src/ZF/Constructible/L_axioms.thy
author paulson
Thu, 17 Oct 2002 10:54:11 +0200
changeset 13651 ac80e101306a
parent 13634 99a593b49b04
child 13655 95b95cdb4704
permissions -rw-r--r--
Cosmetic changes suggested by writing the paper. Deleted some redundant arity proofs
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
     1
(*  Title:      ZF/Constructible/L_axioms.thy
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
     2
    ID:         $Id$
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
     4
*)
13429
wenzelm
parents: 13428
diff changeset
     5
wenzelm
parents: 13428
diff changeset
     6
header {* The ZF Axioms (Except Separation) in L *}
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
     7
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
     8
theory L_axioms = Formula + Relative + Reflection + MetaExists:
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
     9
13564
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
    10
text {* The class L satisfies the premises of locale @{text M_trivial} *}
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    11
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    12
lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
13429
wenzelm
parents: 13428
diff changeset
    13
apply (insert Transset_Lset)
wenzelm
parents: 13428
diff changeset
    14
apply (simp add: Transset_def L_def, blast)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    15
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    16
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    17
lemma nonempty: "L(0)"
13429
wenzelm
parents: 13428
diff changeset
    18
apply (simp add: L_def)
wenzelm
parents: 13428
diff changeset
    19
apply (blast intro: zero_in_Lset)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    20
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    21
13563
paulson
parents: 13506
diff changeset
    22
theorem upair_ax: "upair_ax(L)"
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    23
apply (simp add: upair_ax_def upair_def, clarify)
13429
wenzelm
parents: 13428
diff changeset
    24
apply (rule_tac x="{x,y}" in rexI)
wenzelm
parents: 13428
diff changeset
    25
apply (simp_all add: doubleton_in_L)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    26
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    27
13563
paulson
parents: 13506
diff changeset
    28
theorem Union_ax: "Union_ax(L)"
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    29
apply (simp add: Union_ax_def big_union_def, clarify)
13429
wenzelm
parents: 13428
diff changeset
    30
apply (rule_tac x="Union(x)" in rexI)
wenzelm
parents: 13428
diff changeset
    31
apply (simp_all add: Union_in_L, auto)
wenzelm
parents: 13428
diff changeset
    32
apply (blast intro: transL)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    33
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    34
13563
paulson
parents: 13506
diff changeset
    35
theorem power_ax: "power_ax(L)"
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    36
apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
13429
wenzelm
parents: 13428
diff changeset
    37
apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)
13299
3a932abf97e8 More use of relativized quantifiers
paulson
parents: 13298
diff changeset
    38
apply (simp_all add: LPow_in_L, auto)
13429
wenzelm
parents: 13428
diff changeset
    39
apply (blast intro: transL)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    40
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    41
13563
paulson
parents: 13506
diff changeset
    42
text{*We don't actually need @{term L} to satisfy the foundation axiom.*}
paulson
parents: 13506
diff changeset
    43
theorem foundation_ax: "foundation_ax(L)"
paulson
parents: 13506
diff changeset
    44
apply (simp add: foundation_ax_def)
paulson
parents: 13506
diff changeset
    45
apply (rule rallI) 
paulson
parents: 13506
diff changeset
    46
apply (cut_tac A=x in foundation)
paulson
parents: 13506
diff changeset
    47
apply (blast intro: transL)
paulson
parents: 13506
diff changeset
    48
done
paulson
parents: 13506
diff changeset
    49
13506
acc18a5d2b1a Various tweaks of the presentation
paulson
parents: 13505
diff changeset
    50
subsection{*For L to satisfy Replacement *}
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    51
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    52
(*Can't move these to Formula unless the definition of univalent is moved
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    53
there too!*)
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    54
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    55
lemma LReplace_in_Lset:
13429
wenzelm
parents: 13428
diff changeset
    56
     "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    57
      ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
13429
wenzelm
parents: 13428
diff changeset
    58
apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    59
       in exI)
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    60
apply simp
13429
wenzelm
parents: 13428
diff changeset
    61
apply clarify
wenzelm
parents: 13428
diff changeset
    62
apply (rule_tac a=x in UN_I)
wenzelm
parents: 13428
diff changeset
    63
 apply (simp_all add: Replace_iff univalent_def)
wenzelm
parents: 13428
diff changeset
    64
apply (blast dest: transL L_I)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    65
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    66
13429
wenzelm
parents: 13428
diff changeset
    67
lemma LReplace_in_L:
wenzelm
parents: 13428
diff changeset
    68
     "[|L(X); univalent(L,X,Q)|]
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    69
      ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
13429
wenzelm
parents: 13428
diff changeset
    70
apply (drule L_D, clarify)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    71
apply (drule LReplace_in_Lset, assumption+)
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    72
apply (blast intro: L_I Lset_in_Lset_succ)
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    73
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    74
13563
paulson
parents: 13506
diff changeset
    75
theorem replacement: "replacement(L,P)"
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    76
apply (simp add: replacement_def, clarify)
13429
wenzelm
parents: 13428
diff changeset
    77
apply (frule LReplace_in_L, assumption+, clarify)
wenzelm
parents: 13428
diff changeset
    78
apply (rule_tac x=Y in rexI)
wenzelm
parents: 13428
diff changeset
    79
apply (simp_all add: Replace_iff univalent_def, blast)
13223
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    80
done
45be08fbdcff new theory of inner models
paulson
parents:
diff changeset
    81
13564
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
    82
subsection{*Instantiating the locale @{text M_trivial}*}
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
    83
text{*No instances of Separation yet.*}
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    84
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    85
lemma Lset_mono_le: "mono_le_subset(Lset)"
13429
wenzelm
parents: 13428
diff changeset
    86
by (simp add: mono_le_subset_def le_imp_subset Lset_mono)
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    87
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    88
lemma Lset_cont: "cont_Ord(Lset)"
13429
wenzelm
parents: 13428
diff changeset
    89
by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    90
13428
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
    91
lemmas L_nat = Ord_in_L [OF Ord_nat]
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
    92
13564
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
    93
theorem M_trivial_L: "PROP M_trivial(L)"
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
    94
  apply (rule M_trivial.intro)
13628
87482b5e3f2e Various simplifications of the Constructible theories
paulson
parents: 13566
diff changeset
    95
       apply (erule (1) transL)
13428
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
    96
      apply (rule upair_ax)
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
    97
     apply (rule Union_ax)
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
    98
    apply (rule power_ax)
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
    99
   apply (rule replacement)
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
   100
  apply (rule L_nat)
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
   101
  done
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   102
13564
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   103
lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   104
  and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   105
  and ball_iff_equiv = M_trivial.ball_iff_equiv [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   106
  and M_equalityI = M_trivial.M_equalityI [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   107
  and empty_abs = M_trivial.empty_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   108
  and subset_abs = M_trivial.subset_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   109
  and upair_abs = M_trivial.upair_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   110
  and upair_in_M_iff = M_trivial.upair_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   111
  and singleton_in_M_iff = M_trivial.singleton_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   112
  and pair_abs = M_trivial.pair_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   113
  and pair_in_M_iff = M_trivial.pair_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   114
  and pair_components_in_M = M_trivial.pair_components_in_M [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   115
  and cartprod_abs = M_trivial.cartprod_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   116
  and union_abs = M_trivial.union_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   117
  and inter_abs = M_trivial.inter_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   118
  and setdiff_abs = M_trivial.setdiff_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   119
  and Union_abs = M_trivial.Union_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   120
  and Union_closed = M_trivial.Union_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   121
  and Un_closed = M_trivial.Un_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   122
  and cons_closed = M_trivial.cons_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   123
  and successor_abs = M_trivial.successor_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   124
  and succ_in_M_iff = M_trivial.succ_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   125
  and separation_closed = M_trivial.separation_closed [OF M_trivial_L]
13566
52a419210d5c Streamlined proofs of instances of Separation
paulson
parents: 13564
diff changeset
   126
  and strong_replacementI = 
52a419210d5c Streamlined proofs of instances of Separation
paulson
parents: 13564
diff changeset
   127
      M_trivial.strong_replacementI [OF M_trivial_L, rule_format]
13564
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   128
  and strong_replacement_closed = M_trivial.strong_replacement_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   129
  and RepFun_closed = M_trivial.RepFun_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   130
  and lam_closed = M_trivial.lam_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   131
  and image_abs = M_trivial.image_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   132
  and powerset_Pow = M_trivial.powerset_Pow [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   133
  and powerset_imp_subset_Pow = M_trivial.powerset_imp_subset_Pow [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   134
  and nat_into_M = M_trivial.nat_into_M [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   135
  and nat_case_closed = M_trivial.nat_case_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   136
