src/ZF/CardinalArith.ML
 author paulson Wed Dec 24 10:02:30 1997 +0100 (1997-12-24 ago) changeset 4477 b3e5857d8d99 parent 4312 63844406913c child 5067 62b6288e6005 permissions -rw-r--r--
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
```     1 (*  Title:      ZF/CardinalArith.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1994  University of Cambridge
```
```     5
```
```     6 Cardinal arithmetic -- WITHOUT the Axiom of Choice
```
```     7
```
```     8 Note: Could omit proving the algebraic laws for cardinal addition and
```
```     9 multiplication.  On finite cardinals these operations coincide with
```
```    10 addition and multiplication of natural numbers; on infinite cardinals they
```
```    11 coincide with union (maximum).  Either way we get most laws for free.
```
```    12 *)
```
```    13
```
```    14 open CardinalArith;
```
```    15
```
```    16 (*** Cardinal addition ***)
```
```    17
```
```    18 (** Cardinal addition is commutative **)
```
```    19
```
```    20 goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A";
```
```    21 by (rtac exI 1);
```
```    22 by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")]
```
```    23     lam_bijective 1);
```
```    24 by (safe_tac (claset() addSEs [sumE]));
```
```    25 by (ALLGOALS (Asm_simp_tac));
```
```    26 qed "sum_commute_eqpoll";
```
```    27
```
```    28 goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i";
```
```    29 by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
```
```    30 qed "cadd_commute";
```
```    31
```
```    32 (** Cardinal addition is associative **)
```
```    33
```
```    34 goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
```
```    35 by (rtac exI 1);
```
```    36 by (rtac sum_assoc_bij 1);
```
```    37 qed "sum_assoc_eqpoll";
```
```    38
```
```    39 (*Unconditional version requires AC*)
```
```    40 goalw CardinalArith.thy [cadd_def]
```
```    41     "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
```
```    42 \             (i |+| j) |+| k = i |+| (j |+| k)";
```
```    43 by (rtac cardinal_cong 1);
```
```    44 by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
```
```    45           eqpoll_trans) 1);
```
```    46 by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
```
```    47 by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
```
```    48           eqpoll_sym) 2);
```
```    49 by (REPEAT (ares_tac [well_ord_radd] 1));
```
```    50 qed "well_ord_cadd_assoc";
```
```    51
```
```    52 (** 0 is the identity for addition **)
```
```    53
```
```    54 goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A";
```
```    55 by (rtac exI 1);
```
```    56 by (rtac bij_0_sum 1);
```
```    57 qed "sum_0_eqpoll";
```
```    58
```
```    59 goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K";
```
```    60 by (asm_simp_tac (simpset() addsimps [sum_0_eqpoll RS cardinal_cong,
```
```    61 				      Card_cardinal_eq]) 1);
```
```    62 qed "cadd_0";
```
```    63
```
```    64 (** Addition by another cardinal **)
```
```    65
```
```    66 goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A+B";
```
```    67 by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1);
```
```    68 by (asm_simp_tac (simpset() addsimps [lam_type]) 1);
```
```    69 qed "sum_lepoll_self";
```
```    70
```
```    71 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
```
```    72 goalw CardinalArith.thy [cadd_def]
```
```    73     "!!K. [| Card(K);  Ord(L) |] ==> K le (K |+| L)";
```
```    74 by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
```
```    75 by (rtac sum_lepoll_self 3);
```
```    76 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1));
```
```    77 qed "cadd_le_self";
```
```    78
```
```    79 (** Monotonicity of addition **)
```
```    80
```
```    81 goalw CardinalArith.thy [lepoll_def]
```
```    82      "!!A B C D. [| A lepoll C;  B lepoll D |] ==> A + B  lepoll  C + D";
```
```    83 by (REPEAT (etac exE 1));
```
```    84 by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")]
```
```    85     exI 1);
```
```    86 by (res_inst_tac
```
```    87       [("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")]
```
```    88       lam_injective 1);
```
```    89 by (typechk_tac ([inj_is_fun, case_type, InlI, InrI] @ ZF_typechecks));
```
```    90 by (etac sumE 1);
```
```    91 by (ALLGOALS (asm_simp_tac (simpset() addsimps [left_inverse])));
```
```    92 qed "sum_lepoll_mono";
```
```    93
```
```    94 goalw CardinalArith.thy [cadd_def]
```
```    95     "!!K. [| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)";
```
```    96 by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
```
```    97 by (rtac well_ord_lepoll_imp_Card_le 1);
```
```    98 by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2));
```
```    99 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
```
```   100 qed "cadd_le_mono";
```
```   101
```
```   102 (** Addition of finite cardinals is "ordinary" addition **)
```
```   103
```
```   104 goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
```
```   105 by (rtac exI 1);
