src/ZF/ZF.thy
changeset 65464 f3cd78ba687c
parent 65386 e3fb3036a00e
parent 65453 b2562bdda54e
child 68490 eb53f944c8cd
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65438:f556a7a9080c 65464:f3cd78ba687c
     1 (*  Title:      ZF/ZF.thy
     1 section\<open>Main ZF Theory: Everything Except AC\<close>
     2     Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
       
     3     Copyright   1993  University of Cambridge
       
     4 *)
       
     5 
     2 
     6 section \<open>Zermelo-Fraenkel Set Theory\<close>
     3 theory ZF imports List_ZF IntDiv_ZF CardinalArith begin
     7 
     4 
     8 theory ZF
     5 (*The theory of "iterates" logically belongs to Nat, but can't go there because
     9 imports "~~/src/FOL/FOL"
     6   primrec isn't available into after Datatype.*)
    10 begin
       
    11 
     7 
    12 subsection \<open>Signature\<close>
     8 subsection\<open>Iteration of the function @{term F}\<close>
    13 
     9 
    14 declare [[eta_contract = false]]
    10 consts  iterates :: "[i=>i,i,i] => i"   ("(_^_ '(_'))" [60,1000,1000] 60)
    15 
    11 
    16 typedecl i
    12 primrec
    17 instance i :: "term" ..
    13     "F^0 (x) = x"
       
    14     "F^(succ(n)) (x) = F(F^n (x))"
    18 
    15 
    19 axiomatization mem :: "[i, i] \<Rightarrow> o"  (infixl "\<in>" 50)  \<comment> \<open>membership relation\<close>
    16 definition
    20   and zero :: "i"  ("0")  \<comment> \<open>the empty set\<close>
    17   iterates_omega :: "[i=>i,i] => i" ("(_^\<omega> '(_'))" [60,1000] 60) where
    21   and Pow :: "i \<Rightarrow> i"  \<comment> \<open>power sets\<close>
    18     "F^\<omega> (x) == \<Union>n\<in>nat. F^n (x)"
    22   and Inf :: "i"  \<comment> \<open>infinite set\<close>
       
    23   and Union :: "i \<Rightarrow> i"  ("\<Union>_" [90] 90)
       
    24   and PrimReplace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
       
    25 
    19 
    26 abbreviation not_mem :: "[i, i] \<Rightarrow> o"  (infixl "\<notin>" 50)  \<comment> \<open>negated membership relation\<close>
    20 lemma iterates_triv:
    27   where "x \<notin> y \<equiv> \<not> (x \<in> y)"
    21      "[| n\<in>nat;  F(x) = x |] ==> F^n (x) = x"
       
    22 by (induct n rule: nat_induct, simp_all)
       
    23 
       
    24 lemma iterates_type [TC]:
       
    25      "[| n \<in> nat;  a \<in> A; !!x. x \<in> A ==> F(x) \<in> A |]
       
    26       ==> F^n (a) \<in> A"
       
    27 by (induct n rule: nat_induct, simp_all)
       
    28 
       
    29 lemma iterates_omega_triv:
       
    30     "F(x) = x ==> F^\<omega> (x) = x"
       
    31 by (simp add: iterates_omega_def iterates_triv)
       
    32 
       
    33 lemma Ord_iterates [simp]:
       
    34      "[| n\<in>nat;  !!i. Ord(i) ==> Ord(F(i));  Ord(x) |]
       
    35       ==> Ord(F^n (x))"
       
    36 by (induct n rule: nat_induct, simp_all)
       
    37 
       
    38 lemma iterates_commute: "n \<in> nat ==> F(F^n (x)) = F^n (F(x))"
       
    39 by (induct_tac n, simp_all)
    28 
    40 
    29 
    41 
    30 subsection \<open>Bounded Quantifiers\<close>
    42 subsection\<open>Transfinite Recursion\<close>
    31 
    43 
    32 definition Ball :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
    44 text\<open>Transfinite recursion for definitions based on the
    33   where "Ball(A, P) \<equiv> \<forall>x. x\<in>A \<longrightarrow> P(x)"
    45     three cases of ordinals\<close>
    34 
    46 
    35 definition Bex :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
    47 definition
    36   where "Bex(A, P) \<equiv> \<exists>x. x\<in>A \<and> P(x)"
    48   transrec3 :: "[i, i, [i,i]=>i, [i,i]=>i] =>i" where
       
    49     "transrec3(k, a, b, c) ==
       
    50        transrec(k, \<lambda>x r.
       
