src/HOL/Set.ML
author nipkow
Mon, 04 Mar 1996 14:37:33 +0100
changeset 1531 e5eb247ad13c
parent 1465 5d7a7e439cec
child 1548 afe750876848
permissions -rw-r--r--
Added a constant UNIV == {x.True} Added many new rewrite rules for sets. Moved LEAST into Nat. Added cardinality to Finite.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
1465
5d7a7e439cec expanded tabs
clasohm
parents: 923
diff changeset
     1
(*  Title:      HOL/set
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     2
    ID:         $Id$
1465
5d7a7e439cec expanded tabs
clasohm
parents: 923
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
    Copyright   1991  University of Cambridge
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     5
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     6
For set.thy.  Set theory for higher-order logic.  A set is simply a predicate.
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     7
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     8
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     9
open Set;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    10
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    11
val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    12
by (rtac (mem_Collect_eq RS ssubst) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    13
by (rtac prem 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    14
qed "CollectI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    15
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    16
val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    17
by (resolve_tac (prems RL [mem_Collect_eq  RS subst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    18
qed "CollectD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    19
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    20
val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    21
by (rtac (prem RS ext RS arg_cong RS box_equals) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    22
by (rtac Collect_mem_eq 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    23
by (rtac Collect_mem_eq 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    24
qed "set_ext";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    25
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    26
val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    27
by (rtac (prem RS ext RS arg_cong) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    28
qed "Collect_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    29
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    30
val CollectE = make_elim CollectD;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    31
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    32
(*** Bounded quantifiers ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    33
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    34
val prems = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    35
    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    36
by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    37
qed "ballI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    38
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    39
val [major,minor] = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    40
    "[| ! x:A. P(x);  x:A |] ==> P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    41
by (rtac (minor RS (major RS spec RS mp)) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    42
qed "bspec";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    43
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    44
val major::prems = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    45
    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    46
by (rtac (major RS spec RS impCE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    47
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    48
qed "ballE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    49
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    50
(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    51
fun ball_tac i = etac ballE i THEN contr_tac (i+1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    52
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    53
val prems = goalw Set.thy [Bex_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    54
    "[| P(x);  x:A |] ==> ? x:A. P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    55
by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    56
qed "bexI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    57
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    58
qed_goal "bexCI" Set.thy 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    59
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    60
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    61
  [ (rtac classical 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    62
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    63
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    64
val major::prems = goalw Set.thy [Bex_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    65
    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    66
by (rtac (major RS exE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    67
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    68
qed "bexE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    69
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    70
(*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    71
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    72
    "(! x:A. True) = True";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    73
by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    74
qed "ball_rew";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    75
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    76
(** Congruence rules **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    77
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    78
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    79
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    80
\    (! x:A. P(x)) = (! x:B. Q(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    81
by (resolve_tac (prems RL [ssubst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    82
by (REPEAT (ares_tac [ballI,iffI] 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    83
     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    84
qed "ball_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    85
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    86
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    87
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    88
\    (? x:A. P(x)) = (? x:B. Q(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    89
by (resolve_tac (prems RL [ssubst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    90
by (REPEAT (etac bexE 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    91
     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    92
qed "bex_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    93
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    94
(*** Subsets ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    95
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    96
val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    97
by (REPEAT (ares_tac (prems @ [ballI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    98
qed "subsetI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    99
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   100
(*Rule in Modus Ponens style*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   101
val major::prems = goalw Set.thy [subset_def] "[| A <= B;  c:A |] ==> c:B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   102
by (rtac (major RS bspec) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   103
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   104
qed "subsetD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   105
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   106
(*The same, with reversed premises for use with etac -- cf rev_mp*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   107
qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   108
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   109
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   110
(*Classical elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   111
val major::prems = goalw Set.thy [subset_def] 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   112
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   113
by (rtac (major RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   114
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   115
qed "subsetCE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   116
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   117
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   118
fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   119
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   120
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   121
 (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   122
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   123
val prems = goal Set.thy "[| A<=B;  B<=C |] ==> A<=(C::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   124
by (cut_facts_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   125
by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   126
qed "subset_trans";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   127
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   128
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   129
(*** Equality ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   130
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   131
(*Anti-symmetry of the subset relation*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   132
val prems = goal Set.thy "[| A <= B;  B <= A |] ==> A = (B::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   133
by (rtac (iffI RS set_ext) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   134
by (REPEAT (ares_tac (prems RL [subsetD]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   135
qed "subset_antisym";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   136
val equalityI = subset_antisym;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   137
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   138
(* Equality rules from ZF set theory -- are they appropriate here? *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   139
val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   140
by (resolve_tac (prems RL [subst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   141
