src/ZF/CardinalArith.thy
author obua
Thu, 16 Feb 2006 04:17:19 +0100
changeset 19067 c0321d7d6b3d
parent 16417 9bc16273c2d4
child 24893 b8ef7afe3a6b
permissions -rw-r--r--
variable counter is now also cached
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
1478
2b8c2a7547ab expanded tabs
clasohm
parents: 1401
diff changeset
     1
(*  Title:      ZF/CardinalArith.thy
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
     2
    ID:         $Id$
1478
2b8c2a7547ab expanded tabs
clasohm
parents: 1401
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
     4
    Copyright   1994  University of Cambridge
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
     5
13328
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
     6
*)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
     7
13328
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
     8
header{*Cardinal Arithmetic Without the Axiom of Choice*}
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
     9
16417
9bc16273c2d4 migrated theory headers to new format
haftmann
parents: 14883
diff changeset
    10
theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
    11
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    12
constdefs
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
    13
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    14
  InfCard       :: "i=>o"
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    15
    "InfCard(i) == Card(i) & nat le i"
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
    16
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    17
  cmult         :: "[i,i]=>i"       (infixl "|*|" 70)
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    18
    "i |*| j == |i*j|"
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    19
  
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    20
  cadd          :: "[i,i]=>i"       (infixl "|+|" 65)
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    21
    "i |+| j == |i+j|"
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
    22
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    23
  csquare_rel   :: "i=>i"
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    24
    "csquare_rel(K) ==   
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    25
	  rvimage(K*K,   
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    26
		  lam <x,y>:K*K. <x Un y, x, y>, 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    27
		  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
    28
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    29
  jump_cardinal :: "i=>i"
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    30
    --{*This def is more complex than Kunen's but it more easily proved to
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    31
        be a cardinal*}
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    32
    "jump_cardinal(K) ==   
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
    33
         \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    34
  
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    35
  csucc         :: "i=>i"
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    36
    --{*needed because @{term "jump_cardinal(K)"} might not be the successor
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    37
        of @{term K}*}
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    38
    "csucc(K) == LEAST L. Card(L) & K<L"
484
70b789956bd3 Axiom of choice, cardinality results, etc.
lcp
parents: 467
diff changeset
    39
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 9964
diff changeset
    40
syntax (xsymbols)
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    41
  "op |+|"     :: "[i,i] => i"          (infixl "\<oplus>" 65)
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    42
  "op |*|"     :: "[i,i] => i"          (infixl "\<otimes>" 70)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 13784
diff changeset
    43
syntax (HTML output)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 13784
diff changeset
    44
  "op |+|"     :: "[i,i] => i"          (infixl "\<oplus>" 65)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 13784
diff changeset
    45
  "op |*|"     :: "[i,i] => i"          (infixl "\<otimes>" 70)
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    46
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    47
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    48
lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    49
apply (rule CardI) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    50
 apply (simp add: Card_is_Ord) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    51
apply (clarify dest!: ltD)
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    52
apply (drule bspec, assumption) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    53
apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    54
apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    55
apply (drule lesspoll_trans1, assumption) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
    56
apply (subgoal_tac "B \<lesssim> \<Union>A")
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    57
 apply (drule lesspoll_trans1, assumption, blast) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    58
apply (blast intro: subset_imp_lepoll) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    59
done
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    60
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    61
lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    62
by (blast intro: Card_Union) 
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    63
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    64
lemma Card_OUN [simp,intro,TC]:
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
    65
     "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
12667
7e6eaaa125f2 Added some simprules proofs.
