author | clasohm |
Fri, 17 Nov 1995 13:22:50 +0100 | |
changeset 1341 | 69fec018854c |
parent 1090 | 8ab69b3e396b |
child 1461 | 6bcb44e4d6e5 |
permissions | -rw-r--r-- |
437 | 1 |
(* 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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846 | 8 |
Note: Could omit proving the algebraic laws for cardinal addition and |
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multiplication. On finite cardinals these operations coincide with |
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addition and multiplication of natural numbers; on infinite cardinals they |
846 | 11 |
coincide with union (maximum). Either way we get most laws for free. |
437 | 12 |
*) |
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open CardinalArith; |
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484 | 16 |
(*** Elementary properties ***) |
467 | 17 |
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437 | 18 |
(*Use AC to discharge first premise*) |
19 |
goal CardinalArith.thy |
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"!!A B. [| well_ord(B,r); A lepoll B |] ==> |A| le |B|"; |
|
21 |
by (res_inst_tac [("i","|A|"),("j","|B|")] Ord_linear_le 1); |
|
22 |
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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29 |
by (rewtac lepoll_def); |
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30 |
by (etac exE 1); |
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31 |
by (rtac well_ord_cardinal_eqpoll 1); |
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32 |
by (etac well_ord_rvimage 1); |
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33 |
by (assume_tac 1); |
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767 | 34 |
qed "well_ord_lepoll_imp_Card_le"; |
437 | 35 |
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484 | 36 |
(*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 (etac Fin_induct 1); |
484 | 40 |
by (fast_tac (ZF_cs addIs [eqpoll_refl, nat_0I]) 1); |
41 |
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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760 | 44 |
qed "Fin_eqpoll"; |
484 | 45 |
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437 | 46 |
(*** Cardinal addition ***) |
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(** Cardinal addition is commutative **) |
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49 |
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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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760 | 56 |
qed "sum_commute_eqpoll"; |
437 | 57 |
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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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760 | 60 |
qed "cadd_commute"; |
437 | 61 |
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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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846 | 66 |
by (resolve_tac [sum_assoc_bij] 1); |
760 | 67 |
qed "sum_assoc_eqpoll"; |
437 | 68 |
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(*Unconditional version requires AC*) |
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goalw CardinalArith.thy [cadd_def] |
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484 | 71 |
"!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \ |
437 | 72 |
\ (i |+| j) |+| k = i |+| (j |+| k)"; |
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by (rtac cardinal_cong 1); |
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by (rtac ([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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by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS |
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eqpoll_sym) 2); |
484 | 79 |
by (REPEAT (ares_tac [well_ord_radd] 1)); |
760 | 80 |
qed "well_ord_cadd_assoc"; |
437 | 81 |
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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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846 | 86 |
by (rtac bij_0_sum 1); |
760 | 87 |
qed "sum_0_eqpoll"; |
437 | 88 |
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484 | 89 |
goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K"; |
437 | 90 |
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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760 | 92 |
qed "cadd_0"; |
437 | 93 |
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767 | 94 |
(** Addition by another cardinal **) |
95 |
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goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A+B"; |
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by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1); |
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by (asm_simp_tac (sum_ss addsimps [lam_type]) 1); |
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qed "sum_lepoll_self"; |
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(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) |
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goalw CardinalArith.thy [cadd_def] |
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"!!K. [| Card(K); Ord(L) |] ==> K le (K |+| L)"; |
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by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); |
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by (rtac sum_lepoll_self 3); |
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1)); |
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qed "cadd_le_self"; |
767 | 108 |
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(** Monotonicity of addition **) |
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110 |
