src/ZF/CardinalArith.thy
changeset 46820 c656222c4dc1
parent 45602 2a858377c3d2
child 46821 ff6b0c1087f2
     1.1 --- a/src/ZF/CardinalArith.thy	Sun Mar 04 23:20:43 2012 +0100
     1.2 +++ b/src/ZF/CardinalArith.thy	Tue Mar 06 15:15:49 2012 +0000
     1.3 @@ -9,30 +9,30 @@
     1.4  
     1.5  definition
     1.6    InfCard       :: "i=>o"  where
     1.7 -    "InfCard(i) == Card(i) & nat le i"
     1.8 +    "InfCard(i) == Card(i) & nat \<le> i"
     1.9  
    1.10  definition
    1.11    cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where
    1.12      "i |*| j == |i*j|"
    1.13 -  
    1.14 +
    1.15  definition
    1.16    cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where
    1.17      "i |+| j == |i+j|"
    1.18  
    1.19  definition
    1.20    csquare_rel   :: "i=>i"  where
    1.21 -    "csquare_rel(K) ==   
    1.22 -          rvimage(K*K,   
    1.23 -                  lam <x,y>:K*K. <x Un y, x, y>, 
    1.24 +    "csquare_rel(K) ==
    1.25 +          rvimage(K*K,
    1.26 +                  lam <x,y>:K*K. <x \<union> y, x, y>,
    1.27                    rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
    1.28  
    1.29  definition
    1.30    jump_cardinal :: "i=>i"  where
    1.31      --{*This def is more complex than Kunen's but it more easily proved to
    1.32          be a cardinal*}
    1.33 -    "jump_cardinal(K) ==   
    1.34 +    "jump_cardinal(K) ==
    1.35           \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
    1.36 -  
    1.37 +
    1.38  definition
    1.39    csucc         :: "i=>i"  where
    1.40      --{*needed because @{term "jump_cardinal(K)"} might not be the successor
    1.41 @@ -48,25 +48,25 @@
    1.42    cmult  (infixl "\<otimes>" 70)
    1.43  
    1.44  
    1.45 -lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
    1.46 -apply (rule CardI) 
    1.47 - apply (simp add: Card_is_Ord) 
    1.48 +lemma Card_Union [simp,intro,TC]: "(\<forall>x\<in>A. Card(x)) ==> Card(\<Union>(A))"
    1.49 +apply (rule CardI)
    1.50 + apply (simp add: Card_is_Ord)
    1.51  apply (clarify dest!: ltD)
    1.52 -apply (drule bspec, assumption) 
    1.53 -apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
    1.54 +apply (drule bspec, assumption)
    1.55 +apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord)
    1.56  apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
    1.57 -apply (drule lesspoll_trans1, assumption) 
    1.58 +apply (drule lesspoll_trans1, assumption)
    1.59  apply (subgoal_tac "B \<lesssim> \<Union>A")
    1.60 - apply (drule lesspoll_trans1, assumption, blast) 
    1.61 -apply (blast intro: subset_imp_lepoll) 
    1.62 + apply (drule lesspoll_trans1, assumption, blast)
    1.63 +apply (blast intro: subset_imp_lepoll)
    1.64  done
    1.65  
    1.66 -lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
    1.67 -by (blast intro: Card_Union) 
    1.68 +lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"
    1.69 +by (blast intro: Card_Union)
    1.70  
    1.71  lemma Card_OUN [simp,intro,TC]:
    1.72       "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
    1.73 -by (simp add: OUnion_def Card_0) 
    1.74 +by (simp add: OUnion_def Card_0)
    1.75  
    1.76  lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
    1.77  apply (unfold lesspoll_def)
    1.78 @@ -75,8 +75,8 @@
    1.79  apply (rule notI)
    1.80  apply (erule eqpollE)
    1.81  apply (rule succ_lepoll_natE)
    1.82 -apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
    1.83 -                    lepoll_trans, assumption) 
    1.84 +apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll]
    1.85 +                    lepoll_trans, assumption)
    1.86  done
    1.87  
    1.88  lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
    1.89 @@ -88,7 +88,7 @@
    1.90  lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
    1.91  apply (unfold lesspoll_def)
    1.92  apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
    1.93 -            intro!: eqpollI elim: notE 
    1.94 +            intro!: eqpollI elim: notE
    1.95              elim!: eqpollE lepoll_trans)
    1.96  done
    1.97  
    1.98 @@ -123,18 +123,18 @@
    1.99  done
   1.100  
   1.101  (*Unconditional version requires AC*)
   1.102 -lemma well_ord_cadd_assoc: 
   1.103 +lemma well_ord_cadd_assoc:
   1.104      "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   1.105       ==> (i |+| j) |+| k = i |+| (j |+| k)"
   1.106  apply (unfold cadd_def)
   1.107  apply (rule cardinal_cong)
   1.108  apply (rule eqpoll_trans)
