src/HOL/Set.thy
author wenzelm
Tue, 03 Jul 2007 22:27:11 +0200
changeset 23555 16e5fd18905c
parent 22845 5f9138bcb3d7
child 23857 ad26287a9ccb
permissions -rw-r--r--
assume basic HOL context for compilation (antiquotations); added dest_cTrueprop; tuned Trueprop_conv; added low-level conj_intr/elim/elis (dire need for @{rule} antiquotation!);
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
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
     9
imports Lattices
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"
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
    27
  "op Int"      :: "'a set => 'a set => 'a set"          (infixl "Int" 70)
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
    28
  "op Un"       :: "'a set => 'a set => 'a set"          (infixl "Un" 65)
11979
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"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
    36
  Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    37
  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    38
  "op :"        :: "'a => 'a set => bool"                -- "membership"
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    39
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    40
notation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    41
  "op :"  ("op :") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    42
  "op :"  ("(_/ : _)" [50, 51] 50)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    43
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    44
local
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    45
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    46
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    47
subsection {* Additional concrete syntax *}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
    48
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    49
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    50
  range :: "('a => 'b) => 'b set" where -- "of function"
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    51
  "range f == f ` UNIV"
19323
ec5cd5b1804c Converted translations to abbbreviations.
nipkow
parents: 19295
diff changeset
    52
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    53
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    54
  "not_mem x A == ~ (x : A)" -- "non-membership"
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    55
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    56
notation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    57
  not_mem  ("op ~:") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    58
  not_mem  ("(_/ ~: _)" [50, 51] 50)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    59
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    60
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    61
  "op Int"  (infixl "\<inter>" 70) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    62
  "op Un"  (infixl "\<union>" 65) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    63
  "op :"  ("op \<in>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    64
  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    65
  not_mem  ("op \<notin>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    66
  not_mem  ("(_/ \<notin> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    67
  Union  ("\<Union>_" [90] 90) and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    68
  Inter  ("\<Inter>_" [90] 90)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    69
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    70
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    71
  "op Int"  (infixl "\<inter>" 70) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    72
  "op Un"  (infixl "\<union>" 65) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    73
  "op :"  ("op \<in>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    74
  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    75
  not_mem  ("op \<notin>") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    76
  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    77
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    78
syntax
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    79
  "@Finset"     :: "args => 'a set"                       ("{(_)}")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    80
  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    81
  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
15535
nipkow
parents: 15524
diff changeset
    82
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
22439
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
    83
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
    84
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
    85
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
    86
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    87
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    88
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
    89
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
22478
110f7f6f8a5d fixed typo
haftmann
parents: 22455
diff changeset
    90
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
    91
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
    92
syntax (HOL)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    93
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    94
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
    95
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    96
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    97
translations
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    98
  "{x, xs}"     == "insert x {xs}"
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    99
  "{x}"         == "insert x {}"
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   100
  "{x. P}"      == "Collect (%x. P)"
15535
nipkow
parents: 15524
diff changeset
   101
  "{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
   102
  "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
   103
  "UN x. B"     == "UNION UNIV (%x. B)"
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   104
  "UN x. B"     == "UN x:UNIV. B"
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   105
  "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
   106
  "INT x. B"    == "INTER UNIV (%x. B)"
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   107
  "INT x. B"    == "INT x:UNIV. B"
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   108
  "UN x:A. B"   == "UNION A (%x. B)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   109
  "INT x:A. B"  == "INTER A (%x. B)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   110
  "ALL x:A. P"  == "Ball A (%x. P)"
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   111
  "EX x:A. P"   == "Bex A (%x. P)"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   112
  "EX! x:A. P"  == "Bex1 A (%x. P)"
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
   113
  "LEAST x:A. P" => "LEAST x. x:A & P"
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
   114
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   115
syntax (xsymbols)
14381
1189a8212a12 Modified UN and INT xsymbol syntax: made index subscript
nipkow
parents: 14335
diff changeset
   116
  "_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
   117
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   118
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
18674
98d380757893 *** empty log message ***
nipkow
parents: 18447
diff changeset
   119
  "_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
   120
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   121
syntax (HTML output)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   122
  "_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
   123
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   124
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14551
diff changeset
   125
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   126
syntax (xsymbols)
15535
nipkow
parents: 15524
diff changeset
   127
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
22439
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   128
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   129
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   130
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   131
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   132
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   133
syntax (latex output)
22439
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   134
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   135
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   136
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
b709739c69e6 syntax: proper body priorty for derived binders;
wenzelm
parents: 22377
diff changeset
   137
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
15120
f0359f75682e undid UN/INT syntax
nipkow
parents: 15102
diff changeset
   138
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   139
text{*
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   140
  Note the difference between ordinary xsymbol syntax of indexed
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   141
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   142
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   143
  former does not make the index expression a subscript of the
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   144
  union/intersection symbol because this leads to problems with nested
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   145
  subscripts in Proof General. *}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   146
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   147
instance set :: (type) ord
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22478
diff changeset
   148
  subset_def:  "A \<le> B \<equiv> \<forall>x\<in>A. x \<in> B"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22478
diff changeset
   149
  psubset_def: "A < B \<equiv> A \<le> B \<and> A \<noteq> B" ..
