src/ZF/OrderType.ML
author lcp
Thu, 12 Jan 1995 03:00:38 +0100
changeset 849 013a16d3addb
parent 831 60d850cc5fe6
child 984 4fb1d099ba45
permissions -rw-r--r--
Proved equivalence of Ord and Ord_alt. Proved ordertype_eq_imp_ord_iso, le_well_ord_Memrel, le_ordertype_Memrel, lt_oadd1, oadd_le_self, bij_0_sum, oadd_0, oadd_assoc, id_ord_iso_Memrel, ordertype_0. Now well_ord_Memrel follows from le_well_ord_Memrel and ordertype_Memrel follows from le_ordertype_Memrel. Proved simpler versions of Krzysztof's theorems Ord_oadd, ordertype_pred_subset, ordertype_pred_lt, ordertype_pred_unfold, bij_sum_0, bij_sum_succ, ordertype_sum_Memrel, lt_oadd_disj, oadd_inject. Deleted ordertype_subset: subsumed by ordertype_pred_unfold. Proved ordinal multiplication theorems Ord_omult, lt_omult, omult_oadd_lt, omult_unfold, omult_0, omult_0_left, omult_1, omult_1_left, oadd_omult_distrib, omult_succ, omult_assoc, omult_UN, omult_Limit, lt_omult1, omult_le_self, omult_le_mono1, omult_lt_mono2, omult_le_mono2, omult_le_mono, omult_lt_mono, omult_le_self2, omult_inject.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
     1
(*  Title: 	ZF/OrderType.ML
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
     2
    ID:         $Id$
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
     3
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
     4
    Copyright   1994  University of Cambridge
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
     5
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
     6
Order types and ordinal arithmetic in Zermelo-Fraenkel Set Theory 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
     7
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
     8
Ordinal arithmetic is traditionally defined in terms of order types, as here.
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
     9
But a definition by transfinite recursion would be much simpler!
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    10
*)
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    11
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    12
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    13
open OrderType;
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    14
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    15
(**** Proofs needing the combination of Ordinal.thy and Order.thy ****)
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    16
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    17
val [prem] = goal OrderType.thy "j le i ==> well_ord(j, Memrel(i))";
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    18
by (rtac well_ordI 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    19
by (rtac (wf_Memrel RS wf_imp_wf_on) 1);
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    20
by (resolve_tac [prem RS ltE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    21
by (asm_simp_tac (ZF_ss addsimps [linear_def, Memrel_iff,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    22
				  [ltI, prem] MRS lt_trans2 RS ltD]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    23
by (REPEAT (resolve_tac [ballI, Ord_linear] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    24
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    25
qed "le_well_ord_Memrel";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    26
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    27
(*"Ord(i) ==> well_ord(i, Memrel(i))"*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    28
bind_thm ("well_ord_Memrel", le_refl RS le_well_ord_Memrel);
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    29
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    30
(*Kunen's Theorem 7.3 (i), page 16;  see also Ordinal/Ord_in_Ord
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    31
  The smaller ordinal is an initial segment of the larger *)
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    32
goalw OrderType.thy [pred_def, lt_def]
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    33
    "!!i j. j<i ==> pred(i, j, Memrel(i)) = j";
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    34
by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    35
by (fast_tac (eq_cs addEs [Ord_trans]) 1);
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    36
qed "lt_pred_Memrel";
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    37
831
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
    38
goalw OrderType.thy [pred_def,Memrel_def] 
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    39
      "!!A x. x:A ==> pred(A, x, Memrel(A)) = A Int x";
831
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
    40
by (fast_tac eq_cs 1);
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
    41
qed "pred_Memrel";
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
    42
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    43
goal OrderType.thy
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    44
    "!!i. [| j<i;  f: ord_iso(i,Memrel(i),j,Memrel(j)) |] ==> R";
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    45
by (forward_tac [lt_pred_Memrel] 1);
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    46
by (etac ltE 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    47
by (rtac (well_ord_Memrel RS well_ord_iso_predE) 1 THEN
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    48
    assume_tac 3 THEN assume_tac 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    49
by (asm_full_simp_tac (ZF_ss addsimps [ord_iso_def]) 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    50
(*Combining the two simplifications causes looping*)
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    51
by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    52
by (fast_tac (ZF_cs addSEs [bij_is_fun RS apply_type] addEs [Ord_trans]) 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    53
qed "Ord_iso_implies_eq_lemma";
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    54
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    55
(*Kunen's Theorem 7.3 (ii), page 16.  Isomorphic ordinals are equal*)
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    56
goal OrderType.thy
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    57
    "!!i. [| Ord(i);  Ord(j);  f:  ord_iso(i,Memrel(i),j,Memrel(j))	\
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    58
\         |] ==> i=j";
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    59
by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    60
by (REPEAT (eresolve_tac [asm_rl, ord_iso_sym, Ord_iso_implies_eq_lemma] 1));
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    61
qed "Ord_iso_implies_eq";
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    62
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    63
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    64
(**** Ordermap and ordertype ****)
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
    65
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    66
goalw OrderType.thy [ordermap_def,ordertype_def]
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    67
    "ordermap(A,r) : A -> ordertype(A,r)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    68
by (rtac lam_type 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    69
by (rtac (lamI RS imageI) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    70
by (REPEAT (assume_tac 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
    71
qed "ordermap_type";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    72
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    73
(*** Unfolding of ordermap ***)
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    74
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    75
(*Useful for cardinality reasoning; see CardinalArith.ML*)
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    76
goalw OrderType.thy [ordermap_def, pred_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    77
    "!!r. [| wf[A](r);  x:A |] ==> \
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    78
\         ordermap(A,r) ` x = ordermap(A,r) `` pred(A,x,r)";
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    79
by (asm_simp_tac ZF_ss 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    80
by (etac (wfrec_on RS trans) 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    81
by (assume_tac 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    82
by (asm_simp_tac (ZF_ss addsimps [subset_iff, image_lam,
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    83
                                  vimage_singleton_iff]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
    84
qed "ordermap_eq_image";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    85
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
    86
(*Useful for rewriting PROVIDED pred is not unfolded until later!*)
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    87
goal OrderType.thy 
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    88
    "!!r. [| wf[A](r);  x:A |] ==> \
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    89
\         ordermap(A,r) ` x = {ordermap(A,r)`y . y : pred(A,x,r)}";
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    90
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, pred_subset, 
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
    91
				  ordermap_type RS image_fun]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
    92
qed "ordermap_pred_unfold";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    93
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    94
(*pred-unfolded version.  NOT suitable for rewriting -- loops!*)
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    95
val ordermap_unfold = rewrite_rule [pred_def] ordermap_pred_unfold;
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    96
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
    97
(*** Showing that ordermap, ordertype yield ordinals ***)
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    98
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
    99
fun ordermap_elim_tac i =
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   100
    EVERY [etac (ordermap_unfold RS equalityD1 RS subsetD RS RepFunE) i,
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   101
	   assume_tac (i+1),
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   102
	   assume_tac i];
