src/ZF/Univ.thy
 changeset 46820 c656222c4dc1 parent 45602 2a858377c3d2 child 46821 ff6b0c1087f2
```     1.1 --- a/src/ZF/Univ.thy	Sun Mar 04 23:20:43 2012 +0100
1.2 +++ b/src/ZF/Univ.thy	Tue Mar 06 15:15:49 2012 +0000
1.3 @@ -15,7 +15,7 @@
1.4
1.5  definition
1.6    Vfrom       :: "[i,i]=>i"  where
1.7 -    "Vfrom(A,i) == transrec(i, %x f. A Un (\<Union>y\<in>x. Pow(f`y)))"
1.8 +    "Vfrom(A,i) == transrec(i, %x f. A \<union> (\<Union>y\<in>x. Pow(f`y)))"
1.9
1.10  abbreviation
1.11    Vset :: "i=>i" where
1.12 @@ -24,13 +24,13 @@
1.13
1.14  definition
1.15    Vrec        :: "[i, [i,i]=>i] =>i"  where
1.16 -    "Vrec(a,H) == transrec(rank(a), %x g. lam z: Vset(succ(x)).
1.17 -                           H(z, lam w:Vset(x). g`rank(w)`w)) ` a"
1.18 +    "Vrec(a,H) == transrec(rank(a), %x g. \<lambda>z\<in>Vset(succ(x)).
1.19 +                           H(z, \<lambda>w\<in>Vset(x). g`rank(w)`w)) ` a"
1.20
1.21  definition
1.22    Vrecursor   :: "[[i,i]=>i, i] =>i"  where
1.23 -    "Vrecursor(H,a) == transrec(rank(a), %x g. lam z: Vset(succ(x)).
1.24 -                                H(lam w:Vset(x). g`rank(w)`w, z)) ` a"
1.25 +    "Vrecursor(H,a) == transrec(rank(a), %x g. \<lambda>z\<in>Vset(succ(x)).
1.26 +                                H(\<lambda>w\<in>Vset(x). g`rank(w)`w, z)) ` a"
1.27
1.28  definition
1.29    univ        :: "i=>i"  where
1.30 @@ -40,30 +40,30 @@
1.31  subsection{*Immediate Consequences of the Definition of @{term "Vfrom(A,i)"}*}
1.32
1.33  text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
1.34 -lemma Vfrom: "Vfrom(A,i) = A Un (\<Union>j\<in>i. Pow(Vfrom(A,j)))"
1.35 +lemma Vfrom: "Vfrom(A,i) = A \<union> (\<Union>j\<in>i. Pow(Vfrom(A,j)))"
1.36  by (subst Vfrom_def [THEN def_transrec], simp)
1.37
1.38  subsubsection{* Monotonicity *}
1.39
1.40  lemma Vfrom_mono [rule_format]:
1.41 -     "A<=B ==> \<forall>j. i<=j --> Vfrom(A,i) <= Vfrom(B,j)"
1.42 +     "A<=B ==> \<forall>j. i<=j \<longrightarrow> Vfrom(A,i) \<subseteq> Vfrom(B,j)"
1.43  apply (rule_tac a=i in eps_induct)
1.44  apply (rule impI [THEN allI])
1.45  apply (subst Vfrom [of A])
1.46  apply (subst Vfrom [of B])
1.47  apply (erule Un_mono)
1.48 -apply (erule UN_mono, blast)
1.49 +apply (erule UN_mono, blast)
1.50  done
1.51
1.52  lemma VfromI: "[| a \<in> Vfrom(A,j);  j<i |] ==> a \<in> Vfrom(A,i)"
1.53 -by (blast dest: Vfrom_mono [OF subset_refl le_imp_subset [OF leI]])
1.54 +by (blast dest: Vfrom_mono [OF subset_refl le_imp_subset [OF leI]])
1.55
1.56
1.57  subsubsection{* A fundamental equality: Vfrom does not require ordinals! *}
1.58
1.59
1.60
1.61 -lemma Vfrom_rank_subset1: "Vfrom(A,x) <= Vfrom(A,rank(x))"
1.62 +lemma Vfrom_rank_subset1: "Vfrom(A,x) \<subseteq> Vfrom(A,rank(x))"
1.63  proof (induct x rule: eps_induct)
1.64    fix x
1.65    assume "\<forall>y\<in>x. Vfrom(A,y) \<subseteq> Vfrom(A,rank(y))"
1.66 @@ -72,7 +72,7 @@
1.67          blast intro!: rank_lt [THEN ltD])
1.68  qed
1.69
1.70 -lemma Vfrom_rank_subset2: "Vfrom(A,rank(x)) <= Vfrom(A,x)"
1.71 +lemma Vfrom_rank_subset2: "Vfrom(A,rank(x)) \<subseteq> Vfrom(A,x)"