   1.109   apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   1.110 - apply (blast intro: well_ord_radd ) 
   1.111 + apply (blast intro: well_ord_radd )
   1.112  apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
   1.113  apply (rule eqpoll_sym)
   1.114  apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   1.115 -apply (blast intro: well_ord_radd ) 
   1.116 +apply (blast intro: well_ord_radd )
   1.117  done
   1.118  
   1.119  subsubsection{*0 is the identity for addition*}
   1.120 @@ -154,14 +154,14 @@
   1.121  
   1.122  lemma sum_lepoll_self: "A \<lesssim> A+B"
   1.123  apply (unfold lepoll_def inj_def)
   1.124 -apply (rule_tac x = "lam x:A. Inl (x) " in exI)
   1.125 +apply (rule_tac x = "\<lambda>x\<in>A. Inl (x) " in exI)
   1.126  apply simp
   1.127  done
   1.128  
   1.129  (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   1.130  
   1.131 -lemma cadd_le_self: 
   1.132 -    "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
   1.133 +lemma cadd_le_self:
   1.134 +    "[| Card(K);  Ord(L) |] ==> K \<le> (K |+| L)"
   1.135  apply (unfold cadd_def)
   1.136  apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
   1.137         assumption)
   1.138 @@ -171,18 +171,18 @@
   1.139  
   1.140  subsubsection{*Monotonicity of addition*}
   1.141  
   1.142 -lemma sum_lepoll_mono: 
   1.143 +lemma sum_lepoll_mono:
   1.144       "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
   1.145  apply (unfold lepoll_def)
   1.146  apply (elim exE)
   1.147 -apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   1.148 +apply (rule_tac x = "\<lambda>z\<in>A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   1.149  apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
   1.150         in lam_injective)
   1.151  apply (typecheck add: inj_is_fun, auto)
   1.152  done
   1.153  
   1.154  lemma cadd_le_mono:
   1.155 -    "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
   1.156 +    "[| K' \<le> K;  L' \<le> L |] ==> (K' |+| L') \<le> (K |+| L)"
   1.157  apply (unfold cadd_def)
   1.158  apply (safe dest!: le_subset_iff [THEN iffD1])
   1.159  apply (rule well_ord_lepoll_imp_Card_le)
   1.160 @@ -195,7 +195,7 @@
   1.161  lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
   1.162  apply (unfold eqpoll_def)
   1.163  apply (rule exI)
   1.164 -apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
   1.165 +apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"
   1.166              and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
   1.167     apply simp_all
   1.168  apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
   1.169 @@ -227,8 +227,8 @@
   1.170  lemma prod_commute_eqpoll: "A*B \<approx> B*A"
   1.171  apply (unfold eqpoll_def)
   1.172  apply (rule exI)
   1.173 -apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
   1.174 -       auto) 
   1.175 +apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective,
   1.176 +       auto)
   1.177  done
   1.178  
   1.179  lemma cmult_commute: "i |*| j = j |*| i"
   1.180 @@ -250,11 +250,11 @@
   1.181       ==> (i |*| j) |*| k = i |*| (j |*| k)"
   1.182  apply (unfold cmult_def)
   1.183  apply (rule cardinal_cong)
   1.184 -apply (rule eqpoll_trans) 
   1.185 +apply (rule eqpoll_trans)
   1.186   apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   1.187   apply (blast intro: well_ord_rmult)
   1.188  apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
   1.189 -apply (rule eqpoll_sym) 
   1.190 +apply (rule eqpoll_sym)
   1.191  apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   1.192  apply (blast intro: well_ord_rmult)
   1.193  done
   1.194 @@ -272,12 +272,12 @@
   1.195       ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
   1.196  apply (unfold cadd_def cmult_def)
   1.197  apply (rule cardinal_cong)
   1.198 -apply (rule eqpoll_trans) 
   1.199 +apply (rule eqpoll_trans)
   1.200   apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   1.201  apply (blast intro: well_ord_radd)
   1.202  apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
   1.203 -apply (rule eqpoll_sym) 
   1.204 -apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
   1.205 +apply (rule eqpoll_sym)
   1.206 +apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll
   1.207                                  well_ord_cardinal_eqpoll])
   1.208  apply (blast intro: well_ord_rmult)+
   1.209  done
   1.210 @@ -310,11 +310,11 @@
   1.211  
   1.212  lemma prod_square_lepoll: "A \<lesssim> A*A"
   1.213  apply (unfold lepoll_def inj_def)
