author paulson Thu Oct 17 10:54:11 2002 +0200 (2002-10-17) changeset 13651 ac80e101306a parent 13650 31bd2a8cdbe2 child 13652 172600c40793
Cosmetic changes suggested by writing the paper. Deleted some
redundant arity proofs
 src/ZF/Constructible/Formula.thy file | annotate | diff | revisions src/ZF/Constructible/Internalize.thy file | annotate | diff | revisions src/ZF/Constructible/L_axioms.thy file | annotate | diff | revisions src/ZF/Constructible/Rec_Separation.thy file | annotate | diff | revisions
```     1.1 --- a/src/ZF/Constructible/Formula.thy	Thu Oct 17 10:52:59 2002 +0200
1.2 +++ b/src/ZF/Constructible/Formula.thy	Thu Oct 17 10:54:11 2002 +0200
1.3 @@ -453,9 +453,9 @@
1.4  apply (simp add: DPow_def, blast)
1.5  done
1.6
1.7 -lemma singleton_in_DPow: "x \<in> A ==> {x} \<in> DPow(A)"
1.8 +lemma singleton_in_DPow: "a \<in> A ==> {a} \<in> DPow(A)"
1.10 -apply (rule_tac x="Cons(x,Nil)" in bexI)
1.11 +apply (rule_tac x="Cons(a,Nil)" in bexI)
1.12   apply (rule_tac x="Equal(0,1)" in bexI)
1.13    apply typecheck
1.14  apply (force simp add: succ_Un_distrib [symmetric])
1.15 @@ -473,16 +473,16 @@
1.16  apply (blast intro: cons_in_DPow)
1.17  done
1.18
1.19 -(*DPow is not monotonic.  For example, let A be some non-constructible set
1.20 -  of natural numbers, and let B be nat.  Then A<=B and obviously A : DPow(A)
1.21 -  but A ~: DPow(B).*)
1.22 -lemma DPow_mono: "A : DPow(B) ==> DPow(A) <= DPow(B)"
1.23 -apply (simp add: DPow_def, auto)
1.24 -(*must use the formula defining A in B to relativize the new formula...*)
1.25 +text{*@{term DPow} is not monotonic.  For example, let @{term A} be some
1.26 +non-constructible set of natural numbers, and let @{term B} be @{term nat}.
1.27 +Then @{term "A<=B"} and obviously @{term "A : DPow(A)"} but @{term "A ~:
1.28 +DPow(B)"}.*}
1.29 +
1.30 +(*This may be true but the proof looks difficult, requiring relativization
1.31 +lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
1.32 +apply (rule equalityI, safe)
1.33  oops
1.34 -
1.35 -lemma DPow_0: "DPow(0) = {0}"
1.36 -by (blast intro: empty_in_DPow dest: DPow_imp_subset)
1.37 +*)
1.38
1.39  lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)"
1.40  by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
1.41 @@ -493,14 +493,16 @@
1.42  apply (erule Finite_Pow_subset_Pow)
1.43  done
1.44
1.45 -(*This may be true but the proof looks difficult, requiring relativization
1.46 -lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
1.47 -apply (rule equalityI, safe)
1.48 -oops
1.49 -*)
1.50 +
1.51 +subsection{*Internalized Formulas for the Ordinals*}
1.52
1.53 -
1.54 -subsection{*Internalized formulas for basic concepts*}
1.55 +text{*The @{text sats} theorems below differ from the usual form in that they
1.56 +include an element of absoluteness.  That is, they relate internalized
1.57 +formulas to real concepts such as the subset relation, rather than to the
1.58 +relativized concepts defined in theory @{text Relative}.  This lets us prove
1.59 +the theorem as @{text Ords_in_DPow} without first having to instantiate the
1.60 +locale @{text M_trivial}.  Note that the present theory does not even take
1.61 +@{text Relative} as a parent.*}
1.62
1.63  subsubsection{*The subset relation*}
1.64
1.65 @@ -563,12 +565,25 @@
1.66  apply (blast intro: nth_type)
1.67  done
1.68
1.69 +text{*The subset consisting of the ordinals is definable.  Essential lemma for
1.70 +@{text Ord_in_Lset}.  This result is the objective of the present subsection.*}
1.71 +theorem Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
1.72 +apply (simp add: DPow_def Collect_subset)
1.73 +apply (rule_tac x=Nil in bexI)
1.74 + apply (rule_tac x="ordinal_fm(0)" in bexI)
1.76 +done
1.77 +
1.78
1.79  subsection{* Constant Lset: Levels of the Constructible Universe *}
1.80
1.81 -constdefs Lset :: "i=>i"
1.82 +constdefs
1.83 +  Lset :: "i=>i"
1.84      "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
1.85
1.86 +  L :: "i=>o" --{*Kunen's definition VI 1.5, page 167*}
1.87 +    "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
1.88 +
1.89  text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
1.90  lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
1.91  by (subst Lset_def [THEN def_transrec], simp)
1.92 @@ -601,7 +616,7 @@
1.93  apply (blast intro: elem_subset_in_DPow dest: DPowD)