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22478
diff changeset
   150
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   151
lemmas [code func del] = subset_def psubset_def
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   152
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   153
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   154
  subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   155
  "subset \<equiv> less"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   156
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   157
abbreviation
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   158
  subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   159
  "subset_eq \<equiv> less_eq"
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   160
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   161
notation (output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   162
  subset  ("op <") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   163
  subset  ("(_/ < _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   164
  subset_eq  ("op <=") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   165
  subset_eq  ("(_/ <= _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   166
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   167
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   168
  subset  ("op \<subset>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   169
  subset  ("(_/ \<subset> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   170
  subset_eq  ("op \<subseteq>") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   171
  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   172
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   173
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   174
  subset  ("op \<subset>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   175
  subset  ("(_/ \<subset> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   176
  subset_eq  ("op \<subseteq>") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   177
  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   178
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   179
abbreviation (input)
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   180
  supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   181
  "supset \<equiv> greater"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   182
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   183
abbreviation (input)
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   184
  supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   185
  "supset_eq \<equiv> greater_eq"
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   186
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   187
notation (xsymbols)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   188
  supset  ("op \<supset>") and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   189
  supset  ("(_/ \<supset> _)" [50, 51] 50) and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   190
  supset_eq  ("op \<supseteq>") and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   191
  supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   192
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   193
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   194
subsubsection "Bounded quantifiers"
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   195
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   196
syntax (output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   197
  "_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
   198
  "_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
   199
  "_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
   200
  "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   201
  "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   202
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   203
syntax (xsymbols)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   204
  "_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
   205
  "_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
   206
  "_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
   207
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   208
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   209
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   210
syntax (HOL output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   211
  "_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
   212
  "_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
   213
  "_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
   214
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   215
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   216
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   217
syntax (HTML output)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   218
  "_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
   219
  "_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
   220
  "_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
   221
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   222
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   223
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   224
translations
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   225
 "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   226
 "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   227
 "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   228
 "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   229
 "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   230
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   231
print_translation {*
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   232
let
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   233
  val Type (set_type, _) = @{typ "'a set"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   234
  val All_binder = Syntax.binder_name @{const_syntax "All"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   235
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   236
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   237
  val conj = @{const_syntax "op &"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   238
  val sbset = @{const_syntax "subset"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   239
  val sbset_eq = @{const_syntax "subset_eq"};
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   240
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   241
  val trans =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   242
   [((All_binder, impl, sbset), "_setlessAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   243
    ((All_binder, impl, sbset_eq), "_setleAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   244
    ((Ex_binder, conj, sbset), "_setlessEx"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   245
    ((Ex_binder, conj, sbset_eq), "_setleEx")];
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   246
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   247
  fun mk v v' c n P =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   248
    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   249
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   250
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   251
  fun tr' q = (q,
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   252
    fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =>
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   253
         if T = (set_type) then case AList.lookup (op =) trans (q, c, d)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   254
          of NONE => raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   255
           | SOME l => mk v v' l n P
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   256
         else raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   257
     | _ => raise Match);
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   258
in
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   259
  [tr' All_binder, tr' Ex_binder]
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   260
end
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   261
*}
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   262
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   263
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   264
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   265
  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   266
  "{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
   267
  only translated if @{text "[0..n] subset bvs(e)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   268
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   269
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   270
parse_translation {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   271
  let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   272
    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
   273
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   274
    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   275
      | nvars _ = 1;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   276
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   277
    fun setcompr_tr [e, idts, b] =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   278
      let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   279
        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   280
        val P = Syntax.const "op &" $ eq $ b;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   281
        val exP = ex_tr [idts, P];
17784
5cbb52f2c478 Term.absdummy;
wenzelm
parents: 17715
diff changeset
   282
      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   283
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   284
  in [("@SetCompr", setcompr_tr)] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   285
*}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   286
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   287
(* To avoid eta-contraction of body: *)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   288
print_translation {*
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   289
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   290
  fun btr' syn [A,Abs abs] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   291
    let val (x,t) = atomic_abs_tr' abs
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   292
    in Syntax.const syn $ x $ A $ t end
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   293
in
13858
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   294
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
a077513c9a07 *** empty log message ***
nipkow
parents: 13831
diff changeset
   295
 ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   296
end
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   297
*}
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   298
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   299
print_translation {*
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   300
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   301
  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
   302
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   303
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   304
    let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   305
      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
   306
        | 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
   307
            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
   308
            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   309
        | check _ = false
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   310
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   311
        fun tr' (_ $ abs) =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   312
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   313
          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
   314
    in if check (P, 0) then tr' P
15535
nipkow
parents: 15524
diff changeset
   315
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
nipkow
parents: 15524
diff changeset
   316
                val M = Syntax.const "@Coll" $ x $ t
nipkow
parents: 15524
diff changeset
   317
            in case t of
nipkow
parents: 15524
diff changeset
   318
                 Const("op &",_)
nipkow
parents: 15524
diff changeset
   319
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
nipkow
parents: 15524
diff changeset
   320
                   $ P =>
nipkow
parents: 15524
diff changeset
   321
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
nipkow
parents: 15524
diff changeset
   322
               | _ => M
nipkow
parents: 15524
diff changeset
   323
            end
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   324
    end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   325
  in [("Collect", setcompr_tr')] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   326
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   327
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   328
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   329
subsection {* Rules and definitions *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   330
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   331
text {* Isomorphisms between predicates and sets. *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   332
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   333
axioms
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   334
  mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   335
  Collect_mem_eq: "{x. x:A} = A"
17702
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   336
finalconsts
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   337
  Collect
ea88ddeafabe more finalconsts;
wenzelm
parents: 17589
diff changeset
   338
  "op :"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   339
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   340
defs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   341
  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   342
  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   343
  Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   344
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   345
instance set :: (type) minus
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   346
  Compl_def:    "- A            == {x. ~x:A}"
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   347
  set_diff_def: "A - B          == {x. x:A & ~x:B}" ..
