Axiom of choice, cardinality results, etc.
(* Title: ZF/CardinalArith.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Cardinal arithmetic -- WITHOUT the Axiom of Choice
*)
open CardinalArith;
(*** Elementary properties ***)
(*Use AC to discharge first premise*)
goal CardinalArith.thy
"!!A B. [| well_ord(B,r); A lepoll B |] ==> |A| le |B|";
by (res_inst_tac [("i","|A|"),("j","|B|")] Ord_linear_le 1);
by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI]));
by (rtac (eqpollI RS cardinal_cong) 1 THEN assume_tac 1);
by (rtac lepoll_trans 1);
by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
by (assume_tac 1);
by (etac (le_imp_subset RS subset_imp_lepoll RS lepoll_trans) 1);
by (rtac eqpoll_imp_lepoll 1);
by (rewtac lepoll_def);
by (etac exE 1);
by (rtac well_ord_cardinal_eqpoll 1);
by (etac well_ord_rvimage 1);
by (assume_tac 1);
val well_ord_lepoll_imp_le = result();
val case_ss = ZF_ss addsimps [Inl_iff, Inl_Inr_iff, Inr_iff, Inr_Inl_iff,
case_Inl, case_Inr, InlI, InrI];
(** Congruence laws for successor, cardinal addition and multiplication **)
val bij_inverse_ss =
case_ss addsimps [bij_is_fun RS apply_type,
bij_converse_bij RS bij_is_fun RS apply_type,
left_inverse_bij, right_inverse_bij];
(*Congruence law for cons under equipollence*)
goalw CardinalArith.thy [eqpoll_def]
"!!A B. [| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)";
by (safe_tac ZF_cs);
by (rtac exI 1);
by (res_inst_tac [("c", "%z.if(z=a,b,f`z)"),
("d", "%z.if(z=b,a,converse(f)`z)")] lam_bijective 1);
by (ALLGOALS
(asm_simp_tac (bij_inverse_ss addsimps [consI2]
setloop (etac consE ORELSE'
split_tac [expand_if]))));
by (fast_tac (ZF_cs addIs [bij_is_fun RS apply_type]) 1);
by (fast_tac (ZF_cs addIs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
val cons_eqpoll_cong = result();
(*Congruence law for succ under equipollence*)
goalw CardinalArith.thy [succ_def]
"!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)";
by (REPEAT (ares_tac [cons_eqpoll_cong, mem_not_refl] 1));
val succ_eqpoll_cong = result();
(*Each element of Fin(A) is equivalent to a natural number*)
goal CardinalArith.thy
"!!X A. X: Fin(A) ==> EX n:nat. X eqpoll n";
by (eresolve_tac [Fin_induct] 1);
by (fast_tac (ZF_cs addIs [eqpoll_refl, nat_0I]) 1);
by (fast_tac (ZF_cs addSIs [cons_eqpoll_cong,
rewrite_rule [succ_def] nat_succI]
addSEs [mem_irrefl]) 1);
val Fin_eqpoll = result();
(*Congruence law for + under equipollence*)
goalw CardinalArith.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
by (safe_tac ZF_cs);
by (rtac exI 1);
by (res_inst_tac [("c", "case(%x. Inl(f`x), %y. Inr(fa`y))"),
("d", "case(%x. Inl(converse(f)`x), %y. Inr(converse(fa)`y))")]
lam_bijective 1);
by (safe_tac (ZF_cs addSEs [sumE]));
by (ALLGOALS (asm_simp_tac bij_inverse_ss));
val sum_eqpoll_cong = result();
(*Congruence law for * under equipollence*)
goalw CardinalArith.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
by (safe_tac ZF_cs);
by (rtac exI 1);
by (res_inst_tac [("c", "split(%x y. <f`x, fa`y>)"),
("d", "split(%x y. <converse(f)`x, converse(fa)`y>)")]
lam_bijective 1);
by (safe_tac ZF_cs);
by (ALLGOALS (asm_simp_tac bij_inverse_ss));
val prod_eqpoll_cong = result();
(*** Cardinal addition ***)
(** Cardinal addition is commutative **)
(*Easier to prove the two directions separately*)
goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A";
by (rtac exI 1);
by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")]
lam_bijective 1);
by (safe_tac (ZF_cs addSEs [sumE]));
by (ALLGOALS (asm_simp_tac case_ss));
val sum_commute_eqpoll = result();
goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i";
by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
val cadd_commute = result();
(** Cardinal addition is associative **)
goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
by (rtac exI 1);
by (res_inst_tac [("c", "case(case(Inl, %y.Inr(Inl(y))), %y. Inr(Inr(y)))"),
("d", "case(%x.Inl(Inl(x)), case(%x.Inl(Inr(x)), Inr))")]
lam_bijective 1);
by (ALLGOALS (asm_simp_tac (case_ss setloop etac sumE)));
val sum_assoc_eqpoll = result();
(*Unconditional version requires AC*)
goalw CardinalArith.thy [cadd_def]
"!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
