src/ZF/Cardinal.thy
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basic integration of graphview into document model;
updated Isabelle/jEdit authors and dependencies etc.;
```     1 (*  Title:      ZF/Cardinal.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1994  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header{*Cardinal Numbers Without the Axiom of Choice*}
```
```     7
```
```     8 theory Cardinal imports OrderType Finite Nat_ZF Sum begin
```
```     9
```
```    10 definition
```
```    11   (*least ordinal operator*)
```
```    12    Least    :: "(i=>o) => i"    (binder "LEAST " 10)  where
```
```    13      "Least(P) == THE i. Ord(i) & P(i) & (\<forall>j. j<i \<longrightarrow> ~P(j))"
```
```    14
```
```    15 definition
```
```    16   eqpoll   :: "[i,i] => o"     (infixl "eqpoll" 50)  where
```
```    17     "A eqpoll B == \<exists>f. f \<in> bij(A,B)"
```
```    18
```
```    19 definition
```
```    20   lepoll   :: "[i,i] => o"     (infixl "lepoll" 50)  where
```
```    21     "A lepoll B == \<exists>f. f \<in> inj(A,B)"
```
```    22
```
```    23 definition
```
```    24   lesspoll :: "[i,i] => o"     (infixl "lesspoll" 50)  where
```
```    25     "A lesspoll B == A lepoll B & ~(A eqpoll B)"
```
```    26
```
```    27 definition
```
```    28   cardinal :: "i=>i"           ("|_|")  where
```
```    29     "|A| == (LEAST i. i eqpoll A)"
```
```    30
```
```    31 definition
```
```    32   Finite   :: "i=>o"  where
```
```    33     "Finite(A) == \<exists>n\<in>nat. A eqpoll n"
```
```    34
```
```    35 definition
```
```    36   Card     :: "i=>o"  where
```
```    37     "Card(i) == (i = |i|)"
```
```    38
```
```    39 notation (xsymbols)
```
```    40   eqpoll    (infixl "\<approx>" 50) and
```
```    41   lepoll    (infixl "\<lesssim>" 50) and
```
```    42   lesspoll  (infixl "\<prec>" 50) and
```
```    43   Least     (binder "\<mu>" 10)
```
```    44
```
```    45 notation (HTML)
```
```    46   eqpoll    (infixl "\<approx>" 50) and
```
```    47   Least     (binder "\<mu>" 10)
```
```    48
```
```    49
```
```    50 subsection{*The Schroeder-Bernstein Theorem*}
```
```    51 text{*See Davey and Priestly, page 106*}
```
```    52
```
```    53 (** Lemma: Banach's Decomposition Theorem **)
```
```    54
```
```    55 lemma decomp_bnd_mono: "bnd_mono(X, %W. X - g``(Y - f``W))"
```
```    56 by (rule bnd_monoI, blast+)
```
```    57
```
```    58 lemma Banach_last_equation:
```
```    59     "g \<in> Y->X
```
```    60      ==> g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) =
```
```    61          X - lfp(X, %W. X - g``(Y - f``W))"
```
```    62 apply (rule_tac P = "%u. ?v = X-u"
```
```    63        in decomp_bnd_mono [THEN lfp_unfold, THEN ssubst])
```
```    64 apply (simp add: double_complement  fun_is_rel [THEN image_subset])
```
```    65 done
```
```    66
```
```    67 lemma decomposition:
```
```    68      "[| f \<in> X->Y;  g \<in> Y->X |] ==>
```
```    69       \<exists>XA XB YA YB. (XA \<inter> XB = 0) & (XA \<union> XB = X) &
```
```    70                       (YA \<inter> YB = 0) & (YA \<union> YB = Y) &
```
```    71                       f``XA=YA & g``YB=XB"
```
```    72 apply (intro exI conjI)
```
```    73 apply (rule_tac  Banach_last_equation)
```
```    74 apply (rule_tac  refl)
```
```    75 apply (assumption |
```
```    76        rule  Diff_disjoint Diff_partition fun_is_rel image_subset lfp_subset)+
```
```    77 done
```
```    78
```
```    79 lemma schroeder_bernstein:
```
```    80     "[| f \<in> inj(X,Y);  g \<in> inj(Y,X) |] ==> \<exists>h. h \<in> bij(X,Y)"
```
```    81 apply (insert decomposition [of f X Y g])
```
```    82 apply (simp add: inj_is_fun)
```
```    83 apply (blast intro!: restrict_bij bij_disjoint_Un intro: bij_converse_bij)
```
```    84 (* The instantiation of exI to @{term"restrict(f,XA) \<union> converse(restrict(g,YB))"}
```
```    85    is forced by the context!! *)
```
```    86 done
```
```    87
```
```    88
```
```    89 (** Equipollence is an equivalence relation **)
```
```    90
```
```    91 lemma bij_imp_eqpoll: "f \<in> bij(A,B) ==> A \<approx> B"
```
```    92 apply (unfold eqpoll_def)
```
```    93 apply (erule exI)
```
```    94 done
```
```    95
```
```    96 (*A eqpoll A*)
```
```    97 lemmas eqpoll_refl = id_bij [THEN bij_imp_eqpoll, simp]
```
```    98
```
```    99 lemma eqpoll_sym: "X \<approx> Y ==> Y \<approx> X"
```
```   100 apply (unfold eqpoll_def)
```
```   101 apply (blast intro: bij_converse_bij)
```
```   102 done
```
```   103
```
```   104 lemma eqpoll_trans [trans]:
```
```   105     "[| X \<approx> Y;  Y \<approx> Z |] ==> X \<approx> Z"
```
```   106 apply (unfold eqpoll_def)
```
```   107 apply (blast intro: comp_bij)
```
```   108 done
```
```   109
```
```   110 (** Le-pollence is a partial ordering **)
```
```   111
```
```   112 lemma subset_imp_lepoll: "X<=Y ==> X \<lesssim> Y"
```
```   113 apply (unfold lepoll_def)
```
```   114 apply (rule exI)
```
```   115 apply (erule id_subset_inj)
```
```   116 done
```
```   117
```
```   118 lemmas lepoll_refl = subset_refl [THEN subset_imp_lepoll, simp]
```
```   119
```
```   120 lemmas le_imp_lepoll = le_imp_subset [THEN subset_imp_lepoll]
```
```   121
```
```   122 lemma eqpoll_imp_lepoll: "X \<approx> Y ==> X \<lesssim> Y"
```
```   123 by (unfold eqpoll_def bij_def lepoll_def, blast)
```
```   124
```
```   125 lemma lepoll_trans [trans]: "[| X \<lesssim> Y;  Y \<lesssim> Z |] ==> X \<lesssim> Z"
```
```   126 apply (unfold lepoll_def)
```
```   127 apply (blast intro: comp_inj)
```
```   128 done
```
```   129
```
```   130 lemma eq_lepoll_trans [trans]: "[| X \<approx> Y;  Y \<lesssim> Z |] ==> X \<lesssim> Z"
```
```   131  by (blast intro: eqpoll_imp_lepoll lepoll_trans)
```
```   132
```
```   133 lemma lepoll_eq_trans [trans]: "[| X \<lesssim> Y;  Y \<approx> Z |] ==> X \<lesssim> Z"
```
```   134  by (blast intro: eqpoll_imp_lepoll lepoll_trans)
```
```   135
```
```   136 (*Asymmetry law*)
```
```   137 lemma eqpollI: "[| X \<lesssim> Y;  Y \<lesssim> X |] ==> X \<approx> Y"
```
```   138 apply (unfold lepoll_def eqpoll_def)
```
```   139 apply (elim exE)
```
```   140 apply (rule schroeder_bernstein, assumption+)
```
```   141 done
```
```   142
```
```   143 lemma eqpollE:
```
```   144     "[| X \<approx> Y; [| X \<lesssim> Y; Y \<lesssim> X |] ==> P |] ==> P"
```
```   145 by (blast intro: eqpoll_imp_lepoll eqpoll_sym)
```
```   146
```
```   147 lemma eqpoll_iff: "X \<approx> Y \<longleftrightarrow> X \<lesssim> Y & Y \<lesssim> X"
```
```   148 by (blast intro: eqpollI elim!: eqpollE)
```
```   149
```
```   150 lemma lepoll_0_is_0: "A \<lesssim> 0 ==> A = 0"
```