    51          if x=0 then a
       
    52          else if Limit(x) then c(x, \<lambda>y\<in>x. r`y)
       
    53          else b(Arith.pred(x), r ` Arith.pred(x)))"
    37 
    54 
    38 syntax
    55 lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a"
    39   "_Ball" :: "[pttrn, i, o] \<Rightarrow> o"  ("(3\<forall>_\<in>_./ _)" 10)
    56 by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
    40   "_Bex" :: "[pttrn, i, o] \<Rightarrow> o"  ("(3\<exists>_\<in>_./ _)" 10)
    57 
    41 translations
    58 lemma transrec3_succ [simp]:
    42   "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball(A, \<lambda>x. P)"
    59      "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))"
    43   "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex(A, \<lambda>x. P)"
    60 by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
       
    61 
       
    62 lemma transrec3_Limit:
       
    63      "Limit(i) ==>
       
    64       transrec3(i,a,b,c) = c(i, \<lambda>j\<in>i. transrec3(j,a,b,c))"
       
    65 by (rule transrec3_def [THEN def_transrec, THEN trans], force)
    44 
    66 
    45 
    67 
    46 subsection \<open>Variations on Replacement\<close>
    68 declaration \<open>fn _ =>
    47 
    69   Simplifier.map_ss (Simplifier.set_mksimps (fn ctxt =>
    48 (* Derived form of replacement, restricting P to its functional part.
    70     map mk_eq o Ord_atomize o Variable.gen_all ctxt))
    49    The resulting set (for functional P) is the same as with
    71 \<close>
    50    PrimReplace, but the rules are simpler. *)
       
    51 definition Replace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
       
    52   where "Replace(A,P) == PrimReplace(A, %x y. (\<exists>!z. P(x,z)) & P(x,y))"
       
    53 
       
    54 syntax
       
    55   "_Replace"  :: "[pttrn, pttrn, i, o] => i"  ("(1{_ ./ _ \<in> _, _})")
       
    56 translations
       
    57   "{y. x\<in>A, Q}" \<rightleftharpoons> "CONST Replace(A, \<lambda>x y. Q)"
       
    58 
       
    59 
       
    60 (* Functional form of replacement -- analgous to ML's map functional *)
       
    61 
       
    62 definition RepFun :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
       
    63   where "RepFun(A,f) == {y . x\<in>A, y=f(x)}"
       
    64 
       
    65 syntax
       
    66   "_RepFun" :: "[i, pttrn, i] => i"  ("(1{_ ./ _ \<in> _})" [51,0,51])
       
    67 translations
       
    68   "{b. x\<in>A}" \<rightleftharpoons> "CONST RepFun(A, \<lambda>x. b)"
       
    69 
       
    70 
       
    71 (* Separation and Pairing can be derived from the Replacement
       
    72    and Powerset Axioms using the following definitions. *)
       
    73 definition Collect :: "[i, i \<Rightarrow> o] \<Rightarrow> i"
       
    74   where "Collect(A,P) == {y . x\<in>A, x=y & P(x)}"
       
    75 
       
    76 syntax
       
    77   "_Collect" :: "[pttrn, i, o] \<Rightarrow> i"  ("(1{_ \<in> _ ./ _})")
       
    78 translations
       
    79   "{x\<in>A. P}" \<rightleftharpoons> "CONST Collect(A, \<lambda>x. P)"
       
    80 
       
    81 
       
    82 subsection \<open>General union and intersection\<close>
       
    83 
       
    84 definition Inter :: "i => i"  ("\<Inter>_" [90] 90)
       
    85   where "\<Inter>(A) == { x\<in>\<Union>(A) . \<forall>y\<in>A. x\<in>y}"
       
    86 
       
    87 syntax
       
    88   "_UNION" :: "[pttrn, i, i] => i"  ("(3\<Union>_\<in>_./ _)" 10)
       
    89   "_INTER" :: "[pttrn, i, i] => i"  ("(3\<Inter>_\<in>_./ _)" 10)
       
    90 translations
       
    91   "\<Union>x\<in>A. B" == "CONST Union({B. x\<in>A})"
       
    92   "\<Inter>x\<in>A. B" == "CONST Inter({B. x\<in>A})"
       
    93 
       
    94 
       
    95 subsection \<open>Finite sets and binary operations\<close>
       
    96 
       
    97 (*Unordered pairs (Upair) express binary union/intersection and cons;
       
    98   set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
       
    99 
       
   100 definition Upair :: "[i, i] => i"
       
   101   where "Upair(a,b) == {y. x\<in>Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
       
   102 
       
   103 definition Subset :: "[i, i] \<Rightarrow> o"  (infixl "\<subseteq>" 50)  \<comment> \<open>subset relation\<close>
       
   104   where subset_def: "A \<subseteq> B \<equiv> \<forall>x\<in>A. x\<in>B"
       
   105 
       
   106 definition Diff :: "[i, i] \<Rightarrow> i"  (infixl "-" 65)  \<comment> \<open>set difference\<close>
       
   107   where "A - B == { x\<in>A . ~(x\<in>B) }"
       
   108 
       
   109 definition Un :: "[i, i] \<Rightarrow> i"  (infixl "\<union>" 65)  \<comment> \<open>binary union\<close>
       
   110   where "A \<union> B == \<Union>(Upair(A,B))"
       