1.94  done
1.95
1.96 -text{*Kunen's VI, 1.6 (a)*}
1.97 +text{*Kunen's VI 1.6 (a)*}
1.98  lemma Transset_Lset: "Transset(Lset(i))"
1.99  apply (rule_tac a=i in eps_induct)
1.100  apply (subst Lset)
1.101 @@ -615,7 +630,7 @@
1.102
1.103  subsubsection{* Monotonicity *}
1.104
1.105 -text{*Kunen's VI, 1.6 (b)*}
1.106 +text{*Kunen's VI 1.6 (b)*}
1.107  lemma Lset_mono [rule_format]:
1.108       "ALL j. i<=j --> Lset(i) <= Lset(j)"
1.109  apply (rule_tac a=i in eps_induct)
1.110 @@ -638,7 +653,7 @@
1.111  lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)"
1.112  by (blast dest: ltD [THEN Lset_mono_mem])
1.113
1.114 -subsubsection{* 0, successor and limit equations fof Lset *}
1.115 +subsubsection{* 0, successor and limit equations for Lset *}
1.116
1.117  lemma Lset_0 [simp]: "Lset(0) = 0"
1.118  by (subst Lset, blast)
1.119 @@ -705,15 +720,7 @@
1.120  done
1.121
1.122
1.123 -subsection{*Constructible Ordinals: Kunen's VI, 1.9 (b)*}
1.124 -
1.125 -text{*The subset consisting of the ordinals is definable.*}
1.126 -lemma Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
1.127 -apply (simp add: DPow_def Collect_subset)
1.128 -apply (rule_tac x=Nil in bexI)
1.129 - apply (rule_tac x="ordinal_fm(0)" in bexI)
1.131 -done
1.132 +subsection{*Constructible Ordinals: Kunen's VI 1.9 (b)*}
1.133
1.134  lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
1.135  apply (erule trans_induct3)
1.136 @@ -744,6 +751,9 @@
1.137         rule Ords_in_DPow [OF Transset_Lset])
1.138  done
1.139
1.140 +lemma Ord_in_L: "Ord(i) ==> L(i)"
1.141 +by (simp add: L_def, blast intro: Ord_in_Lset)
1.142 +
1.143  subsubsection{* Unions *}
1.144
1.145  lemma Union_in_Lset:
1.146 @@ -765,6 +775,12 @@
1.147  apply (blast intro: Limit_has_succ lt_LsetI Union_in_Lset)
1.148  done
1.149
1.150 +theorem Union_in_L: "L(X) ==> L(Union(X))"
1.151 +apply (simp add: L_def, clarify)
1.152 +apply (drule Ord_imp_greater_Limit)
1.153 +apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord)
1.154 +done
1.155 +
1.156  subsubsection{* Finite sets and ordered pairs *}
1.157
1.158  lemma singleton_in_Lset: "a: Lset(i) ==> {a} : Lset(succ(i))"
1.159 @@ -780,13 +796,6 @@
1.160  apply (blast intro: doubleton_in_Lset)
1.161  done
1.162
1.163 -lemma singleton_in_LLimit:
1.164 -    "[| a: Lset(i);  Limit(i) |] ==> {a} : Lset(i)"
1.165 -apply (erule Limit_LsetE, assumption)
1.166 -apply (erule singleton_in_Lset [THEN lt_LsetI])
1.167 -apply (blast intro: Limit_has_succ)
1.168 -done
1.169 -
1.170  lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
1.171  lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
1.172
1.173 @@ -799,6 +808,12 @@
1.174                      Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
1.175  done
1.176
1.177 +theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
1.178 +apply (simp add: L_def, clarify)
1.179 +apply (drule Ord2_imp_greater_Limit, assumption)
1.180 +apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord)
1.181 +done
1.182 +
1.183  lemma Pair_in_LLimit:
1.184      "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> : Lset(i)"
1.185  txt{*Infer that a, b occur at ordinals x,xa < i.*}
1.186 @@ -809,49 +824,11 @@
1.187                      Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
1.188  done
1.189
1.190 -lemma product_LLimit: "Limit(i) ==> Lset(i) * Lset(i) <= Lset(i)"
1.191 -by (blast intro: Pair_in_LLimit)
1.192 -
1.193 -lemmas Sigma_subset_LLimit = subset_trans [OF Sigma_mono product_LLimit]
1.194 -
1.195 -lemma nat_subset_LLimit: "Limit(i) ==> nat \<subseteq> Lset(i)"
1.196 -by (blast dest: Ord_subset_Lset nat_le_Limit le_imp_subset Limit_is_Ord)
1.197 -
1.198 -lemma nat_into_LLimit: "[| n: nat;  Limit(i) |] ==> n : Lset(i)"
1.199 -by (blast intro: nat_subset_LLimit [THEN subsetD])
1.200
1.201
1.202 -subsubsection{* Closure under disjoint union *}
1.203 -
1.204 -lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
1.205 -
1.206 -lemma one_in_LLimit: "Limit(i) ==> 1 : Lset(i)"
1.207 -by (blast intro: nat_into_LLimit)
1.208 -
1.209 -lemma Inl_in_LLimit:
1.210 -    "[| a: Lset(i); Limit(i) |] ==> Inl(a) : Lset(i)"
1.211 -apply (unfold Inl_def)
1.212 -apply (blast intro: zero_in_LLimit Pair_in_LLimit)
1.213 -done
1.214 -
1.215 -lemma Inr_in_LLimit:
1.216 -    "[| b: Lset(i); Limit(i) |] ==> Inr(b) : Lset(i)"