```
```   106 by (res_inst_tac [("c", "%z. if(z=Inl(A),A+B,z)"),
```
```   107                   ("d", "%z. if(z=A+B,Inl(A),z)")]
```
```   108     lam_bijective 1);
```
```   109 by (ALLGOALS
```
```   110     (asm_simp_tac (simpset() addsimps [succI2, mem_imp_not_eq]
```
```   111                              setloop eresolve_tac [sumE,succE])));
```
```   112 qed "sum_succ_eqpoll";
```
```   113
```
```   114 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
```
```   115 (*Unconditional version requires AC*)
```
```   116 goalw CardinalArith.thy [cadd_def]
```
```   117     "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
```
```   118 by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
```
```   119 by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
```
```   120 by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
```
```   121 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
```
```   122 qed "cadd_succ_lemma";
```
```   123
```
```   124 val [mnat,nnat] = goal CardinalArith.thy
```
```   125     "[| m: nat;  n: nat |] ==> m |+| n = m#+n";
```
```   126 by (cut_facts_tac [nnat] 1);
```
```   127 by (nat_ind_tac "m" [mnat] 1);
```
```   128 by (asm_simp_tac (simpset() addsimps [nat_into_Card RS cadd_0]) 1);
```
```   129 by (asm_simp_tac (simpset() addsimps [nat_into_Ord, cadd_succ_lemma,
```
```   130 				      nat_into_Card RS Card_cardinal_eq]) 1);
```
```   131 qed "nat_cadd_eq_add";
```
```   132
```
```   133
```
```   134 (*** Cardinal multiplication ***)
```
```   135
```
```   136 (** Cardinal multiplication is commutative **)
```
```   137
```
```   138 (*Easier to prove the two directions separately*)
```
```   139 goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A";
```
```   140 by (rtac exI 1);
```
```   141 by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")]
```
```   142     lam_bijective 1);
```
```   143 by Safe_tac;
```
```   144 by (ALLGOALS (Asm_simp_tac));
```
```   145 qed "prod_commute_eqpoll";
```
```   146
```
```   147 goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i";
```
```   148 by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
```
```   149 qed "cmult_commute";
```
```   150
```
```   151 (** Cardinal multiplication is associative **)
```
```   152
```
```   153 goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
```
```   154 by (rtac exI 1);
```
```   155 by (rtac prod_assoc_bij 1);
```
```   156 qed "prod_assoc_eqpoll";
```
```   157
```
```   158 (*Unconditional version requires AC*)
```
```   159 goalw CardinalArith.thy [cmult_def]
```
```   160     "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
```
```   161 \             (i |*| j) |*| k = i |*| (j |*| k)";
```
```   162 by (rtac cardinal_cong 1);
```
```   163 by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
```
```   164           eqpoll_trans) 1);
```
```   165 by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
```
```   166 by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
```
```   167           eqpoll_sym) 2);
```
```   168 by (REPEAT (ares_tac [well_ord_rmult] 1));
```
```   169 qed "well_ord_cmult_assoc";
```
```   170
```
```   171 (** Cardinal multiplication distributes over addition **)
```
```   172
```
```   173 goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
```
```   174 by (rtac exI 1);
```
```   175 by (rtac sum_prod_distrib_bij 1);
```
```   176 qed "sum_prod_distrib_eqpoll";
```
```   177
```
```   178 goalw CardinalArith.thy [cadd_def, cmult_def]
```
```   179     "!!i j k. [| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==>  \
```
```   180 \             (i |+| j) |*| k = (i |*| k) |+| (j |*| k)";
```
```   181 by (rtac cardinal_cong 1);
```
```   182 by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
```
```   183           eqpoll_trans) 1);
```
```   184 by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2);
```
```   185 by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS
```
```   186 	  sum_eqpoll_cong RS eqpoll_sym) 2);
```
```   187 by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1));
```
```   188 qed "well_ord_cadd_cmult_distrib";
```
```   189
```
```   190 (** Multiplication by 0 yields 0 **)
```
```   191
```
```   192 goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
```
```   193 by (rtac exI 1);
```
```   194 by (rtac lam_bijective 1);
```
```   195 by Safe_tac;
```
```   196 qed "prod_0_eqpoll";
```
```   197
```
```   198 goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
```
```   199 by (asm_simp_tac (simpset() addsimps [prod_0_eqpoll RS cardinal_cong,
```
```   200 				      Card_0 RS Card_cardinal_eq]) 1);
```
```   201 qed "cmult_0";
```
```   202
```
```   203 (** 1 is the identity for multiplication **)
```
```   204
```
```   205 goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A";
```
```   206 by (rtac exI 1);
```
```   207 by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
```
```   208 qed "prod_singleton_eqpoll";
```
```   209
```
```   210 goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K";
```