  and Inl_in_M_iff = M_trivial.Inl_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   137
  and Inr_in_M_iff = M_trivial.Inr_in_M_iff [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   138
  and lt_closed = M_trivial.lt_closed [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   139
  and transitive_set_abs = M_trivial.transitive_set_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   140
  and ordinal_abs = M_trivial.ordinal_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   141
  and limit_ordinal_abs = M_trivial.limit_ordinal_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   142
  and successor_ordinal_abs = M_trivial.successor_ordinal_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   143
  and finite_ordinal_abs = M_trivial.finite_ordinal_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   144
  and omega_abs = M_trivial.omega_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   145
  and number1_abs = M_trivial.number1_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   146
  and number2_abs = M_trivial.number2_abs [OF M_trivial_L]
1500a2e48d44 renamed M_triv_axioms to M_trivial and M_axioms to M_basic
paulson
parents: 13563
diff changeset
   147
  and number3_abs = M_trivial.number3_abs [OF M_trivial_L]
13429
wenzelm
parents: 13428
diff changeset
   148
wenzelm
parents: 13428
diff changeset
   149
declare rall_abs [simp]
wenzelm
parents: 13428
diff changeset
   150
declare rex_abs [simp]
wenzelm
parents: 13428
diff changeset
   151
declare empty_abs [simp]
wenzelm
parents: 13428
diff changeset
   152
declare subset_abs [simp]
wenzelm
parents: 13428
diff changeset
   153
declare upair_abs [simp]
wenzelm
parents: 13428
diff changeset
   154
declare upair_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   155
declare singleton_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   156
declare pair_abs [simp]
wenzelm
parents: 13428
diff changeset
   157
declare pair_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   158
declare cartprod_abs [simp]
wenzelm
parents: 13428
diff changeset
   159
declare union_abs [simp]
wenzelm
parents: 13428
diff changeset
   160
declare inter_abs [simp]
wenzelm
parents: 13428
diff changeset
   161
declare setdiff_abs [simp]
wenzelm
parents: 13428
diff changeset
   162
declare Union_abs [simp]
wenzelm
parents: 13428
diff changeset
   163
declare Union_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   164
declare Un_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   165
declare cons_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   166
declare successor_abs [simp]
wenzelm
parents: 13428
diff changeset
   167
declare succ_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   168
declare separation_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   169
declare strong_replacement_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   170
declare RepFun_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   171
declare lam_closed [intro, simp]
wenzelm
parents: 13428
diff changeset
   172
declare image_abs [simp]
wenzelm
parents: 13428
diff changeset
   173
declare nat_into_M [intro]
wenzelm
parents: 13428
diff changeset
   174
declare Inl_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   175
declare Inr_in_M_iff [iff]
wenzelm
parents: 13428
diff changeset
   176
declare transitive_set_abs [simp]
wenzelm
parents: 13428
diff changeset
   177
declare ordinal_abs [simp]
wenzelm
parents: 13428
diff changeset
   178
declare limit_ordinal_abs [simp]
wenzelm
parents: 13428
diff changeset
   179
declare successor_ordinal_abs [simp]
wenzelm
parents: 13428
diff changeset
   180
declare finite_ordinal_abs [simp]
wenzelm
parents: 13428
diff changeset
   181
declare omega_abs [simp]
wenzelm
parents: 13428
diff changeset
   182
declare number1_abs [simp]
wenzelm
parents: 13428
diff changeset
   183
declare number2_abs [simp]
wenzelm
parents: 13428
diff changeset
   184
declare number3_abs [simp]
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   185
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   186
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   187
subsection{*Instantiation of the locale @{text reflection}*}
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   188
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   189
text{*instances of locale constants*}
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   190
constdefs
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   191
  L_F0 :: "[i=>o,i] => i"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   192
    "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   193
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   194
  L_FF :: "[i=>o,i] => i"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   195
    "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   196
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   197
  L_ClEx :: "[i=>o,i] => o"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   198
    "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   199
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   200
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   201
text{*We must use the meta-existential quantifier; otherwise the reflection
13429
wenzelm
parents: 13428
diff changeset
   202
      terms become enormous!*}
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   203
constdefs
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   204
  L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   205
    "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   206
                           (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   207
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   208
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   209
theorem Triv_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   210
     "REFLECTS[P, \<lambda>a x. P(x)]"
13429
wenzelm
parents: 13428
diff changeset
   211
apply (simp add: L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   212
apply (rule meta_exI)
wenzelm
parents: 13428
diff changeset
   213
apply (rule Closed_Unbounded_Ord)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   214
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   215
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   216
theorem Not_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   217
     "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
13429
wenzelm
parents: 13428
diff changeset
   218
apply (unfold L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   219
apply (erule meta_exE)
wenzelm
parents: 13428
diff changeset
   220
apply (rule_tac x=Cl in meta_exI, simp)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   221
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   222
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   223
theorem And_reflection:
13429
wenzelm
parents: 13428
diff changeset
   224
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   225
      ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
13429
wenzelm
parents: 13428
diff changeset
   226
apply (unfold L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   227
apply (elim meta_exE)
wenzelm
parents: 13428
diff changeset
   228
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
wenzelm
parents: 13428
diff changeset
   229
apply (simp add: Closed_Unbounded_Int, blast)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   230
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   231
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   232
theorem Or_reflection:
13429
wenzelm
parents: 13428
diff changeset
   233
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   234
      ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
13429
wenzelm
parents: 13428
diff changeset
   235
apply (unfold L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   236
apply (elim meta_exE)
wenzelm
parents: 13428
diff changeset
   237
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
wenzelm
parents: 13428
diff changeset
   238
apply (simp add: Closed_Unbounded_Int, blast)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   239
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   240
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   241
theorem Imp_reflection:
13429
wenzelm
parents: 13428
diff changeset
   242
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   243
      ==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]"
13429
wenzelm
parents: 13428
diff changeset
   244
apply (unfold L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   245
apply (elim meta_exE)
wenzelm
parents: 13428
diff changeset
   246
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
wenzelm
parents: 13428
diff changeset
   247
apply (simp add: Closed_Unbounded_Int, blast)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   248
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   249
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   250
theorem Iff_reflection:
13429
wenzelm
parents: 13428
diff changeset
   251
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   252
      ==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]"
13429
wenzelm
parents: 13428
diff changeset
   253
apply (unfold L_Reflects_def)
wenzelm
parents: 13428
diff changeset
   254
apply (elim meta_exE)
wenzelm
parents: 13428
diff changeset
   255
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
wenzelm
parents: 13428
diff changeset
   256
apply (simp add: Closed_Unbounded_Int, blast)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   257
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   258
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   259
13434
78b93a667c01 better sats rules for higher-order operators
paulson
parents: 13429
diff changeset
   260
lemma reflection_Lset: "reflection(Lset)"
13651
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   261
by (blast intro: reflection.intro Lset_mono_le Lset_cont 
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   262
                 Formula.Pair_in_LLimit)+
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   263
13434
78b93a667c01 better sats rules for higher-order operators
paulson
parents: 13429
diff changeset
   264
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   265
theorem Ex_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   266
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   267
      ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
13429
wenzelm
parents: 13428
diff changeset
   268
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
wenzelm
parents: 13428
diff changeset
   269
apply (elim meta_exE)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   270
apply (rule meta_exI)
13434
78b93a667c01 better sats rules for higher-order operators
paulson
parents: 13429
diff changeset
   271
apply (erule reflection.Ex_reflection [OF reflection_Lset])
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   272
done
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   273
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   274
theorem All_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   275
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
13429
wenzelm
parents: 13428
diff changeset
   276
      ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
wenzelm
parents: 13428
diff changeset
   277
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
wenzelm
parents: 13428
diff changeset
   278
apply (elim meta_exE)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   279
apply (rule meta_exI)
13434
78b93a667c01 better sats rules for higher-order operators
paulson
parents: 13429
diff changeset
   280
apply (erule reflection.All_reflection [OF reflection_Lset])
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   281
done
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   282
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   283
theorem Rex_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   284
     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   285