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   103
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   104
goalw OrderType.thy [well_ord_def, tot_ord_def, part_ord_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   105
    "!!r. [| well_ord(A,r);  x:A |] ==> Ord(ordermap(A,r) ` x)";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   106
by (safe_tac ZF_cs);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   107
by (wf_on_ind_tac "x" [] 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   108
by (asm_simp_tac (ZF_ss addsimps [ordermap_pred_unfold]) 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   109
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   110
by (rewrite_goals_tac [pred_def,Transset_def]);
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   111
by (fast_tac ZF_cs 2);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   112
by (safe_tac ZF_cs);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   113
by (ordermap_elim_tac 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   114
by (fast_tac (ZF_cs addSEs [trans_onD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   115
qed "Ord_ordermap";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   116
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   117
goalw OrderType.thy [ordertype_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   118
    "!!r. well_ord(A,r) ==> Ord(ordertype(A,r))";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   119
by (rtac ([ordermap_type, subset_refl] MRS image_fun RS ssubst) 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   120
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   121
by (fast_tac (ZF_cs addIs [Ord_ordermap]) 2);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   122
by (rewrite_goals_tac [Transset_def,well_ord_def]);
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   123
by (safe_tac ZF_cs);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   124
by (ordermap_elim_tac 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   125
by (fast_tac ZF_cs 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   126
qed "Ord_ordertype";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   127
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   128
(*** ordermap preserves the orderings in both directions ***)
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   129
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   130
goal OrderType.thy
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   131
    "!!r. [| <w,x>: r;  wf[A](r);  w: A; x: A |] ==>	\
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   132
\         ordermap(A,r)`w : ordermap(A,r)`x";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   133
by (eres_inst_tac [("x1", "x")] (ordermap_unfold RS ssubst) 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   134
by (assume_tac 1);
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   135
by (fast_tac ZF_cs 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   136
qed "ordermap_mono";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   137
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   138
(*linearity of r is crucial here*)
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   139
goalw OrderType.thy [well_ord_def, tot_ord_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   140
    "!!r. [| ordermap(A,r)`w : ordermap(A,r)`x;  well_ord(A,r);  \
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   141
\            w: A; x: A |] ==> <w,x>: r";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   142
by (safe_tac ZF_cs);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   143
by (linear_case_tac 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   144
by (fast_tac (ZF_cs addSEs [mem_not_refl RS notE]) 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   145
by (dtac ordermap_mono 1);
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   146
by (REPEAT_SOME assume_tac);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   147
by (etac mem_asym 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   148
by (assume_tac 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   149
qed "converse_ordermap_mono";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   150
803
4c8333ab3eae changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents: 788
diff changeset
   151
bind_thm ("ordermap_surj", 
4c8333ab3eae changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents: 788
diff changeset
   152
	  rewrite_rule [symmetric ordertype_def] 
4c8333ab3eae changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents: 788
diff changeset
   153
	      (ordermap_type RS surj_image));
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   154
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   155
goalw OrderType.thy [well_ord_def, tot_ord_def, bij_def, inj_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   156
    "!!r. well_ord(A,r) ==> ordermap(A,r) : bij(A, ordertype(A,r))";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   157
by (safe_tac ZF_cs);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   158
by (rtac ordermap_type 1);
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   159
by (rtac ordermap_surj 2);
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   160
by (linear_case_tac 1);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   161
(*The two cases yield similar contradictions*)
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   162
by (ALLGOALS (dtac ordermap_mono));
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   163
by (REPEAT_SOME assume_tac);
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   164
by (ALLGOALS (asm_full_simp_tac (ZF_ss addsimps [mem_not_refl])));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   165
qed "ordermap_bij";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   166
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   167
(*** Isomorphisms involving ordertype ***)
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   168
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   169
goalw OrderType.thy [ord_iso_def]
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   170
 "!!r. well_ord(A,r) ==> \
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   171
\      ordermap(A,r) : ord_iso(A,r, ordertype(A,r), Memrel(ordertype(A,r)))";
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   172
by (safe_tac ZF_cs);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   173
by (rtac ordermap_bij 1);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   174
by (assume_tac 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   175
by (fast_tac (ZF_cs addSEs [MemrelE, converse_ordermap_mono]) 2);
437
435875e4b21d modifications for cardinal arithmetic
lcp
parents: 435
diff changeset
   176
by (rewtac well_ord_def);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   177
by (fast_tac (ZF_cs addSIs [MemrelI, ordermap_mono,
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   178
			    ordermap_type RS apply_type]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   179
qed "ordertype_ord_iso";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   180
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   181
goal OrderType.thy
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   182
    "!!f. [| f: ord_iso(A,r,B,s);  well_ord(B,s) |] ==>	\
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   183
\    ordertype(A,r) = ordertype(B,s)";
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   184
by (forward_tac [well_ord_ord_iso] 1 THEN assume_tac 1);
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   185
by (resolve_tac [Ord_iso_implies_eq] 1
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   186
    THEN REPEAT (eresolve_tac [Ord_ordertype] 1));
831
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
   187
by (deepen_tac (ZF_cs addIs  [ord_iso_trans, ord_iso_sym]
60d850cc5fe6 Added Krzysztof's theorem pred_Memrel
lcp
parents: 814
diff changeset
   188
                      addSEs [ordertype_ord_iso]) 0 1);
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   189
qed "ordertype_eq";
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   190
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   191
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   192
    "!!A B. [| ordertype(A,r) = ordertype(B,s);	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   193
\              well_ord(A,r);  well_ord(B,s) \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   194
\           |] ==> EX f. f: ord_iso(A,r,B,s)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   195
by (resolve_tac [exI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   196
by (resolve_tac [ordertype_ord_iso RS ord_iso_trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   197
by (assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   198
by (eresolve_tac [ssubst] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   199
by (eresolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   200
qed "ordertype_eq_imp_ord_iso";
435
ca5356bd315a Addition of cardinals and order types, various tidying
lcp
parents:
diff changeset
   201
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   202
(*** Basic equalities for ordertype ***)
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   203
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   204
(*Ordertype of Memrel*)
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   205
goal OrderType.thy "!!i. j le i ==> ordertype(j,Memrel(i)) = j";
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   206
by (resolve_tac [Ord_iso_implies_eq RS sym] 1);
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   207
by (eresolve_tac [ltE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   208
by (REPEAT (ares_tac [le_well_ord_Memrel, Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   209
by (resolve_tac [ord_iso_trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   210
by (eresolve_tac [le_well_ord_Memrel RS ordertype_ord_iso] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   211
by (resolve_tac [id_bij RS ord_isoI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   212
by (asm_simp_tac (ZF_ss addsimps [id_conv, Memrel_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   213
by (fast_tac (ZF_cs addEs [ltE, Ord_in_Ord, Ord_trans]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   214