   111 
       
   112 definition Int :: "[i, i] \<Rightarrow> i"  (infixl "\<inter>" 70)  \<comment> \<open>binary intersection\<close>
       
   113   where "A \<inter> B == \<Inter>(Upair(A,B))"
       
   114 
       
   115 definition cons :: "[i, i] => i"
       
   116   where "cons(a,A) == Upair(a,a) \<union> A"
       
   117 
       
   118 definition succ :: "i => i"
       
   119   where "succ(i) == cons(i, i)"
       
   120 
       
   121 nonterminal "is"
       
   122 syntax
       
   123   "" :: "i \<Rightarrow> is"  ("_")
       
   124   "_Enum" :: "[i, is] \<Rightarrow> is"  ("_,/ _")
       
   125   "_Finset" :: "is \<Rightarrow> i"  ("{(_)}")
       
   126 translations
       
   127   "{x, xs}" == "CONST cons(x, {xs})"
       
   128   "{x}" == "CONST cons(x, 0)"
       
   129 
       
   130 
       
   131 subsection \<open>Axioms\<close>
       
   132 
       
   133 (* ZF axioms -- see Suppes p.238
       
   134    Axioms for Union, Pow and Replace state existence only,
       
   135    uniqueness is derivable using extensionality. *)
       
   136 
       
   137 axiomatization
       
   138 where
       
   139   extension:     "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A" and
       
   140   Union_iff:     "A \<in> \<Union>(C) \<longleftrightarrow> (\<exists>B\<in>C. A\<in>B)" and
       
   141   Pow_iff:       "A \<in> Pow(B) \<longleftrightarrow> A \<subseteq> B" and
       
   142 
       
   143   (*We may name this set, though it is not uniquely defined.*)
       
   144   infinity:      "0 \<in> Inf \<and> (\<forall>y\<in>Inf. succ(y) \<in> Inf)" and
       
   145 
       
   146   (*This formulation facilitates case analysis on A.*)
       
   147   foundation:    "A = 0 \<or> (\<exists>x\<in>A. \<forall>y\<in>x. y\<notin>A)" and
       
   148 
       
   149   (*Schema axiom since predicate P is a higher-order variable*)
       
   150   replacement:   "(\<forall>x\<in>A. \<forall>y z. P(x,y) \<and> P(x,z) \<longrightarrow> y = z) \<Longrightarrow>
       
   151                          b \<in> PrimReplace(A,P) \<longleftrightarrow> (\<exists>x\<in>A. P(x,b))"
       
   152 
       
   153 
       
   154 subsection \<open>Definite descriptions -- via Replace over the set "1"\<close>
       
   155 
       
   156 definition The :: "(i \<Rightarrow> o) \<Rightarrow> i"  (binder "THE " 10)
       
   157   where the_def: "The(P)    == \<Union>({y . x \<in> {0}, P(y)})"
       
   158 
       
   159 definition If :: "[o, i, i] \<Rightarrow> i"  ("(if (_)/ then (_)/ else (_))" [10] 10)
       
   160   where if_def: "if P then a else b == THE z. P & z=a | ~P & z=b"
       
   161 
       
   162 abbreviation (input)
       
   163   old_if :: "[o, i, i] => i"  ("if '(_,_,_')")
       
   164   where "if(P,a,b) == If(P,a,b)"
       
   165 
       
   166 
       
   167 subsection \<open>Ordered Pairing\<close>
       
   168 
       
   169 (* this "symmetric" definition works better than {{a}, {a,b}} *)
       
   170 definition Pair :: "[i, i] => i"
       
   171   where "Pair(a,b) == {{a,a}, {a,b}}"
       
   172 
       
   173 definition fst :: "i \<Rightarrow> i"
       
   174   where "fst(p) == THE a. \<exists>b. p = Pair(a, b)"
       
   175 
       
   176 definition snd :: "i \<Rightarrow> i"
       
   177   where "snd(p) == THE b. \<exists>a. p = Pair(a, b)"
       
   178 
       
   179 definition split :: "[[i, i] \<Rightarrow> 'a, i] \<Rightarrow> 'a::{}"  \<comment> \<open>for pattern-matching\<close>
       
   180   where "split(c) == \<lambda>p. c(fst(p), snd(p))"
       
   181 
       
   182 (* Patterns -- extends pre-defined type "pttrn" used in abstractions *)
       
   183 nonterminal patterns
       
   184 syntax
       
   185   "_pattern"  :: "patterns => pttrn"         ("\<langle>_\<rangle>")
       
   186   ""          :: "pttrn => patterns"         ("_")
       
   187   "_patterns" :: "[pttrn, patterns] => patterns"  ("_,/_")
       
   188   "_Tuple"    :: "[i, is] => i"              ("\<langle>(_,/ _)\<rangle>")
       
   189 translations
       
   190   "\<langle>x, y, z\<rangle>"   == "\<langle>x, \<langle>y, z\<rangle>\<rangle>"
       