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
13429
wenzelm
parents: 13428
diff changeset
   286
apply (unfold rex_def)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   287
apply (intro And_reflection Ex_reflection, assumption)
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   288
done
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   289
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   290
theorem Rall_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   291
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
13429
wenzelm
parents: 13428
diff changeset
   292
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
wenzelm
parents: 13428
diff changeset
   293
apply (unfold rall_def)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   294
apply (intro Imp_reflection All_reflection, assumption)
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   295
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   296
13440
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   297
text{*This version handles an alternative form of the bounded quantifier
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   298
      in the second argument of @{text REFLECTS}.*}
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   299
theorem Rex_reflection':
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   300
     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   301
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[**Lset(a)]. Q(a,x,z)]"
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   302
apply (unfold setclass_def rex_def)
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   303
apply (erule Rex_reflection [unfolded rex_def Bex_def]) 
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   304
done
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   305
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   306
text{*As above.*}
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   307
theorem Rall_reflection':
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   308
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   309
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[**Lset(a)]. Q(a,x,z)]"
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   310
apply (unfold setclass_def rall_def)
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   311
apply (erule Rall_reflection [unfolded rall_def Ball_def]) 
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   312
done
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   313
13429
wenzelm
parents: 13428
diff changeset
   314
lemmas FOL_reflections =
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   315
        Triv_reflection Not_reflection And_reflection Or_reflection
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   316
        Imp_reflection Iff_reflection Ex_reflection All_reflection
13440
cdde97e1db1c some progress towards "satisfies"
paulson
parents: 13434
diff changeset
   317
        Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   318
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   319
lemma ReflectsD:
13429
wenzelm
parents: 13428
diff changeset
   320
     "[|REFLECTS[P,Q]; Ord(i)|]
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   321
      ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
13429
wenzelm
parents: 13428
diff changeset
   322
apply (unfold L_Reflects_def Closed_Unbounded_def)
wenzelm
parents: 13428
diff changeset
   323
apply (elim meta_exE, clarify)
wenzelm
parents: 13428
diff changeset
   324
apply (blast dest!: UnboundedD)
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   325
done
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   326
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   327
lemma ReflectsE:
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   328
     "[| REFLECTS[P,Q]; Ord(i);
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   329
         !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   330
      ==> R"
13429
wenzelm
parents: 13428
diff changeset
   331
apply (drule ReflectsD, assumption, blast)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   332
done
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   333
13428
99e52e78eb65 eliminate open locales and special ML code;
wenzelm
parents: 13418
diff changeset
   334
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
13291
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   335
by blast
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   336
a73ab154f75c towards proving separation for L
paulson
parents: 13269
diff changeset
   337
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   338
subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   339
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   340
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   341
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   342
subsubsection{*Some numbers to help write de Bruijn indices*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   343
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   344
syntax
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   345
    "3" :: i   ("3")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   346
    "4" :: i   ("4")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   347
    "5" :: i   ("5")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   348
    "6" :: i   ("6")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   349
    "7" :: i   ("7")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   350
    "8" :: i   ("8")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   351
    "9" :: i   ("9")
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   352
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   353
translations
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   354
   "3"  == "succ(2)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   355
   "4"  == "succ(3)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   356
   "5"  == "succ(4)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   357
   "6"  == "succ(5)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   358
   "7"  == "succ(6)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   359
   "8"  == "succ(7)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   360
   "9"  == "succ(8)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   361
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   362
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   363
subsubsection{*The Empty Set, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   364
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   365
constdefs empty_fm :: "i=>i"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   366
    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   367
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   368
lemma empty_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   369
     "x \<in> nat ==> empty_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   370
by (simp add: empty_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   371
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   372
lemma sats_empty_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   373
   "[| x \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   374
    ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   375
by (simp add: empty_fm_def empty_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   376
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   377
lemma empty_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   378
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   379
          i \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   380
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   381
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   382
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   383
theorem empty_reflection:
13429
wenzelm
parents: 13428
diff changeset
   384
     "REFLECTS[\<lambda>x. empty(L,f(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   385
               \<lambda>i x. empty(**Lset(i),f(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   386
apply (simp only: empty_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   387
apply (intro FOL_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   388
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   389
13385
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   390
text{*Not used.  But maybe useful?*}
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   391
lemma Transset_sats_empty_fm_eq_0:
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   392
   "[| n \<in> nat; env \<in> list(A); Transset(A)|]
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   393
    ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   394
apply (simp add: empty_fm_def empty_def Transset_def, auto)
13429
wenzelm
parents: 13428
diff changeset
   395
apply (case_tac "n < length(env)")
wenzelm
parents: 13428
diff changeset
   396
apply (frule nth_type, assumption+, blast)
wenzelm
parents: 13428
diff changeset
   397
apply (simp_all add: not_lt_iff_le nth_eq_0)
13385
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   398
done
31df66ca0780 Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents: 13363
diff changeset
   399
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   400
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   401
subsubsection{*Unordered Pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   402
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   403
constdefs upair_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   404
    "upair_fm(x,y,z) ==
wenzelm
parents: 13428
diff changeset
   405
       And(Member(x,z),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   406
           And(Member(y,z),
13429
wenzelm
parents: 13428
diff changeset
   407
               Forall(Implies(Member(0,succ(z)),
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   408
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   409
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   410
lemma upair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   411
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   412
by (simp add: upair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   413
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   414
lemma sats_upair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   415
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   416
    ==> sats(A, upair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   417
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   418
by (simp add: upair_fm_def upair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   419
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   420
lemma upair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   421
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   422
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   423
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   424
by (simp add: sats_upair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   425
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   426
text{*Useful? At least it refers to "real" unordered pairs*}
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   427
lemma sats_upair_fm2 [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   428
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
13429
wenzelm
parents: 13428
diff changeset
   429
    ==> sats(A, upair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   430
        nth(z,env) = {nth(x,env), nth(y,env)}"
13429
wenzelm
parents: 13428
diff changeset
   431
apply (frule lt_length_in_nat, assumption)
wenzelm
parents: 13428
diff changeset
   432
apply (simp add: upair_fm_def Transset_def, auto)
wenzelm
parents: 13428
diff changeset
   433
apply (blast intro: nth_type)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   434
done
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   435