1.217 -apply (unfold Inr_def)
1.218 -apply (blast intro: one_in_LLimit Pair_in_LLimit)
1.219 -done
1.220 -
1.221 -lemma sum_LLimit: "Limit(i) ==> Lset(i) + Lset(i) <= Lset(i)"
1.222 -by (blast intro!: Inl_in_LLimit Inr_in_LLimit)
1.223 -
1.224 -lemmas sum_subset_LLimit = subset_trans [OF sum_mono sum_LLimit]
1.225 -
1.226 -
1.227 -text{*The constructible universe and its rank function*}
1.228 +text{*The rank function for the constructible universe*}
1.229  constdefs
1.230 -  L :: "i=>o" --{*Kunen's definition VI, 1.5, page 167*}
1.231 -    "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
1.232 -
1.233 -  lrank :: "i=>i" --{*Kunen's definition VI, 1.7*}
1.234 +  lrank :: "i=>i" --{*Kunen's definition VI 1.7*}
1.235      "lrank(x) == \<mu>i. x \<in> Lset(succ(i))"
1.236
1.237  lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
1.238 @@ -872,8 +849,9 @@
1.239  apply (blast intro: ltI Limit_is_Ord lt_trans)
1.240  done
1.241
1.242 -text{*Kunen's VI, 1.8, and the proof is much less trivial than the text
1.243 -would suggest.  For a start it need the previous lemma, proved by induction.*}
1.244 +text{*Kunen's VI 1.8.  The proof is much harder than the text would
1.245 +suggest.  For a start, it needs the previous lemma, which is proved by
1.246 +induction.*}
1.247  lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
1.248  apply (simp add: L_def, auto)
1.249   apply (blast intro: Lset_lrank_lt)
1.250 @@ -886,7 +864,7 @@
1.251  lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
1.253
1.254 -text{*Kunen's VI, 1.9 (a)*}
1.255 +text{*Kunen's VI 1.9 (a)*}
1.256  lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
1.257  apply (unfold lrank_def)
1.258  apply (rule Least_equality)
1.259 @@ -897,13 +875,10 @@
1.260  done
1.261
1.262
1.263 -lemma Ord_in_L: "Ord(i) ==> L(i)"
1.264 -by (blast intro: Ord_in_Lset L_I)
1.265 -
1.266  text{*This is lrank(lrank(a)) = lrank(a) *}
1.267  declare Ord_lrank [THEN lrank_of_Ord, simp]
1.268
1.269 -text{*Kunen's VI, 1.10 *}
1.270 +text{*Kunen's VI 1.10 *}
1.271  lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
1.272  apply (simp add: Lset_succ DPow_def)
1.273  apply (rule_tac x=Nil in bexI)
1.274 @@ -922,7 +897,7 @@
1.275  apply (blast intro!: le_imp_subset Lset_mono)
1.276  done
1.277
1.278 -text{*Kunen's VI, 1.11 *}
1.279 +text{*Kunen's VI 1.11 *}
1.280  lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
1.281  apply (erule trans_induct)
1.282  apply (subst Lset)
1.283 @@ -932,7 +907,7 @@
1.284  apply (rule Pow_mono, blast)
1.285  done
1.286
1.287 -text{*Kunen's VI, 1.12 *}
1.288 +text{*Kunen's VI 1.12 *}
1.289  lemma Lset_subset_Vset': "i \<in> nat ==> Lset(i) = Vset(i)";
1.290  apply (erule nat_induct)
1.292 @@ -950,21 +925,7 @@
1.293        ==> P"
1.294  by (blast dest: subset_Lset)
1.295
1.296 -subsection{*For L to satisfy the ZF axioms*}
1.297 -
1.298 -theorem Union_in_L: "L(X) ==> L(Union(X))"
1.299 -apply (simp add: L_def, clarify)
1.300 -apply (drule Ord_imp_greater_Limit)
1.301 -apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord)
1.302 -done
1.303 -
1.304 -theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
1.305 -apply (simp add: L_def, clarify)
1.306 -apply (drule Ord2_imp_greater_Limit, assumption)
1.307 -apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord)
1.308 -done
1.309 -
1.310 -subsubsection{*For L to satisfy Powerset *}
1.311 +subsubsection{*For L to satisfy the Powerset axiom *}
1.312
1.313  lemma LPow_env_typing:
1.314      "[| y : Lset(i); Ord(i); y \<subseteq> X |]
1.315 @@ -996,7 +957,6 @@
1.316
1.317  subsection{*Eliminating @{term arity} from the Definition of @{term Lset}*}
1.318
1.319 -
1.320  lemma nth_zero_eq_0: "n \<in> nat ==> nth(n,[0]) = 0"
1.321  by (induct_tac n, auto)
1.322
1.323 @@ -1046,7 +1006,7 @@
1.324  lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)"
1.325  apply (drule Transset_0_disj)
1.326  apply (erule disjE)
1.327 - apply (simp add: DPow'_0 DPow_0)
1.328 + apply (simp add: DPow'_0 Finite_DPow_eq_Pow)
1.329  apply (rule equalityI)
1.330   apply (rule DPow_subset_DPow')
1.331  apply (erule DPow'_subset_DPow)
```
```     2.1 --- a/src/ZF/Constructible/Internalize.thy	Thu Oct 17 10:52:59 2002 +0200
2.2 +++ b/src/ZF/Constructible/Internalize.thy	Thu Oct 17 10:54:11 2002 +0200
2.3 @@ -726,11 +726,6 @@
