src/ZF/OrdQuant.thy
 author wenzelm Wed Jul 15 23:48:21 2009 +0200 (2009-07-15) changeset 32010 cb1a1c94b4cd parent 28262 aa7ca36d67fd child 35112 ff6f60e6ab85 permissions -rw-r--r--
more antiquotations;
```     1 (*  Title:      ZF/AC/OrdQuant.thy
```
```     2     Authors:    Krzysztof Grabczewski and L C Paulson
```
```     3 *)
```
```     4
```
```     5 header {*Special quantifiers*}
```
```     6
```
```     7 theory OrdQuant imports Ordinal begin
```
```     8
```
```     9 subsection {*Quantifiers and union operator for ordinals*}
```
```    10
```
```    11 definition
```
```    12   (* Ordinal Quantifiers *)
```
```    13   oall :: "[i, i => o] => o"  where
```
```    14     "oall(A, P) == ALL x. x<A --> P(x)"
```
```    15
```
```    16 definition
```
```    17   oex :: "[i, i => o] => o"  where
```
```    18     "oex(A, P)  == EX x. x<A & P(x)"
```
```    19
```
```    20 definition
```
```    21   (* Ordinal Union *)
```
```    22   OUnion :: "[i, i => i] => i"  where
```
```    23     "OUnion(i,B) == {z: \<Union>x\<in>i. B(x). Ord(i)}"
```
```    24
```
```    25 syntax
```
```    26   "@oall"     :: "[idt, i, o] => o"        ("(3ALL _<_./ _)" 10)
```
```    27   "@oex"      :: "[idt, i, o] => o"        ("(3EX _<_./ _)" 10)
```
```    28   "@OUNION"   :: "[idt, i, i] => i"        ("(3UN _<_./ _)" 10)
```
```    29
```
```    30 translations
```
```    31   "ALL x<a. P"  == "CONST oall(a, %x. P)"
```
```    32   "EX x<a. P"   == "CONST oex(a, %x. P)"
```
```    33   "UN x<a. B"   == "CONST OUnion(a, %x. B)"
```
```    34
```
```    35 syntax (xsymbols)
```
```    36   "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
```
```    37   "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
```
```    38   "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
```
```    39 syntax (HTML output)
```
```    40   "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
```
```    41   "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
```
```    42   "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
```
```    43
```
```    44
```
```    45 subsubsection {*simplification of the new quantifiers*}
```
```    46
```
```    47
```
```    48 (*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
```
```    49   is proved.  Ord_atomize would convert this rule to
```
```    50     x < 0 ==> P(x) == True, which causes dire effects!*)
```
```    51 lemma [simp]: "(ALL x<0. P(x))"
```
```    52 by (simp add: oall_def)
```
```    53
```
```    54 lemma [simp]: "~(EX x<0. P(x))"
```
```    55 by (simp add: oex_def)
```
```    56
```
```    57 lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))"
```
```    58 apply (simp add: oall_def le_iff)
```
```    59 apply (blast intro: lt_Ord2)
```
```    60 done
```
```    61
```
```    62 lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))"
```
```    63 apply (simp add: oex_def le_iff)
```
```    64 apply (blast intro: lt_Ord2)
```
```    65 done
```
```    66
```
```    67 subsubsection {*Union over ordinals*}
```
```    68
```
```    69 lemma Ord_OUN [intro,simp]:
```
```    70      "[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
```
```    71 by (simp add: OUnion_def ltI Ord_UN)
```
```    72
```
```    73 lemma OUN_upper_lt:
```
```    74      "[| a<A;  i < b(a);  Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
```
```    75 by (unfold OUnion_def lt_def, blast )
```
```    76
```
```    77 lemma OUN_upper_le:
```
