src/HOL/ZF/Zet.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 22931 11cc1ccad58e
child 32988 d1d4d7a08a66
permissions -rw-r--r--
moved Finite_Set before Datatype
     1 (*  Title:      HOL/ZF/Zet.thy
     2     ID:         $Id$
     3     Author:     Steven Obua
     4 
     5     Introduces a type 'a zet of ZF representable sets.
     6     See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
     7 *)
     8 
     9 theory Zet 
    10 imports HOLZF
    11 begin
    12 
    13 typedef 'a zet = "{A :: 'a set | A f z. inj_on f A \<and> f ` A \<subseteq> explode z}"
    14   by blast
    15 
    16 constdefs
    17   zin :: "'a \<Rightarrow> 'a zet \<Rightarrow> bool"
    18   "zin x A == x \<in> (Rep_zet A)"
    19 
    20 lemma zet_ext_eq: "(A = B) = (! x. zin x A = zin x B)"
    21   by (auto simp add: Rep_zet_inject[symmetric] zin_def)
    22 
    23 constdefs
    24   zimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a zet \<Rightarrow> 'b zet"
    25   "zimage f A == Abs_zet (image f (Rep_zet A))"
    26 
    27 lemma zet_def': "zet = {A :: 'a set | A f z. inj_on f A \<and> f ` A = explode z}"
    28   apply (rule set_ext)
    29   apply (auto simp add: zet_def)
    30   apply (rule_tac x=f in exI)
    31   apply auto
    32   apply (rule_tac x="Sep z (\<lambda> y. y \<in> (f ` x))" in exI)
    33   apply (auto simp add: explode_def Sep)
    34   done
    35 
    36 lemma image_Inv_f_f: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (Inv B f) ` f ` A = A"
    37   apply (rule set_ext)
    38   apply (auto simp add: Inv_f_f image_def)
    39   apply (rule_tac x="f x" in exI)
    40   apply (auto simp add: Inv_f_f)
    41   done
    42   
    43 lemma image_zet_rep: "A \<in> zet \<Longrightarrow> ? z . g ` A = explode z"
    44   apply (auto simp add: zet_def')
    45   apply (rule_tac x="Repl z (g o (Inv A f))" in exI)
    46   apply (simp add: explode_Repl_eq)
    47   apply (subgoal_tac "explode z = f ` A")
    48   apply (simp_all add: comp_image_eq image_Inv_f_f)  
    49   done
    50 
    51 lemma Inv_f_f_mem:       
    52   assumes "x \<in> A"
    53   shows "Inv A g (g x) \<in> A"
    54   apply (simp add: Inv_def)
    55   apply (rule someI2)
    56   using `x \<in> A` apply auto
    57   done
    58 
    59 lemma zet_image_mem:
    60   assumes Azet: "A \<in> zet"
    61   shows "g ` A \<in> zet"
    62 proof -
    63   from Azet have "? (f :: _ \<Rightarrow> ZF). inj_on f A" 
    64     by (auto simp add: zet_def')
    65   then obtain f where injf: "inj_on (f :: _ \<Rightarrow> ZF) A"  
    66     by auto
    67   let ?w = "f o (Inv A g)"
    68   have subset: "(Inv A g) ` (g ` A) \<subseteq> A"
    69     by (auto simp add: Inv_f_f_mem)
    70   have "inj_on (Inv A g) (g ` A)" by (simp add: inj_on_Inv)
    71   then have injw: "inj_on ?w (g ` A)"
    72     apply (rule comp_inj_on)
    73     apply (rule subset_inj_on[where B=A])
    74     apply (auto simp add: subset injf)
    75     done
    76   show ?thesis
    77     apply (simp add: zet_def' comp_image_eq[symmetric])
    78     apply (rule exI[where x="?w"])
    79     apply (simp add: injw image_zet_rep Azet)
    80     done
    81 qed
    82 
    83 lemma Rep_zimage_eq: "Rep_zet (zimage f A) = image f (Rep_zet A)"
    84   apply (simp add: zimage_def)
    85   apply (subst Abs_zet_inverse)
    86   apply (simp_all add: Rep_zet zet_image_mem)
    87   done
    88 
    89 lemma zimage_iff: "zin y (zimage f A) = (? x. zin x A & y = f x)"
    90   by (auto simp add: zin_def Rep_zimage_eq)
    91 
    92 constdefs
    93   zimplode :: "ZF zet \<Rightarrow> ZF"
    94   "zimplode A == implode (Rep_zet A)"
    95   zexplode :: "ZF \<Rightarrow> ZF zet"
    96   "zexplode z == Abs_zet (explode z)"
    97 
    98 lemma Rep_zet_eq_explode: "? z. Rep_zet A = explode z"
    99   by (rule image_zet_rep[where g="\<lambda> x. x",OF Rep_zet, simplified])
   100 
   101 lemma zexplode_zimplode: "zexplode (zimplode A) = A"
   102   apply (simp add: zimplode_def zexplode_def)
   103   apply (simp add: implode_def)
   104   apply (subst f_inv_f[where y="Rep_zet A"])
   105   apply (auto simp add: Rep_zet_inverse Rep_zet_eq_explode image_def)
   106   done
   107 
   108 lemma explode_mem_zet: "explode z \<in> zet"
   109   apply (simp add: zet_def')
   110   apply (rule_tac x="% x. x" in exI)
   111   apply (auto simp add: inj_on_def)
   112   done
   113 
   114 lemma zimplode_zexplode: "zimplode (zexplode z) = z"
