux_induct:

```  [| finite F; F ⊆ A; P {};
!!F a. [| F ⊆ A; a ∈ A; a ∉ F; P F |] ==> P (insert a F) |]
==> P F```

lemma inj_func_bijR_aux1:

`  [| A ⊆ B; a ∉ A; a ∈ B; inj_on f B |] ==> f a ∉ f ` A`

lemma inj_func_bijR_aux2:

```  [| ∀a. a ∈ A --> P a (f a); inj_on f A; finite A; F ⊆ A |]
==> (F, f ` F) ∈ bijR P```

lemma inj_func_bijR:

`  [| ∀a. a ∈ A --> P a (f a); inj_on f A; finite A |] ==> (A, f ` A) ∈ bijR P`

lemma fin_bijER:

`  A ∈ bijER P ==> finite A`

lemma aux1:

```  [| a ∉ A; a ∉ B; F ⊆ insert a A; F ⊆ insert a B; a ∈ F |]
==> ∃C. F = insert a C ∧ a ∉ C ∧ C ⊆ A ∧ C ⊆ B```

lemma aux2:

```  [| a ≠ b; a ∉ A; b ∉ B; a ∈ F; b ∈ F; F ⊆ insert a A; F ⊆ insert b B |]
==> ∃C. F = insert a (insert b C) ∧ a ∉ C ∧ b ∉ C ∧ C ⊆ A ∧ C ⊆ B```

lemma aux_uniq:

`  [| uniqP P; P a b; P c d |] ==> (a = c) = (b = d)`

lemma aux_sym:

`  symP P ==> P a b = P b a`

lemma aux_in1:

`  [| uniqP P; b ∉ C; P b b; bijP P (insert b C) |] ==> bijP P C`

lemma aux_in2:

```  [| symP P; uniqP P; a ∉ C; b ∉ C; a ≠ b; P a b;
bijP P (insert a (insert b C)) |]
==> bijP P C```

lemma aux_foo:

`  [| ∀a b. Q a ∧ P a b --> R b; P a b; Q a |] ==> R b`

lemma aux_bij:

`  [| bijP P F; symP P; P a b |] ==> (a ∈ F) = (b ∈ F)`

lemma aux_bijRER:

```  [| (A, B) ∈ bijR P; uniqP P; symP P |]
==> ∀F. bijP P F ∧ F ⊆ A ∧ F ⊆ B --> F ∈ bijER P```

lemma bijR_bijER:

`  [| (A, A) ∈ bijR P; bijP P A; uniqP P; symP P |] ==> A ∈ bijER P`