```   151 apply (unfold lepoll_def inj_def)
```
```   152 apply (blast dest: apply_type)
```
```   153 done
```
```   154
```
```   155 (*@{term"0 \<lesssim> Y"}*)
```
```   156 lemmas empty_lepollI = empty_subsetI [THEN subset_imp_lepoll]
```
```   157
```
```   158 lemma lepoll_0_iff: "A \<lesssim> 0 \<longleftrightarrow> A=0"
```
```   159 by (blast intro: lepoll_0_is_0 lepoll_refl)
```
```   160
```
```   161 lemma Un_lepoll_Un:
```
```   162     "[| A \<lesssim> B; C \<lesssim> D; B \<inter> D = 0 |] ==> A \<union> C \<lesssim> B \<union> D"
```
```   163 apply (unfold lepoll_def)
```
```   164 apply (blast intro: inj_disjoint_Un)
```
```   165 done
```
```   166
```
```   167 (*A eqpoll 0 ==> A=0*)
```
```   168 lemmas eqpoll_0_is_0 = eqpoll_imp_lepoll [THEN lepoll_0_is_0]
```
```   169
```
```   170 lemma eqpoll_0_iff: "A \<approx> 0 \<longleftrightarrow> A=0"
```
```   171 by (blast intro: eqpoll_0_is_0 eqpoll_refl)
```
```   172
```
```   173 lemma eqpoll_disjoint_Un:
```
```   174     "[| A \<approx> B;  C \<approx> D;  A \<inter> C = 0;  B \<inter> D = 0 |]
```
```   175      ==> A \<union> C \<approx> B \<union> D"
```
```   176 apply (unfold eqpoll_def)
```
```   177 apply (blast intro: bij_disjoint_Un)
```
```   178 done
```
```   179
```
```   180
```
```   181 subsection{*lesspoll: contributions by Krzysztof Grabczewski *}
```
```   182
```
```   183 lemma lesspoll_not_refl: "~ (i \<prec> i)"
```
```   184 by (simp add: lesspoll_def)
```
```   185
```
```   186 lemma lesspoll_irrefl [elim!]: "i \<prec> i ==> P"
```
```   187 by (simp add: lesspoll_def)
```
```   188
```
```   189 lemma lesspoll_imp_lepoll: "A \<prec> B ==> A \<lesssim> B"
```
```   190 by (unfold lesspoll_def, blast)
```
```   191
```
```   192 lemma lepoll_well_ord: "[| A \<lesssim> B; well_ord(B,r) |] ==> \<exists>s. well_ord(A,s)"
```
```   193 apply (unfold lepoll_def)
```
```   194 apply (blast intro: well_ord_rvimage)
```
```   195 done
```
```   196
```
```   197 lemma lepoll_iff_leqpoll: "A \<lesssim> B \<longleftrightarrow> A \<prec> B | A \<approx> B"
```
```   198 apply (unfold lesspoll_def)
```
```   199 apply (blast intro!: eqpollI elim!: eqpollE)
```
```   200 done
```
```   201
```
```   202 lemma inj_not_surj_succ:
```
```   203   assumes fi: "f \<in> inj(A, succ(m))" and fns: "f \<notin> surj(A, succ(m))"
```
```   204   shows "\<exists>f. f \<in> inj(A,m)"
```
```   205 proof -
```
```   206   from fi [THEN inj_is_fun] fns
```
```   207   obtain y where y: "y \<in> succ(m)" "\<And>x. x\<in>A \<Longrightarrow> f ` x \<noteq> y"
```
```   208     by (auto simp add: surj_def)
```
```   209   show ?thesis
```
```   210     proof
```
```   211       show "(\<lambda>z\<in>A. if f`z = m then y else f`z) \<in> inj(A, m)" using y fi
```
```   212         by (simp add: inj_def)
```
```   213            (auto intro!: if_type [THEN lam_type] intro: Pi_type dest: apply_funtype)
```
```   214       qed
```
```   215 qed
```
```   216
```
```   217 (** Variations on transitivity **)
```
```   218
```
```   219 lemma lesspoll_trans [trans]:
```
```   220       "[| X \<prec> Y; Y \<prec> Z |] ==> X \<prec> Z"
```
```   221 apply (unfold lesspoll_def)
```
```   222 apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
```
```   223 done
```
```   224
```
```   225 lemma lesspoll_trans1 [trans]:
```
```   226       "[| X \<lesssim> Y; Y \<prec> Z |] ==> X \<prec> Z"
```
```   227 apply (unfold lesspoll_def)
```
```   228 apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
```
```   229 done
```
```   230
```
```   231 lemma lesspoll_trans2 [trans]:
```
```   232       "[| X \<prec> Y; Y \<lesssim> Z |] ==> X \<prec> Z"
```
```   233 apply (unfold lesspoll_def)
```
```   234 apply (blast elim!: eqpollE intro: eqpollI lepoll_trans)
```
```   235 done
```
```   236
```
```   237 lemma eq_lesspoll_trans [trans]:
```
```   238       "[| X \<approx> Y; Y \<prec> Z |] ==> X \<prec> Z"
```
```   239   by (blast intro: eqpoll_imp_lepoll lesspoll_trans1)
```
```   240
```
```   241 lemma lesspoll_eq_trans [trans]:
```
```   242       "[| X \<prec> Y; Y \<approx> Z |] ==> X \<prec> Z"
```
```   243   by (blast intro: eqpoll_imp_lepoll lesspoll_trans2)
```
```   244
```
```   245
```
```   246 (** LEAST -- the least number operator [from HOL/Univ.ML] **)
```
```   247
```
```   248 lemma Least_equality:
```
```   249     "[| P(i);  Ord(i);  !!x. x<i ==> ~P(x) |] ==> (\<mu> x. P(x)) = i"
```
```   250 apply (unfold Least_def)
```
```   251 apply (rule the_equality, blast)
```
```   252 apply (elim conjE)
```
```   253 apply (erule Ord_linear_lt, assumption, blast+)
```
```   254 done
```
```   255
```
```   256 lemma LeastI:
```
```   257   assumes P: "P(i)" and i: "Ord(i)" shows "P(\<mu> x. P(x))"
```
```   258 proof -
```
```   259   { from i have "P(i) \<Longrightarrow> P(\<mu> x. P(x))"
```
```   260       proof (induct i rule: trans_induct)
```
```   261         case (step i)
```
```   262         show ?case
```
```   263           proof (cases "P(\<mu> a. P(a))")
```
```   264             case True thus ?thesis .
```
```   265           next
```
```   266             case False
```
```   267             hence "\<And>x. x \<in> i \<Longrightarrow> ~P(x)" using step
```
```   268               by blast
```
```   269             hence "(\<mu> a. P(a)) = i" using step
```
```   270               by (blast intro: Least_equality ltD)
```
```   271             thus ?thesis using step.prems
```
```   272               by simp
```
```   273           qed
```
```   274       qed
```
```   275   }
```
```   276   thus ?thesis using P .
```
```   277 qed
```
```   278
```
```   279 text{*The proof is almost identical to the one above!*}
```
```   280 lemma Least_le:
```
```   281   assumes P: "P(i)" and i: "Ord(i)" shows "(\<mu> x. P(x)) \<le> i"
```
```   282 proof -
```
```   283   { from i have "P(i) \<Longrightarrow> (\<mu> x. P(x)) \<le> i"
```
```   284       proof (induct i rule: trans_induct)
```
```   285         case (step i)
```
```   286         show ?case
```
```   287           proof (cases "(\<mu> a. P(a)) \<le> i")
```
```   288             case True thus ?thesis .
```
```   289           next
```
```   290             case False
```
```   291             hence "\<And>x. x \<in> i \<Longrightarrow> ~ (\<mu> a. P(a)) \<le> i" using step
```
```   292               by blast
```
```   293             hence "(\<mu> a. P(a)) = i" using step
```
```   294               by (blast elim: ltE intro: ltI Least_equality lt_trans1)
```
```   295             thus ?thesis using step
```
```   296               by simp
```
```   297           qed
```
```   298       qed
```
```   299   }
```
```   300   thus ?thesis using P .
