src/ZF/CardinalArith.ML
author paulson
Tue, 26 Mar 1996 11:42:36 +0100
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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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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
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coincide with union (maximum).  Either way we get most laws for free.
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*)
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open CardinalArith;
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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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qed "sum_commute_eqpoll";
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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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qed "cadd_commute";
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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 (rtac sum_assoc_bij 1);
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qed "sum_assoc_eqpoll";
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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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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);
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by (REPEAT (ares_tac [well_ord_radd] 1));
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qed "well_ord_cadd_assoc";
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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 (rtac bij_0_sum 1);
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qed "sum_0_eqpoll";
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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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qed "cadd_0";
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(** Addition by another cardinal **)
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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";
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(** Monotonicity of addition **)
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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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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";
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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);
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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";
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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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qed "sum_succ_eqpoll";
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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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qed "cadd_succ_lemma";
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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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qed "nat_cadd_eq_add";
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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", "%<x,y>.<y,x>"), ("d", "%<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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qed "prod_commute_eqpoll";
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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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parents: 571
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   149
qed "cmult_commute";
437
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   150
435875e4b21d modifications for cardinal arithmetic
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   151
(** Cardinal multiplication is associative **)
435875e4b21d modifications for cardinal arithmetic
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   152
435875e4b21d modifications for cardinal arithmetic
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   153
goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
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parents:
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   154
by (rtac exI 1);
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clasohm
parents: 1090
diff changeset
   155
by (rtac prod_assoc_bij 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   156
qed "prod_assoc_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   157
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   158
(*Unconditional version requires AC*)
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   159
goalw CardinalArith.thy [cmult_def]
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clasohm
parents: 1090
diff changeset
   160
    "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
437
435875e4b21d modifications for cardinal arithmetic
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diff changeset
   161
\             (i |*| j) |*| k = i |*| (j |*| k)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   162
by (rtac cardinal_cong 1);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   163
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
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clasohm
parents: 1090
diff changeset
   164
          eqpoll_trans) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   165
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   166
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
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clasohm
parents: 1090
diff changeset
   167
          eqpoll_sym) 2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   168
by (REPEAT (ares_tac [well_ord_rmult] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   169
qed "well_ord_cmult_assoc";
437
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   170
435875e4b21d modifications for cardinal arithmetic
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   171
(** Cardinal multiplication distributes over addition **)
435875e4b21d modifications for cardinal arithmetic
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parents:
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   172
435875e4b21d modifications for cardinal arithmetic
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   173
goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   174
by (rtac exI 1);
1461
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clasohm
parents: 1090
diff changeset
   175
by (rtac sum_prod_distrib_bij 1);
760
f0200e91b272 added qed and qed_goal[w]
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parents: 571
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   176
qed "sum_prod_distrib_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   177
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
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parents: 823
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   178
goalw CardinalArith.thy [cadd_def, cmult_def]
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clasohm
parents: 1090
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   179
    "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   180
\             (i |+| j) |*| k = (i |*| k) |+| (j |*| k)";
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   181
by (rtac cardinal_cong 1);
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   182
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
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clasohm
parents: 1090
diff changeset
   183
          eqpoll_trans) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   184
by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2);
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   185
by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
1461
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clasohm
parents: 1090
diff changeset
   186
          eqpoll_sym) 2);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   187
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1));
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   188
qed "well_ord_cadd_cmult_distrib";
