--- a/src/ZF/CardinalArith.thy Sat Jun 15 22:57:33 2002 +0200
+++ b/src/ZF/CardinalArith.thy Sun Jun 16 11:58:54 2002 +0200
@@ -3,7 +3,12 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
-Cardinal Arithmetic
+Cardinal arithmetic -- WITHOUT the Axiom of Choice
+
+Note: Could omit proving the algebraic laws for cardinal addition and
+multiplication. On finite cardinals these operations coincide with
+addition and multiplication of natural numbers; on infinite cardinals they
+coincide with union (maximum). Either way we get most laws for free.
*)
theory CardinalArith = Cardinal + OrderArith + ArithSimp + Finite:
@@ -123,7 +128,7 @@
apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord)
apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
apply (drule lesspoll_trans1, assumption)
-apply (subgoal_tac "B lepoll \<Union>A")
+apply (subgoal_tac "B \<lesssim> \<Union>A")
apply (drule lesspoll_trans1, assumption, blast)
apply (blast intro: subset_imp_lepoll)
done
@@ -192,4 +197,985 @@
apply (blast intro: eqpoll_trans eqpoll_sym)
done
+
+(*** Cardinal addition ***)
+
+(** Cardinal addition is commutative **)
+
+lemma sum_commute_eqpoll: "A+B \<approx> B+A"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
+apply auto
+done
+
+lemma cadd_commute: "i |+| j = j |+| i"
+apply (unfold cadd_def)
+apply (rule sum_commute_eqpoll [THEN cardinal_cong])
+done
+
+(** Cardinal addition is associative **)
+
+lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule sum_assoc_bij)
+done
+
+(*Unconditional version requires AC*)
+lemma well_ord_cadd_assoc:
+ "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
+ ==> (i |+| j) |+| k = i |+| (j |+| k)"
+apply (unfold cadd_def)
+apply (rule cardinal_cong)
+apply (rule eqpoll_trans)
+ apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
+ apply (blast intro: well_ord_radd elim:)
+apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
+apply (rule eqpoll_sym)
+apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
+apply (blast intro: well_ord_radd elim:)
+done
+
+(** 0 is the identity for addition **)
+
+lemma sum_0_eqpoll: "0+A \<approx> A"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule bij_0_sum)
+done
+
+lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
+apply (unfold cadd_def)
+apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
+done
+
+(** Addition by another cardinal **)
+
+lemma sum_lepoll_self: "A \<lesssim> A+B"
+apply (unfold lepoll_def inj_def)
+apply (rule_tac x = "lam x:A. Inl (x) " in exI)
+apply (simp (no_asm_simp))
+done
+
+(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
+
+lemma cadd_le_self:
+ "[| Card(K); Ord(L) |] ==> K le (K |+| L)"
+apply (unfold cadd_def)
+apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
+apply assumption;
+apply (rule_tac [2] sum_lepoll_self)
+apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
+done
+
+(** Monotonicity of addition **)
+
+lemma sum_lepoll_mono:
+ "[| A \<lesssim> C; B \<lesssim> D |] ==> A + B \<lesssim> C + D"
+apply (unfold lepoll_def)
+apply (elim exE);
+apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
+apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) `y))"
+ in lam_injective)
+apply (typecheck add: inj_is_fun)
+apply auto
+done
+
+lemma cadd_le_mono:
+ "[| K' le K; L' le L |] ==> (K' |+| L') le (K |+| L)"
+apply (unfold cadd_def)
+apply (safe dest!: le_subset_iff [THEN iffD1])
+apply (rule well_ord_lepoll_imp_Card_le)
+apply (blast intro: well_ord_radd well_ord_Memrel)
+apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
+done
+
+(** Addition of finite cardinals is "ordinary" addition **)
+
+(*????????????????upair.ML*)
+lemma eq_imp_not_mem: "a=A ==> a ~: A"
+apply (blast intro: elim: mem_irrefl);
+done
+
+lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"
+ and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
+ apply (simp_all)
+apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
+done
+
+(*Pulling the succ(...) outside the |...| requires m, n: nat *)
+(*Unconditional version requires AC*)
+lemma cadd_succ_lemma:
+ "[| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
+apply (unfold cadd_def)
+apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
+apply (rule succ_eqpoll_cong [THEN cardinal_cong])
+apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
+apply (blast intro: well_ord_radd well_ord_Memrel)
+done
+
+lemma nat_cadd_eq_add: "[| m: nat; n: nat |] ==> m |+| n = m#+n"
+apply (induct_tac "m")
+apply (simp add: nat_into_Card [THEN cadd_0])
+apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
+done
+
+
+(*** Cardinal multiplication ***)
+
+(** Cardinal multiplication is commutative **)
+
+(*Easier to prove the two directions separately*)
+lemma prod_commute_eqpoll: "A*B \<approx> B*A"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective)
+apply (auto );
+done
+
+lemma cmult_commute: "i |*| j = j |*| i"
+apply (unfold cmult_def)
+apply (rule prod_commute_eqpoll [THEN cardinal_cong])
+done
+
+(** Cardinal multiplication is associative **)
+
+lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule prod_assoc_bij)
+done
+
+(*Unconditional version requires AC*)
+lemma well_ord_cmult_assoc:
+ "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
+ ==> (i |*| j) |*| k = i |*| (j |*| k)"
+apply (unfold cmult_def)
+apply (rule cardinal_cong)
+apply (rule eqpoll_trans);
+ apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
+ apply (blast intro: well_ord_rmult)
+apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
+apply (rule eqpoll_sym);
+apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
+apply (blast intro: well_ord_rmult)
+done
+
+(** Cardinal multiplication distributes over addition **)
+
+lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule sum_prod_distrib_bij)
+done
+
+lemma well_ord_cadd_cmult_distrib:
+ "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
+ ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
+apply (unfold cadd_def cmult_def)
+apply (rule cardinal_cong)
+apply (rule eqpoll_trans);
+ apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
+apply (blast intro: well_ord_radd)
+apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
+apply (rule eqpoll_sym);
+apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll
+ well_ord_cardinal_eqpoll])
+apply (blast intro: well_ord_rmult)+
+done
+
+(** Multiplication by 0 yields 0 **)
+
+lemma prod_0_eqpoll: "0*A \<approx> 0"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule lam_bijective)
+apply safe
+done
+
+lemma cmult_0 [simp]: "0 |*| i = 0"
+apply (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
+done
+
+(** 1 is the identity for multiplication **)
+
+lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (rule singleton_prod_bij [THEN bij_converse_bij])
+done
+
+lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
+apply (unfold cmult_def succ_def)
+apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
+done
+
+(*** Some inequalities for multiplication ***)
+
+lemma prod_square_lepoll: "A \<lesssim> A*A"
+apply (unfold lepoll_def inj_def)
+apply (rule_tac x = "lam x:A. <x,x>" in exI)
+apply (simp (no_asm))
+done
+
+(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
+lemma cmult_square_le: "Card(K) ==> K le K |*| K"
+apply (unfold cmult_def)
+apply (rule le_trans)
+apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
+apply (rule_tac [3] prod_square_lepoll)
+apply (simp (no_asm_simp) add: le_refl Card_is_Ord Card_cardinal_eq)
+apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord);
+done
+
+(** Multiplication by a non-zero cardinal **)
+
+lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
+apply (unfold lepoll_def inj_def)
+apply (rule_tac x = "lam x:A. <x,b>" in exI)
+apply (simp (no_asm_simp))
+done
+
+(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
+lemma cmult_le_self:
+ "[| Card(K); Ord(L); 0<L |] ==> K le (K |*| L)"
+apply (unfold cmult_def)
+apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
+ apply assumption;
+ apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
+apply (blast intro: prod_lepoll_self ltD)
+done
+
+(** Monotonicity of multiplication **)
+
+lemma prod_lepoll_mono:
+ "[| A \<lesssim> C; B \<lesssim> D |] ==> A * B \<lesssim> C * D"
+apply (unfold lepoll_def)
+apply (elim exE);
+apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
+apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
+ in lam_injective)
+apply (typecheck add: inj_is_fun)
+apply auto
+done
+
+lemma cmult_le_mono:
+ "[| K' le K; L' le L |] ==> (K' |*| L') le (K |*| L)"
+apply (unfold cmult_def)
+apply (safe dest!: le_subset_iff [THEN iffD1])
+apply (rule well_ord_lepoll_imp_Card_le)
+ apply (blast intro: well_ord_rmult well_ord_Memrel)
+apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
+done
+
+(*** Multiplication of finite cardinals is "ordinary" multiplication ***)
+
+lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
+apply (unfold eqpoll_def)
+apply (rule exI);
+apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
+ and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
+apply safe
+apply (simp_all add: succI2 if_type mem_imp_not_eq)
+done
+
+(*Unconditional version requires AC*)
+lemma cmult_succ_lemma:
+ "[| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
+apply (unfold cmult_def cadd_def)
+apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
+apply (rule cardinal_cong [symmetric])
+apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
+apply (blast intro: well_ord_rmult well_ord_Memrel)
+done
+
+lemma nat_cmult_eq_mult: "[| m: nat; n: nat |] ==> m |*| n = m#*n"
+apply (induct_tac "m")
+apply (simp (no_asm_simp))
+apply (simp (no_asm_simp) add: cmult_succ_lemma nat_cadd_eq_add)
+done
+
+lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
+apply (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
+done
+
+lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
+apply (rule lepoll_trans);
+apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll])
+apply (erule prod_lepoll_mono)
+apply (rule lepoll_refl);
+done
+
+lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
+apply (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
+done
+
+
+(*** Infinite Cardinals are Limit Ordinals ***)
+
+(*This proof is modelled upon one assuming nat<=A, with injection
+ lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
+ and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \
+ If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
+lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
+apply (unfold lepoll_def)
+apply (erule exE)
+apply (rule_tac x =
+ "lam z:cons (u,A).
+ if z=u then f`0
+ else if z: range (f) then f`succ (converse (f) `z) else z"
+ in exI)
+apply (rule_tac d =
+ "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y)
+ else y"
+ in lam_injective)
+apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
+apply (simp add: inj_is_fun [THEN apply_rangeI]
+ inj_converse_fun [THEN apply_rangeI]
+ inj_converse_fun [THEN apply_funtype])
+done
+
+lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
+apply (erule nat_cons_lepoll [THEN eqpollI])
+apply (rule subset_consI [THEN subset_imp_lepoll])
+done
+
+(*Specialized version required below*)
+lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
+apply (unfold succ_def)
+apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
+done
+
+lemma InfCard_nat: "InfCard(nat)"
+apply (unfold InfCard_def)
+apply (blast intro: Card_nat le_refl Card_is_Ord)
+done
+
+lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
+apply (unfold InfCard_def)
+apply (erule conjunct1)
+done
+
+lemma InfCard_Un:
+ "[| InfCard(K); Card(L) |] ==> InfCard(K Un L)"
+apply (unfold InfCard_def)
+apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord)
+done
+
+(*Kunen's Lemma 10.11*)
+lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
+apply (unfold InfCard_def)
+apply (erule conjE)
+apply (frule Card_is_Ord)
+apply (rule ltI [THEN non_succ_LimitI])
+apply (erule le_imp_subset [THEN subsetD])
+apply (safe dest!: Limit_nat [THEN Limit_le_succD])
+apply (unfold Card_def)
+apply (drule trans)
+apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
+apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
+apply (rule le_eqI)
+apply assumption;
+apply (rule Ord_cardinal)
+done
+
+
+(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
+
+(*A general fact about ordermap*)
+lemma ordermap_eqpoll_pred:
+ "[| well_ord(A,r); x:A |] ==> ordermap(A,r)`x \<approx> pred(A,x,r)"
+apply (unfold eqpoll_def)
+apply (rule exI)
+apply (simp (no_asm_simp) add: ordermap_eq_image well_ord_is_wf)
+apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, THEN bij_converse_bij])
+apply (rule pred_subset)
+done
+
+(** Establishing the well-ordering **)
+