paulson
parents: 12114
diff changeset
    66
by (simp add: OUnion_def Card_0) 
9654
9754ba005b64 X-symbols for ordinal, cardinal, integer arithmetic
paulson
parents: 9548
diff changeset
    67
12776
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    68
lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    69
apply (unfold lesspoll_def)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    70
apply (rule conjI)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    71
apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    72
apply (rule notI)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    73
apply (erule eqpollE)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    74
apply (rule succ_lepoll_natE)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    75
apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
12820
02e2ff3e4d37 lexical tidying
paulson
parents: 12776
diff changeset
    76
                    lepoll_trans, assumption) 
12776
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    77
done
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    78
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    79
lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    80
apply (unfold lesspoll_def)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    81
apply (simp add: Card_iff_initial)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    82
apply (fast intro!: le_imp_lepoll ltI leI)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    83
done
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    84
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
    85
lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
12776
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    86
apply (unfold lesspoll_def)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    87
apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    88
            intro!: eqpollI elim: notE 
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    89
            elim!: eqpollE lepoll_trans)
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    90
done
249600a63ba9 Isar version of AC
paulson
parents: 12667
diff changeset
    91
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
    92
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
    93
subsection{*Cardinal addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
    94
13328
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
    95
text{*Note: Could omit proving the algebraic laws for cardinal addition and
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
    96
multiplication.  On finite cardinals these operations coincide with
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
    97
addition and multiplication of natural numbers; on infinite cardinals they
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
    98
coincide with union (maximum).  Either way we get most laws for free.*}
703de709a64b better document preparation
paulson
parents: 13269
diff changeset
    99
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   100
subsubsection{*Cardinal addition is commutative*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   101
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   102
lemma sum_commute_eqpoll: "A+B \<approx> B+A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   103
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   104
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   105
apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   106
apply auto
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   107
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   108
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   109
lemma cadd_commute: "i |+| j = j |+| i"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   110
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   111
apply (rule sum_commute_eqpoll [THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   112
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   113
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   114
subsubsection{*Cardinal addition is associative*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   115
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   116
lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   117
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   118
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   119
apply (rule sum_assoc_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   120
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   121
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   122
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   123
lemma well_ord_cadd_assoc: 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   124
    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   125
     ==> (i |+| j) |+| k = i |+| (j |+| k)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   126
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   127
apply (rule cardinal_cong)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   128
apply (rule eqpoll_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   129
 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   130
 apply (blast intro: well_ord_radd ) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   131
apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   132
apply (rule eqpoll_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   133
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   134
apply (blast intro: well_ord_radd ) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   135
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   136
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   137
subsubsection{*0 is the identity for addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   138
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   139
lemma sum_0_eqpoll: "0+A \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   140
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   141
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   142
apply (rule bij_0_sum)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   143
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   144
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   145
lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   146
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   147
apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   148
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   149
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   150
subsubsection{*Addition by another cardinal*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   151
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   152
lemma sum_lepoll_self: "A \<lesssim> A+B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   153
apply (unfold lepoll_def inj_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   154
apply (rule_tac x = "lam x:A. Inl (x) " in exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   155
apply simp
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   156
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   157
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   158
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   159
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   160
lemma cadd_le_self: 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   161
    "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   162
apply (unfold cadd_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   163
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   164
       assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   165
apply (rule_tac [2] sum_lepoll_self)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   166
apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   167
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   168
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   169
subsubsection{*Monotonicity of addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   170
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   171
lemma sum_lepoll_mono: 
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   172
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   173
apply (unfold lepoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   174
apply (elim exE)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   175
apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   176
apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   177
       in lam_injective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   178
apply (typecheck add: inj_is_fun, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   179
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   180
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   181
lemma cadd_le_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   182
    "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   183
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   184
apply (safe dest!: le_subset_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   185
apply (rule well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   186
apply (blast intro: well_ord_radd well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   187
apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   188
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   189
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   190
subsubsection{*Addition of finite cardinals is "ordinary" addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   191
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   192
lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   193
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   194
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   195
apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   196
            and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   197
   apply simp_all
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   198
apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   199
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   200
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   201
(*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   202
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   203
lemma cadd_succ_lemma:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   204
    "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   205
apply (unfold cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   206
apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   207
apply (rule succ_eqpoll_cong [THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   208
apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   209
apply (blast intro: well_ord_radd well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   210
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   211
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   212
lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
13244
7b37e218f298 moving some results around
paulson
parents: 13221
diff changeset
   213
apply (induct_tac m)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   214
apply (simp add: nat_into_Card [THEN cadd_0])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   215
apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   216
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   217
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   218
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   219
subsection{*Cardinal multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   220
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   221
subsubsection{*Cardinal multiplication is commutative*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   222
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   223
(*Easier to prove the two directions separately*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   224
lemma prod_commute_eqpoll: "A*B \<approx> B*A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   225
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   226
apply (rule exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   227
apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   228
       auto) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   229
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   230
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   231
lemma cmult_commute: "i |*| j = j |*| i"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   232
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   233
apply (rule prod_commute_eqpoll [THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   234