   191   "\<langle>x, y\<rangle>"      == "CONST Pair(x, y)"
       
   192   "\<lambda>\<langle>x,y,zs\<rangle>.b" == "CONST split(\<lambda>x \<langle>y,zs\<rangle>.b)"
       
   193   "\<lambda>\<langle>x,y\<rangle>.b"    == "CONST split(\<lambda>x y. b)"
       
   194 
       
   195 definition Sigma :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
       
   196   where "Sigma(A,B) == \<Union>x\<in>A. \<Union>y\<in>B(x). {\<langle>x,y\<rangle>}"
       
   197 
       
   198 abbreviation cart_prod :: "[i, i] => i"  (infixr "\<times>" 80)  \<comment> \<open>Cartesian product\<close>
       
   199   where "A \<times> B \<equiv> Sigma(A, \<lambda>_. B)"
       
   200 
       
   201 
       
   202 subsection \<open>Relations and Functions\<close>
       
   203 
       
   204 (*converse of relation r, inverse of function*)
       
   205 definition converse :: "i \<Rightarrow> i"
       
   206   where "converse(r) == {z. w\<in>r, \<exists>x y. w=\<langle>x,y\<rangle> \<and> z=\<langle>y,x\<rangle>}"
       
   207 
       
   208 definition domain :: "i \<Rightarrow> i"
       
   209   where "domain(r) == {x. w\<in>r, \<exists>y. w=\<langle>x,y\<rangle>}"
       
   210 
       
   211 definition range :: "i \<Rightarrow> i"
       
   212   where "range(r) == domain(converse(r))"
       
   213 
       
   214 definition field :: "i \<Rightarrow> i"
       
   215   where "field(r) == domain(r) \<union> range(r)"
       
   216 
       
   217 definition relation :: "i \<Rightarrow> o"  \<comment> \<open>recognizes sets of pairs\<close>
       
   218   where "relation(r) == \<forall>z\<in>r. \<exists>x y. z = \<langle>x,y\<rangle>"
       
   219 
       
   220 definition "function" :: "i \<Rightarrow> o"  \<comment> \<open>recognizes functions; can have non-pairs\<close>
       
   221   where "function(r) == \<forall>x y. \<langle>x,y\<rangle> \<in> r \<longrightarrow> (\<forall>y'. \<langle>x,y'\<rangle> \<in> r \<longrightarrow> y = y')"
       
   222 
       
   223 definition Image :: "[i, i] \<Rightarrow> i"  (infixl "``" 90)  \<comment> \<open>image\<close>
       
   224   where image_def: "r `` A  == {y \<in> range(r). \<exists>x\<in>A. \<langle>x,y\<rangle> \<in> r}"
       
   225 
       
   226 definition vimage :: "[i, i] \<Rightarrow> i"  (infixl "-``" 90)  \<comment> \<open>inverse image\<close>
       
   227   where vimage_def: "r -`` A == converse(r)``A"
       
   228 
       
   229 (* Restrict the relation r to the domain A *)
       
   230 definition restrict :: "[i, i] \<Rightarrow> i"
       
   231   where "restrict(r,A) == {z \<in> r. \<exists>x\<in>A. \<exists>y. z = \<langle>x,y\<rangle>}"
       
   232 
       
   233 
       
   234 (* Abstraction, application and Cartesian product of a family of sets *)
       
   235 
       
   236 definition Lambda :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
       
   237   where lam_def: "Lambda(A,b) == {\<langle>x,b(x)\<rangle>. x\<in>A}"
       
   238 
       
   239 definition "apply" :: "[i, i] \<Rightarrow> i"  (infixl "`" 90)  \<comment> \<open>function application\<close>
       
   240   where "f`a == \<Union>(f``{a})"
       
   241 
       
   242 definition Pi :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
       
   243   where "Pi(A,B) == {f\<in>Pow(Sigma(A,B)). A\<subseteq>domain(f) & function(f)}"
       
   244 
       
   245 abbreviation function_space :: "[i, i] \<Rightarrow> i"  (infixr "->" 60)  \<comment> \<open>function space\<close>
       
   246   where "A -> B \<equiv> Pi(A, \<lambda>_. B)"
       
   247 
       
   248 
       
   249 (* binder syntax *)
       
   250 
       
   251 syntax
       
   252   "_PROD"     :: "[pttrn, i, i] => i"        ("(3\<Prod>_\<in>_./ _)" 10)
       
   253   "_SUM"      :: "[pttrn, i, i] => i"        ("(3\<Sum>_\<in>_./ _)" 10)
       
   254   "_lam"      :: "[pttrn, i, i] => i"        ("(3\<lambda>_\<in>_./ _)" 10)
       
   255 translations
       
   256   "\<Prod>x\<in>A. B"   == "CONST Pi(A, \<lambda>x. B)"
       
   257   "\<Sum>x\<in>A. B"   == "CONST Sigma(A, \<lambda>x. B)"
       