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   436
theorem upair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   437
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
wenzelm
parents: 13428
diff changeset
   438
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   439
apply (simp add: upair_def)
13429
wenzelm
parents: 13428
diff changeset
   440
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   441
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   442
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   443
subsubsection{*Ordered pairs, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   444
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   445
constdefs pair_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   446
    "pair_fm(x,y,z) ==
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   447
       Exists(And(upair_fm(succ(x),succ(x),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   448
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   449
                         upair_fm(1,0,succ(succ(z)))))))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   450
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   451
lemma pair_type [TC]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   452
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   453
by (simp add: pair_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   454
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   455
lemma sats_pair_fm [simp]:
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   456
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   457
    ==> sats(A, pair_fm(x,y,z), env) <->
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   458
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   459
by (simp add: pair_fm_def pair_def)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   460
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   461
lemma pair_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   462
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   463
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   464
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   465
by (simp add: sats_pair_fm)
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   466
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   467
theorem pair_reflection:
13429
wenzelm
parents: 13428
diff changeset
   468
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   469
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   470
apply (simp only: pair_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   471
apply (intro FOL_reflections upair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   472
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   473
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   474
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   475
subsubsection{*Binary Unions, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   476
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   477
constdefs union_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   478
    "union_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   479
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   480
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   481
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   482
lemma union_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   483
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   484
by (simp add: union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   485
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   486
lemma sats_union_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   487
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   488
    ==> sats(A, union_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   489
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   490
by (simp add: union_fm_def union_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   491
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   492
lemma union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   493
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   494
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   495
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   496
by (simp add: sats_union_fm)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   497
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   498
theorem union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   499
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   500
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   501
apply (simp only: union_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   502
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   503
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   504
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   505
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   506
subsubsection{*Set ``Cons,'' Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   507
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   508
constdefs cons_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   509
    "cons_fm(x,y,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   510
       Exists(And(upair_fm(succ(x),succ(x),0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   511
                  union_fm(0,succ(y),succ(z))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   512
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   513
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   514
lemma cons_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   515
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   516
by (simp add: cons_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   517
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   518
lemma sats_cons_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   519
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   520
    ==> sats(A, cons_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   521
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   522
by (simp add: cons_fm_def is_cons_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   523
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   524
lemma cons_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   525
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   526
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   527
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   528
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   529
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   530
theorem cons_reflection:
13429
wenzelm
parents: 13428
diff changeset
   531
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   532
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   533
apply (simp only: is_cons_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   534
apply (intro FOL_reflections upair_reflection union_reflection)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   535
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   536
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   537
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   538
subsubsection{*Successor Function, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   539
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   540
constdefs succ_fm :: "[i,i]=>i"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   541
    "succ_fm(x,y) == cons_fm(x,x,y)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   542
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   543
lemma succ_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   544
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   545
by (simp add: succ_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   546
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   547
lemma sats_succ_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   548
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   549
    ==> sats(A, succ_fm(x,y), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   550
        successor(**A, nth(x,env), nth(y,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   551
by (simp add: succ_fm_def successor_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   552
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   553
lemma successor_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   554
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   555
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   556
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   557
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   558
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   559
theorem successor_reflection:
13429
wenzelm
parents: 13428
diff changeset
   560
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   561
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   562
apply (simp only: successor_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   563
apply (intro cons_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   564
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   565
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   566
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   567
subsubsection{*The Number 1, Internalized*}
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   568
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   569
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   570
constdefs number1_fm :: "i=>i"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   571
    "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   572
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   573
lemma number1_type [TC]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   574
     "x \<in> nat ==> number1_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   575
by (simp add: number1_fm_def)
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   576
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   577
lemma sats_number1_fm [simp]:
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   578
   "[| x \<in> nat; env \<in> list(A)|]
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   579
    ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   580
by (simp add: number1_fm_def number1_def)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   581
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   582
lemma number1_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   583
      "[| nth(i,env) = x; nth(j,env) = y;
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   584
          i \<in> nat; env \<in> list(A)|]
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   585
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   586
by simp
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   587
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   588
theorem number1_reflection:
13429
wenzelm
parents: 13428
diff changeset
   589
     "REFLECTS[\<lambda>x. number1(L,f(x)),
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   590
               \<lambda>i x. number1(**Lset(i),f(x))]"
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   591
apply (simp only: number1_def setclass_simps)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   592
apply (intro FOL_reflections empty_reflection successor_reflection)
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   593
done
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   594
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
   595
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   596
subsubsection{*Big Union, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   597
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   598
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   599
constdefs big_union_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   600
    "big_union_fm(A,z) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   601
       Forall(Iff(Member(0,succ(z)),