1.72  apply (rule_tac a=x in eps_induct)
1.73  apply (subst Vfrom)
1.74  apply (subst Vfrom, rule subset_refl [THEN Un_mono])
1.75 @@ -99,19 +99,19 @@
1.76  lemma zero_in_Vfrom: "y:x ==> 0 \<in> Vfrom(A,x)"
1.77  by (subst Vfrom, blast)
1.78
1.79 -lemma i_subset_Vfrom: "i <= Vfrom(A,i)"
1.80 +lemma i_subset_Vfrom: "i \<subseteq> Vfrom(A,i)"
1.81  apply (rule_tac a=i in eps_induct)
1.82  apply (subst Vfrom, blast)
1.83  done
1.84
1.85 -lemma A_subset_Vfrom: "A <= Vfrom(A,i)"
1.86 +lemma A_subset_Vfrom: "A \<subseteq> Vfrom(A,i)"
1.87  apply (subst Vfrom)
1.88  apply (rule Un_upper1)
1.89  done
1.90
1.91  lemmas A_into_Vfrom = A_subset_Vfrom [THEN subsetD]
1.92
1.93 -lemma subset_mem_Vfrom: "a <= Vfrom(A,i) ==> a \<in> Vfrom(A,succ(i))"
1.94 +lemma subset_mem_Vfrom: "a \<subseteq> Vfrom(A,i) ==> a \<in> Vfrom(A,succ(i))"
1.95  by (subst Vfrom, blast)
1.96
1.97  subsubsection{* Finite sets and ordered pairs *}
1.98 @@ -126,13 +126,13 @@
1.99  lemma Pair_in_Vfrom:
1.100      "[| a \<in> Vfrom(A,i);  b \<in> Vfrom(A,i) |] ==> <a,b> \<in> Vfrom(A,succ(succ(i)))"
1.101  apply (unfold Pair_def)
1.102 -apply (blast intro: doubleton_in_Vfrom)
1.103 +apply (blast intro: doubleton_in_Vfrom)
1.104  done
1.105
1.106 -lemma succ_in_Vfrom: "a <= Vfrom(A,i) ==> succ(a) \<in> Vfrom(A,succ(succ(i)))"
1.107 +lemma succ_in_Vfrom: "a \<subseteq> Vfrom(A,i) ==> succ(a) \<in> Vfrom(A,succ(succ(i)))"
1.108  apply (intro subset_mem_Vfrom succ_subsetI, assumption)
1.109 -apply (erule subset_trans)
1.110 -apply (rule Vfrom_mono [OF subset_refl subset_succI])
1.111 +apply (erule subset_trans)
1.112 +apply (rule Vfrom_mono [OF subset_refl subset_succI])
1.113  done
1.114
1.115  subsection{* 0, Successor and Limit Equations for @{term Vfrom} *}
1.116 @@ -140,9 +140,9 @@
1.117  lemma Vfrom_0: "Vfrom(A,0) = A"
1.118  by (subst Vfrom, blast)
1.119
1.120 -lemma Vfrom_succ_lemma: "Ord(i) ==> Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"
1.121 +lemma Vfrom_succ_lemma: "Ord(i) ==> Vfrom(A,succ(i)) = A \<union> Pow(Vfrom(A,i))"
1.122  apply (rule Vfrom [THEN trans])
1.123 -apply (rule equalityI [THEN subst_context,
1.124 +apply (rule equalityI [THEN subst_context,
1.125                         OF _ succI1 [THEN RepFunI, THEN Union_upper]])
1.126  apply (rule UN_least)
1.127  apply (rule subset_refl [THEN Vfrom_mono, THEN Pow_mono])
1.128 @@ -150,7 +150,7 @@
1.129  apply (erule Ord_succ)
1.130  done
1.131
1.132 -lemma Vfrom_succ: "Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"
1.133 +lemma Vfrom_succ: "Vfrom(A,succ(i)) = A \<union> Pow(Vfrom(A,i))"
1.134  apply (rule_tac x1 = "succ (i)" in Vfrom_rank_eq [THEN subst])
1.135  apply (rule_tac x1 = i in Vfrom_rank_eq [THEN subst])
1.136  apply (subst rank_succ)
1.137 @@ -158,8 +158,8 @@
1.138  done
1.139
1.140  (*The premise distinguishes this from Vfrom(A,0);  allowing X=0 forces
1.141 -  the conclusion to be Vfrom(A,Union(X)) = A Un (\<Union>y\<in>X. Vfrom(A,y)) *)
1.142 -lemma Vfrom_Union: "y:X ==> Vfrom(A,Union(X)) = (\<Union>y\<in>X. Vfrom(A,y))"
1.143 +  the conclusion to be Vfrom(A,\<Union>(X)) = A \<union> (\<Union>y\<in>X. Vfrom(A,y)) *)