qed "le_ordertype_Memrel";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   215
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   216
(*"Ord(i) ==> ordertype(i, Memrel(i)) = i"*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   217
bind_thm ("ordertype_Memrel", le_refl RS le_ordertype_Memrel);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   218
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   219
goal OrderType.thy "ordertype(0,r) = 0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   220
by (resolve_tac [id_bij RS ord_isoI RS ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   221
by (etac emptyE 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   222
by (resolve_tac [well_ord_0] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   223
by (resolve_tac [Ord_0 RS ordertype_Memrel] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   224
qed "ordertype_0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   225
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   226
(*Ordertype of rvimage:  [| f: bij(A,B);  well_ord(B,s) |] ==>
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   227
                         ordertype(A, rvimage(A,f,s)) = ordertype(B,s) *)
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   228
bind_thm ("bij_ordertype_vimage", ord_iso_rvimage RS ordertype_eq);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   229
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   230
(*** A fundamental unfolding law for ordertype. ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   231
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   232
(*Ordermap returns the same result if applied to an initial segment*)
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   233
goal OrderType.thy
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   234
    "!!r. [| well_ord(A,r);  y:A;  z: pred(A,y,r) |] ==>	\
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   235
\	  ordermap(pred(A,y,r), r) ` z = ordermap(A, r) ` z";
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   236
by (forward_tac [[well_ord_is_wf, pred_subset] MRS wf_on_subset_A] 1);
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   237
by (wf_on_ind_tac "z" [] 1);
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   238
by (safe_tac (ZF_cs addSEs [predE]));
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   239
by (asm_simp_tac
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   240
    (ZF_ss addsimps [ordermap_pred_unfold, well_ord_is_wf, pred_iff]) 1);
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   241
(*combining these two simplifications LOOPS! *)
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   242
by (asm_simp_tac (ZF_ss addsimps [pred_pred_eq]) 1);
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   243
by (asm_full_simp_tac (ZF_ss addsimps [pred_def]) 1);
807
3abd026e68a4 ran expandshort script
lcp
parents: 803
diff changeset
   244
by (rtac (refl RSN (2,RepFun_cong)) 1);
3abd026e68a4 ran expandshort script
lcp
parents: 803
diff changeset
   245
by (dtac well_ord_is_trans_on 1);
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   246
by (fast_tac (eq_cs addSEs [trans_onD]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 467
diff changeset
   247
qed "ordermap_pred_eq_ordermap";
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   248
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   249
goalw OrderType.thy [ordertype_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   250
    "ordertype(A,r) = {ordermap(A,r)`y . y : A}";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   251
by (rtac ([ordermap_type, subset_refl] MRS image_fun) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   252
qed "ordertype_unfold";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   253
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   254
(** Theorems by Krzysztof Grabczewski; proofs simplified by lcp **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   255
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   256
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   257
    "!!r. [| well_ord(A,r);  x:A |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   258
\         ordertype(pred(A,x,r),r) <= ordertype(A,r)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   259
by (asm_simp_tac (ZF_ss addsimps [ordertype_unfold, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   260
                  pred_subset RSN (2, well_ord_subset)]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   261
by (fast_tac (ZF_cs addIs [ordermap_pred_eq_ordermap, RepFun_eqI]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   262
	            addEs [predE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   263
qed "ordertype_pred_subset";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   264
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   265
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   266
    "!!r. [| well_ord(A,r);  x:A |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   267
\         ordertype(pred(A,x,r),r) < ordertype(A,r)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   268
by (resolve_tac [ordertype_pred_subset RS subset_imp_le RS leE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   269
by (REPEAT (ares_tac [Ord_ordertype, well_ord_subset, pred_subset] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   270
by (eresolve_tac [sym RS ordertype_eq_imp_ord_iso RS exE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   271
by (eresolve_tac [well_ord_iso_predE] 3);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   272
by (REPEAT (ares_tac [pred_subset, well_ord_subset] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   273
qed "ordertype_pred_lt";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   274
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   275
(*May rewrite with this -- provided no rules are supplied for proving that
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   276
 	well_ord(pred(A,x,r), r) *)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   277
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   278
    "!!A r. well_ord(A,r) ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   279
\           ordertype(A,r) = {ordertype(pred(A,x,r),r). x:A}";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   280
by (safe_tac (eq_cs addSIs [ordertype_pred_lt RS ltD]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   281
by (asm_full_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   282
    (ZF_ss addsimps [ordertype_def, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   283
		     ordermap_bij RS bij_is_fun RS image_fun]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   284
by (eresolve_tac [RepFunE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   285
by (asm_full_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   286
    (ZF_ss addsimps [well_ord_is_wf, ordermap_eq_image, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   287
		     ordermap_type RS image_fun, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   288
		     ordermap_pred_eq_ordermap, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   289
		     pred_subset, subset_refl]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   290
by (eresolve_tac [RepFunI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   291
qed "ordertype_pred_unfold";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   292
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   293
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   294
(**** Alternative definition of ordinal ****)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   295
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   296
(*proof by Krzysztof Grabczewski*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   297
goalw OrderType.thy [Ord_alt_def] "!!i. Ord(i) ==> Ord_alt(i)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   298
by (resolve_tac [conjI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   299
by (eresolve_tac [well_ord_Memrel] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   300
by (rewrite_goals_tac [Ord_def, Transset_def, pred_def, Memrel_def]);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   301
by (fast_tac eq_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   302
qed "Ord_is_Ord_alt";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   303
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   304
(*proof by lcp*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   305
goalw OrderType.thy [Ord_alt_def, Ord_def, Transset_def, well_ord_def, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   306
		     tot_ord_def, part_ord_def, trans_on_def] 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   307
    "!!i. Ord_alt(i) ==> Ord(i)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   308
by (asm_full_simp_tac (ZF_ss addsimps [Memrel_iff, pred_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   309
by (safe_tac ZF_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   310
by (fast_tac (ZF_cs addSDs [equalityD1]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   311
by (subgoal_tac "xa: i" 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   312
by (fast_tac (ZF_cs addSDs [equalityD1]) 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   313
by (fast_tac (ZF_cs addSDs [equalityD1]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   314
                    addSEs [bspec RS bspec RS bspec RS mp RS mp]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   315
qed "Ord_alt_is_Ord";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   316
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   317
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   318
(**** Ordinal Addition ****)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   319
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   320
(*** Order Type calculations for radd ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   321
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   322
(** Addition with 0 **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   323
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   324
goal OrderType.thy "(lam z:A+0. case(%x.x, %y.y, z)) : bij(A+0, A)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   325