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   348
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   349
lemmas [code func del] = Compl_def set_diff_def
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22478
diff changeset
   350
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   351
defs
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   352
  Un_def:       "A Un B         == {x. x:A | x:B}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   353
  Int_def:      "A Int B        == {x. x:A & x:B}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   354
  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   355
  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   356
  Inter_def:    "Inter S        == (INT x:S. x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   357
  Union_def:    "Union S        == (UN x:S. x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   358
  Pow_def:      "Pow A          == {B. B <= A}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   359
  empty_def:    "{}             == {x. False}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   360
  UNIV_def:     "UNIV           == {x. True}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   361
  insert_def:   "insert a B     == {x. x=a} Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   362
  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   363
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   364
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   365
subsection {* Lemmas and proof tool setup *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   366
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   367
subsubsection {* Relating predicates and sets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   368
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   369
declare mem_Collect_eq [iff]  Collect_mem_eq [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   370
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
   371
lemma CollectI: "P(a) ==> a : {x. P(x)}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   372
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   373
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   374
lemma CollectD: "a : {x. P(x)} ==> P(a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   375
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   376
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   377
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   378
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   379
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
   380
lemmas CollectE = CollectD [elim_format]
11979
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
subsubsection {* Bounded quantifiers *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   384
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   385
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   386
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   387
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   388
lemmas strip = impI allI ballI
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   389
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   390
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   391
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   392
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   393
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   394
  by (unfold Ball_def) blast
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   395
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   396
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   397
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   398
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   399
  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   400
  @{prop "a:A"}; creates assumption @{prop "P a"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   401
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   402
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   403
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   404
  fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   405
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   406
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   407
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   408
  Gives better instantiation for bound:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   409
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   410
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   411
ML_setup {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   412
  change_claset (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   413
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   414
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   415
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   416
  -- {* Normally the best argument order: @{prop "P x"} constrains the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   417
    choice of @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   418
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   419
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   420
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   421
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   422
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   423
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   424
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
   425
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   426
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   427
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   428
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   429
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   430
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   431
  -- {* Trival rewrite rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   432
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   433
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   434
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   435
  -- {* Dual form for existentials. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   436
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   437
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   438
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   439
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   440
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   441
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   442
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   443
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   444
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
   445
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   446
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   447
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
   448
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   449
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   450
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
   451
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   452
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   453
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
   454
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   456
ML_setup {*
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   457
  local
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   458
    val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   459
    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
   460
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   461
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   462
    val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   463
    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
   464
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   465
  in
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   466
    val defBEX_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   467
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   468
    val defBALL_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   469
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   470
  end;
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   471
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   472
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   473
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   474
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   476
subsubsection {* Congruence rules *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   477
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   478
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   479
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   483
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   484
  "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
   485
    (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
   486
  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
   487
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   488
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   492
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   493
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   494
  "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
   495
    (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
   496
  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
   497
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   498
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
subsubsection {* Subsets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   500
19295
c5d236fe9668 subsetI is often necessary
paulson
parents: 19277
diff changeset
   501
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   502
  by (simp add: subset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
  \medskip Map the type @{text "'a set => anything"} to just @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
  'a}; for overloading constants whose first argument has type @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   507
  "'a set"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   508
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   509
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   510
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   511
  -- {* Rule in Modus Ponens style. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   512
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   513
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   514
declare subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   515
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   516
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
  -- {* The same, with reversed premises for use with @{text erule} --
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
      cf @{text rev_mp}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
  by (rule subsetD)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   521
declare rev_subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   522
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   523
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   524
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   525
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   526
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   528
  fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   530
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   531
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
   532
  -- {* Classical elimination rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   535
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   536
  \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
   537
  creates the assumption @{prop "c \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   538
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   539
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   540
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   541
  fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   542
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   543
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   544
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   546
19175
c6e4b073f6a5 subset_refl now included using the atp attribute
paulson
parents: 18851
diff changeset
   547
lemma subset_refl [simp,atp]: "A \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   548
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   549
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   550
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   551
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   552
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   553
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   554
subsubsection {* Equality *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   555
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   556
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   557
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   558
   apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   559
  apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   560
  done
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   561
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   562
(* Due to Brian Huffman *)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   563
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   564
by(auto intro:set_ext)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   565
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   566
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   567
  -- {* Anti-symmetry of the subset relation. *}
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
   568
  by (iprover intro: set_ext subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   569
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   570
lemmas equalityI [intro!] = subset_antisym
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   571
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
  \medskip Equality rules from ZF set theory -- are they appropriate
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   574
  here?