```   211 by (asm_simp_tac (simpset() addsimps [prod_singleton_eqpoll RS cardinal_cong,
```
```   212 				      Card_cardinal_eq]) 1);
```
```   213 qed "cmult_1";
```
```   214
```
```   215 (*** Some inequalities for multiplication ***)
```
```   216
```
```   217 goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
```
```   218 by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
```
```   219 by (simp_tac (simpset() addsimps [lam_type]) 1);
```
```   220 qed "prod_square_lepoll";
```
```   221
```
```   222 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
```
```   223 goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K";
```
```   224 by (rtac le_trans 1);
```
```   225 by (rtac well_ord_lepoll_imp_Card_le 2);
```
```   226 by (rtac prod_square_lepoll 3);
```
```   227 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
```
```   228 by (asm_simp_tac (simpset()
```
```   229 		  addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
```
```   230 qed "cmult_square_le";
```
```   231
```
```   232 (** Multiplication by a non-zero cardinal **)
```
```   233
```
```   234 goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B";
```
```   235 by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
```
```   236 by (asm_simp_tac (simpset() addsimps [lam_type]) 1);
```
```   237 qed "prod_lepoll_self";
```
```   238
```
```   239 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
```
```   240 goalw CardinalArith.thy [cmult_def]
```
```   241     "!!K. [| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)";
```
```   242 by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
```
```   243 by (rtac prod_lepoll_self 3);
```
```   244 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1));
```
```   245 qed "cmult_le_self";
```
```   246
```
```   247 (** Monotonicity of multiplication **)
```
```   248
```
```   249 goalw CardinalArith.thy [lepoll_def]
```
```   250      "!!A B C D. [| A lepoll C;  B lepoll D |] ==> A * B  lepoll  C * D";
```
```   251 by (REPEAT (etac exE 1));
```
```   252 by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
```
```   253 by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")]
```
```   254                   lam_injective 1);
```
```   255 by (typechk_tac (inj_is_fun::ZF_typechecks));
```
```   256 by (etac SigmaE 1);
```
```   257 by (asm_simp_tac (simpset() addsimps [left_inverse]) 1);
```
```   258 qed "prod_lepoll_mono";
```
```   259
```
```   260 goalw CardinalArith.thy [cmult_def]
```
```   261     "!!K. [| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)";
```
```   262 by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
```
```   263 by (rtac well_ord_lepoll_imp_Card_le 1);
```
```   264 by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
```
```   265 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
```
```   266 qed "cmult_le_mono";
```
```   267
```
```   268 (*** Multiplication of finite cardinals is "ordinary" multiplication ***)
```
```   269
```
```   270 goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
```
```   271 by (rtac exI 1);
```
```   272 by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"),
```
```   273                   ("d", "case(%y. <A,y>, %z. z)")]
```
```   274     lam_bijective 1);
```
```   275 by (safe_tac (claset() addSEs [sumE]));
```
```   276 by (ALLGOALS
```
```   277     (asm_simp_tac (simpset() addsimps [succI2, if_type, mem_imp_not_eq])));
```
```   278 qed "prod_succ_eqpoll";
```
```   279
```
```   280 (*Unconditional version requires AC*)
```
```   281 goalw CardinalArith.thy [cmult_def, cadd_def]
```
```   282     "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
```
```   283 by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
```
```   284 by (rtac (cardinal_cong RS sym) 1);
```
```   285 by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
```
```   286 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
```
```   287 qed "cmult_succ_lemma";
```
```   288
```
```   289 val [mnat,nnat] = goal CardinalArith.thy
```
```   290     "[| m: nat;  n: nat |] ==> m |*| n = m#*n";
```
```   291 by (cut_facts_tac [nnat] 1);
```
```   292 by (nat_ind_tac "m" [mnat] 1);
```
```   293 by (asm_simp_tac (simpset() addsimps [cmult_0]) 1);
```
```   294 by (asm_simp_tac (simpset() addsimps [nat_into_Ord, cmult_succ_lemma,
```
```   295 				      nat_cadd_eq_add]) 1);
```
```   296 qed "nat_cmult_eq_mult";
```
```   297
```
```   298 goal CardinalArith.thy "!!m n. Card(n) ==> 2 |*| n = n |+| n";
```
```   299 by (asm_simp_tac
```
```   300     (simpset() addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma,
```
```   301 			 Card_is_Ord, cadd_0,
```
```   302 			 read_instantiate [("j","0")] cadd_commute]) 1);
```
```   303 qed "cmult_2";
```
```   304
```
```   305
```
```   306 (*** Infinite Cardinals are Limit Ordinals ***)
```
```   307
```
```   308 (*This proof is modelled upon one assuming nat<=A, with injection
```
```   309   lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z))  and inverse
```
```   310   %y. if(y:nat, nat_case(u,%z.z,y), y).  If f: inj(nat,A) then
```