+lemma csquare_lam_inj:
+ "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
+apply (unfold inj_def)
+apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
+done
+
+lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
+apply (unfold csquare_rel_def)
+apply (rule csquare_lam_inj [THEN well_ord_rvimage])
+apply assumption;
+apply (blast intro: well_ord_rmult well_ord_Memrel)
+done
+
+(** Characterising initial segments of the well-ordering **)
+
+lemma csquareD:
+ "[| <<x,y>, <z,z>> : csquare_rel(K); x<K; y<K; z<K |] ==> x le z & y le z"
+apply (unfold csquare_rel_def)
+apply (erule rev_mp)
+apply (elim ltE)
+apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
+apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
+apply (simp_all (no_asm_simp) add: lt_def succI2)
+done
+
+lemma pred_csquare_subset:
+ "z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
+apply (unfold Order.pred_def)
+apply (safe del: SigmaI succCI)
+apply (erule csquareD [THEN conjE])
+apply (unfold lt_def)
+apply (auto );
+done
+
+lemma csquare_ltI:
+ "[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K)"
+apply (unfold csquare_rel_def)
+apply (subgoal_tac "x<K & y<K")
+ prefer 2 apply (blast intro: lt_trans)
+apply (elim ltE)
+apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
+done
+
+(*Part of the traditional proof. UNUSED since it's harder to prove & apply *)
+lemma csquare_or_eqI:
+ "[| x le z; y le z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
+apply (unfold csquare_rel_def)
+apply (subgoal_tac "x<K & y<K")
+ prefer 2 apply (blast intro: lt_trans1)
+apply (elim ltE)
+apply (simp (no_asm_simp) add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
+apply (elim succE)
+apply (simp_all (no_asm_simp) add: subset_Un_iff [THEN iff_sym] subset_Un_iff2 [THEN iff_sym] OrdmemD)
+done
+
+(** The cardinality of initial segments **)
+
+lemma ordermap_z_lt:
+ "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==>
+ ordermap(K*K, csquare_rel(K)) ` <x,y> <
+ ordermap(K*K, csquare_rel(K)) ` <z,z>"
+apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
+prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
+ Limit_is_Ord [THEN well_ord_csquare])
+apply (clarify );
+apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
+apply (erule_tac [4] well_ord_is_wf)
+apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
+done
+
+(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
+lemma ordermap_csquare_le:
+ "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==>
+ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|"
+apply (unfold cmult_def)
+apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
+apply (rule Ord_cardinal [THEN well_ord_Memrel])+
+apply (subgoal_tac "z<K")
+ prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
+apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans])
+apply assumption +
+apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
+apply (erule Limit_is_Ord [THEN well_ord_csquare])
+apply (blast intro: ltD)
+apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
+ assumption)
+apply (elim ltE)
+apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
+apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
+done
+
+(*Kunen: "... so the order type <= K" *)
+lemma ordertype_csquare_le:
+ "[| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |]
+ ==> ordertype(K*K, csquare_rel(K)) le K"
+apply (frule InfCard_is_Card [THEN Card_is_Ord])
+apply (rule all_lt_imp_le)
+apply assumption
+apply (erule well_ord_csquare [THEN Ord_ordertype])
+apply (rule Card_lt_imp_lt)
+apply (erule_tac [3] InfCard_is_Card)
+apply (erule_tac [2] ltE)
+apply (simp add: ordertype_unfold)
+apply (safe elim!: ltE)
+apply (subgoal_tac "Ord (xa) & Ord (ya)")
+ prefer 2 apply (blast intro: Ord_in_Ord)
+apply (clarify );
+(*??WHAT A MESS!*)
+apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
+ (assumption | rule refl | erule ltI)+)
+apply (rule_tac i = "xa Un ya" and j = "nat" in Ord_linear2,
+ simp_all add: Ord_Un Ord_nat)
+prefer 2 (*case nat le (xa Un ya) *)
+ apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]
+ le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
+ ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
+(*the finite case: xa Un ya < nat *)
