src/ZF/CardinalArith.thy
 author paulson Thu Mar 15 16:35:02 2012 +0000 (2012-03-15) changeset 46953 2b6e55924af3 parent 46952 5e1bcfdcb175 child 47101 ded5cc757bc9 permissions -rw-r--r--
replacing ":" by "\<in>"
```     1 (*  Title:      ZF/CardinalArith.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1994  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header{*Cardinal Arithmetic Without the Axiom of Choice*}
```
```     7
```
```     8 theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin
```
```     9
```
```    10 definition
```
```    11   InfCard       :: "i=>o"  where
```
```    12     "InfCard(i) == Card(i) & nat \<le> i"
```
```    13
```
```    14 definition
```
```    15   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where
```
```    16     "i |*| j == |i*j|"
```
```    17
```
```    18 definition
```
```    19   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where
```
```    20     "i |+| j == |i+j|"
```
```    21
```
```    22 definition
```
```    23   csquare_rel   :: "i=>i"  where
```
```    24     "csquare_rel(K) ==
```
```    25           rvimage(K*K,
```
```    26                   lam <x,y>:K*K. <x \<union> y, x, y>,
```
```    27                   rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
```
```    28
```
```    29 definition
```
```    30   jump_cardinal :: "i=>i"  where
```
```    31     --{*This def is more complex than Kunen's but it more easily proved to
```
```    32         be a cardinal*}
```
```    33     "jump_cardinal(K) ==
```
```    34          \<Union>X\<in>Pow(K). {z. r \<in> Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
```
```    35
```
```    36 definition
```
```    37   csucc         :: "i=>i"  where
```
```    38     --{*needed because @{term "jump_cardinal(K)"} might not be the successor
```
```    39         of @{term K}*}
```
```    40     "csucc(K) == LEAST L. Card(L) & K<L"
```
```    41
```
```    42 notation (xsymbols)
```
```    43   cadd  (infixl "\<oplus>" 65) and
```
```    44   cmult  (infixl "\<otimes>" 70)
```
```    45
```
```    46 notation (HTML)
```
```    47   cadd  (infixl "\<oplus>" 65) and
```
```    48   cmult  (infixl "\<otimes>" 70)
```
```    49
```
```    50
```
```    51 lemma Card_Union [simp,intro,TC]:
```
```    52   assumes A: "\<And>x. x\<in>A \<Longrightarrow> Card(x)" shows "Card(\<Union>(A))"
```
```    53 proof (rule CardI)
```
```    54   show "Ord(\<Union>A)" using A
```
```    55     by (simp add: Card_is_Ord)
```
```    56 next
```
```    57   fix j
```
```    58   assume j: "j < \<Union>A"
```
```    59   hence "\<exists>c\<in>A. j < c & Card(c)" using A
```
```    60     by (auto simp add: lt_def intro: Card_is_Ord)
```
```    61   then obtain c where c: "c\<in>A" "j < c" "Card(c)"
```
```    62     by blast
```
```    63   hence jls: "j \<prec> c"
```
```    64     by (simp add: lt_Card_imp_lesspoll)
```
```    65   { assume eqp: "j \<approx> \<Union>A"
```
```    66     have  "c \<lesssim> \<Union>A" using c
```
```    67       by (blast intro: subset_imp_lepoll)
```
```    68     also have "... \<approx> j"  by (rule eqpoll_sym [OF eqp])
```
```    69     also have "... \<prec> c"  by (rule jls)
```
```    70     finally have "c \<prec> c" .
```
```    71     hence False
```
```    72       by auto
```
```    73   } thus "\<not> j \<approx> \<Union>A" by blast
```
```    74 qed
```
```    75
```
```    76 lemma Card_UN: "(!!x. x \<in> A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"
```
```    77   by blast
```
```    78
```
```    79 lemma Card_OUN [simp,intro,TC]:
```
```    80      "(!!x. x \<in> A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
```
```    81   by (auto simp add: OUnion_def Card_0)
```
```    82
```
```    83 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
```
```    84 apply (unfold lesspoll_def)
```
```    85 apply (simp add: Card_iff_initial)
```
```    86 apply (fast intro!: le_imp_lepoll ltI leI)
```
```    87 done
```
```    88
```
```    89
```
```    90 subsection{*Cardinal addition*}
```
```    91
```
```    92 text{*Note: Could omit proving the algebraic laws for cardinal addition and
```
```    93 multiplication.  On finite cardinals these operations coincide with
```
```    94 addition and multiplication of natural numbers; on infinite cardinals they
```
```    95 coincide with union (maximum).  Either way we get most laws for free.*}
```
```    96
```
```    97 subsubsection{*Cardinal addition is commutative*}
```
```    98
```
```    99 lemma sum_commute_eqpoll: "A+B \<approx> B+A"
```
```   100 proof (unfold eqpoll_def, rule exI)
```
```   101   show "(\<lambda>z\<in>A+B. case(Inr,Inl,z)) \<in> bij(A+B, B+A)"
```
```   102     by (auto intro: lam_bijective [where d = "case(Inr,Inl)"])
```
```   103 qed
```
```   104
```
```   105 lemma cadd_commute: "i \<oplus> j = j \<oplus> i"
```
```   106 apply (unfold cadd_def)
```
```   107 apply (rule sum_commute_eqpoll [THEN cardinal_cong])
```
```   108 done
```
```   109
```
```   110 subsubsection{*Cardinal addition is associative*}
```
```   111
```
```   112 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
```
```   113 apply (unfold eqpoll_def)
```
```   114 apply (rule exI)
```
```   115 apply (rule sum_assoc_bij)
```
```   116 done
```
```   117
```
```   118 text{*Unconditional version requires AC*}
```
```   119 lemma well_ord_cadd_assoc:
```
```   120   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
```
```   121   shows "(i \<oplus> j) \<oplus> k = i \<oplus> (j \<oplus> k)"
```
```   122 proof (unfold cadd_def, rule cardinal_cong)
```
```   123   have "|i + j| + k \<approx> (i + j) + k"
```
```   124     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
```
```   125   also have "...  \<approx> i + (j + k)"
```
```   126     by (rule sum_assoc_eqpoll)
```
```   127   also have "...  \<approx> i + |j + k|"
```
```   128     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd j k eqpoll_sym)
```
```   129   finally show "|i + j| + k \<approx> i + |j + k|" .