1.144 +lemma Vfrom_Union: "y:X ==> Vfrom(A,\<Union>(X)) = (\<Union>y\<in>X. Vfrom(A,y))"
1.145  apply (subst Vfrom)
1.146  apply (rule equalityI)
1.147  txt{*first inclusion*}
1.148 @@ -179,11 +179,11 @@
1.149  subsection{* @{term Vfrom} applied to Limit Ordinals *}
1.150
1.151  (*NB. limit ordinals are non-empty:
1.152 -      Vfrom(A,0) = A = A Un (\<Union>y\<in>0. Vfrom(A,y)) *)
1.153 +      Vfrom(A,0) = A = A \<union> (\<Union>y\<in>0. Vfrom(A,y)) *)
1.154  lemma Limit_Vfrom_eq:
1.155      "Limit(i) ==> Vfrom(A,i) = (\<Union>y\<in>i. Vfrom(A,y))"
1.156  apply (rule Limit_has_0 [THEN ltD, THEN Vfrom_Union, THEN subst], assumption)
1.159  done
1.160
1.161  lemma Limit_VfromE:
1.162 @@ -193,7 +193,7 @@
1.163  apply (rule classical)
1.164  apply (rule Limit_Vfrom_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
1.165    prefer 2 apply assumption
1.166 - apply blast
1.167 + apply blast
1.168  apply (blast intro: ltI Limit_is_Ord)
1.169  done
1.170
1.171 @@ -201,12 +201,12 @@
1.172      "[| a \<in> Vfrom(A,i);  Limit(i) |] ==> {a} \<in> Vfrom(A,i)"
1.173  apply (erule Limit_VfromE, assumption)
1.174  apply (erule singleton_in_Vfrom [THEN VfromI])
1.175 -apply (blast intro: Limit_has_succ)
1.176 +apply (blast intro: Limit_has_succ)
1.177  done
1.178
1.179 -lemmas Vfrom_UnI1 =
1.180 +lemmas Vfrom_UnI1 =
1.181      Un_upper1 [THEN subset_refl [THEN Vfrom_mono, THEN subsetD]]
1.182 -lemmas Vfrom_UnI2 =
1.183 +lemmas Vfrom_UnI2 =
1.184      Un_upper2 [THEN subset_refl [THEN Vfrom_mono, THEN subsetD]]
1.185
1.186  text{*Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)*}
1.187 @@ -223,12 +223,12 @@
1.188  txt{*Infer that a, b occur at ordinals x,xa < i.*}
1.189  apply (erule Limit_VfromE, assumption)
1.190  apply (erule Limit_VfromE, assumption)
1.191 -txt{*Infer that succ(succ(x Un xa)) < i *}
1.192 +txt{*Infer that @{term"succ(succ(x \<union> xa)) < i"} *}
1.193  apply (blast intro: VfromI [OF Pair_in_Vfrom]
1.194                      Vfrom_UnI1 Vfrom_UnI2 Limit_has_succ Un_least_lt)
1.195  done
1.196
1.197 -lemma product_VLimit: "Limit(i) ==> Vfrom(A,i) * Vfrom(A,i) <= Vfrom(A,i)"
1.198 +lemma product_VLimit: "Limit(i) ==> Vfrom(A,i) * Vfrom(A,i) \<subseteq> Vfrom(A,i)"
1.199  by (blast intro: Pair_in_VLimit)
1.200
1.201  lemmas Sigma_subset_VLimit =
1.202 @@ -259,7 +259,7 @@
1.203  apply (blast intro: one_in_VLimit Pair_in_VLimit)
1.204  done
1.205
1.206 -lemma sum_VLimit: "Limit(i) ==> Vfrom(C,i)+Vfrom(C,i) <= Vfrom(C,i)"
1.207 +lemma sum_VLimit: "Limit(i) ==> Vfrom(C,i)+Vfrom(C,i) \<subseteq> Vfrom(C,i)"
1.208  by (blast intro!: Inl_in_VLimit Inr_in_VLimit)
1.209
1.210  lemmas sum_subset_VLimit = subset_trans [OF sum_mono sum_VLimit]
1.211 @@ -283,11 +283,11 @@
1.212  apply (erule Transset_Vfrom [THEN Transset_iff_Pow [THEN iffD1]])
1.213  done
1.214
1.215 -lemma Transset_Pair_subset: "[| <a,b> <= C; Transset(C) |] ==> a: C & b: C"
1.216 +lemma Transset_Pair_subset: "[| <a,b> \<subseteq> C; Transset(C) |] ==> a: C & b: C"
1.217  by (unfold Pair_def Transset_def, blast)
1.218
1.219  lemma Transset_Pair_subset_VLimit:
1.220 -     "[| <a,b> <= Vfrom(A,i);  Transset(A);  Limit(i) |]
1.221 +     "[| <a,b> \<subseteq> Vfrom(A,i);  Transset(A);  Limit(i) |]
1.222        ==> <a,b> \<in> Vfrom(A,i)"
1.223  apply (erule Transset_Pair_subset [THEN conjE])
1.224  apply (erule Transset_Vfrom)
1.225 @@ -295,14 +295,14 @@
1.226  done
1.227
1.228  lemma Union_in_Vfrom:
1.229 -     "[| X \<in> Vfrom(A,j);  Transset(A) |] ==> Union(X) \<in> Vfrom(A, succ(j))"
1.230 +     "[| X \<in> Vfrom(A,j);  Transset(A) |] ==> \<Union>(X) \<in> Vfrom(A, succ(j))"
1.231  apply (drule Transset_Vfrom)
1.232  apply (rule subset_mem_Vfrom)
1.233  apply (unfold Transset_def, blast)
1.234  done
1.235
1.236  lemma Union_in_VLimit:
1.237 -     "[| X \<in> Vfrom(A,i);  Limit(i);  Transset(A) |] ==> Union(X) \<in> Vfrom(A,i)"
1.238 +     "[| X \<in> Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \<Union>(X) \<in> Vfrom(A,i)"
1.239  apply (rule Limit_VfromE, assumption+)
1.240  apply (blast intro: Limit_has_succ VfromI Union_in_Vfrom)
1.241  done
1.242 @@ -317,15 +317,15 @@
1.243  lemma in_VLimit:
1.244    "[| a \<in> Vfrom(A,i);  b \<in> Vfrom(A,i);  Limit(i);
1.245        !!x y j. [| j<i; 1:j; x \<in> Vfrom(A,j); y \<in> Vfrom(A,j) |]
1.246 -               ==> EX k. h(x,y) \<in> Vfrom(A,k) & k<i |]
1.247 +               ==> \<exists>k. h(x,y) \<in> Vfrom(A,k) & k<i |]
1.248     ==> h(a,b) \<in> Vfrom(A,i)"
1.249  txt{*Infer that a, b occur at ordinals x,xa < i.*}
1.250  apply (erule Limit_VfromE, assumption)
1.251  apply (erule Limit_VfromE, assumption, atomize)
1.252 -apply (drule_tac x=a in spec)
1.253 -apply (drule_tac x=b in spec)
1.254 -apply (drule_tac x="x Un xa Un 2" in spec)
1.255 -apply (simp add: Un_least_lt_iff lt_Ord Vfrom_UnI1 Vfrom_UnI2)
1.256 +apply (drule_tac x=a in spec)
1.257 +apply (drule_tac x=b in spec)
1.258 +apply (drule_tac x="x \<union> xa \<union> 2" in spec)
1.259 +apply (simp add: Un_least_lt_iff lt_Ord Vfrom_UnI1 Vfrom_UnI2)
1.260  apply (blast intro: Limit_has_0 Limit_has_succ VfromI)
1.261  done
1.262
1.263 @@ -414,15 +414,15 @@
1.264
1.265  subsubsection{* Characterisation of the elements of @{term "Vset(i)"} *}
1.266
1.267 -lemma VsetD [rule_format]: "Ord(i) ==> \<forall>b. b \<in> Vset(i) --> rank(b) < i"
1.268 +lemma VsetD [rule_format]: "Ord(i) ==> \<forall>b. b \<in> Vset(i) \<longrightarrow> rank(b) < i"
1.269  apply (erule trans_induct)
1.270  apply (subst Vset, safe)
1.271  apply (subst rank)
1.272 -apply (blast intro: ltI UN_succ_least_lt)
1.273 +apply (blast intro: ltI UN_succ_least_lt)
1.274  done
1.275
1.276  lemma VsetI_lemma [rule_format]:
1.277 -     "Ord(i) ==> \<forall>b. rank(b) \<in> i --> b \<in> Vset(i)"
1.278 +     "Ord(i) ==> \<forall>b. rank(b) \<in> i \<longrightarrow> b \<in> Vset(i)"
1.279  apply (erule trans_induct)
1.280  apply (rule allI)
1.281  apply (subst Vset)
1.282 @@ -447,30 +447,30 @@
1.283  lemma rank_Vset: "Ord(i) ==> rank(Vset(i)) = i"
1.284  apply (subst rank)
1.285  apply (rule equalityI, safe)
1.286 -apply (blast intro: VsetD [THEN ltD])
1.287 -apply (blast intro: VsetD [THEN ltD] Ord_trans)
1.288 +apply (blast intro: VsetD [THEN ltD])