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goalw CardinalArith.thy [lepoll_def] |
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"!!A B C D. [| A lepoll C; B lepoll D |] ==> A + B lepoll C + D"; |
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by (REPEAT (etac exE 1)); |
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by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")] |
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exI 1); |
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by (res_inst_tac |
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[("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")] |
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lam_injective 1); |
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846 | 119 |
by (typechk_tac ([inj_is_fun, case_type, InlI, InrI] @ ZF_typechecks)); |
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by (etac sumE 1); |
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by (ALLGOALS (asm_simp_tac (sum_ss addsimps [left_inverse]))); |
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qed "sum_lepoll_mono"; |
767 | 123 |
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goalw CardinalArith.thy [cadd_def] |
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"!!K. [| K' le K; L' le L |] ==> (K' |+| L') le (K |+| L)"; |
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by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); |
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by (rtac well_ord_lepoll_imp_Card_le 1); |
767 | 128 |
by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2)); |
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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); |
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qed "cadd_le_mono"; |
767 | 131 |
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437 | 132 |
(** Addition of finite cardinals is "ordinary" addition **) |
133 |
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134 |
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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760 | 142 |
qed "sum_succ_eqpoll"; |
437 | 143 |
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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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760 | 152 |
qed "cadd_succ_lemma"; |
437 | 153 |
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val [mnat,nnat] = goal CardinalArith.thy |
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155 |
"[| 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); |
|
159 |
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma, |
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160 |
nat_into_Card RS Card_cardinal_eq]) 1); |
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760 | 161 |
qed "nat_cadd_eq_add"; |
437 | 162 |
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163 |
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(*** Cardinal multiplication ***) |
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165 |
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(** Cardinal multiplication is commutative **) |
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167 |
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168 |
(*Easier to prove the two directions separately*) |
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169 |
goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A"; |
|
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by (rtac exI 1); |
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171 |
by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")] |
437 | 172 |
lam_bijective 1); |
173 |
by (safe_tac ZF_cs); |
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174 |
by (ALLGOALS (asm_simp_tac ZF_ss)); |
|
760 | 175 |
qed "prod_commute_eqpoll"; |
437 | 176 |
|
177 |
goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i"; |
|
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by (rtac (prod_commute_eqpoll RS cardinal_cong) 1); |
|
760 | 179 |
qed "cmult_commute"; |
437 | 180 |
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(** Cardinal multiplication is associative **) |
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182 |
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183 |
goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)"; |
|
184 |
by (rtac exI 1); |
|
846 | 185 |
by (resolve_tac [prod_assoc_bij] 1); |
760 | 186 |
qed "prod_assoc_eqpoll"; |
437 | 187 |
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188 |
(*Unconditional version requires AC*) |
|
189 |
goalw CardinalArith.thy [cmult_def] |
|
484 | 190 |
"!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \ |
437 | 191 |
\ (i |*| j) |*| k = i |*| (j |*| k)"; |
192 |
by (rtac cardinal_cong 1); |
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193 |
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS |
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eqpoll_trans) 1); |
437 | 195 |
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2); |
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|
196 |
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS |
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|
197 |
eqpoll_sym) 2); |
484 | 198 |
by (REPEAT (ares_tac [well_ord_rmult] 1)); |
760 | 199 |
qed "well_ord_cmult_assoc"; |
437 | 200 |
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(** Cardinal multiplication distributes over addition **) |
|
202 |
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goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)"; |
|
204 |
by (rtac exI 1); |
|
846 | 205 |
by (resolve_tac [sum_prod_distrib_bij] 1); |
760 | 206 |
