src/HOL/Univ.ML
author paulson
Thu Apr 04 11:45:01 1996 +0200 (1996-04-04)
changeset 1642 21db0cf9a1a4
parent 1563 717f8816eca5
child 1760 6f41a494f3b1
permissions -rw-r--r--
Using new "Times" infix
clasohm@1465
     1
(*  Title:      HOL/univ
clasohm@923
     2
    ID:         $Id$
clasohm@1465
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@923
     4
    Copyright   1991  University of Cambridge
clasohm@923
     5
clasohm@923
     6
For univ.thy
clasohm@923
     7
*)
clasohm@923
     8
clasohm@923
     9
open Univ;
clasohm@923
    10
clasohm@923
    11
(** apfst -- can be used in similar type definitions **)
clasohm@923
    12
clasohm@972
    13
goalw Univ.thy [apfst_def] "apfst f (a,b) = (f(a),b)";
clasohm@923
    14
by (rtac split 1);
clasohm@976
    15
qed "apfst_conv";
clasohm@923
    16
clasohm@923
    17
val [major,minor] = goal Univ.thy
clasohm@972
    18
    "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R \
clasohm@923
    19
\    |] ==> R";
clasohm@923
    20
by (rtac PairE 1);
clasohm@923
    21
by (rtac minor 1);
clasohm@923
    22
by (assume_tac 1);
clasohm@923
    23
by (rtac (major RS trans) 1);
clasohm@923
    24
by (etac ssubst 1);
clasohm@976
    25
by (rtac apfst_conv 1);
clasohm@976
    26
qed "apfst_convE";
clasohm@923
    27
clasohm@923
    28
(** Push -- an injection, analogous to Cons on lists **)
clasohm@923
    29
clasohm@923
    30
val [major] = goalw Univ.thy [Push_def] "Push i f =Push j g  ==> i=j";
clasohm@923
    31
by (rtac (major RS fun_cong RS box_equals RS Suc_inject) 1);
clasohm@923
    32
by (rtac nat_case_0 1);
clasohm@923
    33
by (rtac nat_case_0 1);
clasohm@923
    34
qed "Push_inject1";
clasohm@923
    35
clasohm@923
    36
val [major] = goalw Univ.thy [Push_def] "Push i f =Push j g  ==> f=g";
clasohm@923
    37
by (rtac (major RS fun_cong RS ext RS box_equals) 1);
clasohm@923
    38
by (rtac (nat_case_Suc RS ext) 1);
clasohm@923
    39
by (rtac (nat_case_Suc RS ext) 1);
clasohm@923
    40
qed "Push_inject2";
clasohm@923
    41
clasohm@923
    42
val [major,minor] = goal Univ.thy
clasohm@923
    43
    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P \
clasohm@923
    44
\    |] ==> P";
clasohm@923
    45
by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1);
clasohm@923
    46
qed "Push_inject";
clasohm@923
    47
clasohm@923
    48
val [major] = goalw Univ.thy [Push_def] "Push k f =(%z.0) ==> P";
clasohm@923
    49
by (rtac (major RS fun_cong RS box_equals RS Suc_neq_Zero) 1);
clasohm@923
    50
by (rtac nat_case_0 1);
clasohm@923
    51
by (rtac refl 1);
clasohm@923
    52
qed "Push_neq_K0";
clasohm@923
    53
clasohm@923
    54
(*** Isomorphisms ***)
clasohm@923
    55
clasohm@923
    56
goal Univ.thy "inj(Rep_Node)";
clasohm@1465
    57
by (rtac inj_inverseI 1);       (*cannot combine by RS: multiple unifiers*)
clasohm@923
    58
by (rtac Rep_Node_inverse 1);
clasohm@923
    59
qed "inj_Rep_Node";
clasohm@923
    60
clasohm@923
    61
goal Univ.thy "inj_onto Abs_Node Node";
clasohm@923
    62
by (rtac inj_onto_inverseI 1);
clasohm@923
    63
by (etac Abs_Node_inverse 1);
clasohm@923
    64
qed "inj_onto_Abs_Node";
clasohm@923
    65
clasohm@923
    66
val Abs_Node_inject = inj_onto_Abs_Node RS inj_ontoD;
clasohm@923
    67
clasohm@923
    68
clasohm@923
    69
(*** Introduction rules for Node ***)
clasohm@923
    70
clasohm@972
    71
goalw Univ.thy [Node_def] "(%k. 0,a) : Node";
clasohm@923
