src/ZF/Constructible/Relative.thy
 author wenzelm Mon Mar 23 21:05:17 2015 +0100 (2015-03-23) changeset 59788 6f7b6adac439 parent 58871 c399ae4b836f child 60770 240563fbf41d permissions -rw-r--r--
prefer local fixes;
     1 (*  Title:      ZF/Constructible/Relative.thy

     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     3 *)

     4

     5 section {*Relativization and Absoluteness*}

     6

     7 theory Relative imports Main begin

     8

     9 subsection{* Relativized versions of standard set-theoretic concepts *}

    10

    11 definition

    12   empty :: "[i=>o,i] => o" where

    13     "empty(M,z) == \<forall>x[M]. x \<notin> z"

    14

    15 definition

    16   subset :: "[i=>o,i,i] => o" where

    17     "subset(M,A,B) == \<forall>x[M]. x\<in>A \<longrightarrow> x \<in> B"

    18

    19 definition

    20   upair :: "[i=>o,i,i,i] => o" where

    21     "upair(M,a,b,z) == a \<in> z & b \<in> z & (\<forall>x[M]. x\<in>z \<longrightarrow> x = a | x = b)"

    22

    23 definition

    24   pair :: "[i=>o,i,i,i] => o" where

    25     "pair(M,a,b,z) == \<exists>x[M]. upair(M,a,a,x) &

    26                      (\<exists>y[M]. upair(M,a,b,y) & upair(M,x,y,z))"

    27

    28

    29 definition

    30   union :: "[i=>o,i,i,i] => o" where

    31     "union(M,a,b,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> x \<in> a | x \<in> b"

    32

    33 definition

    34   is_cons :: "[i=>o,i,i,i] => o" where

    35     "is_cons(M,a,b,z) == \<exists>x[M]. upair(M,a,a,x) & union(M,x,b,z)"

    36

    37 definition

    38   successor :: "[i=>o,i,i] => o" where

    39     "successor(M,a,z) == is_cons(M,a,a,z)"

    40

    41 definition

    42   number1 :: "[i=>o,i] => o" where

    43     "number1(M,a) == \<exists>x[M]. empty(M,x) & successor(M,x,a)"

    44

    45 definition

    46   number2 :: "[i=>o,i] => o" where

    47     "number2(M,a) == \<exists>x[M]. number1(M,x) & successor(M,x,a)"

    48

    49 definition

    50   number3 :: "[i=>o,i] => o" where

    51     "number3(M,a) == \<exists>x[M]. number2(M,x) & successor(M,x,a)"

    52

    53 definition

    54   powerset :: "[i=>o,i,i] => o" where

    55     "powerset(M,A,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> subset(M,x,A)"

    56

    57 definition

    58   is_Collect :: "[i=>o,i,i=>o,i] => o" where

    59     "is_Collect(M,A,P,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> x \<in> A & P(x)"

    60

    61 definition

    62   is_Replace :: "[i=>o,i,[i,i]=>o,i] => o" where

    63     "is_Replace(M,A,P,z) == \<forall>u[M]. u \<in> z \<longleftrightarrow> (\<exists>x[M]. x\<in>A & P(x,u))"

    64

    65 definition

    66   inter :: "[i=>o,i,i,i] => o" where

    67     "inter(M,a,b,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> x \<in> a & x \<in> b"

    68

    69 definition

    70   setdiff :: "[i=>o,i,i,i] => o" where

    71     "setdiff(M,a,b,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> x \<in> a & x \<notin> b"

    72

    73 definition

    74   big_union :: "[i=>o,i,i] => o" where

    75     "big_union(M,A,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>y[M]. y\<in>A & x \<in> y)"

    76

    77 definition

    78   big_inter :: "[i=>o,i,i] => o" where

    79     "big_inter(M,A,z) ==

    80              (A=0 \<longrightarrow> z=0) &

    81              (A\<noteq>0 \<longrightarrow> (\<forall>x[M]. x \<in> z \<longleftrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow> x \<in> y)))"

    82

    83 definition

    84   cartprod :: "[i=>o,i,i,i] => o" where

    85     "cartprod(M,A,B,z) ==

    86         \<forall>u[M]. u \<in> z \<longleftrightarrow> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))"

    87

    88 definition

    89   is_sum :: "[i=>o,i,i,i] => o" where

    90     "is_sum(M,A,B,Z) ==

    91        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].

    92        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &

    93        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"

    94

    95 definition

    96   is_Inl :: "[i=>o,i,i] => o" where

    97     "is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z)"

    98

    99 definition

   100   is_Inr :: "[i=>o,i,i] => o" where

   101     "is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z)"

   102

   103 definition

   104   is_converse :: "[i=>o,i,i] => o" where

   105     "is_converse(M,r,z) ==

   106         \<forall>x[M]. x \<in> z \<longleftrightarrow>

   107              (\<exists>w[M]. w\<in>r & (\<exists>u[M]. \<exists>v[M]. pair(M,u,v,w) & pair(M,v,u,x)))"

   108

   109 definition

   110   pre_image :: "[i=>o,i,i,i] => o" where

   111     "pre_image(M,r,A,z) ==

   112         \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))"

   113

   114 definition

   115   is_domain :: "[i=>o,i,i] => o" where

   116     "is_domain(M,r,z) ==

   117         \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w)))"

   118

   119 definition

   120   image :: "[i=>o,i,i,i] => o" where

   121     "image(M,r,A,z) ==

   122         \<forall>y[M]. y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w)))"

   123

   124 definition

   125   is_range :: "[i=>o,i,i] => o" where

   126     --{*the cleaner

   127       @{term "\<exists>r'[M]. is_converse(M,r,r') & is_domain(M,r',z)"}

   128       unfortunately needs an instance of separation in order to prove

   129         @{term "M(converse(r))"}.*}

   130     "is_range(M,r,z) ==

   131         \<forall>y[M]. y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w)))"

   132

   133 definition

   134   is_field :: "[i=>o,i,i] => o" where

   135     "is_field(M,r,z) ==

   136         \<exists>dr[M]. \<exists>rr[M]. is_domain(M,r,dr) & is_range(M,r,rr) &

   137                         union(M,dr,rr,z)"

   138

   139 definition

   140   is_relation :: "[i=>o,i] => o" where

   141     "is_relation(M,r) ==

   142         (\<forall>z[M]. z\<in>r \<longrightarrow> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))"

   143

   144 definition

   145   is_function :: "[i=>o,i] => o" where

   146     "is_function(M,r) ==

   147         \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].

   148            pair(M,x,y,p) \<longrightarrow> pair(M,x,y',p') \<longrightarrow> p\<in>r \<longrightarrow> p'\<in>r \<longrightarrow> y=y'"

   149

   150 definition

   151   fun_apply :: "[i=>o,i,i,i] => o" where

   152     "fun_apply(M,f,x,y) ==

   153         (\<exists>xs[M]. \<exists>fxs[M].

   154          upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))"

   155

   156 definition

   157   typed_function :: "[i=>o,i,i,i] => o" where

   158     "typed_function(M,A,B,r) ==

   159         is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &

   160         (\<forall>u[M]. u\<in>r \<longrightarrow> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) \<longrightarrow> y\<in>B))"

   161

   162 definition

   163   is_funspace :: "[i=>o,i,i,i] => o" where

   164     "is_funspace(M,A,B,F) ==

   165         \<forall>f[M]. f \<in> F \<longleftrightarrow> typed_function(M,A,B,f)"

   166

   167 definition

   168   composition :: "[i=>o,i,i,i] => o" where

   169     "composition(M,r,s,t) ==

   170         \<forall>p[M]. p \<in> t \<longleftrightarrow>

   171                (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].

   172                 pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &

   173                 xy \<in> s & yz \<in> r)"

   174

   175 definition

   176   injection :: "[i=>o,i,i,i] => o" where

   177     "injection(M,A,B,f) ==

   178         typed_function(M,A,B,f) &

   179         (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].

   180           pair(M,x,y,p) \<longrightarrow> pair(M,x',y,p') \<longrightarrow> p\<in>f \<longrightarrow> p'\<in>f \<longrightarrow> x=x')"

   181

   182 definition

   183   surjection :: "[i=>o,i,i,i] => o" where

   184     "surjection(M,A,B,f) ==

   185         typed_function(M,A,B,f) &

   186         (\<forall>y[M]. y\<in>B \<longrightarrow> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))"

   187

   188 definition

   189   bijection :: "[i=>o,i,i,i] => o" where

   190     "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)"

   191

   192 definition

   193   restriction :: "[i=>o,i,i,i] => o" where

   194     "restriction(M,r,A,z) ==

   195         \<forall>x[M]. x \<in> z \<longleftrightarrow> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))"

   196

   197 definition

   198   transitive_set :: "[i=>o,i] => o" where

   199     "transitive_set(M,a) == \<forall>x[M]. x\<in>a \<longrightarrow> subset(M,x,a)"

   200

   201 definition

   202   ordinal :: "[i=>o,i] => o" where

   203      --{*an ordinal is a transitive set of transitive sets*}

   204     "ordinal(M,a) == transitive_set(M,a) & (\<forall>x[M]. x\<in>a \<longrightarrow> transitive_set(M,x))"

   205

   206 definition

   207   limit_ordinal :: "[i=>o,i] => o" where

   208     --{*a limit ordinal is a non-empty, successor-closed ordinal*}

   209     "limit_ordinal(M,a) ==

   210         ordinal(M,a) & ~ empty(M,a) &

   211         (\<forall>x[M]. x\<in>a \<longrightarrow> (\<exists>y[M]. y\<in>a & successor(M,x,y)))"

   212

   213 definition

   214   successor_ordinal :: "[i=>o,i] => o" where

   215     --{*a successor ordinal is any ordinal that is neither empty nor limit*}

   216     "successor_ordinal(M,a) ==

   217         ordinal(M,a) & ~ empty(M,a) & ~ limit_ordinal(M,a)"

   218

   219 definition

   220   finite_ordinal :: "[i=>o,i] => o" where

   221     --{*an ordinal is finite if neither it nor any of its elements are limit*}

   222     "finite_ordinal(M,a) ==

   223         ordinal(M,a) & ~ limit_ordinal(M,a) &

   224         (\<forall>x[M]. x\<in>a \<longrightarrow> ~ limit_ordinal(M,x))"

   225

   226 definition

   227   omega :: "[i=>o,i] => o" where

   228     --{*omega is a limit ordinal none of whose elements are limit*}

   229     "omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a \<longrightarrow> ~ limit_ordinal(M,x))"