by (res_inst_tac [("d", "Inl")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   326
by (safe_tac sum_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   327
by (ALLGOALS (asm_simp_tac sum_ss));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   328
qed "bij_sum_0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   329
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   330
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   331
 "!!A r. well_ord(A,r) ==> ordertype(A+0, radd(A,r,0,s)) = ordertype(A,r)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   332
by (resolve_tac [bij_sum_0 RS ord_isoI RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   333
by (assume_tac 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   334
by (asm_simp_tac ZF_ss 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   335
by (REPEAT_FIRST (eresolve_tac [sumE, emptyE]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   336
by (asm_simp_tac (sum_ss addsimps [radd_Inl_iff, Memrel_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   337
qed "ordertype_sum_0_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   338
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   339
goal OrderType.thy "(lam z:0+A. case(%x.x, %y.y, z)) : bij(0+A, A)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   340
by (res_inst_tac [("d", "Inr")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   341
by (safe_tac sum_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   342
by (ALLGOALS (asm_simp_tac sum_ss));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   343
qed "bij_0_sum";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   344
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   345
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   346
 "!!A r. well_ord(A,r) ==> ordertype(0+A, radd(0,s,A,r)) = ordertype(A,r)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   347
by (resolve_tac [bij_0_sum RS ord_isoI RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   348
by (assume_tac 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   349
by (asm_simp_tac ZF_ss 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   350
by (REPEAT_FIRST (eresolve_tac [sumE, emptyE]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   351
by (asm_simp_tac (sum_ss addsimps [radd_Inr_iff, Memrel_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   352
qed "ordertype_0_sum_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   353
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   354
(** Initial segments of radd.  Statements by Grabczewski **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   355
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   356
(*In fact, pred(A+B, Inl(a), radd(A,r,B,s)) = pred(A,a,r)+0 *)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   357
goalw OrderType.thy [pred_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   358
 "!!A B. a:A ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   359
\        (lam x:pred(A,a,r). Inl(x))	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   360
\        : bij(pred(A,a,r), pred(A+B, Inl(a), radd(A,r,B,s)))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   361
by (res_inst_tac [("d", "case(%x.x, %y.y)")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   362
by (safe_tac sum_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   363
by (ALLGOALS
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   364
    (asm_full_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   365
     (sum_ss addsimps [radd_Inl_iff, radd_Inr_Inl_iff])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   366
qed "pred_Inl_bij";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   367
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   368
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   369
 "!!A B. [| a:A;  well_ord(A,r) |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   370
\        ordertype(pred(A+B, Inl(a), radd(A,r,B,s)), radd(A,r,B,s)) = \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   371
\        ordertype(pred(A,a,r), r)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   372
by (resolve_tac [pred_Inl_bij RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   373
by (REPEAT_FIRST (ares_tac [pred_subset, well_ord_subset]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   374
by (asm_full_simp_tac (ZF_ss addsimps [radd_Inl_iff, pred_def]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   375
qed "ordertype_pred_Inl_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   376
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   377
goalw OrderType.thy [pred_def, id_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   378
 "!!A B. b:B ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   379
\        id(A+pred(B,b,s))	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   380
\        : bij(A+pred(B,b,s), pred(A+B, Inr(b), radd(A,r,B,s)))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   381
by (res_inst_tac [("d", "%z.z")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   382
by (safe_tac sum_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   383
by (ALLGOALS (asm_full_simp_tac radd_ss));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   384
qed "pred_Inr_bij";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   385
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   386
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   387
 "!!A B. [| b:B;  well_ord(A,r);  well_ord(B,s) |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   388
\        ordertype(pred(A+B, Inr(b), radd(A,r,B,s)), radd(A,r,B,s)) = \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   389
\        ordertype(A+pred(B,b,s), radd(A,r,pred(B,b,s),s))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   390
by (resolve_tac [pred_Inr_bij RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   391
by (REPEAT_FIRST (ares_tac [well_ord_radd, pred_subset, well_ord_subset]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   392
by (asm_full_simp_tac (ZF_ss addsimps [pred_def, id_def]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   393
by (REPEAT_FIRST (eresolve_tac [sumE]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   394
by (ALLGOALS (asm_simp_tac radd_ss));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   395
qed "ordertype_pred_Inr_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   396
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   397
(*** Basic laws for ordinal addition ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   398
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   399
goalw OrderType.thy [oadd_def] 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   400
    "!!i j. [| Ord(i);  Ord(j) |] ==> Ord(i++j)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   401
by (REPEAT (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   402
qed "Ord_oadd";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   403
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   404
(** Ordinal addition with zero **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   405
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   406
goalw OrderType.thy [oadd_def] "!!i. Ord(i) ==> i++0 = i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   407
by (asm_simp_tac (ZF_ss addsimps [Memrel_0, ordertype_sum_0_eq, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   408
				  ordertype_Memrel, well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   409
qed "oadd_0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   410
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   411
goalw OrderType.thy [oadd_def] "!!i. Ord(i) ==> 0++i = i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   412
by (asm_simp_tac (ZF_ss addsimps [Memrel_0, ordertype_0_sum_eq, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   413
				  ordertype_Memrel, well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   414
qed "oadd_0_left";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   415
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   416
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   417
(*** Further properties of ordinal addition.  Statements by Grabczewski,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   418
    proofs by lcp. ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   419
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   420
goalw OrderType.thy [oadd_def] "!!i j k. [| k<i;  Ord(j) |] ==> k < i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   421
by (resolve_tac [ltE] 1 THEN assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   422
by (resolve_tac [ltI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   423
by (REPEAT (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel] 2));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   424
by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_unfold, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   425
				  well_ord_radd, well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   426
by (resolve_tac [RepFun_eqI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   427
by (eresolve_tac [InlI] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   428
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   429
    (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   430
		     lt_pred_Memrel, leI RS le_ordertype_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   431
qed "lt_oadd1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   432
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   433
goal OrderType.thy "!!i j. [| Ord(i);  Ord(j) |] ==> i le i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   434
by (resolve_tac [all_lt_imp_le] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   435
by (REPEAT (ares_tac [Ord_oadd, lt_oadd1] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   436
qed "oadd_le_self";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   437
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   438
(** A couple of strange but necessary results! **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   439
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   440
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   441
    "!!A B. A<=B ==> id(A) : ord_iso(A, Memrel(A), A, Memrel(B))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   442