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   189
437
435875e4b21d modifications for cardinal arithmetic
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parents:
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   190
(** Multiplication by 0 yields 0 **)
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   191
435875e4b21d modifications for cardinal arithmetic
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parents:
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   192
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   193
by (rtac exI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   194
by (rtac lam_bijective 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   195
by (safe_tac ZF_cs);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   196
qed "prod_0_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   197
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   198
goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   199
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, 
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clasohm
parents: 1090
diff changeset
   200
                                  Card_0 RS Card_cardinal_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
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parents: 571
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   201
qed "cmult_0";
437
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   202
435875e4b21d modifications for cardinal arithmetic
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parents:
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   203
(** 1 is the identity for multiplication **)
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parents:
diff changeset
   204
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   205
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   206
by (rtac exI 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   207
by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   208
qed "prod_singleton_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   209
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   210
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   211
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   212
                                  Card_cardinal_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   213
qed "cmult_1";
437
435875e4b21d modifications for cardinal arithmetic
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parents:
diff changeset
   214
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
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parents: 760
diff changeset
   215
(*** Some inequalities for multiplication ***)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   216
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   217
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   218
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   219
by (simp_tac (ZF_ss addsimps [lam_type]) 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   220
qed "prod_square_lepoll";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   221
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   222
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   223
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   224
by (rtac le_trans 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   225
by (rtac well_ord_lepoll_imp_Card_le 2);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   226
by (rtac prod_square_lepoll 3);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   227
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   228
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   229
qed "cmult_square_le";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   230
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   231
(** Multiplication by a non-zero cardinal **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   232
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   233
goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   234
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   235
by (asm_simp_tac (ZF_ss addsimps [lam_type]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   236
qed "prod_lepoll_self";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   237
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   238
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   239
goalw CardinalArith.thy [cmult_def]
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   240
    "!!K. [| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   241
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   242
by (rtac prod_lepoll_self 3);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   243
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   244
qed "cmult_le_self";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   245
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   246
(** Monotonicity of multiplication **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   247
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   248
goalw CardinalArith.thy [lepoll_def]
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   249
     "!!A B C D. [| A lepoll C;  B lepoll D |] ==> A * B  lepoll  C * D";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   250
by (REPEAT (etac exE 1));
1090
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   251
by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   252
by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")] 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   253
                  lam_injective 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   254
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
   255
by (etac SigmaE 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   256
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
   257
qed "prod_lepoll_mono";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   258
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   259
goalw CardinalArith.thy [cmult_def]
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   260
    "!!K. [| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   261
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
   262
by (rtac well_ord_lepoll_imp_Card_le 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   263
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   264
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
   265
qed "cmult_le_mono";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   266
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   267
(*** Multiplication of finite cardinals is "ordinary" multiplication ***)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   268
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   269
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   270
by (rtac exI 1);
1090
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   271
by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"), 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   272
                  ("d", "case(%y. <A,y>, %z.z)")] 
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   273
    lam_bijective 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   274
by (safe_tac (ZF_cs addSEs [sumE]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   275
by (ALLGOALS
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   276
    (asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   277
qed "prod_succ_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   278
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   279
(*Unconditional version requires AC*)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   280
goalw CardinalArith.thy [cmult_def, cadd_def]
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   281
    "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   282
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   283
by (rtac (cardinal_cong RS sym) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   284
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   285
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   286
qed "cmult_succ_lemma";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   287