qed "sum_prod_distrib_eqpoll"; |
437 | 207 |
|
846 | 208 |
goalw CardinalArith.thy [cadd_def, cmult_def] |
209 |
"!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \ |
|
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\ (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"; |
|
211 |
by (rtac cardinal_cong 1); |
|
212 |
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS |
|
213 |
eqpoll_trans) 1); |
|
214 |
by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2); |
|
215 |
by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS |
|
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eqpoll_sym) 2); |
|
217 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1)); |
|
218 |
qed "well_ord_cadd_cmult_distrib"; |
|
219 |
||
437 | 220 |
(** Multiplication by 0 yields 0 **) |
221 |
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222 |
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0"; |
|
223 |
by (rtac exI 1); |
|
224 |
by (rtac lam_bijective 1); |
|
225 |
by (safe_tac ZF_cs); |
|
760 | 226 |
qed "prod_0_eqpoll"; |
437 | 227 |
|
228 |
goalw CardinalArith.thy [cmult_def] "0 |*| i = 0"; |
|
229 |
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, |
|
230 |
Card_0 RS Card_cardinal_eq]) 1); |
|
760 | 231 |
qed "cmult_0"; |
437 | 232 |
|
233 |
(** 1 is the identity for multiplication **) |
|
234 |
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235 |
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A"; |
|
236 |
by (rtac exI 1); |
|
846 | 237 |
by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1); |
760 | 238 |
qed "prod_singleton_eqpoll"; |
437 | 239 |
|
484 | 240 |
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K"; |
437 | 241 |
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, |
242 |
Card_cardinal_eq]) 1); |
|
760 | 243 |
qed "cmult_1"; |
437 | 244 |
|
767 | 245 |
(*** Some inequalities for multiplication ***) |
246 |
||
247 |
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A"; |
|
248 |
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1); |
|
249 |
by (simp_tac (ZF_ss addsimps [lam_type]) 1); |
|
250 |
qed "prod_square_lepoll"; |
|
251 |
||
823
33dc37d46296
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diff
changeset
|
252 |
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) |
767 | 253 |
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K"; |
254 |
by (rtac le_trans 1); |
|
255 |
by (rtac well_ord_lepoll_imp_Card_le 2); |
|
256 |
by (rtac prod_square_lepoll 3); |
|
257 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2)); |
|
258 |
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1); |
|
259 |
qed "cmult_square_le"; |
|
260 |
||
261 |
(** Multiplication by a non-zero cardinal **) |
|
262 |
||
263 |
goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B"; |
|
264 |
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1); |
|
265 |
by (asm_simp_tac (ZF_ss addsimps [lam_type]) 1); |
|
782
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767
diff
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|
266 |
qed "prod_lepoll_self"; |
767 | 267 |
|
268 |
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) |
|
269 |
goalw CardinalArith.thy [cmult_def] |
|
270 |
"!!K. [| Card(K); Ord(L); 0<L |] ==> K le (K |*| L)"; |
|
271 |
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); |
|
272 |
by (rtac prod_lepoll_self 3); |
|
273 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1)); |
|
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|
274 |
qed "cmult_le_self"; |
767 | 275 |
|
276 |
(** Monotonicity of multiplication **) |
|
277 |
||
278 |
goalw CardinalArith.thy [lepoll_def] |
|
279 |
"!!A B C D. [| A lepoll C; B lepoll D |] ==> A * B lepoll C * D"; |
|
280 |
by (REPEAT (etac exE 1)); |
|
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|
281 |
by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1); |
8ab69b3e396b
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diff
changeset
|
282 |
by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")] |
767 | 283 |
lam_injective 1); |
284 |
by (typechk_tac (inj_is_fun::ZF_typechecks)); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
285 |
by (etac SigmaE 1); |
767 | 286 |
by (asm_simp_tac (ZF_ss addsimps [left_inverse]) 1); |
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
287 |
qed "prod_lepoll_mono"; |
767 | 288 |
|
289 |
goalw CardinalArith.thy [cmult_def] |
|
290 |
"!!K. [| K' le K; L' le L |] ==> (K' |*| L') le (K |*| L)"; |
|
291 |
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
292 |
by (rtac well_ord_lepoll_imp_Card_le 1); |
767 | 293 |
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2)); |
294 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
295 |
qed "cmult_le_mono"; |
767 | 296 |
|
297 |
(*** Multiplication of finite cardinals is "ordinary" multiplication ***) |
|
437 | 298 |
|
299 |
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B"; |
|
300 |
by (rtac exI 1); |
|
1090
8ab69b3e396b
Changed some definitions and proofs to use pattern-matching.
lcp
parents:
1075
diff
changeset
|
301 |
by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"), |
437 | 302 |
("d", "case(%y. <A,y>, %z.z)")] |
303 |
lam_bijective 1); |
|
304 |
by (safe_tac (ZF_cs addSEs [sumE])); |
|
305 |
by (ALLGOALS |
|
306 |
(asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq]))); |
|
760 | 307 |
qed "prod_succ_eqpoll"; |
437 | 308 |
|
309 |
(*Unconditional version requires AC*) |
|
310 |
goalw CardinalArith.thy [cmult_def, cadd_def] |
|
311 |
"!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"; |
|
312 |
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1); |
|
313 |
by (rtac (cardinal_cong RS sym) 1); |
|
314 |
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1); |
|