```    78      "[| a<A;  i\<le>b(a);  Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
```
```    79 apply (unfold OUnion_def, auto)
```
```    80 apply (rule UN_upper_le )
```
```    81 apply (auto simp add: lt_def)
```
```    82 done
```
```    83
```
```    84 lemma Limit_OUN_eq: "Limit(i) ==> (\<Union>x<i. x) = i"
```
```    85 by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
```
```    86
```
```    87 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
```
```    88 lemma OUN_least:
```
```    89      "(!!x. x<A ==> B(x) \<subseteq> C) ==> (\<Union>x<A. B(x)) \<subseteq> C"
```
```    90 by (simp add: OUnion_def UN_least ltI)
```
```    91
```
```    92 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
```
```    93 lemma OUN_least_le:
```
```    94      "[| Ord(i);  !!x. x<A ==> b(x) \<le> i |] ==> (\<Union>x<A. b(x)) \<le> i"
```
```    95 by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
```
```    96
```
```    97 lemma le_implies_OUN_le_OUN:
```
```    98      "[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (\<Union>x<A. c(x)) \<le> (\<Union>x<A. d(x))"
```
```    99 by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
```
```   100
```
```   101 lemma OUN_UN_eq:
```
```   102      "(!!x. x:A ==> Ord(B(x)))
```
```   103       ==> (\<Union>z < (\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z < B(x). C(z))"
```
```   104 by (simp add: OUnion_def)
```
```   105
```
```   106 lemma OUN_Union_eq:
```
```   107      "(!!x. x:X ==> Ord(x))
```
```   108       ==> (\<Union>z < Union(X). C(z)) = (\<Union>x\<in>X. \<Union>z < x. C(z))"
```
```   109 by (simp add: OUnion_def)
```
```   110
```
```   111 (*So that rule_format will get rid of ALL x<A...*)
```
```   112 lemma atomize_oall [symmetric, rulify]:
```
```   113      "(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))"
```
```   114 by (simp add: oall_def atomize_all atomize_imp)
```
```   115
```
```   116 subsubsection {*universal quantifier for ordinals*}
```
```   117
```
```   118 lemma oallI [intro!]:
```
```   119     "[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)"
```
```   120 by (simp add: oall_def)
```
```   121
```
```   122 lemma ospec: "[| ALL x<A. P(x);  x<A |] ==> P(x)"
```
```   123 by (simp add: oall_def)
```
```   124
```
```   125 lemma oallE:
```
```   126     "[| ALL x<A. P(x);  P(x) ==> Q;  ~x<A ==> Q |] ==> Q"
```
```   127 by (simp add: oall_def, blast)
```
```   128
```
```   129 lemma rev_oallE [elim]:
```
```   130     "[| ALL x<A. P(x);  ~x<A ==> Q;  P(x) ==> Q |] ==> Q"
```
```   131 by (simp add: oall_def, blast)
```
```   132
```
```   133
```
```   134 (*Trival rewrite rule;   (ALL x<a.P)<->P holds only if a is not 0!*)
```
```   135 lemma oall_simp [simp]: "(ALL x<a. True) <-> True"
```
```   136 by blast
```
```   137
```
```   138 (*Congruence rule for rewriting*)
```
```   139 lemma oall_cong [cong]:
```
```   140     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
```
```   141      ==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
```
```   142 by (simp add: oall_def)
```
```   143
```
```   144
```
```   145 subsubsection {*existential quantifier for ordinals*}
```
```   146
```
```   147 lemma oexI [intro]:
```
```   148     "[| P(x);  x<A |] ==> EX x<A. P(x)"
```
```   149 apply (simp add: oex_def, blast)
```
```   150 done
```
```   151
```
```   152 (*Not of the general form for such rules; ~EX has become ALL~ *)
```
```   153 lemma oexCI:
```
```   154    "[| ALL x<A. ~P(x) ==> P(a);  a<A |] ==> EX x<A. P(x)"
