src/HOL/Set.thy
author paulson
Mon, 20 Mar 2006 17:38:22 +0100
changeset 19295 c5d236fe9668
parent 19277 f7602e74d948
child 19323 ec5cd5b1804c
permissions -rw-r--r--
subsetI is often necessary
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     1
(*  Title:      HOL/Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     2
    ID:         $Id$
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
     3
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     5
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
     6
header {* Set theory for higher-order logic *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
     7
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15120
diff changeset
     8
theory Set
17508
c84af7f39a6b tuned theory dependencies;
wenzelm
parents: 17085
diff changeset
     9
imports LOrder
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15120
diff changeset
    10
begin
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    11
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    12
text {* A set in HOL is simply a predicate. *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    13
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
    14
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    15
subsection {* Basic syntax *}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
    16
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    17
global
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    18
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    19
typedecl 'a set
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12257
diff changeset
    20
arities set :: (type) type
3820
46b255e140dc fixed infix syntax;
wenzelm
parents: 3370
diff changeset
    21
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    22
consts
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    23
  "{}"          :: "'a set"                             ("{}")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    24
  UNIV          :: "'a set"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    25
  insert        :: "'a => 'a set => 'a set"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    26
  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    27
  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    28
  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    29
  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    30
  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    31
  Union         :: "'a set set => 'a set"                -- "union of a set"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    32
  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    33
  Pow           :: "'a set => 'a set set"                -- "powerset"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    34
  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    35
  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    36
  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    37
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    38
syntax
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    39
  "op :"        :: "'a => 'a set => bool"                ("op :")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    40
consts
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    41
  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    42
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    43
local
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    44
14692
b8d6c395c9e2 improved subscript syntax;
wenzelm
parents: 14565
diff changeset
    45
instance set :: (type) "{ord, minus}" ..
923
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
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    48
subsection {* Additional concrete syntax *}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
    49
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    50
syntax
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    51
  range         :: "('a => 'b) => 'b set"             -- "of function"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    52
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    53
  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    54
  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
    55
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    56
  "@Finset"     :: "args => 'a set"                       ("{(_)}")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    57
  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    58
  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
15535
nipkow
parents: 15524
diff changeset
    59
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    60
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    61
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    62
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    63
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    64
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    65
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    66
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
    67
  "_Bleast"       :: "id => 'a set => bool => 'a"      ("(3LEAST _:_./ _)" [0, 0, 10] 10)
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
    68
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    69
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
    70
syntax (HOL)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    71
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    72
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    73
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    74
translations
10832
e33b47e4246d `` -> and ``` -> ``
nipkow
parents: 10131
diff changeset
    75
  "range f"     == "f`UNIV"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    76
  "x ~: y"      == "~ (x : y)"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    77
  "{x, xs}"     == "insert x {xs}"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    78
  "{x}"         == "insert x {}"
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    79
  "{x. P}"      == "Collect (%x. P)"
15535
nipkow
parents: 15524
diff changeset
    80
  "{x:A. P}"    => "{x. x:A & P}"
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4151
diff changeset
    81
  "UN x y. B"   == "UN x. UN y. B"
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4151
diff changeset
    82
  "UN x. B"     == "UNION UNIV (%x. B)"
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
    83
  "UN x. B"     == "UN x:UNIV. B"
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
    84
  "INT x y. B"  == "INT x. INT y. B"
4159
4aff9b7e5597 UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents: 4151
diff changeset
    85
  "INT x. B"    == "INTER UNIV (%x. B)"
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
    86
  "INT x. B"    == "INT x:UNIV. B"
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    87
  "UN x:A. B"   == "UNION A (%x. B)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    88
  "INT x:A. B"  == "INTER A (%x. B)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    89
  "ALL x:A. P"  == "Ball A (%x. P)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    90
  "EX x:A. P"   == "Bex A (%x. P)"
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
    91
  "LEAST x:A. P" => "LEAST x. x:A & P"
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
    92
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    93
12633
ad9277743664 tuned ``syntax (output)'';
wenzelm
parents: 12338
diff changeset
    94
syntax (output)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    95
  "_setle"      :: "'a set => 'a set => bool"             ("op <=")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    96
  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    97
  "_setless"    :: "'a set => 'a set => bool"             ("op <")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    98
  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    99
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   100
syntax (xsymbols)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   101
  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   102
  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   103
  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   104
  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   105
  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   106
  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   107
  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   108
  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   109
  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   110
  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
14381
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   111
  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   112
  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   113
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   114
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
   115
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
14381
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   116
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   117
syntax (HTML output)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   118
  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   119
  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   120
  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   121
  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   122
  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   123
  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   124
  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   125
  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   126
  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   127
  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   128
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   129
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   130