315 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); |
|
760 | 316 |
qed "cmult_succ_lemma"; |
437 | 317 |
|
318 |
val [mnat,nnat] = goal CardinalArith.thy |
|
319 |
"[| m: nat; n: nat |] ==> m |*| n = m#*n"; |
|
320 |
by (cut_facts_tac [nnat] 1); |
|
321 |
by (nat_ind_tac "m" [mnat] 1); |
|
322 |
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1); |
|
323 |
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma, |
|
324 |
nat_cadd_eq_add]) 1); |
|
760 | 325 |
qed "nat_cmult_eq_mult"; |
437 | 326 |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
327 |
goal CardinalArith.thy "!!m n. Card(n) ==> 2 |*| n = n |+| n"; |
767 | 328 |
by (asm_simp_tac |
329 |
(ZF_ss addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord, |
|
330 |
read_instantiate [("j","0")] cadd_commute, cadd_0]) 1); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
331 |
qed "cmult_2"; |
767 | 332 |
|
437 | 333 |
|
334 |
(*** Infinite Cardinals are Limit Ordinals ***) |
|
335 |
||
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
336 |
(*This proof is modelled upon one assuming nat<=A, with injection |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
337 |
lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z)) and inverse |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
338 |
%y. if(y:nat, nat_case(u,%z.z,y), y). If f: inj(nat,A) then |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
339 |
range(f) behaves like the natural numbers.*) |
516 | 340 |
goalw CardinalArith.thy [lepoll_def] |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
341 |
"!!i. nat lepoll A ==> cons(u,A) lepoll A"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
342 |
by (etac exE 1); |
516 | 343 |
by (res_inst_tac [("x", |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
344 |
"lam z:cons(u,A). if(z=u, f`0, \ |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
345 |
\ if(z: range(f), f`succ(converse(f)`z), z))")] exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
346 |
by (res_inst_tac [("d", "%y. if(y: range(f), \ |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
347 |
\ nat_case(u, %z.f`z, converse(f)`y), y)")] |
516 | 348 |
lam_injective 1); |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
349 |
by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
350 |
addIs [inj_is_fun, inj_converse_fun]) 1); |
516 | 351 |
by (asm_simp_tac |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
352 |
(ZF_ss addsimps [inj_is_fun RS apply_rangeI, |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
353 |
inj_converse_fun RS apply_rangeI, |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
354 |
inj_converse_fun RS apply_funtype, |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
355 |
left_inverse, right_inverse, nat_0I, nat_succI, |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
356 |
nat_case_0, nat_case_succ] |
516 | 357 |
setloop split_tac [expand_if]) 1); |
760 | 358 |
qed "nat_cons_lepoll"; |
516 | 359 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
360 |
goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
361 |
by (etac (nat_cons_lepoll RS eqpollI) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
362 |
by (rtac (subset_consI RS subset_imp_lepoll) 1); |
760 | 363 |
qed "nat_cons_eqpoll"; |
437 | 364 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
365 |
(*Specialized version required below*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
366 |
goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset
|
367 |
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1); |
760 | 368 |
qed "nat_succ_eqpoll"; |
437 | 369 |
|
488 | 370 |
goalw CardinalArith.thy [InfCard_def] "InfCard(nat)"; |
371 |
by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1); |
|
760 | 372 |
qed "InfCard_nat"; |
488 | 373 |
|
484 | 374 |
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)"; |
437 | 375 |
by (etac conjunct1 1); |
760 | 376 |
qed "InfCard_is_Card"; |
437 | 377 |
|
523 | 378 |
goalw CardinalArith.thy [InfCard_def] |
379 |
"!!K L. [| InfCard(K); Card(L) |] ==> InfCard(K Un L)"; |
|
380 |
by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), |
|
381 |
Card_is_Ord]) 1); |
|
760 | 382 |
qed "InfCard_Un"; |
523 | 383 |
|
437 | 384 |
(*Kunen's Lemma 10.11*) |
484 | 385 |
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)"; |
437 | 386 |
by (etac conjE 1); |
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
387 |
by (forward_tac [Card_is_Ord] 1); |
437 | 388 |
by (rtac (ltI RS non_succ_LimitI) 1); |
389 |
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1); |
|
390 |
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD])); |
|
391 |
by (rewtac Card_def); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
392 |
by (dtac trans 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
393 |
by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
394 |
by (etac (Ord_succD RS Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
395 |
by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1)); |
760 | 396 |
qed "InfCard_is_Limit"; |
437 | 397 |
|
398 |
||
399 |
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) |
|
400 |
||
401 |
(*A general fact about ordermap*) |
|
402 |
goalw Cardinal.thy [eqpoll_def] |
|
403 |
"!!A. [| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)"; |
|
404 |
by (rtac exI 1); |
|
405 |
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1); |
|
467 | 406 |
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1); |
437 | 407 |
by (rtac pred_subset 1); |
760 | 408 |
qed "ordermap_eqpoll_pred"; |
437 | 409 |
|
410 |
(** Establishing the well-ordering **) |
|
411 |
||
412 |
goalw CardinalArith.thy [inj_def] |
|
1090
8ab69b3e396b
Changed some definitions and proofs to use pattern-matching.