by (resolve_tac [id_bij RS ord_isoI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   443
by (asm_simp_tac (ZF_ss addsimps [id_conv, Memrel_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   444
by (fast_tac ZF_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   445
qed "id_ord_iso_Memrel";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   446
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   447
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   448
    "!!k. [| well_ord(A,r);  k<j |] ==>			\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   449
\            ordertype(A+k, radd(A, r, k, Memrel(j))) =	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   450
\            ordertype(A+k, radd(A, r, k, Memrel(k)))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   451
by (eresolve_tac [ltE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   452
by (resolve_tac [ord_iso_refl RS sum_ord_iso_cong RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   453
by (eresolve_tac [OrdmemD RS id_ord_iso_Memrel RS ord_iso_sym] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   454
by (REPEAT_FIRST (ares_tac [well_ord_radd, well_ord_Memrel]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   455
qed "ordertype_sum_Memrel";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   456
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   457
goalw OrderType.thy [oadd_def] "!!i j k. [| k<j;  Ord(i) |] ==> i++k < i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   458
by (resolve_tac [ltE] 1 THEN assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   459
by (resolve_tac [ordertype_pred_unfold RS equalityD2 RS subsetD RS ltI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   460
by (REPEAT_FIRST (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   461
by (resolve_tac [RepFun_eqI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   462
by (eresolve_tac [InrI] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   463
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   464
    (ZF_ss addsimps [ordertype_pred_Inr_eq, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   465
		     lt_pred_Memrel, leI RS le_ordertype_Memrel,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   466
		     ordertype_sum_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   467
qed "oadd_lt_mono2";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   468
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   469
goal OrderType.thy "!!i j. [| i++j = i++k; Ord(i); Ord(j); Ord(k) |] ==> j=k";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   470
by (rtac Ord_linear_lt 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   471
by (REPEAT_SOME assume_tac);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   472
by (ALLGOALS
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   473
    (dresolve_tac [oadd_lt_mono2] THEN' assume_tac THEN'
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   474
     asm_full_simp_tac (ZF_ss addsimps [lt_not_refl])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   475
qed "oadd_inject";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   476
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   477
goalw OrderType.thy [oadd_def] 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   478
    "!!i j k. [| k < i++j;  Ord(i);  Ord(j) |] ==> k<i | (EX l:j. k = i++l )";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   479
(*Rotate the hypotheses so that simplification will work*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   480
by (etac revcut_rl 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   481
by (asm_full_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   482
    (ZF_ss addsimps [ordertype_pred_unfold, well_ord_radd,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   483
		     well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   484
by (eresolve_tac [ltD RS RepFunE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   485
by (eresolve_tac [sumE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   486
by (asm_simp_tac
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   487
    (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   488
		     ltI, lt_pred_Memrel, le_ordertype_Memrel, leI]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   489
by (asm_simp_tac
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   490
    (ZF_ss addsimps [ordertype_pred_Inr_eq, well_ord_Memrel,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   491
		     ltI, lt_pred_Memrel, ordertype_sum_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   492
by (fast_tac ZF_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   493
qed "lt_oadd_disj";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   494
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   495
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   496
(*** Ordinal addition with successor -- via associativity! ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   497
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   498
goalw OrderType.thy [oadd_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   499
    "!!i j k. [| Ord(i);  Ord(j);  Ord(k) |] ==> (i++j)++k = i++(j++k)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   500
by (resolve_tac [ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   501
by (rtac ([ordertype_ord_iso RS ord_iso_sym, ord_iso_refl] MRS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   502
	  sum_ord_iso_cong) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   503
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   504
by (resolve_tac [sum_assoc_ord_iso RS ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   505
by (rtac ([ord_iso_refl, ordertype_ord_iso] MRS sum_ord_iso_cong RS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   506
	  ordertype_eq) 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   507
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   508
qed "oadd_assoc";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   509
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   510
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   511
    "!!i j. [| Ord(i);  Ord(j) |] ==> i++j = i Un (UN k:j. {i++k})";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   512
by (rtac (subsetI RS equalityI) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   513
by (eresolve_tac [ltI RS lt_oadd_disj RS disjE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   514
by (REPEAT (ares_tac [Ord_oadd] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   515
by (fast_tac (ZF_cs addSEs [ltE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   516
by (fast_tac ZF_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   517
by (safe_tac ZF_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   518
by (ALLGOALS
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   519
    (asm_full_simp_tac (ZF_ss addsimps [Ord_mem_iff_lt, Ord_oadd])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   520
by (fast_tac (ZF_cs addIs [lt_oadd1]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   521
by (fast_tac (ZF_cs addIs [oadd_lt_mono2]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   522
qed "oadd_unfold";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   523
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   524
goal OrderType.thy "!!i. Ord(i) ==> i++1 = succ(i)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   525
by (asm_simp_tac (ZF_ss addsimps [oadd_unfold, Ord_1, oadd_0]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   526
by (fast_tac eq_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   527
qed "oadd_1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   528
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   529
goal OrderType.thy
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   530
    "!!i. [| Ord(i);  Ord(j) |] ==> i++succ(j) = succ(i++j)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   531
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   532
    (ZF_ss addsimps [oadd_1 RS sym, Ord_oadd, oadd_assoc, Ord_1]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   533
qed "oadd_succ";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   534
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   535
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   536
(** Ordinal addition with limit ordinals **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   537
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   538
val prems = goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   539
    "[| Ord(i);  !!x. x:A ==> Ord(j(x));  a:A |] ==> \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   540
\    i ++ (UN x:A. j(x)) = (UN x:A. i++j(x))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   541
by (fast_tac (eq_cs addIs (prems @ [ltI, Ord_UN, Ord_oadd, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   542
				    lt_oadd1 RS ltD, oadd_lt_mono2 RS ltD])
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   543
                     addSEs [ltE, ltI RS lt_oadd_disj RS disjE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   544
qed "oadd_UN";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   545
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   546
goal OrderType.thy 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   547
    "!!i j. [| Ord(i);  Limit(j) |] ==> i++j = (UN k:j. i++k)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   548
by (forward_tac [Limit_has_0 RS ltD] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   549
by (asm_simp_tac (ZF_ss addsimps [Limit_is_Ord RS Ord_in_Ord,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   550
				  oadd_UN RS sym, Union_eq_UN RS sym, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   551
				  Limit_Union_eq]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   552
qed "oadd_Limit";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   553
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   554
(** Order/monotonicity properties of ordinal addition **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   555
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   556
goal OrderType.thy "!!i j. [| Ord(i);  Ord(j) |] ==> i le j++i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   557
by (eres_inst_tac [("i","i")] trans_induct3 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   558