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   577
lemma equalityD1: "A = B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   580
lemma equalityD2: "A = B ==> B \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
  \medskip Be careful when adding this to the claset as @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   585
  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
   586
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   589
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   590
  by (simp add: subset_refl)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   591
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
lemma equalityCE [elim]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   593
    "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
   594
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   596
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   597
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   599
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   600
  by simp
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   601
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   603
subsubsection {* The universal set -- UNIV *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   605
lemma UNIV_I [simp]: "x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
  by (simp add: UNIV_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   607
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
18144
4edcb5fdc3b0 duplicate axioms in ATP linkup, and general fixes
paulson
parents: 17875
diff changeset
   613
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   614
  by (rule subsetI) (rule UNIV_I)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   615
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   616
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
  \medskip Eta-contracting these two rules (to remove @{text P})
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   618
  causes them to be ignored because of their interaction with
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   619
  congruence rules.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   621
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   622
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   625
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   627
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   628
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   629
subsubsection {* The empty set *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   630
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   631
lemma empty_iff [simp]: "(c : {}) = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
  by (simp add: empty_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   633
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   634
lemma emptyE [elim!]: "a : {} ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   636
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   637
lemma empty_subsetI [iff]: "{} \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   638
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
  by blast
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 equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   642
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   643
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   644
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   647
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   648
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   650
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   652
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   653
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   656
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
12023
wenzelm
parents: 12020
diff changeset
   658
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   659
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   660
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   661
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   662
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   663
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   664
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   665
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   666
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   667
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   668
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   669
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   671
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   672
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   674
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   675
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   678
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   681
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   685
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   686
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   689
lemma ComplD [dest!]: "c : -A ==> c~:A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   690
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   692
lemmas ComplE = ComplD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   695
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   696
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   697
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   705
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   706
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   708
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   709
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   714
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
12023
wenzelm
parents: 12020
diff changeset
   718
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   719
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   720
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   726
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   728
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   729
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   730
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   732
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   733
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
12023
wenzelm
parents: 12020
diff changeset
   736
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   740
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   742
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   749
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   750
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
  by simp
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 {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   755
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   758
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   760
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   763
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   764
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   765
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   772
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
   773
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   774
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   775
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   777
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   780
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   782
lemma singletonD [dest!]: "b : {a} ==> b = a"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   784
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   785
lemmas singletonE = singletonD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   786
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   787
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   789
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   793
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
   794
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   795
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   796
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
   797
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   798
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   799
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   800
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   801
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   802
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   804
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   807
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   808
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
   809
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   810
19870
ef037d1b32d1 new results
paulson
parents: 19656
diff changeset
   811
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
ef037d1b32d1 new results
paulson
parents: 19656
diff changeset
   812
  by (blast elim: equalityE)
ef037d1b32d1 new results
paulson
parents: 19656
diff changeset
   813
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   816
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   817
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   818
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   819
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
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
   822
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
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
   830
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   831
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   832
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   833
    "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
   834
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
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
   842
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   843
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   844
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
   845
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
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
   851
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
    "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
   856
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   857
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   858
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   875
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   876
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   879
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   880
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   885
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   886
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   890
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   894
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   895
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   896
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   897
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   900
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   901
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   903
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   904
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   906
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   907
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   909
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   910
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   912
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   913
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   914
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   915
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   916
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   917
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   918
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   919
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   920
  by blast
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 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
   923
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   924
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   925
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   926
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
   927
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   928
   prefer 2 apply fast
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
   929
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   930
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   931
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   932
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
   933
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   934
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   935
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   936
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   938
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   939
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   940
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   941
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   942
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   943
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   944
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   946
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   947
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
   948
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   949
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   950
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   951
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   952
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   953
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   954
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   955
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   956
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   957
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   958
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
   959
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   960
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   961
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
   962
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   963
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   964
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   965
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   968
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   969
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
   970
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   971
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   972
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
   973
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   974
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   975
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
   976
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   977
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   978
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   979
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   980
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   981
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   982
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   983
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   984
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   985
  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
   986
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   987
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   988
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   989
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   990
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   991
ML_setup {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   992
  val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs;
17875
d81094515061 change_claset/simpset;
wenzelm
parents: 17784
diff changeset
   993
  change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs));
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   994
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   995
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   996
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   997
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   998
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   999
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1000
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1001
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
  1002
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
  1003
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
  1004
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
  1005
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1006
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1007
  "(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
  1008
  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
  1009
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1010
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1011
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1012
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1013
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1014
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1015
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1016
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
  1017
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1018
apply (auto dest: subset_antisym)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1019
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1020
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1021
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1022
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1023
