src/ZF/CardinalArith.ML
author lcp
Mon, 22 Aug 1994 11:11:17 +0200
changeset 571 0b03ce5b62f7
parent 523 972119df615e
child 760 f0200e91b272
permissions -rw-r--r--
ZF/Cardinal: some results moved here from CardinalArith ZF/CardinalArith/nat_succ_lepoll: removed obsolete proof ZF/CardinalArith/nat_cons_eqpoll: new
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(*  Title: 	ZF/CardinalArith.ML
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Cardinal arithmetic -- WITHOUT the Axiom of Choice
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*)
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open CardinalArith;
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(*** Elementary properties ***)
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(*Use AC to discharge first premise*)
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goal CardinalArith.thy
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    "!!A B. [| well_ord(B,r);  A lepoll B |] ==> |A| le |B|";
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by (res_inst_tac [("i","|A|"),("j","|B|")] Ord_linear_le 1);
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by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI]));
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by (rtac (eqpollI RS cardinal_cong) 1 THEN assume_tac 1);
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by (rtac lepoll_trans 1);
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by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
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by (assume_tac 1);
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by (etac (le_imp_subset RS subset_imp_lepoll RS lepoll_trans) 1);
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by (rtac eqpoll_imp_lepoll 1);
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by (rewtac lepoll_def);
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by (etac exE 1);
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by (rtac well_ord_cardinal_eqpoll 1);
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by (etac well_ord_rvimage 1);
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by (assume_tac 1);
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val well_ord_lepoll_imp_le = result();
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(*Each element of Fin(A) is equivalent to a natural number*)
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goal CardinalArith.thy
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    "!!X A. X: Fin(A) ==> EX n:nat. X eqpoll n";
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by (eresolve_tac [Fin_induct] 1);
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by (fast_tac (ZF_cs addIs [eqpoll_refl, nat_0I]) 1);
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by (fast_tac (ZF_cs addSIs [cons_eqpoll_cong, 
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			    rewrite_rule [succ_def] nat_succI] 
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                            addSEs [mem_irrefl]) 1);
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val Fin_eqpoll = result();
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(*** Cardinal addition ***)
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(** Cardinal addition is commutative **)
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goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A";
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by (rtac exI 1);
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by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] 
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    lam_bijective 1);
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by (safe_tac (ZF_cs addSEs [sumE]));
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by (ALLGOALS (asm_simp_tac case_ss));
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val sum_commute_eqpoll = result();
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goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i";
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by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
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val cadd_commute = result();
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(** Cardinal addition is associative **)
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goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
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by (rtac exI 1);
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by (res_inst_tac [("c", "case(case(Inl, %y.Inr(Inl(y))), %y. Inr(Inr(y)))"),
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		  ("d", "case(%x.Inl(Inl(x)), case(%x.Inl(Inr(x)), Inr))")] 
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    lam_bijective 1);
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by (ALLGOALS (asm_simp_tac (case_ss setloop etac sumE)));
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val sum_assoc_eqpoll = result();
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(*Unconditional version requires AC*)
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goalw CardinalArith.thy [cadd_def]
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    "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>	\
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\             (i |+| j) |+| k = i |+| (j |+| k)";
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by (rtac cardinal_cong 1);
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br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
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    eqpoll_trans) 1;
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by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
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br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
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    eqpoll_sym) 2;
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by (REPEAT (ares_tac [well_ord_radd] 1));
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val well_ord_cadd_assoc = result();
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(** 0 is the identity for addition **)
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goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A";
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by (rtac exI 1);
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by (res_inst_tac [("c", "case(%x.x, %y.y)"), ("d", "Inr")] 
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    lam_bijective 1);
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by (ALLGOALS (asm_simp_tac (case_ss setloop eresolve_tac [sumE,emptyE])));
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val sum_0_eqpoll = result();
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goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K";
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by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong, 
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				  Card_cardinal_eq]) 1);
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val cadd_0 = result();
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(** Addition of finite cardinals is "ordinary" addition **)
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goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
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by (rtac exI 1);
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by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"), 
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		  ("d", "%z.if(z=A+B,Inl(A),z)")] 
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    lam_bijective 1);
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by (ALLGOALS
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    (asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq]
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		           setloop eresolve_tac [sumE,succE])));
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val sum_succ_eqpoll = result();
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(*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