1.289 +apply (blast intro: VsetD [THEN ltD] Ord_trans)
1.290  apply (blast intro: i_subset_Vfrom [THEN subsetD]
1.291                      Ord_in_Ord [THEN rank_of_Ord, THEN ssubst])
1.292  done
1.293
1.294  lemma Finite_Vset: "i \<in> nat ==> Finite(Vset(i))";
1.295  apply (erule nat_induct)
1.296 - apply (simp add: Vfrom_0)
1.298 + apply (simp add: Vfrom_0)
1.300  done
1.301
1.302  subsubsection{* Reasoning about Sets in Terms of Their Elements' Ranks *}
1.303
1.304 -lemma arg_subset_Vset_rank: "a <= Vset(rank(a))"
1.305 +lemma arg_subset_Vset_rank: "a \<subseteq> Vset(rank(a))"
1.306  apply (rule subsetI)
1.307  apply (erule rank_lt [THEN VsetI])
1.308  done
1.309
1.310  lemma Int_Vset_subset:
1.311 -    "[| !!i. Ord(i) ==> a Int Vset(i) <= b |] ==> a <= b"
1.312 -apply (rule subset_trans)
1.313 +    "[| !!i. Ord(i) ==> a \<inter> Vset(i) \<subseteq> b |] ==> a \<subseteq> b"
1.314 +apply (rule subset_trans)
1.315  apply (rule Int_greatest [OF subset_refl arg_subset_Vset_rank])
1.316 -apply (blast intro: Ord_rank)
1.317 +apply (blast intro: Ord_rank)
1.318  done
1.319
1.320  subsubsection{* Set Up an Environment for Simplification *}
1.321 @@ -490,7 +490,7 @@
1.322  subsubsection{* Recursion over Vset Levels! *}
1.323
1.324  text{*NOT SUITABLE FOR REWRITING: recursive!*}
1.325 -lemma Vrec: "Vrec(a,H) = H(a, lam x:Vset(rank(a)). Vrec(x,H))"
1.326 +lemma Vrec: "Vrec(a,H) = H(a, \<lambda>x\<in>Vset(rank(a)). Vrec(x,H))"
1.327  apply (unfold Vrec_def)
1.328  apply (subst transrec, simp)
1.329  apply (rule refl [THEN lam_cong, THEN subst_context], simp add: lt_def)
1.330 @@ -499,14 +499,14 @@
1.331  text{*This form avoids giant explosions in proofs.  NOTE USE OF == *}
1.332  lemma def_Vrec:
1.333      "[| !!x. h(x)==Vrec(x,H) |] ==>
1.334 -     h(a) = H(a, lam x: Vset(rank(a)). h(x))"
1.335 -apply simp
1.336 +     h(a) = H(a, \<lambda>x\<in>Vset(rank(a)). h(x))"
1.337 +apply simp
1.338  apply (rule Vrec)
1.339  done
1.340
1.341  text{*NOT SUITABLE FOR REWRITING: recursive!*}
1.342  lemma Vrecursor:
1.343 -     "Vrecursor(H,a) = H(lam x:Vset(rank(a)). Vrecursor(H,x),  a)"
1.344 +     "Vrecursor(H,a) = H(\<lambda>x\<in>Vset(rank(a)). Vrecursor(H,x),  a)"
1.345  apply (unfold Vrecursor_def)
1.346  apply (subst transrec, simp)
1.347  apply (rule refl [THEN lam_cong, THEN subst_context], simp add: lt_def)
1.348 @@ -514,7 +514,7 @@
1.349
1.350  text{*This form avoids giant explosions in proofs.  NOTE USE OF == *}
1.351  lemma def_Vrecursor:
1.352 -     "h == Vrecursor(H) ==> h(a) = H(lam x: Vset(rank(a)). h(x),  a)"
1.353 +     "h == Vrecursor(H) ==> h(a) = H(\<lambda>x\<in>Vset(rank(a)). h(x),  a)"
1.354  apply simp
1.355  apply (rule Vrecursor)
1.356  done
1.357 @@ -522,7 +522,7 @@
1.358
1.359  subsection{* The Datatype Universe: @{term "univ(A)"} *}
1.360
1.361 -lemma univ_mono: "A<=B ==> univ(A) <= univ(B)"
1.362 +lemma univ_mono: "A<=B ==> univ(A) \<subseteq> univ(B)"
1.363  apply (unfold univ_def)
1.364  apply (erule Vfrom_mono)
1.365  apply (rule subset_refl)
1.366 @@ -540,28 +540,28 @@
1.367  apply (rule Limit_nat [THEN Limit_Vfrom_eq])