```
```   301 qed
```
```   302
```
```   303 (*LEAST really is the smallest*)
```
```   304 lemma less_LeastE: "[| P(i);  i < (\<mu> x. P(x)) |] ==> Q"
```
```   305 apply (rule Least_le [THEN  lt_trans2, THEN lt_irrefl], assumption+)
```
```   306 apply (simp add: lt_Ord)
```
```   307 done
```
```   308
```
```   309 (*Easier to apply than LeastI: conclusion has only one occurrence of P*)
```
```   310 lemma LeastI2:
```
```   311     "[| P(i);  Ord(i);  !!j. P(j) ==> Q(j) |] ==> Q(\<mu> j. P(j))"
```
```   312 by (blast intro: LeastI )
```
```   313
```
```   314 (*If there is no such P then LEAST is vacuously 0*)
```
```   315 lemma Least_0:
```
```   316     "[| ~ (\<exists>i. Ord(i) & P(i)) |] ==> (\<mu> x. P(x)) = 0"
```
```   317 apply (unfold Least_def)
```
```   318 apply (rule the_0, blast)
```
```   319 done
```
```   320
```
```   321 lemma Ord_Least [intro,simp,TC]: "Ord(\<mu> x. P(x))"
```
```   322 proof (cases "\<exists>i. Ord(i) & P(i)")
```
```   323   case True
```
```   324   then obtain i where "P(i)" "Ord(i)"  by auto
```
```   325   hence " (\<mu> x. P(x)) \<le> i"  by (rule Least_le)
```
```   326   thus ?thesis
```
```   327     by (elim ltE)
```
```   328 next
```
```   329   case False
```
```   330   hence "(\<mu> x. P(x)) = 0"  by (rule Least_0)
```
```   331   thus ?thesis
```
```   332     by auto
```
```   333 qed
```
```   334
```
```   335
```
```   336 subsection{*Basic Properties of Cardinals*}
```
```   337
```
```   338 (*Not needed for simplification, but helpful below*)
```
```   339 lemma Least_cong: "(!!y. P(y) \<longleftrightarrow> Q(y)) ==> (\<mu> x. P(x)) = (\<mu> x. Q(x))"
```
```   340 by simp
```
```   341
```
```   342 (*Need AC to get @{term"X \<lesssim> Y ==> |X| \<le> |Y|"};  see well_ord_lepoll_imp_Card_le
```
```   343   Converse also requires AC, but see well_ord_cardinal_eqE*)
```
```   344 lemma cardinal_cong: "X \<approx> Y ==> |X| = |Y|"
```
```   345 apply (unfold eqpoll_def cardinal_def)
```
```   346 apply (rule Least_cong)
```
```   347 apply (blast intro: comp_bij bij_converse_bij)
```
```   348 done
```
```   349
```
```   350 (*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*)
```
```   351 lemma well_ord_cardinal_eqpoll:
```
```   352   assumes r: "well_ord(A,r)" shows "|A| \<approx> A"
```
```   353 proof (unfold cardinal_def)
```
```   354   show "(\<mu> i. i \<approx> A) \<approx> A"
```
```   355     by (best intro: LeastI Ord_ordertype ordermap_bij bij_converse_bij bij_imp_eqpoll r)
```
```   356 qed
```
```   357
```
```   358 (* @{term"Ord(A) ==> |A| \<approx> A"} *)
```
```   359 lemmas Ord_cardinal_eqpoll = well_ord_Memrel [THEN well_ord_cardinal_eqpoll]
```
```   360
```
```   361 lemma Ord_cardinal_idem: "Ord(A) \<Longrightarrow> ||A|| = |A|"
```
```   362  by (rule Ord_cardinal_eqpoll [THEN cardinal_cong])
```
```   363
```
```   364 lemma well_ord_cardinal_eqE:
```
```   365   assumes woX: "well_ord(X,r)" and woY: "well_ord(Y,s)" and eq: "|X| = |Y|"
```
```   366 shows "X \<approx> Y"
```
```   367 proof -
```
```   368   have "X \<approx> |X|" by (blast intro: well_ord_cardinal_eqpoll [OF woX] eqpoll_sym)
```
```   369   also have "... = |Y|" by (rule eq)
```
```   370   also have "... \<approx> Y" by (rule well_ord_cardinal_eqpoll [OF woY])
```
```   371   finally show ?thesis .
```
```   372 qed
```
```   373
```
```   374 lemma well_ord_cardinal_eqpoll_iff:
```
```   375      "[| well_ord(X,r);  well_ord(Y,s) |] ==> |X| = |Y| \<longleftrightarrow> X \<approx> Y"
```
```   376 by (blast intro: cardinal_cong well_ord_cardinal_eqE)
```
```   377
```
```   378
```
```   379 (** Observations from Kunen, page 28 **)
```
```   380
```
```   381 lemma Ord_cardinal_le: "Ord(i) ==> |i| \<le> i"
```
```   382 apply (unfold cardinal_def)
```
```   383 apply (erule eqpoll_refl [THEN Least_le])
```
```   384 done
```
```   385
```
```   386 lemma Card_cardinal_eq: "Card(K) ==> |K| = K"
```
```   387 apply (unfold Card_def)
```
```   388 apply (erule sym)
```
```   389 done
```
```   390
```
```   391 (* Could replace the  @{term"~(j \<approx> i)"}  by  @{term"~(i \<preceq> j)"}. *)
```
```   392 lemma CardI: "[| Ord(i);  !!j. j<i ==> ~(j \<approx> i) |] ==> Card(i)"
```
```   393 apply (unfold Card_def cardinal_def)
```
```   394 apply (subst Least_equality)
```
```   395 apply (blast intro: eqpoll_refl)+
```
```   396 done
```
```   397
```
```   398 lemma Card_is_Ord: "Card(i) ==> Ord(i)"
```
```   399 apply (unfold Card_def cardinal_def)
```
```   400 apply (erule ssubst)
```
```   401 apply (rule Ord_Least)
```
```   402 done
```
```   403
```
```   404 lemma Card_cardinal_le: "Card(K) ==> K \<le> |K|"
```
```   405 apply (simp (no_asm_simp) add: Card_is_Ord Card_cardinal_eq)
```
```   406 done
```
```   407
```
```   408 lemma Ord_cardinal [simp,intro!]: "Ord(|A|)"
```
```   409 apply (unfold cardinal_def)
```
```   410 apply (rule Ord_Least)
```
```   411 done
```
```   412
```
```   413 text{*The cardinals are the initial ordinals.*}
```
```   414 lemma Card_iff_initial: "Card(K) \<longleftrightarrow> Ord(K) & (\<forall>j. j<K \<longrightarrow> ~ j \<approx> K)"
```
```   415 proof -
```
```   416   { fix j
```
```   417     assume K: "Card(K)" "j \<approx> K"
```
```   418     assume "j < K"
```
```   419     also have "... = (\<mu> i. i \<approx> K)" using K
```
```   420       by (simp add: Card_def cardinal_def)
```
```   421     finally have "j < (\<mu> i. i \<approx> K)" .
```
```   422     hence "False" using K
```
```   423       by (best dest: less_LeastE)
```
```   424   }
```
```   425   then show ?thesis
```
```   426     by (blast intro: CardI Card_is_Ord)
```
```   427 qed
```
```   428
```
```   429 lemma lt_Card_imp_lesspoll: "[| Card(a); i<a |] ==> i \<prec> a"
```
```   430 apply (unfold lesspoll_def)
```
```   431 apply (drule Card_iff_initial [THEN iffD1])
```
```   432 apply (blast intro!: leI [THEN le_imp_lepoll])
```
```   433 done
```
```   434
```
```   435 lemma Card_0: "Card(0)"
```
```   436 apply (rule Ord_0 [THEN CardI])
```
```   437 apply (blast elim!: ltE)
```
```   438 done
```
```   439
```
```   440 lemma Card_Un: "[| Card(K);  Card(L) |] ==> Card(K \<union> L)"
```
```   441 apply (rule Ord_linear_le [of K L])
```
```   442 apply (simp_all add: subset_Un_iff [THEN iffD1]  Card_is_Ord le_imp_subset
```
```   443                      subset_Un_iff2 [THEN iffD1])
```
```   444 done
```
```   445
```
```   446 (*Infinite unions of cardinals?  See Devlin, Lemma 6.7, page 98*)
```
```   447
```
```   448 lemma Card_cardinal [iff]: "Card(|A|)"
```
```   449 proof (unfold cardinal_def)
```
```   450   show "Card(\<mu> i. i \<approx> A)"
```
```   451     proof (cases "\<exists>i. Ord (i) & i \<approx> A")
```
```   452       case False thus ?thesis           --{*degenerate case*}
```
```   453         by (simp add: Least_0 Card_0)
```
```   454     next
```
```   455       case True                         --{*real case: @{term A} is isomorphic to some ordinal*}
```
```   456       then obtain i where i: "Ord(i)" "i \<approx> A" by blast
```
```   457       show ?thesis
```
```   458         proof (rule CardI [OF Ord_Least], rule notI)
```
```   459           fix j
```
```   460           assume j: "j < (\<mu> i. i \<approx> A)"
```
```   461           assume "j \<approx> (\<mu> i. i \<approx> A)"
```
```   462           also have "... \<approx> A" using i by (auto intro: LeastI)
```
```   463           finally have "j \<approx> A" .
```
```   464           thus False
```
```   465             by (rule less_LeastE [OF _ j])
```
```   466         qed
```
```   467     qed
```
```   468 qed
```
```   469
```
```   470 (*Kunen's Lemma 10.5*)
```
```   471 lemma cardinal_eq_lemma:
```
```   472   assumes i:"|i| \<le> j" and j: "j \<le> i" shows "|j| = |i|"
```
```   473 proof (rule eqpollI [THEN cardinal_cong])
```
```   474   show "j \<lesssim> i" by (rule le_imp_lepoll [OF j])
```
```   475 next
```
```   476   have Oi: "Ord(i)" using j by (rule le_Ord2)
```
```   477   hence "i \<approx> |i|"
```
```   478     by (blast intro: Ord_cardinal_eqpoll eqpoll_sym)
```
```   479   also have "... \<lesssim> j"
```
```   480     by (blast intro: le_imp_lepoll i)
```
```   481   finally show "i \<lesssim> j" .