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   602
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   603
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   604
lemma big_union_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   605
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   606
by (simp add: big_union_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   607
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   608
lemma sats_big_union_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   609
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   610
    ==> sats(A, big_union_fm(x,y), env) <->
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   611
        big_union(**A, nth(x,env), nth(y,env))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   612
by (simp add: big_union_fm_def big_union_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   613
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   614
lemma big_union_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   615
      "[| nth(i,env) = x; nth(j,env) = y;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   616
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   617
       ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   618
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   619
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   620
theorem big_union_reflection:
13429
wenzelm
parents: 13428
diff changeset
   621
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   622
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   623
apply (simp only: big_union_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   624
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   625
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   626
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   627
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   628
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   629
13651
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   630
text{*The @{text sats} theorems below are standard versions of the ones proved
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   631
in theory @{text Formula}.  They relate elements of type @{term formula} to
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   632
relativized concepts such as @{term subset} or @{term ordinal} rather than to
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   633
real concepts such as @{term Ord}.  Now that we have instantiated the locale
ac80e101306a Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents: 13634
diff changeset
   634
@{text M_trivial}, we no longer require the earlier versions.*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   635
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   636
lemma sats_subset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   637
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   638
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))"
wenzelm
parents: 13428
diff changeset
   639
by (simp add: subset_fm_def Relative.subset_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   640
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   641
theorem subset_reflection:
13429
wenzelm
parents: 13428
diff changeset
   642
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
wenzelm
parents: 13428
diff changeset
   643
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   644
apply (simp only: Relative.subset_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   645
apply (intro FOL_reflections)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   646
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   647
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   648
lemma sats_transset_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   649
   "[|x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   650
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
13429
wenzelm
parents: 13428
diff changeset
   651
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   652
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   653
theorem transitive_set_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   654
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   655
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   656
apply (simp only: transitive_set_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   657
apply (intro FOL_reflections subset_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   658
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   659
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   660
lemma sats_ordinal_fm':
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   661
   "[|x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   662
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   663
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   664
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   665
lemma ordinal_iff_sats:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   666
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   667
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   668
by (simp add: sats_ordinal_fm')
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   669
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   670
theorem ordinal_reflection:
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   671
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   672
apply (simp only: ordinal_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   673
apply (intro FOL_reflections transitive_set_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   674
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   675
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   676
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   677
subsubsection{*Membership Relation, Internalized*}
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   678
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   679
constdefs Memrel_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   680
    "Memrel_fm(A,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   681
       Forall(Iff(Member(0,succ(r)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   682
                  Exists(And(Member(0,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   683
                             Exists(And(Member(0,succ(succ(succ(A)))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   684
                                        And(Member(1,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   685
                                            pair_fm(1,0,2))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   686
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   687
lemma Memrel_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   688
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   689
by (simp add: Memrel_fm_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   690
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   691
lemma sats_Memrel_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   692
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   693
    ==> sats(A, Memrel_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   694
        membership(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   695
by (simp add: Memrel_fm_def membership_def)
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   696
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   697
lemma Memrel_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   698
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   699
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   700
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   701
by simp
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   702
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   703
theorem membership_reflection:
13429
wenzelm
parents: 13428
diff changeset
   704
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   705
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   706
apply (simp only: membership_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   707
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   708
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   709
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   710
subsubsection{*Predecessor Set, Internalized*}
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   711
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   712
constdefs pred_set_fm :: "[i,i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   713
    "pred_set_fm(A,x,r,B) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   714
       Forall(Iff(Member(0,succ(B)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   715
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   716
                             And(Member(1,succ(succ(A))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   717
                                 pair_fm(1,succ(succ(x)),0))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   718
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   719
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   720
lemma pred_set_type [TC]:
13429
wenzelm
parents: 13428
diff changeset
   721
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   722
      ==> pred_set_fm(A,x,r,B) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   723
by (simp add: pred_set_fm_def)
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   724
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   725
lemma sats_pred_set_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   726
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   727
    ==> sats(A, pred_set_fm(U,x,r,B), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   728
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   729
by (simp add: pred_set_fm_def pred_set_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   730
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   731
lemma pred_set_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   732
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   733
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   734
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   735
by (simp add: sats_pred_set_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   736
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   737
theorem pred_set_reflection:
13429
wenzelm
parents: 13428
diff changeset
   738
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
wenzelm
parents: 13428
diff changeset
   739
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]"
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   740
apply (simp only: pred_set_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   741
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   742
done
13304
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   743
43ef6c6dd906 more separation instances
paulson
parents: 13299
diff changeset
   744
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
   745
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   746
subsubsection{*Domain of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   747
13429
wenzelm
parents: 13428
diff changeset
   748
(* "is_domain(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   749
        \<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   750
constdefs domain_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   751
    "domain_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   752
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   753
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   754
                             Exists(pair_fm(2,0,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   755
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   756
lemma domain_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   757
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   758
by (simp add: domain_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   759