```
```   155 apply (simp add: oex_def, blast)
```
```   156 done
```
```   157
```
```   158 lemma oexE [elim!]:
```
```   159     "[| EX x<A. P(x);  !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
```
```   160 apply (simp add: oex_def, blast)
```
```   161 done
```
```   162
```
```   163 lemma oex_cong [cong]:
```
```   164     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
```
```   165      ==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
```
```   166 apply (simp add: oex_def cong add: conj_cong)
```
```   167 done
```
```   168
```
```   169
```
```   170 subsubsection {*Rules for Ordinal-Indexed Unions*}
```
```   171
```
```   172 lemma OUN_I [intro]: "[| a<i;  b: B(a) |] ==> b: (\<Union>z<i. B(z))"
```
```   173 by (unfold OUnion_def lt_def, blast)
```
```   174
```
```   175 lemma OUN_E [elim!]:
```
```   176     "[| b : (\<Union>z<i. B(z));  !!a.[| b: B(a);  a<i |] ==> R |] ==> R"
```
```   177 apply (unfold OUnion_def lt_def, blast)
```
```   178 done
```
```   179
```
```   180 lemma OUN_iff: "b : (\<Union>x<i. B(x)) <-> (EX x<i. b : B(x))"
```
```   181 by (unfold OUnion_def oex_def lt_def, blast)
```
```   182
```
```   183 lemma OUN_cong [cong]:
```
```   184     "[| i=j;  !!x. x<j ==> C(x)=D(x) |] ==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))"
```
```   185 by (simp add: OUnion_def lt_def OUN_iff)
```
```   186
```
```   187 lemma lt_induct:
```
```   188     "[| i<k;  !!x.[| x<k;  ALL y<x. P(y) |] ==> P(x) |]  ==>  P(i)"
```
```   189 apply (simp add: lt_def oall_def)
```
```   190 apply (erule conjE)
```
```   191 apply (erule Ord_induct, assumption, blast)
```
```   192 done
```
```   193
```
```   194
```
```   195 subsection {*Quantification over a class*}
```
```   196
```
```   197 definition
```
```   198   "rall"     :: "[i=>o, i=>o] => o"  where
```
```   199     "rall(M, P) == ALL x. M(x) --> P(x)"
```
```   200
```
```   201 definition
```
```   202   "rex"      :: "[i=>o, i=>o] => o"  where
```
```   203     "rex(M, P) == EX x. M(x) & P(x)"
```
```   204
```
```   205 syntax
```
```   206   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3ALL _[_]./ _)" 10)
```
```   207   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3EX _[_]./ _)" 10)
```
```   208
```
```   209 syntax (xsymbols)
```
```   210   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
```
```   211   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
```
```   212 syntax (HTML output)
```
```   213   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
```
```   214   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
```
```   215
```
```   216 translations
```
```   217   "ALL x[M]. P"  == "CONST rall(M, %x. P)"
```
```   218   "EX x[M]. P"   == "CONST rex(M, %x. P)"
```
```   219
```
```   220
```
```   221 subsubsection{*Relativized universal quantifier*}
```
```   222
```
```   223 lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)"
```
```   224 by (simp add: rall_def)
```
```   225
```
```   226 lemma rspec: "[| ALL x[M]. P(x); M(x) |] ==> P(x)"
```
```   227 by (simp add: rall_def)
```
```   228
```
```   229 (*Instantiates x first: better for automatic theorem proving?*)
```
```   230 lemma rev_rallE [elim]:
```
```   231     "[| ALL x[M]. P(x);  ~ M(x) ==> Q;  P(x) ==> Q |] ==> Q"
```
```   232 by (simp add: rall_def, blast)
```
```   233
```
```   234 lemma rallE: "[| ALL x[M]. P(x);  P(x) ==> Q;  ~ M(x) ==> Q |] ==> Q"
```
```   235 by blast
```
```   236
