src/HOL/ZF/Zet.thy
 changeset 19203 778507520684 child 22931 11cc1ccad58e
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ZF/Zet.thy	Tue Mar 07 16:03:31 2006 +0100
@@ -0,0 +1,221 @@
+(*  Title:      HOL/ZF/Zet.thy
+    ID:         \$Id\$
+    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
+
+typedef 'a zet = "{A :: 'a set | A f z. inj_on f A \<and> f ` A \<subseteq> explode z}"
+  by blast
+
+constdefs
+  zin :: "'a \<Rightarrow> 'a zet \<Rightarrow> bool"
+  "zin x A == x \<in> (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)
+
+constdefs
+  zimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a zet \<Rightarrow> 'b zet"
+  "zimage f A == Abs_zet (image f (Rep_zet A))"
+
+lemma zet_def': "zet = {A :: 'a set | A f z. inj_on f A \<and> f ` A = explode z}"
+  apply (rule set_ext)
+  apply (auto simp add: zet_def)
+  apply (rule_tac x=f in exI)
+  apply auto
+  apply (rule_tac x="Sep z (\<lambda> y. y \<in> (f ` x))" in exI)
+  apply (auto simp add: explode_def Sep)
+  done
+
+lemma image_Inv_f_f: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (Inv B f) ` f ` A = A"
+  apply (rule set_ext)
+  apply (auto simp add: Inv_f_f image_def)
+  apply (rule_tac x="f x" in exI)
+  apply (auto simp add: Inv_f_f)
+  done
+
+lemma image_zet_rep: "A \<in> zet \<Longrightarrow> ? z . g ` A = explode z"
+  apply (auto simp add: zet_def')
+  apply (rule_tac x="Repl z (g o (Inv A f))" in exI)
+  apply (subgoal_tac "explode z = f ` A")
+  apply (simp_all add: comp_image_eq image_Inv_f_f)
+  done
+
+lemma Inv_f_f_mem:
+  assumes "x \<in> A"
+  shows "Inv A g (g x) \<in> A"
+  apply (rule someI2)
+  apply (auto!)
+  done
+
+lemma zet_image_mem:
+  assumes Azet: "A \<in> zet"
+  shows "g ` A \<in> zet"
+proof -
+  from Azet have "? (f :: _ \<Rightarrow> ZF). inj_on f A"
+    by (auto simp add: zet_def')
+  then obtain f where injf: "inj_on (f :: _ \<Rightarrow> ZF) A"
+    by auto
+  let ?w = "f o (Inv A g)"
+  have subset: "(Inv A g) ` (g ` A) \<subseteq> A"
+    by (auto simp add: Inv_f_f_mem)
+  have "inj_on (Inv A g) (g ` A)" by (simp add: inj_on_Inv)
+  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' comp_image_eq[symmetric])
+    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 (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)
+
+constdefs
+  zimplode :: "ZF zet \<Rightarrow> ZF"
+  "zimplode A == implode (Rep_zet A)"
+  zexplode :: "ZF \<Rightarrow> ZF zet"
+  "zexplode z == Abs_zet (explode z)"
+
+lemma Rep_zet_eq_explode: "? z. Rep_zet A = explode z"
+  by (rule image_zet_rep[where g="\<lambda> x. x",OF Rep_zet, simplified])
+
+lemma zexplode_zimplode: "zexplode (zimplode A) = A"
+  apply (simp add: zimplode_def zexplode_def)
+  apply (subst f_inv_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 \<in> zet"
+  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 (subst Abs_zet_inverse)
+  apply (simp_all add: comp_image_eq zet_image_mem Rep_zet)
+  done
+
+constdefs
+  zunion :: "'a zet \<Rightarrow> 'a zet \<Rightarrow> 'a zet"
+  "zunion a b \<equiv> Abs_zet ((Rep_zet a) \<union> (Rep_zet b))"
+  zsubset :: "'a zet \<Rightarrow> 'a zet \<Rightarrow> bool"
+  "zsubset a b \<equiv> ! x. zin x a \<longrightarrow> zin x b"
+
+lemma explode_union: "explode (union a b) = (explode a) \<union> (explode b)"
+  apply (rule set_ext)
+  apply (simp add: explode_def union)
+  done
+
+lemma Rep_zet_zunion: "Rep_zet (zunion a b) = (Rep_zet a) \<union> (Rep_zet b)"
+proof -
+  from Rep_zet[of a] have "? f z. inj_on f (Rep_zet a) \<and> f ` (Rep_zet a) = explode z"
+    by (auto simp add: zet_def')
+  then obtain fa za where a:"inj_on fa (Rep_zet a) \<and> 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) \<and> f ` (Rep_zet b) = explode z"
+    by (auto simp add: zet_def')
+  then obtain fb zb where b:"inj_on fb (Rep_zet b) \<and> 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 = "(\<lambda> x. if x \<in> (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 \<noteq> Empty"
+    apply (auto simp add: Ext Singleton)
+    apply (rule exI[where x=Empty])
+    done
+  show ?thesis
+    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) \<or> (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_ext)
+  apply (auto simp add: explode_mem_zet)
+  apply (drule image_zet_rep)
+  apply auto
+  apply (rule_tac x=z in exI)
+  apply auto
+  done
+
+lemma Elem_zimplode: "(Elem x (zimplode z)) = (zin x z)"
+  apply (subst Elem_implode)
+  apply (simp_all add: zin_def Rep_zet range_explode_eq_zet)
+  done
+
+constdefs
+  zempty :: "'a zet"
+  "zempty \<equiv> Abs_zet {}"
+
+lemma zempty[simp]: "\<not> (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[recdef_cong]: "\<lbrakk> M = N; !! x. zin x N \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow> zimage f M = zimage g N"
+  by (auto simp add: zet_ext_eq zimage_iff)
+
+end```