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   131
syntax (xsymbols)
15535
nipkow
parents: 15524
diff changeset
   132
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   133
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   134
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   135
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   136
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   137
(*
14381
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   138
syntax (xsymbols)
14845
345934d5bc1a \<^bsub>/\<^esub> syntax: unbreakable block;
wenzelm
parents: 14812
diff changeset
   139
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" 10)
345934d5bc1a \<^bsub>/\<^esub> syntax: unbreakable block;
wenzelm
parents: 14812
diff changeset
   140
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10)
345934d5bc1a \<^bsub>/\<^esub> syntax: unbreakable block;
wenzelm
parents: 14812
diff changeset
   141
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
345934d5bc1a \<^bsub>/\<^esub> syntax: unbreakable block;
wenzelm
parents: 14812
diff changeset
   142
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   143
*)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   144
syntax (latex output)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   145
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" 10)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   146
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   147
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   148
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   149
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   150
text{* Note the difference between ordinary xsymbol syntax of indexed
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   151
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   152
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   153
former does not make the index expression a subscript of the
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   154
union/intersection symbol because this leads to problems with nested
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   155
subscripts in Proof General.  *}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   156
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   157
2412
025e80ed698d fixed \<subseteq> input;
wenzelm
parents: 2393
diff changeset
   158
translations
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   159
  "op \<subseteq>" => "op <= :: _ set => _ set => bool"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   160
  "op \<subset>" => "op <  :: _ set => _ set => bool"
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   161
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   162
typed_print_translation {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   163
  let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   164
    fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   165
          list_comb (Syntax.const "_setle", ts)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   166
      | le_tr' _ _ _ = raise Match;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   167
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   168
    fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   169
          list_comb (Syntax.const "_setless", ts)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   170
      | less_tr' _ _ _ = raise Match;
19277
f7602e74d948 renamed op < <= to Orderings.less(_eq)
haftmann
parents: 19233
diff changeset
   171
  in [("Orderings.less_eq", le_tr'), ("Orderings.less", less_tr')] end
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   172
*}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   173
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   174
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   175
subsubsection "Bounded quantifiers"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   176
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   177
syntax
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   178
  "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   179
  "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   180
  "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   181
  "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   182
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   183
syntax (xsymbols)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   184
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   185
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   186
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   187
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   188
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   189
syntax (HOL)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   190
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   191
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   192
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   193
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   194
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   195
syntax (HTML output)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   196
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   197
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   198
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   199
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   200
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   201
translations
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   202
 "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   203
 "\<exists>A\<subset>B. P"    =>  "EX A. A \<subset> B & P"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   204
 "\<forall>A\<subseteq>B. P"  =>  "ALL A. A \<subseteq> B --> P"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   205
 "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   206
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   207
print_translation {*
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   208
let
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   209
  fun
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   210
    all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19277
f7602e74d948 renamed op < <= to Orderings.less(_eq)
haftmann
parents: 19233
diff changeset
   211
             Const("op -->",_) $ (Const ("Orderings.less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   212
  (if v=v' andalso T="set"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   213
   then Syntax.const "_setlessAll" $ Syntax.mark_bound v' $ n $ P
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   214
   else raise Match)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   215
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   216
  | all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19277
f7602e74d948 renamed op < <= to Orderings.less(_eq)
haftmann
parents: 19233
diff changeset
   217
             Const("op -->",_) $ (Const ("Orderings.less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   218
  (if v=v' andalso T="set"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   219
   then Syntax.const "_setleAll" $ Syntax.mark_bound v' $ n $ P
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   220
   else raise Match);
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   221
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   222
  fun
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   223
    ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19277
f7602e74d948 renamed op < <= to Orderings.less(_eq)
haftmann
parents: 19233
diff changeset
   224
            Const("op &",_) $ (Const ("Orderings.less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   225
  (if v=v' andalso T="set"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   226
   then Syntax.const "_setlessEx" $ Syntax.mark_bound v' $ n $ P
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   227
   else raise Match)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   228
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   229
  | ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19277
f7602e74d948 renamed op < <= to Orderings.less(_eq)
haftmann
parents: 19233
diff changeset
   230
            Const("op &",_) $ (Const ("Orderings.less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   231
  (if v=v' andalso T="set"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   232
   then Syntax.const "_setleEx" $ Syntax.mark_bound v' $ n $ P
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   233
   else raise Match)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   234
in
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   235
[("ALL ", all_tr'), ("EX ", ex_tr')]
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   236
end
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   237
*}
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   238
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   239
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   240
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   241
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   242
  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   243
  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   244
  only translated if @{text "[0..n] subset bvs(e)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   245
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   246
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   247
parse_translation {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   248
  let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   249
    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
   250
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   251
    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   252
      | nvars _ = 1;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   253