1.368  done
1.369
1.370 -lemma subset_univ_eq_Int: "c <= univ(A) ==> c = (\<Union>i\<in>nat. c Int Vfrom(A,i))"
1.371 +lemma subset_univ_eq_Int: "c \<subseteq> univ(A) ==> c = (\<Union>i\<in>nat. c \<inter> Vfrom(A,i))"
1.372  apply (rule subset_UN_iff_eq [THEN iffD1])
1.373  apply (erule univ_eq_UN [THEN subst])
1.374  done
1.375
1.376  lemma univ_Int_Vfrom_subset:
1.377 -    "[| a <= univ(X);
1.378 -        !!i. i:nat ==> a Int Vfrom(X,i) <= b |]
1.379 -     ==> a <= b"
1.380 +    "[| a \<subseteq> univ(X);
1.381 +        !!i. i:nat ==> a \<inter> Vfrom(X,i) \<subseteq> b |]
1.382 +     ==> a \<subseteq> b"
1.383  apply (subst subset_univ_eq_Int, assumption)
1.384 -apply (rule UN_least, simp)
1.385 +apply (rule UN_least, simp)
1.386  done
1.387
1.388  lemma univ_Int_Vfrom_eq:
1.389 -    "[| a <= univ(X);   b <= univ(X);
1.390 -        !!i. i:nat ==> a Int Vfrom(X,i) = b Int Vfrom(X,i)
1.391 +    "[| a \<subseteq> univ(X);   b \<subseteq> univ(X);
1.392 +        !!i. i:nat ==> a \<inter> Vfrom(X,i) = b \<inter> Vfrom(X,i)
1.393       |] ==> a = b"
1.394  apply (rule equalityI)
1.395  apply (rule univ_Int_Vfrom_subset, assumption)
1.396 -apply (blast elim: equalityCE)
1.397 +apply (blast elim: equalityCE)
1.398  apply (rule univ_Int_Vfrom_subset, assumption)
1.399 -apply (blast elim: equalityCE)
1.400 +apply (blast elim: equalityCE)
1.401  done
1.402
1.403  subsection{* Closure Properties for @{term "univ(A)"}*}
1.404 @@ -571,10 +571,10 @@
1.405  apply (rule nat_0I [THEN zero_in_Vfrom])
1.406  done
1.407
1.408 -lemma zero_subset_univ: "{0} <= univ(A)"
1.409 +lemma zero_subset_univ: "{0} \<subseteq> univ(A)"
1.410  by (blast intro: zero_in_univ)
1.411
1.412 -lemma A_subset_univ: "A <= univ(A)"
1.413 +lemma A_subset_univ: "A \<subseteq> univ(A)"
1.414  apply (unfold univ_def)
1.415  apply (rule A_subset_Vfrom)
1.416  done
1.417 @@ -601,12 +601,12 @@
1.418  done
1.419
1.420  lemma Union_in_univ:
1.421 -     "[| X: univ(A);  Transset(A) |] ==> Union(X) \<in> univ(A)"
1.422 +     "[| X: univ(A);  Transset(A) |] ==> \<Union>(X) \<in> univ(A)"
1.423  apply (unfold univ_def)
1.424  apply (blast intro: Union_in_VLimit Limit_nat)
1.425  done
1.426
1.427 -lemma product_univ: "univ(A)*univ(A) <= univ(A)"
1.428 +lemma product_univ: "univ(A)*univ(A) \<subseteq> univ(A)"
1.429  apply (unfold univ_def)
1.430  apply (rule Limit_nat [THEN product_VLimit])
1.431  done
1.432 @@ -614,7 +614,7 @@
1.433
1.434  subsubsection{* The Natural Numbers *}
1.435
1.436 -lemma nat_subset_univ: "nat <= univ(A)"
1.437 +lemma nat_subset_univ: "nat \<subseteq> univ(A)"
1.438  apply (unfold univ_def)
1.439  apply (rule i_subset_Vfrom)
1.440  done
1.441 @@ -633,7 +633,7 @@
1.442  lemma two_in_univ: "2 \<in> univ(A)"
1.443  by (blast intro: nat_into_univ)
1.444
1.445 -lemma bool_subset_univ: "bool <= univ(A)"
1.446 +lemma bool_subset_univ: "bool \<subseteq> univ(A)"
1.447  apply (unfold bool_def)
1.448  apply (blast intro!: zero_in_univ one_in_univ)
1.449  done
1.450 @@ -653,7 +653,7 @@
1.451  apply (erule Inr_in_VLimit [OF _ Limit_nat])
1.452  done
1.453