apply (auto dest: subsetD)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1024
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1025
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1027
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1028
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1029
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1030
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1031
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1032
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1033
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1034
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1035
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1036
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1037
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1038
18832
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18674
diff changeset
  1039
lemmas [symmetric, rulify] = atomize_ball
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18674
diff changeset
  1040
  and [symmetric, defn] = atomize_ball
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1041
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1042
22455
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  1043
subsection {* Order on sets *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1044
21384
2b58b308300c moved transitivity rules to Orderings.thy
haftmann
parents: 21333
diff changeset
  1045
instance set :: (type) order
2b58b308300c moved transitivity rules to Orderings.thy
haftmann
parents: 21333
diff changeset
  1046
  by (intro_classes,
2b58b308300c moved transitivity rules to Orderings.thy
haftmann
parents: 21333
diff changeset
  1047
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
2b58b308300c moved transitivity rules to Orderings.thy
haftmann
parents: 21333
diff changeset
  1048
22455
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  1049
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  1050
subsection {* Further set-theory lemmas *}
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  1051
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1052
subsubsection {* Derived rules involving subsets. *}
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 {* @{text insert}. *}
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 subset_insertI: "B \<subseteq> insert a B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1059
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1060
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1061
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1062
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1063
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1064
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
  1065
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1067
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1068
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
  1069
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1070
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1071
  by (iprover intro: subsetI UnionI)
12897
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
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
  1074
  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
  1075
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1077
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1078
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1079
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
  1080
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1081
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1082
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
  1083
  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
  1084
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1085
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
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
  1087
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1088
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
  1089
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1090
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1091
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1092
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1093
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1094
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1095
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
  1096
  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
  1097
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1098
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
  1099
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1100
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1101
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
  1102
  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
  1103
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1104
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1105
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
  1106
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1107
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
  1108
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1109
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1110
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
  1111
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
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
  1114
  by blast
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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1117
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
  1118
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
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
  1120
  by blast
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 Int_lower2: "A \<inter> B \<subseteq> B"
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 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
  1126
  by 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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1129
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1130
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1131
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1132
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1133
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1134
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1135
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1136
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1138
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1139
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1140
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1141
  by (auto simp add: mono_def)
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1142
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1143
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1144
  by (auto simp add: mono_def)
12897
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
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1148
text {* @{text "{}"}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1149
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1150
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
  1151
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1152
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1154
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1157
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
18423
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1161
by blast
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1162
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1163
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
d7859164447f new lemmas
nipkow
parents: 18413
diff changeset
  1164
by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1166
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
  1167
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1168
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1169
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
  1170
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
14812
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1172
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
  1173
  by blast
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1174
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1175
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
  1176
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
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
  1179
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1180
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
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
  1182
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1183
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1184
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
  1185
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1186
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1187
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
  1188
  by blast
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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
text {* \medskip @{text insert}. *}
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 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
  1194
  -- {* 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
  1195
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1196
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1197
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
  1198
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1199
17715
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1200
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1201
declare empty_not_insert [simp]
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1202
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1203
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
  1204
  -- {* @{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
  1205
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1206
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1207
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1208
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
  1209
  by blast
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 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
  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 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
  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 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
  1218
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1219
  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
  1220
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1221
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1222
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
  1223
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1224
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1225
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
  1226
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1227
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1228
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
  1229
  by blast
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1230
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1231
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1232
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1233
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1234
  by auto
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1235
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1236
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1237
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1238
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1239
  by auto
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1240
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1241
text {* \medskip @{text image}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1242
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1243
lemma image_empty [simp]: "f`{} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1244
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1245
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1246
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
  1247
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1248
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1249
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1250
  by auto
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1251
21316
4d913b8bccf1 image_constant_conv no longer [simp]
nipkow
parents: 21312
diff changeset
  1252
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21210
diff changeset
  1253
by auto
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21210
diff changeset
  1254
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
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
  1256
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1257
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
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
  1259
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
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
  1262
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1264
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1266
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1267
      with its implicit quantifier and conjunction.  Also image enjoys better
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1268
      equational properties than does the RHS. *}
12897
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
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
  "(\<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
  1273
    = (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
  1274
  by (auto simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1276
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
  1277
  by (simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
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
text {* \medskip @{text range}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1282
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
  1283
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1285
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1286
by (subst image_image, simp)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
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
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1291
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
  1292
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
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
  1295
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1296
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1297
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
  1298
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1300
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
  1301
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1302
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1303
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
  1304
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1306
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
  1307
  -- {* Intersection is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1309
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
  1310
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1312
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
  1313
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1315
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1316
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1318
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1319
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1320
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1321
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
  1322
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1323
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1324
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
  1325
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1326
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1327
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
  1328
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
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
  1331
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1333
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
  1334
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1335
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1336
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
  1337
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1338
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1339
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
  1340
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1341
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1342
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
  1343
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1344
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1345
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
  1346
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1347
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1348
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
  1349
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1350
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
text {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1353
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1354
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
  1355
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1356
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1357
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
  1358
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1359
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1360
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
  1361