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   254
    fun setcompr_tr [e, idts, b] =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   255
      let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   256
        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   257
        val P = Syntax.const "op &" $ eq $ b;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   258
        val exP = ex_tr [idts, P];
17784
5cbb52f2c478 Term.absdummy;
wenzelm
parents: 17715
diff changeset
   259
      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   260
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   261
  in [("@SetCompr", setcompr_tr)] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   262
*}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   263
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   264
(* To avoid eta-contraction of body: *)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   265
print_translation {*
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   266
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   267
  fun btr' syn [A,Abs abs] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   268
    let val (x,t) = atomic_abs_tr' abs
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   269
    in Syntax.const syn $ x $ A $ t end
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   270
in
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   271
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   272
 ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   273
end
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   274
*}
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   275
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   276
print_translation {*
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   277
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   278
  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   279
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   280
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   281
    let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   282
      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   283
        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   284
            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   285
            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   286
        | check _ = false
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   287
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   288
        fun tr' (_ $ abs) =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   289
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   290
          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   291
    in if check (P, 0) then tr' P
15535
nipkow
parents: 15524
diff changeset
   292
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
nipkow
parents: 15524
diff changeset
   293
                val M = Syntax.const "@Coll" $ x $ t
nipkow
parents: 15524
diff changeset
   294
            in case t of
nipkow
parents: 15524
diff changeset
   295
                 Const("op &",_)
nipkow
parents: 15524
diff changeset
   296
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
nipkow
parents: 15524
diff changeset
   297
                   $ P =>
nipkow
parents: 15524
diff changeset
   298
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
nipkow
parents: 15524
diff changeset
   299
               | _ => M
nipkow
parents: 15524
diff changeset
   300
            end
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   301
    end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   302
  in [("Collect", setcompr_tr')] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   303
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   304
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   305
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   306
subsection {* Rules and definitions *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   307
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   308
text {* Isomorphisms between predicates and sets. *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   309
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   310
axioms
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   311
  mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   312
  Collect_mem_eq: "{x. x:A} = A"
17702
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   313
finalconsts
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   314
  Collect
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   315
  "op :"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   316
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   317
defs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   318
  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   319
  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   320
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   321
defs (overloaded)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   322
  subset_def:   "A <= B         == ALL x:A. x:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   323
  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   324
  Compl_def:    "- A            == {x. ~x:A}"
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
   325
  set_diff_def: "A - B          == {x. x:A & ~x:B}"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   326
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   327
defs
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   328
  Un_def:       "A Un B         == {x. x:A | x:B}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   329
  Int_def:      "A Int B        == {x. x:A & x:B}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   330
  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   331
  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   332
  Inter_def:    "Inter S        == (INT x:S. x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   333
  Union_def:    "Union S        == (UN x:S. x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   334
  Pow_def:      "Pow A          == {B. B <= A}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   335
  empty_def:    "{}             == {x. False}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   336
  UNIV_def:     "UNIV           == {x. True}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   337
  insert_def:   "insert a B     == {x. x=a} Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   338
  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   339
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   340
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   341
subsection {* Lemmas and proof tool setup *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   342
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   343
subsubsection {* Relating predicates and sets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   344
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   345
declare mem_Collect_eq [iff]  Collect_mem_eq [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   346
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
   347
lemma CollectI: "P(a) ==> a : {x. P(x)}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   348
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   349
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   350
lemma CollectD: "a : {x. P(x)} ==> P(a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   351
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   352
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   353
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   354
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   355
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
   356
lemmas CollectE = CollectD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   357
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   358
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   359
subsubsection {* Bounded quantifiers *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   360
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   361
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   362
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   363
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   364
lemmas strip = impI allI ballI
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   365
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   366
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   367
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   368
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   369
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   370
  by (unfold Ball_def) blast
14098
54f130df1136 added rev_ballE
oheimb
parents: 13865
diff changeset
   371
ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   372
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   373
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   374
  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   375
  @{prop "a:A"}; creates assumption @{prop "P a"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   376
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   377
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   378
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   379
  local val ballE = thm "ballE"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   380
  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   381
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   382
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   383
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   384
  Gives better instantiation for bound:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   385
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   386
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   387
ML_setup {*
17875
d81094515061 change_claset/simpset;
wenzelm
parents: 17784
diff changeset
   388