2.4       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
2.6
2.7 -lemma arity_cartprod_fm [simp]:
2.8 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
2.9 -      ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
2.10 -by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
2.11 -
2.12  lemma sats_cartprod_fm [simp]:
2.13     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
2.14      ==> sats(A, cartprod_fm(x,y,z), env) <->
2.15 @@ -770,11 +765,6 @@
2.16       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
2.18
2.19 -lemma arity_sum_fm [simp]:
2.20 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
2.21 -      ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
2.22 -by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
2.23 -
2.24  lemma sats_sum_fm [simp]:
2.25     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
2.26      ==> sats(A, sum_fm(x,y,z), env) <->
2.27 @@ -805,10 +795,6 @@
2.28       "x \<in> nat ==> quasinat_fm(x) \<in> formula"
2.30
2.31 -lemma arity_quasinat_fm [simp]:
2.32 -     "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
2.33 -by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
2.34 -
2.35  lemma sats_quasinat_fm [simp]:
2.36     "[| x \<in> nat; env \<in> list(A)|]
2.37      ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
2.38 @@ -1081,11 +1067,6 @@
2.39       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
2.41
2.42 -lemma arity_list_functor_fm [simp]:
2.43 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
2.44 -      ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
2.45 -by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
2.46 -
2.47  lemma sats_list_functor_fm [simp]:
2.48     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
2.49      ==> sats(A, list_functor_fm(x,y,z), env) <->
```
```     3.1 --- a/src/ZF/Constructible/L_axioms.thy	Thu Oct 17 10:52:59 2002 +0200
3.2 +++ b/src/ZF/Constructible/L_axioms.thy	Thu Oct 17 10:54:11 2002 +0200
3.3 @@ -88,8 +88,6 @@
3.4  lemma Lset_cont: "cont_Ord(Lset)"
3.5  by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
3.6
3.7 -lemmas Pair_in_Lset = Formula.Pair_in_LLimit
3.8 -
3.9  lemmas L_nat = Ord_in_L [OF Ord_nat]
3.10
3.11  theorem M_trivial_L: "PROP M_trivial(L)"
3.12 @@ -260,8 +258,9 @@
3.13
3.14
3.15  lemma reflection_Lset: "reflection(Lset)"
3.16 -apply (blast intro: reflection.intro Lset_mono_le Lset_cont Pair_in_Lset) +
3.17 -done
3.18 +by (blast intro: reflection.intro Lset_mono_le Lset_cont
3.19 +                 Formula.Pair_in_LLimit)+
3.20 +
3.21
3.22  theorem Ex_reflection:
3.23       "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
3.24 @@ -370,10 +369,6 @@
3.25       "x \<in> nat ==> empty_fm(x) \<in> formula"
3.27
3.28 -lemma arity_empty_fm [simp]:
3.29 -     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
3.30 -by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac)
3.31 -
3.32  lemma sats_empty_fm [simp]:
3.33     "[| x \<in> nat; env \<in> list(A)|]
3.34      ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
3.35 @@ -416,11 +411,6 @@
3.36       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
3.38
3.39 -lemma arity_upair_fm [simp]:
3.40 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.41 -      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.42 -by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
3.43 -
3.44  lemma sats_upair_fm [simp]:
3.45     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.46      ==> sats(A, upair_fm(x,y,z), env) <->
3.47 @@ -462,11 +452,6 @@
3.48       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
3.50
3.51 -lemma arity_pair_fm [simp]:
3.52 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.53 -      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.54 -by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
3.55 -
3.56  lemma sats_pair_fm [simp]:
3.57     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.58      ==> sats(A, pair_fm(x,y,z), env) <->
3.59 @@ -498,11 +483,6 @@
3.60       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
3.62
3.63 -lemma arity_union_fm [simp]:
3.64 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.65 -      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.66 -by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac)
3.67 -