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   288
val [mnat,nnat] = goal CardinalArith.thy
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   289
    "[| m: nat;  n: nat |] ==> m |*| n = m#*n";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   290
by (cut_facts_tac [nnat] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   291
by (nat_ind_tac "m" [mnat] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   292
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   293
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   294
                                     nat_cadd_eq_add]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   295
qed "nat_cmult_eq_mult";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   296
823
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   297
goal CardinalArith.thy "!!m n. Card(n) ==> 2 |*| n = n |+| n";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   298
by (asm_simp_tac 
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   299
    (ZF_ss addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   300
                     read_instantiate [("j","0")] cadd_commute, cadd_0]) 1);
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   301
qed "cmult_2";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   302
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   303
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   304
(*** Infinite Cardinals are Limit Ordinals ***)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   305
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   306
(*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
   307
  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
   308
  %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
   309
  range(f) behaves like the natural numbers.*)
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   310
goalw CardinalArith.thy [lepoll_def]
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   311
    "!!i. nat lepoll A ==> cons(u,A) lepoll A";
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   312
by (etac exE 1);
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   313
by (res_inst_tac [("x",
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   314
    "lam z:cons(u,A). if(z=u, f`0,      \
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   315
\                        if(z: range(f), f`succ(converse(f)`z), z))")] exI 1);
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   316
by (res_inst_tac [("d", "%y. if(y: range(f),    \
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   317
\                               nat_case(u, %z.f`z, converse(f)`y), y)")] 
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   318
    lam_injective 1);
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   319
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
   320
                    addIs  [inj_is_fun, inj_converse_fun]) 1);
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   321
by (asm_simp_tac 
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   322
    (ZF_ss addsimps [inj_is_fun RS apply_rangeI,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   323
                     inj_converse_fun RS apply_rangeI,
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   324
                     inj_converse_fun RS apply_funtype,
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   325
                     left_inverse, right_inverse, nat_0I, nat_succI, 
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   326
                     nat_case_0, nat_case_succ]
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   327
           setloop split_tac [expand_if]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   328
qed "nat_cons_lepoll";
516
1957113f0d7d installation of new inductive/datatype sections
lcp
parents: 488
diff changeset
   329
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   330
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
   331
by (etac (nat_cons_lepoll RS eqpollI) 1);
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   332
by (rtac (subset_consI RS subset_imp_lepoll) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   333
qed "nat_cons_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   334
571
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   335
(*Specialized version required below*)
0b03ce5b62f7 ZF/Cardinal: some results moved here from CardinalArith
lcp
parents: 523
diff changeset
   336
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
   337
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   338
qed "nat_succ_eqpoll";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   339
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   340
goalw CardinalArith.thy [InfCard_def] "InfCard(nat)";
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   341
by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   342
qed "InfCard_nat";
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   343
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   344
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   345
by (etac conjunct1 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   346
qed "InfCard_is_Card";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   347
523
972119df615e ZF/CardinalArith/InfCard_Un: new
lcp
parents: 517
diff changeset
   348
goalw CardinalArith.thy [InfCard_def]
972119df615e ZF/CardinalArith/InfCard_Un: new
lcp
parents: 517
diff changeset
   349
    "!!K L. [| InfCard(K);  Card(L) |] ==> InfCard(K Un L)";
972119df615e ZF/CardinalArith/InfCard_Un: new
lcp
parents: 517
diff changeset
   350
by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   351
                                  Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   352
qed "InfCard_Un";
523
972119df615e ZF/CardinalArith/InfCard_Un: new
lcp
parents: 517
diff changeset
   353
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   354
(*Kunen's Lemma 10.11*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   355
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   356
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
   357
by (forward_tac [Card_is_Ord] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   358
by (rtac (ltI RS non_succ_LimitI) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   359
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   360
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   361
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
   362
by (dtac trans 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   363
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
   364
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
   365
by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   366
qed "InfCard_is_Limit";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   367
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   368
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   369
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   370
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   371
(*A general fact about ordermap*)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   372
goalw Cardinal.thy [eqpoll_def]
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   373
    "!!A. [| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   374
by (rtac exI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   375
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   376
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   377
by (rtac pred_subset 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   378
qed "ordermap_eqpoll_pred";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   379
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   380
(** Establishing the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   381
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   382
goalw CardinalArith.thy [inj_def]
1090
8ab69b3e396b Changed some definitions and proofs to use pattern-matching.
lcp
parents: 1075
diff changeset
   383
 "!!K. Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
989
deb999e33c62 Tried the new addss in a proof.