  change_claset (fn cs => cs addbefore ("bspec", datac (thm "bspec") 1));
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   389
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   390
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   391
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   392
  -- {* Normally the best argument order: @{prop "P x"} constrains the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   393
    choice of @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   394
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   395
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   396
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   397
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   398
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   399
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   400
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   401
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   402
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   403
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   404
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   405
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   406
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   407
  -- {* Trival rewrite rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   408
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   409
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   410
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   411
  -- {* Dual form for existentials. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   412
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   413
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   414
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   415
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   416
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   417
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   418
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   419
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   420
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   421
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   422
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   423
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   424
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   425
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   426
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   427
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   428
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   429
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   430
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   431
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   432
ML_setup {*
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   433
  local
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   434
    val unfold_bex_tac = unfold_tac [thm "Bex_def"];
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   435
    fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   436
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   437
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   438
    val unfold_ball_tac = unfold_tac [thm "Ball_def"];
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   439
    fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   440
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   441
  in
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   442
    val defBEX_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   443
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   444
    val defBALL_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   445
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   446
  end;
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   447
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   448
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   449
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   450
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   451
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   452
subsubsection {* Congruence rules *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   453
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   454
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   456
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   458
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   459
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   460
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   461
    (ALL x:A. P x) = (ALL x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   462
  by (simp add: simp_implies_def Ball_def)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   463
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   464
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   465
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   466
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   467
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   468
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   469
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   470
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   471
    (EX x:A. P x) = (EX x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   472
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   473
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   474
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
subsubsection {* Subsets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   476
19295
c5d236fe9668 subsetI is often necessary
paulson
parents: 19277
diff changeset
   477
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   478
  by (simp add: subset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   479
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
  \medskip Map the type @{text "'a set => anything"} to just @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
  'a}; for overloading constants whose first argument has type @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
  "'a set"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   485
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   486
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
  -- {* Rule in Modus Ponens style. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   488
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
declare subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   492
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   493
  -- {* The same, with reversed premises for use with @{text erule} --
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   494
      cf @{text rev_mp}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   495
  by (rule subsetD)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
declare rev_subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   498
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   500
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   502
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
  local val rev_subsetD = thm "rev_subsetD"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   507
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   508
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   509
  -- {* Classical elimination rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   510
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   511
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   512
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   513
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   514
  creates the assumption @{prop "c \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   515
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   516
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
  local val subsetCE = thm "subsetCE"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   521
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   522
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   523
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   524
19175
c6e4b073f6a5 subset_refl now included using the atp attribute
paulson
parents: 18851
diff changeset
   525
lemma subset_refl [simp,atp]: "A \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   526
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   528
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   530
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   531
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   532
subsubsection {* Equality *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   534
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   535
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   536
   apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   537
  apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   538
  done
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   539
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   540
(* Due to Brian Huffman *)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   541
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   542
by(auto intro:set_ext)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   543
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   544
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
  -- {* Anti-symmetry of the subset relation. *}
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
   546
  by (iprover intro: set_ext subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   547
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   548
lemmas equalityI [intro!] = subset_antisym
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   549
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   550
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   551
  \medskip Equality rules from ZF set theory -- are they appropriate
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   552
  here?