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   235
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   236
subsubsection{*Cardinal multiplication is associative*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   237
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   238
lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   239
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   240
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   241
apply (rule prod_assoc_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   242
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   243
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   244
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   245
lemma well_ord_cmult_assoc:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   246
    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   247
     ==> (i |*| j) |*| k = i |*| (j |*| k)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   248
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   249
apply (rule cardinal_cong)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   250
apply (rule eqpoll_trans) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   251
 apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   252
 apply (blast intro: well_ord_rmult)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   253
apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   254
apply (rule eqpoll_sym) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   255
apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   256
apply (blast intro: well_ord_rmult)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   257
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   258
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   259
subsubsection{*Cardinal multiplication distributes over addition*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   260
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   261
lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   262
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   263
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   264
apply (rule sum_prod_distrib_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   265
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   266
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   267
lemma well_ord_cadd_cmult_distrib:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   268
    "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   269
     ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   270
apply (unfold cadd_def cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   271
apply (rule cardinal_cong)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   272
apply (rule eqpoll_trans) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   273
 apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   274
apply (blast intro: well_ord_radd)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   275
apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   276
apply (rule eqpoll_sym) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   277
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   278
                                well_ord_cardinal_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   279
apply (blast intro: well_ord_rmult)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   280
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   281
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   282
subsubsection{*Multiplication by 0 yields 0*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   283
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   284
lemma prod_0_eqpoll: "0*A \<approx> 0"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   285
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   286
apply (rule exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   287
apply (rule lam_bijective, safe)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   288
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   289
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   290
lemma cmult_0 [simp]: "0 |*| i = 0"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   291
by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   292
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   293
subsubsection{*1 is the identity for multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   294
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   295
lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   296
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   297
apply (rule exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   298
apply (rule singleton_prod_bij [THEN bij_converse_bij])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   299
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   300
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   301
lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   302
apply (unfold cmult_def succ_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   303
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   304
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   305
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   306
subsection{*Some inequalities for multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   307
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   308
lemma prod_square_lepoll: "A \<lesssim> A*A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   309
apply (unfold lepoll_def inj_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   310
apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   311
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   312
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   313
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   314
lemma cmult_square_le: "Card(K) ==> K le K |*| K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   315
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   316
apply (rule le_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   317
apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   318
apply (rule_tac [3] prod_square_lepoll)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   319
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   320
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   321
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   322
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   323
subsubsection{*Multiplication by a non-zero cardinal*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   324
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   325
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   326
apply (unfold lepoll_def inj_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   327
apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   328
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   329
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   330
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   331
lemma cmult_le_self:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   332
    "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   333
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   334
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   335
  apply assumption
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   336
 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   337
apply (blast intro: prod_lepoll_self ltD)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   338
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   339
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   340
subsubsection{*Monotonicity of multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   341
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   342
lemma prod_lepoll_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   343
     "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   344
apply (unfold lepoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   345
apply (elim exE)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   346
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   347
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   348
       in lam_injective)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   349
apply (typecheck add: inj_is_fun, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   350
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   351
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   352
lemma cmult_le_mono:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   353
    "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   354
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   355
apply (safe dest!: le_subset_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   356
apply (rule well_ord_lepoll_imp_Card_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   357
 apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   358
apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   359
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   360
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   361
subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   362
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   363
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   364
apply (unfold eqpoll_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   365
apply (rule exI)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   366
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   367
            and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   368
apply safe
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   369
apply (simp_all add: succI2 if_type mem_imp_not_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   370
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   371
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   372
(*Unconditional version requires AC*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   373
lemma cmult_succ_lemma:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   374
    "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   375
apply (unfold cmult_def cadd_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   376
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   377
apply (rule cardinal_cong [symmetric])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   378
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   379
apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   380
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   381
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   382
lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
13244
7b37e218f298 moving some results around
paulson
parents: 13221
diff changeset
   383
apply (induct_tac m)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   384
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   385
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   386
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   387
lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   388
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   389
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   390
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   391
apply (rule lepoll_trans) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   392
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   393
apply (erule prod_lepoll_mono) 
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   394
apply (rule lepoll_refl) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   395
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   396
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   397
lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   398
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   399
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   400
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   401
subsection{*Infinite Cardinals are Limit Ordinals*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   402
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   403
(*This proof is modelled upon one assuming nat<=A, with injection
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   404
  lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   405
  and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   406
  If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   407
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   408
apply (unfold lepoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   409
apply (erule exE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   410
apply (rule_tac x = 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   411
          "lam z:cons (u,A).