+apply (rule_tac j = "nat" in lt_trans2)
+ apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
+ nat_into_Card [THEN Card_cardinal_eq] Ord_nat)
+apply (simp add: InfCard_def)
+done
+
+(*Main result: Kunen's Theorem 10.12*)
+lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
+apply (frule InfCard_is_Card [THEN Card_is_Ord])
+apply (erule rev_mp)
+apply (erule_tac i=K in trans_induct)
+apply (rule impI)
+apply (rule le_anti_sym)
+apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
+apply (rule ordertype_csquare_le [THEN [2] le_trans])
+prefer 2 apply (assumption)
+prefer 2 apply (assumption)
+apply (simp (no_asm_simp) add: cmult_def Ord_cardinal_le well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, THEN cardinal_cong] well_ord_csquare [THEN Ord_ordertype])
+done
+
+(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
+lemma well_ord_InfCard_square_eq:
+ "[| well_ord(A,r); InfCard(|A|) |] ==> A*A \<approx> A"
+apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
+apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
+apply (rule well_ord_cardinal_eqE)
+apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel)
+apply assumption;
+apply (simp (no_asm_simp) add: cmult_def [symmetric] InfCard_csquare_eq)
+done
+
+(** Toward's Kunen's Corollary 10.13 (1) **)
+
+lemma InfCard_le_cmult_eq: "[| InfCard(K); L le K; 0<L |] ==> K |*| L = K"
+apply (rule le_anti_sym)
+ prefer 2
+ apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
+apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
+apply (rule cmult_le_mono [THEN le_trans], assumption+)
+apply (simp add: InfCard_csquare_eq)
+done
+
+(*Corollary 10.13 (1), for cardinal multiplication*)
+lemma InfCard_cmult_eq: "[| InfCard(K); InfCard(L) |] ==> K |*| L = K Un L"
+apply (rule_tac i = "K" and j = "L" in Ord_linear_le)
+apply (typecheck add: InfCard_is_Card Card_is_Ord)
+apply (rule cmult_commute [THEN ssubst])
+apply (rule Un_commute [THEN ssubst])
+apply (simp_all (no_asm_simp) add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
+done
+
+lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
+apply (simp (no_asm_simp) add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
+apply (simp (no_asm_simp) add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
+done
+
+(*Corollary 10.13 (1), for cardinal addition*)
+lemma InfCard_le_cadd_eq: "[| InfCard(K); L le K |] ==> K |+| L = K"
+apply (rule le_anti_sym)
+ prefer 2
+ apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
+apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
+apply (rule cadd_le_mono [THEN le_trans], assumption+)
+apply (simp add: InfCard_cdouble_eq)
+done
+
+lemma InfCard_cadd_eq: "[| InfCard(K); InfCard(L) |] ==> K |+| L = K Un L"
+apply (rule_tac i = "K" and j = "L" in Ord_linear_le)
+apply (typecheck add: InfCard_is_Card Card_is_Ord)
+apply (rule cadd_commute [THEN ssubst])
+apply (rule Un_commute [THEN ssubst])
+apply (simp_all (no_asm_simp) add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
+done
+
+(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
+ of all n-tuples of elements of K. A better version for the Isabelle theory
+ might be InfCard(K) ==> |list(K)| = K.
+*)
+
+(*** For every cardinal number there exists a greater one
+ [Kunen's Theorem 10.16, which would be trivial using AC] ***)
+
+lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
+apply (unfold jump_cardinal_def)
+apply (rule Ord_is_Transset [THEN [2] OrdI])
+ prefer 2 apply (blast intro!: Ord_ordertype)
+apply (unfold Transset_def)
+apply (safe del: subsetI)
+apply (simp add: ordertype_pred_unfold)
+apply safe
+apply (rule UN_I)
+apply (rule_tac [2] ReplaceI)
+ prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
+done
+
+(*Allows selective unfolding. Less work than deriving intro/elim rules*)
+lemma jump_cardinal_iff:
+ "i : jump_cardinal(K) <->
+ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
+apply (unfold jump_cardinal_def)
+apply (blast del: subsetI)
+done
+
+(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
+lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
+apply (rule Ord_jump_cardinal [THEN [2] ltI])
+apply (rule jump_cardinal_iff [THEN iffD2])
+apply (rule_tac x="Memrel(K)" in exI)
+apply (rule_tac x=K in exI)