   1.214 -apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
   1.215 +apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp)
   1.216  done
   1.217  
   1.218  (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
   1.219 -lemma cmult_square_le: "Card(K) ==> K le K |*| K"
   1.220 +lemma cmult_square_le: "Card(K) ==> K \<le> K |*| K"
   1.221  apply (unfold cmult_def)
   1.222  apply (rule le_trans)
   1.223  apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
   1.224 @@ -327,12 +327,12 @@
   1.225  
   1.226  lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
   1.227  apply (unfold lepoll_def inj_def)
   1.228 -apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
   1.229 +apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)
   1.230  done
   1.231  
   1.232  (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   1.233  lemma cmult_le_self:
   1.234 -    "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
   1.235 +    "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K |*| L)"
   1.236  apply (unfold cmult_def)
   1.237  apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
   1.238    apply assumption
   1.239 @@ -347,13 +347,13 @@
   1.240  apply (unfold lepoll_def)
   1.241  apply (elim exE)
   1.242  apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
   1.243 -apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
   1.244 +apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
   1.245         in lam_injective)
   1.246  apply (typecheck add: inj_is_fun, auto)
   1.247  done
   1.248  
   1.249  lemma cmult_le_mono:
   1.250 -    "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
   1.251 +    "[| K' \<le> K;  L' \<le> L |] ==> (K' |*| L') \<le> (K |*| L)"
   1.252  apply (unfold cmult_def)
   1.253  apply (safe dest!: le_subset_iff [THEN iffD1])
   1.254  apply (rule well_ord_lepoll_imp_Card_le)
   1.255 @@ -391,10 +391,10 @@
   1.256  by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
   1.257  
   1.258  lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
   1.259 -apply (rule lepoll_trans) 
   1.260 -apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
   1.261 -apply (erule prod_lepoll_mono) 
   1.262 -apply (rule lepoll_refl) 
   1.263 +apply (rule lepoll_trans)
   1.264 +apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll])
   1.265 +apply (erule prod_lepoll_mono)
   1.266 +apply (rule lepoll_refl)
   1.267  done
   1.268  
   1.269  lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
   1.270 @@ -404,20 +404,20 @@
   1.271  subsection{*Infinite Cardinals are Limit Ordinals*}
   1.272  
   1.273  (*This proof is modelled upon one assuming nat<=A, with injection
   1.274 -  lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
   1.275 +  \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z
   1.276    and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
   1.277    If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
   1.278  lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
   1.279  apply (unfold lepoll_def)
   1.280  apply (erule exE)
   1.281 -apply (rule_tac x = 
   1.282 -          "lam z:cons (u,A).
   1.283 -             if z=u then f`0 
   1.284 -             else if z: range (f) then f`succ (converse (f) `z) else z" 
   1.285 +apply (rule_tac x =
   1.286 +          "\<lambda>z\<in>cons (u,A).
   1.287 +             if z=u then f`0
   1.288 +             else if z: range (f) then f`succ (converse (f) `z) else z"
   1.289         in exI)
   1.290  apply (rule_tac d =
   1.291 -          "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
   1.292 -                              else y" 
   1.293 +          "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y)
   1.294 +                              else y"
   1.295         in lam_injective)
   1.296  apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
   1.297  apply (simp add: inj_is_fun [THEN apply_rangeI]
   1.298 @@ -431,7 +431,7 @@
   1.299  done
   1.300  
   1.301  (*Specialized version required below*)
   1.302 -lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
   1.303 +lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"
   1.304  apply (unfold succ_def)
   1.305  apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
   1.306  done
   1.307 @@ -447,7 +447,7 @@
   1.308  done
   1.309  
   1.310  lemma InfCard_Un:
   1.311 -    "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
   1.312 +    "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"
   1.313  apply (unfold InfCard_def)