   258   "\<lambda>x\<in>A. f"    == "CONST Lambda(A, \<lambda>x. f)"
       
   259 
       
   260 
       
   261 subsection \<open>ASCII syntax\<close>
       
   262 
       
   263 notation (ASCII)
       
   264   cart_prod       (infixr "*" 80) and
       
   265   Int             (infixl "Int" 70) and
       
   266   Un              (infixl "Un" 65) and
       
   267   function_space  (infixr "\<rightarrow>" 60) and
       
   268   Subset          (infixl "<=" 50) and
       
   269   mem             (infixl ":" 50) and
       
   270   not_mem         (infixl "~:" 50)
       
   271 
       
   272 syntax (ASCII)
       
   273   "_Ball"     :: "[pttrn, i, o] => o"        ("(3ALL _:_./ _)" 10)
       
   274   "_Bex"      :: "[pttrn, i, o] => o"        ("(3EX _:_./ _)" 10)
       
   275   "_Collect"  :: "[pttrn, i, o] => i"        ("(1{_: _ ./ _})")
       
   276   "_Replace"  :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _: _, _})")
       
   277   "_RepFun"   :: "[i, pttrn, i] => i"        ("(1{_ ./ _: _})" [51,0,51])
       
   278   "_UNION"    :: "[pttrn, i, i] => i"        ("(3UN _:_./ _)" 10)
       
   279   "_INTER"    :: "[pttrn, i, i] => i"        ("(3INT _:_./ _)" 10)
       
   280   "_PROD"     :: "[pttrn, i, i] => i"        ("(3PROD _:_./ _)" 10)
       
   281   "_SUM"      :: "[pttrn, i, i] => i"        ("(3SUM _:_./ _)" 10)
       
   282   "_lam"      :: "[pttrn, i, i] => i"        ("(3lam _:_./ _)" 10)
       
   283   "_Tuple"    :: "[i, is] => i"              ("<(_,/ _)>")
       
   284   "_pattern"  :: "patterns => pttrn"         ("<_>")
       
   285 
       
   286 
       
   287 subsection \<open>Substitution\<close>
       
   288 
       
   289 (*Useful examples:  singletonI RS subst_elem,  subst_elem RSN (2,IntI) *)
       
   290 lemma subst_elem: "[| b\<in>A;  a=b |] ==> a\<in>A"
       
   291 by (erule ssubst, assumption)
       
   292 
       
   293 
       
   294 subsection\<open>Bounded universal quantifier\<close>
       
   295 
       
   296 lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
       
   297 by (simp add: Ball_def)
       
   298 
       
   299 lemmas strip = impI allI ballI
       
   300 
       
   301 lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x);  x: A |] ==> P(x)"
       
   302 by (simp add: Ball_def)
       
   303 
       
   304 (*Instantiates x first: better for automatic theorem proving?*)
       
   305 lemma rev_ballE [elim]:
       
   306     "[| \<forall>x\<in>A. P(x);  x\<notin>A ==> Q;  P(x) ==> Q |] ==> Q"
       
   307 by (simp add: Ball_def, blast)
       
   308 
       
   309 lemma ballE: "[| \<forall>x\<in>A. P(x);  P(x) ==> Q;  x\<notin>A ==> Q |] ==> Q"
       
   310 by blast
       
   311 
       
   312 (*Used in the datatype package*)
       
   313 lemma rev_bspec: "[| x: A;  \<forall>x\<in>A. P(x) |] ==> P(x)"
       
   314 by (simp add: Ball_def)
       
   315 
       
   316 (*Trival rewrite rule;   @{term"(\<forall>x\<in>A.P)<->P"} holds only if A is nonempty!*)
       
   317 lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) \<longrightarrow> P)"
       
   318 by (simp add: Ball_def)
       
   319 
       
   320 (*Congruence rule for rewriting*)
       
   321 lemma ball_cong [cong]:
       
   322     "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
       
   323 by (simp add: Ball_def)
       
   324 
       
   325 lemma atomize_ball:
       
   326     "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
       
   327   by (simp only: Ball_def atomize_all atomize_imp)
       
   328 
       
   329 lemmas [symmetric, rulify] = atomize_ball
       
   330   and [symmetric, defn] = atomize_ball
       
   331 
       
   332 
       
   333 subsection\<open>Bounded existential quantifier\<close>
       
   334 
       
   335 lemma bexI [intro]: "[| P(x);  x: A |] ==> \<exists>x\<in>A. P(x)"
       
   336 by (simp add: Bex_def, blast)
       
   337 
       
   338 (*The best argument order when there is only one @{term"x\<in>A"}*)
       
   339 lemma rev_bexI: "[| x\<in>A;  P(x) |] ==> \<exists>x\<in>A. P(x)"
       
   340 by blast
       
   341 
       
   342 (*Not of the general form for such rules. The existential quanitifer becomes universal. *)
       