    72
by (fast_tac set_cs 1);
clasohm@923
    73
qed "Node_K0_I";
clasohm@923
    74
clasohm@923
    75
goalw Univ.thy [Node_def,Push_def]
clasohm@923
    76
    "!!p. p: Node ==> apfst (Push i) p : Node";
clasohm@976
    77
by (fast_tac (set_cs addSIs [apfst_conv, nat_case_Suc RS trans]) 1);
clasohm@923
    78
qed "Node_Push_I";
clasohm@923
    79
clasohm@923
    80
clasohm@923
    81
(*** Distinctness of constructors ***)
clasohm@923
    82
clasohm@923
    83
(** Scons vs Atom **)
clasohm@923
    84
clasohm@923
    85
goalw Univ.thy [Atom_def,Scons_def,Push_Node_def] "(M$N) ~= Atom(a)";
clasohm@923
    86
by (rtac notI 1);
clasohm@923
    87
by (etac (equalityD2 RS subsetD RS UnE) 1);
clasohm@923
    88
by (rtac singletonI 1);
clasohm@976
    89
by (REPEAT (eresolve_tac [imageE, Abs_Node_inject RS apfst_convE, 
clasohm@1465
    90
                          Pair_inject, sym RS Push_neq_K0] 1
clasohm@923
    91
     ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1));
clasohm@923
    92
qed "Scons_not_Atom";
clasohm@923
    93
bind_thm ("Atom_not_Scons", (Scons_not_Atom RS not_sym));
clasohm@923
    94
clasohm@923
    95
bind_thm ("Scons_neq_Atom", (Scons_not_Atom RS notE));
clasohm@923
    96
val Atom_neq_Scons = sym RS Scons_neq_Atom;
clasohm@923
    97
clasohm@923
    98
(*** Injectiveness ***)
clasohm@923
    99
clasohm@923
   100
(** Atomic nodes **)
clasohm@923
   101
paulson@1563
   102
goalw Univ.thy [Atom_def, inj_def] "inj(Atom)";
paulson@1563
   103
by (fast_tac (prod_cs addSIs [Node_K0_I] addSDs [Abs_Node_inject]) 1);
clasohm@923
   104
qed "inj_Atom";
clasohm@923
   105
val Atom_inject = inj_Atom RS injD;
clasohm@923
   106
clasohm@923
   107
goalw Univ.thy [Leaf_def,o_def] "inj(Leaf)";
clasohm@923
   108
by (rtac injI 1);
clasohm@923
   109
by (etac (Atom_inject RS Inl_inject) 1);
clasohm@923
   110
qed "inj_Leaf";
clasohm@923
   111
clasohm@923
   112
val Leaf_inject = inj_Leaf RS injD;
clasohm@923
   113
clasohm@923
   114
goalw Univ.thy [Numb_def,o_def] "inj(Numb)";
clasohm@923
   115
by (rtac injI 1);
clasohm@923
   116
by (etac (Atom_inject RS Inr_inject) 1);
clasohm@923
   117
qed "inj_Numb";
clasohm@923
   118
clasohm@923
   119
val Numb_inject = inj_Numb RS injD;
clasohm@923
   120
clasohm@923
   121
(** Injectiveness of Push_Node **)
clasohm@923
   122
clasohm@923
   123
val [major,minor] = goalw Univ.thy [Push_Node_def]
clasohm@923
   124
    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P \
clasohm@923
   125
\    |] ==> P";
clasohm@976
   126
by (rtac (major RS Abs_Node_inject RS apfst_convE) 1);
clasohm@923
   127
by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
clasohm@976
   128
by (etac (sym RS apfst_convE) 1);
clasohm@923
   129
by (rtac minor 1);
clasohm@923
   130
by (etac Pair_inject 1);
clasohm@923
   131
by (etac (Push_inject1 RS sym) 1);
clasohm@923
   132
by (rtac (inj_Rep_Node RS injD) 1);
clasohm@923
   133
by (etac trans 1);
clasohm@923
   134
by (safe_tac (HOL_cs addSEs [Pair_inject,Push_inject,sym]));
clasohm@923
   135
qed "Push_Node_inject";
clasohm@923
   136
clasohm@923
   137
clasohm@923
   138
(** Injectiveness of Scons **)
clasohm@923
   139
clasohm@923
   140
val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> M<=M'";
clasohm@923
   141
by (cut_facts_tac [major] 1);
clasohm@923
   142
by (fast_tac (set_cs addSDs [Suc_inject]
clasohm@1465
   143
                     addSEs [Push_Node_inject, Zero_neq_Suc]) 1);
