src/HOL/Set.ML
author nipkow
Mon, 27 Apr 1998 16:45:11 +0200
changeset 4830 bd73675adbed
parent 4770 3e026ace28da
child 5069 3ea049f7979d
permissions -rw-r--r--
Added a few lemmas. Renamed expand_const -> split_const.
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
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1937
diff changeset
     6
Set theory for higher-order logic.  A set is simply a predicate.
923
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
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
    11
section "Relating predicates and sets";
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
    12
3469
61d927bd57ec Now Collect_mem_eq is a default simprule (how could it have ever been omitted?
paulson
parents: 3420
diff changeset
    13
Addsimps [Collect_mem_eq];
61d927bd57ec Now Collect_mem_eq is a default simprule (how could it have ever been omitted?
paulson
parents: 3420
diff changeset
    14
AddIffs  [mem_Collect_eq];
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    15
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
    16
goal Set.thy "!!a. P(a) ==> a : {x. P(x)}";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    17
by (Asm_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    18
qed "CollectI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    19
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
    20
val prems = goal Set.thy "!!a. a : {x. P(x)} ==> P(a)";
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    21
by (Asm_full_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    22
qed "CollectD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    23
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    24
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
    25
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
    26
by (rtac Collect_mem_eq 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    27
by (rtac Collect_mem_eq 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    28
qed "set_ext";
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 [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
    31
by (rtac (prem RS ext RS arg_cong) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    32
qed "Collect_cong";
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 CollectE = make_elim CollectD;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    35
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    36
AddSIs [CollectI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    37
AddSEs [CollectE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    38
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    39
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
    40
section "Bounded quantifiers";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    41
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    42
val prems = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    43
    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    44
by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    45
qed "ballI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    46
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    47
val [major,minor] = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    48
    "[| ! x:A. P(x);  x:A |] ==> P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    49
by (rtac (minor RS (major RS spec RS mp)) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    50
qed "bspec";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    51
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    52
val major::prems = goalw Set.thy [Ball_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    53
    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    54
by (rtac (major RS spec RS impCE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    55
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    56
qed "ballE";
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
(*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
    59
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
    60
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    61
AddSIs [ballI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    62
AddEs  [ballE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    63
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    64
val prems = goalw Set.thy [Bex_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    65
    "[| P(x);  x:A |] ==> ? x:A. P(x)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    66
by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    67
qed "bexI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    68
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    69
qed_goal "bexCI" Set.thy 
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
    70
   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    71
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    72
  [ (rtac classical 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    73
    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    74
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    75
val major::prems = goalw Set.thy [Bex_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    76
    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    77
by (rtac (major RS exE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    78
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    79
qed "bexE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    80
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    81
AddIs  [bexI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    82
AddSEs [bexE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
    83
3420
02dc9c5b035f New miniscoping rules ball_triv and bex_triv
paulson
parents: 3222
diff changeset
    84
(*Trival rewrite rule*)
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
    85
goal Set.thy "(! x:A. P) = ((? x. x:A) --> P)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
    86
by (simp_tac (simpset() addsimps [Ball_def]) 1);
3420
02dc9c5b035f New miniscoping rules ball_triv and bex_triv
paulson
parents: 3222
diff changeset
    87
qed "ball_triv";
1816
b03dba9116d4 New rewrites for vacuous quantification
paulson
parents: 1776
diff changeset
    88
1882
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
    89
(*Dual form for existentials*)
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
    90
goal Set.thy "(? x:A. P) = ((? x. x:A) & P)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
    91
by (simp_tac (simpset() addsimps [Bex_def]) 1);
3420
02dc9c5b035f New miniscoping rules ball_triv and bex_triv
paulson
parents: 3222
diff changeset
    92
qed "bex_triv";
1882
67f49e8c4355 Proved bex_False
paulson
parents: 1841
diff changeset
    93
3420
02dc9c5b035f New miniscoping rules ball_triv and bex_triv
paulson
parents: 3222
diff changeset
    94
Addsimps [ball_triv, bex_triv];
923
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
(** Congruence rules **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    97
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    98
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    99
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   100
\    (! x:A. P(x)) = (! x:B. Q(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   101
by (resolve_tac (prems RL [ssubst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   102
by (REPEAT (ares_tac [ballI,iffI] 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   103
     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   104
qed "ball_cong";
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
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   107