```   311   range(f) behaves like the natural numbers.*)
```
```   312 goalw CardinalArith.thy [lepoll_def]
```
```   313     "!!i. nat lepoll A ==> cons(u,A) lepoll A";
```
```   314 by (etac exE 1);
```
```   315 by (res_inst_tac [("x",
```
```   316     "lam z:cons(u,A). if(z=u, f`0,      \
```
```   317 \                        if(z: range(f), f`succ(converse(f)`z), z))")] exI 1);
```
```   318 by (res_inst_tac [("d", "%y. if(y: range(f),    \
```
```   319 \                               nat_case(u, %z. f`z, converse(f)`y), y)")]
```
```   320     lam_injective 1);
```
```   321 by (fast_tac (claset() addSIs [if_type, nat_succI, apply_type]
```
```   322                       addIs  [inj_is_fun, inj_converse_fun]) 1);
```
```   323 by (asm_simp_tac
```
```   324     (simpset() addsimps [inj_is_fun RS apply_rangeI,
```
```   325 			 inj_converse_fun RS apply_rangeI,
```
```   326 			 inj_converse_fun RS apply_funtype,
```
```   327 			 left_inverse, right_inverse, nat_0I, nat_succI,
```
```   328 			 nat_case_0, nat_case_succ]
```
```   329                setloop split_tac [expand_if]) 1);
```
```   330 qed "nat_cons_lepoll";
```
```   331
```
```   332 goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A";
```
```   333 by (etac (nat_cons_lepoll RS eqpollI) 1);
```
```   334 by (rtac (subset_consI RS subset_imp_lepoll) 1);
```
```   335 qed "nat_cons_eqpoll";
```
```   336
```
```   337 (*Specialized version required below*)
```
```   338 goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A";
```
```   339 by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
```
```   340 qed "nat_succ_eqpoll";
```
```   341
```
```   342 goalw CardinalArith.thy [InfCard_def] "InfCard(nat)";
```
```   343 by (blast_tac (claset() addIs [Card_nat, le_refl, Card_is_Ord]) 1);
```
```   344 qed "InfCard_nat";
```
```   345
```
```   346 goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)";
```
```   347 by (etac conjunct1 1);
```
```   348 qed "InfCard_is_Card";
```
```   349
```
```   350 goalw CardinalArith.thy [InfCard_def]
```
```   351     "!!K L. [| InfCard(K);  Card(L) |] ==> InfCard(K Un L)";
```
```   352 by (asm_simp_tac (simpset() addsimps [Card_Un, Un_upper1_le RSN (2,le_trans),
```
```   353 				      Card_is_Ord]) 1);
```
```   354 qed "InfCard_Un";
```
```   355
```
```   356 (*Kunen's Lemma 10.11*)
```
```   357 goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)";
```
```   358 by (etac conjE 1);
```
```   359 by (forward_tac [Card_is_Ord] 1);
```
```   360 by (rtac (ltI RS non_succ_LimitI) 1);
```
```   361 by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
```
```   362 by (safe_tac (claset() addSDs [Limit_nat RS Limit_le_succD]));
```
```   363 by (rewtac Card_def);
```
```   364 by (dtac trans 1);
```
```   365 by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
```
```   366 by (etac (Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1);
```
```   367 by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));
```
```   368 qed "InfCard_is_Limit";
```
```   369
```
```   370
```
```   371 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
```
```   372
```
```   373 (*A general fact about ordermap*)
```
```   374 goalw Cardinal.thy [eqpoll_def]
```
```   375     "!!A. [| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
```
```   376 by (rtac exI 1);
```
```   377 by (asm_simp_tac (simpset() addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
```
```   378 by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
```
```   379 by (rtac pred_subset 1);
```
```   380 qed "ordermap_eqpoll_pred";
```
```   381
```
```   382 (** Establishing the well-ordering **)
```
```   383
```
```   384 goalw CardinalArith.thy [inj_def]
```
```   385  "!!K. Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
```
```   386 by (fast_tac (claset() addss (simpset())
```
```   387                        addIs [lam_type, Un_least_lt RS ltD, ltI]) 1);
```
```   388 qed "csquare_lam_inj";
```
```   389
```
```   390 goalw CardinalArith.thy [csquare_rel_def]
```
```   391  "!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))";
```
```   392 by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
```
```   393 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
```
```   394 qed "well_ord_csquare";
```
```   395
```
```   396 (** Characterising initial segments of the well-ordering **)
```
```   397
```
```   398 goalw CardinalArith.thy [csquare_rel_def]
```
```   399  "!!K. [| x<K;  y<K;  z<K |] ==> \
```
```   400 \      <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
```
```   401 by (REPEAT (etac ltE 1));
```
```   402 by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
```
```   403 				      Un_absorb, Un_least_mem_iff, ltD]) 1);
```
```   404 by (safe_tac (claset() addSEs [mem_irrefl]
```
```   405                        addSIs [Un_upper1_le, Un_upper2_le]));
```
```   406 by (ALLGOALS (asm_simp_tac (simpset() addsimps [lt_def, succI2, Ord_succ])));
```
```   407 qed_spec_mp "csquareD";
```
```   408
```
```   409 goalw CardinalArith.thy [pred_def]
```
```   410  "!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