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   412
             if z=u then f`0 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   413
             else if z: range (f) then f`succ (converse (f) `z) else z" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   414
       in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   415
apply (rule_tac d =
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   416
          "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   417
                              else y" 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   418
       in lam_injective)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   419
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   420
apply (simp add: inj_is_fun [THEN apply_rangeI]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   421
                 inj_converse_fun [THEN apply_rangeI]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   422
                 inj_converse_fun [THEN apply_funtype])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   423
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   424
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   425
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   426
apply (erule nat_cons_lepoll [THEN eqpollI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   427
apply (rule subset_consI [THEN subset_imp_lepoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   428
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   429
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   430
(*Specialized version required below*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   431
lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   432
apply (unfold succ_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   433
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   434
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   435
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   436
lemma InfCard_nat: "InfCard(nat)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   437
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   438
apply (blast intro: Card_nat le_refl Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   439
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   440
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   441
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   442
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   443
apply (erule conjunct1)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   444
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   445
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   446
lemma InfCard_Un:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   447
    "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   448
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   449
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   450
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   451
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   452
(*Kunen's Lemma 10.11*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   453
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   454
apply (unfold InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   455
apply (erule conjE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   456
apply (frule Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   457
apply (rule ltI [THEN non_succ_LimitI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   458
apply (erule le_imp_subset [THEN subsetD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   459
apply (safe dest!: Limit_nat [THEN Limit_le_succD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   460
apply (unfold Card_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   461
apply (drule trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   462
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   463
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   464
apply (rule le_eqI, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   465
apply (rule Ord_cardinal)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   466
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   467
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   468
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   469
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   470
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   471
(*A general fact about ordermap*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   472
lemma ordermap_eqpoll_pred:
13269
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13244
diff changeset
   473
    "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   474
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   475
apply (rule exI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   476
apply (simp add: ordermap_eq_image well_ord_is_wf)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   477
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   478
                           THEN bij_converse_bij])
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   479
apply (rule pred_subset)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   480
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   481
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   482
subsubsection{*Establishing the well-ordering*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   483
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   484
lemma csquare_lam_inj:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   485
     "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   486
apply (unfold inj_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   487
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   488
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   489
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   490
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   491
apply (unfold csquare_rel_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   492
apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   493
apply (blast intro: well_ord_rmult well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   494
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   495
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   496
subsubsection{*Characterising initial segments of the well-ordering*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   497
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   498
lemma csquareD:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   499
 "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   500
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   501
apply (erule rev_mp)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   502
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   503
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   504
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   505
apply (simp_all add: lt_def succI2)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   506
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   507
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   508
lemma pred_csquare_subset: 
13269
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13244
diff changeset
   509
    "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   510
apply (unfold Order.pred_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   511
apply (safe del: SigmaI succCI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   512
apply (erule csquareD [THEN conjE])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   513
apply (unfold lt_def, auto) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   514
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   515
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   516
lemma csquare_ltI:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   517
 "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   518
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   519
apply (subgoal_tac "x<K & y<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   520
 prefer 2 apply (blast intro: lt_trans) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   521
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   522
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   523
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   524
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   525
(*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   526
lemma csquare_or_eqI:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   527
 "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   528
apply (unfold csquare_rel_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   529
apply (subgoal_tac "x<K & y<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   530
 prefer 2 apply (blast intro: lt_trans1) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   531
apply (elim ltE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   532
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   533
apply (elim succE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   534
apply (simp_all add: subset_Un_iff [THEN iff_sym] 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   535
                     subset_Un_iff2 [THEN iff_sym] OrdmemD)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   536
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   537
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   538
subsubsection{*The cardinality of initial segments*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   539
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   540
lemma ordermap_z_lt:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   541
      "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   542
          ordermap(K*K, csquare_rel(K)) ` <x,y> <
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   543
          ordermap(K*K, csquare_rel(K)) ` <z,z>"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   544
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   545
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   546
                              Limit_is_Ord [THEN well_ord_csquare], clarify) 
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   547
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   548