clasohm@923
   144
qed "Scons_inject_lemma1";
clasohm@923
   145
clasohm@923
   146
val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> N<=N'";
clasohm@923
   147
by (cut_facts_tac [major] 1);
clasohm@923
   148
by (fast_tac (set_cs addSDs [Suc_inject]
clasohm@1465
   149
                     addSEs [Push_Node_inject, Suc_neq_Zero]) 1);
clasohm@923
   150
qed "Scons_inject_lemma2";
clasohm@923
   151
clasohm@923
   152
val [major] = goal Univ.thy "M$N = M'$N' ==> M=M'";
clasohm@923
   153
by (rtac (major RS equalityE) 1);
clasohm@923
   154
by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1));
clasohm@923
   155
qed "Scons_inject1";
clasohm@923
   156
clasohm@923
   157
val [major] = goal Univ.thy "M$N = M'$N' ==> N=N'";
clasohm@923
   158
by (rtac (major RS equalityE) 1);
clasohm@923
   159
by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1));
clasohm@923
   160
qed "Scons_inject2";
clasohm@923
   161
clasohm@923
   162
val [major,minor] = goal Univ.thy
clasohm@923
   163
    "[| M$N = M'$N';  [| M=M';  N=N' |] ==> P \
clasohm@923
   164
\    |] ==> P";
clasohm@923
   165
by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
clasohm@923
   166
qed "Scons_inject";
clasohm@923
   167
clasohm@923
   168
(*rewrite rules*)
clasohm@923
   169
goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)";
clasohm@923
   170
by (fast_tac (HOL_cs addSEs [Atom_inject]) 1);
clasohm@923
   171
qed "Atom_Atom_eq";
clasohm@923
   172
clasohm@923
   173
goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
clasohm@923
   174
by (fast_tac (HOL_cs addSEs [Scons_inject]) 1);
clasohm@923
   175
qed "Scons_Scons_eq";
clasohm@923
   176
clasohm@923
   177
(*** Distinctness involving Leaf and Numb ***)
clasohm@923
   178
clasohm@923
   179
(** Scons vs Leaf **)
clasohm@923
   180
clasohm@923
   181
goalw Univ.thy [Leaf_def,o_def] "(M$N) ~= Leaf(a)";
clasohm@923
   182
by (rtac Scons_not_Atom 1);
clasohm@923
   183
qed "Scons_not_Leaf";
clasohm@923
   184
bind_thm ("Leaf_not_Scons", (Scons_not_Leaf RS not_sym));
clasohm@923
   185
clasohm@923
   186
bind_thm ("Scons_neq_Leaf", (Scons_not_Leaf RS notE));
clasohm@923
   187
val Leaf_neq_Scons = sym RS Scons_neq_Leaf;
clasohm@923
   188
clasohm@923
   189
(** Scons vs Numb **)
clasohm@923
   190
clasohm@923
   191
goalw Univ.thy [Numb_def,o_def] "(M$N) ~= Numb(k)";
clasohm@923
   192
by (rtac Scons_not_Atom 1);
clasohm@923
   193
qed "Scons_not_Numb";
clasohm@923
   194
bind_thm ("Numb_not_Scons", (Scons_not_Numb RS not_sym));
clasohm@923
   195
clasohm@923
   196
bind_thm ("Scons_neq_Numb", (Scons_not_Numb RS notE));
clasohm@923
   197
val Numb_neq_Scons = sym RS Scons_neq_Numb;
clasohm@923
   198
clasohm@923
   199
(** Leaf vs Numb **)
clasohm@923
   200
clasohm@923
   201
goalw Univ.thy [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)";
clasohm@1264
   202
by (simp_tac (!simpset addsimps [Atom_Atom_eq,Inl_not_Inr]) 1);
clasohm@923
   203
qed "Leaf_not_Numb";
clasohm@923
   204
bind_thm ("Numb_not_Leaf", (Leaf_not_Numb RS not_sym));
clasohm@923
   205
clasohm@923
   206
bind_thm ("Leaf_neq_Numb", (Leaf_not_Numb RS notE));
clasohm@923
   207
val Numb_neq_Leaf = sym RS Leaf_neq_Numb;
clasohm@923
   208
clasohm@923
   209
clasohm@923
   210
(*** ndepth -- the depth of a node ***)
clasohm@923
   211
clasohm@1264
   212
Addsimps [apfst_conv,Scons_not_Atom,Atom_not_Scons,Scons_Scons_eq];
clasohm@923