   1.314  apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
   1.315  done
   1.316 @@ -477,7 +477,7 @@
   1.317  apply (unfold eqpoll_def)
   1.318  apply (rule exI)
   1.319  apply (simp add: ordermap_eq_image well_ord_is_wf)
   1.320 -apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
   1.321 +apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,
   1.322                             THEN bij_converse_bij])
   1.323  apply (rule pred_subset)
   1.324  done
   1.325 @@ -485,7 +485,7 @@
   1.326  subsubsection{*Establishing the well-ordering*}
   1.327  
   1.328  lemma csquare_lam_inj:
   1.329 -     "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
   1.330 +     "Ord(K) ==> (lam <x,y>:K*K. <x \<union> y, x, y>) \<in> inj(K*K, K*K*K)"
   1.331  apply (unfold inj_def)
   1.332  apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
   1.333  done
   1.334 @@ -499,7 +499,7 @@
   1.335  subsubsection{*Characterising initial segments of the well-ordering*}
   1.336  
   1.337  lemma csquareD:
   1.338 - "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
   1.339 + "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"
   1.340  apply (unfold csquare_rel_def)
   1.341  apply (erule rev_mp)
   1.342  apply (elim ltE)
   1.343 @@ -508,45 +508,45 @@
   1.344  apply (simp_all add: lt_def succI2)
   1.345  done
   1.346  
   1.347 -lemma pred_csquare_subset: 
   1.348 -    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
   1.349 +lemma pred_csquare_subset:
   1.350 +    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"
   1.351  apply (unfold Order.pred_def)
   1.352  apply (safe del: SigmaI succCI)
   1.353  apply (erule csquareD [THEN conjE])
   1.354 -apply (unfold lt_def, auto) 
   1.355 +apply (unfold lt_def, auto)
   1.356  done
   1.357  
   1.358  lemma csquare_ltI:
   1.359 - "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
   1.360 + "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"
   1.361  apply (unfold csquare_rel_def)
   1.362  apply (subgoal_tac "x<K & y<K")
   1.363 - prefer 2 apply (blast intro: lt_trans) 
   1.364 + prefer 2 apply (blast intro: lt_trans)
   1.365  apply (elim ltE)
   1.366  apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   1.367  done
   1.368  
   1.369  (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
   1.370  lemma csquare_or_eqI:
   1.371 - "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
   1.372 + "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"
   1.373  apply (unfold csquare_rel_def)
   1.374  apply (subgoal_tac "x<K & y<K")
   1.375 - prefer 2 apply (blast intro: lt_trans1) 
   1.376 + prefer 2 apply (blast intro: lt_trans1)
   1.377  apply (elim ltE)
   1.378  apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   1.379  apply (elim succE)
   1.380 -apply (simp_all add: subset_Un_iff [THEN iff_sym] 
   1.381 +apply (simp_all add: subset_Un_iff [THEN iff_sym]
   1.382                       subset_Un_iff2 [THEN iff_sym] OrdmemD)
   1.383  done
   1.384  
   1.385  subsubsection{*The cardinality of initial segments*}
   1.386  
   1.387  lemma ordermap_z_lt:
   1.388 -      "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
   1.389 +      "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>
   1.390            ordermap(K*K, csquare_rel(K)) ` <x,y> <
   1.391            ordermap(K*K, csquare_rel(K)) ` <z,z>"
   1.392  apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
   1.393  prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
   1.394 -                              Limit_is_Ord [THEN well_ord_csquare], clarify) 
   1.395 +                              Limit_is_Ord [THEN well_ord_csquare], clarify)
   1.396  apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
   1.397  apply (erule_tac [4] well_ord_is_wf)
   1.398  apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
   1.399 @@ -554,14 +554,14 @@
   1.400  
   1.401  (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
   1.402  lemma ordermap_csquare_le:
   1.403 -  "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
   1.404 -   ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
   1.405 +  "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |]
   1.406 +   ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | \<le> |succ(z)| |*| |succ(z)|"
   1.407  apply (unfold cmult_def)
   1.408  apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
   1.409  apply (rule Ord_cardinal [THEN well_ord_Memrel])+
   1.410  apply (subgoal_tac "z<K")
   1.411   prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