1.454 -lemma sum_univ: "univ(C)+univ(C) <= univ(C)"
1.455 +lemma sum_univ: "univ(C)+univ(C) \<subseteq> univ(C)"
1.456  apply (unfold univ_def)
1.457  apply (rule Limit_nat [THEN sum_VLimit])
1.458  done
1.459 @@ -663,7 +663,7 @@
1.460  lemma Sigma_subset_univ:
1.461    "[|A \<subseteq> univ(D); \<And>x. x \<in> A \<Longrightarrow> B(x) \<subseteq> univ(D)|] ==> Sigma(A,B) \<subseteq> univ(D)"
1.463 -apply (blast intro: Sigma_subset_VLimit del: subsetI)
1.464 +apply (blast intro: Sigma_subset_VLimit del: subsetI)
1.465  done
1.466
1.467
1.468 @@ -677,14 +677,14 @@
1.469  subsubsection{* Closure under Finite Powerset *}
1.470
1.471  lemma Fin_Vfrom_lemma:
1.472 -     "[| b: Fin(Vfrom(A,i));  Limit(i) |] ==> EX j. b <= Vfrom(A,j) & j<i"
1.473 +     "[| b: Fin(Vfrom(A,i));  Limit(i) |] ==> \<exists>j. b \<subseteq> Vfrom(A,j) & j<i"
1.474  apply (erule Fin_induct)
1.475  apply (blast dest!: Limit_has_0, safe)
1.476  apply (erule Limit_VfromE, assumption)
1.477  apply (blast intro!: Un_least_lt intro: Vfrom_UnI1 Vfrom_UnI2)
1.478  done
1.479
1.480 -lemma Fin_VLimit: "Limit(i) ==> Fin(Vfrom(A,i)) <= Vfrom(A,i)"
1.481 +lemma Fin_VLimit: "Limit(i) ==> Fin(Vfrom(A,i)) \<subseteq> Vfrom(A,i)"
1.482  apply (rule subsetI)
1.483  apply (drule Fin_Vfrom_lemma, safe)
1.484  apply (rule Vfrom [THEN ssubst])
1.485 @@ -693,7 +693,7 @@
1.486
1.487  lemmas Fin_subset_VLimit = subset_trans [OF Fin_mono Fin_VLimit]
1.488
1.489 -lemma Fin_univ: "Fin(univ(A)) <= univ(A)"
1.490 +lemma Fin_univ: "Fin(univ(A)) \<subseteq> univ(A)"
1.491  apply (unfold univ_def)
1.492  apply (rule Limit_nat [THEN Fin_VLimit])
1.493  done
1.494 @@ -701,7 +701,7 @@
1.495  subsubsection{* Closure under Finite Powers: Functions from a Natural Number *}
1.496
1.497  lemma nat_fun_VLimit:
1.498 -     "[| n: nat;  Limit(i) |] ==> n -> Vfrom(A,i) <= Vfrom(A,i)"
1.499 +     "[| n: nat;  Limit(i) |] ==> n -> Vfrom(A,i) \<subseteq> Vfrom(A,i)"
1.500  apply (erule nat_fun_subset_Fin [THEN subset_trans])
1.501  apply (blast del: subsetI
1.502      intro: subset_refl Fin_subset_VLimit Sigma_subset_VLimit nat_subset_VLimit)
1.503 @@ -709,7 +709,7 @@
1.504
1.505  lemmas nat_fun_subset_VLimit = subset_trans [OF Pi_mono nat_fun_VLimit]
1.506
1.507 -lemma nat_fun_univ: "n: nat ==> n -> univ(A) <= univ(A)"
1.508 +lemma nat_fun_univ: "n: nat ==> n -> univ(A) \<subseteq> univ(A)"
1.509  apply (unfold univ_def)
1.510  apply (erule nat_fun_VLimit [OF _ Limit_nat])
1.511  done
1.512 @@ -719,36 +719,36 @@
1.513
1.514  text{*General but seldom-used version; normally the domain is fixed*}
1.515  lemma FiniteFun_VLimit1:
1.516 -     "Limit(i) ==> Vfrom(A,i) -||> Vfrom(A,i) <= Vfrom(A,i)"
1.517 +     "Limit(i) ==> Vfrom(A,i) -||> Vfrom(A,i) \<subseteq> Vfrom(A,i)"
1.518  apply (rule FiniteFun.dom_subset [THEN subset_trans])
1.519  apply (blast del: subsetI
1.520               intro: Fin_subset_VLimit Sigma_subset_VLimit subset_refl)
1.521  done
1.522
1.523 -lemma FiniteFun_univ1: "univ(A) -||> univ(A) <= univ(A)"
1.524 +lemma FiniteFun_univ1: "univ(A) -||> univ(A) \<subseteq> univ(A)"
1.525  apply (unfold univ_def)