   230

   231 definition

   232   is_quasinat :: "[i=>o,i] => o" where

   233     "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))"

   234

   235 definition

   236   is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o" where

   237     "is_nat_case(M, a, is_b, k, z) ==

   238        (empty(M,k) \<longrightarrow> z=a) &

   239        (\<forall>m[M]. successor(M,m,k) \<longrightarrow> is_b(m,z)) &

   240        (is_quasinat(M,k) | empty(M,z))"

   241

   242 definition

   243   relation1 :: "[i=>o, [i,i]=>o, i=>i] => o" where

   244     "relation1(M,is_f,f) == \<forall>x[M]. \<forall>y[M]. is_f(x,y) \<longleftrightarrow> y = f(x)"

   245

   246 definition

   247   Relation1 :: "[i=>o, i, [i,i]=>o, i=>i] => o" where

   248     --{*as above, but typed*}

   249     "Relation1(M,A,is_f,f) ==

   250         \<forall>x[M]. \<forall>y[M]. x\<in>A \<longrightarrow> is_f(x,y) \<longleftrightarrow> y = f(x)"

   251

   252 definition

   253   relation2 :: "[i=>o, [i,i,i]=>o, [i,i]=>i] => o" where

   254     "relation2(M,is_f,f) == \<forall>x[M]. \<forall>y[M]. \<forall>z[M]. is_f(x,y,z) \<longleftrightarrow> z = f(x,y)"

   255

   256 definition

   257   Relation2 :: "[i=>o, i, i, [i,i,i]=>o, [i,i]=>i] => o" where

   258     "Relation2(M,A,B,is_f,f) ==

   259         \<forall>x[M]. \<forall>y[M]. \<forall>z[M]. x\<in>A \<longrightarrow> y\<in>B \<longrightarrow> is_f(x,y,z) \<longleftrightarrow> z = f(x,y)"

   260

   261 definition

   262   relation3 :: "[i=>o, [i,i,i,i]=>o, [i,i,i]=>i] => o" where

   263     "relation3(M,is_f,f) ==

   264        \<forall>x[M]. \<forall>y[M]. \<forall>z[M]. \<forall>u[M]. is_f(x,y,z,u) \<longleftrightarrow> u = f(x,y,z)"

   265

   266 definition

   267   Relation3 :: "[i=>o, i, i, i, [i,i,i,i]=>o, [i,i,i]=>i] => o" where

   268     "Relation3(M,A,B,C,is_f,f) ==

   269        \<forall>x[M]. \<forall>y[M]. \<forall>z[M]. \<forall>u[M].

   270          x\<in>A \<longrightarrow> y\<in>B \<longrightarrow> z\<in>C \<longrightarrow> is_f(x,y,z,u) \<longleftrightarrow> u = f(x,y,z)"

   271

   272 definition

   273   relation4 :: "[i=>o, [i,i,i,i,i]=>o, [i,i,i,i]=>i] => o" where

   274     "relation4(M,is_f,f) ==

   275        \<forall>u[M]. \<forall>x[M]. \<forall>y[M]. \<forall>z[M]. \<forall>a[M]. is_f(u,x,y,z,a) \<longleftrightarrow> a = f(u,x,y,z)"

   276

   277

   278 text{*Useful when absoluteness reasoning has replaced the predicates by terms*}

   279 lemma triv_Relation1:

   280      "Relation1(M, A, \<lambda>x y. y = f(x), f)"

   281 by (simp add: Relation1_def)

   282

   283 lemma triv_Relation2:

   284      "Relation2(M, A, B, \<lambda>x y a. a = f(x,y), f)"

   285 by (simp add: Relation2_def)

   286

   287

   288 subsection {*The relativized ZF axioms*}

   289

   290 definition

   291   extensionality :: "(i=>o) => o" where

   292     "extensionality(M) ==

   293         \<forall>x[M]. \<forall>y[M]. (\<forall>z[M]. z \<in> x \<longleftrightarrow> z \<in> y) \<longrightarrow> x=y"

   294

   295 definition

   296   separation :: "[i=>o, i=>o] => o" where

   297     --{*The formula @{text P} should only involve parameters

   298         belonging to @{text M} and all its quantifiers must be relativized

   299         to @{text M}.  We do not have separation as a scheme; every instance

   300         that we need must be assumed (and later proved) separately.*}

   301     "separation(M,P) ==

   302         \<forall>z[M]. \<exists>y[M]. \<forall>x[M]. x \<in> y \<longleftrightarrow> x \<in> z & P(x)"

   303

   304 definition

   305   upair_ax :: "(i=>o) => o" where

   306     "upair_ax(M) == \<forall>x[M]. \<forall>y[M]. \<exists>z[M]. upair(M,x,y,z)"

   307

   308 definition

   309   Union_ax :: "(i=>o) => o" where

   310     "Union_ax(M) == \<forall>x[M]. \<exists>z[M]. big_union(M,x,z)"

   311

   312 definition

   313   power_ax :: "(i=>o) => o" where

   314     "power_ax(M) == \<forall>x[M]. \<exists>z[M]. powerset(M,x,z)"

   315

   316 definition

   317   univalent :: "[i=>o, i, [i,i]=>o] => o" where

   318     "univalent(M,A,P) ==

   319         \<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. \<forall>z[M]. P(x,y) & P(x,z) \<longrightarrow> y=z)"

   320

   321 definition

   322   replacement :: "[i=>o, [i,i]=>o] => o" where

   323     "replacement(M,P) ==

   324       \<forall>A[M]. univalent(M,A,P) \<longrightarrow>

   325       (\<exists>Y[M]. \<forall>b[M]. (\<exists>x[M]. x\<in>A & P(x,b)) \<longrightarrow> b \<in> Y)"

   326

   327 definition

   328   strong_replacement :: "[i=>o, [i,i]=>o] => o" where

   329     "strong_replacement(M,P) ==

   330       \<forall>A[M]. univalent(M,A,P) \<longrightarrow>

   331       (\<exists>Y[M]. \<forall>b[M]. b \<in> Y \<longleftrightarrow> (\<exists>x[M]. x\<in>A & P(x,b)))"

   332

   333 definition

   334   foundation_ax :: "(i=>o) => o" where

   335     "foundation_ax(M) ==

   336         \<forall>x[M]. (\<exists>y[M]. y\<in>x) \<longrightarrow> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & z \<in> y))"

   337

   338

   339 subsection{*A trivial consistency proof for $V_\omega$ *}

   340

   341 text{*We prove that $V_\omega$

   342       (or @{text univ} in Isabelle) satisfies some ZF axioms.

   343      Kunen, Theorem IV 3.13, page 123.*}

   344

   345 lemma univ0_downwards_mem: "[| y \<in> x; x \<in> univ(0) |] ==> y \<in> univ(0)"

   346 apply (insert Transset_univ [OF Transset_0])

   347 apply (simp add: Transset_def, blast)

   348 done

   349

   350 lemma univ0_Ball_abs [simp]:

   351      "A \<in> univ(0) ==> (\<forall>x\<in>A. x \<in> univ(0) \<longrightarrow> P(x)) \<longleftrightarrow> (\<forall>x\<in>A. P(x))"

   352 by (blast intro: univ0_downwards_mem)

   353

   354 lemma univ0_Bex_abs [simp]:

   355      "A \<in> univ(0) ==> (\<exists>x\<in>A. x \<in> univ(0) & P(x)) \<longleftrightarrow> (\<exists>x\<in>A. P(x))"

   356 by (blast intro: univ0_downwards_mem)

   357

   358 text{*Congruence rule for separation: can assume the variable is in @{text M}*}

   359 lemma separation_cong [cong]:

   360      "(!!x. M(x) ==> P(x) \<longleftrightarrow> P'(x))

   361       ==> separation(M, %x. P(x)) \<longleftrightarrow> separation(M, %x. P'(x))"

   362 by (simp add: separation_def)

   363

   364 lemma univalent_cong [cong]:

   365      "[| A=A'; !!x y. [| x\<in>A; M(x); M(y) |] ==> P(x,y) \<longleftrightarrow> P'(x,y) |]

   366       ==> univalent(M, A, %x y. P(x,y)) \<longleftrightarrow> univalent(M, A', %x y. P'(x,y))"

   367 by (simp add: univalent_def)

   368

   369 lemma univalent_triv [intro,simp]:

   370      "univalent(M, A, \<lambda>x y. y = f(x))"

   371 by (simp add: univalent_def)

   372

   373 lemma univalent_conjI2 [intro,simp]:

   374      "univalent(M,A,Q) ==> univalent(M, A, \<lambda>x y. P(x,y) & Q(x,y))"

   375 by (simp add: univalent_def, blast)

   376

   377 text{*Congruence rule for replacement*}

   378 lemma strong_replacement_cong [cong]:

   379      "[| !!x y. [| M(x); M(y) |] ==> P(x,y) \<longleftrightarrow> P'(x,y) |]

   380       ==> strong_replacement(M, %x y. P(x,y)) \<longleftrightarrow>

   381           strong_replacement(M, %x y. P'(x,y))"

   382 by (simp add: strong_replacement_def)

   383

   384 text{*The extensionality axiom*}

   385 lemma "extensionality(\<lambda>x. x \<in> univ(0))"

   386 apply (simp add: extensionality_def)

   387 apply (blast intro: univ0_downwards_mem)

   388 done

   389

   390 text{*The separation axiom requires some lemmas*}

   391 lemma Collect_in_Vfrom:

   392      "[| X \<in> Vfrom(A,j);  Transset(A) |] ==> Collect(X,P) \<in> Vfrom(A, succ(j))"

   393 apply (drule Transset_Vfrom)

   394 apply (rule subset_mem_Vfrom)

   395 apply (unfold Transset_def, blast)

   396 done

   397

   398 lemma Collect_in_VLimit:

   399      "[| X \<in> Vfrom(A,i);  Limit(i);  Transset(A) |]

   400       ==> Collect(X,P) \<in> Vfrom(A,i)"

   401 apply (rule Limit_VfromE, assumption+)

   402 apply (blast intro: Limit_has_succ VfromI Collect_in_Vfrom)

   403 done

   404

   405 lemma Collect_in_univ:

   406      "[| X \<in> univ(A);  Transset(A) |] ==> Collect(X,P) \<in> univ(A)"

   407 by (simp add: univ_def Collect_in_VLimit Limit_nat)

   408

   409 lemma "separation(\<lambda>x. x \<in> univ(0), P)"