   1.412 -apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
   1.413 +apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans],
   1.414         assumption+)
   1.415  apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   1.416  apply (erule Limit_is_Ord [THEN well_ord_csquare])
   1.417 @@ -573,10 +573,10 @@
   1.418  apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
   1.419  done
   1.420  
   1.421 -(*Kunen: "... so the order type <= K" *)
   1.422 +(*Kunen: "... so the order type is @{text"\<le>"} K" *)
   1.423  lemma ordertype_csquare_le:
   1.424 -     "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
   1.425 -      ==> ordertype(K*K, csquare_rel(K)) le K"
   1.426 +     "[| InfCard(K);  \<forall>y\<in>K. InfCard(y) \<longrightarrow> y |*| y = y |]
   1.427 +      ==> ordertype(K*K, csquare_rel(K)) \<le> K"
   1.428  apply (frule InfCard_is_Card [THEN Card_is_Ord])
   1.429  apply (rule all_lt_imp_le, assumption)
   1.430  apply (erule well_ord_csquare [THEN Ord_ordertype])
   1.431 @@ -587,16 +587,16 @@
   1.432  apply (safe elim!: ltE)
   1.433  apply (subgoal_tac "Ord (xa) & Ord (ya)")
   1.434   prefer 2 apply (blast intro: Ord_in_Ord, clarify)
   1.435 -(*??WHAT A MESS!*)  
   1.436 +(*??WHAT A MESS!*)
   1.437  apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
   1.438 -       (assumption | rule refl | erule ltI)+) 
   1.439 -apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
   1.440 +       (assumption | rule refl | erule ltI)+)
   1.441 +apply (rule_tac i = "xa \<union> ya" and j = nat in Ord_linear2,
   1.442         simp_all add: Ord_Un Ord_nat)
   1.443 -prefer 2 (*case nat le (xa Un ya) *)
   1.444 - apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
   1.445 +prefer 2 (*case @{term"nat \<le> (xa \<union> ya)"} *)
   1.446 + apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]
   1.447                    le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
   1.448                  ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
   1.449 -(*the finite case: xa Un ya < nat *)
   1.450 +(*the finite case: @{term"xa \<union> ya < nat"} *)
   1.451  apply (rule_tac j = nat in lt_trans2)
   1.452   apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
   1.453                    nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
   1.454 @@ -607,14 +607,14 @@
   1.455  lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
   1.456  apply (frule InfCard_is_Card [THEN Card_is_Ord])
   1.457  apply (erule rev_mp)
   1.458 -apply (erule_tac i=K in trans_induct) 
   1.459 +apply (erule_tac i=K in trans_induct)
   1.460  apply (rule impI)
   1.461  apply (rule le_anti_sym)
   1.462  apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
   1.463  apply (rule ordertype_csquare_le [THEN [2] le_trans])
   1.464 -apply (simp add: cmult_def Ord_cardinal_le   
   1.465 +apply (simp add: cmult_def Ord_cardinal_le
   1.466                   well_ord_csquare [THEN Ord_ordertype]
   1.467 -                 well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
   1.468 +                 well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll,
   1.469                                     THEN cardinal_cong], assumption+)
   1.470  done
   1.471  
   1.472 @@ -629,9 +629,9 @@
   1.473  done
   1.474  
   1.475  lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
   1.476 -apply (rule well_ord_InfCard_square_eq)  
   1.477 - apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
   1.478 -apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
   1.479 +apply (rule well_ord_InfCard_square_eq)
   1.480 + apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])
   1.481 +apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])
   1.482  done
   1.483  
   1.484  lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
   1.485 @@ -639,7 +639,7 @@
   1.486  
   1.487  subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
   1.488  
   1.489 -lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
   1.490 +lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K |*| L = K"
   1.491  apply (rule le_anti_sym)
   1.492   prefer 2
   1.493   apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
   1.494 @@ -649,12 +649,12 @@
   1.495  done
   1.496  
   1.497  (*Corollary 10.13 (1), for cardinal multiplication*)
   1.498 -lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
   1.499 +lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K \<union> L"
   1.500  apply (rule_tac i = K and j = L in Ord_linear_le)
   1.501  apply (typecheck add: InfCard_is_Card Card_is_Ord)
