src/ZF/OrdQuant.thy
 author wenzelm Sat Mar 29 19:14:00 2008 +0100 (2008-03-29) changeset 26480 544cef16045b parent 26339 7825c83c9eff child 28262 aa7ca36d67fd permissions -rw-r--r--
replaced 'ML_setup' by 'ML';
```     1 (*  Title:      ZF/AC/OrdQuant.thy
```
```     2     ID:         \$Id\$
```
```     3     Authors:    Krzysztof Grabczewski and L C Paulson
```
```     4 *)
```
```     5
```
```     6 header {*Special quantifiers*}
```
```     7
```
```     8 theory OrdQuant imports Ordinal begin
```
```     9
```
```    10 subsection {*Quantifiers and union operator for ordinals*}
```
```    11
```
```    12 definition
```
```    13   (* Ordinal Quantifiers *)
```
```    14   oall :: "[i, i => o] => o"  where
```
```    15     "oall(A, P) == ALL x. x<A --> P(x)"
```
```    16
```
```    17 definition
```
```    18   oex :: "[i, i => o] => o"  where
```
```    19     "oex(A, P)  == EX x. x<A & P(x)"
```
```    20
```
```    21 definition
```
```    22   (* Ordinal Union *)
```
```    23   OUnion :: "[i, i => i] => i"  where
```
```    24     "OUnion(i,B) == {z: \<Union>x\<in>i. B(x). Ord(i)}"
```
```    25
```
```    26 syntax
```
```    27   "@oall"     :: "[idt, i, o] => o"        ("(3ALL _<_./ _)" 10)
```
```    28   "@oex"      :: "[idt, i, o] => o"        ("(3EX _<_./ _)" 10)
```
```    29   "@OUNION"   :: "[idt, i, i] => i"        ("(3UN _<_./ _)" 10)
```
```    30
```
```    31 translations
```
```    32   "ALL x<a. P"  == "CONST oall(a, %x. P)"
```
```    33   "EX x<a. P"   == "CONST oex(a, %x. P)"
```
```    34   "UN x<a. B"   == "CONST OUnion(a, %x. B)"
```
```    35
```
```    36 syntax (xsymbols)
```
```    37   "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
```
```    38   "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
```
```    39   "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
```
```    40 syntax (HTML output)
```
```    41   "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
```
```    42   "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
```
```    43   "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
```
```    44
```
```    45
```
```    46 subsubsection {*simplification of the new quantifiers*}
```
```    47
```
```    48
```
```    49 (*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
```
```    50   is proved.  Ord_atomize would convert this rule to
```
```    51     x < 0 ==> P(x) == True, which causes dire effects!*)
```
```    52 lemma [simp]: "(ALL x<0. P(x))"
```
```    53 by (simp add: oall_def)
```
```    54
```
```    55 lemma [simp]: "~(EX x<0. P(x))"
```
```    56 by (simp add: oex_def)
```
```    57
```
```    58 lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))"
```
```    59 apply (simp add: oall_def le_iff)
```
```    60 apply (blast intro: lt_Ord2)
```
```    61 done
```
```    62
```
```    63 lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))"
```
```    64 apply (simp add: oex_def le_iff)
```
```    65 apply (blast intro: lt_Ord2)
```
```    66 done
```
```    67
```
```    68 subsubsection {*Union over ordinals*}
```
```    69
```
```    70 lemma Ord_OUN [intro,simp]:
```
```    71      "[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
```
```    72 by (simp add: OUnion_def ltI Ord_UN)
```
```    73
```
```    74 lemma OUN_upper_lt:
```
```    75      "[| a<A;  i < b(a);  Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
```
```    76 by (unfold OUnion_def lt_def, blast )
```
```    77
```
```    78 lemma OUN_upper_le:
```