   410 apply (simp add: separation_def, clarify)

   411 apply (rule_tac x = "Collect(z,P)" in bexI)

   412 apply (blast intro: Collect_in_univ Transset_0)+

   413 done

   414

   415 text{*Unordered pairing axiom*}

   416 lemma "upair_ax(\<lambda>x. x \<in> univ(0))"

   417 apply (simp add: upair_ax_def upair_def)

   418 apply (blast intro: doubleton_in_univ)

   419 done

   420

   421 text{*Union axiom*}

   422 lemma "Union_ax(\<lambda>x. x \<in> univ(0))"

   423 apply (simp add: Union_ax_def big_union_def, clarify)

   424 apply (rule_tac x="\<Union>x" in bexI)

   425  apply (blast intro: univ0_downwards_mem)

   426 apply (blast intro: Union_in_univ Transset_0)

   427 done

   428

   429 text{*Powerset axiom*}

   430

   431 lemma Pow_in_univ:

   432      "[| X \<in> univ(A);  Transset(A) |] ==> Pow(X) \<in> univ(A)"

   433 apply (simp add: univ_def Pow_in_VLimit Limit_nat)

   434 done

   435

   436 lemma "power_ax(\<lambda>x. x \<in> univ(0))"

   437 apply (simp add: power_ax_def powerset_def subset_def, clarify)

   438 apply (rule_tac x="Pow(x)" in bexI)

   439  apply (blast intro: univ0_downwards_mem)

   440 apply (blast intro: Pow_in_univ Transset_0)

   441 done

   442

   443 text{*Foundation axiom*}

   444 lemma "foundation_ax(\<lambda>x. x \<in> univ(0))"

   445 apply (simp add: foundation_ax_def, clarify)

   446 apply (cut_tac A=x in foundation)

   447 apply (blast intro: univ0_downwards_mem)

   448 done

   449

   450 lemma "replacement(\<lambda>x. x \<in> univ(0), P)"

   451 apply (simp add: replacement_def, clarify)

   452 oops

   453 text{*no idea: maybe prove by induction on the rank of A?*}

   454

   455 text{*Still missing: Replacement, Choice*}

   456

   457 subsection{*Lemmas Needed to Reduce Some Set Constructions to Instances

   458       of Separation*}

   459

   460 lemma image_iff_Collect: "r  A = {y \<in> \<Union>(\<Union>(r)). \<exists>p\<in>r. \<exists>x\<in>A. p=<x,y>}"

   461 apply (rule equalityI, auto)

   462 apply (simp add: Pair_def, blast)

   463 done

   464

   465 lemma vimage_iff_Collect:

   466      "r - A = {x \<in> \<Union>(\<Union>(r)). \<exists>p\<in>r. \<exists>y\<in>A. p=<x,y>}"

   467 apply (rule equalityI, auto)

   468 apply (simp add: Pair_def, blast)

   469 done

   470

   471 text{*These two lemmas lets us prove @{text domain_closed} and

   472       @{text range_closed} without new instances of separation*}

   473

   474 lemma domain_eq_vimage: "domain(r) = r - Union(Union(r))"

   475 apply (rule equalityI, auto)

   476 apply (rule vimageI, assumption)

   477 apply (simp add: Pair_def, blast)

   478 done

   479

   480 lemma range_eq_image: "range(r) = r  Union(Union(r))"

   481 apply (rule equalityI, auto)

   482 apply (rule imageI, assumption)

   483 apply (simp add: Pair_def, blast)

   484 done

   485

   486 lemma replacementD:

   487     "[| replacement(M,P); M(A);  univalent(M,A,P) |]

   488      ==> \<exists>Y[M]. (\<forall>b[M]. ((\<exists>x[M]. x\<in>A & P(x,b)) \<longrightarrow> b \<in> Y))"

   489 by (simp add: replacement_def)

   490

   491 lemma strong_replacementD:

   492     "[| strong_replacement(M,P); M(A);  univalent(M,A,P) |]

   493      ==> \<exists>Y[M]. (\<forall>b[M]. (b \<in> Y \<longleftrightarrow> (\<exists>x[M]. x\<in>A & P(x,b))))"

   494 by (simp add: strong_replacement_def)

   495

   496 lemma separationD:

   497     "[| separation(M,P); M(z) |] ==> \<exists>y[M]. \<forall>x[M]. x \<in> y \<longleftrightarrow> x \<in> z & P(x)"

   498 by (simp add: separation_def)

   499

   500

   501 text{*More constants, for order types*}

   502

   503 definition

   504   order_isomorphism :: "[i=>o,i,i,i,i,i] => o" where

   505     "order_isomorphism(M,A,r,B,s,f) ==

   506         bijection(M,A,B,f) &

   507         (\<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow>

   508           (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].

   509             pair(M,x,y,p) \<longrightarrow> fun_apply(M,f,x,fx) \<longrightarrow> fun_apply(M,f,y,fy) \<longrightarrow>

   510             pair(M,fx,fy,q) \<longrightarrow> (p\<in>r \<longleftrightarrow> q\<in>s))))"

   511

   512 definition

   513   pred_set :: "[i=>o,i,i,i,i] => o" where

   514     "pred_set(M,A,x,r,B) ==

   515         \<forall>y[M]. y \<in> B \<longleftrightarrow> (\<exists>p[M]. p\<in>r & y \<in> A & pair(M,y,x,p))"

   516

   517 definition

   518   membership :: "[i=>o,i,i] => o" where --{*membership relation*}

   519     "membership(M,A,r) ==

   520         \<forall>p[M]. p \<in> r \<longleftrightarrow> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>A & x\<in>y & pair(M,x,y,p)))"

   521

   522

   523 subsection{*Introducing a Transitive Class Model*}

   524

   525 text{*The class M is assumed to be transitive and to satisfy some

   526       relativized ZF axioms*}

   527 locale M_trivial =

   528   fixes M

   529   assumes transM:           "[| y\<in>x; M(x) |] ==> M(y)"

   530       and upair_ax:         "upair_ax(M)"

   531       and Union_ax:         "Union_ax(M)"

   532       and power_ax:         "power_ax(M)"

   533       and replacement:      "replacement(M,P)"

   534       and M_nat [iff]:      "M(nat)"           (*i.e. the axiom of infinity*)

   535

   536

   537 text{*Automatically discovers the proof using @{text transM}, @{text nat_0I}

   538 and @{text M_nat}.*}

   539 lemma (in M_trivial) nonempty [simp]: "M(0)"

   540 by (blast intro: transM)

   541

   542 lemma (in M_trivial) rall_abs [simp]:

   543      "M(A) ==> (\<forall>x[M]. x\<in>A \<longrightarrow> P(x)) \<longleftrightarrow> (\<forall>x\<in>A. P(x))"

   544 by (blast intro: transM)

   545

   546 lemma (in M_trivial) rex_abs [simp]:

   547      "M(A) ==> (\<exists>x[M]. x\<in>A & P(x)) \<longleftrightarrow> (\<exists>x\<in>A. P(x))"

   548 by (blast intro: transM)

   549

   550 lemma (in M_trivial) ball_iff_equiv:

   551      "M(A) ==> (\<forall>x[M]. (x\<in>A \<longleftrightarrow> P(x))) \<longleftrightarrow>

   552                (\<forall>x\<in>A. P(x)) & (\<forall>x. P(x) \<longrightarrow> M(x) \<longrightarrow> x\<in>A)"

   553 by (blast intro: transM)

   554

   555 text{*Simplifies proofs of equalities when there's an iff-equality

   556       available for rewriting, universally quantified over M.

   557       But it's not the only way to prove such equalities: its

   558       premises @{term "M(A)"} and  @{term "M(B)"} can be too strong.*}

   559 lemma (in M_trivial) M_equalityI:

   560      "[| !!x. M(x) ==> x\<in>A \<longleftrightarrow> x\<in>B; M(A); M(B) |] ==> A=B"

   561 by (blast intro!: equalityI dest: transM)

   562

   563

   564 subsubsection{*Trivial Absoluteness Proofs: Empty Set, Pairs, etc.*}

   565

   566 lemma (in M_trivial) empty_abs [simp]:

   567      "M(z) ==> empty(M,z) \<longleftrightarrow> z=0"

   568 apply (simp add: empty_def)

   569 apply (blast intro: transM)

   570 done

   571

   572 lemma (in M_trivial) subset_abs [simp]:

   573      "M(A) ==> subset(M,A,B) \<longleftrightarrow> A \<subseteq> B"

   574 apply (simp add: subset_def)

   575 apply (blast intro: transM)

   576 done

   577

   578 lemma (in M_trivial) upair_abs [simp]:

   579      "M(z) ==> upair(M,a,b,z) \<longleftrightarrow> z={a,b}"

   580 apply (simp add: upair_def)

   581 apply (blast intro: transM)

   582 done

   583

   584 lemma (in M_trivial) upair_in_M_iff [iff]:

   585      "M({a,b}) \<longleftrightarrow> M(a) & M(b)"

   586 apply (insert upair_ax, simp add: upair_ax_def)

   587 apply (blast intro: transM)

   588 done

   589

   590 lemma (in M_trivial) singleton_in_M_iff [iff]:

   591      "M({a}) \<longleftrightarrow> M(a)"

   592 by (insert upair_in_M_iff [of a a], simp)

   593

   594 lemma (in M_trivial) pair_abs [simp]:

   595      "M(z) ==> pair(M,a,b,z) \<longleftrightarrow> z=<a,b>"

   596 apply (simp add: pair_def ZF.Pair_def)

   597 apply (blast intro: transM)

   598 done

   599

   600 lemma (in M_trivial) pair_in_M_iff [iff]:

   601      "M(<a,b>) \<longleftrightarrow> M(a) & M(b)"

   602 by (simp add: ZF.Pair_def)

   603

   604 lemma (in M_trivial) pair_components_in_M:

   605      "[| <x,y> \<in> A; M(A) |] ==> M(x) & M(y)"

   606 apply (simp add: Pair_def)

   607 apply (blast dest: transM)

   608 done

   609

   610 lemma (in M_trivial) cartprod_abs [simp]:

   611      "[| M(A); M(B); M(z) |] ==> cartprod(M,A,B,z) \<longleftrightarrow> z = A*B"

   612 apply (simp add: cartprod_def)

   613 apply (rule iffI)

   614  apply (blast intro!: equalityI intro: transM dest!: rspec)

   615 apply (blast dest: transM)