   343 lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a);  a: A |] ==> \<exists>x\<in>A. P(x)"
       
   344 by blast
       
   345 
       
   346 lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x);  !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
       
   347 by (simp add: Bex_def, blast)
       
   348 
       
   349 (*We do not even have @{term"(\<exists>x\<in>A. True) <-> True"} unless @{term"A" is nonempty!!*)
       
   350 lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
       
   351 by (simp add: Bex_def)
       
   352 
       
   353 lemma bex_cong [cong]:
       
   354     "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |]
       
   355      ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
       
   356 by (simp add: Bex_def cong: conj_cong)
       
   357 
       
   358 
       
   359 
       
   360 subsection\<open>Rules for subsets\<close>
       
   361 
       
   362 lemma subsetI [intro!]:
       
   363     "(!!x. x\<in>A ==> x\<in>B) ==> A \<subseteq> B"
       
   364 by (simp add: subset_def)
       
   365 
       
   366 (*Rule in Modus Ponens style [was called subsetE] *)
       
   367 lemma subsetD [elim]: "[| A \<subseteq> B;  c\<in>A |] ==> c\<in>B"
       
   368 apply (unfold subset_def)
       
   369 apply (erule bspec, assumption)
       
   370 done
       
   371 
       
   372 (*Classical elimination rule*)
       
   373 lemma subsetCE [elim]:
       
   374     "[| A \<subseteq> B;  c\<notin>A ==> P;  c\<in>B ==> P |] ==> P"
       
   375 by (simp add: subset_def, blast)
       
   376 
       
   377 (*Sometimes useful with premises in this order*)
       
   378 lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
       
   379 by blast
       
   380 
       
   381 lemma contra_subsetD: "[| A \<subseteq> B; c \<notin> B |] ==> c \<notin> A"
       
   382 by blast
       
   383 
       
   384 lemma rev_contra_subsetD: "[| c \<notin> B;  A \<subseteq> B |] ==> c \<notin> A"
       
   385 by blast
       
   386 
       
   387 lemma subset_refl [simp]: "A \<subseteq> A"
       
   388 by blast
       
   389 
       
   390 lemma subset_trans: "[| A<=B;  B<=C |] ==> A<=C"
       
   391 by blast
       
   392 
       
   393 (*Useful for proving A<=B by rewriting in some cases*)
       
   394 lemma subset_iff:
       
   395      "A<=B <-> (\<forall>x. x\<in>A \<longrightarrow> x\<in>B)"
       
   396 apply (unfold subset_def Ball_def)
       
   397 apply (rule iff_refl)
       
   398 done
       
   399 
       
   400 text\<open>For calculations\<close>
       
   401 declare subsetD [trans] rev_subsetD [trans] subset_trans [trans]
       
   402 
       
   403 
       
   404 subsection\<open>Rules for equality\<close>
       
   405 
       
   406 (*Anti-symmetry of the subset relation*)
       
   407 lemma equalityI [intro]: "[| A \<subseteq> B;  B \<subseteq> A |] ==> A = B"
       
   408 by (rule extension [THEN iffD2], rule conjI)
       
   409 
       
   410 
       
   411 lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
       
   412 by (rule equalityI, blast+)
       
   413 
       
   414 lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1]
       
   415 lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2]
       
   416 
       
   417 lemma equalityE: "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P"
       
   418 by (blast dest: equalityD1 equalityD2)
       
   419 
       
   420 lemma equalityCE:
       
   421     "[| A = B;  [| c\<in>A; c\<in>B |] ==> P;  [| c\<notin>A; c\<notin>B |] ==> P |]  ==>  P"
       
   422 by (erule equalityE, blast)
       
   423 
       
   424 lemma equality_iffD:
       
   425   "A = B ==> (!!x. x \<in> A <-> x \<in> B)"
       
   426   by auto
       
   427 
       
   428 
       
   429 subsection\<open>Rules for Replace -- the derived form of replacement\<close>
       
   430 
       
   431 lemma Replace_iff:
       
   432     "b \<in> {y. x\<in>A, P(x,y)}  <->  (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) \<longrightarrow> y=b))"
       
   433 apply (unfold Replace_def)
       
   434 apply (rule replacement [THEN iff_trans], blast+)
       
   435 done
       
   436 
       
   437 (*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
       
   438 lemma ReplaceI [intro]:
       
   439     "[| P(x,b);  x: A;  !!y. P(x,y) ==> y=b |] ==>
       
   440      b \<in> {y. x\<in>A, P(x,y)}"
       
   441 by (rule Replace_iff [THEN iffD2], blast)
       
   442 
       
   443 (*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
       
   444 lemma ReplaceE:
       
   445     "[| b \<in> {y. x\<in>A, P(x,y)};
       
   446         !!x. [| x: A;  P(x,b);  \<forall>y. P(x,y)\<longrightarrow>y=b |] ==> R
       
   447      |] ==> R"
       