+apply (simp add: ordertype_Memrel well_ord_Memrel)
+apply (simp add: Memrel_def subset_iff)
+done
+
+(*The proof by contradiction: the bijection f yields a wellordering of X
+ whose ordertype is jump_cardinal(K). *)
+lemma Card_jump_cardinal_lemma:
+ "[| well_ord(X,r); r <= K * K; X <= K;
+ f : bij(ordertype(X,r), jump_cardinal(K)) |]
+ ==> jump_cardinal(K) : jump_cardinal(K)"
+apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
+ prefer 2 apply (blast intro: comp_bij ordermap_bij)
+apply (rule jump_cardinal_iff [THEN iffD2])
+apply (intro exI conjI)
+apply (rule subset_trans [OF rvimage_type Sigma_mono])
+apply assumption+
+apply (erule bij_is_inj [THEN well_ord_rvimage])
+apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
+apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
+ ordertype_Memrel Ord_jump_cardinal)
+done
+
+(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
+lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
+apply (rule Ord_jump_cardinal [THEN CardI])
+apply (unfold eqpoll_def)
+apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
+apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
+done
+
+(*** Basic properties of successor cardinals ***)
+
+lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
+apply (unfold csucc_def)
+apply (rule LeastI)
+apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
+done
+
+lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
+
+lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
+
+lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
+apply (blast intro: Ord_0_le lt_csucc lt_trans1)
+done
+
+lemma csucc_le: "[| Card(L); K<L |] ==> csucc(K) le L"
+apply (unfold csucc_def)
+apply (rule Least_le)
+apply (blast intro: Card_is_Ord)+
+done
+
+lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
+apply (rule iffI)
+apply (rule_tac [2] Card_lt_imp_lt)
+apply (erule_tac [2] lt_trans1)
+apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
+apply (rule notI [THEN not_lt_imp_le])
+apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl])
+apply assumption
+apply (rule Ord_cardinal_le [THEN lt_trans1])
+apply (simp_all add: Ord_cardinal Card_is_Ord)
+done
+
+lemma Card_lt_csucc_iff:
+ "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
+by (simp (no_asm_simp) add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
+
+lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
+by (simp add: InfCard_def Card_csucc Card_is_Ord
+ lt_csucc [THEN leI, THEN [2] le_trans])
+
+
+(*** Finite sets ***)
+
+lemma Fin_lemma [rule_format]: "n: nat ==> ALL A. A \<approx> n --> A : Fin(A)"
+apply (induct_tac "n")
+apply (simp (no_asm) add: eqpoll_0_iff)
+apply clarify
+apply (subgoal_tac "EX u. u:A")
+apply (erule exE)
+apply (rule Diff_sing_eqpoll [THEN revcut_rl])
+prefer 2 apply (assumption)
+apply assumption
+apply (rule_tac b = "A" in cons_Diff [THEN subst])
+apply assumption
+apply (rule Fin.consI)
+apply blast
+apply (blast intro: subset_consI [THEN Fin_mono, THEN subsetD])
+(*Now for the lemma assumed above*)
+apply (unfold eqpoll_def)
+apply (blast intro: bij_converse_bij [THEN bij_is_fun, THEN apply_type])
+done
+
+lemma Finite_into_Fin: "Finite(A) ==> A : Fin(A)"
+apply (unfold Finite_def)
+apply (blast intro: Fin_lemma)
+done
+
+lemma Fin_into_Finite: "A : Fin(U) ==> Finite(A)"
+apply (fast intro!: Finite_0 Finite_cons elim: Fin_induct)
+done
+
+lemma Finite_Fin_iff: "Finite(A) <-> A : Fin(A)"
+apply (blast intro: Finite_into_Fin Fin_into_Finite)
+done
+
+lemma Finite_Un: "[| Finite(A); Finite(B) |] ==> Finite(A Un B)"
+by (blast intro!: Fin_into_Finite Fin_UnI
+ dest!: Finite_into_Fin
+ intro: Un_upper1 [THEN Fin_mono, THEN subsetD]
+ Un_upper2 [THEN Fin_mono, THEN subsetD])
+
+lemma Finite_Union: "[| ALL y:X. Finite(y); Finite(X) |] ==> Finite(Union(X))"
+apply (simp add: Finite_Fin_iff)
+apply (rule Fin_UnionI)
+apply (erule Fin_induct)
+apply (simp (no_asm))
+apply (blast intro: Fin.consI Fin_mono [THEN [2] rev_subsetD])
+done
+
+(* Induction principle for Finite(A), by Sidi Ehmety *)
+lemma Finite_induct:
+"[| Finite(A); P(0);
+ !! x B. [| Finite(B); x ~: B; P(B) |] ==> P(cons(x, B)) |]
+ ==> P(A)"
+apply (erule Finite_into_Fin [THEN Fin_induct])
+apply (blast intro: Fin_into_Finite)+
+done
+
+