```
```   482 qed
```
```   483
```
```   484 lemma cardinal_mono:
```
```   485   assumes ij: "i \<le> j" shows "|i| \<le> |j|"
```
```   486 using Ord_cardinal [of i] Ord_cardinal [of j]
```
```   487 proof (cases rule: Ord_linear_le)
```
```   488   case le thus ?thesis .
```
```   489 next
```
```   490   case ge
```
```   491   have i: "Ord(i)" using ij
```
```   492     by (simp add: lt_Ord)
```
```   493   have ci: "|i| \<le> j"
```
```   494     by (blast intro: Ord_cardinal_le ij le_trans i)
```
```   495   have "|i| = ||i||"
```
```   496     by (auto simp add: Ord_cardinal_idem i)
```
```   497   also have "... = |j|"
```
```   498     by (rule cardinal_eq_lemma [OF ge ci])
```
```   499   finally have "|i| = |j|" .
```
```   500   thus ?thesis by simp
```
```   501 qed
```
```   502
```
```   503 text{*Since we have @{term"|succ(nat)| \<le> |nat|"}, the converse of @{text cardinal_mono} fails!*}
```
```   504 lemma cardinal_lt_imp_lt: "[| |i| < |j|;  Ord(i);  Ord(j) |] ==> i < j"
```
```   505 apply (rule Ord_linear2 [of i j], assumption+)
```
```   506 apply (erule lt_trans2 [THEN lt_irrefl])
```
```   507 apply (erule cardinal_mono)
```
```   508 done
```
```   509
```
```   510 lemma Card_lt_imp_lt: "[| |i| < K;  Ord(i);  Card(K) |] ==> i < K"
```
```   511   by (simp (no_asm_simp) add: cardinal_lt_imp_lt Card_is_Ord Card_cardinal_eq)
```
```   512
```
```   513 lemma Card_lt_iff: "[| Ord(i);  Card(K) |] ==> (|i| < K) \<longleftrightarrow> (i < K)"
```
```   514 by (blast intro: Card_lt_imp_lt Ord_cardinal_le [THEN lt_trans1])
```
```   515
```
```   516 lemma Card_le_iff: "[| Ord(i);  Card(K) |] ==> (K \<le> |i|) \<longleftrightarrow> (K \<le> i)"
```
```   517 by (simp add: Card_lt_iff Card_is_Ord Ord_cardinal not_lt_iff_le [THEN iff_sym])
```
```   518
```
```   519 (*Can use AC or finiteness to discharge first premise*)
```
```   520 lemma well_ord_lepoll_imp_Card_le:
```
```   521   assumes wB: "well_ord(B,r)" and AB: "A \<lesssim> B"
```
```   522   shows "|A| \<le> |B|"
```
```   523 using Ord_cardinal [of A] Ord_cardinal [of B]
```
```   524 proof (cases rule: Ord_linear_le)
```
```   525   case le thus ?thesis .
```
```   526 next
```
```   527   case ge
```
```   528   from lepoll_well_ord [OF AB wB]
```
```   529   obtain s where s: "well_ord(A, s)" by blast
```
```   530   have "B  \<approx> |B|" by (blast intro: wB eqpoll_sym well_ord_cardinal_eqpoll)
```
```   531   also have "... \<lesssim> |A|" by (rule le_imp_lepoll [OF ge])
```
```   532   also have "... \<approx> A" by (rule well_ord_cardinal_eqpoll [OF s])
```
```   533   finally have "B \<lesssim> A" .
```
```   534   hence "A \<approx> B" by (blast intro: eqpollI AB)
```
```   535   hence "|A| = |B|" by (rule cardinal_cong)
```
```   536   thus ?thesis by simp
```
```   537 qed
```
```   538
```
```   539 lemma lepoll_cardinal_le: "[| A \<lesssim> i; Ord(i) |] ==> |A| \<le> i"
```
```   540 apply (rule le_trans)
```
```   541 apply (erule well_ord_Memrel [THEN well_ord_lepoll_imp_Card_le], assumption)
```
```   542 apply (erule Ord_cardinal_le)
```
```   543 done
```
```   544
```
```   545 lemma lepoll_Ord_imp_eqpoll: "[| A \<lesssim> i; Ord(i) |] ==> |A| \<approx> A"
```
```   546 by (blast intro: lepoll_cardinal_le well_ord_Memrel well_ord_cardinal_eqpoll dest!: lepoll_well_ord)
```
```   547
```
```   548 lemma lesspoll_imp_eqpoll: "[| A \<prec> i; Ord(i) |] ==> |A| \<approx> A"
```
```   549 apply (unfold lesspoll_def)
```
```   550 apply (blast intro: lepoll_Ord_imp_eqpoll)
```
```   551 done
```
```   552
```
```   553 lemma cardinal_subset_Ord: "[|A<=i; Ord(i)|] ==> |A| \<subseteq> i"
```
```   554 apply (drule subset_imp_lepoll [THEN lepoll_cardinal_le])
```
```   555 apply (auto simp add: lt_def)
```
```   556 apply (blast intro: Ord_trans)
```
```   557 done
```
```   558
```
```   559 subsection{*The finite cardinals *}
```
```   560
```
```   561 lemma cons_lepoll_consD:
```
```   562  "[| cons(u,A) \<lesssim> cons(v,B);  u\<notin>A;  v\<notin>B |] ==> A \<lesssim> B"
```
```   563 apply (unfold lepoll_def inj_def, safe)
```
```   564 apply (rule_tac x = "\<lambda>x\<in>A. if f`x=v then f`u else f`x" in exI)
```
```   565 apply (rule CollectI)
```
```   566 (*Proving it's in the function space A->B*)
```
```   567 apply (rule if_type [THEN lam_type])
```
```   568 apply (blast dest: apply_funtype)
```
```   569 apply (blast elim!: mem_irrefl dest: apply_funtype)
```
```   570 (*Proving it's injective*)
```
```   571 apply (simp (no_asm_simp))
```
```   572 apply blast
```
```   573 done
```
```   574
```
```   575 lemma cons_eqpoll_consD: "[| cons(u,A) \<approx> cons(v,B);  u\<notin>A;  v\<notin>B |] ==> A \<approx> B"
```
```   576 apply (simp add: eqpoll_iff)
```
```   577 apply (blast intro: cons_lepoll_consD)
```
```   578 done
```
```   579
```
```   580 (*Lemma suggested by Mike Fourman*)
```
```   581 lemma succ_lepoll_succD: "succ(m) \<lesssim> succ(n) ==> m \<lesssim> n"
```
```   582 apply (unfold succ_def)
```
```   583 apply (erule cons_lepoll_consD)
```
```   584 apply (rule mem_not_refl)+
```
```   585 done
```
```   586
```
```   587
```
```   588 lemma nat_lepoll_imp_le:
```
```   589      "m \<in> nat ==> n \<in> nat \<Longrightarrow> m \<lesssim> n \<Longrightarrow> m \<le> n"
```
```   590 proof (induct m arbitrary: n rule: nat_induct)
```
```   591   case 0 thus ?case by (blast intro!: nat_0_le)
```
```   592 next
```
```   593   case (succ m)
```
```   594   show ?case  using `n \<in> nat`
```
```   595     proof (cases rule: natE)
```
```   596       case 0 thus ?thesis using succ
```
```   597         by (simp add: lepoll_def inj_def)
```
```   598     next
```
```   599       case (succ n') thus ?thesis using succ.hyps ` succ(m) \<lesssim> n`
```
```   600         by (blast intro!: succ_leI dest!: succ_lepoll_succD)
```
```   601     qed
```
```   602 qed
```
```   603
```
```   604 lemma nat_eqpoll_iff: "[| m \<in> nat; n \<in> nat |] ==> m \<approx> n \<longleftrightarrow> m = n"
```
```   605 apply (rule iffI)
```
```   606 apply (blast intro: nat_lepoll_imp_le le_anti_sym elim!: eqpollE)
```
```   607 apply (simp add: eqpoll_refl)
```
```   608 done
```
```   609
```
```   610 (*The object of all this work: every natural number is a (finite) cardinal*)
```
```   611 lemma nat_into_Card:
```
```   612   assumes n: "n \<in> nat" shows "Card(n)"
```
```   613 proof (unfold Card_def cardinal_def, rule sym)
```
```   614   have "Ord(n)" using n  by auto
```
```   615   moreover
```
```   616   { fix i
```
```   617     assume "i < n" "i \<approx> n"
```
```   618     hence False using n
```
```   619       by (auto simp add: lt_nat_in_nat [THEN nat_eqpoll_iff])
```
```   620   }
```
```   621   ultimately show "(\<mu> i. i \<approx> n) = n" by (auto intro!: Least_equality)
```
```   622 qed
```
```   623
```
```   624 lemmas cardinal_0 = nat_0I [THEN nat_into_Card, THEN Card_cardinal_eq, iff]
```
```   625 lemmas cardinal_1 = nat_1I [THEN nat_into_Card, THEN Card_cardinal_eq, iff]
```
```   626
```
```   627
```
```   628 (*Part of Kunen's Lemma 10.6*)
```
```   629 lemma succ_lepoll_natE: "[| succ(n) \<lesssim> n;  n \<in> nat |] ==> P"
```
```   630 by (rule nat_lepoll_imp_le [THEN lt_irrefl], auto)
```
```   631
```
```   632 lemma nat_lepoll_imp_ex_eqpoll_n:
```
```   633      "[| n \<in> nat;  nat \<lesssim> X |] ==> \<exists>Y. Y \<subseteq> X & n \<approx> Y"
```
```   634 apply (unfold lepoll_def eqpoll_def)
```
```   635 apply (fast del: subsetI subsetCE
```
```   636             intro!: subset_SIs
```
```   637             dest!: Ord_nat [THEN  OrdmemD, THEN  restrict_inj]
```
```   638             elim!: restrict_bij
```
```   639                    inj_is_fun [THEN fun_is_rel, THEN image_subset])
```
```   640 done
```
```   641
```
```   642
```
```   643 (** lepoll, \<prec> and natural numbers **)
```
```   644
```
```   645 lemma lepoll_succ: "i \<lesssim> succ(i)"
```
```   646   by (blast intro: subset_imp_lepoll)
```
```   647
```
```   648 lemma lepoll_imp_lesspoll_succ:
```
```   649   assumes A: "A \<lesssim> m" and m: "m \<in> nat"
```
```   650   shows "A \<prec> succ(m)"
```
```   651 proof -
```
```   652   { assume "A \<approx> succ(m)"
```
```   653     hence "succ(m) \<approx> A" by (rule eqpoll_sym)
```
```   654     also have "... \<lesssim> m" by (rule A)
```
```   655     finally have "succ(m) \<lesssim> m" .