   213
clasohm@923
   214
clasohm@972
   215
goalw Univ.thy [ndepth_def] "ndepth (Abs_Node((%k.0, x))) = 0";
nipkow@1485
   216
by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split]);
clasohm@923
   217
by (rtac Least_equality 1);
clasohm@923
   218
by (rtac refl 1);
clasohm@923
   219
by (etac less_zeroE 1);
clasohm@923
   220
qed "ndepth_K0";
clasohm@923
   221
clasohm@923
   222
goal Univ.thy "k < Suc(LEAST x. f(x)=0) --> nat_case (Suc i) f k ~= 0";
clasohm@923
   223
by (nat_ind_tac "k" 1);
clasohm@1264
   224
by (ALLGOALS Simp_tac);
clasohm@923
   225
by (rtac impI 1);
clasohm@923
   226
by (etac not_less_Least 1);
clasohm@923
   227
qed "ndepth_Push_lemma";
clasohm@923
   228
clasohm@923
   229
goalw Univ.thy [ndepth_def,Push_Node_def]
clasohm@923
   230
    "ndepth (Push_Node i n) = Suc(ndepth(n))";
clasohm@923
   231
by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1);
clasohm@923
   232
by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1);
clasohm@923
   233
by (safe_tac set_cs);
clasohm@1465
   234
by (etac ssubst 1);  (*instantiates type variables!*)
clasohm@1264
   235
by (Simp_tac 1);
clasohm@923
   236
by (rtac Least_equality 1);
clasohm@923
   237
by (rewtac Push_def);
clasohm@923
   238
by (rtac (nat_case_Suc RS trans) 1);
clasohm@923
   239
by (etac LeastI 1);
clasohm@923
   240
by (etac (ndepth_Push_lemma RS mp) 1);
clasohm@923
   241
qed "ndepth_Push_Node";
clasohm@923
   242
clasohm@923
   243
clasohm@923
   244
(*** ntrunc applied to the various node sets ***)
clasohm@923
   245
clasohm@923
   246
goalw Univ.thy [ntrunc_def] "ntrunc 0 M = {}";
clasohm@923
   247
by (safe_tac (set_cs addSIs [equalityI] addSEs [less_zeroE]));
clasohm@923
   248
qed "ntrunc_0";
clasohm@923
   249
clasohm@923
   250
goalw Univ.thy [Atom_def,ntrunc_def] "ntrunc (Suc k) (Atom a) = Atom(a)";
clasohm@923
   251
by (safe_tac (set_cs addSIs [equalityI]));
clasohm@923
   252
by (stac ndepth_K0 1);
clasohm@923
   253
by (rtac zero_less_Suc 1);
clasohm@923
   254
qed "ntrunc_Atom";
clasohm@923
   255
clasohm@923
   256
goalw Univ.thy [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)";
clasohm@923
   257
by (rtac ntrunc_Atom 1);
clasohm@923
   258
qed "ntrunc_Leaf";
clasohm@923
   259
clasohm@923
   260
goalw Univ.thy [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)";
clasohm@923
   261
by (rtac ntrunc_Atom 1);
clasohm@923
   262
qed "ntrunc_Numb";
clasohm@923
   263
clasohm@923
   264
goalw Univ.thy [Scons_def,ntrunc_def]
clasohm@923
   265
    "ntrunc (Suc k) (M$N) = ntrunc k M $ ntrunc k N";
clasohm@923
   266
by (safe_tac (set_cs addSIs [equalityI,imageI]));
clasohm@923
   267
by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3));
clasohm@923
   268
by (REPEAT (rtac Suc_less_SucD 1 THEN 
clasohm@1465
   269
            rtac (ndepth_Push_Node RS subst) 1 THEN 
clasohm@1465
   270
            assume_tac 1));
clasohm@923
   271
qed "ntrunc_Scons";
clasohm@923
   272
clasohm@923
   273
(** Injection nodes **)
clasohm@923
   274
clasohm@923
   275
goalw Univ.thy [In0_def] "ntrunc (Suc 0) (In0 M) = {}";
clasohm@1264
   276
by (simp_tac (!simpset addsimps [ntrunc_Scons,ntrunc_0]) 1);
clasohm@923
   277
by (rewtac Scons_def);
clasohm@923
   278
by (safe_tac (set_cs addSIs [equalityI]));
clasohm@923
   279
qed "ntrunc_one_In0";
clasohm@923
   280
clasohm@923
   281
goalw Univ.thy [In0_def]
clasohm@923
   282
    "ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)";