```
```   130 qed
```
```   131
```
```   132
```
```   133 subsubsection{*0 is the identity for addition*}
```
```   134
```
```   135 lemma sum_0_eqpoll: "0+A \<approx> A"
```
```   136 apply (unfold eqpoll_def)
```
```   137 apply (rule exI)
```
```   138 apply (rule bij_0_sum)
```
```   139 done
```
```   140
```
```   141 lemma cadd_0 [simp]: "Card(K) ==> 0 \<oplus> K = K"
```
```   142 apply (unfold cadd_def)
```
```   143 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
```
```   144 done
```
```   145
```
```   146 subsubsection{*Addition by another cardinal*}
```
```   147
```
```   148 lemma sum_lepoll_self: "A \<lesssim> A+B"
```
```   149 proof (unfold lepoll_def, rule exI)
```
```   150   show "(\<lambda>x\<in>A. Inl (x)) \<in> inj(A, A + B)"
```
```   151     by (simp add: inj_def)
```
```   152 qed
```
```   153
```
```   154 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
```
```   155
```
```   156 lemma cadd_le_self:
```
```   157   assumes K: "Card(K)" and L: "Ord(L)" shows "K \<le> (K \<oplus> L)"
```
```   158 proof (unfold cadd_def)
```
```   159   have "K \<le> |K|"
```
```   160     by (rule Card_cardinal_le [OF K])
```
```   161   moreover have "|K| \<le> |K + L|" using K L
```
```   162     by (blast intro: well_ord_lepoll_imp_Card_le sum_lepoll_self
```
```   163                      well_ord_radd well_ord_Memrel Card_is_Ord)
```
```   164   ultimately show "K \<le> |K + L|"
```
```   165     by (blast intro: le_trans)
```
```   166 qed
```
```   167
```
```   168 subsubsection{*Monotonicity of addition*}
```
```   169
```
```   170 lemma sum_lepoll_mono:
```
```   171      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
```
```   172 apply (unfold lepoll_def)
```
```   173 apply (elim exE)
```
```   174 apply (rule_tac x = "\<lambda>z\<in>A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
```
```   175 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
```
```   176        in lam_injective)
```
```   177 apply (typecheck add: inj_is_fun, auto)
```
```   178 done
```
```   179
```
```   180 lemma cadd_le_mono:
```
```   181     "[| K' \<le> K;  L' \<le> L |] ==> (K' \<oplus> L') \<le> (K \<oplus> L)"
```
```   182 apply (unfold cadd_def)
```
```   183 apply (safe dest!: le_subset_iff [THEN iffD1])
```
```   184 apply (rule well_ord_lepoll_imp_Card_le)
```
```   185 apply (blast intro: well_ord_radd well_ord_Memrel)
```
```   186 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
```
```   187 done
```
```   188
```
```   189 subsubsection{*Addition of finite cardinals is "ordinary" addition*}
```
```   190
```
```   191 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
```
```   192 apply (unfold eqpoll_def)
```
```   193 apply (rule exI)
```
```   194 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"
```
```   195             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
```
```   196    apply simp_all
```
```   197 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
```
```   198 done
```
```   199
```
```   200 (*Pulling the  succ(...)  outside the |...| requires m, n \<in> nat  *)
```
```   201 (*Unconditional version requires AC*)
```
```   202 lemma cadd_succ_lemma:
```
```   203   assumes "Ord(m)" "Ord(n)" shows "succ(m) \<oplus> n = |succ(m \<oplus> n)|"
```
```   204 proof (unfold cadd_def)
```
```   205   have [intro]: "m + n \<approx> |m + n|" using assms
```
```   206     by (blast intro: eqpoll_sym well_ord_cardinal_eqpoll well_ord_radd well_ord_Memrel)
```
```   207
```
```   208   have "|succ(m) + n| = |succ(m + n)|"
```
```   209     by (rule sum_succ_eqpoll [THEN cardinal_cong])
```
```   210   also have "... = |succ(|m + n|)|"
```
```   211     by (blast intro: succ_eqpoll_cong cardinal_cong)
```
```   212   finally show "|succ(m) + n| = |succ(|m + n|)|" .