\ (i |+| j) |+| k = i |+| (j |+| k)";
by (rtac cardinal_cong 1);
br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
eqpoll_trans) 1;
by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
eqpoll_sym) 2;
by (REPEAT (ares_tac [well_ord_radd] 1));
val well_ord_cadd_assoc = result();
(** 0 is the identity for addition **)
goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A";
by (rtac exI 1);
by (res_inst_tac [("c", "case(%x.x, %y.y)"), ("d", "Inr")]
lam_bijective 1);
by (ALLGOALS (asm_simp_tac (case_ss setloop eresolve_tac [sumE,emptyE])));
val sum_0_eqpoll = result();
goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K";
by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong,
Card_cardinal_eq]) 1);
val cadd_0 = result();
(** Addition of finite cardinals is "ordinary" addition **)
goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
by (rtac exI 1);
by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"),
("d", "%z.if(z=A+B,Inl(A),z)")]
lam_bijective 1);
by (ALLGOALS
(asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq]
setloop eresolve_tac [sumE,succE])));
val sum_succ_eqpoll = result();
(*Pulling the succ(...) outside the |...| requires m, n: nat *)
(*Unconditional version requires AC*)
goalw CardinalArith.thy [cadd_def]
"!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
val cadd_succ_lemma = result();
val [mnat,nnat] = goal CardinalArith.thy
"[| m: nat; n: nat |] ==> m |+| n = m#+n";
by (cut_facts_tac [nnat] 1);
by (nat_ind_tac "m" [mnat] 1);
by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1);
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma,
nat_into_Card RS Card_cardinal_eq]) 1);
val nat_cadd_eq_add = result();
(*** Cardinal multiplication ***)
(** Cardinal multiplication is commutative **)
(*Easier to prove the two directions separately*)
goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A";
by (rtac exI 1);
by (res_inst_tac [("c", "split(%x y.<y,x>)"), ("d", "split(%x y.<y,x>)")]
lam_bijective 1);
by (safe_tac ZF_cs);
by (ALLGOALS (asm_simp_tac ZF_ss));
val prod_commute_eqpoll = result();
goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i";
by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
val cmult_commute = result();
(** Cardinal multiplication is associative **)
goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
by (rtac exI 1);
by (res_inst_tac [("c", "split(%w z. split(%x y. <x,<y,z>>, w))"),
("d", "split(%x. split(%y z. <<x,y>, z>))")]
lam_bijective 1);
by (safe_tac ZF_cs);
by (ALLGOALS (asm_simp_tac ZF_ss));
val prod_assoc_eqpoll = result();
(*Unconditional version requires AC*)
goalw CardinalArith.thy [cmult_def]
"!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
\ (i |*| j) |*| k = i |*| (j |*| k)";
by (rtac cardinal_cong 1);
br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
eqpoll_trans) 1;
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
eqpoll_sym) 2;
by (REPEAT (ares_tac [well_ord_rmult] 1));
val well_ord_cmult_assoc = result();
(** Cardinal multiplication distributes over addition **)
goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
by (rtac exI 1);
by (res_inst_tac
[("c", "split(%x z. case(%y.Inl(<y,z>), %y.Inr(<y,z>), x))"),
("d", "case(split(%x y.<Inl(x),y>), split(%x y.<Inr(x),y>))")]
lam_bijective 1);
by (safe_tac (ZF_cs addSEs [sumE]));
by (ALLGOALS (asm_simp_tac case_ss));
val sum_prod_distrib_eqpoll = result();
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
by (simp_tac (ZF_ss addsimps [lam_type]) 1);
val prod_square_lepoll = result();
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K";
by (rtac le_trans 1);
by (rtac well_ord_lepoll_imp_le 2);
by (rtac prod_square_lepoll 3);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
val cmult_square_le = result();
(** Multiplication by 0 yields 0 **)
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
by (rtac exI 1);
by (rtac lam_bijective 1);
by (safe_tac ZF_cs);
val prod_0_eqpoll = result();
goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong,
Card_0 RS Card_cardinal_eq]) 1);
val cmult_0 = result();
(** 1 is the identity for multiplication **)
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A";