lcp
parents:
1075
diff
changeset
|
413 |
"!!K. Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"; |
989 | 414 |
by (fast_tac (ZF_cs addss ZF_ss |
415 |
addIs [lam_type, Un_least_lt RS ltD, ltI]) 1); |
|
760 | 416 |
qed "csquare_lam_inj"; |
437 | 417 |
|
418 |
goalw CardinalArith.thy [csquare_rel_def] |
|
484 | 419 |
"!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))"; |
437 | 420 |
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1); |
421 |
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); |
|
760 | 422 |
qed "well_ord_csquare"; |
437 | 423 |
|
424 |
(** Characterising initial segments of the well-ordering **) |
|
425 |
||
426 |
goalw CardinalArith.thy [csquare_rel_def] |
|
484 | 427 |
"!!K. [| x<K; y<K; z<K |] ==> \ |
428 |
\ <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z"; |
|
437 | 429 |
by (REPEAT (etac ltE 1)); |
430 |
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, |
|
431 |
Un_absorb, Un_least_mem_iff, ltD]) 1); |
|
432 |
by (safe_tac (ZF_cs addSEs [mem_irrefl] |
|
433 |
addSIs [Un_upper1_le, Un_upper2_le])); |
|
434 |
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ]))); |
|
800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset
|
435 |
val csquareD_lemma = result(); |
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset
|
436 |
|
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset
|
437 |
bind_thm ("csquareD", csquareD_lemma RS mp); |
437 | 438 |
|
439 |
goalw CardinalArith.thy [pred_def] |
|
484 | 440 |
"!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"; |
437 | 441 |
by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*) |
442 |
by (rtac (csquareD RS conjE) 1); |
|
443 |
by (rewtac lt_def); |
|
444 |
by (assume_tac 4); |
|
445 |
by (ALLGOALS (fast_tac ZF_cs)); |
|
760 | 446 |
qed "pred_csquare_subset"; |
437 | 447 |
|
448 |
goalw CardinalArith.thy [csquare_rel_def] |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
449 |
"!!K. [| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K)"; |
484 | 450 |
by (subgoals_tac ["x<K", "y<K"] 1); |
437 | 451 |
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2)); |
452 |
by (REPEAT (etac ltE 1)); |
|
453 |
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, |
|
454 |
Un_absorb, Un_least_mem_iff, ltD]) 1); |
|
760 | 455 |
qed "csquare_ltI"; |
437 | 456 |
|
457 |
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) |
|
458 |
goalw CardinalArith.thy [csquare_rel_def] |
|
484 | 459 |
"!!K. [| x le z; y le z; z<K |] ==> \ |
460 |
\ <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"; |
|
461 |
by (subgoals_tac ["x<K", "y<K"] 1); |
|
437 | 462 |
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2)); |
463 |
by (REPEAT (etac ltE 1)); |
|
464 |
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, |
|
465 |
Un_absorb, Un_least_mem_iff, ltD]) 1); |
|
466 |
by (REPEAT_FIRST (etac succE)); |
|
467 |
by (ALLGOALS |
|
468 |
(asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, |
|
469 |
subset_Un_iff2 RS iff_sym, OrdmemD]))); |
|
760 | 470 |
qed "csquare_or_eqI"; |
437 | 471 |
|
472 |
(** The cardinality of initial segments **) |
|
473 |
||
474 |
goal CardinalArith.thy |
|
846 | 475 |
"!!K. [| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \ |
870 | 476 |
\ ordermap(K*K, csquare_rel(K)) ` <x,y> < \ |
484 | 477 |
\ ordermap(K*K, csquare_rel(K)) ` <z,z>"; |
478 |
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1); |
|
846 | 479 |
by (etac (Limit_is_Ord RS well_ord_csquare) 2); |
480 |
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2); |
|
870 | 481 |
by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1); |
437 | 482 |
by (etac well_ord_is_wf 4); |
483 |
by (ALLGOALS |
|
484 |
(fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] |
|
485 |
addSEs [ltE]))); |
|
870 | 486 |
qed "ordermap_z_lt"; |
437 | 487 |
|
484 | 488 |
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) |
437 | 489 |
goalw CardinalArith.thy [cmult_def] |
846 | 490 |
"!!K. [| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \ |
484 | 491 |
\ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|"; |
767 | 492 |
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1); |
437 | 493 |
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1)); |
484 | 494 |
by (subgoals_tac ["z<K"] 1); |
846 | 495 |
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2); |
