Theory Zet

(*  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_image cong: image_cong_simp)
  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)
  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