```
```   213 qed
```
```   214
```
```   215 lemma nat_cadd_eq_add:
```
```   216   assumes m: "m \<in> nat" and [simp]: "n \<in> nat" shows"m \<oplus> n = m #+ n"
```
```   217 using m
```
```   218 proof (induct m)
```
```   219   case 0 thus ?case by (simp add: nat_into_Card cadd_0)
```
```   220 next
```
```   221   case (succ m) thus ?case by (simp add: cadd_succ_lemma nat_into_Card Card_cardinal_eq)
```
```   222 qed
```
```   223
```
```   224
```
```   225 subsection{*Cardinal multiplication*}
```
```   226
```
```   227 subsubsection{*Cardinal multiplication is commutative*}
```
```   228
```
```   229 lemma prod_commute_eqpoll: "A*B \<approx> B*A"
```
```   230 apply (unfold eqpoll_def)
```
```   231 apply (rule exI)
```
```   232 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective,
```
```   233        auto)
```
```   234 done
```
```   235
```
```   236 lemma cmult_commute: "i \<otimes> j = j \<otimes> i"
```
```   237 apply (unfold cmult_def)
```
```   238 apply (rule prod_commute_eqpoll [THEN cardinal_cong])
```
```   239 done
```
```   240
```
```   241 subsubsection{*Cardinal multiplication is associative*}
```
```   242
```
```   243 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
```
```   244 apply (unfold eqpoll_def)
```
```   245 apply (rule exI)
```
```   246 apply (rule prod_assoc_bij)
```
```   247 done
```
```   248
```
```   249 text{*Unconditional version requires AC*}
```
```   250 lemma well_ord_cmult_assoc:
```
```   251   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
```
```   252   shows "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
```
```   253 proof (unfold cmult_def, rule cardinal_cong)
```
```   254   have "|i * j| * k \<approx> (i * j) * k"
```
```   255     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult i j)
```
```   256   also have "...  \<approx> i * (j * k)"
```
```   257     by (rule prod_assoc_eqpoll)
```
```   258   also have "...  \<approx> i * |j * k|"
```
```   259     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult j k eqpoll_sym)
```
```   260   finally show "|i * j| * k \<approx> i * |j * k|" .
```
```   261 qed
```
```   262
```
```   263 subsubsection{*Cardinal multiplication distributes over addition*}
```
```   264
```
```   265 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
```
```   266 apply (unfold eqpoll_def)
```
```   267 apply (rule exI)
```
```   268 apply (rule sum_prod_distrib_bij)
```
```   269 done
```
```   270
```
```   271 lemma well_ord_cadd_cmult_distrib:
```
```   272   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
```
```   273   shows "(i \<oplus> j) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)"
```
```   274 proof (unfold cadd_def cmult_def, rule cardinal_cong)
```
```   275   have "|i + j| * k \<approx> (i + j) * k"
```
```   276     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
```
```   277   also have "...  \<approx> i * k + j * k"
```
```   278     by (rule sum_prod_distrib_eqpoll)
```
```   279   also have "...  \<approx> |i * k| + |j * k|"
```
```   280     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll well_ord_rmult i j k eqpoll_sym)
```
```   281   finally show "|i + j| * k \<approx> |i * k| + |j * k|" .
```
```   282 qed
```
```   283
```
```   284 subsubsection{*Multiplication by 0 yields 0*}
```
```   285
```
```   286 lemma prod_0_eqpoll: "0*A \<approx> 0"
```
```   287 apply (unfold eqpoll_def)
```
```   288 apply (rule exI)
```
```   289 apply (rule lam_bijective, safe)
```
```   290 done
```
```   291
```
```   292 lemma cmult_0 [simp]: "0 \<otimes> i = 0"
```
```   293 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
```
```   294
```
```   295 subsubsection{*1 is the identity for multiplication*}
```
```   296
```
```   297 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
```
```   298 apply (unfold eqpoll_def)
```
```   299 apply (rule exI)
```
```   300 apply (rule singleton_prod_bij [THEN bij_converse_bij])
```
```   301 done
```
```   302
```
```   303 lemma cmult_1 [simp]: "Card(K) ==> 1 \<otimes> K = K"
```
```   304 apply (unfold cmult_def succ_def)
```
```   305 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
```
```   306 done
```
```   307
```
```   308 subsection{*Some inequalities for multiplication*}
```
```   309
```
```   310 lemma prod_square_lepoll: "A \<lesssim> A*A"
```
```   311 apply (unfold lepoll_def inj_def)
```
```   312 apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp)
```
```   313 done
```
```   314
```
```   315 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
```
```   316 lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K"
```
```   317 apply (unfold cmult_def)
```
```   318 apply (rule le_trans)
```
```   319 apply (rule_tac  well_ord_lepoll_imp_Card_le)
```
```   320 apply (rule_tac  prod_square_lepoll)
```
```   321 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
```
```   322 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
```
```   323 done
```
```   324
```
```   325 subsubsection{*Multiplication by a non-zero cardinal*}
```
```   326
```
```   327 lemma prod_lepoll_self: "b \<in> B ==> A \<lesssim> A*B"
```
```   328 apply (unfold lepoll_def inj_def)
```
```   329 apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)
```
```   330 done
```
```   331
```
```   332 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
```
```   333 lemma cmult_le_self:
```
```   334     "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K \<otimes> L)"
```
```   335 apply (unfold cmult_def)
```
```   336 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
```
```   337   apply assumption
```
```   338  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
```
```   339 apply (blast intro: prod_lepoll_self ltD)
```
```   340 done
```
```   341
```
```   342 subsubsection{*Monotonicity of multiplication*}
```
```   343
```
```   344 lemma prod_lepoll_mono:
```
```   345      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
```
```   346 apply (unfold lepoll_def)
```
```   347 apply (elim exE)
```
```   348 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
```
```   349 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"
```
```   350        in lam_injective)
```
```   351 apply (typecheck add: inj_is_fun, auto)
```
```   352 done
```
```   353
```
```   354 lemma cmult_le_mono:
```
```   355     "[| K' \<le> K;  L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)"
```
```   356 apply (unfold cmult_def)
```
```   357 apply (safe dest!: le_subset_iff [THEN iffD1])
```
```   358 apply (rule well_ord_lepoll_imp_Card_le)
```
```   359  apply (blast intro: well_ord_rmult well_ord_Memrel)
```
```   360 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
```
```   361 done
```
```   362
```
```   363 subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
```
```   364
```
```   365 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
```
```   366 apply (unfold eqpoll_def)
```
```   367 apply (rule exI)
```
```   368 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
```
```   369             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
```
```   370 apply safe
```
```   371 apply (simp_all add: succI2 if_type mem_imp_not_eq)
```
```   372 done
```
```   373
```
```   374 (*Unconditional version requires AC*)
```
```   375 lemma cmult_succ_lemma:
```
```   376     "[| Ord(m);  Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)"
```
```   377 apply (unfold cmult_def cadd_def)
```
```   378 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
```
```   379 apply (rule cardinal_cong [symmetric])
```
```   380 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
```
```   381 apply (blast intro: well_ord_rmult well_ord_Memrel)
```
```   382 done
```
```   383
```
```   384 lemma nat_cmult_eq_mult: "[| m \<in> nat;  n \<in> nat |] ==> m \<otimes> n = m#*n"
```
```   385 apply (induct_tac m)
```
```   386 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
```
```   387 done
```
```   388
```
```   389 lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n"
```
```   390 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
```
```   391
```
```   392 lemma sum_lepoll_prod:
```
```   393   assumes C: "2 \<lesssim> C" shows "B+B \<lesssim> C*B"
```
```   394 proof -
```
```   395   have "B+B \<lesssim> 2*B"
```
```   396     by (simp add: sum_eq_2_times)
```
```   397   also have "... \<lesssim> C*B"
```
```   398     by (blast intro: prod_lepoll_mono lepoll_refl C)
```
```   399   finally show "B+B \<lesssim> C*B" .