870 | 496 |
by (rtac (ordermap_z_lt RS leI RS le_imp_subset RS subset_imp_lepoll RS |
497 |
lepoll_trans) 1); |
|
437 | 498 |
by (REPEAT_SOME assume_tac); |
499 |
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1); |
|
846 | 500 |
by (etac (Limit_is_Ord RS well_ord_csquare) 1); |
437 | 501 |
by (fast_tac (ZF_cs addIs [ltD]) 1); |
502 |
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN |
|
503 |
assume_tac 1); |
|
504 |
by (REPEAT_FIRST (etac ltE)); |
|
505 |
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1); |
|
506 |
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll))); |
|
760 | 507 |
qed "ordermap_csquare_le"; |
437 | 508 |
|
484 | 509 |
(*Kunen: "... so the order type <= K" *) |
437 | 510 |
goal CardinalArith.thy |
484 | 511 |
"!!K. [| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] ==> \ |
512 |
\ ordertype(K*K, csquare_rel(K)) le K"; |
|
437 | 513 |
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); |
514 |
by (rtac all_lt_imp_le 1); |
|
515 |
by (assume_tac 1); |
|
516 |
by (etac (well_ord_csquare RS Ord_ordertype) 1); |
|
517 |
by (rtac Card_lt_imp_lt 1); |
|
518 |
by (etac InfCard_is_Card 3); |
|
519 |
by (etac ltE 2 THEN assume_tac 2); |
|
520 |
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1); |
|
521 |
by (safe_tac (ZF_cs addSEs [ltE])); |
|
522 |
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1); |
|
523 |
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2)); |
|
846 | 524 |
by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1 THEN |
437 | 525 |
REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1)); |
526 |
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN |
|
527 |
REPEAT (ares_tac [Ord_Un, Ord_nat] 1)); |
|
528 |
(*the finite case: xb Un y < nat *) |
|
529 |
by (res_inst_tac [("j", "nat")] lt_trans2 1); |
|
530 |
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2); |
|
531 |
by (asm_full_simp_tac |
|
532 |
(ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type, |
|
533 |
nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1); |
|
846 | 534 |
(*case nat le (xb Un y) *) |
437 | 535 |
by (asm_full_simp_tac |
536 |
(ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong, |
|
537 |
le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, |
|
538 |
Ord_Un, ltI, nat_le_cardinal, |
|
539 |
Ord_cardinal_le RS lt_trans1 RS ltD]) 1); |
|
760 | 540 |
qed "ordertype_csquare_le"; |
437 | 541 |
|
542 |
(*Main result: Kunen's Theorem 10.12*) |
|
484 | 543 |
goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K"; |
437 | 544 |
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); |
545 |
by (etac rev_mp 1); |
|
484 | 546 |
by (trans_ind_tac "K" [] 1); |
437 | 547 |
by (rtac impI 1); |
548 |
by (rtac le_anti_sym 1); |
|
549 |
by (etac (InfCard_is_Card RS cmult_square_le) 2); |
|
550 |
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1); |
|
551 |
by (assume_tac 2); |
|
552 |
by (assume_tac 2); |
|
553 |
by (asm_simp_tac |
|
846 | 554 |
(ZF_ss addsimps [cmult_def, Ord_cardinal_le, |
555 |
well_ord_csquare RS ordermap_bij RS |
|
556 |
bij_imp_eqpoll RS cardinal_cong, |
|
437 | 557 |
well_ord_csquare RS Ord_ordertype]) 1); |
760 | 558 |
qed "InfCard_csquare_eq"; |
484 | 559 |
|
767 | 560 |
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*) |
484 | 561 |
goal CardinalArith.thy |
562 |
"!!A. [| well_ord(A,r); InfCard(|A|) |] ==> A*A eqpoll A"; |
|
563 |
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1); |
|
564 |
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1)); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
565 |
by (rtac well_ord_cardinal_eqE 1); |
484 | 566 |
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1)); |
567 |
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1); |
|
760 | 568 |
qed "well_ord_InfCard_square_eq"; |
484 | 569 |
|
767 | 570 |
(** Toward's Kunen's Corollary 10.13 (1) **) |
571 |
||
572 |
goal CardinalArith.thy "!!K. [| InfCard(K); L le K; 0<L |] ==> K |*| L = K"; |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
573 |
by (rtac le_anti_sym 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
574 |
by (etac ltE 2 THEN |
767 | 575 |
REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2)); |
576 |
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); |
|
577 |
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); |
