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
```