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   760
lemma sats_domain_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   761
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   762
    ==> sats(A, domain_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   763
        is_domain(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   764
by (simp add: domain_fm_def is_domain_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   765
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   766
lemma domain_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   767
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   768
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   769
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   770
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   771
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   772
theorem domain_reflection:
13429
wenzelm
parents: 13428
diff changeset
   773
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   774
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   775
apply (simp only: is_domain_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   776
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   777
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   778
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   779
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   780
subsubsection{*Range of a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   781
13429
wenzelm
parents: 13428
diff changeset
   782
(* "is_range(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   783
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   784
constdefs range_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   785
    "range_fm(r,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   786
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   787
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   788
                             Exists(pair_fm(0,2,1))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   789
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   790
lemma range_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   791
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   792
by (simp add: range_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   793
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   794
lemma sats_range_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   795
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   796
    ==> sats(A, range_fm(x,y), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   797
        is_range(**A, nth(x,env), nth(y,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   798
by (simp add: range_fm_def is_range_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   799
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   800
lemma range_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   801
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   802
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   803
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   804
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   805
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   806
theorem range_reflection:
13429
wenzelm
parents: 13428
diff changeset
   807
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   808
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   809
apply (simp only: is_range_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   810
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   811
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   812
13429
wenzelm
parents: 13428
diff changeset
   813
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   814
subsubsection{*Field of a Relation, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   815
13429
wenzelm
parents: 13428
diff changeset
   816
(* "is_field(M,r,z) ==
wenzelm
parents: 13428
diff changeset
   817
        \<exists>dr[M]. is_domain(M,r,dr) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   818
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   819
constdefs field_fm :: "[i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   820
    "field_fm(r,z) ==
wenzelm
parents: 13428
diff changeset
   821
       Exists(And(domain_fm(succ(r),0),
wenzelm
parents: 13428
diff changeset
   822
              Exists(And(range_fm(succ(succ(r)),0),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   823
                         union_fm(1,0,succ(succ(z)))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   824
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   825
lemma field_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   826
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   827
by (simp add: field_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   828
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   829
lemma sats_field_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   830
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   831
    ==> sats(A, field_fm(x,y), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   832
        is_field(**A, nth(x,env), nth(y,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   833
by (simp add: field_fm_def is_field_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   834
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   835
lemma field_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   836
      "[| nth(i,env) = x; nth(j,env) = y;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   837
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   838
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   839
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   840
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   841
theorem field_reflection:
13429
wenzelm
parents: 13428
diff changeset
   842
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   843
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   844
apply (simp only: is_field_def setclass_simps)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   845
apply (intro FOL_reflections domain_reflection range_reflection
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   846
             union_reflection)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   847
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   848
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   849
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   850
subsubsection{*Image under a Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   851
13429
wenzelm
parents: 13428
diff changeset
   852
(* "image(M,r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   853
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   854
constdefs image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   855
    "image_fm(r,A,z) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   856
       Forall(Iff(Member(0,succ(z)),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   857
                  Exists(And(Member(0,succ(succ(r))),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   858
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   859
                                        pair_fm(0,2,1)))))))"
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   860
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   861
lemma image_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   862
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   863
by (simp add: image_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   864
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   865
lemma sats_image_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   866
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   867
    ==> sats(A, image_fm(x,y,z), env) <->
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   868
        image(**A, nth(x,env), nth(y,env), nth(z,env))"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   869
by (simp add: image_fm_def Relative.image_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   870
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   871
lemma image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   872
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   873
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   874
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   875
by (simp add: sats_image_fm)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   876
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   877
theorem image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   878
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   879
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
   880
apply (simp only: Relative.image_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   881
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   882
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   883
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   884
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   885
subsubsection{*Pre-Image under a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   886
13429
wenzelm
parents: 13428
diff changeset
   887
(* "pre_image(M,r,A,z) ==
wenzelm
parents: 13428
diff changeset
   888
        \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   889
constdefs pre_image_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   890
    "pre_image_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   891
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   892
                  Exists(And(Member(0,succ(succ(r))),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   893
                             Exists(And(Member(0,succ(succ(succ(A)))),
13429
wenzelm
parents: 13428
diff changeset
   894
                                        pair_fm(2,0,1)))))))"
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   895
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   896
lemma pre_image_type [TC]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   897
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   898
by (simp add: pre_image_fm_def)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   899
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   900
lemma sats_pre_image_fm [simp]:
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   901
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   902
    ==> sats(A, pre_image_fm(x,y,z), env) <->
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   903
        pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   904
by (simp add: pre_image_fm_def Relative.pre_image_def)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   905
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   906
lemma pre_image_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   907
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   908
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   909
       ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   910
by (simp add: sats_pre_image_fm)
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   911
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   912
theorem pre_image_reflection:
13429
wenzelm
parents: 13428
diff changeset
   913
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   914
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   915
apply (simp only: Relative.pre_image_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   916
apply (intro FOL_reflections pair_reflection)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   917
done
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   918
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
   919
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   920
subsubsection{*Function Application, Internalized*}
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   921
13429
wenzelm
parents: 13428
diff changeset
   922
(* "fun_apply(M,f,x,y) ==
wenzelm
parents: 13428
diff changeset
   923
        (\<exists>xs[M]. \<exists>fxs[M].