   448 by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
       
   449 
       
   450 (*As above but without the (generally useless) 3rd assumption*)
       
   451 lemma ReplaceE2 [elim!]:
       
   452     "[| b \<in> {y. x\<in>A, P(x,y)};
       
   453         !!x. [| x: A;  P(x,b) |] ==> R
       
   454      |] ==> R"
       
   455 by (erule ReplaceE, blast)
       
   456 
       
   457 lemma Replace_cong [cong]:
       
   458     "[| A=B;  !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>
       
   459      Replace(A,P) = Replace(B,Q)"
       
   460 apply (rule equality_iffI)
       
   461 apply (simp add: Replace_iff)
       
   462 done
       
   463 
       
   464 
       
   465 subsection\<open>Rules for RepFun\<close>
       
   466 
       
   467 lemma RepFunI: "a \<in> A ==> f(a) \<in> {f(x). x\<in>A}"
       
   468 by (simp add: RepFun_def Replace_iff, blast)
       
   469 
       
   470 (*Useful for coinduction proofs*)
       
   471 lemma RepFun_eqI [intro]: "[| b=f(a);  a \<in> A |] ==> b \<in> {f(x). x\<in>A}"
       
   472 apply (erule ssubst)
       
   473 apply (erule RepFunI)
       
   474 done
       
   475 
       
   476 lemma RepFunE [elim!]:
       
   477     "[| b \<in> {f(x). x\<in>A};
       
   478         !!x.[| x\<in>A;  b=f(x) |] ==> P |] ==>
       
   479      P"
       
   480 by (simp add: RepFun_def Replace_iff, blast)
       
   481 
       
   482 lemma RepFun_cong [cong]:
       
   483     "[| A=B;  !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
       
   484 by (simp add: RepFun_def)
       
   485 
       
   486 lemma RepFun_iff [simp]: "b \<in> {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
       
   487 by (unfold Bex_def, blast)
       
   488 
       
   489 lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
       
   490 by blast
       
   491 
       
   492 
       
   493 subsection\<open>Rules for Collect -- forming a subset by separation\<close>
       
   494 
       
   495 (*Separation is derivable from Replacement*)
       
   496 lemma separation [simp]: "a \<in> {x\<in>A. P(x)} <-> a\<in>A & P(a)"
       
   497 by (unfold Collect_def, blast)
       
   498 
       
   499 lemma CollectI [intro!]: "[| a\<in>A;  P(a) |] ==> a \<in> {x\<in>A. P(x)}"
       
   500 by simp
       
   501 
       
   502 lemma CollectE [elim!]: "[| a \<in> {x\<in>A. P(x)};  [| a\<in>A; P(a) |] ==> R |] ==> R"
       
   503 by simp
       
   504 
       
   505 lemma CollectD1: "a \<in> {x\<in>A. P(x)} ==> a\<in>A"
       
   506 by (erule CollectE, assumption)
       
   507 
       
   508 lemma CollectD2: "a \<in> {x\<in>A. P(x)} ==> P(a)"
       
   509 by (erule CollectE, assumption)
       
   510 
       
   511 lemma Collect_cong [cong]:
       
   512     "[| A=B;  !!x. x\<in>B ==> P(x) <-> Q(x) |]
       
   513      ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
       
   514 by (simp add: Collect_def)
       
   515 
       
   516 
       
   517 subsection\<open>Rules for Unions\<close>
       
   518 
       
   519 declare Union_iff [simp]
       
   520 
       
   521 (*The order of the premises presupposes that C is rigid; A may be flexible*)
       
   522 lemma UnionI [intro]: "[| B: C;  A: B |] ==> A: \<Union>(C)"
       
   523 by (simp, blast)
       
   524 
       
   525 lemma UnionE [elim!]: "[| A \<in> \<Union>(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R"
       
   526 by (simp, blast)
       
   527 
       
   528 
       
   529 subsection\<open>Rules for Unions of families\<close>
       
   530 (* @{term"\<Union>x\<in>A. B(x)"} abbreviates @{term"\<Union>({B(x). x\<in>A})"} *)
       
   531 
       
   532 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
       
   533 by (simp add: Bex_def, blast)
       
   534 
       
   535 (*The order of the premises presupposes that A is rigid; b may be flexible*)
       
   536 lemma UN_I: "[| a: A;  b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
       
   537 by (simp, blast)
       
   538 
       
   539 
       
   540 lemma UN_E [elim!]:
       
   541     "[| b \<in> (\<Union>x\<in>A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R |] ==> R"
       
   542 by blast
       
   543 
       
   544 lemma UN_cong:
       
   545     "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
       
   546 by simp
       
   547 
       
   548 
       
   549 (*No "Addcongs [UN_cong]" because @{term\<Union>} is a combination of constants*)
       
   550 
       
   551 (* UN_E appears before UnionE so that it is tried first, to avoid expensive
       
   552   calls to hyp_subst_tac.  Cannot include UN_I as it is unsafe: would enlarge
       