lcp
parents: 870
diff changeset
   384
by (fast_tac (ZF_cs addss ZF_ss
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   385
                    addIs [lam_type, Un_least_lt RS ltD, ltI]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   386
qed "csquare_lam_inj";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   387
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   388
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   389
 "!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   390
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   391
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   392
qed "well_ord_csquare";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   393
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   394
(** Characterising initial segments of the well-ordering **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   395
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   396
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   397
 "!!K. [| x<K;  y<K;  z<K |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   398
\      <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   399
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   400
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   401
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   402
by (safe_tac (ZF_cs addSEs [mem_irrefl] 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   403
                    addSIs [Un_upper1_le, Un_upper2_le]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   404
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
   405
val csquareD_lemma = result();
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   406
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   407
bind_thm ("csquareD", csquareD_lemma RS mp);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   408
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   409
goalw CardinalArith.thy [pred_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   410
 "!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   411
by (safe_tac (lemmas_cs addSEs [SigmaE]));      (*avoids using succCI*)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   412
by (rtac (csquareD RS conjE) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   413
by (rewtac lt_def);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   414
by (assume_tac 4);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   415
by (ALLGOALS (fast_tac ZF_cs));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   416
qed "pred_csquare_subset";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   417
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   418
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
   419
 "!!K. [| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   420
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   421
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   422
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   423
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   424
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   425
qed "csquare_ltI";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   426
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   427
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   428
goalw CardinalArith.thy [csquare_rel_def]
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   429
 "!!K. [| x le z;  y le z;  z<K |] ==> \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   430
\      <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   431
by (subgoals_tac ["x<K", "y<K"] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   432
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   433
by (REPEAT (etac ltE 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   434
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   435
                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   436
by (REPEAT_FIRST (etac succE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   437
by (ALLGOALS
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   438
    (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   439
                                   subset_Un_iff2 RS iff_sym, OrdmemD])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   440
qed "csquare_or_eqI";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   441
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   442
(** The cardinality of initial segments **)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   443
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   444
goal CardinalArith.thy
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   445
    "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   446
\         ordermap(K*K, csquare_rel(K)) ` <x,y> <               \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   447
\         ordermap(K*K, csquare_rel(K)) ` <z,z>";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   448
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   449
by (etac (Limit_is_Ord RS well_ord_csquare) 2);
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   450
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2);
870
ef6faaa415dc Replaced ordermap_z_lepoll by ordermap_z_lt, which is
lcp
parents: 846
diff changeset
   451
by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   452
by (etac well_ord_is_wf 4);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   453
by (ALLGOALS 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   454
    (fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   455
                     addSEs [ltE])));
870
ef6faaa415dc Replaced ordermap_z_lepoll by ordermap_z_lt, which is
lcp
parents: 846
diff changeset
   456
qed "ordermap_z_lt";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   457
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   458
(*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
   459
goalw CardinalArith.thy [cmult_def]
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   460
  "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   461
\       | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   462
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   463
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   464
by (subgoals_tac ["z<K"] 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   465
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2);
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   466
by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   467
by (REPEAT_SOME assume_tac);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   468
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   469
by (etac (Limit_is_Ord RS well_ord_csquare) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   470
by (fast_tac (ZF_cs addIs [ltD]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   471
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   472
    assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   473
by (REPEAT_FIRST (etac ltE));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   474
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   475
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   476
qed "ordermap_csquare_le";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   477
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   478
(*Kunen: "... so the order type <= K" *)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   479
goal CardinalArith.thy
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   480
    "!!K. [| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |]  ==>  \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   481
\         ordertype(K*K, csquare_rel(K)) le K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   482
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   483
by (rtac all_lt_imp_le 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   484
by (assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   485
by (etac (well_ord_csquare RS Ord_ordertype) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   486
by (rtac Card_lt_imp_lt 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   487
by (etac InfCard_is_Card 3);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   488
by (etac ltE 2 THEN assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   489
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   490
by (safe_tac (ZF_cs addSEs [ltE]));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   491
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   492
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   493
by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1  THEN
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   494
    REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   495
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1  THEN
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   496
    REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   497
(*the finite case: xb Un y < nat *)
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   498
by (res_inst_tac [("j", "nat")] lt_trans2 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   499
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   500
by (asm_full_simp_tac
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   501
    (ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   502
                     nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   503
(*case nat le (xb Un y) *)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   504
by (asm_full_simp_tac
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   505
    (ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   506
                     le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   507
                     Ord_Un, ltI, nat_le_cardinal,
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   508
                     Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   509
qed "ordertype_csquare_le";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   510
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   511
(*Main result: Kunen's Theorem 10.12*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   512
goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   513
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   514
by (etac rev_mp 1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   515
by (trans_ind_tac "K" [] 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   516
by (rtac impI 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   517
by (rtac le_anti_sym 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   518
by (etac (InfCard_is_Card RS cmult_square_le) 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   519
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   520
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   521
by (assume_tac 2);
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   522
by (asm_simp_tac 
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   523
    (ZF_ss addsimps [cmult_def, Ord_cardinal_le,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   524
                     well_ord_csquare RS ordermap_bij RS 
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   525
                          bij_imp_eqpoll RS cardinal_cong,
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   526
                     well_ord_csquare RS Ord_ordertype]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   527
qed "InfCard_csquare_eq";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   528
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   529
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   530
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   531
    "!!A. [| well_ord(A,r);  InfCard(|A|) |] ==> A*A eqpoll A";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   532
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   533
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
   534
by (rtac well_ord_cardinal_eqE 1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   535
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
   536
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   537
qed "well_ord_InfCard_square_eq";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   538
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   539
(** Toward's Kunen's Corollary 10.13 (1) **)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   540
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   541
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
   542
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
   543
by (etac ltE 2 THEN
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   544
    REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   545
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   546
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   547
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
   548
qed "InfCard_le_cmult_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   549
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   550
(*Corollary 10.13 (1), for cardinal multiplication*)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   551
goal CardinalArith.thy
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   552
    "!!K. [| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   553
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   554
by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   555
by (resolve_tac [cmult_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   556
by (resolve_tac [Un_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   557
by (ALLGOALS
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   558
    (asm_simp_tac 
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   559
     (ZF_ss addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   560
                      subset_Un_iff2 RS iffD1, le_imp_subset])));
782
200a16083201 added bind_thm for theorems defined by "standard ..."