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(*Unconditional version requires AC*)
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goalw CardinalArith.thy [cadd_def]
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    "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
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by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
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by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
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by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
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val cadd_succ_lemma = result();
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val [mnat,nnat] = goal CardinalArith.thy
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    "[| m: nat;  n: nat |] ==> m |+| n = m#+n";
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by (cut_facts_tac [nnat] 1);
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by (nat_ind_tac "m" [mnat] 1);
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by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1);
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by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma,
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				     nat_into_Card RS Card_cardinal_eq]) 1);
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val nat_cadd_eq_add = result();
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(*** Cardinal multiplication ***)
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(** Cardinal multiplication is commutative **)
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(*Easier to prove the two directions separately*)
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goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A";
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by (rtac exI 1);
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by (res_inst_tac [("c", "split(%x y.<y,x>)"), ("d", "split(%x y.<y,x>)")] 
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    lam_bijective 1);
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by (safe_tac ZF_cs);
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by (ALLGOALS (asm_simp_tac ZF_ss));
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val prod_commute_eqpoll = result();
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goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i";
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by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
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val cmult_commute = result();
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(** Cardinal multiplication is associative **)
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goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
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by (rtac exI 1);
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by (res_inst_tac [("c", "split(%w z. split(%x y. <x,<y,z>>, w))"),
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		  ("d", "split(%x.   split(%y z. <<x,y>, z>))")] 
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    lam_bijective 1);
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by (safe_tac ZF_cs);
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by (ALLGOALS (asm_simp_tac ZF_ss));
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val prod_assoc_eqpoll = result();
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(*Unconditional version requires AC*)
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goalw CardinalArith.thy [cmult_def]
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    "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>	\
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\             (i |*| j) |*| k = i |*| (j |*| k)";
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by (rtac cardinal_cong 1);
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br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
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    eqpoll_trans) 1;
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by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
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br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
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    eqpoll_sym) 2;
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by (REPEAT (ares_tac [well_ord_rmult] 1));
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val well_ord_cmult_assoc = result();
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(** Cardinal multiplication distributes over addition **)
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goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
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by (rtac exI 1);
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   171
by (res_inst_tac
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    [("c", "split(%x z. case(%y.Inl(<y,z>), %y.Inr(<y,z>), x))"),
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     ("d", "case(split(%x y.<Inl(x),y>), split(%x y.<Inr(x),y>))")] 
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    lam_bijective 1);
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by (safe_tac (ZF_cs addSEs [sumE]));
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by (ALLGOALS (asm_simp_tac case_ss));
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val sum_prod_distrib_eqpoll = result();
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   178
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   179
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
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by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
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by (simp_tac (ZF_ss addsimps [lam_type]) 1);
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val prod_square_lepoll = result();
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   183
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   184
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K";
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by (rtac le_trans 1);
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   186
by (rtac well_ord_lepoll_imp_le 2);
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   187
by (rtac prod_square_lepoll 3);
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   188
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
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   189
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
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val cmult_square_le = result();
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   191
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(** Multiplication by 0 yields 0 **)
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   193
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   194
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
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by (rtac exI 1);
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   196
by (rtac lam_bijective 1);
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   197
by (safe_tac ZF_cs);
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val prod_0_eqpoll = result();
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   199
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   200
goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
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   201
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, 
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				  Card_0 RS Card_cardinal_eq]) 1);
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val cmult_0 = result();
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   204
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   205
(** 1 is the identity for multiplication **)
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   206
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   207
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A";