   616 done

   617

   618 subsubsection{*Absoluteness for Unions and Intersections*}

   619

   620 lemma (in M_trivial) union_abs [simp]:

   621      "[| M(a); M(b); M(z) |] ==> union(M,a,b,z) \<longleftrightarrow> z = a \<union> b"

   622 apply (simp add: union_def)

   623 apply (blast intro: transM)

   624 done

   625

   626 lemma (in M_trivial) inter_abs [simp]:

   627      "[| M(a); M(b); M(z) |] ==> inter(M,a,b,z) \<longleftrightarrow> z = a \<inter> b"

   628 apply (simp add: inter_def)

   629 apply (blast intro: transM)

   630 done

   631

   632 lemma (in M_trivial) setdiff_abs [simp]:

   633      "[| M(a); M(b); M(z) |] ==> setdiff(M,a,b,z) \<longleftrightarrow> z = a-b"

   634 apply (simp add: setdiff_def)

   635 apply (blast intro: transM)

   636 done

   637

   638 lemma (in M_trivial) Union_abs [simp]:

   639      "[| M(A); M(z) |] ==> big_union(M,A,z) \<longleftrightarrow> z = \<Union>(A)"

   640 apply (simp add: big_union_def)

   641 apply (blast intro!: equalityI dest: transM)

   642 done

   643

   644 lemma (in M_trivial) Union_closed [intro,simp]:

   645      "M(A) ==> M(\<Union>(A))"

   646 by (insert Union_ax, simp add: Union_ax_def)

   647

   648 lemma (in M_trivial) Un_closed [intro,simp]:

   649      "[| M(A); M(B) |] ==> M(A \<union> B)"

   650 by (simp only: Un_eq_Union, blast)

   651

   652 lemma (in M_trivial) cons_closed [intro,simp]:

   653      "[| M(a); M(A) |] ==> M(cons(a,A))"

   654 by (subst cons_eq [symmetric], blast)

   655

   656 lemma (in M_trivial) cons_abs [simp]:

   657      "[| M(b); M(z) |] ==> is_cons(M,a,b,z) \<longleftrightarrow> z = cons(a,b)"

   658 by (simp add: is_cons_def, blast intro: transM)

   659

   660 lemma (in M_trivial) successor_abs [simp]:

   661      "[| M(a); M(z) |] ==> successor(M,a,z) \<longleftrightarrow> z = succ(a)"

   662 by (simp add: successor_def, blast)

   663

   664 lemma (in M_trivial) succ_in_M_iff [iff]:

   665      "M(succ(a)) \<longleftrightarrow> M(a)"

   666 apply (simp add: succ_def)

   667 apply (blast intro: transM)

   668 done

   669

   670 subsubsection{*Absoluteness for Separation and Replacement*}

   671

   672 lemma (in M_trivial) separation_closed [intro,simp]:

   673      "[| separation(M,P); M(A) |] ==> M(Collect(A,P))"

   674 apply (insert separation, simp add: separation_def)

   675 apply (drule rspec, assumption, clarify)

   676 apply (subgoal_tac "y = Collect(A,P)", blast)

   677 apply (blast dest: transM)

   678 done

   679

   680 lemma separation_iff:

   681      "separation(M,P) \<longleftrightarrow> (\<forall>z[M]. \<exists>y[M]. is_Collect(M,z,P,y))"

   682 by (simp add: separation_def is_Collect_def)

   683

   684 lemma (in M_trivial) Collect_abs [simp]:

   685      "[| M(A); M(z) |] ==> is_Collect(M,A,P,z) \<longleftrightarrow> z = Collect(A,P)"

   686 apply (simp add: is_Collect_def)

   687 apply (blast intro!: equalityI dest: transM)

   688 done

   689

   690 text{*Probably the premise and conclusion are equivalent*}

   691 lemma (in M_trivial) strong_replacementI [rule_format]:

   692     "[| \<forall>B[M]. separation(M, %u. \<exists>x[M]. x\<in>B & P(x,u)) |]

   693      ==> strong_replacement(M,P)"

   694 apply (simp add: strong_replacement_def, clarify)

   695 apply (frule replacementD [OF replacement], assumption, clarify)

   696 apply (drule_tac x=A in rspec, clarify)

   697 apply (drule_tac z=Y in separationD, assumption, clarify)

   698 apply (rule_tac x=y in rexI, force, assumption)

   699 done

   700

   701 subsubsection{*The Operator @{term is_Replace}*}

   702

   703

   704 lemma is_Replace_cong [cong]:

   705      "[| A=A';

   706          !!x y. [| M(x); M(y) |] ==> P(x,y) \<longleftrightarrow> P'(x,y);

   707          z=z' |]

   708       ==> is_Replace(M, A, %x y. P(x,y), z) \<longleftrightarrow>

   709           is_Replace(M, A', %x y. P'(x,y), z')"

   710 by (simp add: is_Replace_def)

   711

   712 lemma (in M_trivial) univalent_Replace_iff:

   713      "[| M(A); univalent(M,A,P);

   714          !!x y. [| x\<in>A; P(x,y) |] ==> M(y) |]

   715       ==> u \<in> Replace(A,P) \<longleftrightarrow> (\<exists>x. x\<in>A & P(x,u))"

   716 apply (simp add: Replace_iff univalent_def)

   717 apply (blast dest: transM)

   718 done

   719

   720 (*The last premise expresses that P takes M to M*)

   721 lemma (in M_trivial) strong_replacement_closed [intro,simp]:

   722      "[| strong_replacement(M,P); M(A); univalent(M,A,P);

   723          !!x y. [| x\<in>A; P(x,y) |] ==> M(y) |] ==> M(Replace(A,P))"

   724 apply (simp add: strong_replacement_def)

   725 apply (drule_tac x=A in rspec, safe)

   726 apply (subgoal_tac "Replace(A,P) = Y")

   727  apply simp

   728 apply (rule equality_iffI)

   729 apply (simp add: univalent_Replace_iff)

   730 apply (blast dest: transM)

   731 done

   732

   733 lemma (in M_trivial) Replace_abs:

   734      "[| M(A); M(z); univalent(M,A,P);

   735          !!x y. [| x\<in>A; P(x,y) |] ==> M(y)  |]

   736       ==> is_Replace(M,A,P,z) \<longleftrightarrow> z = Replace(A,P)"

   737 apply (simp add: is_Replace_def)

   738 apply (rule iffI)

   739  apply (rule equality_iffI)

   740  apply (simp_all add: univalent_Replace_iff)

   741  apply (blast dest: transM)+

   742 done

   743

   744

   745 (*The first premise can't simply be assumed as a schema.

   746   It is essential to take care when asserting instances of Replacement.

   747   Let K be a nonconstructible subset of nat and define

   748   f(x) = x if x \<in> K and f(x)=0 otherwise.  Then RepFun(nat,f) = cons(0,K), a

   749   nonconstructible set.  So we cannot assume that M(X) implies M(RepFun(X,f))

   750   even for f \<in> M -> M.

   751 *)

   752 lemma (in M_trivial) RepFun_closed:

   753      "[| strong_replacement(M, \<lambda>x y. y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |]

   754       ==> M(RepFun(A,f))"

   755 apply (simp add: RepFun_def)

   756 done

   757

   758 lemma Replace_conj_eq: "{y . x \<in> A, x\<in>A & y=f(x)} = {y . x\<in>A, y=f(x)}"

   759 by simp

   760

   761 text{*Better than @{text RepFun_closed} when having the formula @{term "x\<in>A"}

   762       makes relativization easier.*}

   763 lemma (in M_trivial) RepFun_closed2:

   764      "[| strong_replacement(M, \<lambda>x y. x\<in>A & y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |]

   765       ==> M(RepFun(A, %x. f(x)))"

   766 apply (simp add: RepFun_def)

   767 apply (frule strong_replacement_closed, assumption)

   768 apply (auto dest: transM  simp add: Replace_conj_eq univalent_def)

   769 done

   770

   771 subsubsection {*Absoluteness for @{term Lambda}*}

   772

   773 definition

   774  is_lambda :: "[i=>o, i, [i,i]=>o, i] => o" where

   775     "is_lambda(M, A, is_b, z) ==

   776        \<forall>p[M]. p \<in> z \<longleftrightarrow>

   777         (\<exists>u[M]. \<exists>v[M]. u\<in>A & pair(M,u,v,p) & is_b(u,v))"

   778

   779 lemma (in M_trivial) lam_closed:

   780      "[| strong_replacement(M, \<lambda>x y. y = <x,b(x)>); M(A); \<forall>x\<in>A. M(b(x)) |]

   781       ==> M(\<lambda>x\<in>A. b(x))"

   782 by (simp add: lam_def, blast intro: RepFun_closed dest: transM)

   783

   784 text{*Better than @{text lam_closed}: has the formula @{term "x\<in>A"}*}

   785 lemma (in M_trivial) lam_closed2:

   786   "[|strong_replacement(M, \<lambda>x y. x\<in>A & y = \<langle>x, b(x)\<rangle>);

   787      M(A); \<forall>m[M]. m\<in>A \<longrightarrow> M(b(m))|] ==> M(Lambda(A,b))"

   788 apply (simp add: lam_def)

   789 apply (blast intro: RepFun_closed2 dest: transM)

   790 done

   791

   792 lemma (in M_trivial) lambda_abs2:

   793      "[| Relation1(M,A,is_b,b); M(A); \<forall>m[M]. m\<in>A \<longrightarrow> M(b(m)); M(z) |]

   794       ==> is_lambda(M,A,is_b,z) \<longleftrightarrow> z = Lambda(A,b)"

   795 apply (simp add: Relation1_def is_lambda_def)

   796 apply (rule iffI)

   797  prefer 2 apply (simp add: lam_def)

   798 apply (rule equality_iffI)

   799 apply (simp add: lam_def)

   800 apply (rule iffI)

   801  apply (blast dest: transM)

   802 apply (auto simp add: transM [of _ A])

   803 done

   804

   805 lemma is_lambda_cong [cong]:

   806      "[| A=A';  z=z';

   807          !!x y. [| x\<in>A; M(x); M(y) |] ==> is_b(x,y) \<longleftrightarrow> is_b'(x,y) |]

   808       ==> is_lambda(M, A, %x y. is_b(x,y), z) \<longleftrightarrow>

   809           is_lambda(M, A', %x y. is_b'(x,y), z')"

   810 by (simp add: is_lambda_def)

   811

   812 lemma (in M_trivial) image_abs [simp]:

   813      "[| M(r); M(A); M(z) |] ==> image(M,r,A,z) \<longleftrightarrow> z = rA"

   814 apply (simp add: image_def)

   815 apply (rule iffI)

   816  apply (blast intro!: equalityI dest: transM, blast)

   817 done

   818

   819 text{*What about @{text Pow_abs}?  Powerset is NOT absolute!