   115   apply (simp add: zimplode_def zexplode_def)
   116   apply (subst Abs_zet_inverse)
   117   apply (auto simp add: explode_mem_zet implode_explode)
   118   done  
   119 
   120 lemma zin_zexplode_eq: "zin x (zexplode A) = Elem x A"
   121   apply (simp add: zin_def zexplode_def)
   122   apply (subst Abs_zet_inverse)
   123   apply (simp_all add: explode_Elem explode_mem_zet) 
   124   done
   125 
   126 lemma comp_zimage_eq: "zimage g (zimage f A) = zimage (g o f) A"
   127   apply (simp add: zimage_def)
   128   apply (subst Abs_zet_inverse)
   129   apply (simp_all add: comp_image_eq zet_image_mem Rep_zet)
   130   done
   131     
   132 constdefs
   133   zunion :: "'a zet \<Rightarrow> 'a zet \<Rightarrow> 'a zet"
   134   "zunion a b \<equiv> Abs_zet ((Rep_zet a) \<union> (Rep_zet b))"
   135   zsubset :: "'a zet \<Rightarrow> 'a zet \<Rightarrow> bool"
   136   "zsubset a b \<equiv> ! x. zin x a \<longrightarrow> zin x b"
   137 
   138 lemma explode_union: "explode (union a b) = (explode a) \<union> (explode b)"
   139   apply (rule set_ext)
   140   apply (simp add: explode_def union)
   141   done
   142 
   143 lemma Rep_zet_zunion: "Rep_zet (zunion a b) = (Rep_zet a) \<union> (Rep_zet b)"
   144 proof -
   145   from Rep_zet[of a] have "? f z. inj_on f (Rep_zet a) \<and> f ` (Rep_zet a) = explode z"
   146     by (auto simp add: zet_def')
   147   then obtain fa za where a:"inj_on fa (Rep_zet a) \<and> fa ` (Rep_zet a) = explode za"
   148     by blast
   149   from a have fa: "inj_on fa (Rep_zet a)" by blast
   150   from a have za: "fa ` (Rep_zet a) = explode za" by blast
   151   from Rep_zet[of b] have "? f z. inj_on f (Rep_zet b) \<and> f ` (Rep_zet b) = explode z"
   152     by (auto simp add: zet_def')
   153   then obtain fb zb where b:"inj_on fb (Rep_zet b) \<and> fb ` (Rep_zet b) = explode zb"
   154     by blast
   155   from b have fb: "inj_on fb (Rep_zet b)" by blast
   156   from b have zb: "fb ` (Rep_zet b) = explode zb" by blast 
   157   let ?f = "(\<lambda> x. if x \<in> (Rep_zet a) then Opair (fa x) (Empty) else Opair (fb x) (Singleton Empty))" 
   158   let ?z = "CartProd (union za zb) (Upair Empty (Singleton Empty))"
   159   have se: "Singleton Empty \<noteq> Empty"
   160     apply (auto simp add: Ext Singleton)
   161     apply (rule exI[where x=Empty])
   162     apply (simp add: Empty)
   163     done
   164   show ?thesis
   165     apply (simp add: zunion_def)
   166     apply (subst Abs_zet_inverse)
   167     apply (auto simp add: zet_def)
   168     apply (rule exI[where x = ?f])
   169     apply (rule conjI)
   170     apply (auto simp add: inj_on_def Opair inj_onD[OF fa] inj_onD[OF fb] se se[symmetric])
   171     apply (rule exI[where x = ?z])
   172     apply (insert za zb)
   173     apply (auto simp add: explode_def CartProd union Upair Opair)
   174     done
   175 qed
   176 
   177 lemma zunion: "zin x (zunion a b) = ((zin x a) \<or> (zin x b))"
   178   by (auto simp add: zin_def Rep_zet_zunion)
   179 
   180 lemma zimage_zexplode_eq: "zimage f (zexplode z) = zexplode (Repl z f)"
   181   by (simp add: zet_ext_eq zin_zexplode_eq Repl zimage_iff)
   182 
   183 lemma range_explode_eq_zet: "range explode = zet"
   184   apply (rule set_ext)
   185   apply (auto simp add: explode_mem_zet)
   186   apply (drule image_zet_rep)
   187   apply (simp add: image_def)
   188   apply auto
   189   apply (rule_tac x=z in exI)
   190   apply auto
   191   done
   192 
   193 lemma Elem_zimplode: "(Elem x (zimplode z)) = (zin x z)"
   194   apply (simp add: zimplode_def)
   195   apply (subst Elem_implode)
   196   apply (simp_all add: zin_def Rep_zet range_explode_eq_zet)
   197   done
   198 
   199 constdefs
   200   zempty :: "'a zet"
   201   "zempty \<equiv> Abs_zet {}"
   202 
   203 lemma zempty[simp]: "\<not> (zin x zempty)"
   204   by (auto simp add: zin_def zempty_def Abs_zet_inverse zet_def)
   205 
   206 lemma zimage_zempty[simp]: "zimage f zempty = zempty"
   207   by (auto simp add: zet_ext_eq zimage_iff)
   208 
   209 lemma zunion_zempty_left[simp]: "zunion zempty a = a"
   210   by (simp add: zet_ext_eq zunion)
   211 
   212 lemma zunion_zempty_right[simp]: "zunion a zempty = a"
   213   by (simp add: zet_ext_eq zunion)
   214 
   215 lemma zimage_id[simp]: "zimage id A = A"
   216   by (simp add: zet_ext_eq zimage_iff)
   217 
   218 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"
   219   by (auto simp add: zet_ext_eq zimage_iff)
   220 
   221 end