# Theory Zet

theory Zet
imports HOLZF
```(*  Title:      HOL/ZF/Zet.thy
Author:     Steven Obua

Introduces a type 'a zet of ZF representable sets.
See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
*)

theory Zet
imports HOLZF
begin

definition "zet = {A :: 'a set | A f z. inj_on f A ∧ f ` A ⊆ explode z}"

typedef 'a zet = "zet :: 'a set set"
unfolding zet_def by blast

definition zin :: "'a ⇒ 'a zet ⇒ bool" where
"zin x A == x ∈ (Rep_zet A)"

lemma zet_ext_eq: "(A = B) = (! x. zin x A = zin x B)"
by (auto simp add: Rep_zet_inject[symmetric] zin_def)

definition zimage :: "('a ⇒ 'b) ⇒ 'a zet ⇒ 'b zet" where
"zimage f A == Abs_zet (image f (Rep_zet A))"

lemma zet_def': "zet = {A :: 'a set | A f z. inj_on f A ∧ f ` A = explode z}"
apply (rule set_eqI)
apply (auto simp add: zet_def)
apply (rule_tac x=f in exI)
apply auto
apply (rule_tac x="Sep z (λ y. y ∈ (f ` x))" in exI)
apply (auto simp add: explode_def Sep)
done

lemma image_zet_rep: "A ∈ zet ⟹ ? z . g ` A = explode z"
apply (auto simp add: zet_def')
apply (rule_tac x="Repl z (g o (inv_into A f))" in exI)
apply (simp add: explode_Repl_eq)
apply (subgoal_tac "explode z = f ` A")
apply (simp_all add: image_comp [symmetric])
done

lemma zet_image_mem:
assumes Azet: "A ∈ zet"
shows "g ` A ∈ zet"
proof -
from Azet have "? (f :: _ ⇒ ZF). inj_on f A"
by (auto simp add: zet_def')
then obtain f where injf: "inj_on (f :: _ ⇒ ZF) A"
by auto
let ?w = "f o (inv_into A g)"
have subset: "(inv_into A g) ` (g ` A) ⊆ A"
by (auto simp add: inv_into_into)
have "inj_on (inv_into A g) (g ` A)" by (simp add: inj_on_inv_into)
then have injw: "inj_on ?w (g ` A)"
apply (rule comp_inj_on)
apply (rule subset_inj_on[where B=A])
apply (auto simp add: subset injf)
done
show ?thesis
apply (simp add: zet_def' image_comp)
apply (rule exI[where x="?w"])
apply (simp add: injw image_zet_rep Azet)
done
qed

lemma Rep_zimage_eq: "Rep_zet (zimage f A) = image f (Rep_zet A)"
apply (simp add: zimage_def)
apply (subst Abs_zet_inverse)
apply (simp_all add: Rep_zet zet_image_mem)
done

lemma zimage_iff: "zin y (zimage f A) = (? x. zin x A & y = f x)"
by (auto simp add: zin_def Rep_zimage_eq)

definition zimplode :: "ZF zet ⇒ ZF" where
"zimplode A == implode (Rep_zet A)"

definition zexplode :: "ZF ⇒ ZF zet" where
"zexplode z == Abs_zet (explode z)"

lemma Rep_zet_eq_explode: "? z. Rep_zet A = explode z"
by (rule image_zet_rep[where g="λ x. x",OF Rep_zet, simplified])

lemma zexplode_zimplode: "zexplode (zimplode A) = A"
apply (simp add: zimplode_def zexplode_def)
apply (simp add: implode_def)
apply (subst f_inv_into_f[where y="Rep_zet A"])
apply (auto simp add: Rep_zet_inverse Rep_zet_eq_explode image_def)
done

lemma explode_mem_zet: "explode z ∈ zet"
apply (simp add: zet_def')
apply (rule_tac x="% x. x" in exI)
apply (auto simp add: inj_on_def)
done

lemma zimplode_zexplode: "zimplode (zexplode z) = z"
apply (simp add: zimplode_def zexplode_def)
apply (subst Abs_zet_inverse)
apply (auto simp add: explode_mem_zet implode_explode)
done

lemma zin_zexplode_eq: "zin x (zexplode A) = Elem x A"
apply (simp add: zin_def zexplode_def)
apply (subst Abs_zet_inverse)
apply (simp_all add: explode_Elem explode_mem_zet)