clasohm@1264
   283
by (simp_tac (!simpset addsimps [ntrunc_Scons,ntrunc_Numb]) 1);
clasohm@923
   284
qed "ntrunc_In0";
clasohm@923
   285
clasohm@923
   286
goalw Univ.thy [In1_def] "ntrunc (Suc 0) (In1 M) = {}";
clasohm@1264
   287
by (simp_tac (!simpset addsimps [ntrunc_Scons,ntrunc_0]) 1);
clasohm@923
   288
by (rewtac Scons_def);
clasohm@923
   289
by (safe_tac (set_cs addSIs [equalityI]));
clasohm@923
   290
qed "ntrunc_one_In1";
clasohm@923
   291
clasohm@923
   292
goalw Univ.thy [In1_def]
clasohm@923
   293
    "ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)";
clasohm@1264
   294
by (simp_tac (!simpset addsimps [ntrunc_Scons,ntrunc_Numb]) 1);
clasohm@923
   295
qed "ntrunc_In1";
clasohm@923
   296
clasohm@923
   297
clasohm@923
   298
(*** Cartesian Product ***)
clasohm@923
   299
clasohm@923
   300
goalw Univ.thy [uprod_def] "!!M N. [| M:A;  N:B |] ==> (M$N) : A<*>B";
clasohm@923
   301
by (REPEAT (ares_tac [singletonI,UN_I] 1));
clasohm@923
   302
qed "uprodI";
clasohm@923
   303
clasohm@923
   304
(*The general elimination rule*)
clasohm@923
   305
val major::prems = goalw Univ.thy [uprod_def]
clasohm@923
   306
    "[| c : A<*>B;  \
clasohm@923
   307
\       !!x y. [| x:A;  y:B;  c=x$y |] ==> P \
clasohm@923
   308
\    |] ==> P";
clasohm@923
   309
by (cut_facts_tac [major] 1);
clasohm@923
   310
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
clasohm@923
   311
     ORELSE resolve_tac prems 1));
clasohm@923
   312
qed "uprodE";
clasohm@923
   313
clasohm@923
   314
(*Elimination of a pair -- introduces no eigenvariables*)
clasohm@923
   315
val prems = goal Univ.thy
clasohm@923
   316
    "[| (M$N) : A<*>B;      [| M:A;  N:B |] ==> P   \
clasohm@923
   317
\    |] ==> P";
clasohm@923
   318
by (rtac uprodE 1);
clasohm@923
   319
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
clasohm@923
   320
qed "uprodE2";
clasohm@923
   321
clasohm@923
   322
clasohm@923
   323
(*** Disjoint Sum ***)
clasohm@923
   324
clasohm@923
   325
goalw Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
clasohm@923
   326
by (fast_tac set_cs 1);
clasohm@923
   327
qed "usum_In0I";
clasohm@923
   328
clasohm@923
   329
goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
clasohm@923
   330
by (fast_tac set_cs 1);
clasohm@923
   331
qed "usum_In1I";
clasohm@923
   332
clasohm@923
   333
val major::prems = goalw Univ.thy [usum_def]
clasohm@923
   334
    "[| u : A<+>B;  \
clasohm@923
   335
\       !!x. [| x:A;  u=In0(x) |] ==> P; \
clasohm@923
   336
\       !!y. [| y:B;  u=In1(y) |] ==> P \
clasohm@923
   337
\    |] ==> P";
clasohm@923
   338
by (rtac (major RS UnE) 1);
clasohm@923
   339
by (REPEAT (rtac refl 1 
clasohm@923
   340
     ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
clasohm@923
   341
qed "usumE";
clasohm@923
   342
clasohm@923
   343
clasohm@923
   344
(** Injection **)
clasohm@923
   345
clasohm@923
   346
goalw Univ.thy [In0_def,In1_def] "In0(M) ~= In1(N)";
clasohm@923
   347
by (rtac notI 1);
clasohm@923
   348
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
clasohm@923
   349
qed "In0_not_In1";
clasohm@923
   350
clasohm@923
   351
bind_thm ("In1_not_In0", (In0_not_In1 RS not_sym));
clasohm@923
   352
bind_thm ("In0_neq_In1", (In0_not_In1 RS notE));
clasohm@923
   353
val In1_neq_In0 = sym RS In0_neq_In1;
clasohm@923
   354
clasohm@923
   355
val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==>  M=N";