by (rtac subset_refl 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   142
qed "equalityD1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   143
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   144
val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   145
by (resolve_tac (prems RL [subst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   146
by (rtac subset_refl 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   147
qed "equalityD2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   148
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   149
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   150
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   151
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   152
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   153
qed "equalityE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   154
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   155
val major::prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   156
    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   157
by (rtac (major RS equalityE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   158
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   159
qed "equalityCE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   160
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   161
(*Lemma for creating induction formulae -- for "pattern matching" on p
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   162
  To make the induction hypotheses usable, apply "spec" or "bspec" to
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   163
  put universal quantifiers over the free variables in p. *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   164
val prems = goal Set.thy 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   165
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   166
by (rtac mp 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   167
by (REPEAT (resolve_tac (refl::prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   168
qed "setup_induction";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   169
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   170
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   171
(*** Set complement -- Compl ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   172
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   173
val prems = goalw Set.thy [Compl_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   174
    "[| c:A ==> False |] ==> c : Compl(A)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   175
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   176
qed "ComplI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   177
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   178
(*This form, with negated conclusion, works well with the Classical prover.
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   179
  Negated assumptions behave like formulae on the right side of the notional
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   180
  turnstile...*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   181
val major::prems = goalw Set.thy [Compl_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   182
    "[| c : Compl(A) |] ==> c~:A";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   183
by (rtac (major RS CollectD) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   184
qed "ComplD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   185
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   186
val ComplE = make_elim ComplD;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   187
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   188
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   189
(*** Binary union -- Un ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   190
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   191
val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   192
by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   193
qed "UnI1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   194
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   195
val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   196
by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   197
qed "UnI2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   198
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   199
(*Classical introduction rule: no commitment to A vs B*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   200
qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   201
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   202
  [ (rtac classical 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   203
    (REPEAT (ares_tac (prems@[UnI1,notI]) 1)),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   204
    (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   205
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   206
val major::prems = goalw Set.thy [Un_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   207
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   208
by (rtac (major RS CollectD RS disjE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   209
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   210
qed "UnE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   211
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   212
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   213
(*** Binary intersection -- Int ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   214
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   215
val prems = goalw Set.thy [Int_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   216
    "[| c:A;  c:B |] ==> c : A Int B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   217
by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   218
qed "IntI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   219
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   220
val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   221
by (rtac (major RS CollectD RS conjunct1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   222
qed "IntD1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   223
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   224
val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   225
by (rtac (major RS CollectD RS conjunct2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   226
qed "IntD2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   227
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   228
val [major,minor] = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   229
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   230
by (rtac minor 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   231
by (rtac (major RS IntD1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   232
by (rtac (major RS IntD2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   233
qed "IntE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   234
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   235
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   236
(*** Set difference ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   237
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   238
qed_goalw "DiffI" Set.thy [set_diff_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   239
    "[| c : A;  c ~: B |] ==> c : A - B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   240
 (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   241
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   242
qed_goalw "DiffD1" Set.thy [set_diff_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   243
    "c : A - B ==> c : A"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   244
 (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   245
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   246
qed_goalw "DiffD2" Set.thy [set_diff_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   247
    "[| c : A - B;  c : B |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   248
 (fn [major,minor]=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   249
     [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   250
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   251
qed_goal "DiffE" Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   252
    "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   253
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   254
  [ (resolve_tac prems 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   255
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   256
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   257
qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   258
 (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   259
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   260
(*** The empty set -- {} ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   261
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   262
qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   263
 (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   264
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   265
qed_goal "empty_subsetI" Set.thy "{} <= A"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   266
 (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   267
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   268
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   269
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   270
  [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   271
      ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   272
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   273
qed_goal "equals0D" Set.thy "[| A={};  a:A |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   274
 (fn [major,minor]=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   275
  [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   276
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   277
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   278
(*** Augmenting a set -- insert ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   279
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   280
qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   281
 (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   282
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   283
qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   284
 (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   285
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   286
qed_goalw "insertE" Set.thy [insert_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   287
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   288
 (fn major::prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   289