   820       This result is one direction of absoluteness.*}

   821

   822 lemma (in M_trivial) powerset_Pow:

   823      "powerset(M, x, Pow(x))"

   824 by (simp add: powerset_def)

   825

   826 text{*But we can't prove that the powerset in @{text M} includes the

   827       real powerset.*}

   828 lemma (in M_trivial) powerset_imp_subset_Pow:

   829      "[| powerset(M,x,y); M(y) |] ==> y \<subseteq> Pow(x)"

   830 apply (simp add: powerset_def)

   831 apply (blast dest: transM)

   832 done

   833

   834 subsubsection{*Absoluteness for the Natural Numbers*}

   835

   836 lemma (in M_trivial) nat_into_M [intro]:

   837      "n \<in> nat ==> M(n)"

   838 by (induct n rule: nat_induct, simp_all)

   839

   840 lemma (in M_trivial) nat_case_closed [intro,simp]:

   841   "[|M(k); M(a); \<forall>m[M]. M(b(m))|] ==> M(nat_case(a,b,k))"

   842 apply (case_tac "k=0", simp)

   843 apply (case_tac "\<exists>m. k = succ(m)", force)

   844 apply (simp add: nat_case_def)

   845 done

   846

   847 lemma (in M_trivial) quasinat_abs [simp]:

   848      "M(z) ==> is_quasinat(M,z) \<longleftrightarrow> quasinat(z)"

   849 by (auto simp add: is_quasinat_def quasinat_def)

   850

   851 lemma (in M_trivial) nat_case_abs [simp]:

   852      "[| relation1(M,is_b,b); M(k); M(z) |]

   853       ==> is_nat_case(M,a,is_b,k,z) \<longleftrightarrow> z = nat_case(a,b,k)"

   854 apply (case_tac "quasinat(k)")

   855  prefer 2

   856  apply (simp add: is_nat_case_def non_nat_case)

   857  apply (force simp add: quasinat_def)

   858 apply (simp add: quasinat_def is_nat_case_def)

   859 apply (elim disjE exE)

   860  apply (simp_all add: relation1_def)

   861 done

   862

   863 (*NOT for the simplifier.  The assumption M(z') is apparently necessary, but

   864   causes the error "Failed congruence proof!"  It may be better to replace

   865   is_nat_case by nat_case before attempting congruence reasoning.*)

   866 lemma is_nat_case_cong:

   867      "[| a = a'; k = k';  z = z';  M(z');

   868        !!x y. [| M(x); M(y) |] ==> is_b(x,y) \<longleftrightarrow> is_b'(x,y) |]

   869       ==> is_nat_case(M, a, is_b, k, z) \<longleftrightarrow> is_nat_case(M, a', is_b', k', z')"

   870 by (simp add: is_nat_case_def)

   871

   872

   873 subsection{*Absoluteness for Ordinals*}

   874 text{*These results constitute Theorem IV 5.1 of Kunen (page 126).*}

   875

   876 lemma (in M_trivial) lt_closed:

   877      "[| j<i; M(i) |] ==> M(j)"

   878 by (blast dest: ltD intro: transM)

   879

   880 lemma (in M_trivial) transitive_set_abs [simp]:

   881      "M(a) ==> transitive_set(M,a) \<longleftrightarrow> Transset(a)"

   882 by (simp add: transitive_set_def Transset_def)

   883

   884 lemma (in M_trivial) ordinal_abs [simp]:

   885      "M(a) ==> ordinal(M,a) \<longleftrightarrow> Ord(a)"

   886 by (simp add: ordinal_def Ord_def)

   887

   888 lemma (in M_trivial) limit_ordinal_abs [simp]:

   889      "M(a) ==> limit_ordinal(M,a) \<longleftrightarrow> Limit(a)"

   890 apply (unfold Limit_def limit_ordinal_def)

   891 apply (simp add: Ord_0_lt_iff)

   892 apply (simp add: lt_def, blast)

   893 done

   894

   895 lemma (in M_trivial) successor_ordinal_abs [simp]:

   896      "M(a) ==> successor_ordinal(M,a) \<longleftrightarrow> Ord(a) & (\<exists>b[M]. a = succ(b))"

   897 apply (simp add: successor_ordinal_def, safe)

   898 apply (drule Ord_cases_disj, auto)

   899 done

   900

   901 lemma finite_Ord_is_nat:

   902       "[| Ord(a); ~ Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a \<in> nat"

   903 by (induct a rule: trans_induct3, simp_all)

   904

   905 lemma (in M_trivial) finite_ordinal_abs [simp]:

   906      "M(a) ==> finite_ordinal(M,a) \<longleftrightarrow> a \<in> nat"

   907 apply (simp add: finite_ordinal_def)

   908 apply (blast intro: finite_Ord_is_nat intro: nat_into_Ord

   909              dest: Ord_trans naturals_not_limit)

   910 done

   911

   912 lemma Limit_non_Limit_implies_nat:

   913      "[| Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a = nat"

   914 apply (rule le_anti_sym)

   915 apply (rule all_lt_imp_le, blast, blast intro: Limit_is_Ord)

   916  apply (simp add: lt_def)

   917  apply (blast intro: Ord_in_Ord Ord_trans finite_Ord_is_nat)

   918 apply (erule nat_le_Limit)

   919 done

   920

   921 lemma (in M_trivial) omega_abs [simp]:

   922      "M(a) ==> omega(M,a) \<longleftrightarrow> a = nat"

   923 apply (simp add: omega_def)

   924 apply (blast intro: Limit_non_Limit_implies_nat dest: naturals_not_limit)

   925 done

   926

   927 lemma (in M_trivial) number1_abs [simp]:

   928      "M(a) ==> number1(M,a) \<longleftrightarrow> a = 1"

   929 by (simp add: number1_def)

   930

   931 lemma (in M_trivial) number2_abs [simp]:

   932      "M(a) ==> number2(M,a) \<longleftrightarrow> a = succ(1)"

   933 by (simp add: number2_def)

   934

   935 lemma (in M_trivial) number3_abs [simp]:

   936      "M(a) ==> number3(M,a) \<longleftrightarrow> a = succ(succ(1))"

   937 by (simp add: number3_def)

   938

   939 text{*Kunen continued to 20...*}

   940

   941 (*Could not get this to work.  The \<lambda>x\<in>nat is essential because everything

   942   but the recursion variable must stay unchanged.  But then the recursion

   943   equations only hold for x\<in>nat (or in some other set) and not for the

   944   whole of the class M.

   945   consts

   946     natnumber_aux :: "[i=>o,i] => i"

   947

   948   primrec

   949       "natnumber_aux(M,0) = (\<lambda>x\<in>nat. if empty(M,x) then 1 else 0)"

   950       "natnumber_aux(M,succ(n)) =

   951            (\<lambda>x\<in>nat. if (\<exists>y[M]. natnumber_aux(M,n)y=1 & successor(M,y,x))

   952                      then 1 else 0)"

   953

   954   definition

   955     natnumber :: "[i=>o,i,i] => o"

   956       "natnumber(M,n,x) == natnumber_aux(M,n)x = 1"

   957

   958   lemma (in M_trivial) [simp]:

   959        "natnumber(M,0,x) == x=0"

   960 *)

   961

   962 subsection{*Some instances of separation and strong replacement*}

   963

   964 locale M_basic = M_trivial +

   965 assumes Inter_separation:

   966      "M(A) ==> separation(M, \<lambda>x. \<forall>y[M]. y\<in>A \<longrightarrow> x\<in>y)"

   967   and Diff_separation:

   968      "M(B) ==> separation(M, \<lambda>x. x \<notin> B)"

   969   and cartprod_separation:

   970      "[| M(A); M(B) |]

   971       ==> separation(M, \<lambda>z. \<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,z)))"

   972   and image_separation:

   973      "[| M(A); M(r) |]

   974       ==> separation(M, \<lambda>y. \<exists>p[M]. p\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,p)))"

   975   and converse_separation:

   976      "M(r) ==> separation(M,

   977          \<lambda>z. \<exists>p[M]. p\<in>r & (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,p) & pair(M,y,x,z)))"

   978   and restrict_separation:

   979      "M(A) ==> separation(M, \<lambda>z. \<exists>x[M]. x\<in>A & (\<exists>y[M]. pair(M,x,y,z)))"

   980   and comp_separation:

   981      "[| M(r); M(s) |]

   982       ==> separation(M, \<lambda>xz. \<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].

   983                   pair(M,x,z,xz) & pair(M,x,y,xy) & pair(M,y,z,yz) &

   984                   xy\<in>s & yz\<in>r)"

   985   and pred_separation:

   986      "[| M(r); M(x) |] ==> separation(M, \<lambda>y. \<exists>p[M]. p\<in>r & pair(M,y,x,p))"

   987   and Memrel_separation:

   988      "separation(M, \<lambda>z. \<exists>x[M]. \<exists>y[M]. pair(M,x,y,z) & x \<in> y)"

   989   and funspace_succ_replacement:

   990      "M(n) ==>

   991       strong_replacement(M, \<lambda>p z. \<exists>f[M]. \<exists>b[M]. \<exists>nb[M]. \<exists>cnbf[M].

   992                 pair(M,f,b,p) & pair(M,n,b,nb) & is_cons(M,nb,f,cnbf) &

   993                 upair(M,cnbf,cnbf,z))"

   994   and is_recfun_separation:

   995      --{*for well-founded recursion: used to prove @{text is_recfun_equal}*}

   996      "[| M(r); M(f); M(g); M(a); M(b) |]

   997      ==> separation(M,

   998             \<lambda>x. \<exists>xa[M]. \<exists>xb[M].