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   208
by (rtac exI 1);
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   209
by (res_inst_tac [("c", "snd"), ("d", "%z.<x,z>")] lam_bijective 1);
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   210
by (safe_tac ZF_cs);
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   211
by (ALLGOALS (asm_simp_tac ZF_ss));
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   212
val prod_singleton_eqpoll = result();
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   213
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   214
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K";
437
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   215
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, 
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   216
				  Card_cardinal_eq]) 1);
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val cmult_1 = result();
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   218
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   219
(** Multiplication of finite cardinals is "ordinary" multiplication **)
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   220
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   221
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
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   222
by (rtac exI 1);
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   223
by (res_inst_tac [("c", "split(%x y. if(x=A, Inl(y), Inr(<x,y>)))"), 
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   224
		  ("d", "case(%y. <A,y>, %z.z)")] 
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   225
    lam_bijective 1);
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   226
by (safe_tac (ZF_cs addSEs [sumE]));
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   227
by (ALLGOALS
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   228
    (asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq])));
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   229
val prod_succ_eqpoll = result();
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   230
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   231
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(*Unconditional version requires AC*)
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   233
goalw CardinalArith.thy [cmult_def, cadd_def]
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    "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
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   235
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
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   236
by (rtac (cardinal_cong RS sym) 1);
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   237
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
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   238
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
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   239
val cmult_succ_lemma = result();
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   240
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   241
val [mnat,nnat] = goal CardinalArith.thy
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   242
    "[| m: nat;  n: nat |] ==> m |*| n = m#*n";
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   243
by (cut_facts_tac [nnat] 1);
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   244
by (nat_ind_tac "m" [mnat] 1);
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   245
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1);
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   246
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma,
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   247
				     nat_cadd_eq_add]) 1);
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   248
val nat_cmult_eq_mult = result();
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   249
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   250
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   251
(*** Infinite Cardinals are Limit Ordinals ***)
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   252
571
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   253
(*This proof is modelled upon one assuming nat<=A, with injection
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   254
  lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z))  and inverse
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   255
  %y. if(y:nat, nat_case(u,%z.z,y), y).  If f: inj(nat,A) then
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   256
  range(f) behaves like the natural numbers.*)
516
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   257
goalw CardinalArith.thy [lepoll_def]
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   258
    "!!i. nat lepoll A ==> cons(u,A) lepoll A";
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   259
by (etac exE 1);
516
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   260
by (res_inst_tac [("x",
571
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   261
    "lam z:cons(u,A). if(z=u, f`0, 	\
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   262
\                        if(z: range(f), f`succ(converse(f)`z), z))")] exI 1);
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parents: 523
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   263
by (res_inst_tac [("d", "%y. if(y: range(f), 	\
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   264
\                               nat_case(u, %z.f`z, converse(f)`y), y)")] 
516
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   265
    lam_injective 1);
571
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   266
by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type]
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   267
                    addIs  [inj_is_fun, inj_converse_fun]) 1);
516
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   268
by (asm_simp_tac 
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   269
    (ZF_ss addsimps [inj_is_fun RS apply_rangeI,
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   270
		     inj_converse_fun RS apply_rangeI,
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   271
		     inj_converse_fun RS apply_funtype,
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   272
		     left_inverse, right_inverse, nat_0I, nat_succI, 
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   273
		     nat_case_0, nat_case_succ]
516
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   274
           setloop split_tac [expand_if]) 1);
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   275
val nat_cons_lepoll = result();
516
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   276
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   277
goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A";
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   278
by (etac (nat_cons_lepoll RS eqpollI) 1);
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   279
by (rtac (subset_consI RS subset_imp_lepoll) 1);
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   280
val nat_cons_eqpoll = result();
437
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   281
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   282
(*Specialized version required below*)
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   283
goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A";
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parents: 523
diff changeset
   284
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
437
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   285
val nat_succ_eqpoll = result();
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   286
488
52f7447d4f1b Addition of infinite branching datatypes
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   287
goalw CardinalArith.thy [InfCard_def] "InfCard(nat)";
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   288
by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1);
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   289
val InfCard_nat = result();
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   290
484
70b789956bd3 Axiom of choice, cardinality results, etc.