3.68  lemma sats_union_fm [simp]:
3.69     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.70      ==> sats(A, union_fm(x,y,z), env) <->
3.71 @@ -535,11 +515,6 @@
3.72       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
3.74
3.75 -lemma arity_cons_fm [simp]:
3.76 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.77 -      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.78 -by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac)
3.79 -
3.80  lemma sats_cons_fm [simp]:
3.81     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.82      ==> sats(A, cons_fm(x,y,z), env) <->
3.83 @@ -569,11 +544,6 @@
3.84       "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
3.86
3.87 -lemma arity_succ_fm [simp]:
3.88 -     "[| x \<in> nat; y \<in> nat |]
3.89 -      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
3.91 -
3.92  lemma sats_succ_fm [simp]:
3.93     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.94      ==> sats(A, succ_fm(x,y), env) <->
3.95 @@ -604,10 +574,6 @@
3.96       "x \<in> nat ==> number1_fm(x) \<in> formula"
3.98
3.99 -lemma arity_number1_fm [simp]:
3.100 -     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
3.101 -by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac)
3.102 -
3.103  lemma sats_number1_fm [simp]:
3.104     "[| x \<in> nat; env \<in> list(A)|]
3.105      ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
3.106 @@ -639,11 +605,6 @@
3.107       "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
3.109
3.110 -lemma arity_big_union_fm [simp]:
3.111 -     "[| x \<in> nat; y \<in> nat |]
3.112 -      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
3.113 -by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
3.114 -
3.115  lemma sats_big_union_fm [simp]:
3.116     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.117      ==> sats(A, big_union_fm(x,y), env) <->
3.118 @@ -666,8 +627,11 @@
3.119
3.120  subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
3.121
3.122 -text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
3.123 -
3.124 +text{*The @{text sats} theorems below are standard versions of the ones proved
3.125 +in theory @{text Formula}.  They relate elements of type @{term formula} to
3.126 +relativized concepts such as @{term subset} or @{term ordinal} rather than to
3.127 +real concepts such as @{term Ord}.  Now that we have instantiated the locale
3.128 +@{text M_trivial}, we no longer require the earlier versions.*}
3.129
3.130  lemma sats_subset_fm':
3.131     "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.132 @@ -724,11 +688,6 @@
3.133       "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
3.135
3.136 -lemma arity_Memrel_fm [simp]:
3.137 -     "[| x \<in> nat; y \<in> nat |]
3.138 -      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
3.139 -by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac)
3.140 -
3.141  lemma sats_Memrel_fm [simp]:
3.142     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.143      ==> sats(A, Memrel_fm(x,y), env) <->
3.144 @@ -763,11 +722,6 @@
3.145        ==> pred_set_fm(A,x,r,B) \<in> formula"
3.147
3.148 -lemma arity_pred_set_fm [simp]:
3.149 -   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
3.150 -    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
3.151 -by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac)
3.152 -
3.153  lemma sats_pred_set_fm [simp]:
3.154     "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
3.155      ==> sats(A, pred_set_fm(U,x,r,B), env) <->
3.156 @@ -803,11 +757,6 @@
3.157       "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
3.159
3.160 -lemma arity_domain_fm [simp]:
3.161 -     "[| x \<in> nat; y \<in> nat |]
3.162 -      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
3.163 -by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac)
3.164 -
3.165  lemma sats_domain_fm [simp]:
3.166     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.167      ==> sats(A, domain_fm(x,y), env) <->
3.168 @@ -842,11 +791,6 @@
3.169       "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
3.171
3.172 -lemma arity_range_fm [simp]:
3.173 -     "[| x \<in> nat; y \<in> nat |]
3.174 -      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
3.175 -by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac)
3.176 -
