src/ZF/OrdQuant.thy
author wenzelm
Wed, 27 Mar 2013 16:38:25 +0100
changeset 51553 63327f679cff
parent 46953 2b6e55924af3
child 51717 9e7d1c139569
permissions -rw-r--r--
more ambitious Goal.skip_proofs: covers Goal.prove forms as well, and do not insist in quick_and_dirty (for the sake of Isabelle/jEdit);
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
41777
1f7cbe39d425 more precise headers;
wenzelm
parents: 38715
diff changeset
     1
(*  Title:      ZF/OrdQuant.thy
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
     2
    Authors:    Krzysztof Grabczewski and L C Paulson
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
     3
*)
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
     4
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
     5
header {*Special quantifiers*}
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
     6
16417
9bc16273c2d4 migrated theory headers to new format
haftmann
parents: 14565
diff changeset
     7
theory OrdQuant imports Ordinal begin
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
     8
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
     9
subsection {*Quantifiers and union operator for ordinals*}
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
    10
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    11
definition
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    12
  (* Ordinal Quantifiers *)
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    13
  oall :: "[i, i => o] => o"  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    14
    "oall(A, P) == \<forall>x. x<A \<longrightarrow> P(x)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    15
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    16
definition
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    17
  oex :: "[i, i => o] => o"  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    18
    "oex(A, P)  == \<exists>x. x<A & P(x)"
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    19
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    20
definition
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    21
  (* Ordinal Union *)
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    22
  OUnion :: "[i, i => i] => i"  where
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
    23
    "OUnion(i,B) == {z: \<Union>x\<in>i. B(x). Ord(i)}"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    24
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    25
syntax
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    26
  "_oall"     :: "[idt, i, o] => o"        ("(3ALL _<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    27
  "_oex"      :: "[idt, i, o] => o"        ("(3EX _<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    28
  "_OUNION"   :: "[idt, i, i] => i"        ("(3UN _<_./ _)" 10)
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    29
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    30
translations
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    31
  "ALL x<a. P"  == "CONST oall(a, %x. P)"
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    32
  "EX x<a. P"   == "CONST oex(a, %x. P)"
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
    33
  "UN x<a. B"   == "CONST OUnion(a, %x. B)"
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    34
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 6093
diff changeset
    35
syntax (xsymbols)
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    36
  "_oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    37
  "_oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    38
  "_OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 13807
diff changeset
    39
syntax (HTML output)
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    40
  "_oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    41
  "_oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
    42
  "_OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    43
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    44
13302
98ce70e7d1f7 fixed precedences of **
paulson
parents: 13298
diff changeset
    45
subsubsection {*simplification of the new quantifiers*}
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    46
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    47
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
    48
(*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    49
  is proved.  Ord_atomize would convert this rule to
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    50
    x < 0 ==> P(x) == True, which causes dire effects!*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    51
lemma [simp]: "(\<forall>x<0. P(x))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    52
by (simp add: oall_def)
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    53
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    54
lemma [simp]: "~(\<exists>x<0. P(x))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    55
by (simp add: oex_def)
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    56
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    57
lemma [simp]: "(\<forall>x<succ(i). P(x)) <-> (Ord(i) \<longrightarrow> P(i) & (\<forall>x<i. P(x)))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    58
apply (simp add: oall_def le_iff)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    59
apply (blast intro: lt_Ord2)
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    60
done
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    61
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    62
lemma [simp]: "(\<exists>x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (\<exists>x<i. P(x))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    63
apply (simp add: oex_def le_iff)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    64
apply (blast intro: lt_Ord2)
12825
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    65
done
f1f7964ed05c new simprules and classical rules
paulson
parents: 12820
diff changeset
    66
13302
98ce70e7d1f7 fixed precedences of **
paulson
parents: 13298
diff changeset
    67
subsubsection {*Union over ordinals*}
13118
336b0bcbd27c new lemmas
paulson
parents: 12825
diff changeset
    68
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    69
lemma Ord_OUN [intro,simp]:
13162
660a71e712af New theorems from Constructible, and moving some Isar material from Main
paulson
parents: 13149
diff changeset
    70
     "[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    71
by (simp add: OUnion_def ltI Ord_UN)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    72
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    73
lemma OUN_upper_lt:
13162
660a71e712af New theorems from Constructible, and moving some Isar material from Main
paulson
parents: 13149
diff changeset
    74
     "[| a<A;  i < b(a);  Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    75
by (unfold OUnion_def lt_def, blast )
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    76
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    77
lemma OUN_upper_le:
13162
660a71e712af New theorems from Constructible, and moving some Isar material from Main
paulson
parents: 13149
diff changeset
    78
     "[| a<A;  i\<le>b(a);  Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
12820
02e2ff3e4d37 lexical tidying
paulson
parents: 12763
diff changeset
    79
apply (unfold OUnion_def, auto)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    80
apply (rule UN_upper_le )
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
    81
apply (auto simp add: lt_def)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    82
done
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
    83
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
    84
lemma Limit_OUN_eq: "Limit(i) ==> (\<Union>x<i. x) = i"
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    85
by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    86
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
    87
(* No < version of this theorem: consider that @{term"(\<Union>i\<in>nat.i)=nat"}! *)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    88
lemma OUN_least:
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
    89
     "(!!x. x<A ==> B(x) \<subseteq> C) ==> (\<Union>x<A. B(x)) \<subseteq> C"
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    90
by (simp add: OUnion_def UN_least ltI)
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    91
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    92
lemma OUN_least_le:
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
    93
     "[| Ord(i);  !!x. x<A ==> b(x) \<le> i |] ==> (\<Union>x<A. b(x)) \<le> i"
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    94
by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    95
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    96
lemma le_implies_OUN_le_OUN:
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
    97
     "[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (\<Union>x<A. c(x)) \<le> (\<Union>x<A. d(x))"
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    98
by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
    99
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
   100
lemma OUN_UN_eq:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46820
diff changeset
   101
     "(!!x. x \<in> A ==> Ord(B(x)))
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
   102
      ==> (\<Union>z < (\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z < B(x). C(z))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   103
by (simp add: OUnion_def)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
   104
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
   105
lemma OUN_Union_eq:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46820
diff changeset
   106
     "(!!x. x \<in> X ==> Ord(x))
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   107
      ==> (\<Union>z < \<Union>(X). C(z)) = (\<Union>x\<in>X. \<Union>z < x. C(z))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   108
by (simp add: OUnion_def)
12620
4e6626725e21 Some new theorems for ordinals
paulson
parents: 12552
diff changeset
   109
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   110
(*So that rule_format will get rid of this quantifier...*)
12763
6cecd9dfd53f now [rule_format] knows about ospec
paulson
parents: 12667
diff changeset
   111
lemma atomize_oall [symmetric, rulify]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   112
     "(!!x. x<A ==> P(x)) == Trueprop (\<forall>x<A. P(x))"
12763
6cecd9dfd53f now [rule_format] knows about ospec
paulson
parents: 12667
diff changeset
   113
by (simp add: oall_def atomize_all atomize_imp)
6cecd9dfd53f now [rule_format] knows about ospec
paulson
parents: 12667
diff changeset
   114
13302
98ce70e7d1f7 fixed precedences of **
paulson
parents: 13298
diff changeset
   115
subsubsection {*universal quantifier for ordinals*}
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   116
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   117
lemma oallI [intro!]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   118
    "[| !!x. x<A ==> P(x) |] ==> \<forall>x<A. P(x)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   119
by (simp add: oall_def)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   120
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   121
lemma ospec: "[| \<forall>x<A. P(x);  x<A |] ==> P(x)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   122
by (simp add: oall_def)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   123
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   124
lemma oallE:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   125
    "[| \<forall>x<A. P(x);  P(x) ==> Q;  ~x<A ==> Q |] ==> Q"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   126
by (simp add: oall_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   127
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   128
lemma rev_oallE [elim]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   129
    "[| \<forall>x<A. P(x);  ~x<A ==> Q;  P(x) ==> Q |] ==> Q"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   130
by (simp add: oall_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   131
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   132
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   133
(*Trival rewrite rule.  @{term"(\<forall>x<a.P)<->P"} holds only if a is not 0!*)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   134
lemma oall_simp [simp]: "(\<forall>x<a. True) <-> True"
13170
paulson
parents: 13169
diff changeset
   135
by blast
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   136
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   137
(*Congruence rule for rewriting*)
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   138
lemma oall_cong [cong]:
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   139
    "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
13289
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   140
     ==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   141
by (simp add: oall_def)
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   142
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   143
13302
98ce70e7d1f7 fixed precedences of **
paulson
parents: 13298
diff changeset
   144
subsubsection {*existential quantifier for ordinals*}
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   145
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   146
lemma oexI [intro]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   147
    "[| P(x);  x<A |] ==> \<exists>x<A. P(x)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   148
apply (simp add: oex_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   149
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   150
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   151
(*Not of the general form for such rules... *)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   152
lemma oexCI:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   153
   "[| \<forall>x<A. ~P(x) ==> P(a);  a<A |] ==> \<exists>x<A. P(x)"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   154