   999                 pair(M,x,a,xa) & xa \<in> r & pair(M,x,b,xb) & xb \<in> r &

  1000                 (\<exists>fx[M]. \<exists>gx[M]. fun_apply(M,f,x,fx) & fun_apply(M,g,x,gx) &

  1001                                    fx \<noteq> gx))"

  1002

  1003 lemma (in M_basic) cartprod_iff_lemma:

  1004      "[| M(C);  \<forall>u[M]. u \<in> C \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}});

  1005          powerset(M, A \<union> B, p1); powerset(M, p1, p2);  M(p2) |]

  1006        ==> C = {u \<in> p2 . \<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}}}"

  1007 apply (simp add: powerset_def)

  1008 apply (rule equalityI, clarify, simp)

  1009  apply (frule transM, assumption)

  1010  apply (frule transM, assumption, simp (no_asm_simp))

  1011  apply blast

  1012 apply clarify

  1013 apply (frule transM, assumption, force)

  1014 done

  1015

  1016 lemma (in M_basic) cartprod_iff:

  1017      "[| M(A); M(B); M(C) |]

  1018       ==> cartprod(M,A,B,C) \<longleftrightarrow>

  1019           (\<exists>p1[M]. \<exists>p2[M]. powerset(M,A \<union> B,p1) & powerset(M,p1,p2) &

  1020                    C = {z \<in> p2. \<exists>x\<in>A. \<exists>y\<in>B. z = <x,y>})"

  1021 apply (simp add: Pair_def cartprod_def, safe)

  1022 defer 1

  1023   apply (simp add: powerset_def)

  1024  apply blast

  1025 txt{*Final, difficult case: the left-to-right direction of the theorem.*}

  1026 apply (insert power_ax, simp add: power_ax_def)

  1027 apply (frule_tac x="A \<union> B" and P="\<lambda>x. rex(M,Q(x))" for Q in rspec)

  1028 apply (blast, clarify)

  1029 apply (drule_tac x=z and P="\<lambda>x. rex(M,Q(x))" for Q in rspec)

  1030 apply assumption

  1031 apply (blast intro: cartprod_iff_lemma)

  1032 done

  1033

  1034 lemma (in M_basic) cartprod_closed_lemma:

  1035      "[| M(A); M(B) |] ==> \<exists>C[M]. cartprod(M,A,B,C)"

  1036 apply (simp del: cartprod_abs add: cartprod_iff)

  1037 apply (insert power_ax, simp add: power_ax_def)

  1038 apply (frule_tac x="A \<union> B" and P="\<lambda>x. rex(M,Q(x))" for Q in rspec)

  1039 apply (blast, clarify)

  1040 apply (drule_tac x=z and P="\<lambda>x. rex(M,Q(x))" for Q in rspec, auto)

  1041 apply (intro rexI conjI, simp+)

  1042 apply (insert cartprod_separation [of A B], simp)

  1043 done

  1044

  1045 text{*All the lemmas above are necessary because Powerset is not absolute.

  1046       I should have used Replacement instead!*}

  1047 lemma (in M_basic) cartprod_closed [intro,simp]:

  1048      "[| M(A); M(B) |] ==> M(A*B)"

  1049 by (frule cartprod_closed_lemma, assumption, force)

  1050

  1051 lemma (in M_basic) sum_closed [intro,simp]:

  1052      "[| M(A); M(B) |] ==> M(A+B)"

  1053 by (simp add: sum_def)

  1054

  1055 lemma (in M_basic) sum_abs [simp]:

  1056      "[| M(A); M(B); M(Z) |] ==> is_sum(M,A,B,Z) \<longleftrightarrow> (Z = A+B)"

  1057 by (simp add: is_sum_def sum_def singleton_0 nat_into_M)

  1058

  1059 lemma (in M_trivial) Inl_in_M_iff [iff]:

  1060      "M(Inl(a)) \<longleftrightarrow> M(a)"

  1061 by (simp add: Inl_def)

  1062

  1063 lemma (in M_trivial) Inl_abs [simp]:

  1064      "M(Z) ==> is_Inl(M,a,Z) \<longleftrightarrow> (Z = Inl(a))"

  1065 by (simp add: is_Inl_def Inl_def)

  1066

  1067 lemma (in M_trivial) Inr_in_M_iff [iff]:

  1068      "M(Inr(a)) \<longleftrightarrow> M(a)"

  1069 by (simp add: Inr_def)

  1070

  1071 lemma (in M_trivial) Inr_abs [simp]:

  1072      "M(Z) ==> is_Inr(M,a,Z) \<longleftrightarrow> (Z = Inr(a))"

  1073 by (simp add: is_Inr_def Inr_def)

  1074

  1075

  1076 subsubsection {*converse of a relation*}

  1077

  1078 lemma (in M_basic) M_converse_iff:

  1079      "M(r) ==>

  1080       converse(r) =

  1081       {z \<in> \<Union>(\<Union>(r)) * \<Union>(\<Union>(r)).

  1082        \<exists>p\<in>r. \<exists>x[M]. \<exists>y[M]. p = \<langle>x,y\<rangle> & z = \<langle>y,x\<rangle>}"

  1083 apply (rule equalityI)

  1084  prefer 2 apply (blast dest: transM, clarify, simp)

  1085 apply (simp add: Pair_def)

  1086 apply (blast dest: transM)

  1087 done

  1088

  1089 lemma (in M_basic) converse_closed [intro,simp]:

  1090      "M(r) ==> M(converse(r))"

  1091 apply (simp add: M_converse_iff)

  1092 apply (insert converse_separation [of r], simp)

  1093 done

  1094

  1095 lemma (in M_basic) converse_abs [simp]:

  1096      "[| M(r); M(z) |] ==> is_converse(M,r,z) \<longleftrightarrow> z = converse(r)"

  1097 apply (simp add: is_converse_def)

  1098 apply (rule iffI)

  1099  prefer 2 apply blast

  1100 apply (rule M_equalityI)

  1101   apply simp

  1102   apply (blast dest: transM)+

  1103 done

  1104

  1105

  1106 subsubsection {*image, preimage, domain, range*}

  1107

  1108 lemma (in M_basic) image_closed [intro,simp]:

  1109      "[| M(A); M(r) |] ==> M(rA)"

  1110 apply (simp add: image_iff_Collect)

  1111 apply (insert image_separation [of A r], simp)

  1112 done

  1113

  1114 lemma (in M_basic) vimage_abs [simp]:

  1115      "[| M(r); M(A); M(z) |] ==> pre_image(M,r,A,z) \<longleftrightarrow> z = r-A"

  1116 apply (simp add: pre_image_def)

  1117 apply (rule iffI)

  1118  apply (blast intro!: equalityI dest: transM, blast)

  1119 done

  1120

  1121 lemma (in M_basic) vimage_closed [intro,simp]:

  1122      "[| M(A); M(r) |] ==> M(r-A)"

  1123 by (simp add: vimage_def)

  1124

  1125

  1126 subsubsection{*Domain, range and field*}

  1127

  1128 lemma (in M_basic) domain_abs [simp]:

  1129      "[| M(r); M(z) |] ==> is_domain(M,r,z) \<longleftrightarrow> z = domain(r)"

  1130 apply (simp add: is_domain_def)

  1131 apply (blast intro!: equalityI dest: transM)

  1132 done

  1133

  1134 lemma (in M_basic) domain_closed [intro,simp]:

  1135      "M(r) ==> M(domain(r))"

  1136 apply (simp add: domain_eq_vimage)

  1137 done

  1138

  1139 lemma (in M_basic) range_abs [simp]:

  1140      "[| M(r); M(z) |] ==> is_range(M,r,z) \<longleftrightarrow> z = range(r)"

  1141 apply (simp add: is_range_def)

  1142 apply (blast intro!: equalityI dest: transM)

  1143 done

  1144

  1145 lemma (in M_basic) range_closed [intro,simp]:

  1146      "M(r) ==> M(range(r))"

  1147 apply (simp add: range_eq_image)

  1148 done

  1149

  1150 lemma (in M_basic) field_abs [simp]:

  1151      "[| M(r); M(z) |] ==> is_field(M,r,z) \<longleftrightarrow> z = field(r)"

  1152 by (simp add: domain_closed range_closed is_field_def field_def)

  1153

  1154 lemma (in M_basic) field_closed [intro,simp]:

  1155      "M(r) ==> M(field(r))"

  1156 by (simp add: domain_closed range_closed Un_closed field_def)

  1157

  1158

  1159 subsubsection{*Relations, functions and application*}

  1160

  1161 lemma (in M_basic) relation_abs [simp]:

  1162      "M(r) ==> is_relation(M,r) \<longleftrightarrow> relation(r)"

  1163 apply (simp add: is_relation_def relation_def)

  1164 apply (blast dest!: bspec dest: pair_components_in_M)+

  1165 done

  1166

  1167 lemma (in M_basic) function_abs [simp]:

  1168      "M(r) ==> is_function(M,r) \<longleftrightarrow> function(r)"

  1169 apply (simp add: is_function_def function_def, safe)

  1170    apply (frule transM, assumption)

  1171   apply (blast dest: pair_components_in_M)+

  1172 done

  1173

  1174 lemma (in M_basic) apply_closed [intro,simp]:

  1175      "[|M(f); M(a)|] ==> M(fa)"

  1176 by (simp add: apply_def)

  1177

  1178 lemma (in M_basic) apply_abs [simp]:

  1179      "[| M(f); M(x); M(y) |] ==> fun_apply(M,f,x,y) \<longleftrightarrow> fx = y"

  1180 apply (simp add: fun_apply_def apply_def, blast)

  1181 done

  1182

  1183 lemma (in M_basic) typed_function_abs [simp]:

  1184      "[| M(A); M(f) |] ==> typed_function(M,A,B,f) \<longleftrightarrow> f \<in> A -> B"

  1185 apply (auto simp add: typed_function_def relation_def Pi_iff)

  1186 apply (blast dest: pair_components_in_M)+

  1187 done

  1188

  1189 lemma (in M_basic) injection_abs [simp]:

  1190      "[| M(A); M(f) |] ==> injection(M,A,B,f) \<longleftrightarrow> f \<in> inj(A,B)"

  1191 apply (simp add: injection_def apply_iff inj_def apply_closed)

  1192 apply (blast dest: transM [of _ A])

  1193 done

  1194

  1195 lemma (in M_basic) surjection_abs [simp]:

  1196      "[| M(A); M(B); M(f) |] ==> surjection(M,A,B,f) \<longleftrightarrow> f \<in> surj(A,B)"

  1197 by (simp add: surjection_def surj_def)

  1198

  1199 lemma (in M_basic) bijection_abs [simp]:

  1200      "[| M(A); M(B); M(f) |] ==> bijection(M,A,B,f) \<longleftrightarrow> f \<in> bij(A,B)"

  1201 by (simp add: bijection_def bij_def)

  1202

  1203

  1204 subsubsection{*Composition of relations*}

  1205

  1206 lemma (in M_basic) M_comp_iff:

  1207      "[| M(r); M(s) |]

  1208       ==> r O s =

  1209           {xz \<in> domain(s) * range(r).