|
578 |
by (asm_simp_tac (ZF_ss addsimps [InfCard_csquare_eq]) 1); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
579 |
qed "InfCard_le_cmult_eq"; |
767 | 580 |
|
581 |
(*Corollary 10.13 (1), for cardinal multiplication*) |
|
582 |
goal CardinalArith.thy |
|
583 |
"!!K. [| InfCard(K); InfCard(L) |] ==> K |*| L = K Un L"; |
|
584 |
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); |
|
585 |
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); |
|
586 |
by (resolve_tac [cmult_commute RS ssubst] 1); |
|
587 |
by (resolve_tac [Un_commute RS ssubst] 1); |
|
588 |
by (ALLGOALS |
|
589 |
(asm_simp_tac |
|
590 |
(ZF_ss addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq, |
|
591 |
subset_Un_iff2 RS iffD1, le_imp_subset]))); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
592 |
qed "InfCard_cmult_eq"; |
767 | 593 |
|
594 |
(*This proof appear to be the simplest!*) |
|
595 |
goal CardinalArith.thy "!!K. InfCard(K) ==> K |+| K = K"; |
|
596 |
by (asm_simp_tac |
|
597 |
(ZF_ss addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
598 |
by (rtac InfCard_le_cmult_eq 1); |
767 | 599 |
by (typechk_tac [Ord_0, le_refl, leI]); |
600 |
by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
601 |
qed "InfCard_cdouble_eq"; |
767 | 602 |
|
603 |
(*Corollary 10.13 (1), for cardinal addition*) |
|
604 |
goal CardinalArith.thy "!!K. [| InfCard(K); L le K |] ==> K |+| L = K"; |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
605 |
by (rtac le_anti_sym 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
606 |
by (etac ltE 2 THEN |
767 | 607 |
REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2)); |
608 |
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); |
|
609 |
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); |
|
610 |
by (asm_simp_tac (ZF_ss addsimps [InfCard_cdouble_eq]) 1); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
611 |
qed "InfCard_le_cadd_eq"; |
767 | 612 |
|
613 |
goal CardinalArith.thy |
|
614 |
"!!K. [| InfCard(K); InfCard(L) |] ==> K |+| L = K Un L"; |
|
615 |
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); |
|
616 |
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); |
|
617 |
by (resolve_tac [cadd_commute RS ssubst] 1); |
|
618 |
by (resolve_tac [Un_commute RS ssubst] 1); |
|
619 |
by (ALLGOALS |
|
620 |
(asm_simp_tac |
|
621 |
(ZF_ss addsimps [InfCard_le_cadd_eq, |
|
622 |
subset_Un_iff2 RS iffD1, le_imp_subset]))); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset
|
623 |
qed "InfCard_cadd_eq"; |
767 | 624 |
|
625 |
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set |
|
626 |
of all n-tuples of elements of K. A better version for the Isabelle theory |
|
627 |
might be InfCard(K) ==> |list(K)| = K. |
|
628 |
*) |
|
484 | 629 |
|
630 |
(*** For every cardinal number there exists a greater one |
|
631 |
[Kunen's Theorem 10.16, which would be trivial using AC] ***) |
|
632 |
||
633 |
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))"; |
|
634 |
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1); |
|
1075
848bf2e18dff
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
989
diff
changeset
|
635 |
by (fast_tac (ZF_cs addSIs [Ord_ordertype]) 2); |
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
636 |
by (rewtac Transset_def); |
1075
848bf2e18dff
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
989
diff
changeset
|
637 |
by (safe_tac subset_cs); |
846 | 638 |
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold]) 1); |
639 |
by (safe_tac ZF_cs); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
640 |
by (rtac UN_I 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
641 |
by (rtac ReplaceI 2); |
846 | 642 |
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset, predE]))); |
760 | 643 |
qed "Ord_jump_cardinal"; |
484 | 644 |
|
645 |
(*Allows selective unfolding. Less work than deriving intro/elim rules*) |
|
646 |
goalw CardinalArith.thy [jump_cardinal_def] |
|
647 |
"i : jump_cardinal(K) <-> \ |
|
648 |
\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"; |
|
649 |
by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*) |
|
760 | 650 |
qed "jump_cardinal_iff"; |
484 | 651 |
|
652 |
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) |
|
653 |
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)"; |
|
654 |
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1); |