1.526  apply (rule Limit_nat [THEN FiniteFun_VLimit1])
1.527  done
1.528
1.529  text{*Version for a fixed domain*}
1.530  lemma FiniteFun_VLimit:
1.531 -     "[| W <= Vfrom(A,i); Limit(i) |] ==> W -||> Vfrom(A,i) <= Vfrom(A,i)"
1.532 -apply (rule subset_trans)
1.533 +     "[| W \<subseteq> Vfrom(A,i); Limit(i) |] ==> W -||> Vfrom(A,i) \<subseteq> Vfrom(A,i)"
1.534 +apply (rule subset_trans)
1.535  apply (erule FiniteFun_mono [OF _ subset_refl])
1.536  apply (erule FiniteFun_VLimit1)
1.537  done
1.538
1.539  lemma FiniteFun_univ:
1.540 -    "W <= univ(A) ==> W -||> univ(A) <= univ(A)"
1.541 +    "W \<subseteq> univ(A) ==> W -||> univ(A) \<subseteq> univ(A)"
1.542  apply (unfold univ_def)
1.543  apply (erule FiniteFun_VLimit [OF _ Limit_nat])
1.544  done
1.545
1.546  lemma FiniteFun_in_univ:
1.547 -     "[| f: W -||> univ(A);  W <= univ(A) |] ==> f \<in> univ(A)"
1.548 +     "[| f: W -||> univ(A);  W \<subseteq> univ(A) |] ==> f \<in> univ(A)"
1.549  by (erule FiniteFun_univ [THEN subsetD], assumption)
1.550
1.551 -text{*Remove <= from the rule above*}
1.552 +text{*Remove @{text "\<subseteq>"} from the rule above*}
1.553  lemmas FiniteFun_in_univ' = FiniteFun_in_univ [OF _ subsetI]
1.554
1.555
1.556 @@ -760,16 +760,16 @@
1.557  lemma doubleton_in_Vfrom_D:
1.558       "[| {a,b} \<in> Vfrom(X,succ(i));  Transset(X) |]
1.559        ==> a \<in> Vfrom(X,i)  &  b \<in> Vfrom(X,i)"
1.560 -by (drule Transset_Vfrom_succ [THEN equalityD1, THEN subsetD, THEN PowD],
1.561 +by (drule Transset_Vfrom_succ [THEN equalityD1, THEN subsetD, THEN PowD],
1.562      assumption, fast)
1.563
1.564  text{*This weaker version says a, b exist at the same level*}
1.565  lemmas Vfrom_doubleton_D = Transset_Vfrom [THEN Transset_doubleton_D]
1.566
1.567 -(** Using only the weaker theorem would prove <a,b> \<in> Vfrom(X,i)
1.568 -      implies a, b \<in> Vfrom(X,i), which is useless for induction.
1.569 -    Using only the stronger theorem would prove <a,b> \<in> Vfrom(X,succ(succ(i)))
1.570 -      implies a, b \<in> Vfrom(X,i), leaving the succ(i) case untreated.
1.571 +(** Using only the weaker theorem would prove <a,b> : Vfrom(X,i)
1.572 +      implies a, b : Vfrom(X,i), which is useless for induction.
1.573 +    Using only the stronger theorem would prove <a,b> : Vfrom(X,succ(succ(i)))
1.574 +      implies a, b : Vfrom(X,i), leaving the succ(i) case untreated.
1.575      The combination gives a reduction by precisely one level, which is
1.576        most convenient for proofs.
1.577  **)
1.578 @@ -783,13 +783,13 @@
1.579
1.580  lemma product_Int_Vfrom_subset:
1.581       "Transset(X) ==>
1.582 -      (a*b) Int Vfrom(X, succ(i)) <= (a Int Vfrom(X,i)) * (b Int Vfrom(X,i))"
1.583 +      (a*b) \<inter> Vfrom(X, succ(i)) \<subseteq> (a \<inter> Vfrom(X,i)) * (b \<inter> Vfrom(X,i))"
1.584  by (blast dest!: Pair_in_Vfrom_D)
1.585
1.586
1.587  ML
1.588  {*
1.589 -val rank_ss = @{simpset} addsimps [@{thm VsetI}]
1.590 +val rank_ss = @{simpset} addsimps [@{thm VsetI}]
1.591                addsimps @{thms rank_rls} @ (@{thms rank_rls} RLN (2, [@{thm lt_trans}]));
1.592  *}
1.593
```