   1.502  apply (rule cmult_commute [THEN ssubst])
   1.503  apply (rule Un_commute [THEN ssubst])
   1.504 -apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
   1.505 +apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq
   1.506                       subset_Un_iff2 [THEN iffD1] le_imp_subset)
   1.507  done
   1.508  
   1.509 @@ -664,7 +664,7 @@
   1.510  done
   1.511  
   1.512  (*Corollary 10.13 (1), for cardinal addition*)
   1.513 -lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
   1.514 +lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K |+| L = K"
   1.515  apply (rule le_anti_sym)
   1.516   prefer 2
   1.517   apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
   1.518 @@ -673,7 +673,7 @@
   1.519  apply (simp add: InfCard_cdouble_eq)
   1.520  done
   1.521  
   1.522 -lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
   1.523 +lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K \<union> L"
   1.524  apply (rule_tac i = K and j = L in Ord_linear_le)
   1.525  apply (typecheck add: InfCard_is_Card Card_is_Ord)
   1.526  apply (rule cadd_commute [THEN ssubst])
   1.527 @@ -704,10 +704,10 @@
   1.528  
   1.529  (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
   1.530  lemma jump_cardinal_iff:
   1.531 -     "i : jump_cardinal(K) <->
   1.532 -      (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
   1.533 +     "i \<in> jump_cardinal(K) <->
   1.534 +      (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"
   1.535  apply (unfold jump_cardinal_def)
   1.536 -apply (blast del: subsetI) 
   1.537 +apply (blast del: subsetI)
   1.538  done
   1.539  
   1.540  (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
   1.541 @@ -715,7 +715,7 @@
   1.542  apply (rule Ord_jump_cardinal [THEN [2] ltI])
   1.543  apply (rule jump_cardinal_iff [THEN iffD2])
   1.544  apply (rule_tac x="Memrel(K)" in exI)
   1.545 -apply (rule_tac x=K in exI)  
   1.546 +apply (rule_tac x=K in exI)
   1.547  apply (simp add: ordertype_Memrel well_ord_Memrel)
   1.548  apply (simp add: Memrel_def subset_iff)
   1.549  done
   1.550 @@ -723,10 +723,10 @@
   1.551  (*The proof by contradiction: the bijection f yields a wellordering of X
   1.552    whose ordertype is jump_cardinal(K).  *)
   1.553  lemma Card_jump_cardinal_lemma:
   1.554 -     "[| well_ord(X,r);  r <= K * K;  X <= K;
   1.555 -         f : bij(ordertype(X,r), jump_cardinal(K)) |]
   1.556 -      ==> jump_cardinal(K) : jump_cardinal(K)"
   1.557 -apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
   1.558 +     "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;
   1.559 +         f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]
   1.560 +      ==> jump_cardinal(K) \<in> jump_cardinal(K)"
   1.561 +apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")
   1.562   prefer 2 apply (blast intro: comp_bij ordermap_bij)
   1.563  apply (rule jump_cardinal_iff [THEN iffD2])
   1.564  apply (intro exI conjI)
   1.565 @@ -760,13 +760,13 @@
   1.566  lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
   1.567  by (blast intro: Ord_0_le lt_csucc lt_trans1)
   1.568  
   1.569 -lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
   1.570 +lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"
   1.571  apply (unfold csucc_def)
   1.572  apply (rule Least_le)
   1.573  apply (blast intro: Card_is_Ord)+
   1.574  done
   1.575  
   1.576 -lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
   1.577 +lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| \<le> K"
   1.578  apply (rule iffI)
   1.579  apply (rule_tac [2] Card_lt_imp_lt)
   1.580  apply (erule_tac [2] lt_trans1)
   1.581 @@ -774,21 +774,21 @@
   1.582  apply (rule notI [THEN not_lt_imp_le])
   1.583  apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
   1.584  apply (rule Ord_cardinal_le [THEN lt_trans1])
   1.585 -apply (simp_all add: Ord_cardinal Card_is_Ord) 
   1.586 +apply (simp_all add: Ord_cardinal Card_is_Ord)
   1.587  done
   1.588  
   1.589  lemma Card_lt_csucc_iff:
   1.590 -     "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
   1.591 +     "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' \<le> K"
   1.592  by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
   1.593  
   1.594  lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
   1.595 -by (simp add: InfCard_def Card_csucc Card_is_Ord 
   1.596 +by (simp add: InfCard_def Card_csucc Card_is_Ord
   1.597                lt_csucc [THEN leI, THEN [2] le_trans])
   1.598  
   1.599  