```
```   411 by (safe_tac (claset_of ZF.thy addSEs [SigmaE]));  (*avoids using succCI,...*)
```
```   412 by (rtac (csquareD RS conjE) 1);
```
```   413 by (rewtac lt_def);
```
```   414 by (assume_tac 4);
```
```   415 by (ALLGOALS Blast_tac);
```
```   416 qed "pred_csquare_subset";
```
```   417
```
```   418 goalw CardinalArith.thy [csquare_rel_def]
```
```   419  "!!K. [| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)";
```
```   420 by (subgoals_tac ["x<K", "y<K"] 1);
```
```   421 by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
```
```   422 by (REPEAT (etac ltE 1));
```
```   423 by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
```
```   424 				      Un_absorb, Un_least_mem_iff, ltD]) 1);
```
```   425 qed "csquare_ltI";
```
```   426
```
```   427 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
```
```   428 goalw CardinalArith.thy [csquare_rel_def]
```
```   429  "!!K. [| x le z;  y le z;  z<K |] ==> \
```
```   430 \      <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
```
```   431 by (subgoals_tac ["x<K", "y<K"] 1);
```
```   432 by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
```
```   433 by (REPEAT (etac ltE 1));
```
```   434 by (asm_simp_tac (simpset() addsimps [rvimage_iff, rmult_iff, Memrel_iff,
```
```   435 				      Un_absorb, Un_least_mem_iff, ltD]) 1);
```
```   436 by (REPEAT_FIRST (etac succE));
```
```   437 by (ALLGOALS
```
```   438     (asm_simp_tac (simpset() addsimps [subset_Un_iff RS iff_sym,
```
```   439 				       subset_Un_iff2 RS iff_sym, OrdmemD])));
```
```   440 qed "csquare_or_eqI";
```
```   441
```
```   442 (** The cardinality of initial segments **)
```
```   443
```
```   444 goal CardinalArith.thy
```
```   445     "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
```
```   446 \         ordermap(K*K, csquare_rel(K)) ` <x,y> <               \
```
```   447 \         ordermap(K*K, csquare_rel(K)) ` <z,z>";
```
```   448 by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
```
```   449 by (etac (Limit_is_Ord RS well_ord_csquare) 2);
```
```   450 by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
```
```   451 by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
```
```   452 by (etac well_ord_is_wf 4);
```
```   453 by (ALLGOALS
```
```   454     (blast_tac (claset() addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap]
```
```   455                          addSEs [ltE])));
```
```   456 qed "ordermap_z_lt";
```
```   457
```
```   458 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
```
```   459 goalw CardinalArith.thy [cmult_def]
```
```   460   "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
```
```   461 \       | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|";
```
```   462 by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
```
```   463 by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
```
```   464 by (subgoals_tac ["z<K"] 1);
```
```   465 by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
```
```   466 by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
```
```   467 by (REPEAT_SOME assume_tac);
```
```   468 by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
```
```   469 by (etac (Limit_is_Ord RS well_ord_csquare) 1);
```
```   470 by (blast_tac (claset() addIs [ltD]) 1);
```
```   471 by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
```
```   472     assume_tac 1);
```
```   473 by (REPEAT_FIRST (etac ltE));
```
```   474 by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
```
```   475 by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
```
```   476 qed "ordermap_csquare_le";
```
```   477
```
```   478 (*Kunen: "... so the order type <= K" *)
```
```   479 goal CardinalArith.thy
```
```   480     "!!K. [| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |]  ==>  \
```
```   481 \         ordertype(K*K, csquare_rel(K)) le K";
```
```   482 by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
```
```   483 by (rtac all_lt_imp_le 1);
```
```   484 by (assume_tac 1);
```
```   485 by (etac (well_ord_csquare RS Ord_ordertype) 1);
```
```   486 by (rtac Card_lt_imp_lt 1);
```
```   487 by (etac InfCard_is_Card 3);
```
```   488 by (etac ltE 2 THEN assume_tac 2);
```
```   489 by (asm_full_simp_tac (simpset() addsimps [ordertype_unfold]) 1);
```
```   490 by (safe_tac (claset() addSEs [ltE]));
```
```   491 by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
```
```   492 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
```
```   493 by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1  THEN
```
```   494     REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
```
```   495 by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1  THEN
```
```   496     REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
```
```   497 (*the finite case: xb Un y < nat *)
```
```   498 by (res_inst_tac [("j", "nat")] lt_trans2 1);
```
```   499 by (asm_full_simp_tac (simpset() addsimps [InfCard_def]) 2);
```
```   500 by (asm_full_simp_tac
```
```   501     (simpset() addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