apply (simp add: oex_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   155
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   156
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   157
lemma oexE [elim!]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   158
    "[| \<exists>x<A. P(x);  !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   159
apply (simp add: oex_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   160
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   161
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   162
lemma oex_cong [cong]:
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   163
    "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
13289
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   164
     ==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   165
apply (simp add: oex_def cong add: conj_cong)
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   166
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   167
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   168
13302
98ce70e7d1f7 fixed precedences of **
paulson
parents: 13298
diff changeset
   169
subsubsection {*Rules for Ordinal-Indexed Unions*}
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   170
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46820
diff changeset
   171
lemma OUN_I [intro]: "[| a<i;  b \<in> B(a) |] ==> b: (\<Union>z<i. B(z))"
13170
paulson
parents: 13169
diff changeset
   172
by (unfold OUnion_def lt_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   173
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   174
lemma OUN_E [elim!]:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46820
diff changeset
   175
    "[| b \<in> (\<Union>z<i. B(z));  !!a.[| b \<in> B(a);  a<i |] ==> R |] ==> R"
13170
paulson
parents: 13169
diff changeset
   176
apply (unfold OUnion_def lt_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   177
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   178
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   179
lemma OUN_iff: "b \<in> (\<Union>x<i. B(x)) <-> (\<exists>x<i. b \<in> B(x))"
13170
paulson
parents: 13169
diff changeset
   180
by (unfold OUnion_def oex_def lt_def, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   181
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   182
lemma OUN_cong [cong]:
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13462
diff changeset
   183
    "[| i=j;  !!x. x<j ==> C(x)=D(x) |] ==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))"
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   184
by (simp add: OUnion_def lt_def OUN_iff)
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   185
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   186
lemma lt_induct:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   187
    "[| i<k;  !!x.[| x<k;  \<forall>y<x. P(y) |] ==> P(x) |]  ==>  P(i)"
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   188
apply (simp add: lt_def oall_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   189
apply (erule conjE)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   190
apply (erule Ord_induct, assumption, blast)
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   191
done
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   192
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   193
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   194
subsection {*Quantification over a class*}
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   195
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   196
definition
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   197
  "rall"     :: "[i=>o, i=>o] => o"  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   198
    "rall(M, P) == \<forall>x. M(x) \<longrightarrow> P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   199
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   200
definition
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   201
  "rex"      :: "[i=>o, i=>o] => o"  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   202
    "rex(M, P) == \<exists>x. M(x) & P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   203
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   204
syntax
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   205
  "_rall"     :: "[pttrn, i=>o, o] => o"        ("(3ALL _[_]./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   206
  "_rex"      :: "[pttrn, i=>o, o] => o"        ("(3EX _[_]./ _)" 10)
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   207
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   208
syntax (xsymbols)
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   209
  "_rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   210
  "_rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 13807
diff changeset
   211
syntax (HTML output)
35112
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   212
  "_rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
ff6f60e6ab85 numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents: 32010
diff changeset
   213
  "_rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   214
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   215
translations
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   216
  "ALL x[M]. P"  == "CONST rall(M, %x. P)"
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   217
  "EX x[M]. P"   == "CONST rex(M, %x. P)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   218
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   219
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   220
subsubsection{*Relativized universal quantifier*}
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   221
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   222
lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> \<forall>x[M]. P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   223
by (simp add: rall_def)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   224
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   225
lemma rspec: "[| \<forall>x[M]. P(x); M(x) |] ==> P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   226
by (simp add: rall_def)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   227
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   228
(*Instantiates x first: better for automatic theorem proving?*)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   229
lemma rev_rallE [elim]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   230
    "[| \<forall>x[M]. P(x);  ~ M(x) ==> Q;  P(x) ==> Q |] ==> Q"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   231
by (simp add: rall_def, blast)
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   232
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   233
lemma rallE: "[| \<forall>x[M]. P(x);  P(x) ==> Q;  ~ M(x) ==> Q |] ==> Q"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   234
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   235
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   236