clasohm@923
   356
by (rtac (major RS Scons_inject2) 1);
clasohm@923
   357
qed "In0_inject";
clasohm@923
   358
clasohm@923
   359
val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==>  M=N";
clasohm@923
   360
by (rtac (major RS Scons_inject2) 1);
clasohm@923
   361
qed "In1_inject";
clasohm@923
   362
clasohm@923
   363
clasohm@923
   364
(*** proving equality of sets and functions using ntrunc ***)
clasohm@923
   365
clasohm@923
   366
goalw Univ.thy [ntrunc_def] "ntrunc k M <= M";
clasohm@923
   367
by (fast_tac set_cs 1);
clasohm@923
   368
qed "ntrunc_subsetI";
clasohm@923
   369
clasohm@923
   370
val [major] = goalw Univ.thy [ntrunc_def]
clasohm@923
   371
    "(!!k. ntrunc k M <= N) ==> M<=N";
clasohm@923
   372
by (fast_tac (set_cs addIs [less_add_Suc1, less_add_Suc2, 
clasohm@1465
   373
                            major RS subsetD]) 1);
clasohm@923
   374
qed "ntrunc_subsetD";
clasohm@923
   375
clasohm@923
   376
(*A generalized form of the take-lemma*)
clasohm@923
   377
val [major] = goal Univ.thy "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
clasohm@923
   378
by (rtac equalityI 1);
clasohm@923
   379
by (ALLGOALS (rtac ntrunc_subsetD));
clasohm@923
   380
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
clasohm@923
   381
by (rtac (major RS equalityD1) 1);
clasohm@923
   382
by (rtac (major RS equalityD2) 1);
clasohm@923
   383
qed "ntrunc_equality";
clasohm@923
   384
clasohm@923
   385
val [major] = goalw Univ.thy [o_def]
clasohm@923
   386
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
clasohm@923
   387
by (rtac (ntrunc_equality RS ext) 1);
clasohm@923
   388
by (rtac (major RS fun_cong) 1);
clasohm@923
   389
qed "ntrunc_o_equality";
clasohm@923
   390
clasohm@923
   391
(*** Monotonicity ***)
clasohm@923
   392
clasohm@923
   393
goalw Univ.thy [uprod_def] "!!A B. [| A<=A';  B<=B' |] ==> A<*>B <= A'<*>B'";
clasohm@923
   394
by (fast_tac set_cs 1);
clasohm@923
   395
qed "uprod_mono";
clasohm@923
   396
clasohm@923
   397
goalw Univ.thy [usum_def] "!!A B. [| A<=A';  B<=B' |] ==> A<+>B <= A'<+>B'";
clasohm@923
   398
by (fast_tac set_cs 1);
clasohm@923
   399
qed "usum_mono";
clasohm@923
   400
clasohm@923
   401
goalw Univ.thy [Scons_def] "!!M N. [| M<=M';  N<=N' |] ==> M$N <= M'$N'";
clasohm@923
   402
by (fast_tac set_cs 1);
clasohm@923
   403
qed "Scons_mono";
clasohm@923
   404
clasohm@923
   405
goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
clasohm@923
   406
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
clasohm@923
   407
qed "In0_mono";
clasohm@923
   408
clasohm@923
   409
goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
clasohm@923
   410
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
clasohm@923
   411
qed "In1_mono";
clasohm@923
   412
clasohm@923
   413
clasohm@923
   414
(*** Split and Case ***)
clasohm@923
   415
clasohm@923
   416
goalw Univ.thy [Split_def] "Split c (M$N) = c M N";
clasohm@923
   417
by (fast_tac (set_cs addIs [select_equality] addEs [Scons_inject]) 1);
clasohm@923
   418
qed "Split";
clasohm@923
   419
clasohm@923
   420
goalw Univ.thy [Case_def] "Case c d (In0 M) = c(M)";
clasohm@923
   421
by (fast_tac (set_cs addIs [select_equality] 
clasohm@1465
   422
                     addEs [make_elim In0_inject, In0_neq_In1]) 1);
clasohm@923
   423
qed "Case_In0";
clasohm@923
   424
clasohm@923
   425
goalw Univ.thy [Case_def] "Case c d (In1 N) = d(N)";