+(** Removing elements from a finite set decreases its cardinality **)
+
+lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
+apply (erule Fin_induct)
+apply (simp (no_asm) add: lepoll_0_iff)
+apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
+apply (simp (no_asm_simp))
+apply (blast dest!: cons_lepoll_consD)
+apply blast
+done
+
+lemma Finite_imp_cardinal_cons: "[| Finite(A); a~:A |] ==> |cons(a,A)| = succ(|A|)"
+apply (unfold cardinal_def)
+apply (rule Least_equality)
+apply (fold cardinal_def)
+apply (simp (no_asm) add: succ_def)
+apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
+ elim!: mem_irrefl dest!: Finite_imp_well_ord)
+apply (blast intro: Card_cardinal Card_is_Ord)
+apply (rule notI)
+apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE])
+apply assumption
+apply assumption
+apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
+apply (erule le_imp_lepoll [THEN lepoll_trans])
+apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
+ dest!: Finite_imp_well_ord)
+done
+
+
+lemma Finite_imp_succ_cardinal_Diff: "[| Finite(A); a:A |] ==> succ(|A-{a}|) = |A|"
+apply (rule_tac b = "A" in cons_Diff [THEN subst])
+apply assumption
+apply (simp (no_asm_simp) add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
+apply (simp (no_asm_simp) add: cons_Diff)
+done
+
+lemma Finite_imp_cardinal_Diff: "[| Finite(A); a:A |] ==> |A-{a}| < |A|"
+apply (rule succ_leE)
+apply (simp (no_asm_simp) add: Finite_imp_succ_cardinal_Diff)
+done
+
+
+(** Theorems by Krzysztof Grabczewski, proofs by lcp **)
+
+lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
+
+lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
+apply (rule eqpoll_trans)
+apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
+apply (erule nat_implies_well_ord)+
+apply (simp (no_asm_simp) add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
+done
+
+
+(*** Theorems by Sidi Ehmety ***)
+
+(*The contrapositive says ~Finite(A) ==> ~Finite(A-{a}) *)
+lemma Diff_sing_Finite: "Finite(A - {a}) ==> Finite(A)"
+apply (unfold Finite_def)
+apply (case_tac "a:A")
+apply (subgoal_tac [2] "A-{a}=A")
+apply auto
+apply (rule_tac x = "succ (n) " in bexI)
+apply (subgoal_tac "cons (a, A - {a}) = A & cons (n, n) = succ (n) ")
+apply (drule_tac a = "a" and b = "n" in cons_eqpoll_cong)
+apply (auto dest: mem_irrefl)
+done
+
+(*And the contrapositive of this says
+ [| ~Finite(A); Finite(B) |] ==> ~Finite(A-B) *)
+lemma Diff_Finite [rule_format (no_asm)]: "Finite(B) ==> Finite(A-B) --> Finite(A)"
+apply (erule Finite_induct)
+apply auto
+apply (case_tac "x:A")
+ apply (subgoal_tac [2] "A-cons (x, B) = A - B")
+apply (subgoal_tac "A - cons (x, B) = (A - B) - {x}")
+apply (rotate_tac -1)
+apply simp
+apply (drule Diff_sing_Finite)
+apply auto
+done
+
+lemma Ord_subset_natD [rule_format (no_asm)]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
+apply (erule trans_induct3)
+apply auto
+apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
+done
+
+lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
+apply (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
+done
+
+lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
+apply (erule Finite_induct)
+apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
+done
+
+lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
+apply (rule succ_inject)
+apply (rule_tac b = "|A|" in trans)
+apply (simp (no_asm_simp) add: Finite_imp_succ_cardinal_Diff)
+apply (subgoal_tac "1 \<lesssim> A")
+prefer 2 apply (blast intro: not_0_is_lepoll_1)
+apply (frule Finite_imp_well_ord)
+apply clarify
+apply (rotate_tac -1)
+apply (drule well_ord_lepoll_imp_Card_le)
+apply (auto simp add: cardinal_1)
+apply (rule trans)
+apply (rule_tac [2] diff_succ)
+apply (auto simp add: Finite_cardinal_in_nat)
+done
+
+lemma cardinal_lt_imp_Diff_not_0 [rule_format (no_asm)]: "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
+apply (erule Finite_induct)
+apply auto
+apply (simp_all add: Finite_imp_cardinal_cons)
+apply (case_tac "Finite (A) ")
+apply (subgoal_tac [2] "Finite (cons (x, B))")
+apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
+apply (auto simp add: Finite_0 Finite_cons)
+apply (subgoal_tac "|B|<|A|")
+prefer 2 apply (blast intro: lt_trans Ord_cardinal)
+apply (case_tac "x:A")