(*Trival rewrite rule;   (ALL x[M].P)<->P holds only if A is nonempty!*)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   237
lemma rall_triv [simp]: "(ALL x[M]. P) <-> ((EX x. M(x)) --> P)"
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   238
by (simp add: rall_def)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   239
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   240
(*Congruence rule for rewriting*)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   241
lemma rall_cong [cong]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   242
    "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (\<forall>x[M]. P(x)) <-> (\<forall>x[M]. P'(x))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   243
by (simp add: rall_def)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   244
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   245
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   246
subsubsection{*Relativized existential quantifier*}
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   247
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   248
lemma rexI [intro]: "[| P(x); M(x) |] ==> \<exists>x[M]. P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   249
by (simp add: rex_def, blast)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   250
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   251
(*The best argument order when there is only one M(x)*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   252
lemma rev_rexI: "[| M(x);  P(x) |] ==> \<exists>x[M]. P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   253
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   254
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   255
(*Not of the general form for such rules... *)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   256
lemma rexCI: "[| \<forall>x[M]. ~P(x) ==> P(a); M(a) |] ==> \<exists>x[M]. P(x)"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   257
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   258
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   259
lemma rexE [elim!]: "[| \<exists>x[M]. P(x);  !!x. [| M(x); P(x) |] ==> Q |] ==> Q"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   260
by (simp add: rex_def, blast)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   261
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   262
(*We do not even have (EX x[M]. True) <-> True unless A is nonempty!!*)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   263
lemma rex_triv [simp]: "(EX x[M]. P) <-> ((EX x. M(x)) & P)"
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   264
by (simp add: rex_def)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   265
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   266
lemma rex_cong [cong]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   267
    "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (\<exists>x[M]. P(x)) <-> (\<exists>x[M]. P'(x))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   268
by (simp add: rex_def cong: conj_cong)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   269
13289
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   270
lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   271
by blast
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   272
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   273
lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   274
by blast
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   275
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   276
lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (\<forall>x[M]. P(x))";
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   277
by (simp add: rall_def atomize_all atomize_imp)
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   278
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   279
declare atomize_rall [symmetric, rulify]
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   280
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   281
lemma rall_simps1:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   282
     "(\<forall>x[M]. P(x) & Q)   <-> (\<forall>x[M]. P(x)) & ((\<forall>x[M]. False) | Q)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   283
     "(\<forall>x[M]. P(x) | Q)   <-> ((\<forall>x[M]. P(x)) | Q)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   284
     "(\<forall>x[M]. P(x) \<longrightarrow> Q) <-> ((\<exists>x[M]. P(x)) \<longrightarrow> Q)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   285
     "(~(\<forall>x[M]. P(x))) <-> (\<exists>x[M]. ~P(x))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   286
by blast+
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   287
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   288
lemma rall_simps2:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   289
     "(\<forall>x[M]. P & Q(x))   <-> ((\<forall>x[M]. False) | P) & (\<forall>x[M]. Q(x))"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   290
     "(\<forall>x[M]. P | Q(x))   <-> (P | (\<forall>x[M]. Q(x)))"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   291
     "(\<forall>x[M]. P \<longrightarrow> Q(x)) <-> (P \<longrightarrow> (\<forall>x[M]. Q(x)))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   292
by blast+
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   293
13289
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   294
lemmas rall_simps [simp] = rall_simps1 rall_simps2
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   295
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   296
lemma rall_conj_distrib:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   297
    "(\<forall>x[M]. P(x) & Q(x)) <-> ((\<forall>x[M]. P(x)) & (\<forall>x[M]. Q(x)))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   298
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   299
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   300
lemma rex_simps1:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   301
     "(\<exists>x[M]. P(x) & Q) <-> ((\<exists>x[M]. P(x)) & Q)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   302
     "(\<exists>x[M]. P(x) | Q) <-> (\<exists>x[M]. P(x)) | ((\<exists>x[M]. True) & Q)"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   303
     "(\<exists>x[M]. P(x) \<longrightarrow> Q) <-> ((\<forall>x[M]. P(x)) \<longrightarrow> ((\<exists>x[M]. True) & Q))"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   304
     "(~(\<exists>x[M]. P(x))) <-> (\<forall>x[M]. ~P(x))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   305
by blast+
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   306
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   307
lemma rex_simps2:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   308
     "(\<exists>x[M]. P & Q(x)) <-> (P & (\<exists>x[M]. Q(x)))"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   309
     "(\<exists>x[M]. P | Q(x)) <-> ((\<exists>x[M]. True) & P) | (\<exists>x[M]. Q(x))"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   310