clasohm
parents: 767
diff changeset
   561
qed "InfCard_cmult_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   562
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   563
(*This proof appear to be the simplest!*)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   564
goal CardinalArith.thy "!!K. InfCard(K) ==> K |+| K = K";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   565
by (asm_simp_tac
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   566
    (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
   567
by (rtac InfCard_le_cmult_eq 1);
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   568
by (typechk_tac [Ord_0, le_refl, leI]);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   569
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
   570
qed "InfCard_cdouble_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   571
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   572
(*Corollary 10.13 (1), for cardinal addition*)
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   573
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
   574
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
   575
by (etac ltE 2 THEN
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   576
    REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   577
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   578
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   579
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
   580
qed "InfCard_le_cadd_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   581
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   582
goal CardinalArith.thy
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   583
    "!!K. [| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L";
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   584
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   585
by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   586
by (resolve_tac [cadd_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   587
by (resolve_tac [Un_commute RS ssubst] 1);
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   588
by (ALLGOALS
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   589
    (asm_simp_tac 
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   590
     (ZF_ss addsimps [InfCard_le_cadd_eq,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   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_cadd_eq";
767
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   593
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   594
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   595
  of all n-tuples of elements of K.  A better version for the Isabelle theory
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   596
  might be  InfCard(K) ==> |list(K)| = K.
a4fce3b94065 sum_lepoll_self, cadd_le_self, prod_lepoll_self,
lcp
parents: 760
diff changeset
   597
*)
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   598
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   599
(*** For every cardinal number there exists a greater one
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   600
     [Kunen's Theorem 10.16, which would be trivial using AC] ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   601
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   602
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   603
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
   604
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
   605
by (rewtac Transset_def);
1075
848bf2e18dff Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents: 989
diff changeset
   606
by (safe_tac subset_cs);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   607
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold]) 1);
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   608
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
   609
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
   610
by (rtac ReplaceI 2);
846
c4566750dc12 Now proof of Ord_jump_cardinal uses
lcp
parents: 823
diff changeset
   611
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset, predE])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   612
qed "Ord_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   613
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   614
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   615
goalw CardinalArith.thy [jump_cardinal_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   616
     "i : jump_cardinal(K) <->   \
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   617
\         (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   618
by (fast_tac subset_cs 1);      (*It's vital to avoid reasoning about <=*)
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   619
qed "jump_cardinal_iff";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   620
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   621
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   622
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   623
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   624
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   625
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
   626
by (rtac subset_refl 2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   627
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   628
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   629
qed "K_lt_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   630
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   631
(*The proof by contradiction: the bijection f yields a wellordering of X
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   632
  whose ordertype is jump_cardinal(K).  *)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   633
goal CardinalArith.thy
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   634
    "!!K. [| well_ord(X,r);  r <= K * K;  X <= K;       \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   635
\            f : bij(ordertype(X,r), jump_cardinal(K))  \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   636
\         |] ==> jump_cardinal(K) : jump_cardinal(K)";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   637
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
   638
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   639
by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   640
by (REPEAT_FIRST (resolve_tac [exI, conjI]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   641
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   642
by (REPEAT (assume_tac 1));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   643
by (etac (bij_is_inj RS well_ord_rvimage) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   644
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   645
by (asm_simp_tac
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   646
    (ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   647
                     ordertype_Memrel, Ord_jump_cardinal]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   648