  [ (rtac (major RS UnE) 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   290
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   291
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   292
qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   293
 (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   294
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   295
(*Classical introduction rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   296
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   297
 (fn [prem]=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   298
  [ (rtac (disjCI RS (insert_iff RS iffD2)) 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   299
    (etac prem 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   300
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   301
(*** Singletons, using insert ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   302
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   303
qed_goal "singletonI" Set.thy "a : {a}"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   304
 (fn _=> [ (rtac insertI1 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   305
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   306
qed_goal "singletonE" Set.thy "[| a: {b};  a=b ==> P |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   307
 (fn major::prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   308
  [ (rtac (major RS insertE) 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   309
    (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   310
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   311
goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   312
by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   313
qed "singletonD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   314
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   315
val singletonE = make_elim singletonD;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   316
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   317
val [major] = goal Set.thy "{a}={b} ==> a=b";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   318
by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   319
by (rtac singletonI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   320
qed "singleton_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   321
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   322
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   323
(*** UNIV - The universal set ***)
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   324
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   325
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   326
  (fn _ => [rtac subsetI 1, rtac ComplI 1, etac emptyE 1]);
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   327
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   328
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   329
(*** Unions of families -- UNION x:A. B(x) is Union(B``A)  ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   330
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   331
(*The order of the premises presupposes that A is rigid; b may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   332
val prems = goalw Set.thy [UNION_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   333
    "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   334
by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   335
qed "UN_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   336
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   337
val major::prems = goalw Set.thy [UNION_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   338
    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   339
by (rtac (major RS CollectD RS bexE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   340
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   341
qed "UN_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   342
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   343
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   344
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   345
\    (UN x:A. C(x)) = (UN x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   346
by (REPEAT (etac UN_E 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   347
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
1465
5d7a7e439cec expanded tabs
clasohm
parents: 923
diff changeset
   348
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   349
qed "UN_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   350
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   351
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   352
(*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   353
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   354
val prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   355
    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   356
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   357
qed "INT_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   358
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   359
val major::prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   360
    "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   361
by (rtac (major RS CollectD RS bspec) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   362
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   363
qed "INT_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   364
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   365
(*"Classical" elimination -- by the Excluded Middle on a:A *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   366
val major::prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   367
    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   368
by (rtac (major RS CollectD RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   370
qed "INT_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   371
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   372
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   373
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
\    (INT x:A. C(x)) = (INT x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   375
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
by (REPEAT (dtac INT_D 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
qed "INT_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   379
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
(*** Unions over a type; UNION1(B) = Union(range(B)) ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
(*The order of the premises presupposes that A is rigid; b may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
val prems = goalw Set.thy [UNION1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
    "b: B(x) ==> b: (UN x. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   387
qed "UN1_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   388
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
val major::prems = goalw Set.thy [UNION1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
    "[| b : (UN x. B(x));  !!x. b: B(x) ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   391
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   393
qed "UN1_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   394
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   395
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   396
(*** Intersections over a type; INTER1(B) = Inter(range(B)) *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   397
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
val prems = goalw Set.thy [INTER1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
    "(!!x. b: B(x)) ==> b : (INT x. B(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
by (REPEAT (ares_tac (INT_I::prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   401
qed "INT1_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
val [major] = goalw Set.thy [INTER1_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
    "b : (INT x. B(x)) ==> b: B(a)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   406
qed "INT1_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   408
(*** Unions ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
(*The order of the premises presupposes that C is rigid; A may be flexible*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   411
val prems = goalw Set.thy [Union_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   412
    "[| X:C;  A:X |] ==> A : Union(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
qed "UnionI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   415
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   416
val major::prems = goalw Set.thy [Union_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   420
qed "UnionE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   422
(*** Inter ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   423
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
val prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   427
qed "InterI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   429
(*A "destruct" rule -- every X in C contains A as an element, but
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   430
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   431
val major::prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
    "[| A : Inter(C);  X:C |] ==> A:X";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
by (rtac (major RS INT_D) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   434
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   435
qed "InterD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   436
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
(*"Classical" elimination rule -- does not require proving X:C *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
val major::prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   439
    "[| A : Inter(C);  A:X ==> R;  X~:C ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
by (rtac (major RS INT_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   442
qed "InterE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   443
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
(*** Powerset ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   446
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
 (fn _ => [ (etac CollectI 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   448
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   449
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   450
 (fn _=> [ (etac CollectD 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   451
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   452
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   453
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)