by (rtac exI 1);
by (res_inst_tac [("c", "snd"), ("d", "%z.<x,z>")] lam_bijective 1);
by (safe_tac ZF_cs);
by (ALLGOALS (asm_simp_tac ZF_ss));
val prod_singleton_eqpoll = result();
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K";
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong,
Card_cardinal_eq]) 1);
val cmult_1 = result();
(** Multiplication of finite cardinals is "ordinary" multiplication **)
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
by (rtac exI 1);
by (res_inst_tac [("c", "split(%x y. if(x=A, Inl(y), Inr(<x,y>)))"),
("d", "case(%y. <A,y>, %z.z)")]
lam_bijective 1);
by (safe_tac (ZF_cs addSEs [sumE]));
by (ALLGOALS
(asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq])));
val prod_succ_eqpoll = result();
(*Unconditional version requires AC*)
goalw CardinalArith.thy [cmult_def, cadd_def]
"!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
by (rtac (cardinal_cong RS sym) 1);
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
val cmult_succ_lemma = result();
val [mnat,nnat] = goal CardinalArith.thy
"[| m: nat; n: nat |] ==> m |*| n = m#*n";
by (cut_facts_tac [nnat] 1);
by (nat_ind_tac "m" [mnat] 1);
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1);
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma,
nat_cadd_eq_add]) 1);
val nat_cmult_eq_mult = result();
(*** Infinite Cardinals are Limit Ordinals ***)
(*Using lam_injective might simplify this proof!*)
goalw CardinalArith.thy [lepoll_def, inj_def]
"!!i. nat <= A ==> succ(A) lepoll A";
by (res_inst_tac [("x",
"lam z:succ(A). if(z=A, 0, if(z:nat, succ(z), z))")] exI 1);
by (rtac (lam_type RS CollectI) 1);
by (rtac if_type 1);
by (etac ([asm_rl, nat_0I] MRS subsetD) 1);
by (etac succE 1);
by (contr_tac 1);
by (rtac if_type 1);
by (assume_tac 2);
by (etac ([asm_rl, nat_succI] MRS subsetD) 1 THEN assume_tac 1);
by (REPEAT (rtac ballI 1));
by (asm_simp_tac
(ZF_ss addsimps [succ_inject_iff, succ_not_0, succ_not_0 RS not_sym]
setloop split_tac [expand_if]) 1);
by (safe_tac (ZF_cs addSIs [nat_0I, nat_succI]));
val nat_succ_lepoll = result();
goal CardinalArith.thy "!!i. nat <= A ==> succ(A) eqpoll A";
by (etac (nat_succ_lepoll RS eqpollI) 1);
by (rtac (subset_succI RS subset_imp_lepoll) 1);
val nat_succ_eqpoll = result();
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)";
by (etac conjunct1 1);
val InfCard_is_Card = result();
(*Kunen's Lemma 10.11*)
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)";
by (etac conjE 1);
by (rtac (ltI RS non_succ_LimitI) 1);
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
by (etac Card_is_Ord 1);
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD]));
by (forward_tac [Card_is_Ord RS Ord_succD] 1);
by (rewtac Card_def);
by (res_inst_tac [("i", "succ(y)")] lt_irrefl 1);
by (dtac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
(*Tricky combination of substitutions; backtracking needed*)
by (etac ssubst 1 THEN etac ssubst 1 THEN rtac Ord_cardinal_le 1);
by (assume_tac 1);
val InfCard_is_Limit = result();
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
(*A general fact about ordermap*)
goalw Cardinal.thy [eqpoll_def]
"!!A. [| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
by (rtac exI 1);
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
by (rtac pred_subset 1);
val ordermap_eqpoll_pred = result();
(** Establishing the well-ordering **)
goalw CardinalArith.thy [inj_def]
"!!K. Ord(K) ==> \
\ (lam z:K*K. split(%x y. <x Un y, <x, y>>, z)) : inj(K*K, K*K*K)";
by (safe_tac ZF_cs);
by (fast_tac (ZF_cs addIs [lam_type, Un_least_lt RS ltD, ltI]
addSEs [split_type]) 1);
by (asm_full_simp_tac ZF_ss 1);
val csquare_lam_inj = result();
goalw CardinalArith.thy [csquare_rel_def]
"!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))";
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
val well_ord_csquare = result();
(** Characterising initial segments of the well-ordering **)
goalw CardinalArith.thy [csquare_rel_def]
"!!K. [| x<K; y<K; z<K |] ==> \
\ <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
by (REPEAT (etac ltE 1));
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
by (safe_tac (ZF_cs addSEs [mem_irrefl]
addSIs [Un_upper1_le, Un_upper2_le]));