   553   the search space.*)
       
   554 
       
   555 
       
   556 subsection\<open>Rules for the empty set\<close>
       
   557 
       
   558 (*The set @{term"{x\<in>0. False}"} is empty; by foundation it equals 0
       
   559   See Suppes, page 21.*)
       
   560 lemma not_mem_empty [simp]: "a \<notin> 0"
       
   561 apply (cut_tac foundation)
       
   562 apply (best dest: equalityD2)
       
   563 done
       
   564 
       
   565 lemmas emptyE [elim!] = not_mem_empty [THEN notE]
       
   566 
       
   567 
       
   568 lemma empty_subsetI [simp]: "0 \<subseteq> A"
       
   569 by blast
       
   570 
       
   571 lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
       
   572 by blast
       
   573 
       
   574 lemma equals0D [dest]: "A=0 ==> a \<notin> A"
       
   575 by blast
       
   576 
       
   577 declare sym [THEN equals0D, dest]
       
   578 
       
   579 lemma not_emptyI: "a\<in>A ==> A \<noteq> 0"
       
   580 by blast
       
   581 
       
   582 lemma not_emptyE:  "[| A \<noteq> 0;  !!x. x\<in>A ==> R |] ==> R"
       
   583 by blast
       
   584 
       
   585 
       
   586 subsection\<open>Rules for Inter\<close>
       
   587 
       
   588 (*Not obviously useful for proving InterI, InterD, InterE*)
       
   589 lemma Inter_iff: "A \<in> \<Inter>(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
       
   590 by (simp add: Inter_def Ball_def, blast)
       
   591 
       
   592 (* Intersection is well-behaved only if the family is non-empty! *)
       
   593 lemma InterI [intro!]:
       
   594     "[| !!x. x: C ==> A: x;  C\<noteq>0 |] ==> A \<in> \<Inter>(C)"
       
   595 by (simp add: Inter_iff)
       
   596 
       
   597 (*A "destruct" rule -- every B in C contains A as an element, but
       
   598   A\<in>B can hold when B\<in>C does not!  This rule is analogous to "spec". *)
       
   599 lemma InterD [elim, Pure.elim]: "[| A \<in> \<Inter>(C);  B \<in> C |] ==> A \<in> B"
       
   600 by (unfold Inter_def, blast)
       
   601 
       
   602 (*"Classical" elimination rule -- does not require exhibiting @{term"B\<in>C"} *)
       
   603 lemma InterE [elim]:
       
   604     "[| A \<in> \<Inter>(C);  B\<notin>C ==> R;  A\<in>B ==> R |] ==> R"
       
   605 by (simp add: Inter_def, blast)
       
   606 
       
   607 
       
   608 subsection\<open>Rules for Intersections of families\<close>
       
   609 
       
   610 (* @{term"\<Inter>x\<in>A. B(x)"} abbreviates @{term"\<Inter>({B(x). x\<in>A})"} *)
       
   611 
       
   612 lemma INT_iff: "b \<in> (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
       
   613 by (force simp add: Inter_def)
       
   614 
       
   615 lemma INT_I: "[| !!x. x: A ==> b: B(x);  A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
       
   616 by blast
       
   617 
       
   618 lemma INT_E: "[| b \<in> (\<Inter>x\<in>A. B(x));  a: A |] ==> b \<in> B(a)"
       
   619 by blast
       
   620 
       
   621 lemma INT_cong:
       
   622     "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
       
   623 by simp
       
   624 
       
   625 (*No "Addcongs [INT_cong]" because @{term\<Inter>} is a combination of constants*)
       
   626 
       
   627 
       
   628 subsection\<open>Rules for Powersets\<close>
       
   629 
       
   630 lemma PowI: "A \<subseteq> B ==> A \<in> Pow(B)"
       
   631 by (erule Pow_iff [THEN iffD2])
       
   632 
       
   633 lemma PowD: "A \<in> Pow(B)  ==>  A<=B"
       
   634 by (erule Pow_iff [THEN iffD1])
       
   635 
       
   636 declare Pow_iff [iff]
       
   637 
       
   638 lemmas Pow_bottom = empty_subsetI [THEN PowI]    \<comment>\<open>@{term"0 \<in> Pow(B)"}\<close>
       
   639 lemmas Pow_top = subset_refl [THEN PowI]         \<comment>\<open>@{term"A \<in> Pow(A)"}\<close>
       
   640 
       
   641 
       
   642 subsection\<open>Cantor's Theorem: There is no surjection from a set to its powerset.\<close>
       
   643 
       
   644 (*The search is undirected.  Allowing redundant introduction rules may
       
   645   make it diverge.  Variable b represents ANY map, such as
       
   646   (lam x\<in>A.b(x)): A->Pow(A). *)
       
   647 lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) \<noteq> S"
       
   648 by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
       
   649 
    72 
   650 end
    73 end