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1362
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1363
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
  1364
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1365
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1366
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
  1367
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1368
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1369
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
  1370
  -- {* Union is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1371
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1372
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
  1373
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1374
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1375
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
  1376
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1377
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1378
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
  1379
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1380
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1381
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
  1382
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1383
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1384
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
  1385
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1386
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1387
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1388
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1389
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1390
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1391
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1392
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1393
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1394
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1395
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1396
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1397
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1398
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1399
lemma Int_insert_left:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1400
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1401
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1402
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1403
lemma Int_insert_right:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1404
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1405
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1406
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1407
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1408
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1409
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1410
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1411
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1412
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1413
lemma Un_Int_crazy:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1414
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1415
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1416
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1417
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1418
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1419
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1420
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1421
  by blast
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1422
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1423
lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1424
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1425
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1426
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1427
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1428
22172
e7d6cb237b5e some new lemmas
paulson
parents: 22139
diff changeset
  1429
lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
e7d6cb237b5e some new lemmas
paulson
parents: 22139
diff changeset
  1430
  by blast
e7d6cb237b5e some new lemmas
paulson
parents: 22139
diff changeset
  1431
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1432
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1433
text {* \medskip Set complement *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1434
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1435
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1436
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1437
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1438
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1439
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1440
13818
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1441
lemma Compl_partition: "A \<union> -A = UNIV"
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1442
  by blast
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1443
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1444
lemma Compl_partition2: "-A \<union> A = UNIV"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1445
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1446
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1447
lemma double_complement [simp]: "- (-A) = (A::'a set)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1448
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1449
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1450
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1451
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1452
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1453
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1454
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1455
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1456
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1457
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1458
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1459
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1460
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1461
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1462
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1463
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1464
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1465
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1466
  -- {* Halmos, Naive Set Theory, page 16. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1467
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1468
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1469
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1470
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1471
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1472
lemma Compl_empty_eq [simp]: "-{} = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1473
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1474
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1475
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1476
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1477
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1478
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1479
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1480
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1481
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1482
text {* \medskip @{text Union}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1483
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1484
lemma Union_empty [simp]: "Union({}) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1485
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1486
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1487
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1488
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1489
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1490
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1491
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1492
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1493
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1494
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1495
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1496
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1497
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1498
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1499
lemma Union_empty_conv [simp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1500
  by blast
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1501
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1502
lemma empty_Union_conv [simp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1503
  by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1504
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1505
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1506
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1507
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1508
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1509
text {* \medskip @{text Inter}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1510
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1511
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1512
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1513
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1514
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1515
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1516
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1517
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1518
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1519
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1520
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1521
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1522
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1523
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1524
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1525
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1526
lemma Inter_UNIV_conv [simp]:
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1527
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1528
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1529
  by blast+
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1530
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1531
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1532
text {*
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1533
  \medskip @{text UN} and @{text INT}.
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1534
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1535
  Basic identities: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1536
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1537
lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1538
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1539
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1540
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1541
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1542
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1543
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1544
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1545
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1546
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1547
  by auto
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1548
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1549
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1550
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1551
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1552
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1553
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1554
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1555
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1556
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1557
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1558
lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1559
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1560
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1561
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1562
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1563
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1564
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1565
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1566
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1567
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1568
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1569
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1570
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1571
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1572
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1573
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1574
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1575
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1576
lemma INT_insert_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1577
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1578
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1579
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1580
lemma Union_image_eq [simp]: "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1581
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1582
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1583
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1584
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1585
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1586
lemma Inter_image_eq [simp]: "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1587
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1588
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1589
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1590
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1591
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1592
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1593
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1594
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1595
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1596
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1597
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1598
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1599
  -- {* Look: it has an \emph{existential} quantifier *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1600
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1601
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1602
lemma UNION_empty_conv[simp]:
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1603
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1604
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1605
by blast+
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1606
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1607
lemma INTER_UNIV_conv[simp]:
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1608
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1609
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1610
by blast+
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1611
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1612
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1613
text {* \medskip Distributive laws: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1614
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1615
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1616
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1617
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1618
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1619
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1620
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1621
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1622
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1623
  -- {* Union of a family of unions *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1624
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1625
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1626
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1627
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1628
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1629
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1630
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1631
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1632
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1633
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1634
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1635
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1636
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1637
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1638
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1639
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1640
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1641
  -- {* Halmos, Naive Set Theory, page 35. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1642
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1643
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1644
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1645
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1646
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1647
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1648
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1649
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1650
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1651
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1652
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1653
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1654
text {* \medskip Bounded quantifiers.
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1655
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1656
  The following are not added to the default simpset because