    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   108
\    (? x:A. P(x)) = (? x:B. Q(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   109
by (resolve_tac (prems RL [ssubst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   110
by (REPEAT (etac bexE 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   111
     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   112
qed "bex_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   113
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   114
section "Subsets";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   115
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   116
val prems = goalw Set.thy [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   117
by (REPEAT (ares_tac (prems @ [ballI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   118
qed "subsetI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   119
4240
8ba60a4cd380 Renamed "overload" to "overloaded" for sml/nj compatibility
paulson
parents: 4231
diff changeset
   120
Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*)
4059
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   121
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   122
(*While (:) is not, its type must be kept
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   123
  for overloading of = to work.*)
4240
8ba60a4cd380 Renamed "overload" to "overloaded" for sml/nj compatibility
paulson
parents: 4231
diff changeset
   124
Blast.overloaded ("op :", domain_type);
8ba60a4cd380 Renamed "overload" to "overloaded" for sml/nj compatibility
paulson
parents: 4231
diff changeset
   125
seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))
4059
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   126
    ["Ball", "Bex"];
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   127
(*need UNION, INTER also?*)
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   128
4469
399868bf8444 Overloading info for image
paulson
parents: 4434
diff changeset
   129
(*Image: retain the type of the set being expressed*)
399868bf8444 Overloading info for image
paulson
parents: 4434
diff changeset
   130
Blast.overloaded ("op ``", domain_type o domain_type);
2881
62ecde1015ae Declares overloading for set-theoretic constants
paulson
parents: 2858
diff changeset
   131
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   132
(*Rule in Modus Ponens style*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   133
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
   134
by (rtac (major RS bspec) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   135
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   136
qed "subsetD";
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
(*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
   139
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
   140
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   141
1920
df683ce7aad8 Added impOfSubs
paulson
parents: 1882
diff changeset
   142
(*Converts A<=B to x:A ==> x:B*)
df683ce7aad8 Added impOfSubs
paulson
parents: 1882
diff changeset
   143
fun impOfSubs th = th RSN (2, rev_subsetD);
df683ce7aad8 Added impOfSubs
paulson
parents: 1882
diff changeset
   144
1841
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   145
qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   146
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   147
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   148
qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   149
 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);
8e5e2fef6d26 Added contra_subsetD and rev_contra_subsetD
paulson
parents: 1816
diff changeset
   150
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   151
(*Classical elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   152
val major::prems = goalw Set.thy [subset_def] 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   153
    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   154
by (rtac (major RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   155
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   156
qed "subsetCE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   157
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   158
(*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
   159
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
   160
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   161
AddSIs [subsetI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   162
AddEs  [subsetD, subsetCE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   163
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   164
qed_goal "subset_refl" Set.thy "A <= (A::'a set)"
4059
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   165
 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   166
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   167
val prems = goal Set.thy "!!B. [| A<=B;  B<=C |] ==> A<=(C::'a set)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   168
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   169
qed "subset_trans";
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
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   172
section "Equality";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   173
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   174
(*Anti-symmetry of the subset relation*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   175
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
   176
by (rtac (iffI RS set_ext) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   177
by (REPEAT (ares_tac (prems RL [subsetD]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   178
qed "subset_antisym";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   179
val equalityI = subset_antisym;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   180
1762
6e481897a811 equalityI is now added to default claset
berghofe
parents: 1760
diff changeset
   181
AddSIs [equalityI];
6e481897a811 equalityI is now added to default claset
berghofe
parents: 1760
diff changeset
   182
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   183
(* Equality rules from ZF set theory -- are they appropriate here? *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   184
val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   185
by (resolve_tac (prems RL [subst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   186
by (rtac subset_refl 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   187
qed "equalityD1";
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
val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   190
by (resolve_tac (prems RL [subst]) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   191
by (rtac subset_refl 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   192
qed "equalityD2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   193
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   194
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   195
    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   196
by (resolve_tac prems 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   197
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   198
qed "equalityE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   199
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   200
val major::prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   201
    "[| 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
   202
by (rtac (major RS equalityE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   203
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
   204
qed "equalityCE";
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
(*Lemma for creating induction formulae -- for "pattern matching" on p