by (asm_simp_tac (ZF_ss addsimps [oadd_0, Ord_0_le]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   559
by (asm_simp_tac (ZF_ss addsimps [oadd_succ, succ_leI]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   560
by (asm_simp_tac (ZF_ss addsimps [oadd_Limit]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   561
by (resolve_tac [le_trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   562
by (resolve_tac [le_implies_UN_le_UN] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   563
by (fast_tac ZF_cs 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   564
by (asm_simp_tac (ZF_ss addsimps [Union_eq_UN RS sym, Limit_Union_eq, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   565
				  le_refl, Limit_is_Ord]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   566
qed "oadd_le_self2";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   567
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   568
goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> k++i le j++i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   569
by (forward_tac [lt_Ord] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   570
by (forward_tac [le_Ord2] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   571
by (eresolve_tac [trans_induct3] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   572
by (asm_simp_tac (ZF_ss addsimps [oadd_0]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   573
by (asm_simp_tac (ZF_ss addsimps [oadd_succ, succ_le_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   574
by (asm_simp_tac (ZF_ss addsimps [oadd_Limit]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   575
by (resolve_tac [le_implies_UN_le_UN] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   576
by (fast_tac ZF_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   577
qed "oadd_le_mono1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   578
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   579
goal OrderType.thy "!!i j. [| i' le i;  j'<j |] ==> i'++j' < i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   580
by (resolve_tac [lt_trans1] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   581
by (REPEAT (eresolve_tac [asm_rl, oadd_le_mono1, oadd_lt_mono2, ltE,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   582
			  Ord_succD] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   583
qed "oadd_lt_mono";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   584
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   585
goal OrderType.thy "!!i j. [| i' le i;  j' le j |] ==> i'++j' le i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   586
by (asm_simp_tac (ZF_ss addsimps [oadd_succ RS sym, le_Ord2, oadd_lt_mono]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   587
qed "oadd_le_mono";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   588
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   589
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   590
(** Ordinal subtraction; the difference is ordertype(j-i, Memrel(j)). 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   591
    Probably simpler to define the difference recursively!
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   592
**)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   593
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   594
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   595
    "!!A B. A<=B ==> (lam y:B. if(y:A, Inl(y), Inr(y))) : bij(B, A+(B-A))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   596
by (res_inst_tac [("d", "case(%x.x, %y.y)")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   597
by (fast_tac (sum_cs addSIs [if_type]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   598
by (fast_tac (ZF_cs addSIs [case_type]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   599
by (eresolve_tac [sumE] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   600
by (ALLGOALS (asm_simp_tac (sum_ss setloop split_tac [expand_if])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   601
qed "bij_sum_Diff";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   602
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   603
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   604
    "!!i j. i le j ==>	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   605
\           ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j))) = 	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   606
\           ordertype(j, Memrel(j))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   607
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   608
by (resolve_tac [bij_sum_Diff RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   609
by (eresolve_tac [well_ord_Memrel] 3);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   610
by (assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   611
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   612
     (radd_ss setloop split_tac [expand_if] addsimps [Memrel_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   613
by (forw_inst_tac [("j", "y")] Ord_in_Ord 1 THEN assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   614
by (forw_inst_tac [("j", "x")] Ord_in_Ord 1 THEN assume_tac 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   615
by (asm_simp_tac (ZF_ss addsimps [Ord_mem_iff_lt, lt_Ord, not_lt_iff_le]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   616
by (fast_tac (ZF_cs addEs [lt_trans2, lt_trans]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   617
qed "ordertype_sum_Diff";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   618
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   619
goalw OrderType.thy [oadd_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   620
    "!!i j. i le j ==> 	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   621
\           i ++ ordertype(j-i, Memrel(j)) = \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   622
\           ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j)))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   623
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   624
by (resolve_tac [sum_ord_iso_cong RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   625
by (eresolve_tac [id_ord_iso_Memrel] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   626
by (resolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   627
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel RS well_ord_subset,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   628
		      Diff_subset] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   629
qed "oadd_ordertype_Diff";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   630
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   631
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   632
    "!!i j. i le j ==> i ++ ordertype(j-i, Memrel(j)) = j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   633
by (asm_simp_tac (ZF_ss addsimps [oadd_ordertype_Diff, ordertype_sum_Diff, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   634
				  ordertype_Memrel, lt_Ord2 RS Ord_succD]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   635
qed "oadd_inverse";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   636
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   637
(*By oadd_inject, the difference between i and j is unique.*)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   638
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   639
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   640
(**** Ordinal Multiplication ****)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   641
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   642
goalw OrderType.thy [omult_def] 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   643
    "!!i j. [| Ord(i);  Ord(j) |] ==> Ord(i**j)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   644
by (REPEAT (ares_tac [Ord_ordertype, well_ord_rmult, well_ord_Memrel] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   645
qed "Ord_omult";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   646
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   647
(*** A useful unfolding law ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   648
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   649
goalw OrderType.thy [pred_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   650
 "!!A B. [| a:A;  b:B |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   651
\        pred(A*B, <a,b>, rmult(A,r,B,s)) =	\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   652
\        pred(A,a,r)*B Un ({a} * pred(B,b,s))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   653
by (safe_tac eq_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   654
by (ALLGOALS (asm_full_simp_tac (ZF_ss addsimps [rmult_iff])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   655
by (ALLGOALS (fast_tac ZF_cs));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   656
qed "pred_Pair_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   657
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   658
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   659
 "!!A B. [| a:A;  b:B;  well_ord(A,r);  well_ord(B,s) |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   660
\        ordertype(pred(A*B, <a,b>, rmult(A,r,B,s)), rmult(A,r,B,s)) = \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   661
\        ordertype(pred(A,a,r)*B + pred(B,b,s), 		\
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   662
\                 radd(A*B, rmult(A,r,B,s), B, s))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   663
by (asm_simp_tac (ZF_ss addsimps [pred_Pair_eq]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   664
by (resolve_tac [ordertype_eq RS sym] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   665
by (resolve_tac [prod_sum_singleton_ord_iso] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   666
by (REPEAT_FIRST (ares_tac [pred_subset, well_ord_rmult RS  well_ord_subset]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   667
by (fast_tac (ZF_cs addSEs [predE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   668
qed "ordertype_pred_Pair_eq";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   669
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   670
goalw OrderType.thy [oadd_def, omult_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   671