qed "Card_jump_cardinal_lemma";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   649
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   650
(*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
   651
goal CardinalArith.thy "Card(jump_cardinal(K))";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   652
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
   653
by (rewtac eqpoll_def);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   654
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1]));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   655
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   656
qed "Card_jump_cardinal";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   657
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   658
(*** Basic properties of successor cardinals ***)
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   659
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   660
goalw CardinalArith.thy [csucc_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   661
    "!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   662
by (rtac LeastI 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   663
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   664
                      Ord_jump_cardinal] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   665
qed "csucc_basic";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   666
800
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   667
bind_thm ("Card_csucc", csucc_basic RS conjunct1);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   668
800
23f55b829ccb Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents: 782
diff changeset
   669
bind_thm ("lt_csucc", csucc_basic RS conjunct2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   670
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   671
goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)";
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   672
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
   673
by (REPEAT (assume_tac 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   674
qed "Ord_0_lt_csucc";
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   675
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   676
goalw CardinalArith.thy [csucc_def]
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   677
    "!!K L. [| Card(L);  K<L |] ==> csucc(K) le L";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   678
by (rtac Least_le 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   679
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   680
qed "csucc_le";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   681
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   682
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   683
    "!!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
   684
by (rtac iffI 1);
33dc37d46296 Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents: 800
diff changeset
   685
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
   686
by (etac lt_trans1 2);
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   687
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   688
by (resolve_tac [notI RS not_lt_imp_le] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   689
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
   690
by (assume_tac 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   691
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   692
by (REPEAT (ares_tac [Ord_cardinal] 1
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   693
     ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   694
qed "lt_csucc_iff";
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   695
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   696
goal CardinalArith.thy
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   697
    "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   698
by (asm_simp_tac 
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
   699
    (ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   700
qed "Card_lt_csucc_iff";
488
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   701
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   702
goalw CardinalArith.thy [InfCard_def]
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   703
    "!!K. InfCard(K) ==> InfCard(csucc(K))";
52f7447d4f1b Addition of infinite branching datatypes
lcp
parents: 484
diff changeset
   704
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, 
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1090
diff changeset
   705
                                  lt_csucc RS leI RSN (2,le_trans)]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 571
diff changeset
   706
qed "InfCard_csucc";
517
a9f93400f307 for infinite datatypes with arbitrary index sets
lcp
parents: 516
diff changeset
   707
1609
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   708
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   709
(*** Finite sets ***)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   710
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   711
goal CardinalArith.thy
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   712
    "!!n. n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   713
by (eresolve_tac [nat_induct] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   714
by (simp_tac (ZF_ss addsimps (eqpoll_0_iff::Fin.intrs)) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   715
by (step_tac ZF_cs 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   716
by (subgoal_tac "EX u. u:A" 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   717
by (eresolve_tac [exE] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   718
by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   719
by (assume_tac 2);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   720
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   721
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   722
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   723
by (resolve_tac [Fin.consI] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   724
by (fast_tac ZF_cs 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   725
by (deepen_tac (ZF_cs addIs [Fin_mono RS subsetD]) 0 1);  (*SLOW*)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   726
(*Now for the lemma assumed above*)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   727
by (rewrite_goals_tac [eqpoll_def]);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   728
by (fast_tac (ZF_cs addSEs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   729
val lemma = result();
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   730
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   731
goalw CardinalArith.thy [Finite_def] "!!A. Finite(A) ==> A : Fin(A)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   732
by (fast_tac (ZF_cs addIs [lemma RS spec RS mp]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   733
qed "Finite_into_Fin";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   734
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   735