done

lemma comp_zimage_eq: "zimage g (zimage f A) = zimage (g o f) A"
apply (simp add: zimage_def)
apply (subst Abs_zet_inverse)
apply (simp_all add: image_comp zet_image_mem Rep_zet)
done

definition zunion :: "'a zet ⇒ 'a zet ⇒ 'a zet" where
"zunion a b ≡ Abs_zet ((Rep_zet a) ∪ (Rep_zet b))"

definition zsubset :: "'a zet ⇒ 'a zet ⇒ bool" where
"zsubset a b ≡ ! x. zin x a ⟶ zin x b"

lemma explode_union: "explode (union a b) = (explode a) ∪ (explode b)"
apply (rule set_eqI)
apply (simp add: explode_def union)
done

lemma Rep_zet_zunion: "Rep_zet (zunion a b) = (Rep_zet a) ∪ (Rep_zet b)"
proof -
from Rep_zet[of a] have "? f z. inj_on f (Rep_zet a) ∧ f ` (Rep_zet a) = explode z"
by (auto simp add: zet_def')
then obtain fa za where a:"inj_on fa (Rep_zet a) ∧ fa ` (Rep_zet a) = explode za"
by blast
from a have fa: "inj_on fa (Rep_zet a)" by blast
from a have za: "fa ` (Rep_zet a) = explode za" by blast
from Rep_zet[of b] have "? f z. inj_on f (Rep_zet b) ∧ f ` (Rep_zet b) = explode z"
by (auto simp add: zet_def')
then obtain fb zb where b:"inj_on fb (Rep_zet b) ∧ fb ` (Rep_zet b) = explode zb"
by blast
from b have fb: "inj_on fb (Rep_zet b)" by blast
from b have zb: "fb ` (Rep_zet b) = explode zb" by blast
let ?f = "(λ x. if x ∈ (Rep_zet a) then Opair (fa x) (Empty) else Opair (fb x) (Singleton Empty))"
let ?z = "CartProd (union za zb) (Upair Empty (Singleton Empty))"
have se: "Singleton Empty ≠ Empty"
apply (auto simp add: Ext Singleton)
apply (rule exI[where x=Empty])
apply (simp add: Empty)
done
show ?thesis
apply (simp add: zunion_def)
apply (subst Abs_zet_inverse)
apply (auto simp add: zet_def)
apply (rule exI[where x = ?f])
apply (rule conjI)
apply (auto simp add: inj_on_def Opair inj_onD[OF fa] inj_onD[OF fb] se se[symmetric])
apply (rule exI[where x = ?z])
apply (insert za zb)
apply (auto simp add: explode_def CartProd union Upair Opair)
done
qed

lemma zunion: "zin x (zunion a b) = ((zin x a) ∨ (zin x b))"
by (auto simp add: zin_def Rep_zet_zunion)

lemma zimage_zexplode_eq: "zimage f (zexplode z) = zexplode (Repl z f)"
by (simp add: zet_ext_eq zin_zexplode_eq Repl zimage_iff)

lemma range_explode_eq_zet: "range explode = zet"
apply (rule set_eqI)
apply (auto simp add: explode_mem_zet)
apply (drule image_zet_rep)
apply (simp add: image_def)
apply auto
apply (rule_tac x=z in exI)
apply auto
done

lemma Elem_zimplode: "(Elem x (zimplode z)) = (zin x z)"
apply (simp add: zimplode_def)
apply (subst Elem_implode)
apply (simp_all add: zin_def Rep_zet range_explode_eq_zet)
done

definition zempty :: "'a zet" where
"zempty ≡ Abs_zet {}"

lemma zempty[simp]: "¬ (zin x zempty)"
by (auto simp add: zin_def zempty_def Abs_zet_inverse zet_def)

lemma zimage_zempty[simp]: "zimage f zempty = zempty"
by (auto simp add: zet_ext_eq zimage_iff)

lemma zunion_zempty_left[simp]: "zunion zempty a = a"
by (simp add: zet_ext_eq zunion)

lemma zunion_zempty_right[simp]: "zunion a zempty = a"
by (simp add: zet_ext_eq zunion)

lemma zimage_id[simp]: "zimage id A = A"
by (simp add: zet_ext_eq zimage_iff)

lemma zimage_cong[fundef_cong]: "⟦ M = N; !! x. zin x N ⟹ f x = g x ⟧ ⟹ zimage f M = zimage g N"
by (auto simp add: zet_ext_eq zimage_iff)

end
```