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   291
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)";
437
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   292
by (etac conjunct1 1);
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   293
val InfCard_is_Card = result();
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   294
523
972119df615e ZF/CardinalArith/InfCard_Un: new
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   295
goalw CardinalArith.thy [InfCard_def]
972119df615e ZF/CardinalArith/InfCard_Un: new
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   296
    "!!K L. [| InfCard(K);  Card(L) |] ==> InfCard(K Un L)";
972119df615e ZF/CardinalArith/InfCard_Un: new
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parents: 517
diff changeset
   297
by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 
972119df615e ZF/CardinalArith/InfCard_Un: new
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   298
				  Card_is_Ord]) 1);
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   299
val InfCard_Un = result();
972119df615e ZF/CardinalArith/InfCard_Un: new
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   300
437
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   301
(*Kunen's Lemma 10.11*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
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parents: 467
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   302
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)";
437
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   303
by (etac conjE 1);
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   304
by (rtac (ltI RS non_succ_LimitI) 1);
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parents:
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   305
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
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parents:
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   306
by (etac Card_is_Ord 1);
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   307
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD]));
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parents:
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   308
by (forward_tac [Card_is_Ord RS Ord_succD] 1);
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parents:
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   309
by (rewtac Card_def);
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   310
by (res_inst_tac [("i", "succ(y)")] lt_irrefl 1);
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   311
by (dtac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
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parents:
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   312
(*Tricky combination of substitutions; backtracking needed*)
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   313
by (etac ssubst 1 THEN etac ssubst 1 THEN rtac Ord_cardinal_le 1);
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   314
by (assume_tac 1);
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   315
val InfCard_is_Limit = result();
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   316
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   317
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   318
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   319
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
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   320
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   321
(*A general fact about ordermap*)
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   322
goalw Cardinal.thy [eqpoll_def]
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   323
    "!!A. [| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
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diff changeset
   324
by (rtac exI 1);
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parents:
diff changeset
   325
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
467
92868dab2939 new cardinal arithmetic developments
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parents: 437
diff changeset
   326
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
437
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parents:
diff changeset
   327
by (rtac pred_subset 1);
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   328
val ordermap_eqpoll_pred = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   329
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   330
(** Establishing the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   331
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   332
goalw CardinalArith.thy [inj_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   333
 "!!K. Ord(K) ==>	\
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   334
\ (lam z:K*K. split(%x y. <x Un y, <x, y>>, z)) : inj(K*K, K*K*K)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   335
by (safe_tac ZF_cs);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   336
by (fast_tac (ZF_cs addIs [lam_type, Un_least_lt RS ltD, ltI]
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   337
                    addSEs [split_type]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   338
by (asm_full_simp_tac ZF_ss 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   339
val csquare_lam_inj = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   340
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   341
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   342
 "!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   343
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   344
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   345
val well_ord_csquare = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   346
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   347
(** Characterising initial segments of the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   348
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   349
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   350
 "!!K. [| x<K;  y<K;  z<K |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   351
\      <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   352
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   353
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   354
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   355
by (safe_tac (ZF_cs addSEs [mem_irrefl] 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   356
                    addSIs [Un_upper1_le, Un_upper2_le]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   357
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ])));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   358
val csquareD_lemma = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   359
val csquareD = csquareD_lemma RS mp |> standard;
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   360
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   361
goalw CardinalArith.thy [pred_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   362
 "!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   363
by (safe_tac (lemmas_cs addSEs [SigmaE]));	(*avoids using succCI*)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   364
by (rtac (csquareD RS conjE) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   365
by (rewtac lt_def);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   366
by (assume_tac 4);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   367
by (ALLGOALS (fast_tac ZF_cs));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   368
val pred_csquare_subset = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   369
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   370
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   371
 "!!K. [| x<z;  y<z;  z<K |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   372
\      <<x,y>, <z,z>> : csquare_rel(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   373
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   374
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   375
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   376
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   377
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   378
val csquare_ltI = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   379
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   380
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   381
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   382
 "!!K. [| x le z;  y le z;  z<K |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   383
\      <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   384
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   385
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   386
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   387
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   388