```
```   400 qed
```
```   401
```
```   402 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
```
```   403 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
```
```   404
```
```   405
```
```   406 subsection{*Infinite Cardinals are Limit Ordinals*}
```
```   407
```
```   408 (*This proof is modelled upon one assuming nat<=A, with injection
```
```   409   \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z
```
```   410   and inverse %y. if y \<in> nat then nat_case(u, %z. z, y) else y.  \
```
```   411   If f \<in> inj(nat,A) then range(f) behaves like the natural numbers.*)
```
```   412 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
```
```   413 apply (unfold lepoll_def)
```
```   414 apply (erule exE)
```
```   415 apply (rule_tac x =
```
```   416           "\<lambda>z\<in>cons (u,A).
```
```   417              if z=u then f`0
```
```   418              else if z \<in> range (f) then f`succ (converse (f) `z) else z"
```
```   419        in exI)
```
```   420 apply (rule_tac d =
```
```   421           "%y. if y \<in> range(f) then nat_case (u, %z. f`z, converse(f) `y)
```
```   422                               else y"
```
```   423        in lam_injective)
```
```   424 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
```
```   425 apply (simp add: inj_is_fun [THEN apply_rangeI]
```
```   426                  inj_converse_fun [THEN apply_rangeI]
```
```   427                  inj_converse_fun [THEN apply_funtype])
```
```   428 done
```
```   429
```
```   430 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
```
```   431 apply (erule nat_cons_lepoll [THEN eqpollI])
```
```   432 apply (rule subset_consI [THEN subset_imp_lepoll])
```
```   433 done
```
```   434
```
```   435 (*Specialized version required below*)
```
```   436 lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"
```
```   437 apply (unfold succ_def)
```
```   438 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
```
```   439 done
```
```   440
```
```   441 lemma InfCard_nat: "InfCard(nat)"
```
```   442 apply (unfold InfCard_def)
```
```   443 apply (blast intro: Card_nat le_refl Card_is_Ord)
```
```   444 done
```
```   445
```
```   446 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
```
```   447 apply (unfold InfCard_def)
```
```   448 apply (erule conjunct1)
```
```   449 done
```
```   450
```
```   451 lemma InfCard_Un:
```
```   452     "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"
```
```   453 apply (unfold InfCard_def)
```
```   454 apply (simp add: Card_Un Un_upper1_le [THEN  le_trans]  Card_is_Ord)
```
```   455 done
```
```   456
```
```   457 (*Kunen's Lemma 10.11*)
```
```   458 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
```
```   459 apply (unfold InfCard_def)
```
```   460 apply (erule conjE)
```
```   461 apply (frule Card_is_Ord)
```
```   462 apply (rule ltI [THEN non_succ_LimitI])
```
```   463 apply (erule le_imp_subset [THEN subsetD])
```
```   464 apply (safe dest!: Limit_nat [THEN Limit_le_succD])
```
```   465 apply (unfold Card_def)
```
```   466 apply (drule trans)
```
```   467 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
```
```   468 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
```
```   469 apply (rule le_eqI, assumption)
```
```   470 apply (rule Ord_cardinal)
```
```   471 done
```
```   472
```
```   473
```
```   474 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
```
```   475
```
```   476 (*A general fact about ordermap*)
```
```   477 lemma ordermap_eqpoll_pred:
```
```   478     "[| well_ord(A,r);  x \<in> A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
```
```   479 apply (unfold eqpoll_def)
```
```   480 apply (rule exI)
```
```   481 apply (simp add: ordermap_eq_image well_ord_is_wf)
```
```   482 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,
```
```   483                            THEN bij_converse_bij])
```
```   484 apply (rule pred_subset)
```
```   485 done
```
```   486
```
```   487 subsubsection{*Establishing the well-ordering*}
```
```   488
```
```   489 lemma well_ord_csquare:
```
```   490   assumes K: "Ord(K)" shows "well_ord(K*K, csquare_rel(K))"
```
```   491 proof (unfold csquare_rel_def, rule well_ord_rvimage)
```
```   492   show "(\<lambda>\<langle>x,y\<rangle>\<in>K \<times> K. \<langle>x \<union> y, x, y\<rangle>) \<in> inj(K \<times> K, K \<times> K \<times> K)" using K
```
```   493     by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI)
```
```   494 next
```
```   495   show "well_ord(K \<times> K \<times> K, rmult(K, Memrel(K), K \<times> K, rmult(K, Memrel(K), K, Memrel(K))))"
```
```   496     using K by (blast intro: well_ord_rmult well_ord_Memrel)