```
```   656     hence False by (rule succ_lepoll_natE) (rule m) }
```
```   657   moreover have "A \<lesssim> succ(m)" by (blast intro: lepoll_trans A lepoll_succ)
```
```   658   ultimately show ?thesis by (auto simp add: lesspoll_def)
```
```   659 qed
```
```   660
```
```   661 lemma lesspoll_succ_imp_lepoll:
```
```   662      "[| A \<prec> succ(m); m \<in> nat |] ==> A \<lesssim> m"
```
```   663 apply (unfold lesspoll_def lepoll_def eqpoll_def bij_def)
```
```   664 apply (auto dest: inj_not_surj_succ)
```
```   665 done
```
```   666
```
```   667 lemma lesspoll_succ_iff: "m \<in> nat ==> A \<prec> succ(m) \<longleftrightarrow> A \<lesssim> m"
```
```   668 by (blast intro!: lepoll_imp_lesspoll_succ lesspoll_succ_imp_lepoll)
```
```   669
```
```   670 lemma lepoll_succ_disj: "[| A \<lesssim> succ(m);  m \<in> nat |] ==> A \<lesssim> m | A \<approx> succ(m)"
```
```   671 apply (rule disjCI)
```
```   672 apply (rule lesspoll_succ_imp_lepoll)
```
```   673 prefer 2 apply assumption
```
```   674 apply (simp (no_asm_simp) add: lesspoll_def)
```
```   675 done
```
```   676
```
```   677 lemma lesspoll_cardinal_lt: "[| A \<prec> i; Ord(i) |] ==> |A| < i"
```
```   678 apply (unfold lesspoll_def, clarify)
```
```   679 apply (frule lepoll_cardinal_le, assumption)
```
```   680 apply (blast intro: well_ord_Memrel well_ord_cardinal_eqpoll [THEN eqpoll_sym]
```
```   681              dest: lepoll_well_ord  elim!: leE)
```
```   682 done
```
```   683
```
```   684
```
```   685 subsection{*The first infinite cardinal: Omega, or nat *}
```
```   686
```
```   687 (*This implies Kunen's Lemma 10.6*)
```
```   688 lemma lt_not_lepoll:
```
```   689   assumes n: "n<i" "n \<in> nat" shows "~ i \<lesssim> n"
```
```   690 proof -
```
```   691   { assume i: "i \<lesssim> n"
```
```   692     have "succ(n) \<lesssim> i" using n
```
```   693       by (elim ltE, blast intro: Ord_succ_subsetI [THEN subset_imp_lepoll])
```
```   694     also have "... \<lesssim> n" by (rule i)
```
```   695     finally have "succ(n) \<lesssim> n" .
```
```   696     hence False  by (rule succ_lepoll_natE) (rule n) }
```
```   697   thus ?thesis by auto
```
```   698 qed
```
```   699
```
```   700 text{*A slightly weaker version of @{text nat_eqpoll_iff}*}
```
```   701 lemma Ord_nat_eqpoll_iff:
```
```   702   assumes i: "Ord(i)" and n: "n \<in> nat" shows "i \<approx> n \<longleftrightarrow> i=n"
```
```   703 using i nat_into_Ord [OF n]
```
```   704 proof (cases rule: Ord_linear_lt)
```
```   705   case lt
```
```   706   hence  "i \<in> nat" by (rule lt_nat_in_nat) (rule n)
```
```   707   thus ?thesis by (simp add: nat_eqpoll_iff n)
```
```   708 next
```
```   709   case eq
```
```   710   thus ?thesis by (simp add: eqpoll_refl)
```
```   711 next
```
```   712   case gt
```
```   713   hence  "~ i \<lesssim> n" using n  by (rule lt_not_lepoll)
```
```   714   hence  "~ i \<approx> n" using n  by (blast intro: eqpoll_imp_lepoll)
```
```   715   moreover have "i \<noteq> n" using `n<i` by auto
```
```   716   ultimately show ?thesis by blast
```
```   717 qed
```
```   718
```
```   719 lemma Card_nat: "Card(nat)"
```
```   720 proof -
```
```   721   { fix i
```
```   722     assume i: "i < nat" "i \<approx> nat"
```
```   723     hence "~ nat \<lesssim> i"
```
```   724       by (simp add: lt_def lt_not_lepoll)
```
```   725     hence False using i
```
```   726       by (simp add: eqpoll_iff)
```
```   727   }
```
```   728   hence "(\<mu> i. i \<approx> nat) = nat" by (blast intro: Least_equality eqpoll_refl)
```
```   729   thus ?thesis
```
```   730     by (auto simp add: Card_def cardinal_def)
```
```   731 qed
```
```   732
```
```   733 (*Allows showing that |i| is a limit cardinal*)
```
```   734 lemma nat_le_cardinal: "nat \<le> i ==> nat \<le> |i|"
```
```   735 apply (rule Card_nat [THEN Card_cardinal_eq, THEN subst])
```
```   736 apply (erule cardinal_mono)
```
```   737 done
```
```   738
```
```   739 lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
```
```   740   by (blast intro: Ord_nat Card_nat ltI lt_Card_imp_lesspoll)
```
```   741
```
```   742
```
```   743 subsection{*Towards Cardinal Arithmetic *}
```
```   744 (** Congruence laws for successor, cardinal addition and multiplication **)
```
```   745
```
```   746 (*Congruence law for  cons  under equipollence*)
```
```   747 lemma cons_lepoll_cong:
```
```   748     "[| A \<lesssim> B;  b \<notin> B |] ==> cons(a,A) \<lesssim> cons(b,B)"
```
```   749 apply (unfold lepoll_def, safe)
```
```   750 apply (rule_tac x = "\<lambda>y\<in>cons (a,A) . if y=a then b else f`y" in exI)
```
```   751 apply (rule_tac d = "%z. if z \<in> B then converse (f) `z else a" in lam_injective)
```
```   752 apply (safe elim!: consE')
```
```   753    apply simp_all
```
```   754 apply (blast intro: inj_is_fun [THEN apply_type])+
```
```   755 done
```
```   756
```
```   757 lemma cons_eqpoll_cong:
```
```   758      "[| A \<approx> B;  a \<notin> A;  b \<notin> B |] ==> cons(a,A) \<approx> cons(b,B)"
```
```   759 by (simp add: eqpoll_iff cons_lepoll_cong)
```
```   760
```
```   761 lemma cons_lepoll_cons_iff:
```
```   762      "[| a \<notin> A;  b \<notin> B |] ==> cons(a,A) \<lesssim> cons(b,B)  \<longleftrightarrow>  A \<lesssim> B"
```
```   763 by (blast intro: cons_lepoll_cong cons_lepoll_consD)
```
```   764
```
```   765 lemma cons_eqpoll_cons_iff:
```
```   766      "[| a \<notin> A;  b \<notin> B |] ==> cons(a,A) \<approx> cons(b,B)  \<longleftrightarrow>  A \<approx> B"
```
```   767 by (blast intro: cons_eqpoll_cong cons_eqpoll_consD)
```
```   768
```
```   769 lemma singleton_eqpoll_1: "{a} \<approx> 1"
```
```   770 apply (unfold succ_def)
```
```   771 apply (blast intro!: eqpoll_refl [THEN cons_eqpoll_cong])
```
```   772 done
```
```   773
```
```   774 lemma cardinal_singleton: "|{a}| = 1"
```
```   775 apply (rule singleton_eqpoll_1 [THEN cardinal_cong, THEN trans])
```
```   776 apply (simp (no_asm) add: nat_into_Card [THEN Card_cardinal_eq])
```
```   777 done
```
```   778
```
```   779 lemma not_0_is_lepoll_1: "A \<noteq> 0 ==> 1 \<lesssim> A"
```
```   780 apply (erule not_emptyE)
```
```   781 apply (rule_tac a = "cons (x, A-{x}) " in subst)
```
```   782 apply (rule_tac  a = "cons(0,0)" and P= "%y. y \<lesssim> cons (x, A-{x})" in subst)
```
```   783 prefer 3 apply (blast intro: cons_lepoll_cong subset_imp_lepoll, auto)