clasohm@923
   426
by (fast_tac (set_cs addIs [select_equality] 
clasohm@1465
   427
                     addEs [make_elim In1_inject, In1_neq_In0]) 1);
clasohm@923
   428
qed "Case_In1";
clasohm@923
   429
clasohm@923
   430
(**** UN x. B(x) rules ****)
clasohm@923
   431
clasohm@923
   432
goalw Univ.thy [ntrunc_def] "ntrunc k (UN x.f(x)) = (UN x. ntrunc k (f x))";
clasohm@923
   433
by (fast_tac (set_cs addIs [equalityI]) 1);
clasohm@923
   434
qed "ntrunc_UN1";
clasohm@923
   435
clasohm@923
   436
goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
clasohm@923
   437
by (fast_tac (set_cs addIs [equalityI]) 1);
clasohm@923
   438
qed "Scons_UN1_x";
clasohm@923
   439
clasohm@923
   440
goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
clasohm@923
   441
by (fast_tac (set_cs addIs [equalityI]) 1);
clasohm@923
   442
qed "Scons_UN1_y";
clasohm@923
   443
clasohm@923
   444
goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
clasohm@1465
   445
by (rtac Scons_UN1_y 1);
clasohm@923
   446
qed "In0_UN1";
clasohm@923
   447
clasohm@923
   448
goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
clasohm@1465
   449
by (rtac Scons_UN1_y 1);
clasohm@923
   450
qed "In1_UN1";
clasohm@923
   451
clasohm@923
   452
clasohm@923
   453
(*** Equality : the diagonal relation ***)
clasohm@923
   454
clasohm@972
   455
goalw Univ.thy [diag_def] "!!a A. [| a=b;  a:A |] ==> (a,b) : diag(A)";
clasohm@923
   456
by (fast_tac set_cs 1);
clasohm@923
   457
qed "diag_eqI";
clasohm@923
   458
clasohm@923
   459
val diagI = refl RS diag_eqI |> standard;
clasohm@923
   460
clasohm@923
   461
(*The general elimination rule*)
clasohm@923
   462
val major::prems = goalw Univ.thy [diag_def]
clasohm@923
   463
    "[| c : diag(A);  \
clasohm@972
   464
\       !!x y. [| x:A;  c = (x,x) |] ==> P \
clasohm@923
   465
\    |] ==> P";
clasohm@923
   466
by (rtac (major RS UN_E) 1);
clasohm@923
   467
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
clasohm@923
   468
qed "diagE";
clasohm@923
   469
clasohm@923
   470
(*** Equality for Cartesian Product ***)
clasohm@923
   471
clasohm@923
   472
goalw Univ.thy [dprod_def]
clasohm@972
   473
    "!!r s. [| (M,M'):r;  (N,N'):s |] ==> (M$N, M'$N') : r<**>s";
clasohm@923
   474
by (fast_tac prod_cs 1);
clasohm@923
   475
qed "dprodI";
clasohm@923
   476
clasohm@923
   477
(*The general elimination rule*)
clasohm@923
   478
val major::prems = goalw Univ.thy [dprod_def]
clasohm@923
   479
    "[| c : r<**>s;  \
clasohm@972
   480
\       !!x y x' y'. [| (x,x') : r;  (y,y') : s;  c = (x$y,x'$y') |] ==> P \
clasohm@923
   481
\    |] ==> P";
clasohm@923
   482
by (cut_facts_tac [major] 1);
clasohm@923
   483
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
clasohm@923
   484
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
clasohm@923
   485
qed "dprodE";
clasohm@923
   486
clasohm@923
   487
clasohm@923
   488
(*** Equality for Disjoint Sum ***)
clasohm@923
   489
clasohm@972
   490
goalw Univ.thy [dsum_def]  "!!r. (M,M'):r ==> (In0(M), In0(M')) : r<++>s";
clasohm@923
   491
by (fast_tac prod_cs 1);
clasohm@923
   492
qed "dsum_In0I";
clasohm@923
   493
clasohm@972
   494
goalw Univ.thy [dsum_def]  "!!r. (N,N'):s ==> (In1(N), In1(N')) : r<++>s";
clasohm@923
   495
by (fast_tac prod_cs 1);
clasohm@923
   496
qed "dsum_In1I";
clasohm@923
   497
clasohm@923
   498
val major::prems = goalw Univ.thy [dsum_def]