   1.600  subsubsection{*Removing elements from a finite set decreases its cardinality*}
   1.601  
   1.602 -lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
   1.603 +lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x\<notin>A \<longrightarrow> ~ cons(x,A) \<lesssim> A"
   1.604  apply (erule Fin_induct)
   1.605  apply (simp add: lepoll_0_iff)
   1.606  apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
   1.607 @@ -797,7 +797,7 @@
   1.608  done
   1.609  
   1.610  lemma Finite_imp_cardinal_cons [simp]:
   1.611 -     "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
   1.612 +     "[| Finite(A);  a\<notin>A |] ==> |cons(a,A)| = succ(|A|)"
   1.613  apply (unfold cardinal_def)
   1.614  apply (rule Least_equality)
   1.615  apply (fold cardinal_def)
   1.616 @@ -827,29 +827,29 @@
   1.617  apply (simp add: Finite_imp_succ_cardinal_Diff)
   1.618  done
   1.619  
   1.620 -lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
   1.621 +lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"
   1.622  apply (erule Finite_induct)
   1.623  apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
   1.624  done
   1.625  
   1.626  lemma card_Un_Int:
   1.627 -     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
   1.628 -apply (erule Finite_induct, simp) 
   1.629 +     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"
   1.630 +apply (erule Finite_induct, simp)
   1.631  apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
   1.632  done
   1.633  
   1.634 -lemma card_Un_disjoint: 
   1.635 -     "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
   1.636 +lemma card_Un_disjoint:
   1.637 +     "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"
   1.638  by (simp add: Finite_Un card_Un_Int)
   1.639  
   1.640  lemma card_partition [rule_format]:
   1.641 -     "Finite(C) ==>  
   1.642 -        Finite (\<Union> C) -->  
   1.643 -        (\<forall>c\<in>C. |c| = k) -->   
   1.644 -        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
   1.645 +     "Finite(C) ==>
   1.646 +        Finite (\<Union> C) \<longrightarrow>
   1.647 +        (\<forall>c\<in>C. |c| = k) \<longrightarrow>
   1.648 +        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<longrightarrow>
   1.649          k #* |C| = |\<Union> C|"
   1.650  apply (erule Finite_induct, auto)
   1.651 -apply (subgoal_tac " x \<inter> \<Union>B = 0")  
   1.652 +apply (subgoal_tac " x \<inter> \<Union>B = 0")
   1.653  apply (auto simp add: card_Un_disjoint Finite_Union
   1.654         subset_Finite [of _ "\<Union> (cons(x,F))"])
   1.655  done
   1.656 @@ -866,12 +866,12 @@
   1.657  apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
   1.658  done
   1.659  
   1.660 -lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
   1.661 +lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<longrightarrow> i \<in> nat | i=nat"
   1.662  apply (erule trans_induct3, auto)
   1.663  apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
   1.664  done
   1.665  
   1.666 -lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
   1.667 +lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"
   1.668  by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
   1.669  
   1.670  lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
   1.671 @@ -889,7 +889,7 @@
   1.672  done
   1.673  
   1.674  lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
   1.675 -     "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
   1.676 +     "Finite(B) ==> \<forall>A. |B|<|A| \<longrightarrow> A - B \<noteq> 0"
   1.677  apply (erule Finite_induct, auto)
   1.678  apply (case_tac "Finite (A)")
   1.679   apply (subgoal_tac [2] "Finite (cons (x, B))")
   1.680 @@ -900,100 +900,13 @@
   1.681  apply (case_tac "x:A")
   1.682   apply (subgoal_tac [2] "A - cons (x, B) = A - B")
   1.683    apply auto
   1.684 -apply (subgoal_tac "|A| le |cons (x, B) |")
   1.685 +apply (subgoal_tac "|A| \<le> |cons (x, B) |")
   1.686   prefer 2
   1.687 - apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
   1.688 + apply (blast dest: Finite_cons [THEN Finite_imp_well_ord]
   1.689                intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
   1.690  apply (auto simp add: Finite_imp_cardinal_cons)
   1.691  apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
   1.692  apply (blast intro: lt_trans)
   1.693  done
   1.694  
   1.695 -
   1.696 -ML{*
   1.697 -val InfCard_def = @{thm InfCard_def};
   1.698 -val cmult_def = @{thm cmult_def};
   1.699 -val cadd_def = @{thm cadd_def};