```
```   784 done
```
```   785
```
```   786 (*Congruence law for  succ  under equipollence*)
```
```   787 lemma succ_eqpoll_cong: "A \<approx> B ==> succ(A) \<approx> succ(B)"
```
```   788 apply (unfold succ_def)
```
```   789 apply (simp add: cons_eqpoll_cong mem_not_refl)
```
```   790 done
```
```   791
```
```   792 (*Congruence law for + under equipollence*)
```
```   793 lemma sum_eqpoll_cong: "[| A \<approx> C;  B \<approx> D |] ==> A+B \<approx> C+D"
```
```   794 apply (unfold eqpoll_def)
```
```   795 apply (blast intro!: sum_bij)
```
```   796 done
```
```   797
```
```   798 (*Congruence law for * under equipollence*)
```
```   799 lemma prod_eqpoll_cong:
```
```   800     "[| A \<approx> C;  B \<approx> D |] ==> A*B \<approx> C*D"
```
```   801 apply (unfold eqpoll_def)
```
```   802 apply (blast intro!: prod_bij)
```
```   803 done
```
```   804
```
```   805 lemma inj_disjoint_eqpoll:
```
```   806     "[| f \<in> inj(A,B);  A \<inter> B = 0 |] ==> A \<union> (B - range(f)) \<approx> B"
```
```   807 apply (unfold eqpoll_def)
```
```   808 apply (rule exI)
```
```   809 apply (rule_tac c = "%x. if x \<in> A then f`x else x"
```
```   810             and d = "%y. if y \<in> range (f) then converse (f) `y else y"
```
```   811        in lam_bijective)
```
```   812 apply (blast intro!: if_type inj_is_fun [THEN apply_type])
```
```   813 apply (simp (no_asm_simp) add: inj_converse_fun [THEN apply_funtype])
```
```   814 apply (safe elim!: UnE')
```
```   815    apply (simp_all add: inj_is_fun [THEN apply_rangeI])
```
```   816 apply (blast intro: inj_converse_fun [THEN apply_type])+
```
```   817 done
```
```   818
```
```   819
```
```   820 subsection{*Lemmas by Krzysztof Grabczewski*}
```
```   821
```
```   822 (*New proofs using cons_lepoll_cons. Could generalise from succ to cons.*)
```
```   823
```
```   824 text{*If @{term A} has at most @{term"n+1"} elements and @{term"a \<in> A"}
```
```   825       then @{term"A-{a}"} has at most @{term n}.*}
```
```   826 lemma Diff_sing_lepoll:
```
```   827       "[| a \<in> A;  A \<lesssim> succ(n) |] ==> A - {a} \<lesssim> n"
```
```   828 apply (unfold succ_def)
```
```   829 apply (rule cons_lepoll_consD)
```
```   830 apply (rule_tac  mem_not_refl)
```
```   831 apply (erule cons_Diff [THEN ssubst], safe)
```
```   832 done
```
```   833
```
```   834 text{*If @{term A} has at least @{term"n+1"} elements then @{term"A-{a}"} has at least @{term n}.*}
```
```   835 lemma lepoll_Diff_sing:
```
```   836   assumes A: "succ(n) \<lesssim> A" shows "n \<lesssim> A - {a}"
```
```   837 proof -
```
```   838   have "cons(n,n) \<lesssim> A" using A
```
```   839     by (unfold succ_def)
```
```   840   also have "... \<lesssim> cons(a, A-{a})"
```
```   841     by (blast intro: subset_imp_lepoll)
```
```   842   finally have "cons(n,n) \<lesssim> cons(a, A-{a})" .
```
```   843   thus ?thesis
```
```   844     by (blast intro: cons_lepoll_consD mem_irrefl)
```
```   845 qed
```
```   846
```
```   847 lemma Diff_sing_eqpoll: "[| a \<in> A; A \<approx> succ(n) |] ==> A - {a} \<approx> n"
```
```   848 by (blast intro!: eqpollI
```
```   849           elim!: eqpollE
```
```   850           intro: Diff_sing_lepoll lepoll_Diff_sing)
```
```   851
```
```   852 lemma lepoll_1_is_sing: "[| A \<lesssim> 1; a \<in> A |] ==> A = {a}"
```
```   853 apply (frule Diff_sing_lepoll, assumption)
```
```   854 apply (drule lepoll_0_is_0)
```
```   855 apply (blast elim: equalityE)
```
```   856 done
```
```   857
```
```   858 lemma Un_lepoll_sum: "A \<union> B \<lesssim> A+B"
```
```   859 apply (unfold lepoll_def)
```
```   860 apply (rule_tac x = "\<lambda>x\<in>A \<union> B. if x\<in>A then Inl (x) else Inr (x)" in exI)
```
```   861 apply (rule_tac d = "%z. snd (z)" in lam_injective)
```
```   862 apply force
```
```   863 apply (simp add: Inl_def Inr_def)
```
```   864 done
```
```   865
```
```   866 lemma well_ord_Un:
```
```   867      "[| well_ord(X,R); well_ord(Y,S) |] ==> \<exists>T. well_ord(X \<union> Y, T)"
```
```   868 by (erule well_ord_radd [THEN Un_lepoll_sum [THEN lepoll_well_ord]],
```
```   869     assumption)
```
```   870
```
```   871 (*Krzysztof Grabczewski*)
```
```   872 lemma disj_Un_eqpoll_sum: "A \<inter> B = 0 ==> A \<union> B \<approx> A + B"
```
```   873 apply (unfold eqpoll_def)
```
```   874 apply (rule_tac x = "\<lambda>a\<in>A \<union> B. if a \<in> A then Inl (a) else Inr (a)" in exI)
```
```   875 apply (rule_tac d = "%z. case (%x. x, %x. x, z)" in lam_bijective)
```
```   876 apply auto
```
```   877 done
```
```   878
```
```   879
```
```   880 subsection {*Finite and infinite sets*}
```
```   881
```
```   882 lemma eqpoll_imp_Finite_iff: "A \<approx> B ==> Finite(A) \<longleftrightarrow> Finite(B)"
```
```   883 apply (unfold Finite_def)
```
```   884 apply (blast intro: eqpoll_trans eqpoll_sym)
```
```   885 done
```
```   886
```
```   887 lemma Finite_0 [simp]: "Finite(0)"
```
```   888 apply (unfold Finite_def)
```
```   889 apply (blast intro!: eqpoll_refl nat_0I)
```
```   890 done
```
```   891
```
```   892 lemma Finite_cons: "Finite(x) ==> Finite(cons(y,x))"
```
```   893 apply (unfold Finite_def)
```
```   894 apply (case_tac "y \<in> x")
```
```   895 apply (simp add: cons_absorb)
```
```   896 apply (erule bexE)
```
```   897 apply (rule bexI)
```
```   898 apply (erule_tac  nat_succI)
```
```   899 apply (simp (no_asm_simp) add: succ_def cons_eqpoll_cong mem_not_refl)
```
```   900 done
```
```   901
```
```   902 lemma Finite_succ: "Finite(x) ==> Finite(succ(x))"
```
```   903 apply (unfold succ_def)
```
```   904 apply (erule Finite_cons)
```
```   905 done
```
```   906
```
```   907 lemma lepoll_nat_imp_Finite:
```
```   908   assumes A: "A \<lesssim> n" and n: "n \<in> nat" shows "Finite(A)"
```
```   909 proof -
```
```   910   have "A \<lesssim> n \<Longrightarrow> Finite(A)" using n
```
```   911     proof (induct n)
```
```   912       case 0
```
```   913       hence "A = 0" by (rule lepoll_0_is_0)
```
```   914       thus ?case by simp
```
```   915     next
```
```   916       case (succ n)
```
```   917       hence "A \<lesssim> n \<or> A \<approx> succ(n)" by (blast dest: lepoll_succ_disj)
```
```   918       thus ?case using succ by (auto simp add: Finite_def)
```
```   919     qed
```
```   920   thus ?thesis using A .