```
```   502 			 nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
```
```   503 (*case nat le (xb Un y) *)
```
```   504 by (asm_full_simp_tac
```
```   505     (simpset() addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
```
```   506 			 le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt,
```
```   507 			 Ord_Un, ltI, nat_le_cardinal,
```
```   508 			 Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
```
```   509 qed "ordertype_csquare_le";
```
```   510
```
```   511 (*Main result: Kunen's Theorem 10.12*)
```
```   512 goal CardinalArith.thy "!!K. InfCard(K) ==> K |*| K = K";
```
```   513 by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
```
```   514 by (etac rev_mp 1);
```
```   515 by (trans_ind_tac "K" [] 1);
```
```   516 by (rtac impI 1);
```
```   517 by (rtac le_anti_sym 1);
```
```   518 by (etac (InfCard_is_Card RS cmult_square_le) 2);
```
```   519 by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
```
```   520 by (assume_tac 2);
```
```   521 by (assume_tac 2);
```
```   522 by (asm_simp_tac
```
```   523     (simpset() addsimps [cmult_def, Ord_cardinal_le,
```
```   524 			 well_ord_csquare RS ordermap_bij RS
```
```   525 			 bij_imp_eqpoll RS cardinal_cong,
```
```   526 			 well_ord_csquare RS Ord_ordertype]) 1);
```
```   527 qed "InfCard_csquare_eq";
```
```   528
```
```   529 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
```
```   530 goal CardinalArith.thy
```
```   531     "!!A. [| well_ord(A,r);  InfCard(|A|) |] ==> A*A eqpoll A";
```
```   532 by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
```
```   533 by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
```
```   534 by (rtac well_ord_cardinal_eqE 1);
```
```   535 by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
```
```   536 by (asm_simp_tac (simpset()
```
```   537 		  addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
```
```   538 qed "well_ord_InfCard_square_eq";
```
```   539
```
```   540 (** Toward's Kunen's Corollary 10.13 (1) **)
```
```   541
```
```   542 goal CardinalArith.thy "!!K. [| InfCard(K);  L le K;  0<L |] ==> K |*| L = K";
```
```   543 by (rtac le_anti_sym 1);
```
```   544 by (etac ltE 2 THEN
```
```   545     REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
```
```   546 by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
```
```   547 by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
```
```   548 by (asm_simp_tac (simpset() addsimps [InfCard_csquare_eq]) 1);
```
```   549 qed "InfCard_le_cmult_eq";
```
```   550
```
```   551 (*Corollary 10.13 (1), for cardinal multiplication*)
```
```   552 goal CardinalArith.thy
```
```   553     "!!K. [| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L";
```
```   554 by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
```
```   555 by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
```
```   556 by (resolve_tac [cmult_commute RS ssubst] 1);
```
```   557 by (resolve_tac [Un_commute RS ssubst] 1);
```
```   558 by (ALLGOALS
```
```   559     (asm_simp_tac
```
```   560      (simpset() addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
```
```   561 			  subset_Un_iff2 RS iffD1, le_imp_subset])));
```
```   562 qed "InfCard_cmult_eq";
```
```   563
```
```   564 (*This proof appear to be the simplest!*)
```
```   565 goal CardinalArith.thy "!!K. InfCard(K) ==> K |+| K = K";
```
```   566 by (asm_simp_tac
```
```   567     (simpset() addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
```
```   568 by (rtac InfCard_le_cmult_eq 1);
```
```   569 by (typechk_tac [Ord_0, le_refl, leI]);
```
```   570 by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]);
```
```   571 qed "InfCard_cdouble_eq";
```
```   572
```
```   573 (*Corollary 10.13 (1), for cardinal addition*)
```
```   574 goal CardinalArith.thy "!!K. [| InfCard(K);  L le K |] ==> K |+| L = K";
```
```   575 by (rtac le_anti_sym 1);
```
```   576 by (etac ltE 2 THEN
```
```   577     REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
```
```   578 by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
```
```   579 by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
```
```   580 by (asm_simp_tac (simpset() addsimps [InfCard_cdouble_eq]) 1);
```
```   581 qed "InfCard_le_cadd_eq";
```
```   582
```
```   583 goal CardinalArith.thy
```
```   584     "!!K. [| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L";
```
```   585 by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
```
```   586 by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
```
```   587 by (resolve_tac [cadd_commute RS ssubst] 1);
```
```   588 by (resolve_tac [Un_commute RS ssubst] 1);
```
```   589 by (ALLGOALS
```
```   590     (asm_simp_tac
```
```   591      (simpset() addsimps [InfCard_le_cadd_eq,
```
```   592 			  subset_Un_iff2 RS iffD1, le_imp_subset])));
```
```   593 qed "InfCard_cadd_eq";
```
```   594
```
```   595 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
```
```   596   of all n-tuples of elements of K.  A better version for the Isabelle theory
```
```   597   might be  InfCard(K) ==> |list(K)| = K.