apply (erule_tac [4] well_ord_is_wf)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   549
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   550
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   551
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   552
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   553
lemma ordermap_csquare_le:
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   554
  "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   555
   ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   556
apply (unfold cmult_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   557
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   558
apply (rule Ord_cardinal [THEN well_ord_Memrel])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   559
apply (subgoal_tac "z<K")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   560
 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   561
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   562
       assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   563
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   564
apply (erule Limit_is_Ord [THEN well_ord_csquare])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   565
apply (blast intro: ltD)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   566
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   567
            assumption)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   568
apply (elim ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   569
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   570
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   571
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   572
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   573
(*Kunen: "... so the order type <= K" *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   574
lemma ordertype_csquare_le:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   575
     "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   576
      ==> ordertype(K*K, csquare_rel(K)) le K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   577
apply (frule InfCard_is_Card [THEN Card_is_Ord])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   578
apply (rule all_lt_imp_le, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   579
apply (erule well_ord_csquare [THEN Ord_ordertype])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   580
apply (rule Card_lt_imp_lt)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   581
apply (erule_tac [3] InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   582
apply (erule_tac [2] ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   583
apply (simp add: ordertype_unfold)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   584
apply (safe elim!: ltE)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   585
apply (subgoal_tac "Ord (xa) & Ord (ya)")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   586
 prefer 2 apply (blast intro: Ord_in_Ord, clarify)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   587
(*??WHAT A MESS!*)  
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   588
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   589
       (assumption | rule refl | erule ltI)+) 
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   590
apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   591
       simp_all add: Ord_Un Ord_nat)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   592
prefer 2 (*case nat le (xa Un ya) *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   593
 apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   594
                  le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   595
                ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   596
(*the finite case: xa Un ya < nat *)
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   597
apply (rule_tac j = nat in lt_trans2)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   598
 apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   599
                  nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   600
apply (simp add: InfCard_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   601
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   602
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   603
(*Main result: Kunen's Theorem 10.12*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   604
lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   605
apply (frule InfCard_is_Card [THEN Card_is_Ord])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   606
apply (erule rev_mp)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   607
apply (erule_tac i=K in trans_induct) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   608
apply (rule impI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   609
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   610
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   611
apply (rule ordertype_csquare_le [THEN [2] le_trans])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   612
apply (simp add: cmult_def Ord_cardinal_le   
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   613
                 well_ord_csquare [THEN Ord_ordertype]
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   614
                 well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   615
                                   THEN cardinal_cong], assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   616
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   617
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   618
(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   619
lemma well_ord_InfCard_square_eq:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   620
     "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   621
apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   622
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   623
apply (rule well_ord_cardinal_eqE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   624
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   625
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   626
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   627
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   628
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   629
apply (rule well_ord_InfCard_square_eq)  
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   630
 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   631
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   632
done
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   633
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   634
lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   635
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   636
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   637
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   638
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   639
lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   640
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   641
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   642
 apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   643
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   644
apply (rule cmult_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   645
apply (simp add: InfCard_csquare_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   646
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   647
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   648
(*Corollary 10.13 (1), for cardinal multiplication*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   649
lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   650
apply (rule_tac i = K and j = L in Ord_linear_le)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   651
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   652
apply (rule cmult_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   653
apply (rule Un_commute [THEN ssubst])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   654
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   655
                     subset_Un_iff2 [THEN iffD1] le_imp_subset)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   656
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   657
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   658
lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   659
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   660
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   661
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   662
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   663
(*Corollary 10.13 (1), for cardinal addition*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   664
lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   665
apply (rule le_anti_sym)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   666
 prefer 2
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   667
 apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   668
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   669
apply (rule cadd_le_mono [THEN le_trans], assumption+)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   670
apply (simp add: InfCard_cdouble_eq)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   671
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   672
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   673
lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   674
apply (rule_tac i = K and j = L in Ord_linear_le)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   675
apply (typecheck add: InfCard_is_Card Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   676
apply (rule cadd_commute [THEN ssubst])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   677
apply (rule Un_commute [THEN ssubst])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   678
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   679
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   680
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   681
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   682
  of all n-tuples of elements of K.  A better version for the Isabelle theory
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   683
  might be  InfCard(K) ==> |list(K)| = K.