```
```   497 qed
```
```   498
```
```   499 subsubsection{*Characterising initial segments of the well-ordering*}
```
```   500
```
```   501 lemma csquareD:
```
```   502  "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"
```
```   503 apply (unfold csquare_rel_def)
```
```   504 apply (erule rev_mp)
```
```   505 apply (elim ltE)
```
```   506 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
```
```   507 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
```
```   508 apply (simp_all add: lt_def succI2)
```
```   509 done
```
```   510
```
```   511 lemma pred_csquare_subset:
```
```   512     "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"
```
```   513 apply (unfold Order.pred_def)
```
```   514 apply (safe del: SigmaI dest!: csquareD)
```
```   515 apply (unfold lt_def, auto)
```
```   516 done
```
```   517
```
```   518 lemma csquare_ltI:
```
```   519  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"
```
```   520 apply (unfold csquare_rel_def)
```
```   521 apply (subgoal_tac "x<K & y<K")
```
```   522  prefer 2 apply (blast intro: lt_trans)
```
```   523 apply (elim ltE)
```
```   524 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
```
```   525 done
```
```   526
```
```   527 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
```
```   528 lemma csquare_or_eqI:
```
```   529  "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"
```
```   530 apply (unfold csquare_rel_def)
```
```   531 apply (subgoal_tac "x<K & y<K")
```
```   532  prefer 2 apply (blast intro: lt_trans1)
```
```   533 apply (elim ltE)
```
```   534 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
```
```   535 apply (elim succE)
```
```   536 apply (simp_all add: subset_Un_iff [THEN iff_sym]
```
```   537                      subset_Un_iff2 [THEN iff_sym] OrdmemD)
```
```   538 done
```
```   539
```
```   540 subsubsection{*The cardinality of initial segments*}
```
```   541
```
```   542 lemma ordermap_z_lt:
```
```   543       "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>
```
```   544           ordermap(K*K, csquare_rel(K)) ` <x,y> <
```
```   545           ordermap(K*K, csquare_rel(K)) ` <z,z>"
```
```   546 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
```
```   547 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
```
```   548                               Limit_is_Ord [THEN well_ord_csquare], clarify)
```
```   549 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
```
```   550 apply (erule_tac  well_ord_is_wf)
```
```   551 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
```
```   552 done
```
```   553
```
```   554 text{*Kunen: "each @{term"\<langle>x,y\<rangle> \<in> K \<times> K"} has no more than @{term"z \<times> z"} predecessors..." (page 29) *}
```
```   555 lemma ordermap_csquare_le:
```
```   556   assumes K: "Limit(K)" and x: "x<K" and y: " y<K"
```
```   557   defines "z \<equiv> succ(x \<union> y)"
```
```   558   shows "|ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>| \<le> |succ(z)| \<otimes> |succ(z)|"
```
```   559 proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le)
```
```   560   show "well_ord(|succ(z)| \<times> |succ(z)|,
```
```   561                  rmult(|succ(z)|, Memrel(|succ(z)|), |succ(z)|, Memrel(|succ(z)|)))"
```
```   562     by (blast intro: Ord_cardinal well_ord_Memrel well_ord_rmult)
```
```   563 next
```
```   564   have zK: "z<K" using x y K z_def
```
```   565     by (blast intro: Un_least_lt Limit_has_succ)
```
```   566   hence oz: "Ord(z)" by (elim ltE)
```
```   567   have "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> ordermap(K \<times> K, csquare_rel(K)) ` \<langle>z,z\<rangle>"
```
```   568     using z_def
```
```   569     by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y)
```
```   570   also have "... \<approx>  Order.pred(K \<times> K, \<langle>z,z\<rangle>, csquare_rel(K))"
```
```   571     proof (rule ordermap_eqpoll_pred)
```
```   572       show "well_ord(K \<times> K, csquare_rel(K))" using K
```
```   573         by (rule Limit_is_Ord [THEN well_ord_csquare])
```
```   574     next
```
```   575       show "\<langle>z, z\<rangle> \<in> K \<times> K" using zK
```
```   576         by (blast intro: ltD)
```
```   577     qed
```
```   578   also have "...  \<lesssim> succ(z) \<times> succ(z)" using zK
```
```   579     by (rule pred_csquare_subset [THEN subset_imp_lepoll])
```
```   580   also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz
```
```   581     by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym)
```
```   582   finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" .