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   924
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   925
constdefs fun_apply_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
   926
    "fun_apply_fm(f,x,y) ==
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   927
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
13429
wenzelm
parents: 13428
diff changeset
   928
                         And(image_fm(succ(succ(f)), 1, 0),
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   929
                             big_union_fm(0,succ(succ(y)))))))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   930
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   931
lemma fun_apply_type [TC]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   932
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   933
by (simp add: fun_apply_fm_def)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   934
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   935
lemma sats_fun_apply_fm [simp]:
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   936
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
   937
    ==> sats(A, fun_apply_fm(x,y,z), env) <->
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   938
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   939
by (simp add: fun_apply_fm_def fun_apply_def)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   940
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   941
lemma fun_apply_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   942
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   943
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   944
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   945
by simp
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   946
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   947
theorem fun_apply_reflection:
13429
wenzelm
parents: 13428
diff changeset
   948
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
wenzelm
parents: 13428
diff changeset
   949
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]"
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   950
apply (simp only: fun_apply_def setclass_simps)
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   951
apply (intro FOL_reflections upair_reflection image_reflection
13429
wenzelm
parents: 13428
diff changeset
   952
             big_union_reflection)
13352
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   953
done
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   954
3cd767f8d78b new definitions of fun_apply and M_is_recfun
paulson
parents: 13348
diff changeset
   955
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   956
subsubsection{*The Concept of Relation, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   957
13429
wenzelm
parents: 13428
diff changeset
   958
(* "is_relation(M,r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   959
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   960
constdefs relation_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
   961
    "relation_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   962
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   963
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   964
lemma relation_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   965
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
   966
by (simp add: relation_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   967
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   968
lemma sats_relation_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   969
   "[| x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   970
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   971
by (simp add: relation_fm_def is_relation_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   972
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   973
lemma relation_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
   974
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   975
          i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   976
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   977
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   978
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   979
theorem is_relation_reflection:
13429
wenzelm
parents: 13428
diff changeset
   980
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   981
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   982
apply (simp only: is_relation_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
   983
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
   984
done
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   985
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   986
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
   987
subsubsection{*The Concept of Function, Internalized*}
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   988
13429
wenzelm
parents: 13428
diff changeset
   989
(* "is_function(M,r) ==
wenzelm
parents: 13428
diff changeset
   990
        \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   991
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   992
constdefs function_fm :: "i=>i"
13429
wenzelm
parents: 13428
diff changeset
   993
    "function_fm(r) ==
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   994
       Forall(Forall(Forall(Forall(Forall(
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   995
         Implies(pair_fm(4,3,1),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   996
                 Implies(pair_fm(4,2,0),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   997
                         Implies(Member(1,r#+5),
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   998
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
   999
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1000
lemma function_type [TC]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1001
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1002
by (simp add: function_fm_def)
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1003
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1004
lemma sats_function_fm [simp]:
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1005
   "[| x \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1006
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1007
by (simp add: function_fm_def is_function_def)
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1008
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1009
lemma is_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1010
      "[| nth(i,env) = x; nth(j,env) = y;
13306
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1011
          i \<in> nat; env \<in> list(A)|]
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1012
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1013
by simp
6eebcddee32b more internalized formulas and separation proofs
paulson
parents: 13304
diff changeset
  1014
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1015
theorem is_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1016
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1017
               \<lambda>i x. is_function(**Lset(i),f(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1018
apply (simp only: is_function_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1019
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1020
done
13298
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
  1021
b4f370679c65 Constructible: some separation axioms
paulson
parents: 13291
diff changeset
  1022
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1023
subsubsection{*Typed Functions, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1024
13429
wenzelm
parents: 13428
diff changeset
  1025
(* "typed_function(M,A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1026
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1027
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1028
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1029
constdefs typed_function_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1030
    "typed_function_fm(A,B,r) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1031
       And(function_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1032
         And(relation_fm(r),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1033
           And(domain_fm(r,A),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1034
             Forall(Implies(Member(0,succ(r)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1035
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1036
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1037
lemma typed_function_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1038
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1039
by (simp add: typed_function_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1040
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1041
lemma sats_typed_function_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1042
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1043
    ==> sats(A, typed_function_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1044
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1045
by (simp add: typed_function_fm_def typed_function_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1046
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1047
lemma typed_function_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1048
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1049
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1050
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1051
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1052
13429
wenzelm
parents: 13428
diff changeset
  1053
lemmas function_reflections =
13363
c26eeb000470 instantiation of locales M_trancl and M_wfrank;
paulson
parents: 13352
diff changeset
  1054
        empty_reflection number1_reflection
13429
wenzelm
parents: 13428
diff changeset
  1055
        upair_reflection pair_reflection union_reflection
wenzelm
parents: 13428
diff changeset
  1056
        big_union_reflection cons_reflection successor_reflection
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1057
        fun_apply_reflection subset_reflection
13429
wenzelm
parents: 13428
diff changeset
  1058
        transitive_set_reflection membership_reflection
wenzelm
parents: 13428
diff changeset
  1059
        pred_set_reflection domain_reflection range_reflection field_reflection
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1060
        image_reflection pre_image_reflection
13429
wenzelm
parents: 13428
diff changeset
  1061
        is_relation_reflection is_function_reflection
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1062
13429
wenzelm
parents: 13428
diff changeset
  1063
lemmas function_iff_sats =
wenzelm
parents: 13428
diff changeset
  1064
        empty_iff_sats number1_iff_sats
wenzelm
parents: 13428
diff changeset
  1065
        upair_iff_sats pair_iff_sats union_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1066
        big_union_iff_sats cons_iff_sats successor_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1067
        fun_apply_iff_sats  Memrel_iff_sats
13429
wenzelm
parents: 13428
diff changeset
  1068
        pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
wenzelm
parents: 13428
diff changeset
  1069
        image_iff_sats pre_image_iff_sats
13505
52a16cb7fefb Relativized right up to L satisfies V=L!