  1210             \<exists>x[M]. \<exists>y[M]. \<exists>z[M]. xz = \<langle>x,z\<rangle> & \<langle>x,y\<rangle> \<in> s & \<langle>y,z\<rangle> \<in> r}"

  1211 apply (simp add: comp_def)

  1212 apply (rule equalityI)

  1213  apply clarify

  1214  apply simp

  1215  apply  (blast dest:  transM)+

  1216 done

  1217

  1218 lemma (in M_basic) comp_closed [intro,simp]:

  1219      "[| M(r); M(s) |] ==> M(r O s)"

  1220 apply (simp add: M_comp_iff)

  1221 apply (insert comp_separation [of r s], simp)

  1222 done

  1223

  1224 lemma (in M_basic) composition_abs [simp]:

  1225      "[| M(r); M(s); M(t) |] ==> composition(M,r,s,t) \<longleftrightarrow> t = r O s"

  1226 apply safe

  1227  txt{*Proving @{term "composition(M, r, s, r O s)"}*}

  1228  prefer 2

  1229  apply (simp add: composition_def comp_def)

  1230  apply (blast dest: transM)

  1231 txt{*Opposite implication*}

  1232 apply (rule M_equalityI)

  1233   apply (simp add: composition_def comp_def)

  1234   apply (blast del: allE dest: transM)+

  1235 done

  1236

  1237 text{*no longer needed*}

  1238 lemma (in M_basic) restriction_is_function:

  1239      "[| restriction(M,f,A,z); function(f); M(f); M(A); M(z) |]

  1240       ==> function(z)"

  1241 apply (simp add: restriction_def ball_iff_equiv)

  1242 apply (unfold function_def, blast)

  1243 done

  1244

  1245 lemma (in M_basic) restriction_abs [simp]:

  1246      "[| M(f); M(A); M(z) |]

  1247       ==> restriction(M,f,A,z) \<longleftrightarrow> z = restrict(f,A)"

  1248 apply (simp add: ball_iff_equiv restriction_def restrict_def)

  1249 apply (blast intro!: equalityI dest: transM)

  1250 done

  1251

  1252

  1253 lemma (in M_basic) M_restrict_iff:

  1254      "M(r) ==> restrict(r,A) = {z \<in> r . \<exists>x\<in>A. \<exists>y[M]. z = \<langle>x, y\<rangle>}"

  1255 by (simp add: restrict_def, blast dest: transM)

  1256

  1257 lemma (in M_basic) restrict_closed [intro,simp]:

  1258      "[| M(A); M(r) |] ==> M(restrict(r,A))"

  1259 apply (simp add: M_restrict_iff)

  1260 apply (insert restrict_separation [of A], simp)

  1261 done

  1262

  1263 lemma (in M_basic) Inter_abs [simp]:

  1264      "[| M(A); M(z) |] ==> big_inter(M,A,z) \<longleftrightarrow> z = \<Inter>(A)"

  1265 apply (simp add: big_inter_def Inter_def)

  1266 apply (blast intro!: equalityI dest: transM)

  1267 done

  1268

  1269 lemma (in M_basic) Inter_closed [intro,simp]:

  1270      "M(A) ==> M(\<Inter>(A))"

  1271 by (insert Inter_separation, simp add: Inter_def)

  1272

  1273 lemma (in M_basic) Int_closed [intro,simp]:

  1274      "[| M(A); M(B) |] ==> M(A \<inter> B)"

  1275 apply (subgoal_tac "M({A,B})")

  1276 apply (frule Inter_closed, force+)

  1277 done

  1278

  1279 lemma (in M_basic) Diff_closed [intro,simp]:

  1280      "[|M(A); M(B)|] ==> M(A-B)"

  1281 by (insert Diff_separation, simp add: Diff_def)

  1282

  1283 subsubsection{*Some Facts About Separation Axioms*}

  1284

  1285 lemma (in M_basic) separation_conj:

  1286      "[|separation(M,P); separation(M,Q)|] ==> separation(M, \<lambda>z. P(z) & Q(z))"

  1287 by (simp del: separation_closed

  1288          add: separation_iff Collect_Int_Collect_eq [symmetric])

  1289

  1290 (*???equalities*)

  1291 lemma Collect_Un_Collect_eq:

  1292      "Collect(A,P) \<union> Collect(A,Q) = Collect(A, %x. P(x) | Q(x))"

  1293 by blast

  1294

  1295 lemma Diff_Collect_eq:

  1296      "A - Collect(A,P) = Collect(A, %x. ~ P(x))"

  1297 by blast

  1298

  1299 lemma (in M_trivial) Collect_rall_eq:

  1300      "M(Y) ==> Collect(A, %x. \<forall>y[M]. y\<in>Y \<longrightarrow> P(x,y)) =

  1301                (if Y=0 then A else (\<Inter>y \<in> Y. {x \<in> A. P(x,y)}))"

  1302 apply simp

  1303 apply (blast intro!: equalityI dest: transM)

  1304 done

  1305

  1306 lemma (in M_basic) separation_disj:

  1307      "[|separation(M,P); separation(M,Q)|] ==> separation(M, \<lambda>z. P(z) | Q(z))"

  1308 by (simp del: separation_closed

  1309          add: separation_iff Collect_Un_Collect_eq [symmetric])

  1310

  1311 lemma (in M_basic) separation_neg:

  1312      "separation(M,P) ==> separation(M, \<lambda>z. ~P(z))"

  1313 by (simp del: separation_closed

  1314          add: separation_iff Diff_Collect_eq [symmetric])

  1315

  1316 lemma (in M_basic) separation_imp:

  1317      "[|separation(M,P); separation(M,Q)|]

  1318       ==> separation(M, \<lambda>z. P(z) \<longrightarrow> Q(z))"

  1319 by (simp add: separation_neg separation_disj not_disj_iff_imp [symmetric])

  1320

  1321 text{*This result is a hint of how little can be done without the Reflection

  1322   Theorem.  The quantifier has to be bounded by a set.  We also need another

  1323   instance of Separation!*}

  1324 lemma (in M_basic) separation_rall:

  1325      "[|M(Y); \<forall>y[M]. separation(M, \<lambda>x. P(x,y));

  1326         \<forall>z[M]. strong_replacement(M, \<lambda>x y. y = {u \<in> z . P(u,x)})|]

  1327       ==> separation(M, \<lambda>x. \<forall>y[M]. y\<in>Y \<longrightarrow> P(x,y))"

  1328 apply (simp del: separation_closed rall_abs

  1329          add: separation_iff Collect_rall_eq)

  1330 apply (blast intro!: Inter_closed RepFun_closed dest: transM)

  1331 done

  1332

  1333

  1334 subsubsection{*Functions and function space*}

  1335

  1336 text{*The assumption @{term "M(A->B)"} is unusual, but essential: in

  1337 all but trivial cases, A->B cannot be expected to belong to @{term M}.*}

  1338 lemma (in M_basic) is_funspace_abs [simp]:

  1339      "[|M(A); M(B); M(F); M(A->B)|] ==> is_funspace(M,A,B,F) \<longleftrightarrow> F = A->B"

  1340 apply (simp add: is_funspace_def)

  1341 apply (rule iffI)

  1342  prefer 2 apply blast

  1343 apply (rule M_equalityI)

  1344   apply simp_all

  1345 done

  1346

  1347 lemma (in M_basic) succ_fun_eq2:

  1348      "[|M(B); M(n->B)|] ==>

  1349       succ(n) -> B =

  1350       \<Union>{z. p \<in> (n->B)*B, \<exists>f[M]. \<exists>b[M]. p = <f,b> & z = {cons(<n,b>, f)}}"

  1351 apply (simp add: succ_fun_eq)

  1352 apply (blast dest: transM)

  1353 done

  1354

  1355 lemma (in M_basic) funspace_succ:

  1356      "[|M(n); M(B); M(n->B) |] ==> M(succ(n) -> B)"

  1357 apply (insert funspace_succ_replacement [of n], simp)

  1358 apply (force simp add: succ_fun_eq2 univalent_def)

  1359 done

  1360

  1361 text{*@{term M} contains all finite function spaces.  Needed to prove the

  1362 absoluteness of transitive closure.  See the definition of

  1363 @{text rtrancl_alt} in in @{text WF_absolute.thy}.*}

  1364 lemma (in M_basic) finite_funspace_closed [intro,simp]:

  1365      "[|n\<in>nat; M(B)|] ==> M(n->B)"

  1366 apply (induct_tac n, simp)

  1367 apply (simp add: funspace_succ nat_into_M)

  1368 done

  1369

  1370

  1371 subsection{*Relativization and Absoluteness for Boolean Operators*}

  1372

  1373 definition

  1374   is_bool_of_o :: "[i=>o, o, i] => o" where

  1375    "is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))"

  1376

  1377 definition

  1378   is_not :: "[i=>o, i, i] => o" where

  1379    "is_not(M,a,z) == (number1(M,a)  & empty(M,z)) |

  1380                      (~number1(M,a) & number1(M,z))"

  1381

  1382 definition

  1383   is_and :: "[i=>o, i, i, i] => o" where

  1384    "is_and(M,a,b,z) == (number1(M,a)  & z=b) |

  1385                        (~number1(M,a) & empty(M,z))"

  1386

  1387 definition

  1388   is_or :: "[i=>o, i, i, i] => o" where

  1389    "is_or(M,a,b,z) == (number1(M,a)  & number1(M,z)) |

  1390                       (~number1(M,a) & z=b)"

  1391

  1392 lemma (in M_trivial) bool_of_o_abs [simp]:

  1393      "M(z) ==> is_bool_of_o(M,P,z) \<longleftrightarrow> z = bool_of_o(P)"

  1394 by (simp add: is_bool_of_o_def bool_of_o_def)

  1395

  1396

  1397 lemma (in M_trivial) not_abs [simp]:

  1398      "[| M(a); M(z)|] ==> is_not(M,a,z) \<longleftrightarrow> z = not(a)"

  1399 by (simp add: Bool.not_def cond_def is_not_def)

  1400

  1401 lemma (in M_trivial) and_abs [simp]:

  1402      "[| M(a); M(b); M(z)|] ==> is_and(M,a,b,z) \<longleftrightarrow> z = a and b"

  1403 by (simp add: Bool.and_def cond_def is_and_def)

  1404

  1405 lemma (in M_trivial) or_abs [simp]:

  1406      "[| M(a); M(b); M(z)|] ==> is_or(M,a,b,z) \<longleftrightarrow> z = a or b"

  1407 by (simp add: Bool.or_def cond_def is_or_def)

  1408

  1409

  1410 lemma (in M_trivial) bool_of_o_closed [intro,simp]:

  1411      "M(bool_of_o(P))"

  1412 by (simp add: bool_of_o_def)

  1413

  1414 lemma (in M_trivial) and_closed [intro,simp]:

  1415      "[| M(p); M(q) |] ==> M(p and q)"

  1416 by (simp add: and_def cond_def)

  1417

  1418 lemma (in M_trivial) or_closed [intro,simp]:

  1419      "[| M(p); M(q) |] ==> M(p or q)"

  1420 by (simp add: or_def cond_def)

  1421

  1422 lemma (in M_trivial) not_closed [intro,simp]:

  1423      "M(p) ==> M(not(p))"

  1424 by (simp add: Bool.not_def cond_def)

  1425

  1426

  1427 subsection{*Relativization and Absoluteness for List Operators*}

  1428

  1429 definition

  1430   is_Nil :: "[i=>o, i] => o" where

  1431      --{* because @{prop "[] \<equiv> Inl(0)"}*}

  1432     "is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs)"

  1433

  1434 definition

  1435   is_Cons :: "[i=>o,i,i,i] => o" where

  1436      --{* because @{prop "Cons(a, l) \<equiv> Inr(\<langle>a,l\<rangle>)"}*}

  1437     "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)"

  1438

  1439

  1440 lemma (in M_trivial) Nil_in_M [intro,simp]: "M(Nil)"

  1441 by (simp add: Nil_def)

  1442

  1443 lemma (in M_trivial) Nil_abs [simp]: "M(Z) ==> is_Nil(M,Z) \<longleftrightarrow> (Z = Nil)"

  1444 by (simp add: is_Nil_def Nil_def)

  1445

  1446 lemma (in M_trivial) Cons_in_M_iff [iff]: "M(Cons(a,l)) \<longleftrightarrow> M(a) & M(l)"

  1447 by (simp add: Cons_def)

  1448

  1449 lemma (in M_trivial) Cons_abs [simp]:

  1450      "[|M(a); M(l); M(Z)|] ==> is_Cons(M,a,l,Z) \<longleftrightarrow> (Z = Cons(a,l))"

  1451 by (simp add: is_Cons_def Cons_def)

  1452

  1453

  1454 definition

  1455   quasilist :: "i => o" where

  1456     "quasilist(xs) == xs=Nil | (\<exists>x l. xs = Cons(x,l))"

  1457

  1458 definition

  1459   is_quasilist :: "[i=>o,i] => o" where

  1460     "is_quasilist(M,z) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))"

  1461

  1462 definition

  1463   list_case' :: "[i, [i,i]=>i, i] => i" where

  1464     --{*A version of @{term list_case} that's always defined.*}

  1465     "list_case'(a,b,xs) ==

  1466        if quasilist(xs) then list_case(a,b,xs) else 0"

  1467

  1468 definition

  1469   is_list_case :: "[i=>o, i, [i,i,i]=>o, i, i] => o" where

  1470     --{*Returns 0 for non-lists*}

  1471     "is_list_case(M, a, is_b, xs, z) ==

  1472        (is_Nil(M,xs) \<longrightarrow> z=a) &

  1473        (\<forall>x[M]. \<forall>l[M]. is_Cons(M,x,l,xs) \<longrightarrow> is_b(x,l,z)) &

  1474        (is_quasilist(M,xs) | empty(M,z))"

  1475

  1476 definition

  1477   hd' :: "i => i" where

  1478     --{*A version of @{term hd} that's always defined.*}

  1479     "hd'(xs) == if quasilist(xs) then hd(xs) else 0"

  1480

  1481 definition

  1482   tl' :: "i => i" where

  1483     --{*A version of @{term tl} that's always defined.*}

  1484     "tl'(xs) == if quasilist(xs) then tl(xs) else 0"

  1485

  1486 definition

  1487   is_hd :: "[i=>o,i,i] => o" where

  1488      --{* @{term "hd([]) = 0"} no constraints if not a list.

  1489           Avoiding implication prevents the simplifier's looping.*}

  1490     "is_hd(M,xs,H) ==

  1491        (is_Nil(M,xs) \<longrightarrow> empty(M,H)) &

  1492        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) &

  1493        (is_quasilist(M,xs) | empty(M,H))"

  1494

  1495 definition

  1496   is_tl :: "[i=>o,i,i] => o" where

  1497      --{* @{term "tl([]) = []"}; see comments about @{term is_hd}*}

  1498     "is_tl(M,xs,T) ==

  1499        (is_Nil(M,xs) \<longrightarrow> T=xs) &

  1500        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &

  1501        (is_quasilist(M,xs) | empty(M,T))"

  1502

  1503 subsubsection{*@{term quasilist}: For Case-Splitting with @{term list_case'}*}

  1504

  1505 lemma [iff]: "quasilist(Nil)"

  1506 by (simp add: quasilist_def)

  1507

  1508 lemma [iff]: "quasilist(Cons(x,l))"

  1509 by (simp add: quasilist_def)

  1510

  1511 lemma list_imp_quasilist: "l \<in> list(A) ==> quasilist(l)"

  1512 by (erule list.cases, simp_all)

  1513

  1514 subsubsection{*@{term list_case'}, the Modified Version of @{term list_case}*}

  1515

  1516 lemma list_case'_Nil [simp]: "list_case'(a,b,Nil) = a"

  1517 by (simp add: list_case'_def quasilist_def)

  1518

  1519 lemma list_case'_Cons [simp]: "list_case'(a,b,Cons(x,l)) = b(x,l)"

  1520 by (simp add: list_case'_def quasilist_def)

  1521

  1522 lemma non_list_case: "~ quasilist(x) ==> list_case'(a,b,x) = 0"

  1523 by (simp add: quasilist_def list_case'_def)

  1524

  1525 lemma list_case'_eq_list_case [simp]:

  1526      "xs \<in> list(A) ==>list_case'(a,b,xs) = list_case(a,b,xs)"

  1527 by (erule list.cases, simp_all)

  1528

  1529 lemma (in M_basic) list_case'_closed [intro,simp]:

  1530   "[|M(k); M(a); \<forall>x[M]. \<forall>y[M]. M(b(x,y))|] ==> M(list_case'(a,b,k))"

  1531 apply (case_tac "quasilist(k)")

  1532  apply (simp add: quasilist_def, force)

  1533 apply (simp add: non_list_case)

  1534 done

  1535

  1536 lemma (in M_trivial) quasilist_abs [simp]:

  1537      "M(z) ==> is_quasilist(M,z) \<longleftrightarrow> quasilist(z)"

  1538 by (auto simp add: is_quasilist_def quasilist_def)

  1539

  1540 lemma (in M_trivial) list_case_abs [simp]:

  1541      "[| relation2(M,is_b,b); M(k); M(z) |]

  1542       ==> is_list_case(M,a,is_b,k,z) \<longleftrightarrow> z = list_case'(a,b,k)"

  1543 apply (case_tac "quasilist(k)")

  1544  prefer 2

  1545  apply (simp add: is_list_case_def non_list_case)

  1546  apply (force simp add: quasilist_def)

  1547 apply (simp add: quasilist_def is_list_case_def)

  1548 apply (elim disjE exE)

  1549  apply (simp_all add: relation2_def)

  1550 done

  1551

  1552

  1553 subsubsection{*The Modified Operators @{term hd'} and @{term tl'}*}

  1554

  1555 lemma (in M_trivial) is_hd_Nil: "is_hd(M,[],Z) \<longleftrightarrow> empty(M,Z)"

  1556 by (simp add: is_hd_def)

  1557

  1558 lemma (in M_trivial) is_hd_Cons:

  1559      "[|M(a); M(l)|] ==> is_hd(M,Cons(a,l),Z) \<longleftrightarrow> Z = a"

  1560 by (force simp add: is_hd_def)

  1561

  1562 lemma (in M_trivial) hd_abs [simp]:

  1563      "[|M(x); M(y)|] ==> is_hd(M,x,y) \<longleftrightarrow> y = hd'(x)"

  1564 apply (simp add: hd'_def)

  1565 apply (intro impI conjI)

  1566  prefer 2 apply (force simp add: is_hd_def)

  1567 apply (simp add: quasilist_def is_hd_def)

  1568 apply (elim disjE exE, auto)

  1569 done

  1570

  1571 lemma (in M_trivial) is_tl_Nil: "is_tl(M,[],Z) \<longleftrightarrow> Z = []"

  1572 by (simp add: is_tl_def)

  1573

  1574 lemma (in M_trivial) is_tl_Cons:

  1575      "[|M(a); M(l)|] ==> is_tl(M,Cons(a,l),Z) \<longleftrightarrow> Z = l"

  1576 by (force simp add: is_tl_def)

  1577

  1578 lemma (in M_trivial) tl_abs [simp]:

  1579      "[|M(x); M(y)|] ==> is_tl(M,x,y) \<longleftrightarrow> y = tl'(x)"

  1580 apply (simp add: tl'_def)

  1581 apply (intro impI conjI)

  1582  prefer 2 apply (force simp add: is_tl_def)

  1583 apply (simp add: quasilist_def is_tl_def)

  1584 apply (elim disjE exE, auto)

  1585 done

  1586

  1587 lemma (in M_trivial) relation1_tl: "relation1(M, is_tl(M), tl')"

  1588 by (simp add: relation1_def)

  1589

  1590 lemma hd'_Nil: "hd'([]) = 0"

  1591 by (simp add: hd'_def)

  1592

  1593 lemma hd'_Cons: "hd'(Cons(a,l)) = a"

  1594 by (simp add: hd'_def)

  1595

  1596 lemma tl'_Nil: "tl'([]) = []"

  1597 by (simp add: tl'_def)

  1598

  1599 lemma tl'_Cons: "tl'(Cons(a,l)) = l"

  1600 by (simp add: tl'_def)

  1601

  1602 lemma iterates_tl_Nil: "n \<in> nat ==> tl'^n ([]) = []"

  1603 apply (induct_tac n)

  1604 apply (simp_all add: tl'_Nil)

  1605 done

  1606

  1607 lemma (in M_basic) tl'_closed: "M(x) ==> M(tl'(x))"

  1608 apply (simp add: tl'_def)

  1609 apply (force simp add: quasilist_def)

  1610 done

  1611

  1612

  1613 end