```
```   583 qed
```
```   584
```
```   585 text{*Kunen: "... so the order type is @{text"\<le>"} K" *}
```
```   586 lemma ordertype_csquare_le:
```
```   587   assumes IK: "InfCard(K)" and eq: "\<And>y. y\<in>K \<Longrightarrow> InfCard(y) \<Longrightarrow> y \<otimes> y = y"
```
```   588   shows "ordertype(K*K, csquare_rel(K)) \<le> K"
```
```   589 proof -
```
```   590   have  CK: "Card(K)" using IK by (rule InfCard_is_Card)
```
```   591   hence OK: "Ord(K)"  by (rule Card_is_Ord)
```
```   592   moreover have "Ord(ordertype(K \<times> K, csquare_rel(K)))" using OK
```
```   593     by (rule well_ord_csquare [THEN Ord_ordertype])
```
```   594   ultimately show ?thesis
```
```   595   proof (rule all_lt_imp_le)
```
```   596     fix i
```
```   597     assume i: "i < ordertype(K \<times> K, csquare_rel(K))"
```
```   598     hence Oi: "Ord(i)" by (elim ltE)
```
```   599     obtain x y where x: "x \<in> K" and y: "y \<in> K"
```
```   600                  and ieq: "i = ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>"
```
```   601       using i by (auto simp add: ordertype_unfold elim: ltE)
```
```   602     hence xy: "Ord(x)" "Ord(y)" "x < K" "y < K" using OK
```
```   603       by (blast intro: Ord_in_Ord ltI)+
```
```   604     hence ou: "Ord(x \<union> y)"
```
```   605       by (simp add: Ord_Un)
```
```   606     show "i < K"
```
```   607       proof (rule Card_lt_imp_lt [OF _ Oi CK])
```
```   608         have "|i| \<le> |succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))|" using IK xy
```
```   609           by (auto simp add: ieq intro: InfCard_is_Limit [THEN ordermap_csquare_le])
```
```   610         moreover have "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
```
```   611           proof (cases rule: Ord_linear2 [OF ou Ord_nat])
```
```   612             assume "x \<union> y < nat"
```
```   613             hence "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| \<in> nat"
```
```   614               by (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
```
```   615                          nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
```
```   616             also have "... \<subseteq> K" using IK
```
```   617               by (simp add: InfCard_def le_imp_subset)
```
```   618             finally show "|succ(succ(x \<union> y))| \<otimes> |succ(succ(x \<union> y))| < K"
```
```   619               by (simp add: ltI OK)
```
```   620           next
```
```   621             assume natxy: "nat \<le> x \<union> y"
```
```   622             hence seq: "|succ(succ(x \<union> y))| = |x \<union> y|" using xy
```
```   623               by (simp add: le_imp_subset nat_succ_eqpoll [THEN cardinal_cong] le_succ_iff)
```
```   624             also have "... < K" using xy
```
```   625               by (simp add: Un_least_lt Ord_cardinal_le [THEN lt_trans1])
```
```   626             finally have "|succ(succ(x \<union> y))| < K" .
```
```   627             moreover have "InfCard(|succ(succ(x \<union> y))|)" using xy natxy
```
```   628               by (simp add: seq InfCard_def Card_cardinal nat_le_cardinal)
```
```   629             ultimately show ?thesis  by (simp add: eq ltD)
```
```   630           qed
```
```   631         ultimately show "|i| < K" by (blast intro: lt_trans1)
```
```   632     qed
```
```   633   qed
```
```   634 qed
```
```   635
```
```   636 (*Main result: Kunen's Theorem 10.12*)
```
```   637 lemma InfCard_csquare_eq:
```
```   638   assumes IK: "InfCard(K)" shows "InfCard(K) ==> K \<otimes> K = K"
```
```   639 proof -
```
```   640   have  OK: "Ord(K)" using IK by (simp add: Card_is_Ord InfCard_is_Card)
```
```   641   show "InfCard(K) ==> K \<otimes> K = K" using OK
```
```   642   proof (induct rule: trans_induct)
```
```   643     case (step i)
```
```   644     show "i \<otimes> i = i"
```
```   645     proof (rule le_anti_sym)
```
```   646       have "|i \<times> i| = |ordertype(i \<times> i, csquare_rel(i))|"
```
```   647         by (rule cardinal_cong,
```
```   648           simp add: step.hyps well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll])
```
```   649       hence "i \<otimes> i \<le> ordertype(i \<times> i, csquare_rel(i))"
```
```   650         by (simp add: step.hyps cmult_def Ord_cardinal_le well_ord_csquare [THEN Ord_ordertype])
```
```   651       moreover
```
```   652       have "ordertype(i \<times> i, csquare_rel(i)) \<le> i" using step
```
```   653         by (simp add: ordertype_csquare_le)
```
```   654       ultimately show "i \<otimes> i \<le> i" by (rule le_trans)
```
```   655     next
```
```   656       show "i \<le> i \<otimes> i" using step
```
```   657         by (blast intro: cmult_square_le InfCard_is_Card)
```
```   658     qed
```
```   659   qed
```
```   660 qed
```
```   661
```
```   662 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
```
```   663 lemma well_ord_InfCard_square_eq:
```
```   664   assumes r: "well_ord(A,r)" and I: "InfCard(|A|)" shows "A \<times> A \<approx> A"
```
```   665 proof -
```
```   666   have "A \<times> A \<approx> |A| \<times> |A|"
```
```   667     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_sym r)
```
```   668   also have "... \<approx> A"
```
```   669     proof (rule well_ord_cardinal_eqE [OF _ r])
```
```   670       show "well_ord(|A| \<times> |A|, rmult(|A|, Memrel(|A|), |A|, Memrel(|A|)))"
```
```   671         by (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel r)
```
```   672     next
```
```   673       show "||A| \<times> |A|| = |A|" using InfCard_csquare_eq I
```
```   674         by (simp add: cmult_def)
```
```   675     qed
```
```   676   finally show ?thesis .