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1657
  (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1658
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1659
lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1660
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1661
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1662
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1663
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1664
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1665
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1666
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1667
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1668
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1669
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1670
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1671
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1672
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1673
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1674
lemma Diff_eq: "A - B = A \<inter> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1675
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1676
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1677
lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1678
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1679
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1680
lemma Diff_cancel [simp]: "A - A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1681
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1682
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1683
lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1684
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1685
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1686
lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1687
  by (blast elim: equalityE)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1688
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1689
lemma empty_Diff [simp]: "{} - A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1690
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1691
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1692
lemma Diff_empty [simp]: "A - {} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1693
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1694
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1695
lemma Diff_UNIV [simp]: "A - UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1696
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1697
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1698
lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1699
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1700
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1701
lemma Diff_insert: "A - insert a B = A - B - {a}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1702
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1703
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1704
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1705
lemma Diff_insert2: "A - insert a B = A - {a} - B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1706
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1707
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1708
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1709
lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1710
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1711
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1712
lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1713
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1714
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1715
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1716
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1717
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1718
lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1719
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1720
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1721
lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1722
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1723
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1724
lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1725
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1726
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1727
lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1728
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1729
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1730
lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1731
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1732
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1733
lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1734
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1735
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1736
lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1737
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1738
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1739
lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1740
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1741
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1742
lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1743
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1744
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1745
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1746
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1747
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1748
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1749
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1750
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1751
lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1752
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1753
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1754
lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1755
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1756
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1757
lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1758
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1759
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1760
lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1761
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1762
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1763
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1764
text {* \medskip Quantification over type @{typ bool}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1765
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1766
lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
21549
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1767
  by (cases x) auto
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1768
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1769
lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1770
  by (auto intro: bool_induct)
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1771
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1772
lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1773
  by (cases x) auto
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1774
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1775
lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1776
  by (auto intro: bool_contrapos)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1777
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1778
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1779
  by (auto simp add: split_if_mem2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1780
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1781
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
21549
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1782
  by (auto intro: bool_contrapos)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1783
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1784
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
21549
12eff58b56a0 restructured some proofs
haftmann
parents: 21524
diff changeset
  1785
  by (auto intro: bool_induct)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1786
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1787
text {* \medskip @{text Pow} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1788
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1789
lemma Pow_empty [simp]: "Pow {} = {{}}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1790
  by (auto simp add: Pow_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1791
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1792
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1793
  by (blast intro: image_eqI [where ?x = "u - {a}", standard])
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1794
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1795
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1796
  by (blast intro: exI [where ?x = "- u", standard])
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1797
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1798
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1799
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1800
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1801
lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1802
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1803
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1804
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1805
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1806
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1807
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1808
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1809
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1810
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1811
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1812
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1813
lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1814
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1815
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1816
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1817
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1818
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1819
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1820
text {* \medskip Miscellany. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1821
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1822
lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1823
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1824
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1825
lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1826
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1827
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1828
lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1829
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1830
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1831
lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1832
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1833
13831
ab27b36aba99 new lemma
paulson
parents: 13826
diff changeset
  1834
lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
ab27b36aba99 new lemma
paulson
parents: 13826
diff changeset
  1835
  by blast
ab27b36aba99 new lemma
paulson
parents: 13826
diff changeset
  1836
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1837
lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1838
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1839
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1840
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1841
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1842
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1843
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1844
lemma UN_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1845
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1846
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1847
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1848
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1849
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1850
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1851
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1852
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1853
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1854
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1855
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1856
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1857
lemma INT_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1858
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1859
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1860
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1861
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1862
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1863
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1864
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1865
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1866
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1867
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1868
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1869
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1870
lemma ball_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1871
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1872
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1873
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1874
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1875
  "!!P. (ALL x:{}. P x) = True"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1876
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1877
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1878
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1879
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1880
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1881
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1882
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1883
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1884
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1885
lemma bex_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1886
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1887
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1888
  "!!P. (EX x:{}. P x) = False"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1889
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1890
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1891
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1892
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1893
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1894
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1895
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1896
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1898
lemma ball_conj_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1899
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1900
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1901
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1902
lemma bex_disj_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1903
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1904
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1905
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1906
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1907
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1908
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1909
lemma UN_extend_simps:
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1910
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1911
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1912
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1913
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1914
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1915
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1916
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1917
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1918
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1919
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1920
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1921
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1922
lemma INT_extend_simps:
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1923
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1924
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1925
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1926
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1927
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1928
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1929
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1930
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1931
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1932
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1933
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1934
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1935
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1936
subsubsection {* Monotonicity of various operations *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1937
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1938
lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1939
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1940
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1941
lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1942
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1943
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1944
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1945
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1946
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1947
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1948
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1949