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   553
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   554
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   555
lemma equalityD1: "A = B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   556
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   557
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   558
lemma equalityD2: "A = B ==> B \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   559
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   560
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   561
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
  \medskip Be careful when adding this to the claset as @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   564
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   565
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   566
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   567
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   568
  by (simp add: subset_refl)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   569
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
lemma equalityCE [elim]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   571
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   574
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   577
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   578
  by simp
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   579
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   580
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
subsubsection {* The universal set -- UNIV *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
lemma UNIV_I [simp]: "x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
  by (simp add: UNIV_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   585
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   586
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   589
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   590
18144
4edcb5fdc3b0 duplicate axioms in ATP linkup, and general fixes
paulson
parents: 17875
diff changeset
   591
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
  by (rule subsetI) (rule UNIV_I)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   593
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   594
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
  \medskip Eta-contracting these two rules (to remove @{text P})
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   596
  causes them to be ignored because of their interaction with
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   597
  congruence rules.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   599
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   600
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   601
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   603
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   605
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   607
subsubsection {* The empty set *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
lemma empty_iff [simp]: "(c : {}) = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
  by (simp add: empty_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
lemma emptyE [elim!]: "a : {} ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   613
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   614
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   615
lemma empty_subsetI [iff]: "{} \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   616
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   618
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   619
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   621
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   622
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   625
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   627
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   628
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   629
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   630
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   631
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   633
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   634
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
12023
wenzelm
parents: 12020
diff changeset
   636
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   637
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   638
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   640
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   641
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   642
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   643
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   644
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   647
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   648
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   650
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   652
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   653
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   656
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   658
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   659
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   660
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   661
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   662
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   663
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   664
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   665
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   666
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   667
lemma ComplD [dest!]: "c : -A ==> c~:A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   668
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   670
lemmas ComplE = ComplD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   671
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   672
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   674
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   675
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   678
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   683
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   685
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   686
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   689
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   690
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   692
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   695
12023
wenzelm
parents: 12020
diff changeset
   696
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   697
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   705
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   706
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   708
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   709
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
12023
wenzelm
parents: 12020
diff changeset
   714
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   718
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   719
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   720
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   726
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   728
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   729
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   730
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   732
subsubsection {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   733
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   736
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   740
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   742
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   749
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   750
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   752
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   753
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   754
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   755
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   758
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   760
lemma singletonD [dest!]: "b : {a} ==> b = a"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   763
lemmas singletonE = singletonD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   764
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   765
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   771
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   772
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   773
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   774
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   775
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   777
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   780
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   784
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   785
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   786
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   787
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   789
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   793
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   794
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   795
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   796
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   797
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   798
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   799
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   800
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   801
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   802
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   804
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   806
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   807
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   809
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   810
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   811
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   812
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   813
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   816
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   817
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   818
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   819
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   822
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   830
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   832
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   833
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   834
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   842
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   844
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   845
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   851
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   857
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   858
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   875
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   876
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   879
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   880
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   885
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   886
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   890
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   894
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   895
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   896
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   897
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   900
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   901
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   903
   prefer 2 apply fast
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
   904
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   906
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   907
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   909
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   910
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   912
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   913
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   914
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   915
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   916
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   917
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   918
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   919
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   920
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   921
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   922
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   923
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   924
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   925
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   926
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   927
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   928
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   929
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   930
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   931
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   933
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   934
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   935
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   936
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   938
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   939
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   940
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   941
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   942
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   943
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   944
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   946
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   947
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   948
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   949
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   950
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   951
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   952
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   953
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   954
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   955
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   956
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   957
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   958
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   959
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   960
  apply, then the formula should be kept.