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   207
  To make the induction hypotheses usable, apply "spec" or "bspec" to
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   208
  put universal quantifiers over the free variables in p. *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   209
val prems = goal Set.thy 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   210
    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   211
by (rtac mp 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   212
by (REPEAT (resolve_tac (refl::prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   213
qed "setup_induction";
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
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   216
section "The universal set -- UNIV";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   217
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   218
qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   219
  (fn _ => [rtac CollectI 1, rtac TrueI 1]);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   220
4434
75f38104ff80 UNIV_I no longer counts as safe
paulson
parents: 4423
diff changeset
   221
Addsimps [UNIV_I];
75f38104ff80 UNIV_I no longer counts as safe
paulson
parents: 4423
diff changeset
   222
AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   223
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   224
qed_goal "subset_UNIV" Set.thy "A <= UNIV"
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   225
  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   226
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   227
(** Eta-contracting these two rules (to remove P) causes them to be ignored
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   228
    because of their interaction with congruence rules. **)
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   229
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   230
goalw Set.thy [Ball_def] "Ball UNIV P = All P";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   231
by (Simp_tac 1);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   232
qed "ball_UNIV";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   233
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   234
goalw Set.thy [Bex_def] "Bex UNIV P = Ex P";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   235
by (Simp_tac 1);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   236
qed "bex_UNIV";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   237
Addsimps [ball_UNIV, bex_UNIV];
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   238
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   239
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   240
section "The empty set -- {}";
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   241
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   242
qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   243
 (fn _ => [ (Blast_tac 1) ]);
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   244
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   245
Addsimps [empty_iff];
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   246
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   247
qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   248
 (fn _ => [Full_simp_tac 1]);
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   249
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   250
AddSEs [emptyE];
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   251
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   252
qed_goal "empty_subsetI" Set.thy "{} <= A"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   253
 (fn _ => [ (Blast_tac 1) ]);
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   254
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   255
qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   256
 (fn [prem]=>
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   257
  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   258
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   259
qed_goal "equals0D" Set.thy "!!a. [| A={};  a:A |] ==> P"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   260
 (fn _ => [ (Blast_tac 1) ]);
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   261
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   262
goalw Set.thy [Ball_def] "Ball {} P = True";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   263
by (Simp_tac 1);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   264
qed "ball_empty";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   265
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   266
goalw Set.thy [Bex_def] "Bex {} P = False";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   267
by (Simp_tac 1);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   268
qed "bex_empty";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   269
Addsimps [ball_empty, bex_empty];
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   270
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   271
goal thy "UNIV ~= {}";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   272
by (blast_tac (claset() addEs [equalityE]) 1);
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   273
qed "UNIV_not_empty";
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   274
AddIffs [UNIV_not_empty];
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   275
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   276
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   277
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   278
section "The Powerset operator -- Pow";
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   279
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   280
qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   281
 (fn _ => [ (Asm_simp_tac 1) ]);
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   282
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   283
AddIffs [Pow_iff]; 
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   284
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   285
qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   286
 (fn _ => [ (etac CollectI 1) ]);
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   287
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   288
qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   289
 (fn _=> [ (etac CollectD 1) ]);
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   290
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   291
val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   292
val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   293
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   294
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   295
section "Set complement -- Compl";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   296
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   297
qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   298
 (fn _ => [ (Blast_tac 1) ]);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   299
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   300
Addsimps [Compl_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   301
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   302
val prems = goalw Set.thy [Compl_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   303
    "[| c:A ==> False |] ==> c : Compl(A)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   304
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   305
qed "ComplI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   306
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   307
(*This form, with negated conclusion, works well with the Classical prover.