```
```   237 (*Trival rewrite rule;   (ALL x[M].P)<->P holds only if A is nonempty!*)
```
```   238 lemma rall_triv [simp]: "(ALL x[M]. P) <-> ((EX x. M(x)) --> P)"
```
```   239 by (simp add: rall_def)
```
```   240
```
```   241 (*Congruence rule for rewriting*)
```
```   242 lemma rall_cong [cong]:
```
```   243     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (ALL x[M]. P(x)) <-> (ALL x[M]. P'(x))"
```
```   244 by (simp add: rall_def)
```
```   245
```
```   246
```
```   247 subsubsection{*Relativized existential quantifier*}
```
```   248
```
```   249 lemma rexI [intro]: "[| P(x); M(x) |] ==> EX x[M]. P(x)"
```
```   250 by (simp add: rex_def, blast)
```
```   251
```
```   252 (*The best argument order when there is only one M(x)*)
```
```   253 lemma rev_rexI: "[| M(x);  P(x) |] ==> EX x[M]. P(x)"
```
```   254 by blast
```
```   255
```
```   256 (*Not of the general form for such rules; ~EX has become ALL~ *)
```
```   257 lemma rexCI: "[| ALL x[M]. ~P(x) ==> P(a); M(a) |] ==> EX x[M]. P(x)"
```
```   258 by blast
```
```   259
```
```   260 lemma rexE [elim!]: "[| EX x[M]. P(x);  !!x. [| M(x); P(x) |] ==> Q |] ==> Q"
```
```   261 by (simp add: rex_def, blast)
```
```   262
```
```   263 (*We do not even have (EX x[M]. True) <-> True unless A is nonempty!!*)
```
```   264 lemma rex_triv [simp]: "(EX x[M]. P) <-> ((EX x. M(x)) & P)"
```
```   265 by (simp add: rex_def)
```
```   266
```
```   267 lemma rex_cong [cong]:
```
```   268     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (EX x[M]. P(x)) <-> (EX x[M]. P'(x))"
```
```   269 by (simp add: rex_def cong: conj_cong)
```
```   270
```
```   271 lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
```
```   272 by blast
```
```   273
```
```   274 lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
```
```   275 by blast
```
```   276
```
```   277 lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (ALL x[M]. P(x))";
```
```   278 by (simp add: rall_def atomize_all atomize_imp)
```
```   279
```
```   280 declare atomize_rall [symmetric, rulify]
```
```   281
```
```   282 lemma rall_simps1:
```
```   283      "(ALL x[M]. P(x) & Q)   <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)"
```
```   284      "(ALL x[M]. P(x) | Q)   <-> ((ALL x[M]. P(x)) | Q)"
```
```   285      "(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)"
```
```   286      "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
```
```   287 by blast+
```
```   288
```
```   289 lemma rall_simps2:
```
```   290      "(ALL x[M]. P & Q(x))   <-> ((ALL x[M]. False) | P) & (ALL x[M]. Q(x))"
```
```   291      "(ALL x[M]. P | Q(x))   <-> (P | (ALL x[M]. Q(x)))"
```
```   292      "(ALL x[M]. P --> Q(x)) <-> (P --> (ALL x[M]. Q(x)))"
```
```   293 by blast+
```
```   294
```
```   295 lemmas rall_simps [simp] = rall_simps1 rall_simps2
```
```   296
```
```   297 lemma rall_conj_distrib:
```
```   298     "(ALL x[M]. P(x) & Q(x)) <-> ((ALL x[M]. P(x)) & (ALL x[M]. Q(x)))"
```
```   299 by blast
```
```   300
```
```   301 lemma rex_simps1:
```
```   302      "(EX x[M]. P(x) & Q) <-> ((EX x[M]. P(x)) & Q)"
```
```   303      "(EX x[M]. P(x) | Q) <-> (EX x[M]. P(x)) | ((EX x[M]. True) & Q)"
```
```   304      "(EX x[M]. P(x) --> Q) <-> ((ALL x[M]. P(x)) --> ((EX x[M]. True) & Q))"
```
```   305      "(~(EX x[M]. P(x))) <-> (ALL x[M]. ~P(x))"
```
```   306 by blast+
```
```   307
```
```   308 lemma rex_simps2:
```
```   309      "(EX x[M]. P & Q(x)) <-> (P & (EX x[M]. Q(x)))"