```
```   921 qed
```
```   922
```
```   923 lemma lesspoll_nat_is_Finite:
```
```   924      "A \<prec> nat ==> Finite(A)"
```
```   925 apply (unfold Finite_def)
```
```   926 apply (blast dest: ltD lesspoll_cardinal_lt
```
```   927                    lesspoll_imp_eqpoll [THEN eqpoll_sym])
```
```   928 done
```
```   929
```
```   930 lemma lepoll_Finite:
```
```   931   assumes Y: "Y \<lesssim> X" and X: "Finite(X)" shows "Finite(Y)"
```
```   932 proof -
```
```   933   obtain n where n: "n \<in> nat" "X \<approx> n" using X
```
```   934     by (auto simp add: Finite_def)
```
```   935   have "Y \<lesssim> X"         by (rule Y)
```
```   936   also have "... \<approx> n"  by (rule n)
```
```   937   finally have "Y \<lesssim> n" .
```
```   938   thus ?thesis using n by (simp add: lepoll_nat_imp_Finite)
```
```   939 qed
```
```   940
```
```   941 lemmas subset_Finite = subset_imp_lepoll [THEN lepoll_Finite]
```
```   942
```
```   943 lemma Finite_cons_iff [iff]: "Finite(cons(y,x)) \<longleftrightarrow> Finite(x)"
```
```   944 by (blast intro: Finite_cons subset_Finite)
```
```   945
```
```   946 lemma Finite_succ_iff [iff]: "Finite(succ(x)) \<longleftrightarrow> Finite(x)"
```
```   947 by (simp add: succ_def)
```
```   948
```
```   949 lemma Finite_Int: "Finite(A) | Finite(B) ==> Finite(A \<inter> B)"
```
```   950 by (blast intro: subset_Finite)
```
```   951
```
```   952 lemmas Finite_Diff = Diff_subset [THEN subset_Finite]
```
```   953
```
```   954 lemma nat_le_infinite_Ord:
```
```   955       "[| Ord(i);  ~ Finite(i) |] ==> nat \<le> i"
```
```   956 apply (unfold Finite_def)
```
```   957 apply (erule Ord_nat [THEN  Ord_linear2])
```
```   958 prefer 2 apply assumption
```
```   959 apply (blast intro!: eqpoll_refl elim!: ltE)
```
```   960 done
```
```   961
```
```   962 lemma Finite_imp_well_ord:
```
```   963     "Finite(A) ==> \<exists>r. well_ord(A,r)"
```
```   964 apply (unfold Finite_def eqpoll_def)
```
```   965 apply (blast intro: well_ord_rvimage bij_is_inj well_ord_Memrel nat_into_Ord)
```
```   966 done
```
```   967
```
```   968 lemma succ_lepoll_imp_not_empty: "succ(x) \<lesssim> y ==> y \<noteq> 0"
```
```   969 by (fast dest!: lepoll_0_is_0)
```
```   970
```
```   971 lemma eqpoll_succ_imp_not_empty: "x \<approx> succ(n) ==> x \<noteq> 0"
```
```   972 by (fast elim!: eqpoll_sym [THEN eqpoll_0_is_0, THEN succ_neq_0])
```
```   973
```
```   974 lemma Finite_Fin_lemma [rule_format]:
```
```   975      "n \<in> nat ==> \<forall>A. (A\<approx>n & A \<subseteq> X) \<longrightarrow> A \<in> Fin(X)"
```
```   976 apply (induct_tac n)
```
```   977 apply (rule allI)
```
```   978 apply (fast intro!: Fin.emptyI dest!: eqpoll_imp_lepoll [THEN lepoll_0_is_0])
```
```   979 apply (rule allI)
```
```   980 apply (rule impI)
```
```   981 apply (erule conjE)
```
```   982 apply (rule eqpoll_succ_imp_not_empty [THEN not_emptyE], assumption)
```
```   983 apply (frule Diff_sing_eqpoll, assumption)
```
```   984 apply (erule allE)
```
```   985 apply (erule impE, fast)
```
```   986 apply (drule subsetD, assumption)
```
```   987 apply (drule Fin.consI, assumption)
```
```   988 apply (simp add: cons_Diff)
```
```   989 done
```
```   990
```
```   991 lemma Finite_Fin: "[| Finite(A); A \<subseteq> X |] ==> A \<in> Fin(X)"
```
```   992 by (unfold Finite_def, blast intro: Finite_Fin_lemma)
```
```   993
```
```   994 lemma Fin_lemma [rule_format]: "n \<in> nat ==> \<forall>A. A \<approx> n \<longrightarrow> A \<in> Fin(A)"
```
```   995 apply (induct_tac n)
```
```   996 apply (simp add: eqpoll_0_iff, clarify)
```
```   997 apply (subgoal_tac "\<exists>u. u \<in> A")
```
```   998 apply (erule exE)
```
```   999 apply (rule Diff_sing_eqpoll [elim_format])
```
```  1000 prefer 2 apply assumption
```
```  1001 apply assumption
```
```  1002 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
```
```  1003 apply (rule Fin.consI, blast)
```
```  1004 apply (blast intro: subset_consI [THEN Fin_mono, THEN subsetD])
```
```  1005 (*Now for the lemma assumed above*)
```
```  1006 apply (unfold eqpoll_def)
```
```  1007 apply (blast intro: bij_converse_bij [THEN bij_is_fun, THEN apply_type])
```
```  1008 done
```
```  1009
```
```  1010 lemma Finite_into_Fin: "Finite(A) ==> A \<in> Fin(A)"
```
```  1011 apply (unfold Finite_def)
```
```  1012 apply (blast intro: Fin_lemma)
```
```  1013 done
```
```  1014
```
```  1015 lemma Fin_into_Finite: "A \<in> Fin(U) ==> Finite(A)"
```
```  1016 by (fast intro!: Finite_0 Finite_cons elim: Fin_induct)
```
```  1017
```
```  1018 lemma Finite_Fin_iff: "Finite(A) \<longleftrightarrow> A \<in> Fin(A)"
```
```  1019 by (blast intro: Finite_into_Fin Fin_into_Finite)
```
```  1020
```
```  1021 lemma Finite_Un: "[| Finite(A); Finite(B) |] ==> Finite(A \<union> B)"
```
```  1022 by (blast intro!: Fin_into_Finite Fin_UnI
```
```  1023           dest!: Finite_into_Fin
```
```  1024           intro: Un_upper1 [THEN Fin_mono, THEN subsetD]
```
```  1025                  Un_upper2 [THEN Fin_mono, THEN subsetD])
```
```  1026
```
```  1027 lemma Finite_Un_iff [simp]: "Finite(A \<union> B) \<longleftrightarrow> (Finite(A) & Finite(B))"
```
```  1028 by (blast intro: subset_Finite Finite_Un)
```
```  1029
```
```  1030 text{*The converse must hold too.*}
```
```  1031 lemma Finite_Union: "[| \<forall>y\<in>X. Finite(y);  Finite(X) |] ==> Finite(\<Union>(X))"
```
```  1032 apply (simp add: Finite_Fin_iff)
```
```  1033 apply (rule Fin_UnionI)
```
```  1034 apply (erule Fin_induct, simp)
```
```  1035 apply (blast intro: Fin.consI Fin_mono [THEN  rev_subsetD])
```
```  1036 done
```
```  1037
```
```  1038 (* Induction principle for Finite(A), by Sidi Ehmety *)
```
```  1039 lemma Finite_induct [case_names 0 cons, induct set: Finite]:
```
```  1040 "[| Finite(A); P(0);
```
```  1041     !! x B.   [| Finite(B); x \<notin> B; P(B) |] ==> P(cons(x, B)) |]
```
```  1042  ==> P(A)"
```
```  1043 apply (erule Finite_into_Fin [THEN Fin_induct])
```
```  1044 apply (blast intro: Fin_into_Finite)+
```
```  1045 done
```
```  1046
```
```  1047 (*Sidi Ehmety.  The contrapositive says ~Finite(A) ==> ~Finite(A-{a}) *)
```
```  1048 lemma Diff_sing_Finite: "Finite(A - {a}) ==> Finite(A)"
```
```  1049 apply (unfold Finite_def)
```
```  1050 apply (case_tac "a \<in> A")
```
```  1051 apply (subgoal_tac  "A-{a}=A", auto)
```
```  1052 apply (rule_tac x = "succ (n) " in bexI)
```
```  1053 apply (subgoal_tac "cons (a, A - {a}) = A & cons (n, n) = succ (n) ")
```
```  1054 apply (drule_tac a = a and b = n in cons_eqpoll_cong)
```
```  1055 apply (auto dest: mem_irrefl)
```
```  1056 done
```
```  1057
```
```  1058 (*Sidi Ehmety.  And the contrapositive of this says