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   308
  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
   309
  turnstile...*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   310
val major::prems = goalw Set.thy [Compl_def]
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   311
    "c : Compl(A) ==> c~:A";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   312
by (rtac (major RS CollectD) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   313
qed "ComplD";
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 ComplE = make_elim ComplD;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   316
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   317
AddSIs [ComplI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   318
AddSEs [ComplE];
1640
581165679095 Added more _iff rewrites for Compl, Un, Int
paulson
parents: 1618
diff changeset
   319
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   320
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   321
section "Binary union -- Un";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   322
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   323
qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   324
 (fn _ => [ Blast_tac 1 ]);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   325
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   326
Addsimps [Un_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   327
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   328
goal Set.thy "!!c. c:A ==> c : A Un B";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   329
by (Asm_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   330
qed "UnI1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   331
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   332
goal Set.thy "!!c. c:B ==> c : A Un B";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   333
by (Asm_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   334
qed "UnI2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   335
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   336
(*Classical introduction rule: no commitment to A vs B*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   337
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
   338
 (fn prems=>
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   339
  [ (Simp_tac 1),
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   340
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   341
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   342
val major::prems = goalw Set.thy [Un_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   343
    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   344
by (rtac (major RS CollectD RS disjE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   345
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   346
qed "UnE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   347
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   348
AddSIs [UnCI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   349
AddSEs [UnE];
1640
581165679095 Added more _iff rewrites for Compl, Un, Int
paulson
parents: 1618
diff changeset
   350
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   351
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   352
section "Binary intersection -- Int";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   353
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   354
qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   355
 (fn _ => [ (Blast_tac 1) ]);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   356
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   357
Addsimps [Int_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   358
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   359
goal Set.thy "!!c. [| c:A;  c:B |] ==> c : A Int B";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   360
by (Asm_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   361
qed "IntI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   362
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   363
goal Set.thy "!!c. c : A Int B ==> c:A";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   364
by (Asm_full_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   365
qed "IntD1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   366
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   367
goal Set.thy "!!c. c : A Int B ==> c:B";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   368
by (Asm_full_simp_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
qed "IntD2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   370
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   371
val [major,minor] = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   372
    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   373
by (rtac minor 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
by (rtac (major RS IntD1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   375
by (rtac (major RS IntD2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
qed "IntE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   378
AddSIs [IntI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   379
AddSEs [IntE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   381
section "Set difference";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   383
qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   384
 (fn _ => [ (Blast_tac 1) ]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   386
Addsimps [Diff_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   387
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   388
qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   389
 (fn _=> [ Asm_simp_tac 1 ]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   391
qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   392
 (fn _=> [ (Asm_full_simp_tac 1) ]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   393
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   394
qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   395
 (fn _=> [ (Asm_full_simp_tac 1) ]);
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   396
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   397
qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
 (fn prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
  [ (resolve_tac prems 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   401
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   402
AddSIs [DiffI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   403
AddSEs [DiffE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   406
section "Augmenting a set -- insert";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   408
qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   409
 (fn _ => [Blast_tac 1]);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   410
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   411
Addsimps [insert_iff];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   412
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   413
qed_goal "insertI1" Set.thy "a : insert a B"
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   414
 (fn _ => [Simp_tac 1]);
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   415
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   416
qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   417
 (fn _=> [Asm_simp_tac 1]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
qed_goalw "insertE" Set.thy [insert_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   420
    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
 (fn major::prems=>
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   422
  [ (rtac (major RS UnE) 1),
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   423
    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
(*Classical introduction rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   427
 (fn prems=>
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   428
  [ (Simp_tac 1),
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   429
    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   430
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   431
AddSIs [insertCI]; 
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   432
AddSEs [insertE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   434
section "Singletons, using insert";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   435
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   436
qed_goal "singletonI" Set.thy "a : {a}"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
 (fn _=> [ (rtac insertI1 1) ]);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   439
goal Set.thy "!!a. b : {a} ==> b=a";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   440
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
qed "singletonD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   442
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   443
bind_thm ("singletonE", make_elim singletonD);
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   444
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   445
qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   446
(fn _ => [Blast_tac 1]);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   448
goal Set.thy "!!a b. {a}={b} ==> a=b";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   449
by (blast_tac (claset() addEs [equalityE]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   450
qed "singleton_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   451
2858
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   452
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)
1f3f5c44e159 Re-ordering of rules to assist blast_tac
paulson
parents: 2721
diff changeset
   453
AddSIs [singletonI];   
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   454
AddSDs [singleton_inject];
3718
d78cf498a88c Minor tidying to use Clarify_tac, etc.
paulson
parents: 3582
diff changeset
   455
AddSEs [singletonE];
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   456
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   457
goal Set.thy "{x. x=a} = {a}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4240
diff changeset
   458
by (Blast_tac 1);
3582
b87c86b6c291 Added {x.x=a} = a to !simpset.
nipkow
parents: 3469
diff changeset
   459
qed "singleton_conv";
b87c86b6c291 Added {x.x=a} = a to !simpset.