```
```   677 qed
```
```   678
```
```   679 lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
```
```   680 apply (rule well_ord_InfCard_square_eq)
```
```   681  apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])
```
```   682 apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])
```
```   683 done
```
```   684
```
```   685 lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
```
```   686 by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
```
```   687
```
```   688 subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
```
```   689
```
```   690 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K \<otimes> L = K"
```
```   691 apply (rule le_anti_sym)
```
```   692  prefer 2
```
```   693  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
```
```   694 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
```
```   695 apply (rule cmult_le_mono [THEN le_trans], assumption+)
```
```   696 apply (simp add: InfCard_csquare_eq)
```
```   697 done
```
```   698
```
```   699 (*Corollary 10.13 (1), for cardinal multiplication*)
```
```   700 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<otimes> L = K \<union> L"
```
```   701 apply (rule_tac i = K and j = L in Ord_linear_le)
```
```   702 apply (typecheck add: InfCard_is_Card Card_is_Ord)
```
```   703 apply (rule cmult_commute [THEN ssubst])
```
```   704 apply (rule Un_commute [THEN ssubst])
```
```   705 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq
```
```   706                      subset_Un_iff2 [THEN iffD1] le_imp_subset)
```
```   707 done
```
```   708
```
```   709 lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K"
```
```   710 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
```
```   711 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
```
```   712 done
```
```   713
```
```   714 (*Corollary 10.13 (1), for cardinal addition*)
```
```   715 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K \<oplus> L = K"
```
```   716 apply (rule le_anti_sym)
```
```   717  prefer 2
```
```   718  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
```
```   719 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
```
```   720 apply (rule cadd_le_mono [THEN le_trans], assumption+)
```
```   721 apply (simp add: InfCard_cdouble_eq)
```
```   722 done
```
```   723
```
```   724 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<oplus> L = K \<union> L"
```
```   725 apply (rule_tac i = K and j = L in Ord_linear_le)
```
```   726 apply (typecheck add: InfCard_is_Card Card_is_Ord)
```
```   727 apply (rule cadd_commute [THEN ssubst])
```
```   728 apply (rule Un_commute [THEN ssubst])
```
```   729 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
```
```   730 done
```
```   731
```
```   732 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
```
```   733   of all n-tuples of elements of K.  A better version for the Isabelle theory
```
```   734   might be  InfCard(K) ==> |list(K)| = K.
```
```   735 *)
```
```   736
```
```   737 subsection{*For Every Cardinal Number There Exists A Greater One*}
```
```   738
```
```   739 text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
```
```   740
```
```   741 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
```
```   742 apply (unfold jump_cardinal_def)
```
```   743 apply (rule Ord_is_Transset [THEN  OrdI])
```
```   744  prefer 2 apply (blast intro!: Ord_ordertype)
```
```   745 apply (unfold Transset_def)
```
```   746 apply (safe del: subsetI)
```
```   747 apply (simp add: ordertype_pred_unfold, safe)
```
```   748 apply (rule UN_I)
```
```   749 apply (rule_tac  ReplaceI)
```
```   750    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
```
```   751 done
```
```   752
```
```   753 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
```
```   754 lemma jump_cardinal_iff:
```
```   755      "i \<in> jump_cardinal(K) \<longleftrightarrow>
```
```   756       (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"
```
```   757 apply (unfold jump_cardinal_def)
```
```   758 apply (blast del: subsetI)
```
```   759 done
```
```   760
```
```   761 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
```
```   762 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
```
```   763 apply (rule Ord_jump_cardinal [THEN  ltI])
```
```   764 apply (rule jump_cardinal_iff [THEN iffD2])
```
```   765 apply (rule_tac x="Memrel(K)" in exI)
```
```   766 apply (rule_tac x=K in exI)
```
```   767 apply (simp add: ordertype_Memrel well_ord_Memrel)
```
```   768 apply (simp add: Memrel_def subset_iff)
```
```   769 done
```
```   770
```
```   771 (*The proof by contradiction: the bijection f yields a wellordering of X
```
```   772   whose ordertype is jump_cardinal(K).  *)
```
```   773 lemma Card_jump_cardinal_lemma:
```
```   774      "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;
```
```   775          f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]
```
```   776       ==> jump_cardinal(K) \<in> jump_cardinal(K)"
```
```   777 apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")
```
```   778  prefer 2 apply (blast intro: comp_bij ordermap_bij)
```
```   779 apply (rule jump_cardinal_iff [THEN iffD2])
```
```   780 apply (intro exI conjI)
```
```   781 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
```
```   782 apply (erule bij_is_inj [THEN well_ord_rvimage])
```
```   783 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
```
```   784 apply (simp add: well_ord_Memrel [THEN  bij_ordertype_vimage]
```
```   785                  ordertype_Memrel Ord_jump_cardinal)
```
```   786 done
```
```   787
```
```   788 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
```
```   789 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
```
```   790 apply (rule Ord_jump_cardinal [THEN CardI])
```
```   791 apply (unfold eqpoll_def)
```
```   792 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
```
```   793 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
```
```   794 done
```
```   795
```
```   796 subsection{*Basic Properties of Successor Cardinals*}
```
```   797
```
```   798 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
```
```   799 apply (unfold csucc_def)
```
```   800 apply (rule LeastI)
```
```   801 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
```
```   802 done
```
```   803
```
```   804 lemmas Card_csucc = csucc_basic [THEN conjunct1]
```
```   805
```
```   806 lemmas lt_csucc = csucc_basic [THEN conjunct2]
```
```   807
```
```   808 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
```