```
```   598 *)
```
```   599
```
```   600 (*** For every cardinal number there exists a greater one
```
```   601      [Kunen's Theorem 10.16, which would be trivial using AC] ***)
```
```   602
```
```   603 goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))";
```
```   604 by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
```
```   605 by (blast_tac (claset() addSIs [Ord_ordertype]) 2);
```
```   606 by (rewtac Transset_def);
```
```   607 by (safe_tac subset_cs);
```
```   608 by (asm_full_simp_tac (simpset() addsimps [ordertype_pred_unfold]) 1);
```
```   609 by Safe_tac;
```
```   610 by (rtac UN_I 1);
```
```   611 by (rtac ReplaceI 2);
```
```   612 by (ALLGOALS (blast_tac (claset() addIs [well_ord_subset] addSEs [predE])));
```
```   613 qed "Ord_jump_cardinal";
```
```   614
```
```   615 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
```
```   616 goalw CardinalArith.thy [jump_cardinal_def]
```
```   617      "i : jump_cardinal(K) <->   \
```
```   618 \         (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
```
```   619 by (fast_tac subset_cs 1);      (*It's vital to avoid reasoning about <=*)
```
```   620 qed "jump_cardinal_iff";
```
```   621
```
```   622 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
```
```   623 goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)";
```
```   624 by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
```
```   625 by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
```
```   626 by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
```
```   627 by (rtac subset_refl 2);
```
```   628 by (asm_simp_tac (simpset() addsimps [Memrel_def, subset_iff]) 1);
```
```   629 by (asm_simp_tac (simpset() addsimps [ordertype_Memrel]) 1);
```
```   630 qed "K_lt_jump_cardinal";
```
```   631
```
```   632 (*The proof by contradiction: the bijection f yields a wellordering of X
```
```   633   whose ordertype is jump_cardinal(K).  *)
```
```   634 goal CardinalArith.thy
```
```   635     "!!K. [| well_ord(X,r);  r <= K * K;  X <= K;       \
```
```   636 \            f : bij(ordertype(X,r), jump_cardinal(K))  \
```
```   637 \         |] ==> jump_cardinal(K) : jump_cardinal(K)";
```
```   638 by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
```
```   639 by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
```
```   640 by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
```
```   641 by (REPEAT_FIRST (resolve_tac [exI, conjI]));
```
```   642 by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
```
```   643 by (REPEAT (assume_tac 1));
```
```   644 by (etac (bij_is_inj RS well_ord_rvimage) 1);
```
```   645 by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
```
```   646 by (asm_simp_tac
```
```   647     (simpset() addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage),
```
```   648 			 ordertype_Memrel, Ord_jump_cardinal]) 1);
```
```   649 qed "Card_jump_cardinal_lemma";
```
```   650
```
```   651 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
```
```   652 goal CardinalArith.thy "Card(jump_cardinal(K))";
```
```   653 by (rtac (Ord_jump_cardinal RS CardI) 1);
```
```   654 by (rewtac eqpoll_def);
```
```   655 by (safe_tac (claset() addSDs [ltD, jump_cardinal_iff RS iffD1]));
```
```   656 by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
```
```   657 qed "Card_jump_cardinal";
```
```   658
```
```   659 (*** Basic properties of successor cardinals ***)
```
```   660
```
```   661 goalw CardinalArith.thy [csucc_def]
```
```   662     "!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
```
```   663 by (rtac LeastI 1);
```
```   664 by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
```
```   665                       Ord_jump_cardinal] 1));
```
```   666 qed "csucc_basic";
```
```   667
```
```   668 bind_thm ("Card_csucc", csucc_basic RS conjunct1);
```
```   669
```
```   670 bind_thm ("lt_csucc", csucc_basic RS conjunct2);
```
```   671
```
```   672 goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)";
```
```   673 by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
```
```   674 by (REPEAT (assume_tac 1));
```
```   675 qed "Ord_0_lt_csucc";
```
```   676
```
```   677 goalw CardinalArith.thy [csucc_def]
```
```   678     "!!K L. [| Card(L);  K<L |] ==> csucc(K) le L";
```
```   679 by (rtac Least_le 1);
```
```   680 by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
```
```   681 qed "csucc_le";
```
```   682
```
```   683 goal CardinalArith.thy
```
```   684     "!!K. [| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
```
```   685 by (rtac iffI 1);
```
```   686 by (rtac Card_lt_imp_lt 2);
```
```   687 by (etac lt_trans1 2);
```
```   688 by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
```
```   689 by (resolve_tac [notI RS not_lt_imp_le] 1);
```
```   690 by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
```
```   691 by (assume_tac 1);
```
```   692 by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
```
```   693 by (REPEAT (ares_tac [Ord_cardinal] 1
```
```   694      ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
```
```   695 qed "lt_csucc_iff";
```
```   696
```
```   697 goal CardinalArith.thy
```
```   698     "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
```
```   699 by (asm_simp_tac
```
```   700     (simpset() addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
```
```   701 qed "Card_lt_csucc_iff";
```
```   702
```
```   703 goalw CardinalArith.thy [InfCard_def]
```
```   704     "!!K. InfCard(K) ==> InfCard(csucc(K))";
```
```   705 by (asm_simp_tac (simpset() addsimps [Card_csucc, Card_is_Ord,
```
```   706 				      lt_csucc RS leI RSN (2,le_trans)]) 1);