nipkow
parents: 3469
diff changeset
   460
Addsimps [singleton_conv];
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   461
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   462
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   463
section "Unions of families -- UNION x:A. B(x) is Union(B``A)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   464
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   465
goalw Set.thy [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   466
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   467
qed "UN_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   468
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   469
Addsimps [UN_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   470
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   471
(*The order of the premises presupposes that A is rigid; b may be flexible*)
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   472
goal Set.thy "!!b. [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   473
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   474
qed "UN_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   475
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   476
val major::prems = goalw Set.thy [UNION_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   477
    "[| 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
   478
by (rtac (major RS CollectD RS bexE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   480
qed "UN_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   481
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   482
AddIs  [UN_I];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   483
AddSEs [UN_E];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   484
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   485
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   486
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   487
\    (UN x:A. C(x)) = (UN x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   488
by (REPEAT (etac UN_E 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   489
     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 
1465
5d7a7e439cec expanded tabs
clasohm
parents: 923
diff changeset
   490
                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   491
qed "UN_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   492
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   494
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   495
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   496
goalw Set.thy [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   497
by Auto_tac;
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   498
qed "INT_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   500
Addsimps [INT_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   501
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   502
val prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   503
    "(!!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
   504
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   505
qed "INT_I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   506
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   507
goal Set.thy "!!b. [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   508
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   509
qed "INT_D";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   510
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   511
(*"Classical" elimination -- by the Excluded Middle on a:A *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   512
val major::prems = goalw Set.thy [INTER_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   513
    "[| 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
   514
by (rtac (major RS CollectD RS ballE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   515
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   516
qed "INT_E";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   517
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   518
AddSIs [INT_I];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   519
AddEs  [INT_D, INT_E];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   520
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   521
val prems = goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   522
    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   523
\    (INT x:A. C(x)) = (INT x:B. D(x))";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   524
by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   525
by (REPEAT (dtac INT_D 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   526
     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   527
qed "INT_cong";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   528
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   529
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   530
section "Union";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   531
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   532
goalw Set.thy [Union_def] "(A : Union(C)) = (EX X:C. A:X)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   533
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   534
qed "Union_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   535
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   536
Addsimps [Union_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   537
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   538
(*The order of the premises presupposes that C is rigid; A may be flexible*)
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   539
goal Set.thy "!!X. [| X:C;  A:X |] ==> A : Union(C)";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   540
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   541
qed "UnionI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   542
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   543
val major::prems = goalw Set.thy [Union_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   544
    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   545
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   546
by (REPEAT (ares_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   547
qed "UnionE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   548
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   549
AddIs  [UnionI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   550
AddSEs [UnionE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   551
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   552
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   553
section "Inter";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   554
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   555
goalw Set.thy [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2881
diff changeset
   556
by (Blast_tac 1);
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   557
qed "Inter_iff";
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   558
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   559
Addsimps [Inter_iff];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   560
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   561
val prems = goalw Set.thy [Inter_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   562
    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   563
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   564
qed "InterI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   565
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   566
(*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
   567
  A:X can hold when X:C does not!  This rule is analogous to "spec". *)
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   568
goal Set.thy "!!X. [| A : Inter(C);  X:C |] ==> A:X";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   569
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   570
qed "InterD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   571
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   572
(*"Classical" elimination rule -- does not require proving X:C *)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   573
val major::prems = goalw Set.thy [Inter_def]
2721
f08042e18c7d New version of InterE, like its ZF counterpart
paulson
parents: 2608
diff changeset
   574
    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   575
by (rtac (major RS INT_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   576
by (REPEAT (eresolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   577
qed "InterE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   578
2499
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   579
AddSIs [InterI];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   580
AddEs  [InterD, InterE];
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   581
0bc87b063447 Tidying of proofs. New theorems are enterred immediately into the
paulson
parents: 2031
diff changeset
   582
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   583
(*** Image of a set under a function ***)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   584
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   585
(*Frequently b does not have the syntactic form of f(x).*)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   586
val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   587
by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   588
qed "image_eqI";
3909
e48e2fb8b895 Added image_eqI to simpset.