 "!!i j. [| i'<i;  j'<j |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   672
\        ordertype(pred(i*j, <i',j'>, rmult(i,Memrel(i),j,Memrel(j))), \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   673
\                  rmult(i,Memrel(i),j,Memrel(j))) = \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   674
\        j**i' ++ j'";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   675
by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_Pair_eq, lt_pred_Memrel, ltD, lt_Ord2, well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   676
by (resolve_tac [trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   677
by (resolve_tac [ordertype_ord_iso RS sum_ord_iso_cong RS ordertype_eq] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   678
by (resolve_tac [ord_iso_refl] 3);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   679
by (resolve_tac [id_bij RS ord_isoI RS ordertype_eq] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   680
by (REPEAT_FIRST (eresolve_tac [SigmaE, sumE, ltE, ssubst]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   681
by (REPEAT_FIRST (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   682
			    Ord_ordertype]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   683
by (ALLGOALS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   684
    (asm_simp_tac (radd_ss addsimps [rmult_iff, id_conv, Memrel_iff])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   685
by (safe_tac ZF_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   686
by (ALLGOALS (fast_tac (ZF_cs addEs [Ord_trans])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   687
qed "ordertype_pred_Pair_lemma";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   688
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   689
goalw OrderType.thy [omult_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   690
 "!!i j. [| Ord(i);  Ord(j);  k<j**i |] ==>  \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   691
\        EX j' i'. k = j**i' ++ j' & j'<j & i'<i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   692
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   693
				       well_ord_rmult, well_ord_Memrel]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   694
by (step_tac (ZF_cs addSEs [ltE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   695
by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   696
				  symmetric omult_def]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   697
by (fast_tac (ZF_cs addIs [ltI]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   698
qed "lt_omult";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   699
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   700
goalw OrderType.thy [omult_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   701
 "!!i j. [| j'<j;  i'<i |] ==> j**i' ++ j'  <  j**i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   702
by (resolve_tac [ltI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   703
by (asm_full_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   704
    (ZF_ss addsimps [ordertype_pred_unfold, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   705
		     well_ord_rmult, well_ord_Memrel, lt_Ord2]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   706
by (resolve_tac [RepFun_eqI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   707
by (fast_tac (ZF_cs addSEs [ltE]) 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   708
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   709
    (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI, symmetric omult_def]) 1);
814
a32b420c33d4 Moved well_ord_Memrel, lt_eq_pred, Ord_iso_implies_eq_lemma,
lcp
parents: 807
diff changeset
   710
by (asm_simp_tac
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   711
    (ZF_ss addsimps [Ord_ordertype, well_ord_rmult, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   712
		     lt_Ord2]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   713
qed "omult_oadd_lt";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   714
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   715
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   716
 "!!i j. [| Ord(i);  Ord(j) |] ==> j**i = (UN j':j. UN i':i. {j**i' ++ j'})";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   717
by (rtac (subsetI RS equalityI) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   718
by (resolve_tac [lt_omult RS exE] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   719
by (eresolve_tac [ltI] 3);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   720
by (REPEAT (ares_tac [Ord_omult] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   721
by (fast_tac (ZF_cs addSEs [ltE]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   722
by (fast_tac (ZF_cs addIs [omult_oadd_lt RS ltD, ltI]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   723
qed "omult_unfold";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   724
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   725
(*** Basic laws for ordinal multiplication ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   726
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   727
(** Ordinal multiplication by zero **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   728
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   729
goalw OrderType.thy [omult_def] "i**0 = 0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   730
by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   731
qed "omult_0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   732
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   733
goalw OrderType.thy [omult_def] "0**i = 0";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   734
by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   735
qed "omult_0_left";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   736
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   737
(** Ordinal multiplication by 1 **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   738
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   739
goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> i**1 = i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   740
by (resolve_tac [ord_isoI RS ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   741
by (res_inst_tac [("c", "snd"), ("d", "%z.<0,z>")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   742
by (REPEAT_FIRST (eresolve_tac [snd_type, SigmaE, succE, emptyE, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   743
				well_ord_Memrel, ordertype_Memrel]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   744
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   745
qed "omult_1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   746
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   747
goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> 1**i = i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   748
by (resolve_tac [ord_isoI RS ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   749
by (res_inst_tac [("c", "fst"), ("d", "%z.<z,0>")] lam_bijective 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   750
by (REPEAT_FIRST (eresolve_tac [fst_type, SigmaE, succE, emptyE, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   751
				well_ord_Memrel, ordertype_Memrel]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   752
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   753
qed "omult_1_left";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   754
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   755
(** Distributive law for ordinal multiplication and addition **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   756
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   757
goalw OrderType.thy [omult_def, oadd_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   758
    "!!i. [| Ord(i);  Ord(j);  Ord(k) |] ==> i**(j++k) = (i**j)++(i**k)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   759
by (resolve_tac [ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   760
by (rtac ([ordertype_ord_iso RS ord_iso_sym, ord_iso_refl] MRS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   761
	  prod_ord_iso_cong) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   762
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   763
		      Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   764
by (rtac (sum_prod_distrib_ord_iso RS ordertype_eq RS trans) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   765
by (rtac ordertype_eq 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   766
by (rtac ([ordertype_ord_iso, ordertype_ord_iso] MRS sum_ord_iso_cong) 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   767
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   768
		      Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   769
qed "oadd_omult_distrib";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   770
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   771
goal OrderType.thy "!!i. [| Ord(i);  Ord(j) |] ==> i**succ(j) = (i**j)++i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   772
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   773
    (ZF_ss addsimps [oadd_1 RS sym, omult_1, oadd_omult_distrib, Ord_1]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   774
qed "omult_succ";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   775
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   776
(** Associative law **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   777
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   778
goalw OrderType.thy [omult_def]
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   779
    "!!i j k. [| Ord(i);  Ord(j);  Ord(k) |] ==> (i**j)**k = i**(j**k)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   780
by (resolve_tac [ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   781
by (rtac ([ord_iso_refl, ordertype_ord_iso RS ord_iso_sym] MRS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   782
	  prod_ord_iso_cong) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   783
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   784
by (resolve_tac [prod_assoc_ord_iso RS ord_iso_sym RS 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   785
		 ordertype_eq RS trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   786
by (rtac ([ordertype_ord_iso, ord_iso_refl] MRS prod_ord_iso_cong RS