|
655 |
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); |
|
656 |
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel])); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
657 |
by (rtac subset_refl 2); |
484 | 658 |
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1); |
659 |
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1); |
|
760 | 660 |
qed "K_lt_jump_cardinal"; |
484 | 661 |
|
662 |
(*The proof by contradiction: the bijection f yields a wellordering of X |
|
663 |
whose ordertype is jump_cardinal(K). *) |
|
664 |
goal CardinalArith.thy |
|
665 |
"!!K. [| well_ord(X,r); r <= K * K; X <= K; \ |
|
666 |
\ f : bij(ordertype(X,r), jump_cardinal(K)) \ |
|
667 |
\ |] ==> jump_cardinal(K) : jump_cardinal(K)"; |
|
668 |
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1); |
|
669 |
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2)); |
|
670 |
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); |
|
671 |
by (REPEAT_FIRST (resolve_tac [exI, conjI])); |
|
672 |
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1); |
|
673 |
by (REPEAT (assume_tac 1)); |
|
674 |
by (etac (bij_is_inj RS well_ord_rvimage) 1); |
|
675 |
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1); |
|
676 |
by (asm_simp_tac |
|
677 |
(ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), |
|
678 |
ordertype_Memrel, Ord_jump_cardinal]) 1); |
|
760 | 679 |
qed "Card_jump_cardinal_lemma"; |
484 | 680 |
|
681 |
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) |
|
682 |
goal CardinalArith.thy "Card(jump_cardinal(K))"; |
|
683 |
by (rtac (Ord_jump_cardinal RS CardI) 1); |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
684 |
by (rewtac eqpoll_def); |
484 | 685 |
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1])); |
686 |
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1)); |
|
760 | 687 |
qed "Card_jump_cardinal"; |
484 | 688 |
|
689 |
(*** Basic properties of successor cardinals ***) |
|
690 |
||
691 |
goalw CardinalArith.thy [csucc_def] |
|
692 |
"!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)"; |
|
693 |
by (rtac LeastI 1); |
|
694 |
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal, |
|
695 |
Ord_jump_cardinal] 1)); |
|
760 | 696 |
qed "csucc_basic"; |
484 | 697 |
|
800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset
|
698 |
bind_thm ("Card_csucc", csucc_basic RS conjunct1); |
484 | 699 |
|
800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset
|
700 |
bind_thm ("lt_csucc", csucc_basic RS conjunct2); |
484 | 701 |
|
517 | 702 |
goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)"; |
703 |
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1); |
|
704 |
by (REPEAT (assume_tac 1)); |
|
760 | 705 |
qed "Ord_0_lt_csucc"; |
517 | 706 |
|
484 | 707 |
goalw CardinalArith.thy [csucc_def] |
708 |
"!!K L. [| Card(L); K<L |] ==> csucc(K) le L"; |
|
709 |
by (rtac Least_le 1); |
|
710 |
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1)); |
|
760 | 711 |
qed "csucc_le"; |
484 | 712 |
|
713 |
goal CardinalArith.thy |
|
714 |
"!!K. [| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"; |
|
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
715 |
by (rtac iffI 1); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
716 |
by (rtac Card_lt_imp_lt 2); |
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset
|
717 |
by (etac lt_trans1 2); |
484 | 718 |
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2)); |
719 |
by (resolve_tac [notI RS not_lt_imp_le] 1); |
|
720 |
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1); |
|
721 |
by (assume_tac 1); |
|
722 |
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1); |
|
723 |
by (REPEAT (ares_tac [Ord_cardinal] 1 |
|
724 |
ORELSE eresolve_tac [ltE, Card_is_Ord] 1)); |
|
760 | 725 |
qed "lt_csucc_iff"; |
484 | 726 |
|
727 |
goal CardinalArith.thy |
|
728 |
"!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"; |
|
729 |
by (asm_simp_tac |
|
730 |
(ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1); |
|
760 | 731 |
qed "Card_lt_csucc_iff"; |
488 | 732 |
|
733 |
goalw CardinalArith.thy [InfCard_def] |
|
734 |
"!!K. InfCard(K) ==> InfCard(csucc(K))"; |
|
735 |
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, |
|
736 |
lt_csucc RS leI RSN (2,le_trans)]) 1); |
|
760 | 737 |
qed "InfCard_csucc"; |
517 | 738 |