```    79      "[| a<A;  i\<le>b(a);  Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
```
```    80 apply (unfold OUnion_def, auto)
```
```    81 apply (rule UN_upper_le )
```
```    82 apply (auto simp add: lt_def)
```
```    83 done
```
```    84
```
```    85 lemma Limit_OUN_eq: "Limit(i) ==> (\<Union>x<i. x) = i"
```
```    86 by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
```
```    87
```
```    88 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
```
```    89 lemma OUN_least:
```
```    90      "(!!x. x<A ==> B(x) \<subseteq> C) ==> (\<Union>x<A. B(x)) \<subseteq> C"
```
```    91 by (simp add: OUnion_def UN_least ltI)
```
```    92
```
```    93 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
```
```    94 lemma OUN_least_le:
```
```    95      "[| Ord(i);  !!x. x<A ==> b(x) \<le> i |] ==> (\<Union>x<A. b(x)) \<le> i"
```
```    96 by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
```
```    97
```
```    98 lemma le_implies_OUN_le_OUN:
```
```    99      "[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (\<Union>x<A. c(x)) \<le> (\<Union>x<A. d(x))"
```
```   100 by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
```
```   101
```
```   102 lemma OUN_UN_eq:
```
```   103      "(!!x. x:A ==> Ord(B(x)))
```
```   104       ==> (\<Union>z < (\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z < B(x). C(z))"
```
```   105 by (simp add: OUnion_def)
```
```   106
```
```   107 lemma OUN_Union_eq:
```
```   108      "(!!x. x:X ==> Ord(x))
```
```   109       ==> (\<Union>z < Union(X). C(z)) = (\<Union>x\<in>X. \<Union>z < x. C(z))"
```
```   110 by (simp add: OUnion_def)
```
```   111
```
```   112 (*So that rule_format will get rid of ALL x<A...*)
```
```   113 lemma atomize_oall [symmetric, rulify]:
```
```   114      "(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))"
```
```   115 by (simp add: oall_def atomize_all atomize_imp)
```
```   116
```
```   117 subsubsection {*universal quantifier for ordinals*}
```
```   118
```
```   119 lemma oallI [intro!]:
```
```   120     "[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)"
```
```   121 by (simp add: oall_def)
```
```   122
```
```   123 lemma ospec: "[| ALL x<A. P(x);  x<A |] ==> P(x)"
```
```   124 by (simp add: oall_def)
```
```   125
```
```   126 lemma oallE:
```
```   127     "[| ALL x<A. P(x);  P(x) ==> Q;  ~x<A ==> Q |] ==> Q"
```
```   128 by (simp add: oall_def, blast)
```
```   129
```
```   130 lemma rev_oallE [elim]:
```
```   131     "[| ALL x<A. P(x);  ~x<A ==> Q;  P(x) ==> Q |] ==> Q"
```
```   132 by (simp add: oall_def, blast)
```
```   133
```
```   134
```
```   135 (*Trival rewrite rule;   (ALL x<a.P)<->P holds only if a is not 0!*)
```
```   136 lemma oall_simp [simp]: "(ALL x<a. True) <-> True"
```
```   137 by blast
```
```   138
```
```   139 (*Congruence rule for rewriting*)
```
```   140 lemma oall_cong [cong]:
```
```   141     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
```
```   142      ==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
```
```   143 by (simp add: oall_def)
```
```   144
```
```   145
```
```   146 subsubsection {*existential quantifier for ordinals*}
```
```   147
```
```   148 lemma oexI [intro]:
```
```   149     "[| P(x);  x<A |] ==> EX x<A. P(x)"
```
```   150 apply (simp add: oex_def, blast)
```
```   151 done
```
```   152
```
```   153 (*Not of the general form for such rules; ~EX has become ALL~ *)
```
```   154 lemma oexCI:
```