goal CardinalArith.thy "!!A. A : Fin(U) ==> Finite(A)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   736
by (fast_tac (ZF_cs addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   737
qed "Fin_into_Finite";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   738
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   739
goal CardinalArith.thy "Finite(A) <-> A : Fin(A)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   740
by (fast_tac (ZF_cs addIs [Finite_into_Fin] addEs [Fin_into_Finite]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   741
qed "Finite_Fin_iff";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   742
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   743
goal CardinalArith.thy
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   744
    "!!A. [| Finite(A); Finite(B) |] ==> Finite(A Un B)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   745
by (fast_tac (ZF_cs addSIs [Fin_into_Finite, Fin_UnI] 
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   746
                    addSDs [Finite_into_Fin]
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   747
		    addSEs [Un_upper1 RS Fin_mono RS subsetD,
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   748
			    Un_upper2 RS Fin_mono RS subsetD]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   749
qed "Finite_Un";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   750
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   751
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   752
(** Removing elements from a finite set decreases its cardinality **)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   753
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   754
goal CardinalArith.thy
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   755
    "!!A. A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   756
by (eresolve_tac [Fin_induct] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   757
by (simp_tac (ZF_ss addsimps [lepoll_0_iff]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   758
by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   759
by (asm_simp_tac ZF_ss 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   760
by (fast_tac (ZF_cs addSDs [cons_lepoll_consD]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   761
by (fast_tac eq_cs 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   762
qed "Fin_imp_not_cons_lepoll";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   763
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   764
goal CardinalArith.thy
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   765
    "!!a A. [| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   766
by (rewrite_goals_tac [cardinal_def]);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   767
by (resolve_tac [Least_equality] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   768
by (fold_tac [cardinal_def]);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   769
by (simp_tac (ZF_ss addsimps [succ_def]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   770
by (fast_tac (ZF_cs addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll] 
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   771
                    addSEs [mem_irrefl]
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   772
                    addSDs [Finite_imp_well_ord]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   773
by (fast_tac (ZF_cs addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   774
by (resolve_tac [notI] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   775
by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   776
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   777
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   778
by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   779
by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   780
by (fast_tac (ZF_cs addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll] 
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   781
                    addSDs [Finite_imp_well_ord]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   782
qed "Finite_imp_cardinal_cons";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   783
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   784
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   785
goal CardinalArith.thy "!!a A. [| Finite(A);  a:A |] ==> |A-{a}| < |A|";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   786
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   787
by (assume_tac 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   788
by (asm_simp_tac (ZF_ss addsimps [Finite_imp_cardinal_cons,
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   789
				  Diff_subset RS subset_imp_lepoll RS 
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   790
				  lepoll_Finite]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   791
by (asm_simp_tac (ZF_ss addsimps [cons_Diff, Ord_cardinal RS le_refl]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   792
qed "Finite_imp_cardinal_Diff";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   793
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   794
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   795
(** Thanks to Krzysztof Grabczewski **)
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   796
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   797
val nat_implies_well_ord = nat_into_Ord RS well_ord_Memrel;
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   798
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   799
goal CardinalArith.thy "!!m n. [| m:nat; n:nat |] ==> m + n eqpoll m #+ n";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   800
by (rtac eqpoll_trans 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   801
by (eresolve_tac [nat_implies_well_ord RS (
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   802
                  nat_implies_well_ord RSN (2,
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   803
                  well_ord_radd RS well_ord_cardinal_eqpoll)) RS eqpoll_sym] 1 
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   804
    THEN (assume_tac 1));
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   805
by (eresolve_tac [nat_cadd_eq_add RS subst] 1 THEN (assume_tac 1));
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   806
by (asm_full_simp_tac (ZF_ss addsimps [cadd_def, eqpoll_refl]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   807
qed "nat_sum_eqpoll_sum";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   808
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   809
goal Nat.thy "!!m. [| m le n; n:nat |] ==> m:nat";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   810
by (fast_tac (ZF_cs addSDs [nat_succI RS (Ord_nat RSN (2, OrdmemD))]
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   811
        addSEs [ltE]) 1);
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   812
qed "le_in_nat";
5324067d993f New lemmas for Mutilated Checkerboard
paulson
parents: 1461
diff changeset
   813