     "(\<exists>x[M]. P \<longrightarrow> Q(x)) <-> (((\<forall>x[M]. False) | P) \<longrightarrow> (\<exists>x[M]. Q(x)))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   311
by blast+
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   312
13289
53e201efdaa2 miniscoping for class-bounded quantifiers (rall and rex)
paulson
parents: 13253
diff changeset
   313
lemmas rex_simps [simp] = rex_simps1 rex_simps2
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   314
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   315
lemma rex_disj_distrib:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   316
    "(\<exists>x[M]. P(x) | Q(x)) <-> ((\<exists>x[M]. P(x)) | (\<exists>x[M]. Q(x)))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   317
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   318
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   319
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   320
subsubsection{*One-point rule for bounded quantifiers*}
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   321
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   322
lemma rex_triv_one_point1 [simp]: "(\<exists>x[M]. x=a) <-> ( M(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   323
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   324
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   325
lemma rex_triv_one_point2 [simp]: "(\<exists>x[M]. a=x) <-> ( M(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   326
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   327
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   328
lemma rex_one_point1 [simp]: "(\<exists>x[M]. x=a & P(x)) <-> ( M(a) & P(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   329
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   330
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   331
lemma rex_one_point2 [simp]: "(\<exists>x[M]. a=x & P(x)) <-> ( M(a) & P(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   332
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   333
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   334
lemma rall_one_point1 [simp]: "(\<forall>x[M]. x=a \<longrightarrow> P(x)) <-> ( M(a) \<longrightarrow> P(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   335
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   336
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   337
lemma rall_one_point2 [simp]: "(\<forall>x[M]. a=x \<longrightarrow> P(x)) <-> ( M(a) \<longrightarrow> P(a))"
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   338
by blast
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   339
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   340
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   341
subsubsection{*Sets as Classes*}
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   342
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   343
definition
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   344
  setclass :: "[i,i] => o"       ("##_" [40] 40)  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   345
   "setclass(A) == %x. x \<in> A"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   346
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   347
lemma setclass_iff [simp]: "setclass(A,x) <-> x \<in> A"
13362
cd7f9ea58338 tweaked definition of setclass
paulson
parents: 13339
diff changeset
   348
by (simp add: setclass_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   349
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13615
diff changeset
   350
lemma rall_setclass_is_ball [simp]: "(\<forall>x[##A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   351
by auto
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   352
13807
a28a8fbc76d4 changed ** to ## to avoid conflict with new comment syntax
paulson
parents: 13615
diff changeset
   353
lemma rex_setclass_is_bex [simp]: "(\<exists>x[##A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   354
by auto
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   355
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   356
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   357
ML
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   358
{*
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   359
val Ord_atomize =
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 18324
diff changeset
   360
    atomize ([("OrdQuant.oall", [@{thm ospec}]),("OrdQuant.rall", [@{thm rspec}])]@
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13289
diff changeset
   361
                 ZF_conn_pairs,
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   362
             ZF_mem_pairs);
26339
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 24893
diff changeset
   363
*}
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 24893
diff changeset
   364
declaration {* fn _ =>
45625
750c5a47400b modernized some old-style infix operations, which were left over from the time of ML proof scripts;
wenzelm
parents: 42459
diff changeset
   365
  Simplifier.map_ss (Simplifier.set_mksimps (K (map mk_eq o Ord_atomize o gen_all)))
13169
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   366
*}
394a6c649547 conversion of OrdQuant.ML to Isar
paulson
parents: 13162
diff changeset
   367
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13362
diff changeset
   368
text {* Setting up the one-point-rule simproc *}
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   369
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   370
simproc_setup defined_rex ("\<exists>x[M]. P(x) & Q(x)") = {*
42455
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   371
  let
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   372
    val unfold_rex_tac = unfold_tac @{thms rex_def};
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   373
    fun prove_rex_tac ss = unfold_rex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
42459
38b9f023cc34 misc tuning and simplification;
wenzelm
parents: 42456
diff changeset
   374
  in fn _ => fn ss => Quantifier1.rearrange_bex (prove_rex_tac ss) ss end
42455
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   375
*}
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   376
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 45625
diff changeset
   377
simproc_setup defined_rall ("\<forall>x[M]. P(x) \<longrightarrow> Q(x)") = {*
42455
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   378
  let
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   379
    val unfold_rall_tac = unfold_tac @{thms rall_def};
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 41777
diff changeset
   380
    fun prove_rall_tac ss = unfold_rall_tac ss THEN Quantifier1.prove_one_point_all_tac;
42459
38b9f023cc34 misc tuning and simplification;
wenzelm
parents: 42456
diff changeset
   381
  in fn _ => fn ss => Quantifier1.rearrange_ball (prove_rall_tac ss) ss end
13253
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   382
*}
edbf32029d33 added class quantifiers
paulson
parents: 13244
diff changeset
   383
2469
b50b8c0eec01 Implicit simpsets and clasets for FOL and ZF
paulson
parents:
diff changeset
   384
end