paulson
parents: 13496
diff changeset
  1070
        relation_iff_sats is_function_iff_sats
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1071
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1072
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1073
theorem typed_function_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1074
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1075
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1076
apply (simp only: typed_function_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1077
apply (intro FOL_reflections function_reflections)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1078
done
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1079
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1080
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1081
subsubsection{*Composition of Relations, Internalized*}
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1082
13429
wenzelm
parents: 13428
diff changeset
  1083
(* "composition(M,r,s,t) ==
wenzelm
parents: 13428
diff changeset
  1084
        \<forall>p[M]. p \<in> t <->
wenzelm
parents: 13428
diff changeset
  1085
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
wenzelm
parents: 13428
diff changeset
  1086
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1087
                xy \<in> s & yz \<in> r)" *)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1088
constdefs composition_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1089
  "composition_fm(r,s,t) ==
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1090
     Forall(Iff(Member(0,succ(t)),
13429
wenzelm
parents: 13428
diff changeset
  1091
             Exists(Exists(Exists(Exists(Exists(
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1092
              And(pair_fm(4,2,5),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1093
               And(pair_fm(4,3,1),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1094
                And(pair_fm(3,2,0),
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1095
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1096
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1097
lemma composition_type [TC]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1098
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1099
by (simp add: composition_fm_def)
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1100
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1101
lemma sats_composition_fm [simp]:
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1102
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1103
    ==> sats(A, composition_fm(x,y,z), env) <->
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1104
        composition(**A, nth(x,env), nth(y,env), nth(z,env))"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1105
by (simp add: composition_fm_def composition_def)
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1106
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1107
lemma composition_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1108
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1109
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1110
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1111
by simp
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1112
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1113
theorem composition_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1114
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
13323
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1115
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
2c287f50c9f3 More relativization, reflection and proofs of separation
paulson
parents: 13316
diff changeset
  1116
apply (simp only: composition_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1117
apply (intro FOL_reflections pair_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1118
done
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1119
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1120
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1121
subsubsection{*Injections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1122
13429
wenzelm
parents: 13428
diff changeset
  1123
(* "injection(M,A,B,f) ==
wenzelm
parents: 13428
diff changeset
  1124
        typed_function(M,A,B,f) &
wenzelm
parents: 13428
diff changeset
  1125
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1126
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1127
constdefs injection_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1128
 "injection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1129
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1130
       Forall(Forall(Forall(Forall(Forall(
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1131
         Implies(pair_fm(4,2,1),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1132
                 Implies(pair_fm(3,2,0),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1133
                         Implies(Member(1,f#+5),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1134
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1135
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1136
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1137
lemma injection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1138
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1139
by (simp add: injection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1140
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1141
lemma sats_injection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1142
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1143
    ==> sats(A, injection_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1144
        injection(**A, nth(x,env), nth(y,env), nth(z,env))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1145
by (simp add: injection_fm_def injection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1146
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1147
lemma injection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1148
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1149
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1150
   ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1151
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1152
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1153
theorem injection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1154
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1155
               \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1156
apply (simp only: injection_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1157
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1158
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1159
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1160
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1161
subsubsection{*Surjections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1162
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1163
(*  surjection :: "[i=>o,i,i,i] => o"
13429
wenzelm
parents: 13428
diff changeset
  1164
    "surjection(M,A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1165
        typed_function(M,A,B,f) &
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1166
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1167
constdefs surjection_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1168
 "surjection_fm(A,B,f) ==
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1169
    And(typed_function_fm(A,B,f),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1170
       Forall(Implies(Member(0,succ(B)),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1171
                      Exists(And(Member(0,succ(succ(A))),
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1172
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1173
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1174
lemma surjection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1175
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1176
by (simp add: surjection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1177
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1178
lemma sats_surjection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1179
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1180
    ==> sats(A, surjection_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1181
        surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1182
by (simp add: surjection_fm_def surjection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1183
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1184
lemma surjection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1185
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1186
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1187
   ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1188
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1189
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1190
theorem surjection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1191
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1192
               \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1193
apply (simp only: surjection_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1194
apply (intro FOL_reflections function_reflections typed_function_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1195
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1196
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1197
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1198
13339
0f89104dd377 Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents: 13323
diff changeset
  1199
subsubsection{*Bijections, Internalized*}
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1200
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1201
(*   bijection :: "[i=>o,i,i,i] => o"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1202
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1203
constdefs bijection_fm :: "[i,i,i]=>i"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1204
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1205
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1206
lemma bijection_type [TC]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1207
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
13429
wenzelm
parents: 13428
diff changeset
  1208
by (simp add: bijection_fm_def)
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1209
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1210
lemma sats_bijection_fm [simp]:
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1211
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
13429
wenzelm
parents: 13428
diff changeset
  1212
    ==> sats(A, bijection_fm(x,y,z), env) <->
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1213
        bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1214
by (simp add: bijection_fm_def bijection_def)
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1215
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1216
lemma bijection_iff_sats:
13429
wenzelm
parents: 13428
diff changeset
  1217
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1218
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1219
   ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1220
by simp
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1221
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1222
theorem bijection_reflection:
13429
wenzelm
parents: 13428
diff changeset
  1223
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1224
               \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1225
apply (simp only: bijection_def setclass_simps)
13429
wenzelm
parents: 13428
diff changeset
  1226
apply (intro And_reflection injection_reflection surjection_reflection)
13314
84b9de3cbc91 Defining a meta-existential quantifier.
paulson
parents: 13309
diff changeset
  1227
done
13309
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1228
a6adee8ea75a reflection for more internal formulas
paulson
parents: 13306
diff changeset
  1229
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1230
subsubsection{*Restriction of a Relation, Internalized*}
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1231
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1232
13429
wenzelm
parents: 13428
diff changeset
  1233
(* "restriction(M,r,A,z) ==
wenzelm
parents: 13428
diff changeset
  1234
        \<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1235
constdefs restriction_fm :: "[i,i,i]=>i"
13429
wenzelm
parents: 13428
diff changeset
  1236
    "restriction_fm(r,A,z) ==
13348
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1237
       Forall(Iff(Member(0,succ(z)),
374d05460db4 Separation/Replacement up to M_wfrank!
paulson
parents: 13339
diff changeset
  1238
                  And(Member(0,succ(r)),