```   155    "[| ALL x<A. ~P(x) ==> P(a);  a<A |] ==> EX x<A. P(x)"
```
```   156 apply (simp add: oex_def, blast)
```
```   157 done
```
```   158
```
```   159 lemma oexE [elim!]:
```
```   160     "[| EX x<A. P(x);  !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
```
```   161 apply (simp add: oex_def, blast)
```
```   162 done
```
```   163
```
```   164 lemma oex_cong [cong]:
```
```   165     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
```
```   166      ==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
```
```   167 apply (simp add: oex_def cong add: conj_cong)
```
```   168 done
```
```   169
```
```   170
```
```   171 subsubsection {*Rules for Ordinal-Indexed Unions*}
```
```   172
```
```   173 lemma OUN_I [intro]: "[| a<i;  b: B(a) |] ==> b: (\<Union>z<i. B(z))"
```
```   174 by (unfold OUnion_def lt_def, blast)
```
```   175
```
```   176 lemma OUN_E [elim!]:
```
```   177     "[| b : (\<Union>z<i. B(z));  !!a.[| b: B(a);  a<i |] ==> R |] ==> R"
```
```   178 apply (unfold OUnion_def lt_def, blast)
```
```   179 done
```
```   180
```
```   181 lemma OUN_iff: "b : (\<Union>x<i. B(x)) <-> (EX x<i. b : B(x))"
```
```   182 by (unfold OUnion_def oex_def lt_def, blast)
```
```   183
```
```   184 lemma OUN_cong [cong]:
```
```   185     "[| i=j;  !!x. x<j ==> C(x)=D(x) |] ==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))"
```
```   186 by (simp add: OUnion_def lt_def OUN_iff)
```
```   187
```
```   188 lemma lt_induct:
```
```   189     "[| i<k;  !!x.[| x<k;  ALL y<x. P(y) |] ==> P(x) |]  ==>  P(i)"
```
```   190 apply (simp add: lt_def oall_def)
```
```   191 apply (erule conjE)
```
```   192 apply (erule Ord_induct, assumption, blast)
```
```   193 done
```
```   194
```
```   195
```
```   196 subsection {*Quantification over a class*}
```
```   197
```
```   198 definition
```
```   199   "rall"     :: "[i=>o, i=>o] => o"  where
```
```   200     "rall(M, P) == ALL x. M(x) --> P(x)"
```
```   201
```
```   202 definition
```
```   203   "rex"      :: "[i=>o, i=>o] => o"  where
```
```   204     "rex(M, P) == EX x. M(x) & P(x)"
```
```   205
```
```   206 syntax
```
```   207   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3ALL _[_]./ _)" 10)
```
```   208   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3EX _[_]./ _)" 10)
```
```   209
```
```   210 syntax (xsymbols)
```
```   211   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
```
```   212   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
```
```   213 syntax (HTML output)
```
```   214   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
```
```   215   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
```
```   216
```
```   217 translations
```
```   218   "ALL x[M]. P"  == "CONST rall(M, %x. P)"
```
```   219   "EX x[M]. P"   == "CONST rex(M, %x. P)"
```
```   220
```
```   221
```
```   222 subsubsection{*Relativized universal quantifier*}
```
```   223
```
```   224 lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)"
```
```   225 by (simp add: rall_def)
```
```   226
```
```   227 lemma rspec: "[| ALL x[M]. P(x); M(x) |] ==> P(x)"
```
```   228 by (simp add: rall_def)
```
```   229
```
```   230 (*Instantiates x first: better for automatic theorem proving?*)
```
```   231 lemma rev_rallE [elim]:
```
```   232     "[| ALL x[M]. P(x);  ~ M(x) ==> Q;  P(x) ==> Q |] ==> Q"
```
```   233 by (simp add: rall_def, blast)
```
```   234
```
```   235 lemma rallE: "[| ALL x[M]. P(x);  P(x) ==> Q;  ~ M(x) ==> Q |] ==> Q"
```
```   236 by blast