```
```   707 qed "InfCard_csucc";
```
```   708
```
```   709
```
```   710 (*** Finite sets ***)
```
```   711
```
```   712 goal CardinalArith.thy
```
```   713     "!!n. n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
```
```   714 by (etac nat_induct 1);
```
```   715 by (simp_tac (simpset() addsimps (eqpoll_0_iff::Fin.intrs)) 1);
```
```   716 by (Clarify_tac 1);
```
```   717 by (subgoal_tac "EX u. u:A" 1);
```
```   718 by (etac exE 1);
```
```   719 by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
```
```   720 by (assume_tac 2);
```
```   721 by (assume_tac 1);
```
```   722 by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
```
```   723 by (assume_tac 1);
```
```   724 by (resolve_tac [Fin.consI] 1);
```
```   725 by (Blast_tac 1);
```
```   726 by (blast_tac (claset() addIs [subset_consI  RS Fin_mono RS subsetD]) 1);
```
```   727 (*Now for the lemma assumed above*)
```
```   728 by (rewtac eqpoll_def);
```
```   729 by (blast_tac (claset() addIs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
```
```   730 val lemma = result();
```
```   731
```
```   732 goalw CardinalArith.thy [Finite_def] "!!A. Finite(A) ==> A : Fin(A)";
```
```   733 by (blast_tac (claset() addIs [lemma RS spec RS mp]) 1);
```
```   734 qed "Finite_into_Fin";
```
```   735
```
```   736 goal CardinalArith.thy "!!A. A : Fin(U) ==> Finite(A)";
```
```   737 by (fast_tac (claset() addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
```
```   738 qed "Fin_into_Finite";
```
```   739
```
```   740 goal CardinalArith.thy "Finite(A) <-> A : Fin(A)";
```
```   741 by (blast_tac (claset() addIs [Finite_into_Fin, Fin_into_Finite]) 1);
```
```   742 qed "Finite_Fin_iff";
```
```   743
```
```   744 goal CardinalArith.thy
```
```   745     "!!A. [| Finite(A); Finite(B) |] ==> Finite(A Un B)";
```
```   746 by (blast_tac (claset() addSIs [Fin_into_Finite, Fin_UnI]
```
```   747                         addSDs [Finite_into_Fin]
```
```   748                         addIs  [Un_upper1 RS Fin_mono RS subsetD,
```
```   749 	 		        Un_upper2 RS Fin_mono RS subsetD]) 1);
```
```   750 qed "Finite_Un";
```
```   751
```
```   752
```
```   753 (** Removing elements from a finite set decreases its cardinality **)
```
```   754
```
```   755 goal CardinalArith.thy
```
```   756     "!!A. A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
```
```   757 by (etac Fin_induct 1);
```
```   758 by (simp_tac (simpset() addsimps [lepoll_0_iff]) 1);
```
```   759 by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
```
```   760 by (Asm_simp_tac 1);
```
```   761 by (blast_tac (claset() addSDs [cons_lepoll_consD]) 1);
```
```   762 by (Blast_tac 1);
```
```   763 qed "Fin_imp_not_cons_lepoll";
```
```   764
```
```   765 goal CardinalArith.thy
```
```   766     "!!a A. [| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)";
```
```   767 by (rewtac cardinal_def);
```
```   768 by (rtac Least_equality 1);
```
```   769 by (fold_tac [cardinal_def]);
```
```   770 by (simp_tac (simpset() addsimps [succ_def]) 1);
```
```   771 by (blast_tac (claset() addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll]
```
```   772                         addSEs [mem_irrefl]
```
```   773                         addSDs [Finite_imp_well_ord]) 1);
```
```   774 by (blast_tac (claset() addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1);
```
```   775 by (rtac notI 1);
```
```   776 by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
```
```   777 by (assume_tac 1);
```
```   778 by (assume_tac 1);
```
```   779 by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
```
```   780 by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
```
```   781 by (blast_tac (claset() addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll]
```
```   782                     addSDs [Finite_imp_well_ord]) 1);
```
```   783 qed "Finite_imp_cardinal_cons";
```
```   784
```
```   785
```
```   786 goal CardinalArith.thy "!!a A. [| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|";
```
```   787 by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
```
```   788 by (assume_tac 1);
```
```   789 by (asm_simp_tac (simpset() addsimps [Finite_imp_cardinal_cons,
```
```   790                                   Diff_subset RS subset_Finite]) 1);
```
```   791 by (asm_simp_tac (simpset() addsimps [cons_Diff]) 1);
```
```   792 qed "Finite_imp_succ_cardinal_Diff";
```
```   793
```
```   794 goal CardinalArith.thy "!!a A. [| Finite(A);  a:A |] ==> |A-{a}| < |A|";
```
```   795 by (rtac succ_leE 1);
```
```   796 by (asm_simp_tac (simpset() addsimps [Finite_imp_succ_cardinal_Diff,
```
```   797 				      Ord_cardinal RS le_refl]) 1);
```
```   798 qed "Finite_imp_cardinal_Diff";
```
```   799
```
```   800
```
```   801 (** Theorems by Krzysztof Grabczewski, proofs by lcp **)
```
```   802
```
```   803 val nat_implies_well_ord =
```
```   804   (transfer CardinalArith.thy nat_into_Ord) RS well_ord_Memrel;
```
```   805
```
```   806 goal CardinalArith.thy "!!m n. [| m:nat; n:nat |] ==> m + n eqpoll m #+ n";
```
```   807 by (rtac eqpoll_trans 1);
```
```   808 by (resolve_tac [well_ord_radd RS well_ord_cardinal_eqpoll RS eqpoll_sym] 1);
```
```   809 by (REPEAT (etac nat_implies_well_ord 1));
```
```   810 by (asm_simp_tac (simpset()
```
```   811 		  addsimps [nat_cadd_eq_add RS sym, cadd_def, eqpoll_refl]) 1);
```
```   812 qed "nat_sum_eqpoll_sum";
```
```   813
```
```   814 goal Nat.thy "!!m. [| m le n; n:nat |] ==> m:nat";
```
```   815 by (blast_tac (claset() addSIs [nat_succI] addSDs [lt_nat_in_nat]) 1);
```
```   816 qed "le_in_nat";
```
```   817
```