```   809 by (blast intro: Ord_0_le lt_csucc lt_trans1)
```
```   810
```
```   811 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"
```
```   812 apply (unfold csucc_def)
```
```   813 apply (rule Least_le)
```
```   814 apply (blast intro: Card_is_Ord)+
```
```   815 done
```
```   816
```
```   817 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K"
```
```   818 apply (rule iffI)
```
```   819 apply (rule_tac  Card_lt_imp_lt)
```
```   820 apply (erule_tac  lt_trans1)
```
```   821 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
```
```   822 apply (rule notI [THEN not_lt_imp_le])
```
```   823 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
```
```   824 apply (rule Ord_cardinal_le [THEN lt_trans1])
```
```   825 apply (simp_all add: Ord_cardinal Card_is_Ord)
```
```   826 done
```
```   827
```
```   828 lemma Card_lt_csucc_iff:
```
```   829      "[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K"
```
```   830 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
```
```   831
```
```   832 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
```
```   833 by (simp add: InfCard_def Card_csucc Card_is_Ord
```
```   834               lt_csucc [THEN leI, THEN  le_trans])
```
```   835
```
```   836
```
```   837 subsubsection{*Removing elements from a finite set decreases its cardinality*}
```
```   838
```
```   839 lemma Finite_imp_cardinal_cons [simp]:
```
```   840   assumes FA: "Finite(A)" and a: "a\<notin>A" shows "|cons(a,A)| = succ(|A|)"
```
```   841 proof -
```
```   842   { fix X
```
```   843     have "Finite(X) ==> a \<notin> X \<Longrightarrow> cons(a,X) \<lesssim> X \<Longrightarrow> False"
```
```   844       proof (induct X rule: Finite_induct)
```
```   845         case 0 thus False  by (simp add: lepoll_0_iff)
```
```   846       next
```
```   847         case (cons x Y)
```
```   848         hence "cons(x, cons(a, Y)) \<lesssim> cons(x, Y)" by (simp add: cons_commute)
```
```   849         hence "cons(a, Y) \<lesssim> Y" using cons        by (blast dest: cons_lepoll_consD)
```
```   850         thus False using cons by auto
```
```   851       qed
```
```   852   }
```
```   853   hence [simp]: "~ cons(a,A) \<lesssim> A" using a FA by auto
```
```   854   have [simp]: "|A| \<approx> A" using Finite_imp_well_ord [OF FA]
```
```   855     by (blast intro: well_ord_cardinal_eqpoll)
```
```   856   have "(\<mu> i. i \<approx> cons(a, A)) = succ(|A|)"
```
```   857     proof (rule Least_equality [OF _ _ notI])
```
```   858       show "succ(|A|) \<approx> cons(a, A)"
```
```   859         by (simp add: succ_def cons_eqpoll_cong mem_not_refl a)
```
```   860     next
```
```   861       show "Ord(succ(|A|))" by simp
```
```   862     next
```
```   863       fix i
```
```   864       assume i: "i \<le> |A|" "i \<approx> cons(a, A)"
```
```   865       have "cons(a, A) \<approx> i" by (rule eqpoll_sym) (rule i)
```
```   866       also have "... \<lesssim> |A|" by (rule le_imp_lepoll) (rule i)
```
```   867       also have "... \<approx> A"   by simp
```
```   868       finally have "cons(a, A) \<lesssim> A" .
```
```   869       thus False by simp
```
```   870     qed
```
```   871   thus ?thesis by (simp add: cardinal_def)
```
```   872 qed
```
```   873
```
```   874 lemma Finite_imp_succ_cardinal_Diff:
```
```   875      "[| Finite(A);  a \<in> A |] ==> succ(|A-{a}|) = |A|"
```
```   876 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
```
```   877 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
```
```   878 apply (simp add: cons_Diff)
```
```   879 done
```
```   880
```
```   881 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a \<in> A |] ==> |A-{a}| < |A|"
```
```   882 apply (rule succ_leE)
```
```   883 apply (simp add: Finite_imp_succ_cardinal_Diff)
```
```   884 done
```
```   885
```
```   886 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"
```
```   887 proof (induct rule: Finite_induct)
```
```   888   case 0 thus ?case by (simp add: cardinal_0)
```
```   889 next
```
```   890   case (cons x A) thus ?case by (simp add: Finite_imp_cardinal_cons)
```
```   891 qed
```
```   892
```
```   893 lemma card_Un_Int:
```
```   894      "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"
```
```   895 apply (erule Finite_induct, simp)
```
```   896 apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
```
```   897 done
```
```   898
```
```   899 lemma card_Un_disjoint:
```
```   900      "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"
```
```   901 by (simp add: Finite_Un card_Un_Int)
```
```   902
```
```   903 lemma card_partition:
```
```   904   assumes FC: "Finite(C)"
```
```   905   shows
```
```   906      "Finite (\<Union> C) \<Longrightarrow>
```
```   907         (\<forall>c\<in>C. |c| = k) \<Longrightarrow>
```
```   908         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<Longrightarrow>
```
```   909         k #* |C| = |\<Union> C|"
```
```   910 using FC
```
```   911 proof (induct rule: Finite_induct)
```
```   912   case 0 thus ?case by simp
```
```   913 next
```
```   914   case (cons x B)
```
```   915   hence "x \<inter> \<Union>B = 0" by auto
```
```   916   thus ?case using cons
```
```   917     by (auto simp add: card_Un_disjoint)
```
```   918 qed
```
```   919
```
```   920
```
```   921 subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
```
```   922
```
```   923 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel]
```
```   924
```
```   925 lemma nat_sum_eqpoll_sum:
```
```   926   assumes m: "m \<in> nat" and n: "n \<in> nat" shows "m + n \<approx> m #+ n"
```
```   927 proof -
```
```   928   have "m + n \<approx> |m+n|" using m n
```
```   929     by (blast intro: nat_implies_well_ord well_ord_radd well_ord_cardinal_eqpoll eqpoll_sym)
```
```   930   also have "... = m #+ n" using m n
```
```   931     by (simp add: nat_cadd_eq_add [symmetric] cadd_def)
```
```   932   finally show ?thesis .
```
```   933 qed
```
```   934
```
```   935 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<Longrightarrow> i \<in> nat | i=nat"
```
```   936 proof (induct i rule: trans_induct3)
```
```   937   case 0 thus ?case by auto
```
```   938 next
```
```   939   case (succ i) thus ?case by auto
```
```   940 next
```
```   941   case (limit l) thus ?case
```
```   942     by (blast dest: nat_le_Limit le_imp_subset)
```
```   943 qed
```
```   944
```
```   945 lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"
```
```   946 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
```
```   947
```
```   948 end
```