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1950
lemma UN_mono:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1951
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1952
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1953
  by (blast dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1954
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1955
lemma INT_anti_mono:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1956
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1957
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1958
  -- {* The last inclusion is POSITIVE! *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1959
  by (blast dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1960
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1961
lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1962
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1963
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1964
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1965
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1966
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1967
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1968
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1969
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1970
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1971
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1972
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1973
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1974
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1975
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1976
text {* \medskip Monotonicity of implications. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1977
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1978
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1979
  apply (rule impI)
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1980
  apply (erule subsetD, assumption)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1981
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1982
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1983
lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1984
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1985
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1986
lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1987
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1988
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1989
lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1990
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1991
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1992
lemma imp_refl: "P --> P" ..
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1993
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1994
lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1995
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1996
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1997
lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1998
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1999
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2000
lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2001
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2002
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2003
lemma Int_Collect_mono:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2004
    "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2005
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2006
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2007
lemmas basic_monos =
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2008
  subset_refl imp_refl disj_mono conj_mono
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2009
  ex_mono Collect_mono in_mono
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2010
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2011
lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  2012
  by iprover
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2013
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2014
lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  2015
  by iprover
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  2016
11982
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2017
lemma Least_mono:
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2018
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2019
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2020
    -- {* Courtesy of Stephan Merz *}
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2021
  apply clarify
15950
5c067c956a20 Added thms by Brian Huffmann
nipkow
parents: 15554
diff changeset
  2022
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
5c067c956a20 Added thms by Brian Huffmann
nipkow
parents: 15554
diff changeset
  2023
  apply (rule LeastI2_order)
11982
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2024
  apply (auto elim: monoD intro!: order_antisym)
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2025
  done
65e2822d83dd lemma Least_mono moved from Typedef.thy to Set.thy;
wenzelm
parents: 11979
diff changeset
  2026
12020
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2027
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2028
subsection {* Inverse image of a function *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2029
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2030
constdefs
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2031
  vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-`" 90)
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2032
  "f -` B == {x. f x : B}"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2033
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2034
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2035
subsubsection {* Basic rules *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2036
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2037
lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2038
  by (unfold vimage_def) blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2039
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2040
lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2041
  by simp
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2042
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2043
lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2044
  by (unfold vimage_def) blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2045
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2046
lemma vimageI2: "f a : A ==> a : f -` A"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2047
  by (unfold vimage_def) fast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2048
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2049
lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2050
  by (unfold vimage_def) blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2051
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2052
lemma vimageD: "a : f -` A ==> f a : A"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2053
  by (unfold vimage_def) fast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2054
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2055
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2056
subsubsection {* Equations *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2057
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2058
lemma vimage_empty [simp]: "f -` {} = {}"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2059
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2060
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2061
lemma vimage_Compl: "f -` (-A) = -(f -` A)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2062
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2063
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2064
lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2065
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2066
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2067
lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2068
  by fast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2069
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2070
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2071
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2072
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2073
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2074
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2075
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2076
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2077
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2078
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2079
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2080
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2081
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2082
lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2083
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2084
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2085
lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2086
  -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2087
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2088
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2089
lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2090
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2091
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2092
lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2093
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2094
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2095
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2096
  -- {* NOT suitable for rewriting *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2097
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2098
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2099
lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2100
  -- {* monotonicity *}
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2101
  by blast
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2102
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
  2103
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2104
subsection {* Getting the Contents of a Singleton Set *}
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2105
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2106
constdefs
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2107
  contents :: "'a set => 'a"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2108
   "contents X == THE x. X = {x}"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2109
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2110
lemma contents_eq [simp]: "contents {x} = x"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2111
by (simp add: contents_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2112
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14398
diff changeset
  2113
12023
wenzelm
parents: 12020
diff changeset
  2114
subsection {* Transitivity rules for calculational reasoning *}
12020
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2115
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2116
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
12020
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2117
  by (rule subsetD)
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2118
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  2119
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
12020
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2120
  by (rule subsetD)
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2121
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2122
lemmas basic_trans_rules [trans] =
21384
2b58b308300c moved transitivity rules to Orderings.thy
haftmann
parents: 21333
diff changeset
  2123
  order_trans_rules set_rev_mp set_mp
12020
a24373086908 theory Calculation move to Set;
wenzelm
parents: 11982
diff changeset
  2124
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2125
22455
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2126
subsection {* Sets as lattice *}
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2127
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2128
instance set :: (type) distrib_lattice
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2129
  inf_set_eq: "inf A B \<equiv> A \<inter> B"
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2130
  sup_set_eq: "sup A B \<equiv> A \<union> B"
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2131
  by intro_classes (auto simp add: inf_set_eq sup_set_eq)
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2132
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  2133
lemmas [code func del] = inf_set_eq sup_set_eq
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22478
diff changeset
  2134
22455
f6f22aba2e0e added instance of sets as distributive lattices
haftmann
parents: 22439
diff changeset
  2135
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2136
subsection {* Basic ML bindings *}
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2137
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2138
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2139
val Ball_def = @{thm Ball_def}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2140
val Bex_def = @{thm Bex_def}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2141
val CollectD = @{thm CollectD}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2142
val CollectE = @{thm CollectE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2143
val CollectI = @{thm CollectI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2144
val Collect_conj_eq = @{thm Collect_conj_eq}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2145
val Collect_mem_eq = @{thm Collect_mem_eq}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2146
val IntD1 = @{thm IntD1}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2147
val IntD2 = @{thm IntD2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2148
val IntE = @{thm IntE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2149
val IntI = @{thm IntI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2150
val Int_Collect = @{thm Int_Collect}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2151
val UNIV_I = @{thm UNIV_I}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2152
val UNIV_witness = @{thm UNIV_witness}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2153
val UnE = @{thm UnE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2154
val UnI1 = @{thm UnI1}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2155
val UnI2 = @{thm UnI2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2156
val ballE = @{thm ballE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2157
val ballI = @{thm ballI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2158
val bexCI = @{thm bexCI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2159
val bexE = @{thm bexE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2160
val bexI = @{thm bexI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2161
val bex_triv = @{thm bex_triv}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2162
val bspec = @{thm bspec}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2163
val contra_subsetD = @{thm contra_subsetD}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2164
val distinct_lemma = @{thm distinct_lemma}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2165
val eq_to_mono = @{thm eq_to_mono}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2166
val eq_to_mono2 = @{thm eq_to_mono2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2167
val equalityCE = @{thm equalityCE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2168
val equalityD1 = @{thm equalityD1}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2169
val equalityD2 = @{thm equalityD2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2170
val equalityE = @{thm equalityE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2171
val equalityI = @{thm equalityI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2172
val imageE = @{thm imageE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2173
val imageI = @{thm imageI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2174
val image_Un = @{thm image_Un}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2175
val image_insert = @{thm image_insert}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2176
val insert_commute = @{thm insert_commute}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2177
val insert_iff = @{thm insert_iff}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2178
val mem_Collect_eq = @{thm mem_Collect_eq}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2179
val rangeE = @{thm rangeE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2180
val rangeI = @{thm rangeI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2181
val range_eqI = @{thm range_eqI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2182
val subsetCE = @{thm subsetCE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2183
val subsetD = @{thm subsetD}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2184
val subsetI = @{thm subsetI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2185
val subset_refl = @{thm subset_refl}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2186
val subset_trans = @{thm subset_trans}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2187
val vimageD = @{thm vimageD}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2188
val vimageE = @{thm vimageE}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2189
val vimageI = @{thm vimageI}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2190
val vimageI2 = @{thm vimageI2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2191
val vimage_Collect = @{thm vimage_Collect}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2192
val vimage_Int = @{thm vimage_Int}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
  2193
val vimage_Un = @{thm vimage_Un}
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2194
*}
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  2195
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  2196
end