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   787
	  ordertype_eq) 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   788
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Ord_ordertype] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   789
qed "omult_assoc";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   790
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   791
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   792
(** Ordinal multiplication with limit ordinals **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   793
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   794
val prems = goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   795
    "[| Ord(i);  !!x. x:A ==> Ord(j(x)) |] ==> \
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   796
\    i ** (UN x:A. j(x)) = (UN x:A. i**j(x))";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   797
by (asm_simp_tac (ZF_ss addsimps (prems@[Ord_UN, omult_unfold])) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   798
by (fast_tac eq_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   799
qed "omult_UN";
467
92868dab2939 new cardinal arithmetic developments
lcp
parents: 437
diff changeset
   800
849
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   801
goal OrderType.thy 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   802
    "!!i j. [| Ord(i);  Limit(j) |] ==> i**j = (UN k:j. i**k)";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   803
by (asm_simp_tac 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   804
    (ZF_ss addsimps [Limit_is_Ord RS Ord_in_Ord, omult_UN RS sym, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   805
		     Union_eq_UN RS sym, Limit_Union_eq]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   806
qed "omult_Limit";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   807
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   808
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   809
(*** Ordering/monotonicity properties of ordinal multiplication ***)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   810
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   811
(*As a special case we have "[| 0<i;  0<j |] ==> 0 < i**j" *)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   812
goal OrderType.thy "!!i j. [| k<i;  0<j |] ==> k < i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   813
by (safe_tac (ZF_cs addSEs [ltE] addSIs [ltI, Ord_omult]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   814
by (asm_simp_tac (ZF_ss addsimps [omult_unfold]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   815
by (REPEAT (eresolve_tac [UN_I] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   816
by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0_left]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   817
qed "lt_omult1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   818
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   819
goal OrderType.thy "!!i j. [| Ord(i);  0<j |] ==> i le i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   820
by (resolve_tac [all_lt_imp_le] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   821
by (REPEAT (ares_tac [Ord_omult, lt_omult1, lt_Ord2] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   822
qed "omult_le_self";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   823
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   824
goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> k**i le j**i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   825
by (forward_tac [lt_Ord] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   826
by (forward_tac [le_Ord2] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   827
by (eresolve_tac [trans_induct3] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   828
by (asm_simp_tac (ZF_ss addsimps [omult_0, le_refl, Ord_0]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   829
by (asm_simp_tac (ZF_ss addsimps [omult_succ, oadd_le_mono]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   830
by (asm_simp_tac (ZF_ss addsimps [omult_Limit]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   831
by (resolve_tac [le_implies_UN_le_UN] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   832
by (fast_tac ZF_cs 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   833
qed "omult_le_mono1";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   834
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   835
goal OrderType.thy "!!i j k. [| k<j;  0<i |] ==> i**k < i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   836
by (resolve_tac [ltI] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   837
by (asm_simp_tac (ZF_ss addsimps [omult_unfold, lt_Ord2]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   838
by (safe_tac (ZF_cs addSEs [ltE] addSIs [Ord_omult]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   839
by (REPEAT (eresolve_tac [UN_I] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   840
by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0, Ord_omult]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   841
qed "omult_lt_mono2";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   842
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   843
goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> i**k le i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   844
by (resolve_tac [subset_imp_le] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   845
by (safe_tac (ZF_cs addSEs [ltE, make_elim Ord_succD] addSIs [Ord_omult]));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   846
by (asm_full_simp_tac (ZF_ss addsimps [omult_unfold]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   847
by (safe_tac ZF_cs);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   848
by (eresolve_tac [UN_I] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   849
by (deepen_tac (ZF_cs addEs [Ord_trans]) 0 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   850
qed "omult_le_mono2";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   851
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   852
goal OrderType.thy "!!i j. [| i' le i;  j' le j |] ==> i'**j' le i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   853
by (resolve_tac [le_trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   854
by (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_le_mono2, ltE,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   855
			  Ord_succD] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   856
qed "omult_le_mono";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   857
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   858
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   859
      "!!i j. [| i' le i;  j'<j;  0<i |] ==> i'**j' < i**j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   860
by (resolve_tac [lt_trans1] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   861
by (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_lt_mono2, ltE,
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   862
			  Ord_succD] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   863
qed "omult_lt_mono";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   864
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   865
goal OrderType.thy
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   866
      "!!i j. [| i' le i;  j' le j |] ==> i'++j' le i++j";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   867
by (asm_simp_tac (ZF_ss addsimps [oadd_succ RS sym, le_Ord2, oadd_lt_mono]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   868
qed "oadd_le_mono";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   869
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   870
goal OrderType.thy "!!i j. [| Ord(i);  0<j |] ==> i le j**i";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   871
by (forward_tac [lt_Ord2] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   872
by (eres_inst_tac [("i","i")] trans_induct3 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   873
by (asm_simp_tac (ZF_ss addsimps [omult_0, Ord_0 RS le_refl]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   874
by (asm_simp_tac (ZF_ss addsimps [omult_succ, succ_le_iff]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   875
by (eresolve_tac [lt_trans1] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   876
by (res_inst_tac [("b", "j**x")] (oadd_0 RS subst) 1 THEN 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   877
    rtac oadd_lt_mono2 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   878
by (REPEAT (ares_tac [Ord_omult] 1));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   879
by (asm_simp_tac (ZF_ss addsimps [omult_Limit]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   880
by (resolve_tac [le_trans] 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   881
by (resolve_tac [le_implies_UN_le_UN] 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   882
by (fast_tac ZF_cs 2);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   883
by (asm_simp_tac (ZF_ss addsimps [Union_eq_UN RS sym, Limit_Union_eq, 
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   884
				  Limit_is_Ord RS le_refl]) 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   885
qed "omult_le_self2";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   886
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   887
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   888
(** Further properties of ordinal multiplication **)
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   889
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   890
goal OrderType.thy "!!i j. [| i**j = i**k;  0<i;  Ord(j);  Ord(k) |] ==> j=k";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   891
by (rtac Ord_linear_lt 1);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   892
by (REPEAT_SOME assume_tac);
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   893
by (ALLGOALS
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   894
    (dresolve_tac [omult_lt_mono2] THEN' assume_tac THEN'
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   895
     asm_full_simp_tac (ZF_ss addsimps [lt_not_refl])));
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   896
qed "omult_inject";
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   897
013a16d3addb Proved equivalence of Ord and Ord_alt. Proved
lcp
parents: 831
diff changeset
   898