3.177  lemma sats_range_fm [simp]:
3.178     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.179      ==> sats(A, range_fm(x,y), env) <->
3.180 @@ -882,11 +826,6 @@
3.181       "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
3.183
3.184 -lemma arity_field_fm [simp]:
3.185 -     "[| x \<in> nat; y \<in> nat |]
3.186 -      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
3.187 -by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac)
3.188 -
3.189  lemma sats_field_fm [simp]:
3.190     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
3.191      ==> sats(A, field_fm(x,y), env) <->
3.192 @@ -923,11 +862,6 @@
3.193       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
3.195
3.196 -lemma arity_image_fm [simp]:
3.197 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.198 -      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.199 -by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac)
3.200 -
3.201  lemma sats_image_fm [simp]:
3.202     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.203      ==> sats(A, image_fm(x,y,z), env) <->
3.204 @@ -963,11 +897,6 @@
3.205       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
3.207
3.208 -lemma arity_pre_image_fm [simp]:
3.209 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.210 -      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.211 -by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)
3.212 -
3.213  lemma sats_pre_image_fm [simp]:
3.214     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.215      ==> sats(A, pre_image_fm(x,y,z), env) <->
3.216 @@ -1003,11 +932,6 @@
3.217       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
3.219
3.220 -lemma arity_fun_apply_fm [simp]:
3.221 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.222 -      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.223 -by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac)
3.224 -
3.225  lemma sats_fun_apply_fm [simp]:
3.226     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.227      ==> sats(A, fun_apply_fm(x,y,z), env) <->
3.228 @@ -1041,10 +965,6 @@
3.229       "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
3.231
3.232 -lemma arity_relation_fm [simp]:
3.233 -     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
3.234 -by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac)
3.235 -
3.236  lemma sats_relation_fm [simp]:
3.237     "[| x \<in> nat; env \<in> list(A)|]
3.238      ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
3.239 @@ -1081,10 +1001,6 @@
3.240       "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
3.242
3.243 -lemma arity_function_fm [simp]:
3.244 -     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
3.245 -by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac)
3.246 -
3.247  lemma sats_function_fm [simp]:
3.248     "[| x \<in> nat; env \<in> list(A)|]
3.249      ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
3.250 @@ -1122,11 +1038,6 @@
3.251       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
3.253
3.254 -lemma arity_typed_function_fm [simp]:
3.255 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.256 -      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.257 -by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac)
3.258 -
3.259  lemma sats_typed_function_fm [simp]:
3.260     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.261      ==> sats(A, typed_function_fm(x,y,z), env) <->
3.262 @@ -1187,11 +1098,6 @@
3.263       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
3.265
3.266 -lemma arity_composition_fm [simp]:
3.267 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.268 -      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.269 -by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac)
3.270 -
3.271  lemma sats_composition_fm [simp]:
3.272     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.273      ==> sats(A, composition_fm(x,y,z), env) <->
3.274 @@ -1232,11 +1138,6 @@
3.275       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
3.277
3.278 -lemma arity_injection_fm [simp]:
3.279 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.280 -      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.281 -by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac)
3.282 -
3.283  lemma sats_injection_fm [simp]:
3.284     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.285      ==> sats(A, injection_fm(x,y,z), env) <->
3.286 @@ -1274,11 +1175,6 @@
3.287       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
3.289
3.290 -lemma arity_surjection_fm [simp]:
3.291 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.292 -      ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.293 -by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac)
3.294 -
3.295  lemma sats_surjection_fm [simp]:
3.296     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.297      ==> sats(A, surjection_fm(x,y,z), env) <->
3.298 @@ -1311,11 +1207,6 @@
3.299       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
3.301
3.302 -lemma arity_bijection_fm [simp]:
3.303 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.304 -      ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.305 -by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac)
3.306 -
3.307  lemma sats_bijection_fm [simp]:
3.308     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.309      ==> sats(A, bijection_fm(x,y,z), env) <->
3.310 @@ -1352,11 +1243,6 @@
3.311       "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
3.313
3.314 -lemma arity_restriction_fm [simp]:
3.315 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
3.316 -      ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
3.317 -by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac)
3.318 -
3.319  lemma sats_restriction_fm [simp]:
3.320     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
3.321      ==> sats(A, restriction_fm(x,y,z), env) <->
3.322 @@ -1404,12 +1290,6 @@
3.323        ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
3.325
3.326 -lemma arity_order_isomorphism_fm [simp]:
3.327 -     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
3.328 -      ==> arity(order_isomorphism_fm(A,r,B,s,f)) =
3.329 -          succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)"
3.330 -by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac)
3.331 -
3.332  lemma sats_order_isomorphism_fm [simp]:
3.333     "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
3.334      ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <->
3.335 @@ -1452,10 +1332,6 @@
3.336       "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
3.338
3.339 -lemma arity_limit_ordinal_fm [simp]:
3.340 -     "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
3.341 -by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac)
3.342 -
3.343  lemma sats_limit_ordinal_fm [simp]:
3.344     "[| x \<in> nat; env \<in> list(A)|]
3.345      ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
3.346 @@ -1523,10 +1399,6 @@
3.347       "x \<in> nat ==> omega_fm(x) \<in> formula"
3.349
3.350 -lemma arity_omega_fm [simp]:
3.351 -     "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
3.352 -by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac)
3.353 -
3.354  lemma sats_omega_fm [simp]:
3.355     "[| x \<in> nat; env \<in> list(A)|]
3.356      ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
```
```     4.1 --- a/src/ZF/Constructible/Rec_Separation.thy	Thu Oct 17 10:52:59 2002 +0200
4.2 +++ b/src/ZF/Constructible/Rec_Separation.thy	Thu Oct 17 10:54:11 2002 +0200
4.3 @@ -53,11 +53,6 @@
4.4   "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
4.6
4.7 -lemma arity_rtran_closure_mem_fm [simp]:
4.8 -     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
4.9 -      ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
4.10 -by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
4.11 -
4.12  lemma sats_rtran_closure_mem_fm [simp]:
4.13     "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
4.14      ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
4.15 @@ -103,11 +98,6 @@
4.16       "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
4.18
4.19 -lemma arity_rtran_closure_fm [simp]:
4.20 -     "[| x \<in> nat; y \<in> nat |]
4.21 -      ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
4.22 -by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
4.23 -
4.24  lemma sats_rtran_closure_fm [simp]:
4.25     "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
4.26      ==> sats(A, rtran_closure_fm(x,y), env) <->
4.27 @@ -140,11 +130,6 @@
4.28       "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"