19233
77ca20b0ed77 renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents: 19175
diff changeset
   961
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   962
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   963
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   964
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   965
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
ML_setup {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
17875
d81094515061 change_claset/simpset;
wenzelm
parents: 17784
diff changeset
   968
  change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs));
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   969
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   970
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   971
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   972
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   973
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   974
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   975
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   976
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   977
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   978
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   979
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   980
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   981
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   982
  "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   983
  by (auto simp add: psubset_def subset_insert_iff)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   984
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   985
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   986
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   987
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   988
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   989
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   990
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   991
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   992
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   993
apply (auto dest: subset_antisym)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   994
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   995
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   996
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   997
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   998
apply (auto dest: subsetD)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   999
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1000
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1001
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1002
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1003
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1004
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1005
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1006
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1007
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1008
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1009
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1010
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1011
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1012
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1013
18832
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18674
diff changeset
  1014
lemmas [symmetric, rulify] = atomize_ball
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18674
diff changeset
  1015
  and [symmetric, defn] = atomize_ball
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1016
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1017
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1018
subsection {* Further set-theory lemmas *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1019
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1020
subsubsection {* Derived rules involving subsets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1021
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
text {* @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1024
lemma subset_insertI: "B \<subseteq> insert a B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1027
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1028
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1029
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1030
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1031
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1032
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1033
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1034
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1035
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1036
text {* \medskip Big Union -- least upper bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1038
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1039
  by (iprover intro: subsetI UnionI)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1040
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1041
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1042
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1043
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1044
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1045
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1046
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1047
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1048
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1049
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1050
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1051
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1052
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1053
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1054
text {* \medskip Big Intersection -- greatest lower bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1055
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1059
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1060
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1061
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1062
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1063
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1064
  by (iprover intro: InterI subsetI dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1065
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1067
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1068
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1069
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1070
  by (iprover intro: INT_I subsetI dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1071
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1072
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1073
text {* \medskip Finite Union -- the least upper bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1074
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1075
lemma Un_upper1: "A \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1077
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1078
lemma Un_upper2: "B \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1079
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1080
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1081
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1082
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1083
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1084
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1085
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1087
lemma Int_lower1: "A \<inter> B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1088
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1089
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1090
lemma Int_lower2: "A \<inter> B \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1091
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1093
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1094
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1095
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1096
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1097
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1098
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1099
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1100
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1101
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1102
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1103
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1104
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1105
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1106
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1107
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1108
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1109
  by (auto simp add: mono_def)
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1110
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1111
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1112
  by (auto simp add: mono_def)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1114
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1115
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1116
text {* @{text "{}"}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1117
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1118
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1121
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1122
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1123
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1124
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1125
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1127
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1128
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
18423
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1129
by blast
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1130
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1131
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1132
by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1133
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1134
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1135
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1136
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1138
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1139
14812
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1140
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1141
  by blast
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1142
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1143
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1144
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1145
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1146
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1148
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1149
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1150
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1151
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1152
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1154
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1157
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
text {* \medskip @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
lemma insert_is_Un: "insert a A = {a} Un A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1162
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1163
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1166
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1167
17715
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1168
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1169
declare empty_not_insert [simp]
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1170
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
lemma insert_absorb: "a \<in> A ==> insert a A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1172
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1173
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1174
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1175
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1176
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1179
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1180
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1182
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1183
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1184
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1185
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1186
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1187
  apply (rule_tac x = "A - {a}" in exI, blast)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1188
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1189
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1190
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1192
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1193
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1194
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1195
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1196
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1197
  by blast
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1198
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1199
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1200
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1201
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1202
  by auto
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1203
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1204
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1205
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1206
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1207
  by auto
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1208
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1209
text {* \medskip @{text image}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1210
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1211
lemma image_empty [simp]: "f`{} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1212
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1213
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1214
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1215
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1216
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1217
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1218
  by auto
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1219
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1220
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1221
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1222
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1223
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1224
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1225
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1226
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1227
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1228
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1229
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1230
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1231
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1232
      with its implicit quantifier and conjunction.  Also image enjoys better
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1233
      equational properties than does the RHS. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1234
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1235
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1236
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1237
  "(\<lambda>x. if P x then f x else g x) ` S
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1238
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1239
  by (auto simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1240
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1241
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1242
  by (simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1243
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1244
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1245
text {* \medskip @{text range}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1246
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1247
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1248
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1249
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1250
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1251
by (subst image_image, simp)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1252
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1253
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1254
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1256
lemma Int_absorb [simp]: "A \<inter> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1257
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1259
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1262
lemma Int_commute: "A \<inter> B = B \<inter> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1264
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1266
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1267
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1268
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1269
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1270
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1271
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
  -- {* Intersection is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1273
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1274
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1276
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1277
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1279
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1280
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1282
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1283
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1285
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1286
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1288
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1289
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1291
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1292
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1295
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1296
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1297
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1298
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1300
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1301
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1302
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1303
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1304
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1306
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1307
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1309
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1310
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1312
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1313
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1315
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1316
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
text {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1318
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1319
lemma Un_absorb [simp]: "A \<union> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1320
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1321
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1322
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1323
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1324
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1325
lemma Un_commute: "A \<union> B = B \<union> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1326
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1327
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1328
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1331
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1333
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1334
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1335
  -- {* Union is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1336
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1337
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1338
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1339
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1340
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1341
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1342
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1343
lemma Un_empty_left [simp]: "{} \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1344
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1345
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1346
lemma Un_empty_right [simp]: "A \<union> {} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1347
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1348
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1349
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1350
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1351
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1352
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV&quo