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   389
by (REPEAT_FIRST (etac succE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   390
by (ALLGOALS
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   391
    (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   392
				   subset_Un_iff2 RS iff_sym, OrdmemD])));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   393
val csquare_or_eqI = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   394
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   395
(** The cardinality of initial segments **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   396
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   397
goal CardinalArith.thy
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   398
    "!!K. [| InfCard(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   399
\         ordermap(K*K, csquare_rel(K)) ` <x,y> lepoll 		\
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   400
\         ordermap(K*K, csquare_rel(K)) ` <z,z>";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   401
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   402
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   403
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   404
by (rtac (OrdmemD RS subset_imp_lepoll) 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   405
by (res_inst_tac [("z1","z")] (csquare_ltI RS ordermap_mono) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   406
by (etac well_ord_is_wf 4);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   407
by (ALLGOALS 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   408
    (fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   409
                     addSEs [ltE])));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   410
val ordermap_z_lepoll = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   411
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   412
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   413
goalw CardinalArith.thy [cmult_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   414
  "!!K. [| InfCard(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   415
\       | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   416
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_le) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   417
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   418
by (subgoals_tac ["z<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   419
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   420
                            Limit_has_succ]) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   421
by (rtac (ordermap_z_lepoll RS lepoll_trans) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   422
by (REPEAT_SOME assume_tac);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   423
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   424
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   425
by (fast_tac (ZF_cs addIs [ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   426
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   427
    assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   428
by (REPEAT_FIRST (etac ltE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   429
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   430
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   431
val ordermap_csquare_le = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   432
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   433
(*Kunen: "... so the order type <= K" *)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   434
goal CardinalArith.thy
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   435
    "!!K. [| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |]  ==>  \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   436
\         ordertype(K*K, csquare_rel(K)) le K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   437
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   438
by (rtac all_lt_imp_le 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   439
by (assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   440
by (etac (well_ord_csquare RS Ord_ordertype) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   441
by (rtac Card_lt_imp_lt 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   442
by (etac InfCard_is_Card 3);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   443
by (etac ltE 2 THEN assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   444
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   445
by (safe_tac (ZF_cs addSEs [ltE]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   446
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   447
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   448
by (rtac (ordermap_csquare_le RS lt_trans1) 1  THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   449
    REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   450
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1  THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   451
    REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   452
(*the finite case: xb Un y < nat *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   453
by (res_inst_tac [("j", "nat")] lt_trans2 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   454
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   455
by (asm_full_simp_tac
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   456
    (ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   457
		     nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   458
(*case nat le (xb Un y), equivalently InfCard(xb Un y)  *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   459
by (asm_full_simp_tac
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   460
    (ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   461
		     le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   462
		     Ord_Un, ltI, nat_le_cardinal,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   463
		     Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   464
val ordertype_csquare_le = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   465
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   466
(*This lemma can easily be generalized to premise well_ord(A*A,r) *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   467
goalw CardinalArith.thy [cmult_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   468
    "!!K. Ord(K) ==> K |*| K  =  |ordertype(K*K, csquare_rel(K))|";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   469
by (rtac cardinal_cong 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   470
by (rewtac eqpoll_def);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   471
by (rtac exI 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   472
by (etac (well_ord_csquare RS ordermap_bij) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   473
val csquare_eq_ordertype = result();
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   474
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   475
(*Main result: Kunen's Theorem 10.12*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   476
goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   477
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   478
by (etac rev_mp 1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   479
by (trans_ind_tac "K" [] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   480
by (rtac impI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   481
by (rtac le_anti_sym 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   482
by (etac (InfCard_is_Card RS cmult_square_le) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   483
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   484
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   485
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   486
by (asm_simp_tac 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   487
    (ZF_ss addsimps [csquare_eq_ordertype, Ord_cardinal_le,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   488
                     well_ord_csquare RS Ord_ordertype]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   489
val InfCard_csquare_eq = result();
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   490
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   491
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   492
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   493
    "!!A. [| well_ord(A,r);  InfCard(|A|) |] ==> A*A eqpoll A";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   494
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   495
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   496
by (resolve_tac [well_ord_cardinal_eqE] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   497
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   498
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   499