nipkow
parents: 3905
diff changeset
   589
Addsimps [image_eqI];
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   590
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   591
bind_thm ("imageI", refl RS image_eqI);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   592
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   593
(*The eta-expansion gives variable-name preservation.*)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   594
val major::prems = goalw thy [image_def]
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   595
    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   596
by (rtac (major RS CollectD RS bexE) 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   597
by (REPEAT (ares_tac prems 1));
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   598
qed "imageE";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   599
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   600
AddIs  [image_eqI];
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   601
AddSEs [imageE]; 
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   602
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   603
goalw thy [o_def] "(f o g)``r = f``(g``r)";
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2912
diff changeset
   604
by (Blast_tac 1);
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   605
qed "image_compose";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   606
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   607
goal thy "f``(A Un B) = f``A Un f``B";
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2912
diff changeset
   608
by (Blast_tac 1);
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   609
qed "image_Un";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   610
4510
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   611
goal thy "(z : f``A) = (EX x:A. z = f x)";
3960
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   612
by (Blast_tac 1);
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   613
qed "image_iff";
7a38fae985f9 New rewrite rules image_iff
paulson
parents: 3919
diff changeset
   614
4523
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   615
(*This rewrite rule would confuse users if made default.*)
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   616
goal thy "(f``A <= B) = (ALL x:A. f(x): B)";
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   617
by (Blast_tac 1);
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   618
qed "image_subset_iff";
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   619
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   620
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too
16f5efe9812d New rule: image_subset
paulson
parents: 4510
diff changeset
   621
  many existing proofs.*)
4510
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   622
val prems = goal thy "(!!x. x:A ==> f(x) : B) ==> f``A <= B";
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   623
by (blast_tac (claset() addIs prems) 1);
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   624
qed "image_subsetI";
a37f515a0b25 New theorem image_subsetI
paulson
parents: 4477
diff changeset
   625
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   626
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   627
(*** Range of a function -- just a translation for image! ***)
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   628
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   629
goal thy "!!b. b=f(x) ==> b : range(f)";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   630
by (EVERY1 [etac image_eqI, rtac UNIV_I]);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   631
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   632
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   633
bind_thm ("rangeI", UNIV_I RS imageI);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   634
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   635
val [major,minor] = goal thy 
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3718
diff changeset
   636
    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   637
by (rtac (major RS imageE) 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   638
by (etac minor 1);
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   639
qed "rangeE";
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   640
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   641
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   642
(*** Set reasoning tools ***)
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   643
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   644
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   645
(** Rewrite rules for boolean case-splitting: faster than 
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   646
	addsplits[split_if]
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   647
**)
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   648
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   649
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   650
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   651
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   652
bind_thm ("split_if_mem1", 
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   653
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   654
bind_thm ("split_if_mem2", 
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   655
    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   656
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   657
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,
bd73675adbed Added a few lemmas.
nipkow
parents: 4770
diff changeset
   658
		  split_if_mem1, split_if_mem2];
3912
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   659
4ed64ad7fb42 New rewrite rules for simplifying conditionals
paulson
parents: 3909
diff changeset
   660
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   661
(*Each of these has ALREADY been added to simpset() above.*)
2024
909153d8318f Rationalized the rewriting of membership for {} and insert
paulson
parents: 1985
diff changeset
   662
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4135
diff changeset
   663
                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   664
1937
e59ff0eb1e91 Proved mem_if
paulson
parents: 1920
diff changeset
   665
(*Not for Addsimps -- it can cause goals to blow up!*)
e59ff0eb1e91 Proved mem_if
paulson
parents: 1920
diff changeset
   666
goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4523
diff changeset
   667
by (Simp_tac 1);
1937
e59ff0eb1e91 Proved mem_if
paulson
parents: 1920
diff changeset
   668
qed "mem_if";
e59ff0eb1e91 Proved mem_if
paulson
parents: 1920
diff changeset
   669
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   670
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   671
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4059
diff changeset
   672
simpset_ref() := simpset() addcongs [ball_cong,bex_cong]
1776
d7e77cb8ce5c moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents: 1762
diff changeset
   673
                    setmksimps (mksimps mksimps_pairs);
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   674
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   675
Addsimps[subset_UNIV, empty_subsetI, subset_refl];
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   676
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   677
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   678
(*** < ***)
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   679
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   680
goalw Set.thy [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   681
by (Blast_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   682
qed "psubsetI";
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   683
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   684
goalw Set.thy [psubset_def]
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   685
    "!!x. A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";
4477
b3e5857d8d99 New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents: 4469
diff changeset
   686
by Auto_tac;
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2935
diff changeset
   687
qed "psubset_insertD";
4059
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   688
59c1422c9da5 New Blast_tac (and minor tidying...)
paulson
parents: 3960
diff changeset
   689
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);