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   684
*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   685
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   686
subsection{*For Every Cardinal Number There Exists A Greater One}
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   687
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   688
text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   689
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   690
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   691
apply (unfold jump_cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   692
apply (rule Ord_is_Transset [THEN [2] OrdI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   693
 prefer 2 apply (blast intro!: Ord_ordertype)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   694
apply (unfold Transset_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   695
apply (safe del: subsetI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   696
apply (simp add: ordertype_pred_unfold, safe)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   697
apply (rule UN_I)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   698
apply (rule_tac [2] ReplaceI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   699
   prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   700
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   701
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   702
(*Allows selective unfolding.  Less work than deriving intro/elim rules*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   703
lemma jump_cardinal_iff:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   704
     "i : jump_cardinal(K) <->
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   705
      (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   706
apply (unfold jump_cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   707
apply (blast del: subsetI) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   708
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   709
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   710
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   711
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   712
apply (rule Ord_jump_cardinal [THEN [2] ltI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   713
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   714
apply (rule_tac x="Memrel(K)" in exI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   715
apply (rule_tac x=K in exI)  
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   716
apply (simp add: ordertype_Memrel well_ord_Memrel)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   717
apply (simp add: Memrel_def subset_iff)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   718
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   719
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   720
(*The proof by contradiction: the bijection f yields a wellordering of X
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   721
  whose ordertype is jump_cardinal(K).  *)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   722
lemma Card_jump_cardinal_lemma:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   723
     "[| well_ord(X,r);  r <= K * K;  X <= K;
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   724
         f : bij(ordertype(X,r), jump_cardinal(K)) |]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   725
      ==> jump_cardinal(K) : jump_cardinal(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   726
apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   727
 prefer 2 apply (blast intro: comp_bij ordermap_bij)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   728
apply (rule jump_cardinal_iff [THEN iffD2])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   729
apply (intro exI conjI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   730
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   731
apply (erule bij_is_inj [THEN well_ord_rvimage])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   732
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   733
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   734
                 ordertype_Memrel Ord_jump_cardinal)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   735
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   736
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   737
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   738
lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   739
apply (rule Ord_jump_cardinal [THEN CardI])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   740
apply (unfold eqpoll_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   741
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   742
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   743
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   744
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   745
subsection{*Basic Properties of Successor Cardinals*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   746
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   747
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   748
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   749
apply (rule LeastI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   750
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   751
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   752
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   753
lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   754
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   755
lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   756
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   757
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   758
by (blast intro: Ord_0_le lt_csucc lt_trans1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   759
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   760
lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   761
apply (unfold csucc_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   762
apply (rule Least_le)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   763
apply (blast intro: Card_is_Ord)+
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   764
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   765
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   766
lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   767
apply (rule iffI)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   768
apply (rule_tac [2] Card_lt_imp_lt)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   769
apply (erule_tac [2] lt_trans1)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   770
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   771
apply (rule notI [THEN not_lt_imp_le])
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   772
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   773
apply (rule Ord_cardinal_le [THEN lt_trans1])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   774
apply (simp_all add: Ord_cardinal Card_is_Ord) 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   775
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   776
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   777
lemma Card_lt_csucc_iff:
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   778
     "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   779
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   780
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   781
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   782
by (simp add: InfCard_def Card_csucc Card_is_Ord 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   783
              lt_csucc [THEN leI, THEN [2] le_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   784
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   785
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   786
subsubsection{*Removing elements from a finite set decreases its cardinality*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   787
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   788
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   789
apply (erule Fin_induct)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   790
apply (simp add: lepoll_0_iff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   791