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ])));
val csquareD_lemma = result();
val csquareD = csquareD_lemma RS mp |> standard;
goalw CardinalArith.thy [pred_def]
"!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*)
by (rtac (csquareD RS conjE) 1);
by (rewtac lt_def);
by (assume_tac 4);
by (ALLGOALS (fast_tac ZF_cs));
val pred_csquare_subset = result();
goalw CardinalArith.thy [csquare_rel_def]
"!!K. [| x<z; y<z; z<K |] ==> \
\ <<x,y>, <z,z>> : csquare_rel(K)";
by (subgoals_tac ["x<K", "y<K"] 1);
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
by (REPEAT (etac ltE 1));
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
val csquare_ltI = result();
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *)
goalw CardinalArith.thy [csquare_rel_def]
"!!K. [| x le z; y le z; z<K |] ==> \
\ <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
by (subgoals_tac ["x<K", "y<K"] 1);
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
by (REPEAT (etac ltE 1));
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
Un_absorb, Un_least_mem_iff, ltD]) 1);
by (REPEAT_FIRST (etac succE));
by (ALLGOALS
(asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym,
subset_Un_iff2 RS iff_sym, OrdmemD])));
val csquare_or_eqI = result();
(** The cardinality of initial segments **)
goal CardinalArith.thy
"!!K. [| InfCard(K); x<K; y<K; z=succ(x Un y) |] ==> \
\ ordermap(K*K, csquare_rel(K)) ` <x,y> lepoll \
\ ordermap(K*K, csquare_rel(K)) ` <z,z>";
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 2);
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2);
by (rtac (OrdmemD RS subset_imp_lepoll) 1);
by (res_inst_tac [("z1","z")] (csquare_ltI RS ordermap_mono) 1);
by (etac well_ord_is_wf 4);
by (ALLGOALS
(fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap]
addSEs [ltE])));
val ordermap_z_lepoll = result();
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
goalw CardinalArith.thy [cmult_def]
"!!K. [| InfCard(K); x<K; y<K; z=succ(x Un y) |] ==> \
\ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|";
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_le) 1);
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
by (subgoals_tac ["z<K"] 1);
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit,
Limit_has_succ]) 2);
by (rtac (ordermap_z_lepoll RS lepoll_trans) 1);
by (REPEAT_SOME assume_tac);
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 1);
by (fast_tac (ZF_cs addIs [ltD]) 1);
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
assume_tac 1);
by (REPEAT_FIRST (etac ltE));
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
val ordermap_csquare_le = result();
(*Kunen: "... so the order type <= K" *)
goal CardinalArith.thy
"!!K. [| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] ==> \
\ ordertype(K*K, csquare_rel(K)) le K";
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
by (rtac all_lt_imp_le 1);
by (assume_tac 1);
by (etac (well_ord_csquare RS Ord_ordertype) 1);
by (rtac Card_lt_imp_lt 1);
by (etac InfCard_is_Card 3);
by (etac ltE 2 THEN assume_tac 2);
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1);
by (safe_tac (ZF_cs addSEs [ltE]));
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
by (rtac (ordermap_csquare_le RS lt_trans1) 1 THEN
REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN
REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
(*the finite case: xb Un y < nat *)
by (res_inst_tac [("j", "nat")] lt_trans2 1);
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2);
by (asm_full_simp_tac
(ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
(*case nat le (xb Un y), equivalently InfCard(xb Un y) *)
by (asm_full_simp_tac
(ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt,
Ord_Un, ltI, nat_le_cardinal,
Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
val ordertype_csquare_le = result();
(*This lemma can easily be generalized to premise well_ord(A*A,r) *)
goalw CardinalArith.thy [cmult_def]
"!!K. Ord(K) ==> K |*| K = |ordertype(K*K, csquare_rel(K))|";
by (rtac cardinal_cong 1);
by (rewtac eqpoll_def);
by (rtac exI 1);
by (etac (well_ord_csquare RS ordermap_bij) 1);
val csquare_eq_ordertype = result();