```
```   310      "(EX x[M]. P | Q(x)) <-> ((EX x[M]. True) & P) | (EX x[M]. Q(x))"
```
```   311      "(EX x[M]. P --> Q(x)) <-> (((ALL x[M]. False) | P) --> (EX x[M]. Q(x)))"
```
```   312 by blast+
```
```   313
```
```   314 lemmas rex_simps [simp] = rex_simps1 rex_simps2
```
```   315
```
```   316 lemma rex_disj_distrib:
```
```   317     "(EX x[M]. P(x) | Q(x)) <-> ((EX x[M]. P(x)) | (EX x[M]. Q(x)))"
```
```   318 by blast
```
```   319
```
```   320
```
```   321 subsubsection{*One-point rule for bounded quantifiers*}
```
```   322
```
```   323 lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))"
```
```   324 by blast
```
```   325
```
```   326 lemma rex_triv_one_point2 [simp]: "(EX x[M]. a=x) <-> ( M(a))"
```
```   327 by blast
```
```   328
```
```   329 lemma rex_one_point1 [simp]: "(EX x[M]. x=a & P(x)) <-> ( M(a) & P(a))"
```
```   330 by blast
```
```   331
```
```   332 lemma rex_one_point2 [simp]: "(EX x[M]. a=x & P(x)) <-> ( M(a) & P(a))"
```
```   333 by blast
```
```   334
```
```   335 lemma rall_one_point1 [simp]: "(ALL x[M]. x=a --> P(x)) <-> ( M(a) --> P(a))"
```
```   336 by blast
```
```   337
```
```   338 lemma rall_one_point2 [simp]: "(ALL x[M]. a=x --> P(x)) <-> ( M(a) --> P(a))"
```
```   339 by blast
```
```   340
```
```   341
```
```   342 subsubsection{*Sets as Classes*}
```
```   343
```
```   344 definition
```
```   345   setclass :: "[i,i] => o"       ("##_" [40] 40)  where
```
```   346    "setclass(A) == %x. x : A"
```
```   347
```
```   348 lemma setclass_iff [simp]: "setclass(A,x) <-> x : A"
```
```   349 by (simp add: setclass_def)
```
```   350
```
```   351 lemma rall_setclass_is_ball [simp]: "(\<forall>x[##A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
```
```   352 by auto
```
```   353
```
```   354 lemma rex_setclass_is_bex [simp]: "(\<exists>x[##A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
```
```   355 by auto
```
```   356
```
```   357
```
```   358 ML
```
```   359 {*
```
```   360 val Ord_atomize =
```
```   361     atomize ([("OrdQuant.oall", [@{thm ospec}]),("OrdQuant.rall", [@{thm rspec}])]@
```
```   362                  ZF_conn_pairs,
```
```   363              ZF_mem_pairs);
```
```   364 *}
```
```   365 declaration {* fn _ =>
```
```   366   Simplifier.map_ss (fn ss => ss setmksimps (map mk_eq o Ord_atomize o gen_all))
```
```   367 *}
```
```   368
```
```   369 text {* Setting up the one-point-rule simproc *}
```
```   370
```
```   371 ML {*
```
```   372 local
```
```   373
```
```   374 val unfold_rex_tac = unfold_tac [@{thm rex_def}];
```
```   375 fun prove_rex_tac ss = unfold_rex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
```
```   376 val rearrange_bex = Quantifier1.rearrange_bex prove_rex_tac;
```
```   377
```
```   378 val unfold_rall_tac = unfold_tac [@{thm rall_def}];
```
```   379 fun prove_rall_tac ss = unfold_rall_tac ss THEN Quantifier1.prove_one_point_all_tac;
```
```   380 val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac;
```
```   381
```
```   382 in
```
```   383
```
```   384 val defREX_regroup = Simplifier.simproc @{theory}
```
```   385   "defined REX" ["EX x[M]. P(x) & Q(x)"] rearrange_bex;
```
```   386 val defRALL_regroup = Simplifier.simproc @{theory}
```
```   387   "defined RALL" ["ALL x[M]. P(x) --> Q(x)"] rearrange_ball;
```
```   388
```
```   389 end;
```
```   390
```
```   391 Addsimprocs [defRALL_regroup,defREX_regroup];
```
```   392 *}
```
```   393
```
```   394 end
```