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   792
apply simp
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   793
apply (blast dest!: cons_lepoll_consD, blast)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   794
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   795
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   796
lemma Finite_imp_cardinal_cons [simp]:
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   797
     "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   798
apply (unfold cardinal_def)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   799
apply (rule Least_equality)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   800
apply (fold cardinal_def)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   801
apply (simp add: succ_def)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   802
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   803
             elim!: mem_irrefl  dest!: Finite_imp_well_ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   804
apply (blast intro: Card_cardinal Card_is_Ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   805
apply (rule notI)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   806
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   807
       assumption, assumption)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   808
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   809
apply (erule le_imp_lepoll [THEN lepoll_trans])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   810
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   811
             dest!: Finite_imp_well_ord)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   812
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   813
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   814
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   815
lemma Finite_imp_succ_cardinal_Diff:
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   816
     "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13615
diff changeset
   817
apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   818
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   819
apply (simp add: cons_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   820
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   821
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   822
lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   823
apply (rule succ_leE)
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   824
apply (simp add: Finite_imp_succ_cardinal_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   825
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   826
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   827
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   828
apply (erule Finite_induct)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   829
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   830
done
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   831
14883
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   832
lemma card_Un_Int:
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   833
     "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   834
apply (erule Finite_induct, simp) 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   835
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   836
done
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   837
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   838
lemma card_Un_disjoint: 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   839
     "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   840
by (simp add: Finite_Un card_Un_Int)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   841
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   842
lemma card_partition [rule_format]:
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   843
     "Finite(C) ==>  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   844
        Finite (\<Union> C) -->  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   845
        (\<forall>c\<in>C. |c| = k) -->   
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   846
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   847
        k #* |C| = |\<Union> C|"
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   848
apply (erule Finite_induct, auto)
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   849
apply (subgoal_tac " x \<inter> \<Union>B = 0")  
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   850
apply (auto simp add: card_Un_disjoint Finite_Union
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   851
       subset_Finite [of _ "\<Union> (cons(x,F))"])
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   852
done
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   853
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   854
ca000a495448 Groups, Rings and supporting lemmas
paulson
parents: 14565
diff changeset
   855
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   856
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   857
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   858
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   859
lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   860
apply (rule eqpoll_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   861
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   862
apply (erule nat_implies_well_ord)+
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   863
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   864
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   865
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   866
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   867
apply (erule trans_induct3, auto)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   868
apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   869
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   870
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   871
lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   872
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   873
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   874
lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   875
apply (rule succ_inject)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   876
apply (rule_tac b = "|A|" in trans)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   877
 apply (simp add: Finite_imp_succ_cardinal_Diff)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   878
apply (subgoal_tac "1 \<lesssim> A")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   879
 prefer 2 apply (blast intro: not_0_is_lepoll_1)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   880
apply (frule Finite_imp_well_ord, clarify)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   881
apply (drule well_ord_lepoll_imp_Card_le)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   882
 apply (auto simp add: cardinal_1)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   883
apply (rule trans)
13615
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   884
 apply (rule_tac [2] diff_succ)
449a70d88b38 Numerous cosmetic changes, prompted by the new simplifier
paulson
parents: 13356
diff changeset
   885
  apply (auto simp add: Finite_cardinal_in_nat)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   886
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   887
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   888
lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   889
     "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   890
apply (erule Finite_induct, auto)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   891
apply (case_tac "Finite (A)")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   892
 apply (subgoal_tac [2] "Finite (cons (x, B))")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   893
  apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   894
   apply (auto simp add: Finite_0 Finite_cons)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   895
apply (subgoal_tac "|B|<|A|")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   896
 prefer 2 apply (blast intro: lt_trans Ord_cardinal)
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   897
apply (case_tac "x:A")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   898