(*Main result: Kunen's Theorem 10.12*)
goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K";
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
by (etac rev_mp 1);
by (trans_ind_tac "K" [] 1);
by (rtac impI 1);
by (rtac le_anti_sym 1);
by (etac (InfCard_is_Card RS cmult_square_le) 2);
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
by (assume_tac 2);
by (assume_tac 2);
by (asm_simp_tac
(ZF_ss addsimps [csquare_eq_ordertype, Ord_cardinal_le,
well_ord_csquare RS Ord_ordertype]) 1);
val InfCard_csquare_eq = result();
goal CardinalArith.thy
"!!A. [| well_ord(A,r); InfCard(|A|) |] ==> A*A eqpoll A";
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
by (resolve_tac [well_ord_cardinal_eqE] 1);
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
val well_ord_InfCard_square_eq = result();
(*** For every cardinal number there exists a greater one
[Kunen's Theorem 10.16, which would be trivial using AC] ***)
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))";
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
by (safe_tac (ZF_cs addSIs [Ord_ordertype]));
bw Transset_def;
by (safe_tac ZF_cs);
by (rtac (ordertype_subset RS exE) 1 THEN REPEAT (assume_tac 1));
by (resolve_tac [UN_I] 1);
by (resolve_tac [ReplaceI] 2);
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset])));
val Ord_jump_cardinal = result();
(*Allows selective unfolding. Less work than deriving intro/elim rules*)
goalw CardinalArith.thy [jump_cardinal_def]
"i : jump_cardinal(K) <-> \
\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*)
val jump_cardinal_iff = result();
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)";
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
by (resolve_tac [subset_refl] 2);
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1);
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1);
val K_lt_jump_cardinal = result();
(*The proof by contradiction: the bijection f yields a wellordering of X
whose ordertype is jump_cardinal(K). *)
goal CardinalArith.thy
"!!K. [| well_ord(X,r); r <= K * K; X <= K; \
\ f : bij(ordertype(X,r), jump_cardinal(K)) \
\ |] ==> jump_cardinal(K) : jump_cardinal(K)";
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
by (REPEAT_FIRST (resolve_tac [exI, conjI]));
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
by (REPEAT (assume_tac 1));
by (etac (bij_is_inj RS well_ord_rvimage) 1);
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
by (asm_simp_tac
(ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage),
ordertype_Memrel, Ord_jump_cardinal]) 1);
val Card_jump_cardinal_lemma = result();
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
goal CardinalArith.thy "Card(jump_cardinal(K))";
by (rtac (Ord_jump_cardinal RS CardI) 1);
by (rewrite_goals_tac [eqpoll_def]);
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1]));
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
val Card_jump_cardinal = result();
(*** Basic properties of successor cardinals ***)
goalw CardinalArith.thy [csucc_def]
"!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
by (rtac LeastI 1);
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
Ord_jump_cardinal] 1));
val csucc_basic = result();
val Card_csucc = csucc_basic RS conjunct1 |> standard;
val lt_csucc = csucc_basic RS conjunct2 |> standard;
goalw CardinalArith.thy [csucc_def]
"!!K L. [| Card(L); K<L |] ==> csucc(K) le L";
by (rtac Least_le 1);
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
val csucc_le = result();
goal CardinalArith.thy
"!!K. [| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
by (resolve_tac [iffI] 1);
by (resolve_tac [Card_lt_imp_lt] 2);
by (eresolve_tac [lt_trans1] 2);
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
by (resolve_tac [notI RS not_lt_imp_le] 1);
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
by (assume_tac 1);
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
by (REPEAT (ares_tac [Ord_cardinal] 1
ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
val lt_csucc_iff = result();
goal CardinalArith.thy
"!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
by (asm_simp_tac
(ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
val Card_lt_csucc_iff = result();