val well_ord_InfCard_square_eq = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   500
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   501
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   502
(*** For every cardinal number there exists a greater one
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   503
     [Kunen's Theorem 10.16, which would be trivial using AC] ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   504
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   505
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   506
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   507
by (safe_tac (ZF_cs addSIs [Ord_ordertype]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   508
bw Transset_def;
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   509
by (safe_tac ZF_cs);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   510
by (rtac (ordertype_subset RS exE) 1 THEN REPEAT (assume_tac 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   511
by (resolve_tac [UN_I] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   512
by (resolve_tac [ReplaceI] 2);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   513
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset])));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   514
val Ord_jump_cardinal = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   515
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   516
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   517
goalw CardinalArith.thy [jump_cardinal_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   518
     "i : jump_cardinal(K) <->   \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   519
\         (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   520
by (fast_tac subset_cs 1);	(*It's vital to avoid reasoning about <=*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   521
val jump_cardinal_iff = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   522
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   523
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   524
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   525
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   526
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   527
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   528
by (resolve_tac [subset_refl] 2);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   529
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   530
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   531
val K_lt_jump_cardinal = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   532
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   533
(*The proof by contradiction: the bijection f yields a wellordering of X
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   534
  whose ordertype is jump_cardinal(K).  *)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   535
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   536
    "!!K. [| well_ord(X,r);  r <= K * K;  X <= K;	\
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   537
\            f : bij(ordertype(X,r), jump_cardinal(K)) 	\
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   538
\	  |] ==> jump_cardinal(K) : jump_cardinal(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   539
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   540
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   541
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   542
by (REPEAT_FIRST (resolve_tac [exI, conjI]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   543
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   544
by (REPEAT (assume_tac 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   545
by (etac (bij_is_inj RS well_ord_rvimage) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   546
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   547
by (asm_simp_tac
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   548
    (ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   549
		     ordertype_Memrel, Ord_jump_cardinal]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   550
val Card_jump_cardinal_lemma = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   551
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   552
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   553
goal CardinalArith.thy "Card(jump_cardinal(K))";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   554
by (rtac (Ord_jump_cardinal RS CardI) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   555
by (rewrite_goals_tac [eqpoll_def]);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   556
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   557
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   558
val Card_jump_cardinal = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   559
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   560
(*** Basic properties of successor cardinals ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   561
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   562
goalw CardinalArith.thy [csucc_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   563
    "!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   564
by (rtac LeastI 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   565
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   566
		      Ord_jump_cardinal] 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   567
val csucc_basic = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   568
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   569
val Card_csucc = csucc_basic RS conjunct1 |> standard;
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   570
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   571
val lt_csucc = csucc_basic RS conjunct2 |> standard;
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   572
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   573
goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)";
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   574
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   575
by (REPEAT (assume_tac 1));
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   576
val Ord_0_lt_csucc = result();
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   577
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   578
goalw CardinalArith.thy [csucc_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   579
    "!!K L. [| Card(L);  K<L |] ==> csucc(K) le L";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   580
by (rtac Least_le 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   581
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   582
val csucc_le = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   583
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   584
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   585
    "!!K. [| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   586
by (resolve_tac [iffI] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   587
by (resolve_tac [Card_lt_imp_lt] 2);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   588
by (eresolve_tac [lt_trans1] 2);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   589
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   590
by (resolve_tac [notI RS not_lt_imp_le] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   591
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   592
by (assume_tac 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   593
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   594
by (REPEAT (ares_tac [Ord_cardinal] 1
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   595
     ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   596
val lt_csucc_iff = result();
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   597
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   598
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   599
    "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   600
by (asm_simp_tac 
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   601
    (ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   602
val Card_lt_csucc_iff = result();
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   603
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   604
goalw CardinalArith.thy [InfCard_def]
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   605
    "!!K. InfCard(K) ==> InfCard(csucc(K))";
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   606
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, 
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   607
				  lt_csucc RS leI RSN (2,le_trans)]) 1);
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   608
val InfCard_csucc = result();
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   609
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   610
val Limit_csucc = InfCard_csucc RS InfCard_is_Limit |> standard;