```
```  1059    [| ~Finite(A); Finite(B) |] ==> ~Finite(A-B) *)
```
```  1060 lemma Diff_Finite [rule_format]: "Finite(B) ==> Finite(A-B) \<longrightarrow> Finite(A)"
```
```  1061 apply (erule Finite_induct, auto)
```
```  1062 apply (case_tac "x \<in> A")
```
```  1063  apply (subgoal_tac  "A-cons (x, B) = A - B")
```
```  1064 apply (subgoal_tac "A - cons (x, B) = (A - B) - {x}", simp)
```
```  1065 apply (drule Diff_sing_Finite, auto)
```
```  1066 done
```
```  1067
```
```  1068 lemma Finite_RepFun: "Finite(A) ==> Finite(RepFun(A,f))"
```
```  1069 by (erule Finite_induct, simp_all)
```
```  1070
```
```  1071 lemma Finite_RepFun_iff_lemma [rule_format]:
```
```  1072      "[|Finite(x); !!x y. f(x)=f(y) ==> x=y|]
```
```  1073       ==> \<forall>A. x = RepFun(A,f) \<longrightarrow> Finite(A)"
```
```  1074 apply (erule Finite_induct)
```
```  1075  apply clarify
```
```  1076  apply (case_tac "A=0", simp)
```
```  1077  apply (blast del: allE, clarify)
```
```  1078 apply (subgoal_tac "\<exists>z\<in>A. x = f(z)")
```
```  1079  prefer 2 apply (blast del: allE elim: equalityE, clarify)
```
```  1080 apply (subgoal_tac "B = {f(u) . u \<in> A - {z}}")
```
```  1081  apply (blast intro: Diff_sing_Finite)
```
```  1082 apply (thin_tac "\<forall>A. ?P(A) \<longrightarrow> Finite(A)")
```
```  1083 apply (rule equalityI)
```
```  1084  apply (blast intro: elim: equalityE)
```
```  1085 apply (blast intro: elim: equalityCE)
```
```  1086 done
```
```  1087
```
```  1088 text{*I don't know why, but if the premise is expressed using meta-connectives
```
```  1089 then  the simplifier cannot prove it automatically in conditional rewriting.*}
```
```  1090 lemma Finite_RepFun_iff:
```
```  1091      "(\<forall>x y. f(x)=f(y) \<longrightarrow> x=y) ==> Finite(RepFun(A,f)) \<longleftrightarrow> Finite(A)"
```
```  1092 by (blast intro: Finite_RepFun Finite_RepFun_iff_lemma [of _ f])
```
```  1093
```
```  1094 lemma Finite_Pow: "Finite(A) ==> Finite(Pow(A))"
```
```  1095 apply (erule Finite_induct)
```
```  1096 apply (simp_all add: Pow_insert Finite_Un Finite_RepFun)
```
```  1097 done
```
```  1098
```
```  1099 lemma Finite_Pow_imp_Finite: "Finite(Pow(A)) ==> Finite(A)"
```
```  1100 apply (subgoal_tac "Finite({{x} . x \<in> A})")
```
```  1101  apply (simp add: Finite_RepFun_iff )
```
```  1102 apply (blast intro: subset_Finite)
```
```  1103 done
```
```  1104
```
```  1105 lemma Finite_Pow_iff [iff]: "Finite(Pow(A)) \<longleftrightarrow> Finite(A)"
```
```  1106 by (blast intro: Finite_Pow Finite_Pow_imp_Finite)
```
```  1107
```
```  1108 lemma Finite_cardinal_iff:
```
```  1109   assumes i: "Ord(i)" shows "Finite(|i|) \<longleftrightarrow> Finite(i)"
```
```  1110   by (auto simp add: Finite_def) (blast intro: eqpoll_trans eqpoll_sym Ord_cardinal_eqpoll [OF i])+
```
```  1111
```
```  1112
```
```  1113 (*Krzysztof Grabczewski's proof that the converse of a finite, well-ordered
```
```  1114   set is well-ordered.  Proofs simplified by lcp. *)
```
```  1115
```
```  1116 lemma nat_wf_on_converse_Memrel: "n \<in> nat ==> wf[n](converse(Memrel(n)))"
```
```  1117 proof (induct n rule: nat_induct)
```
```  1118   case 0 thus ?case by (blast intro: wf_onI)
```
```  1119 next
```
```  1120   case (succ x)
```
```  1121   hence wfx: "\<And>Z. Z = 0 \<or> (\<exists>z\<in>Z. \<forall>y. z \<in> y \<and> z \<in> x \<and> y \<in> x \<and> z \<in> x \<longrightarrow> y \<notin> Z)"
```
```  1122     by (simp add: wf_on_def wf_def)  --{*not easy to erase the duplicate @{term"z \<in> x"}!*}
```
```  1123   show ?case
```
```  1124     proof (rule wf_onI)
```
```  1125       fix Z u
```
```  1126       assume Z: "u \<in> Z" "\<forall>z\<in>Z. \<exists>y\<in>Z. \<langle>y, z\<rangle> \<in> converse(Memrel(succ(x)))"
```
```  1127       show False
```
```  1128         proof (cases "x \<in> Z")
```
```  1129           case True thus False using Z
```
```  1130             by (blast elim: mem_irrefl mem_asym)
```
```  1131           next
```
```  1132           case False thus False using wfx [of Z] Z
```
```  1133             by blast
```
```  1134         qed
```
```  1135     qed
```
```  1136 qed
```
```  1137
```
```  1138 lemma nat_well_ord_converse_Memrel: "n \<in> nat ==> well_ord(n,converse(Memrel(n)))"
```
```  1139 apply (frule Ord_nat [THEN Ord_in_Ord, THEN well_ord_Memrel])
```
```  1140 apply (simp add: well_ord_def tot_ord_converse nat_wf_on_converse_Memrel)
```
```  1141 done
```
```  1142
```
```  1143 lemma well_ord_converse:
```
```  1144      "[|well_ord(A,r);
```
```  1145         well_ord(ordertype(A,r), converse(Memrel(ordertype(A, r)))) |]
```
```  1146       ==> well_ord(A,converse(r))"
```
```  1147 apply (rule well_ord_Int_iff [THEN iffD1])
```
```  1148 apply (frule ordermap_bij [THEN bij_is_inj, THEN well_ord_rvimage], assumption)
```
```  1149 apply (simp add: rvimage_converse converse_Int converse_prod
```
```  1150                  ordertype_ord_iso [THEN ord_iso_rvimage_eq])
```
```  1151 done
```
```  1152
```
```  1153 lemma ordertype_eq_n:
```
```  1154   assumes r: "well_ord(A,r)" and A: "A \<approx> n" and n: "n \<in> nat"
```
```  1155   shows "ordertype(A,r) = n"
```
```  1156 proof -
```
```  1157   have "ordertype(A,r) \<approx> A"
```
```  1158     by (blast intro: bij_imp_eqpoll bij_converse_bij ordermap_bij r)
```
```  1159   also have "... \<approx> n" by (rule A)
```
```  1160   finally have "ordertype(A,r) \<approx> n" .
```
```  1161   thus ?thesis
```
```  1162     by (simp add: Ord_nat_eqpoll_iff Ord_ordertype n r)
```
```  1163 qed
```
```  1164
```
```  1165 lemma Finite_well_ord_converse:
```
```  1166     "[| Finite(A);  well_ord(A,r) |] ==> well_ord(A,converse(r))"
```
```  1167 apply (unfold Finite_def)
```
```  1168 apply (rule well_ord_converse, assumption)
```
```  1169 apply (blast dest: ordertype_eq_n intro!: nat_well_ord_converse_Memrel)
```
```  1170 done
```
```  1171
```
```  1172 lemma nat_into_Finite: "n \<in> nat ==> Finite(n)"
```
```  1173   by (auto simp add: Finite_def intro: eqpoll_refl)
```
```  1174
```
```  1175 lemma nat_not_Finite: "~ Finite(nat)"
```
```  1176 proof -
```
```  1177   { fix n
```
```  1178     assume n: "n \<in> nat" "nat \<approx> n"
```
```  1179     have "n \<in> nat"    by (rule n)
```
```  1180     also have "... = n" using n
```
```  1181       by (simp add: Ord_nat_eqpoll_iff Ord_nat)
```
```  1182     finally have "n \<in> n" .
```
```  1183     hence False
```
```  1184       by (blast elim: mem_irrefl)
```
```  1185   }
```
```  1186   thus ?thesis
```
```  1187     by (auto simp add: Finite_def)
```
```  1188 qed
```
```  1189
```
```  1190 end
```