```
```   237
```
```   238 (*Trival rewrite rule;   (ALL x[M].P)<->P holds only if A is nonempty!*)
```
```   239 lemma rall_triv [simp]: "(ALL x[M]. P) <-> ((EX x. M(x)) --> P)"
```
```   240 by (simp add: rall_def)
```
```   241
```
```   242 (*Congruence rule for rewriting*)
```
```   243 lemma rall_cong [cong]:
```
```   244     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (ALL x[M]. P(x)) <-> (ALL x[M]. P'(x))"
```
```   245 by (simp add: rall_def)
```
```   246
```
```   247
```
```   248 subsubsection{*Relativized existential quantifier*}
```
```   249
```
```   250 lemma rexI [intro]: "[| P(x); M(x) |] ==> EX x[M]. P(x)"
```
```   251 by (simp add: rex_def, blast)
```
```   252
```
```   253 (*The best argument order when there is only one M(x)*)
```
```   254 lemma rev_rexI: "[| M(x);  P(x) |] ==> EX x[M]. P(x)"
```
```   255 by blast
```
```   256
```
```   257 (*Not of the general form for such rules; ~EX has become ALL~ *)
```
```   258 lemma rexCI: "[| ALL x[M]. ~P(x) ==> P(a); M(a) |] ==> EX x[M]. P(x)"
```
```   259 by blast
```
```   260
```
```   261 lemma rexE [elim!]: "[| EX x[M]. P(x);  !!x. [| M(x); P(x) |] ==> Q |] ==> Q"
```
```   262 by (simp add: rex_def, blast)
```
```   263
```
```   264 (*We do not even have (EX x[M]. True) <-> True unless A is nonempty!!*)
```
```   265 lemma rex_triv [simp]: "(EX x[M]. P) <-> ((EX x. M(x)) & P)"
```
```   266 by (simp add: rex_def)
```
```   267
```
```   268 lemma rex_cong [cong]:
```
```   269     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (EX x[M]. P(x)) <-> (EX x[M]. P'(x))"
```
```   270 by (simp add: rex_def cong: conj_cong)
```
```   271
```
```   272 lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
```
```   273 by blast
```
```   274
```
```   275 lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
```
```   276 by blast
```
```   277
```
```   278 lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (ALL x[M]. P(x))";
```
```   279 by (simp add: rall_def atomize_all atomize_imp)
```
```   280
```
```   281 declare atomize_rall [symmetric, rulify]
```
```   282
```
```   283 lemma rall_simps1:
```
```   284      "(ALL x[M]. P(x) & Q)   <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)"
```
```   285      "(ALL x[M]. P(x) | Q)   <-> ((ALL x[M]. P(x)) | Q)"
```
```   286      "(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)"
```
```   287      "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
```
```   288 by blast+
```
```   289
```
```   290 lemma rall_simps2:
```
```   291      "(ALL x[M]. P & Q(x))   <-> ((ALL x[M]. False) | P) & (ALL x[M]. Q(x))"
```
```   292      "(ALL x[M]. P | Q(x))   <-> (P | (ALL x[M]. Q(x)))"
```
```   293      "(ALL x[M]. P --> Q(x)) <-> (P --> (ALL x[M]. Q(x)))"
```
```   294 by blast+
```
```   295
```
```   296 lemmas rall_simps [simp] = rall_simps1 rall_simps2
```
```   297
```
```   298 lemma rall_conj_distrib:
```
```   299     "(ALL x[M]. P(x) & Q(x)) <-> ((ALL x[M]. P(x)) & (ALL x[M]. Q(x)))"
```
```   300 by blast
```
```   301
```
```   302 lemma rex_simps1:
```
```   303      "(EX x[M]. P(x) & Q) <-> ((EX x[M]. P(x)) & Q)"
```
```   304      "(EX x[M]. P(x) | Q) <-> (EX x[M]. P(x)) | ((EX x[M]. True) & Q)"
```
```   305      "(EX x[M]. P(x) --> Q) <-> ((ALL x[M]. P(x)) --> ((EX x[M]. True) & Q))"
```
```   306      "(~(EX x[M]. P(x))) <-> (ALL x[M]. ~P(x))"
```
```   307 by blast+
```
```   308
```
```   309 lemma rex_simps2:
```
```   310      "(EX x[M]. P & Q(x)) <-> (P & (EX x[M]. Q(x)))"