   1.700 -val jump_cardinal_def = @{thm jump_cardinal_def};
   1.701 -val csucc_def = @{thm csucc_def};
   1.702 -
   1.703 -val sum_commute_eqpoll = @{thm sum_commute_eqpoll};
   1.704 -val cadd_commute = @{thm cadd_commute};
   1.705 -val sum_assoc_eqpoll = @{thm sum_assoc_eqpoll};
   1.706 -val well_ord_cadd_assoc = @{thm well_ord_cadd_assoc};
   1.707 -val sum_0_eqpoll = @{thm sum_0_eqpoll};
   1.708 -val cadd_0 = @{thm cadd_0};
   1.709 -val sum_lepoll_self = @{thm sum_lepoll_self};
   1.710 -val cadd_le_self = @{thm cadd_le_self};
   1.711 -val sum_lepoll_mono = @{thm sum_lepoll_mono};
   1.712 -val cadd_le_mono = @{thm cadd_le_mono};
   1.713 -val eq_imp_not_mem = @{thm eq_imp_not_mem};
   1.714 -val sum_succ_eqpoll = @{thm sum_succ_eqpoll};
   1.715 -val nat_cadd_eq_add = @{thm nat_cadd_eq_add};
   1.716 -val prod_commute_eqpoll = @{thm prod_commute_eqpoll};
   1.717 -val cmult_commute = @{thm cmult_commute};
   1.718 -val prod_assoc_eqpoll = @{thm prod_assoc_eqpoll};
   1.719 -val well_ord_cmult_assoc = @{thm well_ord_cmult_assoc};
   1.720 -val sum_prod_distrib_eqpoll = @{thm sum_prod_distrib_eqpoll};
   1.721 -val well_ord_cadd_cmult_distrib = @{thm well_ord_cadd_cmult_distrib};
   1.722 -val prod_0_eqpoll = @{thm prod_0_eqpoll};
   1.723 -val cmult_0 = @{thm cmult_0};
   1.724 -val prod_singleton_eqpoll = @{thm prod_singleton_eqpoll};
   1.725 -val cmult_1 = @{thm cmult_1};
   1.726 -val prod_lepoll_self = @{thm prod_lepoll_self};
   1.727 -val cmult_le_self = @{thm cmult_le_self};
   1.728 -val prod_lepoll_mono = @{thm prod_lepoll_mono};
   1.729 -val cmult_le_mono = @{thm cmult_le_mono};
   1.730 -val prod_succ_eqpoll = @{thm prod_succ_eqpoll};
   1.731 -val nat_cmult_eq_mult = @{thm nat_cmult_eq_mult};
   1.732 -val cmult_2 = @{thm cmult_2};
   1.733 -val sum_lepoll_prod = @{thm sum_lepoll_prod};
   1.734 -val lepoll_imp_sum_lepoll_prod = @{thm lepoll_imp_sum_lepoll_prod};
   1.735 -val nat_cons_lepoll = @{thm nat_cons_lepoll};
   1.736 -val nat_cons_eqpoll = @{thm nat_cons_eqpoll};
   1.737 -val nat_succ_eqpoll = @{thm nat_succ_eqpoll};
   1.738 -val InfCard_nat = @{thm InfCard_nat};
   1.739 -val InfCard_is_Card = @{thm InfCard_is_Card};
   1.740 -val InfCard_Un = @{thm InfCard_Un};
   1.741 -val InfCard_is_Limit = @{thm InfCard_is_Limit};
   1.742 -val ordermap_eqpoll_pred = @{thm ordermap_eqpoll_pred};
   1.743 -val ordermap_z_lt = @{thm ordermap_z_lt};
   1.744 -val InfCard_le_cmult_eq = @{thm InfCard_le_cmult_eq};
   1.745 -val InfCard_cmult_eq = @{thm InfCard_cmult_eq};
   1.746 -val InfCard_cdouble_eq = @{thm InfCard_cdouble_eq};
   1.747 -val InfCard_le_cadd_eq = @{thm InfCard_le_cadd_eq};
   1.748 -val InfCard_cadd_eq = @{thm InfCard_cadd_eq};
   1.749 -val Ord_jump_cardinal = @{thm Ord_jump_cardinal};
   1.750 -val jump_cardinal_iff = @{thm jump_cardinal_iff};
   1.751 -val K_lt_jump_cardinal = @{thm K_lt_jump_cardinal};
   1.752 -val Card_jump_cardinal = @{thm Card_jump_cardinal};
   1.753 -val csucc_basic = @{thm csucc_basic};
   1.754 -val Card_csucc = @{thm Card_csucc};
   1.755 -val lt_csucc = @{thm lt_csucc};
   1.756 -val Ord_0_lt_csucc = @{thm Ord_0_lt_csucc};
   1.757 -val csucc_le = @{thm csucc_le};
   1.758 -val lt_csucc_iff = @{thm lt_csucc_iff};
   1.759 -val Card_lt_csucc_iff = @{thm Card_lt_csucc_iff};
   1.760 -val InfCard_csucc = @{thm InfCard_csucc};
   1.761 -val Finite_into_Fin = @{thm Finite_into_Fin};
   1.762 -val Fin_into_Finite = @{thm Fin_into_Finite};
   1.763 -val Finite_Fin_iff = @{thm Finite_Fin_iff};
   1.764 -val Finite_Un = @{thm Finite_Un};
   1.765 -val Finite_Union = @{thm Finite_Union};
   1.766 -val Finite_induct = @{thm Finite_induct};
   1.767 -val Fin_imp_not_cons_lepoll = @{thm Fin_imp_not_cons_lepoll};
   1.768 -val Finite_imp_cardinal_cons = @{thm Finite_imp_cardinal_cons};
   1.769 -val Finite_imp_succ_cardinal_Diff = @{thm Finite_imp_succ_cardinal_Diff};
   1.770 -val Finite_imp_cardinal_Diff = @{thm Finite_imp_cardinal_Diff};
   1.771 -val nat_implies_well_ord = @{thm nat_implies_well_ord};
   1.772 -val nat_sum_eqpoll_sum = @{thm nat_sum_eqpoll_sum};
   1.773 -val Diff_sing_Finite = @{thm Diff_sing_Finite};
   1.774 -val Diff_Finite = @{thm Diff_Finite};
   1.775 -val Ord_subset_natD = @{thm Ord_subset_natD};
   1.776 -val Ord_nat_subset_into_Card = @{thm Ord_nat_subset_into_Card};
   1.777 -val Finite_cardinal_in_nat = @{thm Finite_cardinal_in_nat};
   1.778 -val Finite_Diff_sing_eq_diff_1 = @{thm Finite_Diff_sing_eq_diff_1};
   1.779 -val cardinal_lt_imp_Diff_not_0 = @{thm cardinal_lt_imp_Diff_not_0};
   1.780 -*}
   1.781 -
   1.782  end