clasohm@923
   499
    "[| w : r<++>s;  \
clasohm@972
   500
\       !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P; \
clasohm@972
   501
\       !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P \
clasohm@923
   502
\    |] ==> P";
clasohm@923
   503
by (cut_facts_tac [major] 1);
clasohm@923
   504
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
clasohm@923
   505
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
clasohm@923
   506
qed "dsumE";
clasohm@923
   507
clasohm@923
   508
clasohm@923
   509
val univ_cs =
clasohm@923
   510
    prod_cs addSIs [diagI, uprodI, dprodI]
clasohm@923
   511
            addIs  [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I]
clasohm@923
   512
            addSEs [diagE, uprodE, dprodE, usumE, dsumE];
clasohm@923
   513
clasohm@923
   514
clasohm@923
   515
(*** Monotonicity ***)
clasohm@923
   516
clasohm@923
   517
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<**>s <= r'<**>s'";
clasohm@923
   518
by (fast_tac univ_cs 1);
clasohm@923
   519
qed "dprod_mono";
clasohm@923
   520
clasohm@923
   521
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<++>s <= r'<++>s'";
clasohm@923
   522
by (fast_tac univ_cs 1);
clasohm@923
   523
qed "dsum_mono";
clasohm@923
   524
clasohm@923
   525
clasohm@923
   526
(*** Bounding theorems ***)
clasohm@923
   527
paulson@1642
   528
goal Univ.thy "diag(A) <= A Times A";
clasohm@923
   529
by (fast_tac univ_cs 1);
clasohm@923
   530
qed "diag_subset_Sigma";
clasohm@923
   531
paulson@1642
   532
goal Univ.thy "((A Times B) <**> (C Times D)) <= (A<*>C) Times (B<*>D)";
clasohm@923
   533
by (fast_tac univ_cs 1);
clasohm@923
   534
qed "dprod_Sigma";
clasohm@923
   535
clasohm@923
   536
val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard;
clasohm@923
   537
clasohm@923
   538
(*Dependent version*)
clasohm@923
   539
goal Univ.thy
clasohm@923
   540
    "(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))";
clasohm@923
   541
by (safe_tac univ_cs);
clasohm@923
   542
by (stac Split 1);
clasohm@923
   543
by (fast_tac univ_cs 1);
clasohm@923
   544
qed "dprod_subset_Sigma2";
clasohm@923
   545
paulson@1642
   546
goal Univ.thy "(A Times B <++> C Times D) <= (A<+>C) Times (B<+>D)";
clasohm@923
   547
by (fast_tac univ_cs 1);
clasohm@923
   548
qed "dsum_Sigma";
clasohm@923
   549
clasohm@923
   550
val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard;
clasohm@923
   551
clasohm@923
   552
clasohm@923
   553
(*** Domain ***)
clasohm@923
   554
clasohm@923
   555
goal Univ.thy "fst `` diag(A) = A";
clasohm@923
   556
by (fast_tac (prod_cs addIs [equalityI, diagI] addSEs [diagE]) 1);
clasohm@923
   557
qed "fst_image_diag";
clasohm@923
   558
clasohm@923
   559
goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
clasohm@923
   560
by (fast_tac (prod_cs addIs [equalityI, uprodI, dprodI]
clasohm@923
   561
                     addSEs [uprodE, dprodE]) 1);
clasohm@923
   562
qed "fst_image_dprod";
clasohm@923
   563
clasohm@923
   564
goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
clasohm@923
   565
by (fast_tac (prod_cs addIs [equalityI, usum_In0I, usum_In1I, 
clasohm@1465
   566
                             dsum_In0I, dsum_In1I]
clasohm@923
   567
                     addSEs [usumE, dsumE]) 1);
clasohm@923
   568
qed "fst_image_dsum";
clasohm@923
   569
clasohm@1264
   570
Addsimps [fst_image_diag, fst_image_dprod, fst_image_dsum];