```
```   311      "(EX x[M]. P | Q(x)) <-> ((EX x[M]. True) & P) | (EX x[M]. Q(x))"
```
```   312      "(EX x[M]. P --> Q(x)) <-> (((ALL x[M]. False) | P) --> (EX x[M]. Q(x)))"
```
```   313 by blast+
```
```   314
```
```   315 lemmas rex_simps [simp] = rex_simps1 rex_simps2
```
```   316
```
```   317 lemma rex_disj_distrib:
```
```   318     "(EX x[M]. P(x) | Q(x)) <-> ((EX x[M]. P(x)) | (EX x[M]. Q(x)))"
```
```   319 by blast
```
```   320
```
```   321
```
```   322 subsubsection{*One-point rule for bounded quantifiers*}
```
```   323
```
```   324 lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))"
```
```   325 by blast
```
```   326
```
```   327 lemma rex_triv_one_point2 [simp]: "(EX x[M]. a=x) <-> ( M(a))"
```
```   328 by blast
```
```   329
```
```   330 lemma rex_one_point1 [simp]: "(EX x[M]. x=a & P(x)) <-> ( M(a) & P(a))"
```
```   331 by blast
```
```   332
```
```   333 lemma rex_one_point2 [simp]: "(EX x[M]. a=x & P(x)) <-> ( M(a) & P(a))"
```
```   334 by blast
```
```   335
```
```   336 lemma rall_one_point1 [simp]: "(ALL x[M]. x=a --> P(x)) <-> ( M(a) --> P(a))"
```
```   337 by blast
```
```   338
```
```   339 lemma rall_one_point2 [simp]: "(ALL x[M]. a=x --> P(x)) <-> ( M(a) --> P(a))"
```
```   340 by blast
```
```   341
```
```   342
```
```   343 subsubsection{*Sets as Classes*}
```
```   344
```
```   345 definition
```
```   346   setclass :: "[i,i] => o"       ("##_" [40] 40)  where
```
```   347    "setclass(A) == %x. x : A"
```
```   348
```
```   349 lemma setclass_iff [simp]: "setclass(A,x) <-> x : A"
```
```   350 by (simp add: setclass_def)
```
```   351
```
```   352 lemma rall_setclass_is_ball [simp]: "(\<forall>x[##A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
```
```   353 by auto
```
```   354
```
```   355 lemma rex_setclass_is_bex [simp]: "(\<exists>x[##A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
```
```   356 by auto
```
```   357
```
```   358
```
```   359 ML
```
```   360 {*
```
```   361 val Ord_atomize =
```
```   362     atomize ([("OrdQuant.oall", [@{thm ospec}]),("OrdQuant.rall", [@{thm rspec}])]@
```
```   363                  ZF_conn_pairs,
```
```   364              ZF_mem_pairs);
```
```   365 *}
```
```   366 declaration {* fn _ =>
```
```   367   Simplifier.map_ss (fn ss => ss setmksimps (map mk_eq o Ord_atomize o gen_all))
```
```   368 *}
```
```   369
```
```   370 text {* Setting up the one-point-rule simproc *}
```
```   371
```
```   372 ML {*
```
```   373 local
```
```   374
```
```   375 val unfold_rex_tac = unfold_tac [@{thm rex_def}];
```
```   376 fun prove_rex_tac ss = unfold_rex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
```
```   377 val rearrange_bex = Quantifier1.rearrange_bex prove_rex_tac;
```
```   378
```
```   379 val unfold_rall_tac = unfold_tac [@{thm rall_def}];
```
```   380 fun prove_rall_tac ss = unfold_rall_tac ss THEN Quantifier1.prove_one_point_all_tac;
```
```   381 val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac;
```
```   382
```
```   383 in
```
```   384
```
```   385 val defREX_regroup = Simplifier.simproc @{theory}
```
```   386   "defined REX" ["EX x[M]. P(x) & Q(x)"] rearrange_bex;
```
```   387 val defRALL_regroup = Simplifier.simproc @{theory}
```
```   388   "defined RALL" ["ALL x[M]. P(x) --> Q(x)"] rearrange_ball;
```
```   389
```
```   390 end;
```
```   391
```
```   392 Addsimprocs [defRALL_regroup,defREX_regroup];
```
```   393 *}
```
```   394
```
```   395 end
```