+apply (subgoal_tac [2] "A - cons (x, B) = A - B")
+apply auto
+apply (subgoal_tac "|A| le |cons (x, B) |")
+prefer 2
+ apply (blast dest: Finite_cons [THEN Finite_imp_well_ord]
+ intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
+apply (auto simp add: Finite_imp_cardinal_cons)
+apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
+apply (blast intro: lt_trans)
+done
+
+
+ML{*
+val InfCard_def = thm "InfCard_def"
+val cmult_def = thm "cmult_def"
+val cadd_def = thm "cadd_def"
+val jump_cardinal_def = thm "jump_cardinal_def"
+val csucc_def = thm "csucc_def"
+
+val sum_commute_eqpoll = thm "sum_commute_eqpoll";
+val cadd_commute = thm "cadd_commute";
+val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
+val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
+val sum_0_eqpoll = thm "sum_0_eqpoll";
+val cadd_0 = thm "cadd_0";
+val sum_lepoll_self = thm "sum_lepoll_self";
+val cadd_le_self = thm "cadd_le_self";
+val sum_lepoll_mono = thm "sum_lepoll_mono";
+val cadd_le_mono = thm "cadd_le_mono";
+val eq_imp_not_mem = thm "eq_imp_not_mem";
+val sum_succ_eqpoll = thm "sum_succ_eqpoll";
+val nat_cadd_eq_add = thm "nat_cadd_eq_add";
+val prod_commute_eqpoll = thm "prod_commute_eqpoll";
+val cmult_commute = thm "cmult_commute";
+val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
+val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
+val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
+val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
+val prod_0_eqpoll = thm "prod_0_eqpoll";
+val cmult_0 = thm "cmult_0";
+val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
+val cmult_1 = thm "cmult_1";
+val prod_lepoll_self = thm "prod_lepoll_self";
+val cmult_le_self = thm "cmult_le_self";
+val prod_lepoll_mono = thm "prod_lepoll_mono";
+val cmult_le_mono = thm "cmult_le_mono";
+val prod_succ_eqpoll = thm "prod_succ_eqpoll";
+val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
+val cmult_2 = thm "cmult_2";
+val sum_lepoll_prod = thm "sum_lepoll_prod";
+val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
+val nat_cons_lepoll = thm "nat_cons_lepoll";
+val nat_cons_eqpoll = thm "nat_cons_eqpoll";
+val nat_succ_eqpoll = thm "nat_succ_eqpoll";
+val InfCard_nat = thm "InfCard_nat";
+val InfCard_is_Card = thm "InfCard_is_Card";
+val InfCard_Un = thm "InfCard_Un";
+val InfCard_is_Limit = thm "InfCard_is_Limit";
+val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
+val ordermap_z_lt = thm "ordermap_z_lt";
+val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
+val InfCard_cmult_eq = thm "InfCard_cmult_eq";
+val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
+val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
+val InfCard_cadd_eq = thm "InfCard_cadd_eq";
+val Ord_jump_cardinal = thm "Ord_jump_cardinal";
+val jump_cardinal_iff = thm "jump_cardinal_iff";
+val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
+val Card_jump_cardinal = thm "Card_jump_cardinal";
+val csucc_basic = thm "csucc_basic";
+val Card_csucc = thm "Card_csucc";
+val lt_csucc = thm "lt_csucc";
+val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
+val csucc_le = thm "csucc_le";
+val lt_csucc_iff = thm "lt_csucc_iff";
+val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
+val InfCard_csucc = thm "InfCard_csucc";
+val Finite_into_Fin = thm "Finite_into_Fin";
+val Fin_into_Finite = thm "Fin_into_Finite";
+val Finite_Fin_iff = thm "Finite_Fin_iff";
+val Finite_Un = thm "Finite_Un";
+val Finite_Union = thm "Finite_Union";
+val Finite_induct = thm "Finite_induct";
+val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
+val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
+val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
+val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
+val nat_implies_well_ord = thm "nat_implies_well_ord";
+val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
+val Diff_sing_Finite = thm "Diff_sing_Finite";
+val Diff_Finite = thm "Diff_Finite";
+val Ord_subset_natD = thm "Ord_subset_natD";
+val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
+val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
+val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
+val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
+*}
+
end