 apply (subgoal_tac [2] "A - cons (x, B) = A - B")
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   899
  apply auto
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   900
apply (subgoal_tac "|A| le |cons (x, B) |")
13221
e29378f347e4 conversion of Cardinal, CardinalArith
paulson
parents: 13216
diff changeset
   901
 prefer 2
13216
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   902
 apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   903
              intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   904
apply (auto simp add: Finite_imp_cardinal_cons)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   905
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   906
apply (blast intro: lt_trans)
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   907
done
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   908
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   909
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   910
ML{*
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   911
val InfCard_def = thm "InfCard_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   912
val cmult_def = thm "cmult_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   913
val cadd_def = thm "cadd_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   914
val jump_cardinal_def = thm "jump_cardinal_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   915
val csucc_def = thm "csucc_def"
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   916
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   917
val sum_commute_eqpoll = thm "sum_commute_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   918
val cadd_commute = thm "cadd_commute";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   919
val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   920
val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   921
val sum_0_eqpoll = thm "sum_0_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   922
val cadd_0 = thm "cadd_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   923
val sum_lepoll_self = thm "sum_lepoll_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   924
val cadd_le_self = thm "cadd_le_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   925
val sum_lepoll_mono = thm "sum_lepoll_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   926
val cadd_le_mono = thm "cadd_le_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   927
val eq_imp_not_mem = thm "eq_imp_not_mem";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   928
val sum_succ_eqpoll = thm "sum_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   929
val nat_cadd_eq_add = thm "nat_cadd_eq_add";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   930
val prod_commute_eqpoll = thm "prod_commute_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   931
val cmult_commute = thm "cmult_commute";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   932
val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   933
val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   934
val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   935
val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   936
val prod_0_eqpoll = thm "prod_0_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   937
val cmult_0 = thm "cmult_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   938
val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   939
val cmult_1 = thm "cmult_1";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   940
val prod_lepoll_self = thm "prod_lepoll_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   941
val cmult_le_self = thm "cmult_le_self";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   942
val prod_lepoll_mono = thm "prod_lepoll_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   943
val cmult_le_mono = thm "cmult_le_mono";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   944
val prod_succ_eqpoll = thm "prod_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   945
val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   946
val cmult_2 = thm "cmult_2";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   947
val sum_lepoll_prod = thm "sum_lepoll_prod";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   948
val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   949
val nat_cons_lepoll = thm "nat_cons_lepoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   950
val nat_cons_eqpoll = thm "nat_cons_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   951
val nat_succ_eqpoll = thm "nat_succ_eqpoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   952
val InfCard_nat = thm "InfCard_nat";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   953
val InfCard_is_Card = thm "InfCard_is_Card";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   954
val InfCard_Un = thm "InfCard_Un";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   955
val InfCard_is_Limit = thm "InfCard_is_Limit";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   956
val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   957
val ordermap_z_lt = thm "ordermap_z_lt";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   958
val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   959
val InfCard_cmult_eq = thm "InfCard_cmult_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   960
val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   961
val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   962
val InfCard_cadd_eq = thm "InfCard_cadd_eq";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   963
val Ord_jump_cardinal = thm "Ord_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   964
val jump_cardinal_iff = thm "jump_cardinal_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   965
val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   966
val Card_jump_cardinal = thm "Card_jump_cardinal";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   967
val csucc_basic = thm "csucc_basic";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   968
val Card_csucc = thm "Card_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   969
val lt_csucc = thm "lt_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   970
val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   971
val csucc_le = thm "csucc_le";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   972
val lt_csucc_iff = thm "lt_csucc_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   973
val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   974
val InfCard_csucc = thm "InfCard_csucc";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   975
val Finite_into_Fin = thm "Finite_into_Fin";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   976
val Fin_into_Finite = thm "Fin_into_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   977
val Finite_Fin_iff = thm "Finite_Fin_iff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   978
val Finite_Un = thm "Finite_Un";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   979
val Finite_Union = thm "Finite_Union";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   980
val Finite_induct = thm "Finite_induct";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   981
val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   982
val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   983
val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   984
val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   985
val nat_implies_well_ord = thm "nat_implies_well_ord";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   986
val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   987
val Diff_sing_Finite = thm "Diff_sing_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   988
val Diff_Finite = thm "Diff_Finite";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   989
val Ord_subset_natD = thm "Ord_subset_natD";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   990
val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   991
val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   992
val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   993
val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   994
*}
6104bd4088a2 conversion of CardinalArith to Isar script
paulson
parents: 13161
diff changeset
   995
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents:
diff changeset
   996
end